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0704.0987

A kind of prediction from string phenomenology: extra matter at low energy

September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz Modern Physics Letters A c© World Scientific Publishing Company A KIND OF PREDICTION FROM STRING PHENOMENOLOGY: EXTRA MATTER AT LOW ENERGY CARLOS MUÑOZ Departamento de F́ısica Teórica C-XI and Instituto de F́ısica Teórica C-XVI, Universidad Autónoma de Madrid, Cantoblanco, E-28049 Madrid, Spain [email protected] We review the possibility that the Supersymmetric Standard Model arises from orbifold constructions of the E8×E8 Heterotic Superstring, and the phenomenological properties that such a model should have. In particular, trying to solve the discrepancy between the unification scale predicted by the Heterotic Superstring (≈ gGUT × 5.27 · 10 17 GeV) and the value deduced from LEP experiments (≈ 2 · 1016 GeV), we will predict the presence at low energies of three families of Higgses and vector-like colour triplets. Our approach relies on the Fayet-Iliopoulos breaking, and this is also a crucial ingredient, together with having three Higgs families, to obtain in these models an interesting pattern of fermion masses and mixing angles at the renormalizable lebel. Namely, after the gauge breaking some physical particles appear combined with other states, and the Yukawa couplings are modified in a well controlled way. On the other hand, dangerous flavour-changing neutral currents may appear when fermions of a given charge receive their mass through couplings with several Higgs doublets. We will address this potential problem, finding that viable scenarios can be obtained for a reasonable light Higgs spectrum. Keywords: Strings; Orbifolds; Phenomenology. PACS Nos.: 11.25.Wx, 11.25.Mj, 12.60.Jv, 12.60.Fr 1. Introduction In SuperString Theory the elementary particles are not point-like objects but ex- tended, string-like objects. It is surprising that this apparently small change allows us to answer fundamental questions that in the context of the quantum field theory of point-like particles cannot even be posed. For example: Why is the Standard Model gauge group SU(3) × SU(2)L × U(1)Y ? Why are there three families of particles? Why is the mass of the electron me = 0.5 MeV? Why is the fine struc- ture constant α = 1/137? In addition, only SuperString Theory has the potential to unify all gauge interactions with gravity in a consistent way. In this sense, it is a crucial step in the construction of the fundamental theory of particle physics to find a consistent SuperString model in four dimensions accommodating the ob- served Standard Model (SM), i.e. we need to find the SuperString Standard Model (SSSM). Actually, this is the main task of what we call String Phenomenology. In the late eighties, the compactification of the E8 × E8 Heterotic String on http://arxiv.org/abs/0704.0987v4 September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz 2 Carlos Muñoz six-dimensional orbifolds proved to be an interesting method to carry out this taska (for a brief historical account see the Introduction in Ref. 1 and references therein). For example, it was shown that the use of two Wilson lines on the torus defining the symmetric Z3 orbifold can give rise to four-dimensional supersymmetric models with gauge group SU(3)×SU(2)×U(1)5×Ghidden and, automatically, three generations of chiral particles2. In addition, it was also shown that the Fayet–Iliopoulos (FI) D-term, which appears because of the presence of an anomalous U(1), can give rise to the breaking of the extra U(1)’s. In this way it was possible to construct3,4,5 supersymmetric models with gauge group SU(3)×SU(2)×U(1)Y , three generations of particles in the observable sector, and absence of dangerous baryon- and lepton- number-violating operatorsb. Unfortunately, we cannot claim that one of these Z3 orbifold models is the SSSM, since several problems are always present. For example, the initially large number of extra particles, which are generically present in these constructions, is highly reduced through the FI mechanism, since many of them get a high mass (≈ 1016−17 GeV). However, in general, some extra SU(3) triplets, SU(2) doublets and SU(3)× SU(2) singlets still remain at low energy. On the other hand, given the predicted value for the unification scale in the Heterotic String12, MGUT ≈ gGUT ×5.27 ·1017 GeV, the values of the gauge couplings deduced from LEP experiments cannot13 be obtainedc. It was also not possible to obtain in these models the necessary Yukawa couplings reproducing the observed fermion masses5,4,14. At this point, it is fair to say that almost 20 years have gone by since String Phenomenology started, and the SSSM has not been found yetd As acquittal on the charge we should remark that there are thousands of models (vacua) that can be built. Some of them have the gauge group of the SM or GUT groups, three families of particles, and other interesting properties, but many others have a number of families different from three, no appropriate gauge groups, no appropriate matter, etc. A perfect way of solving this problem would be to use a dynamical mechanism to select the correct model (vacuum). Such a mechanism should be able to determine a point in the parameter space of the Heterotic String determining the correct compactification producing the gauge group SU(3)c×SU(2)L×U(1)Y , three families of the known particles, the correct Yukawa couplings, etc. The problem is that such a mechanism has not been discovered yet. So, for the moment, the best we can do is keep trying, i.e. to use the experimental aOther interesting attempts at model building used Calabi–Yau spaces and fermionic constructions. bRecently, other interesting models in the context of the Z3 orbifold 6, as well as in the context of the Z6 orbifold 7,8,9, and Z12 orbifold 10,11, have been analysed. cRecall that this is only possible in the context of the Minimal Supersymmetric Standard Model (MSSM) for MGUT ≈ 2× 10 16 GeV. dAnd this sentence can also be applied to any of the interesting models constructed in more recent years using D-brane technology15. Actually, the probability of obtaining an MSSM like gauge group with three generation in the context of intersecting D-branes in an orientifold background seems to be extremelly small16,17, of about 10−9. September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz A kind of prediction from string phenomenology: extra matter at low energy 3 results available (such as the SM gauge group, three families, fermion masses, mixing angles, etc.), to discard models. Although the model space is in principle huge, a detailed analysis can reduce this to a reasonable size. For example, within the Z3 orbifold with two Wilson lines, one can construct in principle a number of order 50000 of three-generation models with the SU(3) × SU(2) × U(1)5 gauge group associated to the first E8 of the Heterotic String. However, a study implied that most of them are equivalent18, and in fact, at the end of the day, only 192 different models were found19,18. This reduction is remarkable, but we should keep in mind that the analysis of each one of these models is really complicated. Nevertheless, a certain degree of optimism is important when working in String Phenomenology, and one can argue that if the SM arises from SuperString Theory there must exist one model with the right properties. In the present review we will adopt this viewpoint, and will assume that the SM arises from orbifolds construc- tions. Instead of the painful work of searching for the correct orbifold model, we will try to deduce the phenomenological properties that such a model must have in order to solve the crucial problems mentioned above, with the hope that this analysis will allow us to make predictions that can be tested at the LHC. In fact, all those problems, extra matter, gauge coupling unification, and cor- rect Yukawa couplings, are closely related. The first two because the evolution of the gauge couplings from high to low energy through the renormalization group equations (RGEs), depends on the existing matter20,6. In Section 2 we will dis- cuss a solution to the gauge coupling unification problem implying the prediction of three generations of supersymmetric Higgses and vector-like colour triplets at low energies1. In this solution the FI scale plays an important role. Concerning the third problem, how to obtain the observed structure of fermion masses and mixing angles, this is in our opinion the most difficult task in String Phenomenology. For example, the right model must reproduce also the correct mass hierarchy for quarks and leptons, mt ∼ 105, mτ ∼ 103, etc., and this is not a trivial task, although it is true that one can find interesting results in the literature. In particular, orbifold spaces have a beautiful mechanism to generate a mass hierar- chy at the renormalizable level. Namely, Yukawa couplings between twisted matter can be explicitly computed and they get suppression factors, which depend on the distance between the fixed points to which the relevant fields are attached21−26. The couplings can be schematically written as λ ∼ e− Ti , with Re Ti ∼ R2i , and the Ti are the moduli fields associated to the size and shape of the orbifold. The distances can be varied by giving different vacuum expectation values (VEVs) to these moduli, implying that one can span in principle five orders of magnitude the Yukawa couplings24−26. Unfortunately, this is not the end of the story, since Nature tells us that a weak coupling matrix exists with weird magnitudes for the entries, and that therefore we must arrange our up-and down-quark Yukawa cou- plings in order to have specific off diagonal elementse. In Section 3 we will see that e Needless to say, the recent experimental confirmation of neutrino masses makes the task even September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz 4 Carlos Muñoz to obtain this at the renormalizable level is possible if three Higgs families and the FI breaking are present26,27. Thus we have a common solution for the three problems mentioned above. On the other hand, it is well known that dangerous flavour-changing neutral currents (FCNCs) may appear when fermions of a given charge receive their mass through couplings with several Higgs doublets28,29. This situation might be present here since we have three generations of supersymmetric Higgses. In Section 4 we will address this potential problem, finding that viable scenarios can be obtained27. 2. Predictions from gauge coupling unification Since we are interested in the analysis of gauge couplings, we need to first clarify which is the relevant scale for the running between the supersymmetric scaleMS and the unification point. Let us recall that in heterotic compactifications some scalars singlets Ci develop vacuum expectation values (VEVs) in order to cancel the FI D-term, without breaking the SM gauge group. An estimate about their VEVs can be done with the average result 〈Ci〉 ∼ 1016−17 GeV (see e.g. Ref. 6). After the breaking, many particles, say ξ, acquire a high mass because of the generation of effective mass terms. These come for example from operators of the type Ciξξ. In this way extra vector-like triplets and doublets and also singlets become very heavy. We will use the above value as our relevant scale, the so-called FI scale MFI ≈ 1016−17 GeV. As discussed in the Introduction, we are interested in the unification of the gauge couplings at MGUT ≈ gGUT ×5.27 ·1017 GeV. This is not a simple issue, and various approaches towards understanding it have been proposed in the literature30. Some of these proposals consist of using string GUT models, extra matter at intermedi- ate scales, heavy string threshold corrections, non-standard hypercharge normaliza- tions, etc. In our case, we will try to obtain this value by using first the existence of extra matter at the scale MS. We will see that this is not sufficient and, as a conse- quence, the FI scale must be included. Let us concentrate for the moment on α3 and α2. Recalling that three generations appear automatically for all the matter in Z3 orbifold scenarios with two Wilson lines, the most natural possibility is to assume the presence of three light generations of supersymmetric Higgses. This implies that we have four extra Higgs doublets, n2 = 4, with respect to the case of the MSSM. Unfortunately, this goes wrong. Whereas α−13 remains unchanged, since the number of extra triplets is n3 = 0, the line for α 2 is pushed down with respect to the case of the MSSM. As a consequence, the two couplings cross at a very low scale (≈ 1012 GeV). We could try to improve this situation by assuming the presence of extra triplets in addition to the four extra doublets. Then the line for α−13 is also pushed down and therefore the crossing might be obtained for larger scales. However, even more involved. We have to explain also the weak coupling matrix with the charged leptons. Besides, in addition to the hierarchies shown above, we have to explain others such as me September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz A kind of prediction from string phenomenology: extra matter at low energy 5 Fig. 1. Unification of the gauge couplings at MGUT ≈ gGUT × 5.27 · 10 17 GeV with three light generations of supersymmetric Higgses and vector-like colour triplets. In this example we show one of the four possible patterns of heavy matter in eq. (2), in particular that with a) nFI3 = 0. The line corresponding to α1 is just one of the many possible examples. for the minimum number of extra triplets that can be naturally obtained in our scenario, 3 × {(3, 1) + (3̄, 1)}, i.e. n3 = 6, the “unification” scale turns out to be too large (≈ 1021 GeV). One can check that other possibilities including more extra doublets and/or triplets do not work1. Thus, using extra matter at MS we are not able to obtain the Heterotic String unification scale, since α3 never crosses α2 at MGUT ≈ gGUT ×5.27 ·1017 GeV. Fortunately, this is not the end of the story. As we will show now, the FI scale MFI is going to play an important role in the analysis. In order to determine whether or not the Heterotic String unification scale can be obtained, we need to know the number of doublets nFI2 and triplets n 3 in our construction with masses of order the FI scale MFI . It is possible to show that within the Z3 orbifold with two Wilson lines, three-generation standard-like models must fulfil the following relation for the extra matter: 2+n2+n 2 = n3+n 3 +12. Then, it is now straightforward to check that only models with n2 = 4, n3 = 6, and therefore nFI2 − nFI3 = 12, may give rise to the Heterotic String unification scale1 (other possibilities for n2, n3 do not even produce the crossing of α3 and α2). This is shown in Fig. 1 for an example with nFI3 = 0, and assuming MS = 500 GeV. There we are using MFI = 2 · 1016 GeV as will be discussed below. Note that at low energy we then have (excluding singlets) 3× {(3, 2) + 2(3̄, 1) + (1, 2)}+ 3× {(3, 1) + (3̄, 1) + 2(1, 2)} , (1) i.e. the matter content of the Supersymmetric SM with three generations of Higgses and vector-like colour triplets. Let us remark that in these constructions only the following patterns of matter September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz 6 Carlos Muñoz with masses of order MFI are allowed: a) nFI3 = 0 , n 2 = 12 → 3× {4(1, 2)} , b) nFI3 = 6 , n 2 = 18 → 3× {(3, 1) + (3̄, 1) + 6(1, 2)} , c) nFI3 = 12 , n 2 = 24 → 3× {2[(3, 1) + (3̄, 1)] + 8(1, 2)} , d) nFI3 = 18 , n 2 = 30 → 3× {3[(3, 1) + (3̄, 1)] + 10(1, 2)} . (2) Thus for a given FI scale, MFI , each one of the four patterns in eq. (2) will give rise to a different value for gGUT . Adjusting MFI appropriately, we can always get MGUT ≈ gGUT ×5.27 ·1017 GeV. In particular this is so for MFI ≈ 2×1016 GeV as shown in Fig. 1. It is remarkable that this number is within the allowed range for the FI breaking scale as discussed above. For the pattern in Fig. 1 corresponding to case a) we have gGUT ≈ 1.1, and therefore MGUT ≈ 5.8 · 1017 GeV. Of course, we cannot claim to have obtained the Heterotic String unification scale until we have shown that the coupling α1 joins the other two couplings at MGUT . The analysis becomes more involved now and a detailed account of this issue can be found in Ref. 1. Let us just mention that the fact that the normalization constant, C, of the U(1)Y hypercharge generator is not fixed in these constructions as in the case of GUTs (e.g., for SU(5), C2 = 3/5) is crucial in order to obtain the unification with the other couplings. Summarizing, the main characteristic of this scenario is the presence at low energy of extra matter. In particular, we have obtained that three generations of Higgses and vector-like colour triplets are necessary. Since more Higgs particles than in the MSSM are present, there will be of course a much richer phenomenology. Note for instance that the presence of six Higgs doublets implies the existence of sixteen physical Higgs bosons, where eleven of them are neutral and five charged. Concerning the three generations of vector-like colour triplets, sayD andD, they should acquire masses above the experimental limit O(200 GeV). This is possible, in principle, through couplings with some of the extra singlets with vanishing hy- percharge, say Ni, which are usually left at low energies, even after the FI breaking. For example, in the model of Ref. 3, there are 13 of these singlets. Thus couplings NiDD might be present. From the electroweak symmetry breaking, the fields Ni a VEV might develop. Note in this sense that the Giudice–Masiero mechanism to generate a µ term through the Kähler potential is not available in prime orbifolds as Z3. Thus an interesting possibility to generate VEVs, given the large number of singlets present in orbifold models, is to consider couplings of the type NiHuHd, similarly to the Next-to-Minimal Supersymmetric Standard Model (NMSSM). It is also worth noticing that some of these singlets might not have the necessary cou- plings to develop VEVs and then their fermionic partners might be candidates for September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz A kind of prediction from string phenomenology: extra matter at low energy 7 right-handed neutrinosf . For the models studied in Refs. 4, 6 the extra colour triplets have non-standard fractional electric charge, ±1/15 and ±1/6 respectively. In fact, the existence of this kind of matter is a generic property of the massless spectrum of supersymmetric models. This means that they have necessarily colour-neutral fractionally charged states, since the triplets bind with the ordinary quarks. For example, the model with triplets with electric charge ±1/6 will have mesons and baryons with charges ±1/2 and ±3/2. On the other hand, the model studied in Ref. 3 has ‘standard’ extra triplets, i.e. with electric charges ∓1/3 and ±2/3; these will therefore give rise to colour-neutral integrally charged states. For example, a d-like quark D forms states of the type uD, uuD, etc. Let us finally mention that a detailed discussion about the stability of these charged states, how to solve possible conflicts with cosmological bounds, and their production modes can be found in Ref. 1. 3. Quark and lepton masses and mixing angles Crucial ingredients in the above analysis were that all three generations of super- symmetric Higgses remain light (Hui , H i ), i = 1, 2, 3, and the FI breaking. And, precisely, both ingredients favour to obtain the correct Yukawa couplings at the renormalizable levelg. Namely, having three families of Higgses introduces more Yukawa couplings, and after the FI breaking some physical particles appear com- bined with other states, and the Yukawa couplings are modified in a well controlled way. This, of course introduces more flexibility in the computation of the mass matrices. Let us recall that the Z3 orbifold is constructed by dividing R 6 by the [SU(3)]3 root lattice modded by the point group (P ) with generator θ, where the action of θ on the lattice basis is θei = ei+1, θei+1 = −(ei + ei+1), with i = 1, 3, 5. The two-dimensional sublattices associated to [SU(3)]3 are shown in Fig. 2. In orbifold constructions, twisted strings appear attached to fixed points under the point group. In the case of the Z3 orbifold there are 27 fixed points under P , and therefore there are 27 twisted sectors. We will denote the three fixed points of each two-dimensional sublattice as shown in Fig. 2. Thus the three generations arise because in addition to the overall factor of 3 coming from the right-moving part of the untwisted matter, the twisted matter come in 9 sets with 3 equivalent sectors on each one. Let us fLet us remark however, that right-handed neutrino superfields with R-parity breaking couplings of the type NiHuHd have been proposed recently 31 to solve the µ problem. gLet us recall that the major problem that one encounters when trying to obtain models with en- tirely renormalizable Yukawas lies at the phenomenological level, and is deeply related to obtaining the correct quark mixing. Summarizing the analyses of Refs. 24, 25, for prime orbifolds with the minimal Higgs content the space selection rules and the need for a fermion hierarchy forces the fermion mass matrices to be diagonal at the renormalizable level. Thus, in these cases the CKM parameters must arise at the non-renormalizable level. For analyses of non-prime orbifolds see Refs. 24, 25, 32. September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz 8 Carlos Muñoz Fig. 2. Two dimensional sublattices (i = 1, 3, 5) of the Z3 orbifold. The fixed point components are also shown. suppose that the two Wilson lines correspond to the first and second sublattices. The three generations correspond to move the third sublattice component (x · o) of the fixed point keeping the other two fixed. As mentioned in the Introduction, we must arrange our up-and down-quark Yukawa couplings in order to have specific off diagonal elements, HuūLαλ u uRγ +Hdd̄Lαλ dRγ . (3) In principle this property arises naturally in the Z3 orbifold with two Wilson lines23−26. For example, if the SU(2) doublet Hu corresponds to (o o o), the three generations of (3,2) quarks to (o o (o, x, ·)) and the three generations of (3̄, 1) up- quarks to (o o (o, x, ·)), then there are three couplings allowed from the space group selection rule (the components of the three fixed points in each sublattice must be either equal or different): λttHut̄LtR associated to (o o o)(o o o)(o o o) with λtt ∼ 1, λcuHuc̄LuR associated to (o o o)(o o x)(o o ·) with λcu ∼ e−T5 , and λucHuūLcR associated to (o o o)(o o ·)(o o x) with λuc ∼ e−T5 . In this simple example one gets one diagonal Yukawa coupling without suppression factor and two off diagonal degenerate ones ∼ e−T5 , but other more realistic examples producing the observed structure of quark and lepton masses and mixing angles can be obtained using three generations of Higgses26,27. Let us first study the situation before taking into account the effect of the FI breaking. Consider for example the following assignments of observable matter to fixed point components in the first two sublattices; Q o o uc o o dc x o Hu o o Hd · o (4) In this case the up- and down-quark mass matrices, assuming three different radii, are given by Mu = gNAu , Md = gNε1A d , (5) where g is the gauge coupling constant, N is proportional to the square root of September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz A kind of prediction from string phenomenology: extra matter at low energy 9 volume of the unit cell for the Z3 lattice, and vu1 v 3 ε5 v vu3 ε5 v vu2 ε5 v 1 ε5 v  , Ad = vd1 v 3ε5 v vd3ε5 v vd2ε5 v 1ε5 v  . (6) Here vui , v i denote the VEVs of the HiggsesH i , H i respectively, and εi = 3 e For example, for T5 ∼ 1.95 one has ǫ5 ∼ 0.05. The elements in the above matrices can be obtained straightforwardly. For ex- ample, if the Higgs Hu1 corresponds to (o,o,o), then since the three generations of (3,2) quarks Q correspond to (o,o,(o,x,·)) and the three generations of (3̄,1) quarks uc to (o,o,(o,x,·)), there are only three allowed couplings, (o,o,o)(o,o,o)(o,o,o) , (o,o,o)(o,o,x)(o,o,·) , (o,o,o)(o,o,·)(o,o,x) . The corresponding suppression factors are given by 1, ε5, ε5 respectively, and are associated with the elements 11, 23, 32 in the matrix Mu. These matrices clearly improve the result obtained with only one Higgs family. However, it is possible to show that although the observed quark mass ratios and Cabbibo angle can be reproduced correctly, the 13 and 23 elements of the CKM matrix cannot be obtained26. Fortunately, this is not the end of the story because the previous result is modified when one takes into account the FI breaking. In particular, it will be possible to get the right spectrum and a CKM with the right form26,27. As discussed in the Introduction and Section 2, some scalars Ci develop large VEVs in order to cancel the FI D-term generated by the anomalous U(1). Thus many particles ξ are expected to acquire a high mass because of the generation of effective mass terms, and in this way vector-like triplets and doublets and also sin- glets become heavy and disappear from the low-energy spectrum. This is the type of extra matter that typically appears in orbifold constructions. The remarkable point is that the SM matter remain massless, surviving through certain combina- tions with other states3,4,14. Let us consider the simplest example, a model with the Yukawa couplings C1ξ1f , C2ξ1ξ2 , (7) where f denotes a SM field, ξ1,2 denote two extra matter fields (triplets, doublets or singlets), and C1,2 are the fields developing large VEVs denoted by 〈C1,2〉 = c1,2. It is worth noting here that f can be an uc, dc, L, νc or ec field, but not a Q field. This is because in these orbifold models no extra (3,2) representations are present, and therefore the Standard Model field Q cannot mix with other representations through Yukawas. Clearly the ‘old’ physical particle f will combine with ξ1,2. It is now straightfor- ward to diagonalise the mass matrix arising from the mass terms in eq. (7) to find September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz 10 Carlos Muñoz two very massive and one massless combination. The latter is given by f ′ ≡ 1√ |c1|2 + |c2|2 (c∗2f − c∗1ξ2) . (8) Notice for example that the mass terms (7) can be rewritten as |c1|2 + |c2|2 ξ1ξ′2, where ξ′2 ≡ 1√|c1|2+|c2|2 (c1f + c2ξ2). Indeed the unitary combination is the massless field in eq. (8). The Yukawa couplings and hence mass matrices of the effective low energy theory are modified accordingly. For example, consider a model where we begin with a Yukawa coupling HQf . Since we have |c1|2 + |c2|2 ′ + c∗1ξ 2) , (9) then the ‘new’ coupling (involving the light state) will beh |c1|2 + |c2|2 HQf ′ . The situation in realistic models is more involved since the fields appear in three copies. All these effects modify the mass matrices of the low-energy effective theory (see Eq. (5)), which, for the example studied in Ref. 26, are now given by Mu = gNau , Md = gNε1ad , (10) where A B = v1ε5β v3ε5 v2α 5β v2 v1α 5β v1ε5 v3α/ε5  , (11) and the parameters af , α, and β depend oni c1,2 ǫ1,3, and their possible values are discussed in Ref. 26. As shown in Ref. 27, for natural values of those parameters and the VEVs, one can find configurations that obey the electroweak symmetry breaking conditions, and can account for the correct quark masses and mixings. In addition to the magnitudes of the CKM matrix elements we also require a CP violating phase. Although it has been shown that observable CP violation cannot be obtained at the renormalizable level in odd order orbifolds34,33 for a minimal Higgs sector, the above matrices having in addition to the ‘mixing’ of states three families of Higgses avoid this problem35. Thus one possibility here (in addition to the one already mentioned in footnote i) is to assume that the VEVs of the moduli hWe should add that the coupling HQξ2, which would induce another contribution to HQf ′, is not in fact allowed. For this to be the case the fields ξ2 and f would have had to have exactly the same U(1)n charges. This is not possible since different particles all have different gauge quantum numbers. iNote that the ci are in general complex VEVs, and therefore they can give rise to a contribution to the CP phase. This mechanism to generate the CP phase through the VEVs of the fields cancelling the FI D-term was used first, in the context of non-renormalisable couplings, in Ref. 24. For a recent analysis, see Ref. 33. September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz A kind of prediction from string phenomenology: extra matter at low energy 11 (a) (b) Fig. 3. Feynman diagrams contributing to ∆mK at tree-level. h s(p) denote scalar (pseudoscalar) Higgses. have an imaginary phase, which can occur when the flat moduli directions are lifted by supersymmetry breaking and find their minimum where the phases are non- zero36,35,37. Such a phase feeds directly into ε5. It is easy to check that this phase is physically observable, and leads to a non-zero δ phase for the CKM matrix which is of order one. Let us finally mention that the correct masses for charged leptons can be ob- tained following a similar approach, as discussed in Ref. [26]. For neutrinos this turns out to be not sufficient, but a see-saw mechanism arising in a natural way in orbifolds might solve the problem [38]. 4. Phenomenological viability of orbifold models with three Higgs families The most challenging implication of an extended Higgs sector is perhaps the oc- currence of tree-level FCNCs mediated by the exchange of neutral Higgs states. Clearly, having six Higgs doublets (and thus six quark Yukawa couplings) the trans- formations diagonalising the fermion mass matrices do not diagonalise the Yukawa interactions. Since experimental data is in good agreement with the SM predictions, where such an effect is not present, the potentially large contributions arising from the tree-level interactions must be suppressed in order to have a model which is experimentally viable. In general, the most stringent limit on the flavour-changing processes emerges from the small value of the KL −KS mass difference39. A detailed discussion of FCNCs in multi-Higgs doublet models was presented in Ref. 40 (see also the references therein). We summarise here some relevant points and apply the method to the orbifold case27, focusing on the neutral kaon sector and investigating the tree-level contributions to ∆mK . The latter is simply defined as the mass difference between the long- and short-lived kaon masses, ∆mK = mKL −mKS ≃ 2 ∣MK12 ∣ = 2 ∣H∆S=2eff , (12) where H∆S=2eff is the effective Hamiltonian for the diagrams in Fig. 3. Once all September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz 12 Carlos Muñoz the contributions to MK12 have been taken into account, the prediction of this orbifold model regarding ∆mK should be compared with the experimental value, (∆mK)exp ≃ 3.49× 10−12 MeV. In Ref. 27 the numerical approach was divided in two steps. Firstly, one focus on the string sector of the model, and for each point in the space generated by the free parameters of the orbifold (ε5, α f ), one derives the up- and down-quark mass matrices and computes the CKMmatrix. Further imposing the conditions associated with electroweak symmetry breaking, and fixing a value for tanβ, one can then determine the values of g N and ε1. A secondary step requires specifying the several Higgs parameters, which must obey the minimum criteria Finally, the last step comprehends the analysis of how each of the Yukawa patterns constrains the Higgs parameters in order to have compatibility with the FCNC data. In particular, we want to investigate how heavy the scalar and pseudoscalar eigenstates are required to be in order to accommodate the observed value of ∆mK . Let us summarize the analysis of the orbifold parameter space by commenting on the relative number of input parameters and number of observables fitted. Working with the six Higgs VEVs (v i ), and the orbifold parameters ε1, ε5, α and αd one can obtain the correct electroweak symmetry breaking (MZ), as well as the correct quark masses and mixings (six masses and three mixing angles). In order to discuss now the tree-level FCNCs, let us remark that the present orbifold model does not include a specific prediction regarding the Higgs sector. For instance, we have no hint regarding the value of the several bilinear terms, nor towards their origin. Concerning the soft breaking terms, the situation is similar. In the absence of further information, we merely assume that the structure of the soft breaking terms is the usual one (see Ref. 27 for further details), taking the Higgs soft breaking masses and the Bµ-terms as free parameters (provided that the electroweak symmetry breaking and minimisation conditions are verified). In the absence of orbifold predictions for the Higgs sector parameters, and mo- tivated by an argument of simplicity, we begin our analysis by considering textures for the soft parameters as simple as possible. In particular, we arrive to four repre- sentative cases with the following associated scalar and pseudoscalar Higgs spectra (a) ms = {82.5, 190.6, 493.9, 515.9, 744.4, 760.2} GeV ; mp = {186.8, 493.9, 515.9, 744.4, 760.2} GeV . (b) ms = {84.6, 213.9, 387.4, 560.8, 785.9, 879.1} GeV ; mp = {215.2, 387.3, 560.5, 785.9, 878.9} GeV . (c) ms = {83.6, 292.9, 733.6, 785.9, 987.6, 1057.0} GeV ; mp = {291.1, 733.6, 785.9, 987.6, 1057.0} GeV . (d) ms = {79.4, 121.5, 296.9, 354.3, 794.6, 808.8} GeV ; mp = {114.8, 296.9, 353.7, 794.6, 808.8} GeV . In Fig. 4 we plot the ratio ∆mK/(∆mK)exp versus ε5, for cases (a)-(d), and tanβ = 5. All the points displayed comply with the bounds from the CKM matrix. September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz A kind of prediction from string phenomenology: extra matter at low energy 13 0.013 0.014 0.015 0.016 0.017 0.013 0.014 0.015 0.016 0.017 Fig. 4. ∆mK/(∆mK)exp as a function of ε5 for tanβ = 5. The Higgs parameters correspond to textures (a)-(d). From Fig. 4 it is clear that it is quite easy for the orbifold model to accommodate the current experimental values for ∆mK . Even though the model presents the possibility of important tree-level contributions to the kaon mass difference, all the textures considered give rise to contributions very close to the experimental value. Although (a) and (b) are not in agreement with the measured value of ∆mK , their contribution is within order of magnitude of (∆mK)exp. As seen from Fig. 4, with a considerably light Higgs spectrum (i.e. mh0 < 1 TeV), one is safely below the experimental bound, as exhibited by cases (c) and (d). This is not entirely unexpected given the strongly hierarchical structure of the Yukawa couplings (notice from Eq. (11) that λd21 is suppressed by ε Let us finally mention that the analysis for other neutral meson systems, Bd, Bs and D 0, can be carried out in an analogous way27. Additionally, and given the existence of flavour violating neutral Higgs couplings, and the possibility of having complex Yukawa couplings, it is natural to have tree- level contributions to CP violation. In the kaon sector, indirect CP violation is parameterised by εK . From experiment one has εK = (2.284 ± 0.014) × 10−3. A comparison of this quantity with the theoretical result in orbifold models can be found in Ref. 27. 5. Conclusions We have attacked the problem of the unification of gauge couplings in Heterotic String constructions. In particular, we have obtained that due to the Fayet- Iliopoulos scale, α3 and α2 cross at the right scale when a certain type of extra September 15, 2021 7:34 WSPC/INSTRUCTION FILE mplareviewMunoz 14 Carlos Muñoz matter is present. In this sense three families of supersymmetric Higgses and vector- like colour triplets might be observed in forthcoming experiments. The unification with α1 is obtained if the model has the appropriate normalization factor of the hypercharge. Let us recall that although we have been working with explicit orb- ifold examples, our arguments are quite general and can be used for other schemes where the Standard Model gauge group with three generations of particles is ob- tained, since extra matter and anomalous U(1)’s are generically present in string compactifications. Another advantage of these models is that they naturally predict three gener- ations, and also that the three generations of Higgs fields give enough freedom to allow an entirely geometric explanation of masses and mixings. The Fayet-Iliopoulos mechanism plays also an important role here. Namely, after the gauge breaking some physical particles appear combined with other states, and the Yukawa couplings are modified in a well controlled way. On the other hand, the presence of six Higgs doublets poses the potential prob- lem of having tree-level FCNCs. 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0704.0988

Evidence for a Massive Protocluster in S255N

Evidence for a Massive Protocluster in S255N C.J. Cyganowski1, C.L. Brogan2, T.R. Hunter2 [email protected] [email protected] [email protected] ABSTRACT S255N is a luminous far-infrared source that contains many indications of active star formation but lacks a prominent near-infrared stellar cluster. We present mid-infrared through radio observations aimed at exploring the evolution- ary state of this region. Our observations include 1.3 mm continuum and spectral line data from the Submillimeter Array, Very Large Array 3.6 cm continuum and 1.3 cm water maser data, and multicolor IRAC images from the Spitzer Space Telescope. The cometary morphology of the previously-known UCH ii region G192.584-0.041 is clearly revealed in our sensitive, multi-configuration 3.6 cm images. The 1.3 mm continuum emission has been resolved into three compact cores, all of which are dominated by dust emission and have radii < 7000 AU. The mass estimates for these cores range from 6 to 35 M⊙. The centroid of the brightest dust core (SMA1) is offset by 1.1′′ (2800 AU) from the peak of the cometary UCH ii region and exhibits the strongest HC3N, CN, and DCN line emission in the region. SMA1 also exhibits compact CH3OH, SiO, and H2CO emission and likely contains a young hot core. We find spatial and kinematic evidence that SMA1 may contain further multiplicity, with one of the compo- nents coincident with a newly-detected H2O maser. There are no mid-infrared point source counterparts to any of the dust cores, further suggesting an early evolutionary phase for these objects. The dominant mid-infrared emission is a diffuse, broadband component that traces the surface of the cometary UCH ii re- gion but is obscured by foreground material on its southern edge. An additional 4.5 µm linear feature emanating to the northeast of SMA1 is aligned with a cluster of methanol masers and likely traces a outflow from a protostar within 1University of Wisconsin, Madison, WI 53706 2NRAO, 520 Edgemont Rd, Charlottesville, VA 22903 http://arxiv.org/abs/0704.0988v1 – 2 – SMA1. Our observations provide direct evidence that S255N is forming a cluster of intermediate to high-mass stars. Subject headings: stars: formation — infrared: stars — ISM: individual (S255N) — ISM: individual (G192.60-MM1) — ISM: individual (G192.584-0.041) — tech- niques: interferometric — submillimeter 1. Introduction The formation and early evolution of stellar clusters occurs in deeply embedded regions of giant molecular clouds (Lada & Lada 2003). While much has been learned from recent sur- veys in the infrared (Gutermuth et al. 2005; Muench et al. 2002), the earliest stages of clus- ter formation will (at least in many cases) be hidden from all but the longest IR/millimeter wavelengths. Due to the small size scales of young clusters–multiplicity of protostars has been observed on scales of a few thousand AU (e.g. Megeath, Wilson, & Corbin 2005)–high angular resolution is necessary to resolve individual objects, particularly in massive star forming regions which lie at relatively large distances (> 1 kpc). These considerations point to millimeter-wavelength interferometric observations of thermal dust continuum emission as an effective means of searching for clusters of young protostars, as the mm emission remains optically thin up to high column densities (NH∼10 25 cm−2 ). Located ∼ 1′ north of the luminous infrared cluster S255IR, S255N is a promising tar- get in the search for young protoclusters. As illustrated in Figure 1, S255N is located in a complicated region of past and ongoing massive star formation. S255N and S255IR (sat- urated in this mid-IR Spitzer IRAC image) lie between two large H ii regions, S255 and S257. Large-scale 12CO and HCN observations show that S255IR and S255N occupy op- posite ends of a molecular ridge between the H ii regions (Heyer et al. 1989). The total luminosity of S255N (∼ 1× 105 L⊙) is about twice that of S255IR (Minier et al. 2005), and single-dish continuum and spectral line observations at submillimeter and millimeter wave- lengths have established the presence of large column densities of dust and gas toward both regions (e.g. Richardson et al. 1985; Mezger et al. 1988; Zinchenko, Henning, & Schreyer 1997; Minier et al. 2005). Observations at infrared and radio wavelengths, however, sug- gest that S255N is the younger of the two regions. For example, S255IR is bright (> 70 Jy) in all infrared bands of the Midcourse Space Experiment (MSX) and contains a well-developed near-IR cluster of early-B type stars (S255-2 Howard, Pipher & Forrest 1997; Itoh et al. 2001), a cluster of compact H ii regions (Snell & Bally 1986), and a wealth of complex H2 emission features (Miralles et al. 1997). In contrast, S255N (also called S255 FIR1 and G192.60-MM1) is undetected in MSX images at wavelengths shorter than – 3 – 21 µm (Crowther & Conti 2003), and contains only a single cometary UC H ii region, G192.584-0.041 (e.g. Kurtz, Churchwell, & Wood 1994). Additional evidence for protostellar activity in S255N exists in the form of outflow tracers. For example, two small knots of H2 emission bracket the UC H ii region and may be tracing an outflow (Miralles et al. 1997). In beamsizes of ∼ 50′′, redshifted CO emission (Heyer et al. 1989) and highly-blueshifted OH absorption (Ruiz et al. 1992) are also seen toward the UC H ii region. Finally, 44 GHz (70-61) A + Class I methanol masers form a linear feature extending northeast of the UC H ii region (Kurtz, Hofner, & Álvarez 2004); masers of this type have been observed in association with molecular outflows in other objects (Plambeck & Menten 1990; Kurtz, Hofner, & Álvarez 2004). At this stage, further understanding of the state of star formation in S255N requires resolving the mm dust continuum emission in order to search for additional compact sources that may be present in the vicinity of the UC H ii region. The high angular resolution now available with the Submillimeter Array (SMA)1 makes this goal possible, while the wide bandwidth allows simultaneous observation of many spectral lines, which are sensitive to a range of gas conditions across the region. We describe our observations in section 2, present our results in section 3, and discuss our interpretations in section 4. A range of distances to S255 and S255N have been used in the literature. At the extremes, Georgelin, Georglein, & Roux (1973) find an optical distance of 3.2 kpc to each of the adjacent H ii regions S254 and S257 (located west of S255N), while Hunter & Massey (1990) find a distance of 1.1 kpc to the optical H ii region S255 (located east of S255N, see Figure 1) based on spectroscopy of its exciting star. Using a LSR velocity of +9 km s−1 (typical of the centroid velocities of our observed lines), and a Galactic center distance of 8.5 kpc, we find a kinematic distance to S255N of 2.2 kpc using the rotation curve of Brand & Blitz (1993). In this paper we adopt a distance of 2.6 kpc for S255N (also see Moffat et al. 1979; Minier et al. 2005). 1The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica. – 4 – 2. Observations 2.1. Submillimeter Array (SMA) Our SMA observations toward S255N were obtained on December 06, 2003 in the com- pact configuration. Six antennas were operational and the projected baseline lengths ranged from 12 to 62 kλ, resulting in a synthesized beam of 4.′′7 × 2.′′4 (P.A.=−45.65◦). In this configuration, the interferometer is not sensitive to smooth structures larger than 17′′. The phase center was 06h12m53s.56, 18◦00′25′′.0 (J2000), and the double-sideband SIS receivers were tuned to an LO frequency of 222.78 GHz. With a bandwidth of ∼2 GHz, the cor- relator covered 216.796-218.764 GHz in lower sideband (LSB) and 226.796-228.764 GHz in upper sideband (USB). The data were resampled to provide uniform spectral resolution of 1.12 km s−1. The zenith opacity as measured by the 225 GHz tipping radiometer at the Caltech Submillimeter Observatory (CSO) was 0.18 at the beginning of the observations and fell as low as 0.16. Data recorded late in the observing session, when the opacity had climbed to 0.3, were not used. The typical system temperature at source transit was 200 K. The primary beamsize of the 6-meter SMA antennas at this frequency is 56′′. Initial calibration of the data was performed in Miriad. The gain calibrators were J0739+016 and J0423-013. J0423-013 was also used for passband calibration. Subsequent processing was carried out in AIPS. The continuum emission was measured using line-free channels and removed in the u− v plane. The resulting continuum-only data were then self- calibrated; these complex gain solutions were also applied to the continuum-subtracted line data. The absolute position uncertainty is estimated to be 0.′′3 and the amplitude calibration is accurate to 20%. For maximum sensitivity, both the continuum and line data were imaged with natural weighting. The rms sensitivity of the continuum image is 4 mJy beam−1 (8 mK) and the rms sensitivity of the line data is 89 mJy beam−1 (170 mK). 2.2. Very Large Array (VLA) Archival 3.6 cm data from the NRAO2 Very Large Array (VLA) were calibrated and imaged in AIPS. The observation date was 2003 June 15 and the total time on source was ∼70 minutes (project code AH819). The VLA was in A-configuration, in which the interferometer is not sensitive to smooth structures larger than ∼7′′. The bandwidth was 50 MHz in two IFs. The flux calibrator was 3C147, and the gain calibrator was J0613+131. 2The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under agreement by the Associated Universities, Inc. – 5 – The synthesized beam of the 3.6 cm continuum image is 0.′′27× 0.′′23 (P.A.=77.51◦) and the rms sensitivity is 18 µJy beam−1. Archival 3.6 cm data from the B configuration (in which the interferometer is not sensitive to smooth structures larger than ∼20′′), observed on 1990 August 27 (project code AM301), and from the the CnB configuration, observed on 2005 June 21 (project code TSTCJC), were also reduced and combined with the A-configuration data. The resulting image, made with a UV taper at 500 kλ, has a synthesized beam of 1.′′03× 0.′′84 (P.A.=−81.42◦) and a rms sensitivity of 34 µJy beam−1. Archival 1.3 cm data from the VLA A-configuration (project code AC299) were analyzed for water maser emission at 22.235 GHz. The observation date was 1991 August 1, total on-source integration time was ∼10 minutes. The phase center of this observation was toward the IR cluster ∼ 1′ to the south of S255N, hence a correction for the primary beam attenuation has been applied to the data. The bandpass calibrator was 3C84 and the flux calibration was derived assuming a flux density of 2.16 Jy for J0528+134. The synthesized beam is 0.′′18 × 0.′′16 (P.A.=−56.55◦), the spectral line channel spacing is 0.33 km s−1, and the rms noise is 0.13 Jy beam−1. 2.3. Spitzer Space Telescope Mid-infrared images of S255 were obtained with the IRAC camera (Fazio et al. 2004) on the Spitzer Space Telescope as part of Guaranteed Time Observations program 201 (P.I. G. Fazio) on 12 March 2004. Integrations of 0.4 s and 10.4 s were taken in the high dynamic range mode; S255N is not saturated in the longer exposures, and only the 10.4 s exposures are discussed in this paper. Four 10.4 s exposures covered S255N, for a total integration time on S255N of 41.6 s. Mosaiced post-BCD 3.6, 4.5, 5.8, and 8.0 µm images, calibrated and processed using pipeline version S13.2.0, were downloaded from the Spitzer data archive. 2.4. Caltech Submillimeter Observatory Our submillimeter continuum observations were obtained at the CSO using the Submil- limeter High Angular Resolution Camera (SHARC), a 3He-cooled monolithic silicon bolome- ter array of 24 pixels in a linear arrangement (Hunter, Benford & Serabyn 1996; Wang et al. 1996). For a typical dust source with a submillimeter spectral index of ∼ 4, the effective frequency of the broadband 350 µm filter is 852 GHz and the bandwidth is 103 GHz. An on-the-fly (OTF) map of S255 was obtained on 21 December 1995 by scanning the array through the source in azimuth while the secondary mirror was chopping at a rate of 4.1 – 6 – Hz and a throw of 88′′. Successive scans were made after stepping the array in elevation by increments of 5′′. Airmass corrections were applied to each scan using the opacity de- rived from frequent scans of Saturn during the night. The map data were restored with a NOD2 dual beam restoration algorithm (Emerson, Klein & Haslam 1979) and transformed into equatorial coordinates. The resulting image was smoothed with a Gaussian to produce an effective half-power beamsize of 15′′. 3. Results 3.1. Continuum emission Our 1.3 mm SMA continuum data resolve three distinct sources within the previously- observed submm/mm clump of S255N (aka S255 FIR1 and G192.60-MM1: Jaffe et al. 1984; Mezger et al. 1988; Minier et al. 2005). Figure 2 shows the SMA 1.3 mm continuum image with CSO 350 µm contours superposed. As illustrated in Figure 2, the strongest SMA 1.3 mm emission peak coincides with the CSO 350 µm peak (resolution 15′′). The CSO 350 µm integrated flux density is 575±20 Jy, consistent to within 10% of the value predicted by the dust spectral energy distribution (SED) models of Minier et al. (2005). The three mm sources resolved with the SMA are designated SMA1, SMA2, and SMA3 in order of descending peak intensity. The observed properties of each source (peak intensity, brightness temperature, integrated flux density, and source size) are listed in Table 1, and the sources are labeled in Figure 2. The integrated flux densities and source sizes listed in Table 1 were determined by fitting a single Gaussian component to each SMA source. SMA1 was not well fit by a single Gaussian, indicating that the observed continuum emission may arise from multiple sources unresolved by the SMA beam; this issue is discussed further in §4.2. The total flux density of the three compact mm sources is 0.79±0.16 Jy; stated errors include the 20% uncertainty in flux calibration. This total corresponds to 15±3% of the single-dish flux density measured by Minier et al. (2005) with the SEST 15m telescope at 1.2mm (resolution 24′′). Figure 3 compares the morphology of the 1.3 mm dust continuum emission with the 3.6 cm free-free continuum emission from the cometary UC H ii region, G192.584-0.041. Figure 3a shows the lower-resolution (1.′′03 × 0.′′84) 3.6 cm VLA image superposed on the SMA 1.3 mm continuum, while Figure 3b shows the high-resolution VLA 3.6 cm image (0.′′27× 0.′′23), with the positions of the newly-reported water maser (see §3.2) and the Class I methanol masers detected by Kurtz, Hofner, & Álvarez (2004) indicated. The ∼ 1′′ resolution 3.6 cm VLA image presented in Figure 3a provides the most detailed – 7 – view to date of the diffuse cometary “tail” of the UC H ii region. The integrated flux density of G192.584-0.041 measured from this image is 26.0±0.1 mJy. Based on our measurement and published 2 cm integrated flux densities for G192.584-0.041 (Kurtz, Churchwell, & Wood 1994; Rengarajan & Ho 1996), the spectral index from 3.6 cm to 2 cm is ∼-0.1, consistent with optically thin free-free emission. The flux density is consistent with a single exciting star of spectral type B0.5 (as determined by Kurtz, Churchwell, & Wood 1994; Snell & Bally 1986). Extrapolating to 1.3 mm, we estimate the free-free contribution of G192.584-0.041 to the 1.3 mm flux density of SMA1 to be .20 mJy (≤3.5%). The 3.6 cm VLA image presented in Figure 3b is the highest-resolution cm-wavelength image of G192.584-0.041 to date. With a resolution of 0.′′27× 0.′′23, the continuum emission from G192.584-0.041 is resolved into three components (east to west): an arc, a point source, and an extended feature (the extended “tail” is resolved out in the higher resolution data). All three of these components overlap with the eastern side of SMA1, but none is coincident with the mm emission peak, in agreement with the estimate that the free-free contribution at 1.3 mm is quite small. The arc, which is oriented with its convex side towards the mm peak, contains the brightest 3.6 cm emission. The 3.6 cm peak is located in the southern part of the arc, east of the point source, and is offset by 1.′′1 (∼2800 AU) from the location of the SMA1 mm continuum peak determined by fitting a single Gaussian component. The 3.6 cm point source, which is located west of the arc and faces its concave side, is offset by 1.′′6 (∼4,200 AU) from the SMA1 mm continuum peak. The peak brightness temperature of the 3.6 cm point source is only 122 K at the current resolution. Absent a second radio frequency image with comparable resolution, it is not currently possible to ascertain the spectral indices of the individual components. The 3.6 cm images presented in Figure 3(a-b) place strong limits on the presence of any additional H ii regions in S255N. Other than G192.584-0.041, no cm-wavelength emission is detected down to a 5σ limit of 90 µJy beam−1 (high-resolution image). 3.2. Water maser emission Water maser emission was detected at the position 06h12m53s.71, 18◦00′27.′′6 (J2000), offset 0.′′9 (∼2,300 AU at 2.6 kpc) to the northeast of the 1.3 mm continuum emission peak of SMA1, as determined by fitting a single Gaussian component. This is the first report of water maser emission from S255N. The peak intensity is 2.8 Jy beam−1 (corrected for primary beam attenuation) at vLSR=+8.5 km s −1. The line is barely resolved by the 0.33 km s−1 spectral resolution. The positions of the Class I 44 GHz (70-61) A + methanol masers detected by Kurtz, Hofner, & Álvarez – 8 – (2004) are marked with crosses in Fig. 3b, which shows the 1.3 mm and 3.6 cm continuum emission and the newly-reported water maser. Kurtz, Hofner, & Álvarez (2004) estimate an astrometric uncertainty of 0.′′5 for the CH3OH maser spots, while the absolute astrometry of the H2O maser is better than 0. ′′1. The position of one of the CH3OH maser spots is consistent with SMA1, within the astrometric uncertainty. The newly detected water maser is < 1′′ from two CH3OH masers, and falls into the linear pattern of 44 GHz (70-61) A CH3OH maser spots that extends northeast from SMA1. 3.3. Line emission Molecular line emission from H2CO, CH3OH, SiO, CN, DCN, and HC3N is detected in S255N; the specific transitions, frequencies, and upper state energies are listed in Table 2. The lines detected in S255N are the same as those detected in the spectral regions covered by our sidebands by Sutton et al. (1985) in their line survey of the Orion A molecular cloud, which is similar to our data in spectral resolution (1.3 km/s), rms sensitivity (0.2 K), and linear size scale (30′′= ∼13,500 AU at 450 pc). Integrated intensity images for CH3OH, SiO, and H2CO are presented in Figure 4(a-d) and for DCN, HC3N, and CN in Figure 5(a-c). The distributions of molecular emission observed in S255N fall into two main categories: CH3OH, H2CO and SiO exhibit emission from multiple locations, while DCN and HC3N are detected only in the vicinity of SMA1. CN exhibits compact emission towards SMA1, and is also weakly detected toward SMA3. The CN lines have the lowest Eupper of the observed transitions, and the CN images show artifacts from large-scale emission resolved out by the interferometer, suggesting that much of the CN emission originates in an extended, cool envelope around the compact continuum sources. As shown in Figure 4(a-d), the spatial distributions of the integrated emission from H2CO, CH3OH, and SiO are similar to one another. The kinematics of these molecules are complex, as illustrated in Figures 6 and 7, with multiple spatially and kinematically distinct components apparent. A finder chart for the positions of the profiles displayed in Fig. 7 (named after their relative positions with respect to SMA1) is shown in Figure 5(d). The positions are listed in Table 3. Line centroid velocities, ∆vFWHM , and integrated line intensities obtained from Gaussian fits to the line profiles at these positions are listed in Table 4. Unless otherwise noted, the fit parameters in Table 4 are for the strongest component in the spectrum. Fits to SiO line profiles are not included because the SiO line shapes are so complex. The strongest molecular emission in S255N lies toward the “SW” position ∼ 6′′ to the southwest of SMA1 at a peak velocity between ∼ 6 − 8 km s−1 (Figs. 4, 6, 7c, Table 4). – 9 – This molecular emission is not coincident with any mm continuum emission and overlaps the southern edge of the extended “tail” of the UC H ii region (Fig. 4). The line emission toward the “SW” position is broader than at any other location in S255N, with ∆vFWHM ∼ 7 and 9 km s−1 for CH3OH and H2CO, respectively, and shows pronounced blue wings. The velocity of the H2CO peak is slightly more blueshifted than CH3OH, while SiO is significantly blueshifted relative to both H2CO and CH3OH (Fig. 7c, Table 4). The second-brightest region of H2CO, CH3OH, and SiO emission in S255N is located in the vicinity of SMA1, east of the cometary head of the UCH ii region (Fig. 4). DCN, CN, and HC3N have their strongest emission in this area (Fig. 5). Spectral line profiles for DCN, CN, and HC3N are shown in Figure 8 and channel maps for DCN are shown in Figure 9. Two positionally and kinematically distinct components are evident in H2CO, DCN, CN, HC3N, and weakly, CH3OH, in the vicinity of SMA1. One component (denoted SMA1-NE) lies 1.′′21 to the northeast of the SMA1 mm peak at a velocity of ∼ 7 km s−1, and the other (denoted SMA1-SW) lies 1.′′17 southwest of the SMA1 mm peak at ∼ 11.5 km s−1 (Figs. 6, 7a,b, and 8a,b). SMA1-SW is 1.′′08 east of the 3.6 cm point source. Interestingly, SMA1-NE and SMA1-SW also show differences in their chemical proper- ties. For example, H2CO shows nearly equal strength towards both positions, as does HC3N, while CH3OH is much stronger toward SMA1-SW (Fig. 6, 8). In contrast, DCN and CN are both significantly stronger toward SMA1-NE (Fig. 8). Some differences in the peak velocities at the two positions are also apparent amongst species. Relative to the other molecules, SiO is significantly blueshifted (vLSR< 5 km s −1) towards both SMA1-SW and SMA1-NE (Fig. 7, Table 4). DCN is slightly redshifted relative to CN and HC3N toward both positions (Fig. 8, Table 4). The CN and HC3N lines are also more than twice as broad as those of H2CO or CH3OH toward SMA1-SW (Table 4). The H2CO, CH3OH, and SiO integrated intensity peak located ∼5 ′′ north of the mm continuum source SMA2 (Fig. 4, “NW” position in Fig. 5d), is comprised of relatively weak, broad emission (Fig. 7f). The H2CO and CH3OH lines are narrower than at the SW position, but broader than at any of the other positions (Table 4). As at the other positions, the peak of the SiO line profile is blueshifted relative to H2CO and CH3OH (Fig. 7f). The line emission north and northeast of the SMA1 region (“N” and “NE” positions, finding chart Fig. 5d) consists of narrow velocity features in CH3OH and H2CO, and, at the NE position, SiO (Figs. 6 & 7d-e). In contrast to the other positions, at the NE position CH3OH, H2CO, and SiO have the same velocity, vLSR∼8 km s −1(Fig. 7d, Ta- ble 4). The velocity of the H2CO and CH3OH emission at the N position is similar to the vLSR∼11.5 km s −1 component toward SMA1-SW (Table 4). The H2CO and CH3OH lines are narrow, and the CH3OH peak is slightly blueshifted relative to H2CO. Broad and – 10 – weak blueshifted SiO emission is also detected at this position (Table 4, Fig. 7e). 3.4. Spitzer Space Telescope IRAC Observations Figure 10 shows a three-color IRAC image (red 8.0 µm, green 4.5 µm, blue 3.6 µm) of S255N, overlaid with contours of the 1.3 mm continuum emission (yellow) and the 3.6 cm continuum emission (white). The positions of the Class I CH3OH masers reported by Kurtz, Hofner, & Álvarez (2004) and the newly-reported water maser are marked with crosses. Most of the observed mid-IR emission is offset to the northwest of the UC H ii region, and this diffuse emission appears in all IRAC bands. An exception is the linear, green 4.5 µm emis- sion feature that extends NE from the SMA1 mm continuum peak. No mid-IR emission is associated with either SMA2 or SMA3, indeed these two positions are notably absent of IR emission. 4. Discussion 4.1. Mass estimates from the dust emission With an estimate of the dust temperature, we can estimate the masses of the compact dust sources SMA1, SMA2, and SMA3 using a simple isothermal model of optically thin dust emission (Beltrán et al. 2006): Mgas,thin = R Fν D B(ν, Td) κν where R is the gas-to-dust mass ratio (assumed to be 100), Fν is the observed flux density, D is the distance to the source, B(ν, Td) is the Planck function, and κν is the dust mass opacity coefficient. At 1.3 mm, the value of κ for gas densities of 106-108 cm−3 does not differ much for grains with thick or thin ice mantles; we adopt a value of κ1.3mm=1 cm 2 g−1 for all of the compact mm sources (Ossenkopf & Henning 1994). The assumption of low optical depth is justified by the low observed millimeter brightness temperatures (Table 1), however, for highest accuracy we have made the small correction to our derived masses for non-zero optical depth using the formula: Mgas = Mgas,thinτ/(1− e −τ ). Previous determinations of the dust temperature and mass in S255N have relied on fitting multiple components to the (unresolved) mid-IR to mm SED. Minier et al. (2005) fit a hot, compact core (T = 106 K, diameter = 400 AU) and an extended warm envelope (T = 44 K, diameter = 58,000AU) to a SED comprised of MSX, IRAS, SCUBA, and SEST data, – 11 – assuming a distance of 2.6 kpc. The derived luminosity and gas mass are 1.1× 105 L⊙ and 220 M⊙, respectively. While many of the datapoints in the SED constructed by Minier et al. (2005) blend emission from all three SMA sources, the very compact hot core implied by their fits would be unresolved by our SMA beam (∼12,200 × 6,200 AU at 2.6 kpc). Thus, the hot core temperature of 106 K derived from the SED modeling provides an upper limit for the dust temperature of the compact SMA sources. The fitted warm envelope temperature of 44 K is likewise a good lower limit to the temperature of SMA1 since it dominates the 1.3 mm flux, contributing 73% of the total SMA flux density. Several single dish estimates of the gas temperature are also available. For example, Effelsberg 100-m observations of NH3 (1,1) and (2,2) (resolution 40 ′′) suggest that the kinetic temperature of the gas is only Tkin = 23 ± 1 K (Zinchenko, Henning, & Schreyer 1997). Measurements of CH3C2H(6-5) K=0-3 toward S255N with the Onsala 20-m (resolution 38 by Malafeev et al. (2005) yield Trot = 35 ± 1 K, in better agreement with the extended warm component derived for the dust. These authors find significantly higher temperatures using CH3C2H(6-5) than NH3 towards all five sources observed (including S255) and suggest that methyl acetylene may preferentially trace warmer/denser gas. In any case, since beam dilution may play a significant role in these single dish estimates, they can only provide a lower limit to the gas temperatures on SMA sizescales. From our SMA line data it is clear that SMA1 is the warmest of the compact mm sources: the two detected transitions with the highest Eupper (both HC3N, Eupper=131 K and 142 K) are detected only towards SMA1, suggesting this may be a hot core (e.g. Hatchell, Millar, & Rodgers 1998). DCN is also seen at this warm position. Though DCN is formed in cold clouds, in this case it can serve as a young hot core tracer since its presence in this warm region suggests it has recently been liberated from the icy mantles of dust grains (e.g. Mangum et al. 1991). The HC3N emission is consistent with an upper temperature limit of ∼100 K; a more quantitative determination is not possible with only two observed transitions. In contrast, the ratio of the H2CO(30,3–20,2) and H2CO(32,2–22,1) lines is a reli- able density-independent temperature diagnostic for TK . 50 K, and N(para-H2CO)/∆v . 1013.5 cm−2 (km s−1)−1 (Mangum & Wootten 1993). In this regime, the 30,3–20,2/32,2–22,1 ra- tio ranges from 15 to 5 for Tk=20 to 50 K. In contrast, the observed line ratios in S255N are less than 2.5 throughout the imaged region and are smallest (∼ 1.5) toward SMA1, suggesting that the column density (i.e. opacity) and/or temperature is too high for these lines to be diagnostic. For the position of SMA1-NE, assuming a temperature of 75 K, NH2CO∼3.7 × 10 13 cm−2 from the H2CO(30,3–20,2) line and NH2CO∼1.0 × 10 14 cm−2 from the H2CO(32,2–22,1). This comparison suggests that the 30,3–20,2 line is moderately optically thick compared to the 32,2–22,1 line, and that the low line ratios are a combination of both the column density and temperature being higher than the diagnostic range of these two – 12 – transitions. Combining this analysis with the SED models and the single dish line results described above, the allowed temperature range for SMA1 is 40 - 100 K. The resulting ranges of gas mass, column density, and number density computed for SMA1 are shown in Table 5. The high derived gas density (nH2∼3-16×10 6 cm−3), also implied by the presence of the water maser, indicates that the gas and dust temperatures are likely to be well-coupled (e.g. Kaufman, Hollenbach, & Tielens 1998; Ceccarelli, Hollenbach & Tielens 1996). Unlike SMA1, SMA2 and SMA3 are not accompanied by significant line emission. H2CO, CH3OH, and SiO emission are present to the north of SMA2, and CN emission is detected toward SMA3 (§3.3), but the physical relationship (if any) between this line emission and the dust continuum sources is unclear. Also unlike SMA1, SMA2 and SMA3 are not associated with mid-IR emission in any IRAC band (§3.4). Instead, SMA2 and SMA3 appear to be cold, dark, young mm cores, without evidence for current star formation. On the basis of the lack of line and mid-IR emission towards SMA2 and SMA3, we adopt a lower temperature limit of 20 K and an upper temperature limit of 40 K for these sources. The corresponding range of masses, column densities, and number densities for SMA2 and SMA3 are tabulated in Table 5. The mass of each (7-17 M⊙ for SMA2, 6-13 M⊙ for SMA3) is sufficient to form a low to intermediate mass star. No other cores are detected in the field to a 5σ upper limit of M < 3M⊙ (at T = 20 K). 4.2. Velocity Structure and Outflows Figure 10 shows a close-up view of the Spitzer 3-color IRAC image shown in Figure 1. The brightest mid-IR emission is extended along a NE-SW axis, approximately parallel to the axis of the UCH ii region, but with an offset to the northwest of ∼ 2′′. The overall morphology of S255N is consistent with the multi-band bright mid-IR emission tracing the surface of the UCH ii region, which is less dense to the southwest (of SMA1), as indicated by the diffuse “tail” of the cometary UC H ii region extending in this direction. However, the detailed interpretation of the mid-IR emission toward S255N is complicated by the offset described above, and the sharp cutoff of the 3.6 and 8.0 µm emission along the southeast boundary of the UCH ii region. Indeed, mid-IR emission is notably absent toward SMA2 and SMA3, as well as toward much of SMA1. A likely scenario for this behavior is absorption of the mid-IR emission by the high column density mm cores; in this scenario the bulk of the relatively cold mm cores must be in front of the UCH ii region. With the exception of SMA1, the molecular emission in S255N is not obviously associ- ated with any continuum emission, and is therefore unlikely to be centrally heated. However, as described in §4.1 the H2CO line ratios suggest the gas is warm. Thus, it is likely that – 13 – much of this emission is associated with outflow material, although the number of outflows and their driving source(s) are unclear. Published data on large-scale outflows in the region (e.g. Miralles et al. 1997; Richardson et al. 1985; Heyer et al. 1989; Ruiz et al. 1992) are un- fortunately too low in angular resolution to be useful in distinguishing outflows associated with S255N from those associated with S255IR to the south, and/or the maps are swamped by emission from a large outflow flowing north from S255IR. Excluding the “SW” position, the relatively narrow linewidths of these S255N line emission regions suggest that they are density enhancements within a larger extended flow resolved out by the interferometer. The relative similarity of the line center velocities further suggests that the outflows are mostly in the plane of the sky. The linear morphology of the (green) 4.5 µm emission northeast of SMA1 is suggestive of an outflow (Fig. 10). Such 4.5 µm nebulosity is a conspicuous feature of IRAC images of star forming regions. Recent analysis of the massive DR21 outflow, the best-studied ex- ample, has shown that H2 line emission accounts for ∼50% of the observed 4.5 µm IRAC flux, and that the outflow morphology is almost identical in IRAC 4.5 µm and narrow-band 2.122 µm (H2 1-0 S(1) line) images (Davis et al. 2007; Smith et al. 2006). In S255N, the 2.122 µm H2 clump S255:H2-3 lies at the base of the 4.5 µm nebulosity; the H2 clump is also coincident, within reported astrometric uncertainties, with our newly-reported water maser. Both the 2.122 µm H2 1-0 S(1) line and the 4.6947 µm H2 0-0 S(9) line, identified by Smith et al. (2006) as the dominant contributor to IRAC band 2, trace moderate-velocity shocks (Draine, Roberge, & Dalgarno 1983). Recent models by Smith & Rosen (2005) of shocks in dense protostellar molecular jets predict that the integrated H2 line contribution to IRAC band 2 will be 5-14 times greater than to IRAC band 1 (3.6 µm), consistent with the ratio of the emission seen in these bands toward the linear 4.5 µm feature. The 44 GHz Class I methanol masers, five of which lie along the 4.5 µm emission feature, pro- vide further evidence for its identification as an outflow. Kurtz, Hofner, & Álvarez (2004) found that masers of this type are often found in association with such outflow tracers as SiO (IRAS 20126+4104, G31.41+0.31 and G34.26+0.15) and H2 (IRAS 20126+4104). In G31.41+0.31, the 44 GHz methanol masers are also associated with thermal methanol emission (Kurtz, Hofner, & Álvarez 2004), which Liechti & Walmsley (1997) found traced shock/clump interfaces in the DR21 outflow. The parallels between these examples and S255N strongly suggest that the 44 GHz methanol masers, H2CO, CH3OH, SiO, H2, and 4.5 µm emission extending northeast from SMA1-NE trace a molecular outflow from a pro- tostar, probably SMA1-NE. At the SW position, the broad lines with strong blue wings combined with the morphol- ogy of the H2CO and CH3OH channel maps are consistent with a blueshifted outflow lobe driven by SMA1-SW. Notably, no 44 GHz methanol masers coincide with the very strong – 14 – thermal CH3OH emission of the SW line peak, although elsewhere in S255N the methanol maser flux densities are loosely correlated with the strength of thermal CH3OH emission. The absence of masers towards the SW position suggests that the physical conditions are not appropriate for the collisional pumping of Class I methanol masers (Cragg et al. 1992; Plambeck & Menten 1990). 4.3. The Nature of SMA1 The complex kinematic behavior of the molecular line emission in the vicinity of SMA1 including SMA1-NE, SMA1-SW, and position “SW” (§3.3) is difficult to explain in the context of a single protostar. Though SMA1-NE and SMA1-SW could be interpreted as the blue and red-shifted lobes, respectively, of an outflow, this scenario does not explain the very blue-shifted emission further to the southwest at position “SW”. Rotation also seems like an unlikely explanation for the velocity gradient between SMA1-NE and SMA1-SW since the gradient is parallel to the direction of the two probable outflow regions: the 4.5 µm emission to the northeast and “SW” to the southwest. Instead, the combination of the chemical and kinematic differentiation between SMA1-SW and SMA1-NE suggests the presence of two individual sources, one at +7 km s−1 and one at +11.5 km s−1. To investigate this possibility, in Figure 11 we show a uniform weighted SMA millimeter continuum image restored with a beam of 1.′′0 (∼ 3 times smaller than the longest observed baseline), which essentially reveals the location of the clean components. The localization of clean components into two main regions in the vicinity of SMA1 suggests the presence of at least two sources separated by ∼1.′′84 (4800 AU). If these two clean component peaks correspond to real dust sources, their positions are in good agreement with the two kinematically distinct formaldehyde peaks (< 0.′′1 and < 0.′′5). The northeast component of the pair is also within 0.′′2 of the water maser. That two distinct protostars would exist with this separation is reasonable, as multiplicity of protostars has been observed on scales of <6,000 AU (e.g. Megeath, Wilson, & Corbin 2005). Although the presence of two protostars is a plausible interpretation, it clearly requires higher resolution continuum observations for confirmation. In any case, the molecular line emission from SMA1-NE and SMA1-SW is reminiscent of hot molecular cores (HMCs), particularly the detection of DCN and HC3N. These molecules– like CH3OH and H2CO, which also show strong emission towards SMA1-NE and SMA1-SW- are present in the gas phase in HMCs because they have been evaporated from grain mantles (e.g. Caselli 2005; Szczepanski et al. 2005). Complex organic molecules such as HCOOCH3, however, are believed to be “daughter” species, formed in the gas phase by reactions of “parent” species such as H2CO and CH3OH (Caselli 2005). Thus, while SMA1-NE and – 15 – SMA1-SW do not exhibit the truly copious molecular emission observed towards some HMCs (e.g. Hatchell, Millar, & Rodgers 1998; Schilke et al. 2006), this is consistent with SMA1-NE and SMA1-SW being very young sources, in which gas-phase hot-core chemistry has not yet produced abundant complex organic molecules. The line emission from the SMA1 sources is unusual in that the DCN(3-2) emission is stronger than HC3N(24–23). By modeling deuterium chemistry, Roberts & Millar (2000) find that the steady-state abundance of DCN in molecular clouds is a complicated function of temperature and density (see their Figure 7), but generally higher at low metallicity. We note this because S255N is located approximately (l ∼ 192◦) in the direction of the Galactic anticenter, and may have lower metallicity than inner-galaxy star-forming regions (Daflon & Cunha 2004; Afflerbach et al. 1997). Our SMA data, however, do not allow us to disentangle the effects of abundance and excitation on the strengths of DCN and HC3N emission. The geometry of a HMC located a few arcseconds ahead of the vertex of a cometary UCH ii region has been seen in other objects observed at high angular resolution, and this notable configuration has led to much discussion on the energy source responsible for these HMCs. The best-studied case for external heating is G34.26+0.15, in which Watt & Mundy (1999) and Mookerjea et al. (2007) argue that HMC emission (characterized by complex ni- trogen and oxygen-rich molecules) arises in gas heated by component C, the most evolved of three nearby UCH ii regions. In the absence of extinction, a 1.1 × 104 L⊙ source at the location of the 3.6 cm point source could heat SMA1-NE to ∼37 K, and SMA1-SW to ∼51 K, consistent only with the lower end of the range of plausible gas temperatures. In contrast, G29.96-0.02 is the prototype for a HMC located ahead of a cometary UCH ii region and internally heated by a high mass protostellar object (De Buizer, Osorio, & Calvet 2005; Gibb, Wyrowski, & Mundy 2003). In G29.96-0.02, HMC emission is coincident with a re- solved 1.4 mm continuum source, water maser spots, and a mid-IR sub-arcsecond point source (De Buizer, Osorio, & Calvet 2005; Gibb, Wyrowski, & Mundy 2003; Olmi et al. 2003, and references therein). In S255N, the arrangement of the 1.3 mm and 3.6 cm sources along with the presence of water maser emission coincident with the molecular emission closely resembles the case of G29.96-0.02. We favor the interpretation that one or more sources younger than the excitation source of the UCH ii region are present and responsible for the compact dust and molecular line emission from SMA1, consistent with the interpretation of the hot core emission outlined above. – 16 – 5. Conclusions Our multiwavelength observations of S255N reveal significant new details in this lu- minous star-forming region. While the previously-identified UCH ii region dominates the cm continuum and mid-IR emission, the 1.3 mm continuum emission has been resolved into three compact cores (SMA1, SMA2, and SMA3) clustered on scales of 0.1-0.2 pc. Dominated by dust emission, these cores range in mass from 6 to 35 M⊙. There are no mid-infrared point source counterparts to any of the dust cores, suggesting an early evolutionary phase. The spectral line emission at the position of the brightest core, SMA1, is spatially compact and includes HC3N, CN, DCN, CH3OH, SiO, and H2CO. SMA1 appears to be a developing hot core offset by a few thousand AU from the UCH ii region. The chemical and kinematic structure toward SMA1 is suggestive of further multiplicity at these scales. A 4.5 µm linear feature emanating to the northeast of SMA1 is aligned with a cluster of methanol masers and likely traces a outflow from a protostar within SMA1. We conclude that S255N is actively forming a cluster of intermediate to high-mass stars. In addition, we speculate that some of the missing flux in the SMA continuum image could be in the form of additional compact low-mass, cold dust cores that lie below the sensitivity limit of our observations (M ∼ 3M⊙ at T = 20 K). Higher-resolution and more sensitive observations are needed to search for ad- ditional protostars in S255N and other young protoclusters. Resolving individual protostars in regions like these is a necessary task to determine how dense these young protoclusters are, and how interactions among protostars in protoclusters may affect the process of star formation. 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The SMA beam is 4.′′7× 2.′′4 (P.A.=−45.65◦). bUncertainties include 20% calibration uncertainty. cBrightness temperature computed using the Rayleigh-Jeans approximation. – 21 – Table 2. Molecular species and transitions observed in S255N Species Transition Frequency Eupper/k (GHz) (K) SiO 5→4 217.104980 31.2 DCN 3→2 217.238538 20.9 H2CO 30,3 → 20,2 218.222192 21.0 HC3N 24→23 218.324723 131 CH3OH 4+2,2,0 → 3+1,2,0 218.440050 45.5 H2CO 32,2 → 22,1 218.475632 68.1 H2CO 32,1 → 22,0 218.760066 68.1 CNa,b 20,3,3 → 10,2,2 226.874191 16.4 CNa,c 20,3,4 → 10,2,3 226.874781 16.4 CNa 20,3,2 → 10,2,1 226.875896 16.4 CN 20,3,2 → 10,2,2 226.887420 16.4 CN 20,3,3 → 10,2,3 226.892128 16.4 HC3N 25→24 227.418905 142 aThese components are blended in our spectra. bThe CN hyperfine components at frequencies lower than this one lie outside of our observed band- pass. cThis transition is used to set the velocity scale for CN in Fig 8. – 22 – Table 3. Spectral line positions J2000 coordinates Name α (h m s) δ (◦ ′ ′′) SMA1-NE 06 12 53.73 +18 00 27.8 SMA1-SW 06 12 53.64 +18 00 25.4 SW 06 12 53.45 +18 00 23.0 NE 06 12 53.76 +18 00 33.2 N 06 12 53.70 +18 00 39.8 NW 06 12 52.86 +18 00 36.2 Table 4. Fitted line properties H2CO(30,3–20,2) CH3OH DCN HC3N(24-23) Position Center Width Svdv Center Width Svdv Center Width Svdv Center Width km s−1 km s−1 Jy b−1*km s−1 km s−1 km s−1 Jy b−1*km s−1 km s−1 km s−1 Jy b−1*km s−1 km s−1 km s−1 Jy b−1*km s−1 SMA1-NE 6.9(0.3)a 5.9(0.6)a 5.5(0.7)a 6.6(0.3)c 3.4(0.9)c 0.9(0.3)c 8.2(0.1) 3.3(0.2) 5.2(0.4) 6.3(0.3)a 3.5(0.8)a 1.9(0.6)a SMA1-SW 12.1(0.1) 2.9(0.2) 4.2(0.3) 11.2(0.1) 2.4(0.2) 3.1(0.4) 10.7(0.3)b 5.6(0.7)b 3.1(0.5)b 9.5(0.8)a 11.2(2.1)a 3.0(0.7)a SW 6.3(0.1) 9.2(0.4) 24.3(1.2) 7.8(0.1) 6.9(0.3) 13.4(0.8) · · · · · · · · · · · · · · · · · · NE 8.3(0.1) 2.9(0.3) 2.4(0.4) 8.3(0.2) 2.5(0.4) 1.6(0.4) · · · · · · · · · · · · · · · · · · N 11.5(0.1) 2.6(0.3) 2.4(0.3) 10.1(0.2) 2.5(0.5) 1.5(0.4) · · · · · · · · · · · · · · · · · · NW 8.9(0.2) 5.8(0.6) 5.3(0.7) 8.3(0.2)a 5.7(0.5)a 3.7(0.5)a · · · · · · · · · · · · · · · · · · aNot well fit by a single Gaussian. bGaussian fit encompasses two blended components. cParameters for second-strongest velocity component. Strongest component is very similar to that towards SMA1-SW. – 24 – Table 5. Range of estimated masses of dust cores in S255N κ Tdust τdust M NH2 b nH2 Sourcea (cm2 g−1) (K) (1.3mm) (M⊙) (10 23cm−2) (106cm−3) SMA1c 1 40-100 0.04-0.02 35-13 10.5-3.8 15.9-5.8 SMA2 1 20-40 0.02-0.01 17-7 5.0-2.2 7.6-3.3 SMA3 1 20-40 0.01-0.01 13-6 3.9-1.7 5.9-2.5 Total 65-26 aAssumed distance is 2.6 kpc. bBeam-averaged quantities.The SMA beam is 4.′′7× 2.′′4 (P.A.=−45.65◦). cThe mass of SMA1 was calculated using the 1.3 mm flux density less the estimated free-free contribution of 20 mJy. – 25 – Fig. 1.— Three-color Spitzer IRAC image of S255N and its surroundings showing 8.0 µm (red), 4.5 µm (green), and 3.6 µm (blue). S255N lies in a complex region of past and ongoing massive star-formation. – 26 – Fig. 2.— Greyscale and solid contours of the SMA 1.3 mm continuum with dotted contours of the CSO 350 µm continuum superposed. The primary beam of the SMA (56′′ at 218.7 GHz) is indicated with a black circle. Naming conventions for mm and submm sources used in the literature and in this paper are also indicated. The black 1.3 mm contour levels are (-3, 3, 7, 15, 31, 47, 63) × 4 mJy beam−1 (the rms noise), observed with a 4.′′7 × 2.′′4 (P.A.=−45.65◦) beam. The dotted 350 µm contour levels are (2, 2.5, 3, 4, 5) × 16.5 Jy beam−1, resolution 15′′. – 27 – Fig. 3.— (a) Black contours of the SMA 1.3 mm continuum, observed with a 4.′′7 × 2.′′4 (P.A.=−45.65◦) beam, with greyscale and dotted contours of the 3.6 cm continuum, observed with an 1.′′03×0.′′84 beam (P.A.=−81.42◦), superposed. The black 1.3 mm contour levels are (-3, 3, 7, 15, 31, 47, 63) × 4 mJy beam−1 (the rms noise). The dotted 3.6 cm contour levels are (-3, 3, 5, 7, 11, 21, 41, 61, 81, 101, 121, 161) × 34 µJy beam−1 (the rms noise). The SMA beam (black ellipse) and VLA beam (filled black ellipse) are plotted at lower left. (b) Black contours of the SMA 1.3 mm continuum with greyscale of high-resolution (0.′′27×0.′′23, P.A.=77.51◦ beam) 3.6 cm continuum superposed. The positions of methanol masers and of the newly-reported water maser are marked. The VLA beam (filled black ellipse) is plotted at lower left. – 28 – Fig. 4.— (a-d) Colorscale of the 1.3 mm continuum with black integrated intensity contours for molecules that exhibit emission from multiple positions. The molecular species and upper state energies are indicated in the lower right of each panel (also see Table 2). The white contours show 3.6 cm emission from the UC H ii region G192.584-0.041. The blue cross marks the position of the newly detected water maser. The black integrated intensity contour levels are (-3, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29) × the rms noise levels: (a) CH3OH 0.82 Jy beam −1*km s−1, (b) SiO 1.35 Jy beam−1*km s−1, (c) H2CO 30,3 → 20,2 0.8 Jy beam−1*km s−1, (d) H2CO 32,2 → 22,1 0.76 Jy beam −1*km s−1. The white 3.6 cm contour levels are (-3, 3, 5, 7, 11, 21, 41, 61, 81, 101, 121, 161) × 34 µJy beam−1 (the rms noise). The 4.′′7 × 2.′′4 (P.A.=−45.65◦) SMA beam (filled black ellipse) and 1.′′03× 0.′′84 (P.A.=−81.42◦) VLA beam (filled white ellipse) are shown at lower left in each panel. – 29 – Fig. 5.— (a-c) Similar to Figure 4a-d except integrated intensity images are shown for molecules that are only detected in the vicinity of SMA1. The black integrated intensity contour levels are (-3, 3, 5, 7, 9, 11) × the rms noise levels: (a) CN 0.9 Jy beam−1*km s−1, (b) DCN 0.55 Jy beam−1*km s−1, and (c) HC3N 24→23 0.68 Jy beam −1*km s−1. (d) Finding chart for representative line profiles shown in Figures 7 & 8 and discussed in §3.3. – 30 – Fig. 6.— Channel maps of (a) H2CO(30,3–20,2) and (b) CH3OH, showing line emission (contours) overlaid on the 1.3 mm continuum (greyscale). The contours levels are (1, 2, 3, 4, 5, 6, 7, 8, 9) × 0.28 Jy beam−1. Each panel is labeled with the channel velocity. The position of the 3.6 cm point source is marked with a white cross. – 31 – CH3OH (d) NE (e) North (f) NW (c) SW(a) SMA1−NE (b) SMA1−SW Fig. 7.— Representative line profiles for molecules that exhibit emission from multiple positions, demonstrating the wide range of velocity and chemical behavior observed. The positions for which spectra are presented are indicated on Figure 5d and listed in Table 3. A vertical line is drawn at 10 km s−1 for reference. (a) SMA1−NE (b) SMA1−SW Fig. 8.— Representative line profiles for molecules that have their strongest emission towards SMA1. The positions for which spectra are presented are indicated on Figure 5d and listed in Table 3. A vertical line is drawn at 10 km s−1 for reference. Note that the vertical scale is not the same as in Figure 7. For CN, the weaker features offset by -16 and -22 km s−1 from the main feature are due to the hyperfine components (see Table 2). – 32 – Fig. 9.— Channel maps of DCN emission (contours) overlaid on the 1.3 mm continuum (greyscale). Each panel is labeled with the channel velocity. The position of the 3.6 cm point source is marked with a white cross. – 33 – Fig. 10.— Close-up three-color Spitzer IRAC image of S255N showing mid-IR emission offset to the NW of the UC H ii region with yellow SMA 1.3 mm and black 3.6 cm contours superposed. The colorscale correspond to: 8.0 µm (red), 4.5 µm (green), and 3.6 µm (blue). The yellow SMA 1.3 mm continuum contour levels are (3, 7, 15, 31, 47, 63) × 4 mJy beam−1 (the rms noise). The black VLA 3.6 cm continuum contour levels are (3, 5, 7, 11, 21, 41, 61, 81, 101, 121, 161) × 34 µJy beam−1 (the rms noise). Positions of Class I CH3OH masers from Kurtz, Hofner, & Álvarez (2004) are marked with red crosses, and the position of the newly-reported water maser is marked with a blue cross. – 34 – Fig. 11.— Greyscale image of the high resolution VLA 3.6 cm emission shown in Figure 3b with SMA 1.3 mm uniform weighted continuum contours superposed. The 1.3 mm image was restored with a 1′′ beam (∼ 3 times smaller than the longest observed baseline) which essentially shows regions where clean components are concentrated. Red + symbols show the 44 GHz methanol maser positions, the blue △ shows the H2O maser position, the green × symbols show the peak locations of the ∼ 7 (SMA1-NE) and ∼ 11.5 km s−1 (SMA1-SW) H2CO components, and the black ⋄ shows the location of SMA1 reported in Table 1. The localization of clean components into two distinct regions in the vicinity of SMA1 suggests the presence of at least two continuum sources; this result requires confirmation by higher resolution mm/submm data. Introduction Observations Submillimeter Array (SMA) Very Large Array (VLA) Spitzer Space Telescope Caltech Submillimeter Observatory Results Continuum emission Water maser emission Line emission Spitzer Space Telescope IRAC Observations Discussion Mass estimates from the dust emission Velocity Structure and Outflows The Nature of SMA1 Conclusions

0704.0989

Enumerating limit groups

7 Enumerating limit groups Daniel Groves and Henry Wilton 21st May 2007 Abstract We prove that the set of limit groups is recursive, answering a question of Delzant. One ingredient of the proof is the observation that a finitely presented group with local retractions (à la Long and Reid) is coherent and, furthermore, there exists an algorithm that computes presentations for finitely generated subgroups. The other main ingredient is the ability to algorithmically calculate centralizers in relatively hyperbolic groups. Applications include the existence of recognition algorithms for limit groups and free groups. A limit group is a finitely generated, fully residually free group. Recent research into limit groups has been motivated by their role in the theory of the set of homomorphisms from a finitely presented group to a free group, and in the logic of free groups. This research has culminated in the independent solutions to Tarski’s problems on the elementary theory of free groups by Z. Sela (see [21], [22] et seq.) and O. Kharlampovich and A. Miasnikov (see [12], [13] et seq.). Sela’s work extends to the elementary theory of hyperbolic groups [19]. We will be entirely concerned with finitely presentable groups. A class of groups G is recursively enumerable if there exists a Turing machine that outputs a list of presentations for every group G; it is recursive if, furthermore, the Turing machine only outputs one presentation from each isomorphism class of G. T. Delzant asked if the class of limit groups is recursive [20, I.13]. Theorem A (Corollary 3.8) The class of limit groups is recursive. In [4] and [8, 7] it is shown that the isomorphism problem is solvable for the class of limit groups. Therefore, if the class of limit groups is recursively http://arxiv.org/abs/0704.0989v2 enumerable it is recursive. To enumerate limit groups, our approach is to use the structure theory of limit groups developed in [13]. An equivalent structure theory is described in [21], which could also be used. Either way, two problems need to be solved. First, one needs to be able to compute presentations for finitely generated subgroups of limit groups. We call this property effective coherence. Secondly, one needs to be able to compute centralizers of elements in limit groups. To solve the second problem we use the relatively hyperbolic structure on limit groups found in [9] and [1]. Our solution to the first problem relies on local retractions. D. Long and A. Reid [14] defined a group to have local retractions or property LR if every finitely generated subgroup is a retract of a finite-index subgroup. A finitely presented group with local retractions is coherent. Fur- thermore, one can compute presentations for subgroups. Theorem B (Theorem 2.4) There exists an algorithm that, given a finite presentation for a group G with local retractions and a finite set of elements S, outputs a presentation for the subgroup generated by S. It is a remarkable fact that limit groups are finitely presented. It was proved in [23] that limit groups have local retractions. There is a lengthier proof that limit groups are effectively coherent using the theorem, also proved in [23], that iterated centralizer extensions are coset separable with respect to their vertex groups. As an application of Theorem A, in section 4 we prove the following theorem. Theorem C (Theorem 4.1) There exists an algorithm that, given as in- put a presentation for a group G and a solution to the word problem in G, determines whether or not G is a limit group. In Corollary 4.3, we deduce the existence of a similar recognition algo- rithm for free groups (pointed out to us by Gilbert Levitt). This paper is the first of a series, in which we intend to prove algorith- mic versions of Sela’s results. Specifically, enumerating limit groups will be useful in the algorithmic construction of Makanin–Razborov diagrams over free groups. Acknowledgements The authors would like to thank Zlil Sela for many insightful and generous conversations, and also François Dahmani and Vincent Guirardel for pointing out Corollary 4.2 to us. Thanks also to Gilbert Levitt for drawing Corollary 4.3 to our attention, and to Martin Bridson for explaining the ideas of the paragraph before Theorem 3.5. The first author was supported in part by NSF Grant DMS-0504251. 1 Effective coherence A finitely generated group is coherent if all of its finitely generated subgroups are finitely presented. We will be interested in the following algorithmic version of coherence. Definition 1.1 A coherent group G is effectively coherent if there exists an algorithm that, given a finite subset S as input, outputs a presentation for the subgroup generated by S. A class G of coherent groups is uniformly effectively coherent if there exists an algorithm that, given as input a presentation of a group G ∈ G and a finite set S of elements of G, outputs a presentation for the subgroup of G generated by S. An appealing consequence of this property is that, under mild hypothe- ses, one can decide if a homomorphism to an effectively coherent group is injective. Lemma 1.2 If a group G is effectively coherent then there exists an algo- rithm that, given a presentation for a group H, a solution to the word problem in H and a homomorphism f : H → G, determines whether f is injective. Proof. Given a presentation for the image of f and a solution to the word problem in H , it is easy to check whether f has a well-defined inverse and hence is injective. Therefore, if G is effectively coherent it is easy to check if f is injective. � Remark 1.3 Even without a solution to the word problem in H, there exists a Turing machine that will confirm in finite time if the homomorphism f is injective. Indeed, if f is injective then we know what the inverse to f must be. By effective coherence, it is possible to compute a presentation for the image of f , and the inverse homomorphism exists if and only if the relations for f(H) hold in H (under the supposed inverse map). Even though the word problem for H may be unsolvable, it is straightforward to enumerate the words which are equal to 1 in H, and if f is a homomorphism then the relations for f(H) (interpreted as words in the generators for H) will eventually appear on this list. However, if the word problem in H is unsolvable then there will in general be no Turing machine which terminates if the map f is not injective, since we will not be able to tell, for example, if the group H is the trivial group. Of course, a finitely generated subgroup of an effectively coherent group is effectively coherent. If G is a class of groups, denote by S(G) the class of finitely generated subgroups of groups in G. We are interested in effective coherence because it allows the property of being recursively enumerable to pass from G to S(G). Furthermore, uniform effective coherence also passes to subgroups. Lemma 1.4 If G is recursively enumerable and uniformly effectively coher- ent then S(G) is recursively enumerable and uniformly effectively coherent. Proof. Enumerating the presentations of groups G ∈ G and finite subsets S ⊂ G, then using uniform effective coherence to compute presentations for 〈S〉, one enumerates presentations for every group in S(G). So S(G) is recursively enumerable. Given a presentation for a group G ∈ S(G) and a finite subset S of G, we can enumerate groups K ∈ G and homomorphisms f : G → K and check whether f is an injection using the Turing machine described in Remark 1.3. Since G ∈ S(G) one will eventually find such an injection f . Using the effective coherence of K, one can now compute a presentation for 〈f(S)〉. So S(G) is uniformly effectively coherent. � We approach effective coherence through local retractions. 2 Local retractions A group G retracts onto a subgroup H if the inclusion map H →֒ G admits a left-inverse ρ : G → H . The subgroup H is called a retract and the map ρ is a retraction. Following [14], a group has local retractions if every finitely generated subgroup is a retract of a finite-index subgroup. This has immediate consequences for coherence. Lemma 2.1 If H is a retract of a finitely presented group G then H is finitely presented. Proof. The proof of the lemma is a diagram chase. Let ρ : G → H be the retraction. If B generates H then, since G = H ker ρ we can add elements from ker ρ to B to give a (finite) generating set A = B ∪A′ for G. Furthermore, any finite presentation for G can be modified to give a finite presentation with generators of this form. Denote by FX the free group on a set X . Let ρ ′ be the obvious retraction from FA = FB ∗ FA′ to FB that kills FA′. This gives a commutative square −−−→ G −−−→ H where p and q are the natural surjections FA → G and FB → H respectively. Denote the inclusion H →֒ G by i and the inclusion FB →֒ FA by i ′. The lemma follows directly from the claim that ρ′ restricts to a retraction ker p → ker q. If l ∈ ker q then p◦ i′(l) = i◦q(l) = 1 so i′(l) ∈ ker p. Likewise, if k ∈ ker p then q◦ρ′(k) = ρ◦p(k) = 1 so ρ′(k) ∈ ker q. This proves the claim and hence the lemma. � Since finite-index subgroups of finitely presented groups are finitely pre- sented, coherence for finitely presented groups with local retractions follows immediately. Proposition 2.2 If a finitely presented group G has local retractions then G is coherent. Better still, Lemma 2.1 is effective. Lemma 2.3 Let G be a finitely presented group with solvable word problem. There is an algorithm that takes as input a finite presentation for G and a collection of words which are the images of the generators under a homomor- phism ρ : G → G that is a retract onto ρ(G), and outputs a presentation for ρ(G). Proof. Applying Tietze transformations, the given generating set for G will eventually be of the form required in the proof of Lemma 2.1, namely the union of some generators for ρ(G) and some elements of ker ρ, and since G has solvable word problem we can tell when we have found such a presentation. By the proof of Lemma 2.1, a presentation for ρ(G) is then obtained by eliminating all the generators in ker ρ from the presentation of G. � By [14, Theorem 2.4], groups with local retractions are residually finite and hence have (uniformly) solvable word problem. Let LR be the class of finitely presented groups with local retractions. Theorem 2.4 The class LR is uniformly effectively coherent. Proof. Given a finite presentation for a group G ∈ LR and a finite collection of elements S ∈ G, we can enumerate all finite-index subgroups K of G using the Reidemeister–Schreier Process (see, for instance, [15]). Since G ∈ LR, there is a finite-index subgroup K of G so that 〈S〉 ⊆ K and so that there exists a retraction ρ : K → 〈S〉. We find such a retraction as follows. In parallel, consider each of the finite-index subgroups of G. Given such a finite-index subgroup K, look for the elements of S as words in the generators for K. Suppose we have found a finite-index subgroup K so that 〈S〉 ⊆ K, and a finite presentation 〈X | R(X)〉 of K, with S = {s1(X), . . . , sn(X)} written as words in X Now search for a collection of words Y in S± with a bijection ρ : X → Y so that each of the relations of the form R(Y ) holds and so that for each i we have si(Y ) = si(X). Then the map ρ extends to a retraction ρ : K → 〈S〉. Since there is a retraction, we will eventually find such a K and Y . The algorithm of Lemma 2.3 now computes a presentation for 〈S〉. � 3 Enumerating I and L The class of iterated extensions of centralizers is defined inductively. If G is a group, g ∈ G and Z(g) is the centralizer of g then an amalgamated free product G′ = G ∗Z(g) (Z(g)× Z is said to be obtained from G by extension of centralizers. Definition 3.1 The class I of iterated extensions of centralizers is the small- est class of groups containing all finitely generated free groups and closed under extension of centralizers. The class of limit groups is defined to be L = S(I), the class of finitely generated subgroups of iterated extensions of centralizers. The usual definition of limit groups is as finitely generated fully residually free groups. Definition 3.2 A group G is fully residually free if, for every finite subset X ⊂ G r 1, there exists a homomorphism to a free group G → F such that 1 /∈ f(X). A finitely generated group is fully residually free if and only if it is in L, by a theorem of [13]. Fully residually free groups are residually finite (since free groups are) and so have solvable word problem. Using the fact that limit groups are fully residually free, the following fact is well known and easy to prove. Lemma 3.3 If G is a limit group and g ∈ G then Z(g) is a free abelian group. By Theorem B of [23], limit groups have local retractions. It is clear that all groups in I are finitely presented. Corollary 3.4 The class I is uniformly effectively coherent. By Lemma 1.4, to enumerate limit groups it remains only to enumerate I. The crucial step is the ability to calculate centralizers. For this we use the relatively hyperbolic structure of limit groups (found independently by E. Alibegović [1] and F. Dahmani [9]). See [11] for an intro- duction to relatively hyperbolic groups (where in Farb’s language we mean ‘relatively hyperbolic with BCP’). Limit groups are torsion-free and hyper- bolic relative to a finite collection of maximal noncyclic abelian subgroups. Dahmani [6] provides an algorithm which takes as input a finite presentation of such a relatively hyperbolic group and outputs a basis for a representative of each conjugacy class of noncyclic maximal abelian subgroup (Dahmani’s algorithm takes as input an arbitrary finite presentation, and does not need to be given the ‘relatively hyperbolic structure’ of the group). Another important tool will be the universal theory of a group. The ele- mentary theory of a group G is the set of all sentences in first-order predicate logic (possibly with constants) that hold in G. For example, G is abelian if and only if the sentence ∀x, y ∈ G [x, y] = 1 is in the elementary theory of G. A universal sentence is a sentence in the elementary theory with a single universal quantifier. The universal theory of G is the set of universal sentences in the elementary theory ofG. Deciding the truth of universal sentences is equivalent to deciding whether finite systems of equations and inequations (with constants) have solutions. In [16], Makanin proved that the universal theory of a free group F is decidable—that is, there exists an algorithm that, given as input a universal sentence, determines whether or not it lies in the universal theory of F . The universal theory of torsion-free relatively hyperbolic groups with abelian parabolic subgroups is also decidable, by another algorithm of Dahmani [5] (again the input is any finite presentation for the group, along with the universal sentence). There is an alternative approach to calculating centralizers using biauto- matic structures. It follows from work of Rebbechi [18] that limit groups are biautomatic, and the algorithm for finding automatic structures described in [10] can be adapted to find biautomatic structures [3]. In particular, one can calculate the fellow-traveller constant of the bicombing. Using the ideas of [2], it is then easy to compute a presentation for the centralizer of an arbitrary finite subset. Theorem 3.5 There exists an algorithm that, given as input a presentation for a group G ∈ I and an element g ∈ G, outputs a minimal set of generators for Z(g). Proof. Apply Dahmani’s algorithm from [6] to find a basis for a represen- tative of each conjugacy class of maximal noncyclic abelian subgroup. Let g ∈ G. There are two cases to consider: either g is parabolic (which means conjugate into a noncyclic abelian subgroup) or else g is hyperbolic (which means g is not parabolic). It is possible to decide whether or not g is parabolic. This is because the universal theory of G is decidable [5]. The element is parabolic if and only if there exists an element h ∈ G so that hgh−1 commutes with each element of one of the above bases for the noncyclic abelian subgroups. This is a finite system of equations over G, which we can determine the truth of by Dahmani’s algorithm from [5]. If g is parabolic, then we will find such an element h, and the conjugates by h−1 of the basis for the maximal noncyclic abelian subgroup generates the centralizer of g. In this case we have found a minimal generating set for Z(g). If g is hyperbolic then its centralizer is generated by a maximal root of g. According to D. Osin [17, Theorem 1.16.(3)], it is possible to algorithmically extract roots from hyperbolic elements of G. On the face of it, Osin’s algo- rithm needs to be given as input the relatively hyperbolic structure of the group. However, Dahmani’s algorithm from [6] will find this structure, so we can make Osin’s algorithm take only the finite presentation as input. There- fore, if g is hyperbolic we can find a maximal root of g, and this maximal root is a minimal generating set for Z(g). � Corollary 3.6 The set I is recursively enumerable. Combining this with Theorem 3.4 it follows that the set of limit groups L is recursively enumerable, by Lemma 1.4. Corollary 3.7 The set of limit groups L is recursively enumerable and uni- formly effectively coherent. The results of [4] (see also [8, 7]) show that limit groups have solvable iso- morphism problem. Hence we can improve recursively enumerable to recur- sive: we can ensure that the list produced includes at most one presentation for each isomorphism class of limit groups. Corollary 3.8 The set of limit groups L is recursive. On the other hand, by systematically applying Tietze transformations, it is possible to effectively list all of the finite presentations of all limit groups. We give a simple application to recognition algorithms here. 4 Recognition algorithms Theorem 4.1 There exists an algorithm that, given as input a presentation for a group G and a solution to the word problem in G, determines whether or not G is a limit group. Proof. Let P = 〈X | R〉 be the finite presentation defining G. We have already noted that it is possible to enumerate all finite presentations of limit groups. Thus if G is a limit group then P will eventually appear on this list. Suppose then that G is not a limit group. Then G is not fully residually free, so there is a finite set {g1, . . . , gr} of nontrivial elements of G so that for any homomorphism φ from G to a free group F , at least one of the gi is in ker(φ). This property of G can easily be translated into a system of equations and inequations over F as follows. Consider both the elements of R and each gi as a word in X ±, and write R = {r1, . . . , rk}. Then the following sentence encodes the fact that at least one of {g1, . . . , gr} is in the kernel of any homomorphism from G to F : ∀X ⊂ F r1(X) = 1∧· · ·∧rk(X) = 1 g1(X) = 1∨· · ·∨gr(X) = 1 . (1) By Makanin’s algorithm [16], it is possible to decide whether or not universal sentences are true in a free group. Enumerate finite sets of nontrivial elements of G (the solution to the word problem allows us to know that the elements are nontrivial). Now, for each such finite set {r1, . . . , rk}, decide whether the sentence (1) is true or not. If G is not a limit group, we will eventually find a finite set for which (1) is true. � Of course, one cannot recognize limit groups amongst arbitrary finitely presented groups. A cyclically pinched group is an amalgamated free product of two free groups with cyclic amalgamated subgroup. Some, but not all, of these groups are limit groups. In [20, I.3], Sela asks for necessary or sufficient conditions for a cyclically pinched group to be a limit group. We do not have an answer to this question. However, Theorem 4.1 implies that at least the question has an answer. The following result was pointed to us by François Dahmani and Vincent Guirardel (its proof contains the core of the proof of Theorem 4.1). Corollary 4.2 There is an algorithm that takes as input a finite presentation of a cyclically pinched group and decides whether or not the defined group is a limit group. It does not matter whether the input presentation exhibits the cyclically pinched nature of the group, since by applying some finite number of Ti- etze transformations it is possible to find such a presentation. Once such a presentation is found, there is an explicit solution to the word problem. Therefore Corollary 4.2 follows immediately from Theorem 4.1. As remarked above, limit groups are torsion-free and hyperbolic relative to their maximal abelian subgroups. There is an algorithm to distinguish free groups among such relatively hyperbolic groups; indeed, it is proved in [7, Theorem 1.4] that there exists an algorithm that computes the Grushko decomposition from a presentation of such a group. Combining this with Theorem 4.1, we obtain a similar recognition algorithm for free groups. This corollary was pointed out to us by Gilbert Levitt. Corollary 4.3 There exists an algorithm that, given as input a presentation for a group G and a solution to the word problem in G, determines whether or not G is free. One can also deduce a similar result for surfaces. In [8, Theorem D] it is shown that there exists an algorithm that computes a JSJ decomposition for a torsion-free, freely indecomposable group that is hyperbolic relative to its maximal abelian subgroups. In particular, combining this with the algorithm from [7], one can decide whether or not a limit group is a surface group. It follows as before that there exists an algorithm that, given as input a presentation for a group G and a solution to the word problem in G, determines whether or not G is a (fully) residually free surface group. (The only surface groups that are not residually free are the fundamental groups of the non-orientable surfaces of Euler characteristic 1, 0 and -1.) References [1] Emina Alibegović. A combination theorem for relatively hyperbolic groups. Bull. London Math. Soc., 37(3):459–466, 2005. [2] Martin R. Bridson. On the subgroups of semihyperbolic groups. In Essays on geometry and related topics, Vol. 1, 2, volume 38 of Monogr. Enseign. Math., pages 85–111. Enseignement Math., Geneva, 2001. [3] Martin R. Bridson and Lawrence D. Reeves. On the algorithmic con- struction of classifying spaces and the isomorphism problem for biauto- matic groups. Preprint, 2007. [4] I. Bumagin, O. Kharlampovich, and A. Miasnikov. Isomorphism prob- lem for finitely generated fully residually free groups. J. Pure and Ap- plied Algebra, 208(3):961–977, 2007. [5] François Dahmani. Existential questions in (relatively) hyperbolic groups. Preprint, 2006. [6] François Dahmani. Finding relatively hyperbolic structures. Preprint, 2006. [7] François Dahmani and Daniel Groves. Detecting free splittings in rela- tively hyperbolic groups. TAMS, to appear. [8] François Dahmani and Daniel Groves. The isomorphism problem for toral relatively hyperbolic groups. Preprint, 2005. [9] François Dahmani. Combination of convergence groups. Geom. Topol., 7:933–963 (electronic), 2003. [10] David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston. Word processing in groups. Jones and Bartlett Publishers, Boston, MA, 1992. [11] B. Farb. Relatively hyperbolic groups. Geom. Funct. Anal., 8(5):810– 840, 1998. [12] O. Kharlampovich and A. Miasnikov. Irreducible affine varieties over a free group. I. Irreducibility of quadratic equations and Nullstellensatz. J. Algebra, 200(2):472–516, 1998. [13] O. Kharlampovich and A. Miasnikov. Irreducible affine varieties over a free group. II. Systems in triangular quasi-quadratic form and descrip- tion of residually free groups. J. Algebra, 200(2):517–570, 1998. [14] D. D. Long and A. W. Reid. Subgroup separability and virtual retrac- tions of groups. Topology, to appear, 2006. [15] Wilhelm Magnus, Abraham Karrass, and Donald Solitar. Combinato- rial group theory: Presentations of groups in terms of generators and relations. Interscience Publishers [John Wiley & Sons, Inc.], New York- London-Sydney, 1966. [16] G. S. Makanin. Decidability of the universal and positive theories of a free group. Izv. Akad. Nauk SSSR Ser. Mat., 48(4):735–749, 1984. [17] Denis V. Osin. Relatively hyperbolic groups: intrinsic geometry, alge- braic properties, and algorithmic problems. Mem. Amer. Math. Soc., 179(843):vi+100, 2006. [18] D. Y. Rebbechi. Algorithmic properties of relatively hyperbolic groups. Thesis, 2003. ArXiv: math/0302245. [19] Z. Sela. Diophantine geometry over groups VIII: The elementary theory of a hyperbolic group. Preprint. [20] Zlil Sela. Diophantine geometry over groups: a list of research problems. http://www.ma.huji.ac.il/~zlil/problems.dvi. [21] Zlil Sela. Diophantine geometry over groups. I. Makanin-Razborov dia- grams. Publ. Math. Inst. Hautes Études Sci., 93:31–105, 2001. [22] Zlil Sela. Diophantine geometry over groups. II. Completions, closures and formal solutions. Israel J. Math., 134:173–254, 2003. [23] Henry Wilton. Hall’s Theorem for limit groups. Preprint, 2006. Daniel Groves, Mathematics 253-37, California Institute of Tech- nology, Pasadena, CA 91125 E-mail: [email protected] http://arxiv.org/abs/math/0302245 Henry Wilton, Department of Mathematics, 1 University Sta- tion C1200, Austin, TX 78712-0257 E-mail: [email protected] Effective coherence Local retractions Enumerating I and L Recognition algorithms

0704.0990

Dynamics of Size-Selected Gold Nanoparticles Studied by Ultrafast Electron Nanocrystallography

Microsoft Word - Nanocrystallography_w_Figs_Physics_archive.doc Dynamics of Size-Selected Gold Nanoparticles Studied by Ultrafast Electron Nanocrystallography Chong-Yu Ruan*, Yoshie Murooka, Ramani K. Raman, Ryan A. Murdick Department of Physics and Astronomy Michigan State University, East Lansing, MI 48824, USA * corresponding author: [email protected] ABSTRACT We report the studies of ultrafast electron nanocrystallography on size-selected Au nanoparticles (2-20 nm) supported on a molecular interface. Reversible surface melting, melting, and recrystallization were investigated with dynamical full-profile radial distribution functions determined with sub-picosecond and picometer accuracies. In an ultrafast photoinduced melting, the nanoparticles are driven to a non-equilibrium transformation, characterized by the initial lattice deformations, nonequilibrium electron-phonon coupling, and upon melting, the collective bonding and debonding, transforming nanocrystals into shelled nanoliquids. The displasive structural excitation at premelting and the coherent transformation with crystal/liquid coexistence during photomelting differ from the reciprocal behavior of recrystallization, where a hot lattice forms from liquid and then thermally contracts. The degree of structural change and the thermodynamics of melting are found to depend on the size of nanoparticle. Understanding the phases of materials and their transformations is a fundamental problem, especially on the nanometer scale, where thermodynamics and kinetic processes are influenced by local environments, including surfaces, microsctructures, and interfacial chemistry1,2. Their elucidation requires characterization at the atomistic scale, both in space and time; at nanointerfaces molecular sensitivity is necessary. The possibility for a scattering experiment to couple with the high time resolution of a femtosecond laser in a pump-probe arrangement makes it a favorable option to study structural dynamics, as evident from recent developments3-6. Here we report a nanocrystallographic method, based on Ultrafast Electron Crystallography (UEC)7, which allows quantitative studies of local structures and transient dynamics of nanoparticles (NPs) dispersed on a molecular interface. The implementation is general. Specifically demonstrated here are the studies of reversible photoinduced melting and subsequent recrystallization of size-selective Au NPs (2nm, 10nm, 20nm), a prototypical system for studying nanophases1 and catalysis2, supported on a molecular surface. The size-dependent phase transitions are examined using a more bulk-like NP (20nm, melting point ~ 1300K8) and a much smaller one, where surface and confinement play significant roles (2nm, melting point ~ 800K8). By achieving spatial and thermal energy isolation of NPs from their environment and from each other, the normally irreversible phase transformations become reversible, allowing multi-shot pump-probe diffraction to map out their full courses. Such implementation allows the use of a low-density electron pulse to avoid the pulse-broadening effect9 and has high data reproducibility compared with single-shot experiments where a much higher density electron pulse is required10. The spatial and thermal isolation of the NPs from their environment is achieved by implementing a buffer molecular layer, in this case aminosilane, self-assembled on a silicon substrate as shown in Fig. 1A 11,12, with which, substrate scattering is sufficiently suppressed. Since the NPs are dispersed, the diffraction is via transmission, mostly through individual particles rather than multiple particles (or aggregates). To highlight the difference, two cases are presented in Fig. 1 (D-I). Without buffering, NPs tend to aggregate, as visible in the electron micrograph (D). The diffraction pattern (E) is dominated by the substrate, as also evident from the rocking curve (F). With buffering, however, NPs are separated from each other on the surface (G), from which the diffraction (H) is predominantly from the NPs and the buffering molecular layer, with no indication of Si periodicity in the rocking curve (I). Samples were also examined following the laser irradiation experiment and showed no signs of agglomeration or damage. A time-resolved structural study of surface supported Au NPs was conducted earlier using a synchrotron X-ray source13. However, because X-ray is much more penetrating, with 5 orders of magnitude less scattering power than electron, its interfacial structural resolution is limited by the signal-to-noise level, particularly for very small NPs. High-energy electron diffraction, with its short wavelength and high surface sensitivity as demonstrated here, has higher structural resolution than small angle X-ray diffraction. Prior to studying dynamics, the static structure of NPs is analyzed. The static pattern shows Debye- Scherrer diffraction rings from the Au NPs, while the ordered buffer layer produces Bragg spots, primarily in the surface streak regions. The Debye-Scherrer rings are radially averaged into a 1D diffraction intensity curve, shown in Fig. 2A. This curve, obtained from the diffraction pattern of 2nm NPs, shows isolated peaks, reflecting a crystalline structure, which, from inspection, resembles more towards cuboctahedral and/or decahedral structures than an icosahedral one. Here, s = (4π/λ)sin(θ/2) represents the magnitude of the reciprocal space wave-vector of diffracted electrons, with wavelength λ=0.069 Å for 30keV electrons, and θ being the electron scattering angle. The deviation from an ideal structure can be attributed to different possible conformations and the surface strain associated with small particles14. Their cuboctahedra-like crystalline characteristics are more easily seen in the modified radial distribution functions (mRDF)15, deduced from a Fourier analysis of 1D diffraction curves, shown in Fig. 2B. All the major peaks above 2.8Å are in close agreement with the Au-Au distance table based on a face-center cubic (FCC) motif (Fig. 2C), which constitutes the internal lattice repeat of a cuboctahedra. The sensitivity of our technique is sufficient to permit the observation of molecular density peaks as well in the mRDF, such as those at 1.5 - 1.7 Å representing the C-C, C-N, and Si-N bonds and ~1.1 Å for C-H and N-H bonds. To study the dynamics, an ultrashort laser pulse (800 nm, P-polarized, ~40 fs) is used to excite the NPs, while the probing electron pulse is delayed relative to the laser pulse to monitor the structural evolution (Fig. 1B). To improve temporal resolution, a proximity-coupled optical system allows the photogenerated electron beam to be focused to ~5 µm in less than 6 cm from the photocathode with 1000 or less electrons/pulse in order to remediate the space charge induced broadening effects and to reduce the pulse overlap between the pump and probe. Sub-ps accuracy can be readily achieved (Fig. 1C). Using different excitation fluences (tuned to nonmelting, surface melting, and melting), transient responses of atoms in the NPs are determined from dynamical full profile mRDFs, highlighting the local dynamics, compared with the global ones based on analyzing Bragg peaks. The dynamics of bonds following laser irradiation can be extracted from the mRDF maps, shown in Fig. 3, selected here with the surface melting (31 mJ/cm2) and melting (75 mJ/cm2) fluences for 2nm NP. Their differentiation is evident from the rapid change in bond densities. In general, a melting is characterized by the replacement of sharp 2nd nearest neighbor peaks with more diffusive ones7. Uniquely here, the peak density reduction is coupled to the formation of new density peaks at slightly larger distances. This redistribution of 2nd nearest neighbor peaks can be used to determine the onset of melting as well as recrystallization, defined here as a 1/e drop in peak intensity at ~5Å. Based on this criterion, at 31 mJ/cm2, we observe no melting. The laser heating causes the NPs to expand with little adjustment of bond densities below 1nm. At 18 ps, breaking and forming of bonds beyond 1nm are evident, indicative of surface melting. The transition is rapid, within 1-2 ps, and the molten layer lasts for only tens of ps. By 40 ps the newly formed long bonds begin to cool and slowly replace the original broken ones. The surfaces revert to original crystalline structure in ns timescale. However, at 75 mJ/cm2, the bond densities across the full NP length scale are modified and complete melting occurs. Using the change in 2nd nearest neighbor peak, we determine the occurrence of melting and recrystallization at 18 ps and 110 ps respectively. Furthermore, the melting and recrystallization dynamics display vastly different characters, nonreciprocal to each other, as seen in the mRDF map. Following the initial expansion of the lattice, we can clearly see bonding and debonding emerge already at ~ 12 ps. The depletion of the bond density (debonding) around the major peaks (numbered 3,5,7,12 in Fig. 2B) is coupled to the enhancement of bond density (bonding) at longer distances, where bonds may or may not have been present before. The emerging longer distance peaks, which constitute a shoulder region, gain in density towards melting, ultimately smearing out into bands. The coherent bonding and debonding dynamics observed at the premelting period and the continued development of the newly formed long distance peaks into liquid structures suggests that liquid structures are populated while the particle is still relatively cold. This reflects the displasive character of the photo-melting process during which the transformation of crystal into liquid is through breaking old bonds and forming new bonds. In a sense, this photomelting dynamics resembles a conformation change between minimum energy structures on the free energy landscape. In contrast, the recrystallization is much like the reverse of a ‘thermal’ melting in which crystal simply thermally expands and then disorders. The liquid structure of a NP is unique, and can be characterized by the reduced coordination number, judging from the reduction of the direct bond density, and shell-like mRDF densities. The structure of the liquid is compared with an icosahedral NP, which also possesses shell-like structure. The average distances of the liquid shells in the mRDF, taken at 40 ps, match well with those of a 10% expanded icosahedral shells with the pronounced crystalline peaks being smeared out. This suggests the density of the transient hot nanoliquid is reduced compared with a room temperature crystalline structure and the atom-atom correlation between liquid shells is lost. Closer inspection of the NPs expansion before melting reveals anisotropic movement of the lattice depending on the irradiating fluence. By fitting the mRDF density profiles using Gaussian function, we follow the time-dependent evolutions of the bond distance, density, and width for bonds at 2.88, 5.00, and 7.64 Å. At a low fluence (<31 mJ/cm2), the changes are isotropic with thermal expansion being equal for all three distances. However, the deformation of the lattice sets in as the fluence increases. At a threshold fluence of 38 mJ/cm2, right before a full melting occurs, the early time (1-15 ps) anisotropic bond movements are evident, representing a combination of shearing motion along (100) direction and expansion (Fig. 4A, left panel). Generally, prior to lattice disorder, the nearest neighbor bond (2.88 Å) sharpens while the rest of the longer bonds decay and widen. The transient narrowing at direct bond distance suggests a brief reduction of strain in the NP following the impulsive laser excitation. Expansion of direct bond continues till 60 ps, indicating uninterrupted transfer of energy into bond stretching vibrations. However, these slower dynamics do not represent the time scale of electron- lattice equilibration. The intrinsic electron-phonon coupling time for Au NPs is less than 4±1 ps, a limit derived based on fitting the Debye-Waller factor from low fluence (15 mJ/cm2) data where no lattice disorder or coherent motion is evidently present at short times. Thus the longer period for melting (18- 20 ps) and the lattice expansion (60 ps) reflect the time scales for atomic disorder in the crystal and thermal energy relaxation from the initially strongly excited hot phonons to the bond stretching vibrations. Link and El-Sayed16 have found time constant of 30 ps for NP shape change from nanorod to nanosphere, a time scale comparable to phonon-phonon scattering time. Melting of Au NPs has been investigated by other time-resolved techniques also. Plech and coworkers have reported the melting transition of Au NPs (100nm) using synchrotron based time-resolved X-ray powder diffraction, on the 100 ps time scale (their pulse duration)17. The sample was irradiated by a 400 nm fs laser with steadily increasing power, and the phase changes were interpreted based on monitoring the deviation of the integrated area under (111) and (200) Bragg peaks from a constant value, at a fixed delay of 105 ps. The corresponding lattice temperature is derived based on the lattice expansion by monitoring the shift of Bragg peak position. They observe a sub-bulk melting temperature (70% of the bulk value), which is unexpected for particle size larger than 30 nm. They attribute this suppression of melting temperature to possible onset of surface melting. The largest lattice expansion before melting was determined to be 1.2% (1.82% is expected for bulk melting), and there was no indication of any significant crystal anisotropy based on diffraction. In a more recent small-angle X-ray scattering (SAXS) study, Plech and coworkers found 15 mJ/cm2 as the melting threshold for 38 nm Au NPs, and have shown laser alignment effects just below the melting fluence18. Hartland, Hu, and Sader have addressed the melting transition by measuring the vibration frequency of the breathing mode using time-resolved spectroscopy, but have found no discontinuity at the melting point19. They conclude that a saturation of light absorption limits the energy that can be transferred to the lattice. To connect our data to these studies, we also inspect the temporal evolution of Bragg peaks in s-space. At 38 mJ/cm2 (the threshold fluence for 2nm NPs), we find a rapid decay of intensity (~6 ps for (111) peak to drop by 1/e). Such a significant change, however, does not correspond to a melting, as shown from our mRDF analyses, rather it indicates a breaking of lattice symmetry induced by photo-excitation. This deformation is also evident from the anisotropic shifts of different lattice planes – (220) blue-shifts while (311) and (331) peaks red-shift. Their associated peak-widths exhibit instantaneous narrowing followed by widening, again confirming the coherent change at short times. These shifts are consistent with a lattice deformation along (100) direction. The lattice then expands significantly to a maximum of ~1.5% at 60 ps. The ‘lattice’ is found significantly disordered, no longer suitable for s-space analyses. However, the disorder is just below the threshold considered as melting according to our mRDF analyses. At 75 mJ/cm2 (melting fluence), a rapid drop of (111) intensity to 20% of the original level is found to appear at 15 ps, indicating the rapid loss of long-range order. This time scale is close to the onset of bonding and debonding observed in the premelting period according to the mRDF analyses. For 20nm Au NPs, we found the threshold fluence to be between 15-20 mJ/cm2, consistent with the results for 38 nm NPs obtained by Plech and coworkers. To understand the thermal energy redistribution, we use the stretch of bonds to gauge the ‘local temperature’ of bonds in NPs. The term ‘local’ used here suggests a temperature based on the vibrational sampling of local bonding potential between atoms. The anharmonicity of the bonding potential leads to the expansion of the bond, which increases with the vibrational amplitude. Coherent motion resulting from impulsive strain at early times from the fs excitation will cause splitting or broadening (if unresolved) of peaks or driven anisotropic deformation of the lattice. To this end, coarse- graining the dynamics with longer time period should be conducted to reflect the average extension of the lattice due to thermal (stochastic) energy. This method based on the mRDF analysis allows differentiation of inhomogeneity that exists on different length scales, not possible by extracting temperature solely based on following the shift of a Bragg peak 17. However, such a definition of temperature must not be confused with the temperature of the NP as a whole, which can only be defined when the thermal equilibrium is reached. By using the long-time data, where thermalization has been established, we can extract the NPs’ true temperatures under different fluences by comparing them to a two-temperature model (TTM) 20 (Fig. 4A). To convert the thermal expansion into temperature, we use the temperature-dependent thermal expansion coefficient from reference 21 which is valid between 300 and 1300 K. This thermal expansion coefficient is found to apply for 60nm Au NPs17. Comparing the long-time thermal relaxation, which is fit to a TTM, with the short-time heightened lattice expansion reveals the hot phonons effect caused by non-equilibrium electron-phonon coupling (Fig. 4A, right panel). The existence of hot phonons was recently invoked to explain the non-equilibrium electron- phonon coupling in low-dimensional systems, such as graphite22 and nanotube23, as well as molecular systems24. They are the vibrational modes coupled more directly to the de-excitation of electrons, thus gaining higher ‘temperature’ compared with an equilibrated lattice temperature obtained from a TTM. These hot phonons produce large amplitude of vibration, thus leading to heightened lattice expansion. Because the phonon-phonon interaction time is 30-60 ps, these hot phonons are likely responsible for initiating melting and influencing recrystallization, making the photomelting phenomena different from a thermal one. Size-dependent effects in the transient heating of NPs are seen, shown in Fig. 4B. First, the transient maximum bond stretch is significantly higher in the 2nm NPs at 75 mJ/cm2 (6% for 2nm, 3.5% for 20nm), albeit, the thermal temperature is very close in both cases - a fact deduced by comparing the equilibrated (∆R/R) data at longer times (from 1-3 ns data, see insets). Second, 20nm NPs have similar maximum bond stretch at 80 mJ/cm2 and 31 mJ/cm2, both leading to melting, but differing in their liquid residence time. For 2nm NPs however, melting occurs only at 75 mJ/cm2. These results suggest that the particle size plays a role in determining the thermodynamics of NPs. For 20 nm NPs, increasing the fluence does not cause a continuous rise in liquid temperature, leading instead to a longer liquid residence time, suggesting that latent heat already exists at 20 nm. The lack of a sharp transition expected from first-order phase transition reflects the ultrafast nature of transformation. However, for 2nm NPs, the temperature continues to rise significantly after melting, suggesting the transformation being a second-order phase transition25,26. Based on the TTM27, at the irradiating fluence of 31 mJ/cm2, 1.5×1022 e-/cm3 (~24%) are excited in the Au NP, whereas at 75 mJ/cm2, 3.25×1022 e-/cm3 (~57%) are excited. At these high fluences, the interband transition starts to play a role. The lowest interband transition energy in Au is 1.7 eV28, which corresponds to promoting d-electrons (5d) in the vicinity of the X-point of the first Brillouin zone to the conduction band (6sp) and is slightly higher than our excitation energy (1.55 eV). However, as conduction electrons are strongly excited, their Fermi-Dirac distribution is modified, with part of the electronic levels below the Fermi level being emptied, to make way for interband transition. Because of this hot electron effect, the contribution of the interband transition increases with fluence. The effect of interband transition is manifested in the lattice anisotropic deformation observed at the short times (1-15 ps). Although, the d holes relaxation will proceed in tens of fs by hole-hole scattering29, the lattice deformation likely persist to the ps time scale due to the slow collective motion of atoms responding to the modification of the energy landscape caused by the core electron (5d) excitation. The coupling of ps lattice deformation to electronic heating in bulk system was also discussed recently by Guo and Taylor30. In addition, we find that by using 400 nm excitation this anisotropic deformation is replaced by an isotropic one. Because of the high excitation energy, the interband transition is no longer pinned to the X-point. These results suggest that the excited energy landscape can be explored by following the ultrafast lattice dynamics as a function of the excitation energy. In conclusion, using ultrafast electron nanocrystallography, we have mapped out the dynamics of liquid-crystalline and crystalline-liquid phase transformations for Au nanoparticles, at and beyond the thermodynamic limit. The accurate mRDF determinations of nanostructures allow quantitative studies of atomic dynamics with molecular scale resolutions. The size dependence is evident in the change of structures and in the extent of melting. The reversible and coherent transformation on the ultrafast time scale demonstrates the directed dynamics on the energy landscape of finite systems. Abundant details can be further extracted by comparing the dynamical mRDF maps and electron micrographs obtained for fluence far beyond the melting threshold, and under different excitation wavelengths. This methodology is general and could be implemented to study a wide class of phenomena pertaining to nanoscaled materials. ACKNOWLEDGMENT This work is supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Material Sciences and Engineering and the Intramural Research Grant Program at Michigan State University. REFERENCES 1. Marks, L.D. Rep. Prog. Phys. 1994, 57, 603. 2. Haruta, M. Catal. Today 1997, 36, 153. 3. Zewail, A.H. Annu Rev. Phys. Chem. 2006, 57, 65. 4. Rousse, A. ; Rischel, C. ; Gauthier, J. Rev. Mod. Phys. 2001, 73, 17. 5. Bargheer, M. ; Zhavoronkov, N. ; Woerner, M. ; Elsaesser, T. ChemPhysChem. 2006, 7, 783. 6. Chergui, M. ; Bressler, C. Chem. Rev. 2004, 104, 1781. 7. Ruan, C-Y. ; Vigliotti, F. ; Lobastov, V.A. ; Chen, S. ; Zewail, A.H. Proc. Natl. Acad. Sci. USA 2004, 101, 1123. 8. Buffat, Ph. ; Borel, J-P. Phys. Rev. A 1976, 13, 2287. 9. Schelev, M. Ya. ; Richardson, M.C. ; Alcock, A.J. Appl. Phys. Lett. 1971, 18, 354. 10. Siwick, B.J. ; Dwyer, J.R. ; Jordan, R.E. ; Miller, R.J.D. Science 2003, 302, 5649. 11. Sample Preparation: Si(111) wafers were first cleaned by immersing in H2SO4/H2O2 (7:3) for 10 min, followed by immersion in 40% NH4F solution in de-ionized water for 20 min, producing H-terminated Si(111) surface. This was followed by immersion in NH4OH/H2O2/H2O (1:1:5) at 80°C for 10 min, subsequent immersion in HCl/H2O2/H2O (1:1:6) at 80°C for 10 min, followed by drying under high purity (99.9%) dry N2 gas to produce OH-terminated Si(111). After each step, the wafer was thoroughly rinsed in de-ionized water (>18Mohm.cm). The wafer was dried in high purity N2 gas and immersed in [3-(2-Aminoethylamino)propyl]trimethoxysilane (AEAPTMS, Sigma- Aldrich) for 20 min, followed by heating at 80°C over dry N2 cover to allow surface functionalization. Dispersion and immobilization of Au NP on the wafer was finally achieved by immersing the functionalized Si(111) wafer in a colloidal Au NP solution (Ted Pella) and Ethanol/water (2:1) for 2 hrs. Also see Ref. 12. 12. Sato, T. ; Brown, D. ; Johnson, B.F.G. Chem. Commun. 1997, 11, 1007. 13. Plech, A. ; Gresillon, S. ; Plessen, G.von ; Scheidt, K. ; Naylor, G. Chem. Phys. 2004, 299, 183. 14. Cervellino, A. ; Giannini, C. ; Guagliardi, A. J. Appl. Cryst. 2003, 36, 1148. 15. Warren, B.E. X-ray Diffraction; Dover publications, Dover ed.: New York, 1990. 16. Link, S.; El-Sayed, M.A. Int. Rev. Phys. Chem. 2000, 19, 409. 17. Plech, A.; Kotaidis, V. Phys. Rev. B 2004, 70, 195423. 18. Plech, A.;Kotaidis,V.;Lorenc,M.;Boneberg,J. Nature Phys. 2006, 2, 44. 19. Hartland,G.V.;Hu,M.;Sader.J.E. J. Phys. Chem. B 2003, 107, 7472. 20. Kaganov, M.I.; Lifshitz, I.M.; Tanatarov, L.V. Zh. Eksp. Teor. Fiz. 1956, 31, 232. [Sov. Phys. JETP 1957, 4, 173]. 21. Touloukian,Y.S.;Kirby,R.K.;Taylor,R.E.;Desai,P.D. Thermal expansion – Metallic elements and alloys, in Thermophysical Properties of Matter Vol. 12; IFI Plenum: New York, 1975. 22. Kampfrath,T.;Perfetti,L.;Schapper,F.;Frischkorn,C.;Wolf,M. Phys. Rev. Lett. 2005, 95, 187403. 23. Auer,C.;Schurrer,F.;Ertler,C. Phys. Rev. B 2006, 74, 165409. 24. Ruan, C-Y. ; Lobastov, V.A.; Srinivasan, R. ; Goodson, B.M. ; Ihee, H. ; Zewail, A.H. Proc. Natl. Acad. Sci. USA 2001, 98, 7117. 25. Liu, H.B. ; Ascencio, J.A.; Perez-Alvarez, M.; Yacaman, M.J. Surf. Sci. 2001, 491, 88. 26. Jellinek, J. ; Beck, T.L. ; Berry, R.S. J. Chem. Phys. 1986, 84, 2783. 27. Chen, J.K. ; Beraun, J.E.; Tham, C.L. Num.Heat.Trans. A 2003, 44, 705. 28. Hache, F. ; Ricard, D. ; Flytzanis, C. ; Kreibig, U. Applied. Phys A 1988, 47, 347. 29. Petek,H.;Ogawa,S. Prog. Surf. Sci. 1997, 56, 239. 30. Guo,C.;Taylor,A.J. Phys. Rev. B 2000, 62, R11921. Figure 1 Figure 1. (A) Au nanoparticles dispersed on a self-assembled molecular interface (B) Pump-probe arrangement of UEC. (C) Zero-of-time determination using the diffraction signals from the photomechanical responses of graphite multilayers. (D) SEM image of 20 nm Au NPs dispersed on the surface without proper buffering. (E) Diffraction pattern from (D) showing Bragg spots of silicon substrate. (F) Rocking curve analysis gated at central streak in (E) with varying incident angle θi. (G) SEM image of 20 nm Au NPs with proper buffering. (H) Diffraction pattern from (G) showing Debye- Scherrer diffraction rings and Bragg spots from buffer layer (Si, N and C stack layers in self-assembled aminosilane, spacing 2.2Å, tilt angle 31°). (I) Rocking curve of (H). Figure 2 Figure 2. Structure analyses of size-selected Au nanoparticles. (A) 1D diffraction intensity curve (black) obtained from radial averaging of the Debye-Scherrer UEC pattern (inset) of the surface supported 2nm NPs. Also shown are simulations generated from 2nm structures (cuboctahedra, decahedra, and icosahedra) of Au NPs at 300 K. The indices show associated Bragg reflection planes based on an FCC structure. (B) Experimental modified radial distribution functions (mRDFs) of static Au NPs along with theoretical prediction for cuboctahedra. The numeric labels represent the bond order in the FCC distance table (below). (C) FCC coordination shells corresponding to interatomic distances ri, calculated based on the bond order i and the Au lattice constant a=4.08 Å. Figure 3 Figure 3. The melting dynamics of 2nm Au nanoparticles. (Left) mRDF map constructed by stacking mRDFs of UEC patterns at a sequence of delays between 5-2300 ps at irradiation fluence F=31 mJ/cm2. Surface melting (enclosed by the dashed white line) is visible. (Right) mRDF map for F=75 mJ/cm2. Full scale melting is observed. The liquid state (enclosed by dashed white line) is characterized by the drop of 2nd nearest density (at ~ 5Å) to (1-1/e) of the static value (at negative time). Figure 4 Figure 4. (A) The left panel shows the short-time relative distance change ∆R/R of dominant mRDF peaks (numbered 1, 3, and 7 in Fig. 2B) for 2nm Au NPs at fluence of 38 mJ/cm2, from which the lattice deformation can be deduced. The right panel shows the extension of these bonds for longer times and the corresponding local temperature deduced from the bond extensions (see text) compared with a TTM calculation. tD is the thermal relaxation time to the environment obtained by fitting data to a two- temperature model (TTM) after equilibration. (B) Temporal evolution of the relative distance change ∆R/R of nearest neighbor bond (~2.88Å) in 2nm (solid line and symbol) and 20nm (dash-dash-dot line and open symbol) NPs, irradiated under F=31 mJ/cm2 in the left panel, and F = 75 (for 2nm NPs) and 80 (for 20nm NPs) mJ/cm2 in the right panel. The insets show the corresponding ∆R/R dynamics at long times (50-2750 ps).

0704.0991

A Direct Method for Solving Optimal Switching Problems of One-Dimensional Diffusions

A Direct Method for Solving Optimal Switching Problems of One-Dimensional Diffusions Masahiko Egami Abstract In this paper, we propose a direct solution method for optimal switching problems of one-dimensional diffusions. This method is free from conjectures about the form of the value function and switching strategies, or does not require the proof of optimality through quasi-variational inequalities. The direct method uses a general theory of optimal stopping problems for one-dimensional diffusions and charac- terizes the value function as sets of the smallest linear majorants in their respective transformed spaces. 1 Introduction Stochastic optimal switching problems (or starting and stopping problems) are important subjects both in mathematics and economics. Since there are numerous articles about real options in the economic and financial literature in recent years, the importance and applicability of control problems including optimal switching problems cannot be exaggerated. A typical optimal switching problem is described as follows: The controller monitors the price of natural resources for optimizing (in some sense) the operation of an extraction facility. She can choose when to start extracting this resource and when to temporarily stop doing so, based upon price fluctuations she observes. The problem is concerned with finding an optimal switching policy and the corresponding value function. A number of papers on this topic are well worth mentioning : Brennan and Schwarz (1985) in conjunction with convenience yield in the energy market, Dixit (1989) for production facility problems, Brekke and Øksendal (1994) for resource extraction problems, Yushkevich (2001) for positive recurrent countable Markov chain, and Duckworth and Zervos (2001) for reversible investment problems. Hamdadène and Jeanblanc (2004) analyze a general adapted process for finite time horizon using reflected stochastic backward differential equations. Carmona and Ludkovski (2005) apply to energy tolling agreement in a finite time horizon using Monte-Carlo regressions. A basic analytical tool for solving switching problems is quasi-variational inequalities. This method is indirect in the sense that one first conjectures the form of the value function and the switching policy and next verifies the optimality of the candidate function by proving that the candidate satisfies the variational inequalities. In finding the specific form of the candidate function, appropriate boundary conditions includ- ing the smooth-fit principle are employed. This formation shall lead to a system of non-linear equations that are often hard to solve and the existence of the solution to the system is also difficult to prove. Moreover, http://arxiv.org/abs/0704.0991v1 this indirect solution method is specific to the underlying process and reward/cost structure of the prob- lem. Hence a slight change in the original problem often causes a complete overhaul in the highly technical solution procedures. Our solution method is direct in the sense that we first show a new mathematical characterization of the value functions and, based on the characterization, we shall directly find the value function and optimal switching policy. Therefore, it is free from any guesswork and applicable to a larger set of problems (where the underlying process is one-dimensional diffusions) than the conventional methods. Our approach here is similar to Dayanik and Karatzas (2003) and Dayanik and Egami (2005) that propose direct methods of solving optimal stopping problems and stochastic impulse control problems, respectively. The paper is organized in the following way. In the next section, after we introduce our setup of one dimensional optimal switching problems, in section 2.1, we characterize the optimal switching times as exit times from certain intervals through sequential optimal stopping problems equivalent to the original switching problem. In section 2.2, we shall provide a new characterization of the value function, which leads to a direct solution method described in 2.3. We shall illustrate this method through examples in section 3, one of which is a new optimal switching problem. Section 4 concludes with comments on an extension to a further general problem. 2 Optimal Switching Problems We consider the following optimal switching problems for one dimensional diffusions. Let (Ω,F ,P) be a complete probability space with a standard Brownian motion W = {Wt; t ≥ 0}. Let Zt be the indicator vector at time t, Zt ∈ {z1, z2, ..., zm} , Z where each vector zi = (a1, a2, ..., ak) with a is either 0 (closed) or 1 (open), so that m = 2k. In this section, we consider the case of k = 1. That is, Zt takes either 0 or 1. The admissible switching strategy is w = (θ0, θ1, θ2, ..., θk, ...; ζ0, ζ1, ζ2, ..., ζk, ...) with θ0 = 0 where where where 0 ≤ θ1 < θ2 < .... are an increasing sequence of Ft-stopping times and ζ1, ζ2... are Fθi-measurable random variables representing the new value of Zt at the corresponding switching times θi (in this section, ζi = 1 or 0). The state process at time t is denoted by (Xt)t≥0 with state space I = (c, d) ⊆ R and X0 = x ∈ I , and with the following dynamics: If ζ0 = 1 (starting in open state), we have, for m = 0, 1, 2, ....., dXt = dX1t = µ1(X 1)dt+ σ1(X 1)dWt, θ2m ≤ t < θ2m+1, dX0t = µ0(X 0)dt+ σ0(X 0)dWt, θ2m+1 ≤ t < θ2m+2, (2.1) and if ζ0 = 0 (starting in closed state), dXt = dX0t = µ0(X 1)dt+ σ0(X 0)dWt, θ2m ≤ t < θ2m+1, dX1t = µ1(X 1)dt+ σ1(X 1)dWt, θ2m+1 ≤ t < θ2m+2. (2.2) We assume that µi : R → R and σi : R → R are some Borel functions that ensure the existence and uniqueness of the solution of (2.1) for i = 1 and (2.2) for i = 0. Our performance measure, corresponding to starting state i = 0, 1, is Jwi (x) = E e−αsf(Xs)ds − e−αθjH(Xθj− , ζj)  (2.3) where H : R×Z → R+ is the switching cost function and f : R → R is a continuous function that satisfies e−αs|f(Xs)|ds <∞. (2.4) In this section, the cost functions are of the form: H(Xθ−, ζ) = H(Xθ−, 1) opening cost, H(Xθ−, 0) closing cost. The optimal switching problem is to optimize the performance measure for i = 0 (start in closed state) and 1 (start in open state). That is to find, for both i = 1 and i = 0, vi(x) , sup Jw(x) with X0 = x (2.5) where W is the set of all the admissible strategies. 2.1 Characterization of switching times For the remaining part of section 2, we assume that the state space X is I = (c, d) where both c and d are natural boundaries of X. But our characterization of the value function does not rely on this assumption. In fact, it is easily applied to other types of boundaries, for example, absorbing boundary. The first task is to characterize the optimal switching times as exit times from intervals in R. For this purpose, we define two functions g0 and g1 : R+ → R with g1(x) , sup Jw1 (x) and g0(x) , sup Jw0 (x). (2.6) where W0 , {w ∈ W : w = (θ0, ζ0, θ1 = +∞)}. In other words, g1(·) is the discounted expected revenue by starting with ζ0 = 1 and making no switches. Similarly, g0(·) is the discounted expected revenue by staring with ζ0 = 0 and making no switches. We set w0 , g1 and y0 , g0. We consider the following simultaneous sequential optimal stopping problems with wn : R+ → R and yn : R+ → R for n = 1, 2, ....: wn(x) , sup e−αsf(Xs)ds + e −ατ (yn−1(Xτ )−H(Xτ−, 1− Zτ−)) , (2.7) yn(x) , sup e−αsf(Xs)ds+ e −ατ (wn−1(Xτ )−H(Xτ−, 1− Zτ−)) , (2.8) where S is a set of Ft stopping times. Note that for each n, the sequential problem 2.7 (resp. (2.8)) starts in open (resp. closed) state. On the other hand, we define n-time switching problems for ζ0 = 1: q(n)(x) , sup Jw1 (x), (2.9) where Wn , {w ∈W ;w = (θ1, θ2, ...θn+1; ζ1, ζ2, ...ζn); θn+1 = +∞}. In other words, we start with ζ0 = 1 (open) and are allowed to make at most n switches. Similarly, we define another n-time switching problems corresponding to ζ0 = 0: p(n)(x) , sup Jw0 (x). (2.10) We investigate the relationship of these four problems: Lemma 2.1. For any x ∈ R, wn(x) = q(n)(x) and yn(x) = p(n)(x). Proof. We shall prove only the first assertion since the proof of the second is similar. We have set y0(x) = g0(x). Now we consider w1 by using the strong Markov property of X: w1(x) = sup e−αsf(Xs)ds+ e −ατ (g0(Xτ )−H(Xτ−, 0)) = sup e−αsf(Xs)ds − e−αsf(Xs)ds− e−ατ (g0(Xτ )−H(Xτ−, 0)) = sup e−ατ (g0(Xτ )− g1(Xτ )−H(Xτ−, 0)) + g1(x). On the other hand, q(1)(x) = sup e−αsf(Xs)ds− e−αθ1H(Xθ1− , ζ1) = sup [∫ θ1 e−αsf(Xs)ds + e−αsf(Xs)ds− e−αθ1H(Xθ1− , 0) = sup (g1(x)− e−αθ1g1(Xθ1))− e −αθ1(g0(Xθ1)−H(Xθ1− , 0)) = sup e−αθ1(g0(Xθ1)− g1(Xθ1)−H(Xθ1− , 0)) + g1(x). Since both τ and θ1 are Ft stopping times, we have w1(x) = q(1)(x) for all x ∈ R. Moreover, by the theory of the optimal stopping (see Appendix A, especially Proposition A.4), τ and hence θ1 are characterized as an exit time from an interval. Similarly, we can prove y1(x) = p (1)(x). Now we consider q(2)(x) which is the value if we start in open state and make at most 2 switches (open → close → open). For this purpose, we consider the performance measure q̄(2) that starts in an open state and is allowed two switches: For arbitrary switching times θ1, θ2 > θ1 ∈ S , we have q̄(2)(x) , Ex e−αsf(Xs)ds− e−αθjH(Xθj− , ζj) e−αsf(Xs)ds + e−αsf(Xs)ds + e−αsf(Xs)ds − e−αθ1H(Xθ1−, 0)− e −αθ2H(Xθ2−, 1) g1(x)− Ex[e−αθ1g1(Xθ1)] x[e−αθ1g0(Xθ1)− e−αθ2g0(Xθ2)] + Ex[e−αθ2g1(Xθ2)] − Ex[e−αθ1H(Xθ1−, 0) + e−αθ2H(Xθ2−, 1)]. Hence we have the following multiple optimal stopping problems: q̄(2)(x) = sup (θ1,θ2)∈S2 e−αθ1 (g0 − g1)(Xθ1)−H(Xθ1−, 0) + e−αθ2 (g1 − g0)(Xθ2)−H(Xθ2−, 1) + g1(x) where S2 , {(θ1, θ2); θ1 ∈ S; θ2 ∈ Sθ1} and Sσ = {τ ∈ S; τ ≥ σ} for every σ ∈ S . Let us denote h1(x) , g1(x)− g0(x)−H(x, 0), h2(x) , g0(x)− g1(x)−H(x, 1), V1(x) , sup e−ατh1(Xτ ) and V2(x) , sup e−ατ (h2(Xτ ) + V1(Xτ )) We also define Γ1 , {x ∈ I : V1(x) = h1(x)} and Γ2 , {x ∈ I : V2(x) = h2(x) + V1(x)} with σn , inf{t ≥ 0 : Xt ∈ Γn}. By using Proposition 5.4. in Carmona and Dayanik (2003), we conclude that θ1 = σ1 and θ2 = θ1 + σ2 ◦ s(θ1) is optimal strategy where s(·) is the shift operator. Hence we only consider the maximization over the set of admissible strategy W ∗2 where W ∗2 , {w ∈W2 : θ1, θ2 are exit imes from an interval in I}, and can use the relation θ2 − θ1 = θ ◦ s(θ1) with some exit time θ ∈ S . q(2)(x) = sup w∈W ∗2 e−αsf(Xs)ds− e−αθjH(Xθj− , ζj) = sup w∈W ∗2 e−αsf(Xs)ds + e−αsf(Xs)ds + e−αsf(Xs)ds − e−αθ1(H(Xθ1− , 0) + e−α(θ2−θ1)H(Xθ2− , 1)) = sup w∈W ∗2 e−αsf(Xs)ds + e [(∫ θ e−αsf(Xs)ds− e−αθH(Xθ−, 1) − e−αθ1H(Xθ1− , 0) Now by using the result for p(1), we can conclude q(2)(x) = sup w∈W ∗2 [∫ θ1 e−αsf(Xs)ds + e p(1)(Xθ1)−H(Xθ1− , 0) = sup [∫ θ1 e−αsf(Xs)ds+ e y1(Xθ1)−H(Xθ1− , 0) = w2(x) Similarly, we can prove y2(x) = p (2)(x) and we can continue this process inductively to conclude that wn(x) = q (n)(x) and yn(x) = p (n)(x) for all x and n. Lemma 2.2. For all x ∈ R, limn→∞ q(n)(x) = v1(x) and limn→∞ p(n)(x) = v0(x). Proof. Let us define q(x) , limn→∞ q (n)(x). Since Wn ⊂ W , q(n)(x) ≤ v1(x) and hence q(x) ≤ v1(x). To show the reverse inequality, we define W+ to be a set of admissible strategies such that W+ = {w ∈W : Jw1 (x) <∞ for all x ∈ R}. Let us assume that v1(x) < +∞ and consider a strategy w+ ∈ W+ and another strategy wn that coincides with w+ up to and including time θn and then takes no further interventions. 1 (x)− Jw1 (x) = Ex e−αs(f(Xs)− f(Xs−θn))− i≥n+1 e−αθiH(Xθi−, ζi)  , (2.11) which implies |Jw+1 (x)− Jw1 (x)| ≤ Ex e−αθn − i≥n+1 e−αθiH(Xθi−, ζi) As n→ +∞, the right hand side goes to zero by the dominated convergence theorem. Hence it is shown v1(x) = sup Jw1 (x) = sup w∈∪nWn Jw1 (x) so that v1(x) ≤ q(x). Next we consider v1(x) = +∞. Then we have some m ∈ N such that wm(x) = q(m)(x) = ∞. Hence q(n)(x) = ∞ for all n ≥ m. The second assertion is proved similarly. We define an operator L : H → H where H is a set of Borel functions Lu(x) , sup e−αsf(Xs)ds+ e −ατ (u(Xτ )−H(Xτ−, 1 − Zτ−)) Lemma 2.3. The function w(x) , limn→∞wn(x) is the smallest solution, that majorizes g1(x), of the function equation w = Lw. Proof. We renumber the sequence (w0, y1, w2, y3...) as (u0, u1, u2, u3....). Since un is monotone increas- ing, the limit u(x) exists. We have un+1(x) = Lun(x) and apply the monotone convergence theorem by taking n → ∞, we have u(x) = Lu(x). We assume that u′(x) satisfies u′ = Lu′ and majorizes g1(x) = u0(x). Then u ′ = Lu′ ≥ Lu0 = u1. Let us assume, for induction argument that u′ ≥ un, then u′ = Lu′ ≥ Lun = un+1. Hence we have u′ ≥ un for all n, leading to u′ ≥ limn→∞ un = u. Now we take the subsequence in (w0, y1, w2, y3....) to complete the proof. Proposition 2.1. For each x ∈ R, limn→∞wn(x) = v1(x) and limn→∞ yn(x) = v0(x). Moreover, the optimal switching times, θ∗i are exit times from an interval. Proof. We can prove the first assertion by combining the first two lemmas above. Now we concentrate on the sequence of wn(x). For each n, finding wn(x) by solving (2.7) is an optimal stopping problem. By Proposition A.4, the optimal stopping times are characterized as an exit time of X from an interval for all n. This is also true in the limit: Indeed, by Lemma 2.3, in the limit, the value function of optimal switching problem v1(x) = w(x) satisfies w = Lw, implying that v1(x) is the solution of an optimal stopping problem. Hence the optimal switching times are characterized as exit time from an interval. 2.2 Characterization of the value functions We go back to the original problem (2.3) to characterize the value function of the optimal switching prob- lems. By the exit time characterization of the optimal switching times, θ∗i are given by θ∗i = inf{t > θi−1;X1t ∈ Γ1} inf{t > θi−1;X0t ∈ Γ0} (2.12) where Γ1 = R \C1 and Γ0 = R \C0. We define here Ci and Γi to be continuation and stopping region for Xit , respectively. We can simplify the performance measure J w considerably. For ζ0 = 1, we have Jw1 (x) = E e−αsf(Xs)ds − e−αθjH(Xθj− , ζj) e−αsf(Xs)ds+ e−αsf(Xs)ds − e−αθ1 H(Xθ1−, 0) + e−α(θi−θ1)H(Xθj− , ζj) e−αsf(Xs)ds+ e −αθ1E e−αsf(Xs)ds− e−αθjH(Xθj− , ζj) − e−αθ1H(Xθ1−, 0) We notice that in the time interval (0, θ1), the process X is not intervened. The inner expectation is just Jw0 (Xθ1). Hence we further simplify Jw1 (x) = E e−αsf(Xs)ds+ e −αθ1(Jw0 (Xθ1)−H(Xθ1−, 0)) −e−αθ1g1(Xθ1) + e−αθ1(Jw0 (Xθ1)−H(Xθ1−, 0)) + g1(x) −e−αθ1g1(Xθ1) + e−αθ1Jw1 (Xθ1) + g1(x). The third equality is a critical observation. Finally, we define u1 , J1 − g1 and obtain u1(x) = J 1 (x)− g1(x) = Ex e−αθ1u1(Xθ1) . (2.13) Since the switching time θ1 is characterized as a hitting time of a certain point in the state space, we can represent θ1 = τa , inf{t ≥ 0 : Xt = a} for some a ∈ R. Hence equation (2.13) is an optimal stopping problem that maximizes u1(x) = J 1 (x)− g1(x) = Ex e−ατau1(Xτa) . (2.14) among all the τa ∈ S . When θ1 = 0 (i.e., x = Xθ1), Jw1 (x) = E x [−g1(x) + Jw0 (x)−H(x, 0)] + g1(x) and hence u1(x) = J 0 (x)−H(x, 0)− g1(x). In other words, we make a switch from open to closed immediately by paying the switching cost. Similarly, for ζ0 = 0, we can simplify the performance measure J 0 (·) to obtain Jw0 (x) = E −e−αθ1g0(Xθ1) + e −αθ1Jw0 (Xθ1) + g0(x). By defining u0 , J 0 − g0, we have u0(x) = J 0 (x)− g0(x) = Ex e−αθ1u0(Xθ1) Again, by using the characterization of switching times, we replace θ1 with τb, u0(x) = J 0 (x)− g0(x) = Ex e−ατbu0(Xτb) . (2.15) In summary, we have u1(x) = u0(x) + g0(x)−H(x, 0) − g1(x), x ∈ Γ1, x [e−ατau1(Xτa)] = E x [e−ατa(u0(Xτa) + g0(Xτa)− g1(Xτa)−H(Xτa , 0))] , x ∈ C1, (2.16) u0(x) = x [e−ατbu0(Xτb)] = E x [e−ατb(u1(Xτb) + g1(Xτb)− g0(Xτb)−H(Xτb , 1))] , x ∈ C0, u1(x) + g1(x)−H(x, 1) − g0(x), x ∈ Γ0. (2.17) Hence we should solve the following optimal stopping problems simultaneously: v̄1(x) , supτ∈S E x [e−ατ (u1(Xτ )] v̄0(x) , supσ∈S E x [e−ασ(u0(Xσ)] (2.18) Now we let the infinitesimal generators of X1 and X0 be A1 and A0, respectively. We consider (Ai − α)v(x) = 0 for i = 0, 1. This ODE has two fundamental solutions, ψi(·) and ϕi(·). We set ψi(·) is an increasing and ϕi(·) is a decreasing function. Note that ψi(c+) = 0, ϕi(c+) = ∞ and ψi(d−) = ∞, ϕi(d−) = 0. We define Fi(x) , ψi(x) ϕi(x) and Gi(x) , − ϕi(x) ψi(x) for i = 0, 1. By referring to Dayanik and Karatzas (2003), we have the following representation x[e−ατr1{τr<τl}] = ψ(l)ϕ(x) − ψ(x)ϕ(l) ψ(l)ϕ(r) − ψ(r)ϕ(l) , Ex[e−ατr1{τl<τr}] = ψ(x)ϕ(r) − ψ(r)ϕ(x) ψ(l)ϕ(r) − ψ(r)ϕ(l) for x ∈ [l, r] where τl , inf{t > 0;Xt = l} and τr , inf{t > 0;Xt = r}. By defining W1 = (u1/ψ1) ◦G−11 and W0 = (u0/ϕ0) ◦ F the second equation in (2.16) and the first equation in (2.17) become W1(G1(x)) =W1(G1(a)) G1(d)−G1(x) G1(d) −G1(a) +W1(G1(d)) G1(x)−G1(a) G1(d) −G1(a) x ∈ [a, d), (2.19) W0(F0(x)) =W0(F0(c)) F0(b)− F0(x) F0(b)− F0(c) +W0(F0(b)) F0(x)− F0(c) F0(b)− F0(c) , x ∈ (c, b], (2.20) respectively. We should understand that F0(c) , F0(c+) = ψ0(c+)/ϕ0(c+) = 0 and that G1(d) , G1(d−) = −ϕ1(d−)/ψ1(d−) = 0. In the next subsection, we shall explain W1(G1(d−)) and W0(F0(c+)) in details. Both W1 and W0 are a linear function in their respective transformed spaces. Hence under the appropriate transformations, the two value functions are linear functions in the continuation region. 2.3 Direct Method for a Solution We have established a mathematical characterization of the value functions of optimal switching problems. We shall investigate, by using the characterization, a direct solution method that does not require the recur- sive optimal stopping schemes described in section 2.1. Since the two optimal stopping problems (2.18) have to be solved simultaneously, finding u0 in x ∈ C0, for example, requires that we find the smallest F0-concave majorant of (u1(x) + g1(x)− g0(x)−H(x, 1))/ϕ0(x) as in (2.17) that involves u1. There are two cases, depending on whether x ∈ C1 ∩C0 or x ∈ Γ1 ∩C0, as to what u1(·) represents. In the region x ∈ Γ1 ∩C0, u1(·) that shows up in the equation of u0(x) is of the form u1(x) = u0(x) + g0(x)−H(x, 1, 0) − g1(x). In this case, the “obstacle” that should be majorized is in the form u1(x) + g1(x)− g0(x)−H(x, 1) = (u0(x) + g0(x)−H(x, 0)− g1(x)) + g1(x)− g0(x)−H(x, 1) = u0(x)−H(x, 0)−H(x, 1) < u0(x). (2.21) This implies that in x ∈ Γ1∩C0, the u0(x) function always majorizes the obstacle. Similarly, in x ∈ Γ0∩C1, the u1(x) function always majorizes the obstacle. Next, we consider the region x ∈ C0 ∩ C1. The u0(·) term in (2.16) is represented, due to its linear characterization, as W0(F0(x)) = β0(F0(x)) + d0 with some β0 ∈ R and d0 ∈ R+ in the transformed space. (The nonnegativity of d0 will be shown.) In the original space, it has the form of ϕ0(x)(β0F0(x) + d0). Hence by the transformation (u1/ψ1) ◦ G−1, W1(G1(x)) is the smallest linear majorant of K1(x) + ϕ0(x)(β0F0(x) + d0) ψ1(x) K1(x) + β0ψ0(x) + d0ϕ0(x) ψ1(x) on (G1(d−), G1(a∗)) where K1(x) , g0(x)− g1(x)−H(x, 0). (2.22) This linear function passes a point (G1(d−), ld) where G1(d−) = 0 and ld = lim sup (K1(x) + β0ψ0(x) + d0ϕ0(x)) ψ1(x) Let us consider further the quantity ld ≥ 0. By noting lim sup (K1(x) + β0ψ0(x)) ψ1(x) ≤ lim sup (K1(x) + β0ψ0(x) + d0ϕ0(x)) ψ1(x) ≤ lim sup (K1(x) + β0ψ0(x)) ψ1(x) + lim sup d0ϕ0(x) ψ1(x) and lim supx↑d d0ϕ0(x) ψ1(x) = 0, we can redefine ld by ld , lim sup (K1(x) + β0ψ0(x)) ψ1(x) (2.23) to determine the finiteness of the value function of the optimal switching problem, v1(x), based upon Propo- sition A.5-A.7. Let us concentrate on the case ld = 0. Similar analysis applies to (2.17). u1(x) in (2.17) is represented as W1(G1(x)) = β1G1(x) + d1 with some β1 ∈ R and d1 ∈ R+. Note that d1 = ld ≥ 0. In the original space, it has the form of ψ1(x)(β1G1(x) + d1). Hence by the transformation (u0/ϕ0(x)) ◦ F−1, W0(F0(x)) is the smallest linear majorant of K0(x) + ψ1(x)(β1G1(x) + d1) ϕ0(x) K0(x)− β1ϕ1(x) + d1ψ1(x) ϕ0(x) on (F0(c+), F0(b ∗)) where K0(x) , g1(x)− g0(x)−H(x, 1). (2.24) This linear function passes a point (F0(c+), lc) where F0(c+) = 0 and lc = lim sup (K0(x)− β1ϕ1(x) + d1ψ1(x))+ ϕ0(x) Hence we have lc = d0 ≥ 0. By the same argument as for ld, we can redefine lc , lim sup (K0(x)− β1ϕ1(x))+ ϕ0(x) . (2.25) Remark 2.1. (a) Evaluation of ld or lc does not require knowledge of β0 or β1, respectively unless the orders of max(K1(x), ψ1(x)) and ψ0(x) are equal, for example. (For this event, see Proposition 2.4.) Otherwise, we just compare the order of the positive leading terms of the numerator in (2.23) and (2.25) with that of the denominator. (b) A sufficient condition for ld = lc = 0: since we have 0 ≤ ld ≤ lim sup (K1(x)) ψ1(x) + lim sup (β0ψ0(x)) ψ1(x) a sufficient condition for ld = 0 is lim sup (K1(x)) ψ1(x) = 0 and lim sup ψ0(x) ψ1(x) = 0. (2.26) Similarly, 0 ≤ lc ≤ lim sup (K0(x)) ϕ0(x) + lim sup (−β1ϕ1(x))+ ϕ0(x) Hence a sufficient condition for lc = 0 is lim sup (K0(x)) ϕ0(x) = 0 and lim sup ϕ1(x) ϕ0(x) = 0. (2.27) Moreover, it is obvious β1 < 0 and β0 > 0 since the linear majorant passes the origin of each transformed space. Recall a points in the interval (c, d) ∈ R+ will be transformed by G(·) to (G(c), G(d−)) ∈ R−. We summarize the case of lc = ld = 0: Proposition 2.2. Suppose that ld = lc = 0, the quantities being defined by (2.23) and by (2.25), respectively. The value functions in the transformed space are the smallest linear majorants of R1(·) , 1 (·)) 1 (·)) and R0(·) , 0 (·)) 0 (·)) where r1(x) , g0(x)− g1(x) + β0ψ0(x)−H(x, 0) r0(x) , g1(x)− g0(x)− β1ϕ1(x)−H(x, 1) β0 > 0 and β1 < 0. (2.28) Furthermore, Γ1 and Γ0 in (2.16) and (2.17) are given by Γ1 , {x ∈ (c, d) :W1(G1(x)) = R1(G1(x))}, and Γ0 , {x ∈ (c, d) :W0(F0(x)) = R0(F0(x))}. Corollary 2.1. If either of the boundary points c or d is absorbing, then (F0(c),W0(F0(c)) or (G1(d),W1(G1(d))) is obtained directly. We can entirely omit the analysis of lc or ld. The characterization of the value function (2.19) and (2.20) remains exactly the same. Remark 2.2. An algorithm to find (a∗, b∗, β∗0 , β 1) can be described as follows: 1. Start with some β′1 ∈ R. 2. Calculate r0 and then R0 by the transformation R0(·) = 0 (·)) 0 (·)) 3. Find the linear majorant of R0 passing the origin of the transformed space. Call the slope of the linear majorant, β0 and the point, F0(b), where R0 and the linear majorant meet . 4. Plug b and β0 in the equation for r1 and calculate R1 by the transformation R1(·) = 1 (·)) 1 (·)) 5. Find the linear majorant of R1 passing the origin of the transformed space. Call the slope of the linear majorant, β1 and the point, G1(a), where R1 and the linear majorant meet. 6. Iterate step 1 to 5 until β1 = β If both R1 and R0 are differentiable functions with their respective arguments, we can find (a ∗, b∗) analyti- cally. Namely, we solve the following system for a and b: dR0(y) y=F0(b) (F0(b)− F0(c)) = R0(F0(b)) dR1(y) y=G1(a) (G1(a)−G1(d)) = R1(G1(a)) (2.29) where dR0(y) y=F0(b∗) = β∗0 and dR1(y) y=G1(a∗) = β∗1 . Once we find W1(·) and W0(·), then we convert to the original space and add back g1(x) and g0(x) respectively so that v1(x) = ψ1(x)W1(G1(x)) + g1(x) and v0(x) = ϕ0(x)W0(F0(x)) + g0(x). Therefore, by (2.16) and (2.17), the value functions v1(·) and v0(·) are given by: Proposition 2.3. If the optimal continuation regions for both of the value functions are connected and if lc = ld = 0, then the pair of the value functions v1(x) and v0(x) are represented as v1(x) = v̂0(x)−H(x, 0), x ≤ a∗, v̂1(x) , ψ1(x)W1(G1(x)) + g1(x), a ∗ < x, v0(x) = v̂0(x) , ϕ0(x)W0(F0(x)) + g0(x) x < b v̂1(x)−H(x, 1), b∗ ≤ x, for some a∗, b∗ ∈ R with a∗ < b∗. Proof. If the optimal continuation regions for both of the value functions are connected and if ld = lc = 0, then the optimal intervention times (2.30) have the following form: θ∗i = inf{t > θi−1;Xt /∈ (a∗, d)}, Z = 1, inf{t > θi−1;Xt /∈ (c, b∗)}, Z = 0. (2.30) Indeed, since we have lc = ld = 0, the linear majorants W1(·) and W0(·) pass the origins in their respective transformed coordinates. Hence the continuation regions shall necessarily of the form of (2.30). By our construction, both v1(x) and v0(x) are continuous in x ∈ R. Suppose we have a∗ > b∗. In this case, by the form of the value functions, v0(b−)−H(b, 1, 0) = v1(b). Since the cost function H(·) > 0 and continuous, it follows v0(b−) > v1(b). On the other hand, v0(b+) = v1(b)−H(b, 0, 1) implying v0(b+) < v1(b). This contradicts the continuity of v0(x). Also, a ∗ = b∗ will lead to v1(x) = v1(x)−H(x, 1, 0) which is impossible. Hence if the value functions exist, then we must necessarily have a∗ < b∗. In relation to Proposition 2.3, we have the following observations: Remark 2.3. (a) It is obvious that v0(x) = v̂0(x) > v̂0(x)−H(x, 0) = v1(x), x ∈ (c, a∗), v1(x) = v̂1(x) > v̂1(x)−H(x, 1) = v0(x), x ∈ (b∗, d). (b) Since u1(x) is continuous in (c, d), the “obstacle” u1(x) + g1(x)− g0(x)−H(x, 1) to be majorized by u0(x) on x ∈ C0 = (c, b∗) is also continuous, in particular at x = a∗. We proved that u0(x) always majorizes the obstacle on (c, a∗). Hence F (a∗) ∈ {y : W0(y) > R0(y)} if there exists a linear majorant of R0(y) in an interval of the form (F0(q), F0(d)) with some q ∈ (c, d): otherwise, the continuity of u1(x) + g1(x) − g0(x) −H(x, 1) does not hold. Similarly, we have F (b∗) ∈ {y : W1(y) > R1(y)} if there exists a linear majorant of R0(y) in an interval of the form (G1(c), G1(q)). Finally, we summarize other cases than lc = ld = 0: Proposition 2.4. (a) If either ld = +∞ or lc = +∞, then v1(x) = v0(x) ≡ +∞. (b) If both ld and lc are finite, then ld = lc = 0. Proof. (a) The proof is immediate by invoking Proposition A.5. (b) When lc is finite, we know by Proposi- tion A.5 that the value function v0(x) is finite. On x ∈ (c, a∗), u1(x)+g1(x)−g0(x)−H(x, 1) < u0(x) < +∞ is finite (see (2.21)) and thereby lc = lim sup u1(x) + g1(x)− g0(x)−H(x, 1) ϕ0(x) The same argument for ld = 0. Therefore, we can conclude that ld = 0 for the situation where the orders of max(K1(x), ψ1(x)) and ψ0(x) are equal (⇒ ld is finite) as described in Remark 2.1 (a). 3 Examples We recall some useful observations. If h(·) is twice-differentiable at x ∈ I and y , F (x), then we define H(y) , h(F−1(y))/ϕ(F−1(y)) and we obtain H (y) = m(x) and H (y) = m (x)/F (x) with m(x) = (x), and H (y)(A− α)h(x) ≥ 0, y = F (x) (3.1) with strict inequality if H (y) 6= 0. These identities are of practical use in identifying the concavities of H(·) when it is hard to calculate its derivatives explicitly. Using these representations, we can modify (2.29) F ′0(b) (b)(F0(b)− F0(c)) = r0(b)ϕ0(b) G′1(a) (a)(G1(a)−G1(d)) = r1(a)ψ1(a) (3.2) Example 3.1. Brekke and Øksendal (1994): We first illustrate our solution method by using a resource extraction problem solved by Brekke and Øksendal (1994). The price Pt at time t per unit of the resource follows a geometric Brownian motion. Qt denotes the stock of remaining resources in the field that decays exponentially. Hence we have dPt = αPtdt+ βPtdWt and dQt = −λQtdt where α, β, and λ > 0 (extraction rate) are constants. The objective of the problem is to find the optimal switching times of resource extraction: v(x) = sup Jw(x) = sup e−ρt(λPtQt −K)Ztdt− e−ρθiH(Xθi−, Zθi) where rho ∈ R+ is a discount factor with ρ > α, K ∈ R+ is the operating cost and H(x, 0) = C ∈ R+ and H(x, 1) = L ∈ R+ are constant closing and opening costs. Since P and Q always show up in the form of PQ, we reduce the dimension by defining Xt = PtQt with the dynamics: dXt = (α− λZt)Xtdt+ βXtdWt. Solution: (1) We shall calculate all the necessary functions. For Zt = 1 (open state), we solve (A1 − ρ)v(x) = 0 where A1 = (α − λ)xv′(x) + 12β 2x2v′′(x) to obtain ψ1(x) = x ν+ and ϕ1(x) = x ν− where ν+,− = β −α+ λ+ 1 (α− λ− 1 β2)2 + 2ρβ2 . Similarly, for Zt = 0 (closed state), we solve (A0 − ρ)v(x) = 0 where A0 = αxv′(x) + 12β 2x2v′′(x) to obtain ψ0(x) = x µ+ and ϕ0(x) = x µ− where µ+,− = β −α+ 1 (α− 1 β2)2 + 2ρβ2 . Note that under the assumption ρ > α, we have ν+, µ+ > 1 and ν−, ν− < 0. By setting ∆1 = (α− λ− 1 β2)2 + 2ρβ2 and ∆0 = (α− 1 β2)2 + 2ρβ2, we have G1(x) = −ϕ1(x)/ψ1(x) = −x−2∆1/β and F0(x) = ψ0(x)/ϕ0(x) = x 2∆0/β . It follows thatG−11 (y) = (−y)−β 2/2∆1 and F−10 (y) = y β2/2∆0 . In this problem, we can calculate g1(x), g0(x) explicitly: g1(x) = E e−ρs(λXs −K)ds ρ+ λ− α and g(x) = 0. Lastly, K1(x) = g0(x) − g1(x) − H(x, 0) = − ρ+λ−α − C and K0(x) = g1(x)− g0(x)−H(x, 1) = xρ+λ−α − (2) The state space of X is (c, d) = (0,∞) and we evaluate lc and ld. Let us first note that ∆0−∆1+λ > 0. Since limx↓0 ϕ1(x) ϕ0(x) = limx↓0 x ∆0−∆1+λ β2 = 0 and limx↓0(K0(x)) +/ϕ0(x) = 0, we have lc = l0 = 0 by (2.27). Similarly, by noting limx↑+∞ ψ0(x) ψ1(x) = limx↑+∞ x −(∆0−∆1+λ) β2 = 0 and limx↑+∞(K1(x)) +/ϕ0(x) = 0, we have ld = l+∞ = 0 by (2.26). (3) To find the value functions together with continuation regions, we set r1(x) = − ρ+ λ− α − C + β0ψ0(x) and r0(x) = ρ+ λ− α − L− β1ϕ1(x) and make transformations R1(y) = r1(F −1(y))/ψ1(F −1(y)) and R0(y) = r0(F −1(y))/ϕ0(F −1(y)), re- spectively. We examine the shape and behavior of the two functions R1(·) and R0(·) with an aid of (3.1). By calculating (r0/ϕ0) ′(x) explicitly to examine the derivative of R0(y), we can find a critical point x = q, at which R0(F (x)) attains a local minimum and from which R0(F (x)) is increasing monotonically on (F0(q),∞). Moreover, we can confirm that limy→∞R′0(y) = limx→∞ (r0/ϕ0) F ′0(x) = 0, which shows that there exists a finite linear majorant of R0(y). We define p(x) = β1ωx ν− − (ρ− α) ρ+ λ− α + (K + ρL) such that (A0 − ρ)r0(x) = p(x) where ω , β2ν−(ν− − 1)− αν− (∆0 − ∆1 + λ)(∆0 + ∆1−λ) > 0. By the second identity in (3.1), the sign of the second derivative R′′0(y) is the same as the sign of p(x). It is easy to see that p(x) has only one critical point. For any β1 < 0, the first term is dominant as x → 0, so that limx↓0 p(x) < 0. As x gets larger, for |β1| sufficiently small, p(x) can take positive values, providing two positive roots, say x = k1, k2 with k1 < k2. We also have limx→+∞ p(x) = −∞. In this case, R0(y) is concave on (0, F (k1) ∪ (F (k2),+∞) and convex on (F (k1), F (k2)). Since we know that R0(y) attains a local minimum at y = F (q), we have q < k2, and it implies that there is one and only on tangency point of the linear majorant W (y) and R0(y) on (F (q),∞), so that the continuation region is of the form (0, b∗). ¿From this analysis of the derivatives of R0(y), there is only one tangency point of the linear majorant W0(y) and R0(y). (See Figure 3.1-(a)). A similar analysis shows that there is only one tangency point of the linear majorant W1(y) and R1(y). (See Figure 3.1-(b)). 0.5 1 1.5 2 2.5 3 3.5 RHFH.L,WHFH.LL -200 -150 -100 -50 RHGH.L,WHGH.LL 0.5 1 1.5 2 2.5 v0HxL 0.1 0.2 0.3 0.4 0.5 0.6 0.7 v1HxL Figure 1: A numerical example of resource extraction problem. with parameters (α, β, λ, ρ,K, L,C) = (0.01, 0.25, 0.01, 0.05, 0.4, 2, 2)(a) The smallest linear majorant W0(F0(x)) and R0(F0(x)) with b ∗ = 1.15042 and = 10.8125. (b)The smallest linear majorantW1(G1(x)) andR1(G1(x)) with a ∗ = 0.18300 and β∗ = −0.695324. (c) The value function v0(x). (d) The value function v1(x). (4) By solving the system of equations (2.29), we can find (a∗, b∗, β∗0 , β 1). We transform back to the original space to find v̂1(x) = ψ1(x)W1(G1(x)) + g1(x) = ψ1(x)β 1G1(x) + g1(x) = −β∗1ϕ1(x) + g1(x) = −β∗1xν− + ρ+ λ− α v̂0(x) = ϕ0(x)W0(F0(x)) + g0(x) = ϕ0(x)β 0F0(x) + g0(x) = β 0ψ0(x) + g0(x) = β Hence the solution is v1(x) = µ+ − C, x ≤ a∗, −β∗1xν− + ρ+λ−α , x > a∗, v0(x) = µ+ , x ≤ b∗, −β∗1xν− + ρ+λ−α − L, x > b∗, which agrees with Brekke and Økesendal (1994). Example 3.2. Ornstein-Uhrenbeck process: We shall consider a new problem involving an Ornstein- Uhrenbeck process. Consider a firm whose revenue solely depends on the price of one product. Due to its cyclical nature of the prices, the firm does not want to have a large production facilty and decides to rent additional production facility when the price is favorable. The revenue process to the firm is dXt = δ(m −Xt − λZt)dt+ σdWt, where λ = r/δ with r being a rent per unit of time. The firm’s objective is to maximize the incremental revenue generated by renting the facility until the time τ0 when the price is at an intolerably low level. Without loss of generality, we set τ0 = inf{t > 0 : Xt = 0}. We keep assuming constant operating cost K , opening cost, L and closing cost C . Now the value function is defined as v(x) = sup Jw(x) = sup e−αt(Xt −K)Ztdt− θi<τ0 e−αθiH(Xθi−, Zθi) Solution: (1) We denote, by ψ̃(·) and ϕ̃(·), the functions of the fundamental solutions for the auxiliary process Pt , (Xt −m+ λ)/σ, t ≥ 0, which satisfies dPt = −δPtdt+ dWt. For every x ∈ R, ψ̃(x) = eδx 2/2D−α/δ(−x 2δ) and ϕ̃(x) = eδx 2/2D−α/δ(x which leads to ψ1(x) = ψ̃((x −m + λ)/σ), ϕ1(x) = ϕ̃((x −m + λ)/σ), ψ0(x) = ψ̃((x −m)/σ), and ϕ0(x) = ϕ̃((x −m)/σ) where Dν(·) is the parabolic cylinder function; (see Borodin and Salminen (2002, Appendices 1.24 and 2.9) and Carmona and Dayanik (2003, Section 6.3)). By using the relation Dν(z) = 2−ν/2e−z 2/4Hν(z/ 2), z ∈ R (3.3) in terms of the Hermite function Hν of degree ν and its integral representation Hν(z) = Γ(−ν) 2−2tzt−ν−1dt, Re(ν) < 0, (3.4) (see for example, Lebedev(1972, pp 284, 290)). Since Ex[Xt] = e −δtx + (1 − e−δt)(m − λ), we have g0(x) = 0 and g1(x) = x−(m−λ) + m−λ−K (2) The state space of X is (c, d) = (0,+∞). Since the left boundary 0 is the absorbing, the linear majorant passes (0, F0(0)). Since limx→+∞ ψ0(x)/ψ1(x) = 0, we have ld = 0. (3) We formulate r1(x) = − x− (m− λ) δ + α m− λ−K − C + β0ψ0(x) r0(x) = x− (m− λ) δ + α m− λ−K − L− β1ϕ1(x) and make transformations: R1(y) = r1(F −1(y))/ψ1(F −1(y)) and R0(y) = r0(F −1(y))/ϕ0(F −1(y)), respectively. We examine the shape and behavior of the two functions R1(·) and R0(·) with an aid of (3.1). First we check the sign of R′0(y) and find a critical point x = q, at which R0(F (x)) attains a local minimum and from which R0(F (x)) is increasing monotonically on (F0(q),∞). It can be shown that R 0(+∞) = 0 by using (3.3) and (3.4) and the identity H′ν(z) = 2νHν−1(z), z ∈ R (see Lebedev (1972, p.289), for example.) This shows that there must exist a (finite) linear majorant of R0(y) on (F (q),∞). To check convexity of R0(y), we define p(x) = − ϕ′′1(x) + δ(m− x− λ) δ + α − β1ϕ′1(x) − αr0(x) such that (A0−α)r0(x) = p(x). We can show easily limx→+∞ p(x) = −∞ since ϕ1(+∞) = ϕ′1(+∞) = ϕ′′1(+∞) = 0. Due to the monotonicity of ϕ1(x) and its derivatives, p(x) can have at most one critical point and p(x) = 0 can have one or two positive roots depending on the value of β1. In either case, let us call the largest positive root x = k2. We also have limx→+∞ p(x) = −∞. Since we know that R0(y) attains a local minimum at y = F (q) and is increasing thereafter, we have q < k2. It follows that there is one and only on tangency point of the linear majorant W (y) and R0(y) on (F (q),∞), so that the continuation region is of the form (0, b∗). A similar analysis shows that there is only one tangency point of the linear majorant W1(y) and R1(y). (4) Solving (3.2), we we can find (a∗, b∗, β∗0 , β 1). We transform back to the original space to find v̂1(x) = ψ1(x)W1(G1(x)) + g1(x) = ψ1(x)β 1G1(x) + g1(x) = −β∗1ϕ1(x) + g1(x) = −β∗1e δ(x−m+λ)2 2σ2 D−α/δ (x−m+ λ) x− (m− λ) δ + α v̂0(x) = ϕ0(x)W0(F0(x)) + g0(x) = ϕ0(x)β 0 (F0(x)− F0(0)) + g0(x) = β∗0{ψ0(x)− F0(0)ϕ0(x)}+ g0(x) = β∗0e (x−m+λ)2 D−α/δ x−m+ λ − F (0)D−α/δ Hence the solution is, using the above functions, v1(x) = v̂0(x)− C, x ≤ a∗, v̂1(x), x > a v0(x) = v̂0(x), x ≤ b∗, v̂1(x)− L, x > b∗. See Figure 3.2 for a numerical example. 0.5 1 1.5 2 2.5 3 v0HxL 0.5 1 1.5 2 2.5 3 v1HxL Figure 2: A numerical example of leasing production facility problem with parameters (m,α, σ, δ, λ,K, L,C) = (5, 0.105, 0.35, 0.05, 4, 0.4, 0.2, 0.2): (a) The value function v0(x) with b ∗ = 1.66182 and β∗ = 144.313. (b)The value function v1(x) with a ∗ = 0.781797 and β∗ = −2.16941. 4 Extensions and conclusions 4.1 An extension to the case of k ≥ 2 It is not difficult to extend to a general case of k ≥ 2 where more than one switching opportunities are available. But we put a condition that z ∈ Z is of the form z = (a1, a2, ...., ak) where only one element of this vector is 1 with the rest being zero, i.e., z = (0, 0, 0, ...., 1, 0, 0) for example. We should introduce the switching operator M0 on h ∈ H, M0h(u, z) = max ζ∈Z\{z} {h(u, ζ)−H(u, z; ζ)} . (4.1) In words, this operator would calculate which production mode should be chosen by moving from the current production mode z. Now the recursive optimal stopping (2.7) becomes wn+1(x) , sup e−αsf(Xs)ds+ e −ατMwn(Xτ ) Accordingly, the optimization procedure will become two-stage. To illustrate this, we suppose k = 2 so that i = 0, 1, and 2. By eliminating the integral in (4.1), we redefine the switching operator, Mhz(x) , max ζ∈Z\{z} {hζ(x) + gζ(x)− gz(x)−H(x, z, ζ)} , (4.2) where gz(x) , sup Jwz (x) = E e−αsf(Xs)ds Hence (2.13) will be modified to uz(x) = E x[e−ατMuz(Xτ )]. It follows that our system of equations (2.18) is now v̄2(x) , supτ∈S E x [e−ατMv̄2(Xτ )] v̄1(x) , supτ∈S E x [e−ατMv̄1(Xτ )] v̄0(x) , supτ∈S E x [e−ατMv̄0(Xτ )] (4.3) The first stage is optimal stopping problem. One possibility of switching production modes is (0 → 1, 1 → 2, 2 → 0). First, we fix this switching scheme, say c, and solve the system of equations (4.3) as three optimal stopping problems. All the arguments in Section 2.3 hold. This first-stage optimization will give (x∗0(c), x 1(c), x 2(c), β 0 (c), β 1 (c), β 2 (c)), where xi’s are switching boundaries, depending on this switching scheme c. Now we move to another switching scheme c′ and solve the system of optimal stopping problems until we find the optimal scheme. 4.2 Conclusions We have studied optimal switching problems for one-dimensional diffusions. We characterize the value function as linear functions in their respective spaces, and provide a direct method to find the value functions and the opening and switching boundaries at the same time. Using the techniques we developed here as well as the ones in Dayanik and Karazas (2003) and Dayanik and Egami (2005), we solved two specific problems, one of which involves a mean-reverting process. This problem might be hard to solve with just the HJB equation and the related quasi-variational inequalities. Finally, an extension to more general cases is suggested. We believe that this direct method and the new characterization will expand the coverage of solvable problems in the financial engineering and economic analysis. A Summary of Optimal Stopping Theory Let (Ω,F ,P) be a complete probability space with a standard Brownian motion W = {Wt; t ≥ 0} and consider the diffusion process X0 with state pace I ⊆ R and dynamics dX0t = µ(X t )dt+ σ(X t )dWt (A.1) for some Borel functions µ : I → R and σ : I → (0,∞). We emphasize here that X0 is an uncontrolled process. We assume that I is an interval with endpoints −∞ ≤ a < b ≤ +∞, and that X0 is regular in (a, b); in other words, X0 reaches y with positive probability starting at x for every x and y in (a, b). We shall denote by F = {Ft} the natural filtration generated by X0. Let α ≥ 0 be a real constant and h(·) a Borel function such that Ex[e−ατh(X0τ )] is well-defined for every F-stopping time τ and x ∈ I . Let τy be the first hitting time of y ∈ I by X0, and let c ∈ I be a fixed point of the state space. We set: ψ(x) = x[e−ατc1{τc<∞}], x ≤ c, 1/Ec[e−ατx1{τx<∞}], x > c, ϕ(x) = e−ατx1{τx<∞} , x ≤ c, x[e−ατc1{τc<∞}], x > c, F (x) , , x ∈ I. (A.2) Then F (·) is continuous and strictly increasing. It should be noted that ψ(·) and ϕ(·) consist of an increasing and a decreasing solution of the second-order differential equation (A − α)u = 0 in I where A is the infinitesimal generator of X0. They are linearly independent positive solutions and uniquely determined up to multiplication. For the complete characterization of ψ(·) and ϕ(·) corresponding to various types of boundary behavior, refer to Itô and McKean (1974). Let F : [c, d] → R be a strictly increasing function. A real valued function u is called F -concave on [c, d] if, for every a ≤ l < r ≤ b and x ∈ [l, r], u(x) ≥ u(l) F (r)− F (x) F (r)− F (l) + u(r) F (x)− F (l) F (r)− F (l) We denote by V (x) , sup x[e−ατh(X0τ )], x ∈ [c, d] (A.3) the value function of the optimal stopping problem with the reward function h(·) where the supremum is taken over the class S of all F-stopping times. Then we have the following results, the proofs of which we refer to Dayanik and Karatzas (2003). Proposition A.1. For a given function U : [c, d] → [0,+∞) the quotient U(·)/ϕ(·) is an F -concave function if and only if U(·) is α-excessive, i.e., U(x) ≥ Ex[e−ατU(X0τ )],∀τ ∈ S,∀x ∈ [c, d]. (A.4) Proposition A.2. The value function V (·) of (A.3) is the smallest nonnegative majorant of h(·) such that V (·)/ϕ(·) is F -concave on [c, d]. Proposition A.3. Let W (·) be the smallest nonnegative concave majorant of H , (h/ϕ) ◦ F−1 on [F (c), F (d)], where F−1(·) is the inverse of the strictly increasing function F (·) in (A.2). Then V (x) = ϕ(x)W (F (x)) for every x ∈ [c, d]. Proposition A.4. Define S , {x ∈ [c, d] : V (x) = h(x)}, and τ∗ , inf{t ≧ 0 : X0t ∈ S}. (A.5) If h(·) is continuous on [c, d], then τ∗ is an optimal stopping rule. When both boundaries are natural, we have the following results: Proposition A.5. We have either V ≡ 0 in (c, d) or V (x) < +∞ for all (c, d). Moreover, V (x) < +∞ for every x ∈ (c, d) if and only if lc , lim sup h+(x) and ld , lim sup h+(x) (A.6) are both finite. In the finite case, furthermore, Proposition A.6. The value function V (·) is continuous on (c, d). If h : (c, d) → R is continuous and lc = ld = 0, then τ ∗ of (A.5) is an optimal stopping time. Proposition A.7. Suppose that lc and ld are finite and one of them is strictly positive, and h(·) is continuous. Define the continuation region C , (c, d) \ Γ. Then τ∗ of (A.5) is an optimal stopping time, if and only if there is no r ∈ (c, d) such that (c, r) ⊂ C if lc > 0 and there is no l ∈ (c, d) such that (l, d) ⊂ C if ld > 0. References Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian motion - facts and formulae, 2nd Edition. Birkhäuser, Basel. Brekke, K. A. and Øksendal, B. (1994). Optimal switching in an economic activity under uncertainty. SIAM J. Control Optim. 32(4), 1021–1036. Brennan, M. J. and Schwartz, E. S. (1985). Evaluating natural resource investments. J. Business. 58, 135– Carmona, R. and Dayanik, S. (2003). Optimal multiple-stopping of linear diffusions and swing options. Preprint. Princeton University. Carmona, R. and Ludkovski, M. (2005). Optimal switching with applications to energy tolling agreements. Preprint. University of Michigan. Dayanik, S. and Egami, M. (2005). Solving stochastic impulse control problems via optimal stopping for one-dimensional diffusions. Preprint. www.math.lsa.umich.edu/∼ egami. Dayanik, S. and Karatzas, I. (2003). On the optimal stopping problem for one-dimensional diffusions. Stochastic Process. Appl. 107(2), 173–212. Dixit, A. (1989). Entry and exit decisions under uncertainty. J. Political Economy. 97, 620–638. Duckworth, K. (2001). A model for investment decisions with switching costs. Annuls of Appl. Prob. 11(1), 239–260. Hamadène, S. and Jeanblanc, M. (2004). On the starting and stopping problem: Applications in reversible investments. Preprint. Itô, K. and McKean, Jr., H. P. (1974). Diffusions processes and their sample paths. Springer-Verlag, Berlin. Lebedev, N. N. (1972). Special functions and their applications, Dover Publications Inc., New York. Revised edition, translated from the Russian and editied by R. A. Silverman. Yushkevich, A. (2001). Optimal switching problem for countable Markov chains: average reward criterion. Math. Meth. Oper. Res.53, 1–24. Introduction Optimal Switching Problems Characterization of switching times Characterization of the value functions Direct Method for a Solution Examples Extensions and conclusions An extension to the case of k2 Conclusions Summary of Optimal Stopping Theory

0704.0995

Composite Structure and Causality

Composite Structure and Causality Satish D. Joglekar∗ October 30, 2018 Department of Physics, I.I.T. Kanpur, Kanpur 208016 (INDIA) Abstract We study the question of whether a composite structure of elemen- tary particles, with a length scale 1/Λ, can leave observable effects of non-locality and causality violation at higher energies (but . Λ). We formulate a model-independent approach based on Bogoliubov- Shirkov formulation of causality. We analyze the relation between the fundamental theory (of finer constituents) and the derived theory (of composite particles). We assume that the fundamental theory is causal and formulate a condition which must be fulfilled for the derived the- ory to be causal. We analyze the condition and exhibit possibilities which fulfil and which violate the condition. We make comments on how causality violating amplitudes can arise. 1 Introduction The standard model (SM), a local quantum field theory, has served so far as a very good description of elementary particle processes [1]. It is however widely believed that soon, when higher energies are experimentally accessible, new phenomena may emerge that require a description that goes beyond the standard model. Among the various the possibilities, is the possibility that a composite nature of the standard model constituents may be revealed [2] and a possible failure of locality [3]. It is possible that the underlying physics ∗e-mail address:[email protected] http://arxiv.org/abs/0704.0995v2 is nonlocal at shorter distances which could be a result of composite struc- ture of particles, or granularity of space-time, or underlying noncommutative structure of space-time [4]. With a nonlocal interaction, often goes causality violation that can arise because the interaction region, encloses points sep- arated by a space-like interval. Causality violation has been studied in the context of non-local [5] and non-commutative quantum field theories [6, 7]. It has in fact been suggested [8, 5] that non-local quantum field theories [9, 10] may indeed serve as effective field theories for a deeper/more fundamental theory such as a composite model; and the former indeed show causality violation[10, 5]. An effective tool to study causality has been developed by Bogoliubov and Shirkov [11] and has been in particular employed for the causality violation in non-local [5] and non-commutative QFT’s [7]. We wish to consider the following question: in view of the possible composite nature of elementary particles, leading to extended structures, will these leave an observable effect in the form of a violation of causality and locality that can be detected? A similar question regarding a violation of the Pauli exclusion principle on account of the compositeness of particles has been earlier ad- dressed to [12]. This question is particularly interesting since should there be a signal of CV, it will be detected long before an explicit knowledge of composite structure is known. In fact, it is has been suggested [5] that the unknown physics at high energy scales (Λ) from a possible source can effec- tively be represented in a consistent way (a unitary, gauge-invariant, finite (or renormalizable) theory) by a nonlocal theory at energies lower than Λ, but higher than the present ones. In other words, the nonlocal standard model, with a parameter Λ, can serve as such an effective field theory and will afford a model-independent way of consistently reparametrizing the effects beyond standard model . In this model, one finds that there is but a small CV at low energies, which grows rapidly as energies approach Λ and beyond these, the fundamental theory is expected to take over and presumably it leads to no CV again. The aim of the present work is to approach this question in a model-independent way in connection with a composite structure of SM constituents. 2 Preliminary 2.1 Definition of the problem Suppose that the presently known standard model particles are a composite of a set of finer constituents. Suppose that these underlying constituents belong to a local causality-preserving fundamental theory. Suppose, further that at lower energies, one only observes the composite bound states and their scattering processes. These bound state particles are extended objects. A priori, their interaction is expected to be non-local. A nonlocal covariant interaction has, at a given instant, interaction spread over a region in space, which therefore contains spatially separated points. An obvious question arises: will the interactions of the composite theory be such that causality is preserved by this low-energy theory? We need the fundamental theory for energy scales >> Λ, and for energy scales << Λ, we have the set of composite particles described by the ”derived” theory. Then the question, paraphrased differently is, will the phase transition (should there be one) from the fundamental to the composite be causality preserving or it could lead to a breakdown of causality at short enough distances? 2.2 Definition of the system Let, for simplicity, the fundamental theory, denoted by F , be character- ized by a single coupling constant g. For the purpose of formulation of the Bogoliubov-Shirkov (BS) criterion of causality, we shall need to formulate a theory with a variable coupling g(x). Let the low-energy derived theory be characterized by its own coupling constant λ ≡ λ [g], which for identical reasons, we shall need to allow to depend on space-time: λ = λ (x). We shall assume, for simplicity, that the derived theory, denoted by C, is completely described by its scattering states: i.e. we shall assume that the model admits no bound states. A scattering state of the derived theory can be looked upon from two different point of views: ⋆ a scattering state, as t → ±∞, is a state of a certain set of non- interacting composite particles of the low energy theory with certain momenta, polarizations etc. ⋆ a scattering state, as t → ±∞, is a (complicated) configuration of fields of the fundamental theory. 3 Causality formulation for a theory without a well-defined S-matrix Bogoliubov and Shirkov have shown [11] that S−operator is causal only if it satisfies, δg (x) δg (y) = 0, x <∼ y (1) (Here, x < y⇐⇒ x0 < y0 and x ∼ y ⇐⇒ (x − y) 2 < 0 ). The condition is obtained from the primary meaning of causality that a disturbance does not propagate outside the forward light-cone (the disturbance considered is that in g(x)1), and is independent of any specific field theory formulation. The BS causality criterion holds for a theory for which an S-operator is defined. For a theory such as QCD, some of the matrix elements of the S-operator may not exist on account of the infrared divergences. It is nonetheless true that an alternate formulation in terms of the U-operator (i.e. U (−T, T ′)) can be given. This is so because, the U operator is unitary as much as the S-operator and the BS criterion depends on two points x, y with x <∼ y which can always be chosen to be such that they both lie in (−T, T ′). The relation would then read δg (x) δg (y) = 0, x <∼ y; −T < x0, y0 < T ′ (2) It is possible to alternately formulate the causality condition in terms of the following choices of the couplings. [This way results when we suitably integrate (2) over x0 < 0 and y0 > 0]. In this approach, we make a comparison of the following two neighboring theories in the coupling constant space2: 1. Fundamental theory F ′: Coupling constants = g′2 ( a constant value) for x0 > 0 and g1(a constant value) for x0 < 0. Corresponding derived theory is C′. 2. Fundamental theory F ′′: Coupling constants = g′′2 ( a constant value) for x0 > 0 and g1(the same constant value) for x0 < 0. Corresponding derived theory is C′′. 1One may consider varying g(x) an unphysical operation, but one can look alternately upon varying g(x) at a point x0 as insertion of a (specific) local operator ∂g(x) at x0 and study the propagation of its effects. 2The idea of varying the coupling with time over all space is not an entirely unfamiliar one: it is also employed in the LSZ formulation. All the coupling constants are (chosen to be) space-independent. It suffices for our purpose that g′2 differs infinitesimally from g1 and g 2 . (We can in fact assume that the infinitesimal change from g1 to g2 is carried out adiabatically and in an infinitesimal time). Then, we can alternately formulate [13] the causality condition as, U [g1, g 2 ;−T, T ′]U † [g1, g 2;−T, T ′] is independent of g1 (3) This alternate formulation makes mathematics simpler, though it may lead to an unusual-looking Physics. In the following, we shall adopt a ”reductio ad absurdum” approach: We shall let, if possible, that the theory C be causality-preserving and deduce the consequences of causality of F for C and analyze these. 4 Relations between the derived theory and the fundamental theory 4.1 Relations between coupling constants The coupling constant λ is a function of g. If we allow a space-time dependent coupling, then λ = λ [g]. A small change3δg (x) in the coupling g (x) about g(x) = g = constant, will cause a change in λ (x) as given by4 δλ(z) = δλ(z) δg(y) g(y)=g δg (y). For the BS criterion of causality of Eq. (2), we need to know the impact of a localized change g(x) → g (x) + δg (x) ≡ g (x) + εδ4 (x− x̃) on the function λ (x). Now, if causality is valid, λ (x) cannot be affected for any x0 < x̃0. Assuming that the theory has T-invariance λ (x) cannot be affected for any x0 > x̃0. Thus, this together with causality requires that, λ (x) → λ (x) + Cεδ4 (x− x̃) 3For the argument presented subsequently, we shall go back to a general space-time dependent coupling and not confine ourselves to the specific couplings presented in the previous section. 4 We shall assume the existence and non-vanishing of δλ(z) δg(y) g(y)=g . By translational invariance, this quantity is a function of (z − y) and is independent of the point z as such. 5We shall need that the theory F with a variable coupling has a T-invariance. This is possible to formulate a time-reversal transformation for a theory with a variable g (x): we need to define the action of time-reversal as Tg (x, t)T−1 = g (x,−t). + terms having finite order derivatives of delta function Thus, δλ (z) δg (y) = Cδ4 (z − y) + terms having finite order derivatives of delta function. Then, for a constant small change δg = ε, for all x0 > 0, [i.e. δg(x) = ǫθ(x0)]; we find, δλ(z) = δλ(z) δg(y) δg (y) Cδ4 (z − y) + derivatives of delta function εθ (y0) = C ′εθ (z0) for z0 > 0. We shall denote by λ′2 = λ[g 2, g1] and λ 2 = λ[g 2 , g1]. 4.2 Relation between states We shall work in the interaction picture of C. Let the derived theory C′ have as incoming states6 {|c̃m (λ1,−T )〉} which, as −T → −∞, represents scattering states with a number of free composite particles. We shall keep T finite and will let T → ∞ only at the end of the argument. Evidently, as −T → −∞, |c̃m (λ1,−T )〉 depends on λ1 only through the self-interaction of each individual non-interacting particle in the state. Let H̃ denote the Hilbert space of states of C′. Then the hypothesis that the scattering states of C′ forms a complete set implies that the set {|c̃m (λ1,−T )〉} spans H̃: H̃ ≡ sp {|c̃m (λ1,−T )〉}. We shall denote by H, the Hilbert space of states of F ′ (and likewise for F ′′ ). Consider a state |c̃m (λ1,−T )〉 ∈ H̃ in the interaction picture. On physical grounds, we know that there is a corre- sponding state of F ′ in the interaction picture, denoted by |cm (g1,−T )〉. We note that H can, in addition, have states linearly independent of the states {|cm (g1,−T )〉}. We augment this set to complete an orthonormal ba- sis {|cm (g1,−T )〉} ∪ {|βn (g1,−T )〉} ≡ {|αp (g1,−T )〉} for H. We shall call 6For technical simplicity, we shall assume that the set of states is countably infinite. the span of {|cm (g1,−T )〉} by Ĥ ⊂ H. A similar discussion holds for F Let us now consider the time-evolution, from t = −T to t = T ′, of a single particle state of C′′ denoted by |s̃p (λ1,−T )〉, which belongs to the basis of H̃. The unitary time evolution operator Ũ [λ1, λ 2;−T, T ′] as applied to the state leads to Ũ [λ1, λ 2;−T, T ′] |s̃p (λ1,−T )〉 = |s̃p (λ ′)〉 ∈ H̃ (4) This state is a single particle state of slightly different mass, on account of a slightly different self-energy, and interacts with a coupling λ′′2. We shall also introduce interaction picture states d̃m (λ . These states are at t = T ′ and as T ′ → ∞ consist of a set of non-interacting (but self-interacting) parti- cles of a slightly different mass and coupling constant λ′′2. These are analogues of the ”out” states. We shall assume that these also span H̃. We shall further make a convention: Under time reversal, the quantum numbers of particles in the state |c̃m (λ1,−T )〉 become those of d̃m (λ1, T . Now, consider an exclusive process in C′′. The magnitude of the quantum mechanical ampli- tude for it, as seen from C′′and F ′′ are identical, as these are, in principle, experimentally observable: |ũnm| ≡ d̃n (λ Ũ [λ1, λ 2;−T, T ′] |c̃m (λ1,−T )〉 ≡ |〈dn (g 2 , T ′)|U [g1, g 2 ;−T, T ′] |cm (g1,−T )〉| ≡ |unm| (5) Here, we have introduced states |dn (g 2 , T ′)〉 in H analogous to d̃n (λ in H̃. We note that U here is the U−matrix in the interaction picture of F ′′, as the set of states {|cm (g1,−T )〉} evolve according to the interaction Hamiltonian H′I (g) (in the interaction picture) of the F First we note that on account of unitarity of Ũ and Eq. (5), d̃n (λ Ũ [λ1, λ 2;−T, T ′] |c̃m (λ1,−T )〉 |〈dn (g 2 , T ′)|U [g1, g 2 ;−T, T ′] |cm (g1,−T )〉| d̃n (λ Ũ [λ1, λ 2;−T, T ′] |c̃m (λ1,−T )〉 |〈dn (g 2 , T ′)|U [g1, g 2 ;−T, T ′] |cm (g1,−T )〉| So, the unitarity of U implies, 〈dn (g 2 , T ′)|U [g1, g 2 ;−T, T ′] |βm (g1,−T )〉 = 0 〈βn (g 2 , T ′)|U [g1, g 2 ;−T, T ′] |cm (g1,−T )〉 = 0 (8) The relations (8) implies that U is a block-diagonal matrix. The unitarity of U then implies that the block corresponding to the subspace ∧H, viz. Û , is also unitary. We shall now attempt relate these further. In this connection, we recall a result for a finite dimensional matrices: Lemma : Let U and U ′ be two N ×N unitary matrices satisfying: |u′ij| = |uij|; 1 ≤ i, j ≤ N . Then, there exist phases {θi : i = 1, 2, . . . , N} and {φi : i = 2, . . . , N} such that u ij = uij exp [i (θi + φj)] : 1 ≤ i, j ≤ N withφ1 ≡ 0.. Proof : Let the diagonal elements of U ′ and U be related by: u′ii = exp (iΘi)uii. We define U ′′ by u′′ij = exp (−iΘi)u ij. Then, u ii = uii. Now U ′′ is unitary and thus, a priori, has N2 independent parameters. The information on moduli of elements constitutes (N − 1)2 independent conditions, corresponding to an (N − 1)× (N − 1) dimensional submatrix; the rest of the (2N − 1) mod- uli being determined by relations implying that the norm of each row and column is unity. The relations u′′ii = uii imply additional N relations on the phases on uii.This leaves N 2 − (N − 1)2 −N = N − 1 free parameters. The phases of u′′1j , 2 ≤ j ≤ N are unconstrained by |u ij| = |uij| : 1 ≤ i, j ≤ N and u′′ii = uii and we define u 1j = u1j exp (iφj), 2 ≤ j ≤ N . Then, there are no free parameters and must lead to a unique U ′′. Now, U ′′ specified by u′′ij = uij exp (iφj − iφi), 1 ≤ i, j ≤ N (φ1 ≡ 0) is such a solution. This together with u′′ij = exp (−iΘi)u ij leads to the result; with the definition Θi − φi = θi. Thus, in view of the unitarity of Ũ , and Û and (5), we write, d̃n (λ Ũ [λ1, λ 2;−T, T ′] |c̃m (λ1,−T )〉 ≡ 〈dn (g 2 , T ′)|U [g1, g 2 ;−T, T ′] |cm (g1,−T )〉 × exp (iθ n + iφm) (9) We shall assume that F and C are have time-reversal invariance and derive the consequences. Under time reversal, we know then that, 〈β|S |α〉 = 〈T α|S |T β〉 (10) where |T β〉 is the state obtained by time-reversing the quantum numbers of the state |β〉. In this case, it would imply, keeping in mind our choice of definitions, d̃n (λ Ũ [λ1, λ 2;−T, T ′] |c̃m (λ1,−T )〉 d̃m (λ1, T ) Ũ [λ1, λ 2;−T, T ′] |c̃n (λ ′)〉 (11) We write a similar relation for F . Putting T ′ = T , (or equivalently, noting that the matrix elements are insensitive to T ′ and T ), we find, φp(λ2, λ1) = θp(λ1, λ2) (12) 5 Consequence of Causality of F for C We shall assume that the fundamental theory F is causal and deduce the consequences for the derived theory C(C′, C′′). The causality of F implies U [g1, g 2 ;−T, T ′]U † [g1, g 2;−T, T is independent of g1. Hence, Mnm ≡ 〈dn (g 2 , T ′)|U [g1, g 2 ;−T, T ′]U † [g1, g 2;−T, T ′] |dm (g is also independent of g1 since the state vectors 〈dn (g 2 , T ′)| and |dm (g are independent of g1 with g 2 and g 2 fixed. We shall re-express Mnm in terms of the matrix elements of the derived theory C(C′, C′′) and deduce the consequences. We note, Mnm = 〈dn (g 2 , T ′)|U [g1, g 2 ;−T, T ′]U † [g1, g 2;−T, T ′] |dm (g 〈dn (g 2 , T ′)|U [g1, g 2 ;−T, T ′] |αp (g1,−T )〉 × 〈αp (g1,−T )|U † [g1, g 2;−T, T ′] |dm (g ′)〉 (13) 〈dn (g 2 , T ′)|U [g1, g 2 ;−T, T ′] |cp (g1,−T )〉 × 〈cp (g1,−T )|U † [g1, g 2;−T, T ′] |dm (g ′)〉 (14) d̃n (λ Ũ [λ1, λ 2;−T, T ′] |c̃p (λ1,−T )〉 exp [−i(θ̃ p − θ̃ × 〈c̃p (λ1,−T )| Ũ † [λ1, λ 2;−T, T d̃m (λ exp−[i(θ′′n − θ m)](15) ≡ M̃nm(λ 2, λ1) (16) In the 3rd step, we have employed the equations (8) and in the second step, we have employed the closure relation for F . In the above, θ′′n ≡ θn(λ 2, λ1), θ m ≡ θm(λ 2, λ1), and θ̃ p ≡ θp(λ1, λ 2) etc. Thus, the expression (15) is independent of λ1: ∂M̃nm(λ 2, λ1) = 0 (17) 6 Analysis of Causality Condition We shall now analyze the condition (17) obtained as an implication of causal- ity of F . For this purpose, we shall find it useful to Taylor-expand θn as follows7: 2, λ1) = θn(λ 2, λ1(0)) + βn∆1 + γn∆2 + δn∆1∆2 + · · · ≡ αn + βn∆1 + γn∆2 + δn∆1∆2 + · · · 2, λ1) = αm + βm∆1 + · · · (18) Here, ∆1 ≡ λ1 − λ1(0); ∆2 ≡ λ 2 − λ 2; β, γ, δ refer to appropriate partial derivatives at (λ′2, λ1(0)) and λ1(0) is some value near λ1. We note that if θn(λ2, λ1) is a function only of its first argument (I) then, (θ̃′′p − θ̃ p) ≡ θp(λ1, λ 2)− θp(λ1, λ 2) is zero and (θ n − θ m) ≡ θn(λ 2, λ1)− 2, λ1) is independent of λ1. Also, we can then carry out the sum over p using the completeness relation and find that the independence from λ1 of M̃nm(λ 2, λ1) d̃n (λ Ũ [λ1, λ 2;−T, T ′] Ũ † [λ1, λ 2;−T, T d̃m (λ × exp−[i(θ′′n − θ m)] (19) 7Throughout, we have employed only the infinitesimal variations in the couplings. These are sufficient to determine the first order partial derivatives with respect to each λ1 and λ2. Hence, we shall content ourselves with expansion only upto O(∆1∆2) for all m,n implies Ũ [λ1, λ 2;−T, T ′] Ũ † [λ1, λ 2;−T, T ′] is independent of λ1. This condition is indeed necessary for causality of C. In fact, in this case, we can rewrite8 d̃n (λ2, T Ũ [λ1, λ2;−T, T ′] |c̃m (λ1,−T )〉 ≡ 〈dn (g2, T ′)|U [g1, g2;−T, T ′] |cm (g1,−T )〉 × exp (iθn(λ2, λ1) + iθm(λ1, λ2)) (20) d̃∗n (λ2, T Ũ [λ1, λ2;−T, T ∣c̃∗m (λ1,−T ) ≡ 〈dn (g2, T ′)|U [g1, g2;−T, T ′] |cm (g1,−T )〉 (21) by redefining states by absorbing phases: ∣c̃∗m (λ1,−T ) = e−iθm(λ1) |c̃m (λ1,−T )〉) etc. We note that this redefinition of the states is meaningful and compatible with causality when θn is independent of its second argument. If on the other hand, θn is dependent on its second argument (excepting a possibility below), we cannot absorb a phase in a manner compatible with causality : a state ∣c̃∗m at t = −T cannot be made to depend on the value of coupling λ2 it would have at a later time t > 0. We can, in fact, liberalize somewhat the above condition by requiring that, βn = β and δn = 0 ∀ n (II) In this case, (θ̃′′p − θ̃ p) ≡ θp(λ1, λ 2)− θp(λ1, λ = β∆2 + · · · (22) is independent of λ1 and does not depend also on p and thus comes out of the summation in (15). The summation in (15) can be carried out using the completeness relation. Also, (θ′′n − θ m) ≡ θn(λ 2, λ1) − θm(λ 2, λ1) is still independent of λ1. Thus, the entire discussion proceeds as before: in par- ticular, as a little analysis shows, the phases can again be absorbed into the definition of states in a manner compatible with causality. 8We have dropped primes on λ2. While we shall not provide the general analysis of (17), we shall establish examples of a few specific sufficient conditions for causality violation. (These are simple conditions that, in fact, contradict I or II above) We can easily verify the following results: 1. There is causality violation if (i) for some n, δn 6= 0 and (ii) βn = βm ∀ n,m 2. There is causality violation if there be m 6= n such that M̃nm(λ 2, λ1) 6= 0, when evaluated to O(∆), and βm 6= βn. Proof : We shall let, if possible, C be causal. We can then write, Ũ [λ1, λ 2;−T, T ′] = Ũ [λ′′2; 0, T ′] Ũ [λ1;−T, 0] (23) Then, we can write the expression (15) as, M̃nm(λ 2, λ1) = d̃n (λ 2, 0) c̃p (λ1, 0)〉 exp [−i(θ̃ p − θ̃ × 〈c̃p (λ1, 0) d̃m (λ 2, 0) exp−[i(θ′′n − θ d̃n (λ 2, 0) d̃m (λ 2, 0) exp [−i(θ′′n − θ m)] (24) where X ≡ |c̃p (λ1, 0)〉 〈c̃p (λ1, 0)| exp [−i(θ̃ p − θ̃ p)]. We shall now expand the quantities involved to the first order in the infinitesimals as in (18). In addition, we note that to the zeroth order in ∆2, (i.e. λ”2 − λ 2 = 0), we have, (θ̃′′p − θ̃ p) = 0 and the completeness relation leads to X = 1. We further define, d̃n (λ 2, 0) d̃m (λ 2, 0) = δnm + iηnm∆2 + · · · (25) Proof of (i): We define δ0 ≡ max{|δn|}; and let ±δ0 = δq for some q. We now have, (θ̃′′p − θ̃ p) ≡ θp(λ1, λ 2)− θp(λ1, λ = β∆2 + δp∆1∆2 + · · · (26) and thus, |c̃p (λ1, 0)〉 〈c̃p (λ1, 0)| exp [−i(θ̃ p − θ̃ = exp (−iβ∆2) |c̃p (λ1, 0)〉 〈c̃p (λ1, 0)| × exp [−i(δp∆1∆2)] = exp (−iβ∆2) I − i∆1∆2 |c̃p (λ1, 0)〉 〈c̃p (λ1, 0)| δp Thus, exp (iβ∆2) d̃q (λ 2, 0) d̃q (λ 2, 0) = 1 + iηqq∆2 − i∆1∆2 δp|upq| 2 + · · · (28) where upq ≡ d̃q (λ2, 0) c̃p (λ1, 0)〉 (We can ignore primes on λ2 in this term). The multiplicative exponential factor in (24) becomes: exp (−iγq∆2 − iδq∆1∆2 + · · ·) ≈ 1− iγq∆2 − iδq∆1∆2 + · · · . Thus, M̃qq = 1 + iηqq∆2 − iγq∆2 − iδq∆1∆2 − i∆1∆2 δp|upq| 2 + · · · = 1 + iηqq∆2 − iγq∆2 − i∆1∆2 [δq + δp]|upq| 2 + · · · (29) In view of the fact that either δp + δq ≥ 0 ∀ p or δp+ δq ≤ 0 ∀ p the last term is necessarily non-vanishing and dependent on ∆1 Proof of (ii): Consider the matrix element M̃nm(λ 2, λ1) ≡ d̃n (λ 2, 0) d̃m (λ 2, 0) exp−[i(θ′′n − θ m)] 6= 0(30) for n 6= m. To the first order in the infinitesimals, the nonzero matrix element d̃n (λ 2, 0) d̃m (λ 2, 0) is independent of ∆1. The multiplicative phase factor, exp−[i(θ′′n − θ m)] = exp{−i(αn − αm)− i(βn − βm)∆1 − iγn∆2} is necessarily dependent on ∆1, thus implying causality violation. 9There is the obvious exception that δp = −δq for every such p such thatupq 6= 0; and this has to be valid for each such q for which δq = ±δ0. 7 Additional comments We comment in a qualitative way upon how a phase factor depending on both values of the coupling can arise. Suppose that the derived theory C is actually correctly described by a nonlocal covariant theory with a finite non-zero non-locality scale ∆ ∼ 1/Λ. Since the theory is covariant, it is also non-local in time. We write, Ũ(λ1, λ2;−T, T ′) = Ũ(λ2; ∆, T ′)Ũ(λ1, λ2;−∆,∆)Ũ(λ1;−T,−∆) (31) where the first and the third factors on the right hand side depends only on one value of the coupling due to finite size of non-locality in time. The sec- ond factor however depends on both couplings because in this time-interval (−∆,∆), time evolution depends on both values of the coupling λ. On the other hand, the fundamental theory, being local and causal, however has no such analogue . The matrix Ũ(λ1, λ2;−∆,∆) can then give rise to phases depending on both couplings in relation (9). Naively, one may expect that if the fundamental theory is causal, the de- rived theory should be so. Examples are however known where the diagrams of the fundamental theory are associated with a different weight in the actual phenomenology. For example, OZI rule in hadronic phenomenology gives a suppression of a subset of the QCD diagrams. While such a possibility is distinct from what is discussed in this work, generally such a modification of the amplitudes within the fundamental theory may alter the underlying properties of the fundamental theory such as causality. References [1] See e.g. Reviews of Particle Properties in W.-M. Yao et al., Journal of Physics G 33, 1 (2006). [2] See e.g. H. Harari, Phys.Rept.104:159,1984 [3] C. Bourrely , N.N. Khuri , Andre Martin , J. Soffer , Tai Tsun Wu hep-ph/0511135; N.N. Khuri hep-ph/9512386 [4] See e.g. R.J. Szabo, Phys.Rep. 278,207 (2003). http://arxiv.org/abs/hep-ph/0511135 http://arxiv.org/abs/hep-ph/9512386 [5] S. D. Joglekar, and A. Jain, Int. J.Mod. Phys. 19, (2004), S.D. Joglekar, hep-th/0601006; [6] See e.g. M. Chaichian, K. Nishijima, A. Tureanu, Phys.Lett. B568:146- 152,2003; O.W. Greenberg, Phys.Rev.D73:045014,2006 [7] A. Haque and S.D. Joglekar, hep-th/0701171 [8] S.D.Joglekar, J. Phys. A 34, 2765 (2001); S.D.Joglekar,Int.J.Mod. Phys.A 16, (2001). [9] E. D. Evens et al, Phys Rev D43, 499 (1991) [10] G. Kleppe, and R. P. Woodard, Nucl. Phys. B 388, 81(1992). [11] N. N. Bogoliubov, and D. V. Shirkov, Introduction to the theory of quantized fields (John Wiley, New York, 1980) pg. 200-220. [12] K. Akama et al Phys. Rev. Lett. 68, 1826 (1991); O.W. Greenberg, R.N. Mohapatra Phys.Rev.Lett.59:2507,1987, Erratum-ibid.61:1432,1988; Phys.Rev.Lett.62:712,1989, Erratum-ibid.62:1927,1989 [13] For a simpler and intuitive understanding of the causality condition in either form, see e.g. S.D. Joglekar, hep-th/0601006 http://arxiv.org/abs/hep-th/0601006 http://arxiv.org/abs/hep-th/0701171 http://arxiv.org/abs/hep-th/0601006 Introduction Preliminary Definition of the problem Definition of the system Causality formulation for a theory without a well-defined S-matrix Relations between the derived theory and the fundamental theory Relations between coupling constants Relation between states Consequence of Causality of F for C Analysis of Causality Condition Additional comments

0704.0996

Brane World Black Rings

Brane World Black Rings. Anurag Sahay∗, Gautam Sengupta, †, Department of Physics, Indian Institute of Technology, Kanpur 208016, India. Abstract Five dimensional neutral rotating black rings are described from a Randall-Sundrum brane world perspective in the bulk black string frame- work. To this end we consider a rotating black string extension of a five dimensional black ring into the bulk of a six dimensional Randall-Sundrum brane world with a single four brane. The bulk solution intercepts the four brane in a five dimensional black ring with the usual curvature singular- ity on the brane. The bulk geodesics restricted to the plane of rotation of the black ring are constructed and their projections on the four brane match with the usual black ring geodesics restricted to the same plane. The asymptotic nature of the bulk geodesics are elucidated with reference to a bulk singularity at the AdS horizon. We further discuss the descrip- tion of a brane world black ring as a limit of a boosted bulk black 2 brane with periodic identification. April 2007 ∗[email protected] †[email protected] http://arxiv.org/abs/0704.0996v2 1 Introduction. Higher dimensional spacetimes are now an essential aspect of effective field theories arising from fundamental theories of quantum gravity. The general as- sumption implicit in such constructions was that the extra spatial dimensions are compactified to ultrashort length scales. Hence quantum gravity effects were relegated to very high energy scales. However in recent years the exciting possi- bility of low scale quantum gravity effects in the brane world models have inspired considerable interest and interesting phenomenological consequences [1–3]. The brane world scenario envisaged the gauge sector of the fundamental interactions to be restricted on a smooth codimension one hypersurface ( refered to as a brane) embedded in a higher dimensional space-time and the electroweak scale as the fundamental scale. The usual four dimensional Planck scale was then a derived scale. In particular the Randall-Sundrum models and their variants based on a warped non factorable compactification geometry in a bulk Anti deSitter ( AdS) space time offered a partial resolution to the vexing hierarchy problem [4]. Al- though the analysis was valid in a linearized framework a full non linear study from a supergravity perspective confirmed the conclusions and their extension to any Ricci flat geometry on the brane [5, 6]. For consistency the brane world scenario requires generic four dimensional gravitational configurations on the brane to arise from a higher dimensional bulk. The investigation of black hole configurations in this context has been an exciting aspect of the study of brane world gravity [5]. Such a black hole on the brane is expected to be a configuration extended in the bulk. Chamblin, Hawking and Reall [7] attempted the description of a Schwarzschild black hole in a typical single three brane five dimensional Randall-Sundrum brane world as a bulk black string. This reproduced the usual Schwarzschild singularity on the brane but additionaly was also singular at the AdS horizon far away from the three brane. Although a pathology, this singularity was possibly a linearization artifact and could be shown to be a mild p-p curvature singularity. The bulk black string was subject to the usual instabilities against long wavelength perturbations [8,9] and was expected to pinch off to a cigar geometry before reaching the AdS horizon. However the issue of stability is contentious and for sphericaly symmetric solutions it was shown that a more likely scenario is a transition to a non uniform black string [10] In an earlier article [11]we have generalized the construction of Chamblin et. al. [7]to consider rotating black holes in a five dimensional single three brane RS brane world. The bulk configuration proposed was a five dimensional rotat- ing black string which intercepted the three-brane in a four dimensional rotating black hole described by a Kerr metric on the three brane. It was found that the Kerr solution too was singular at the AdS horizon apart from the usual ring singularity on the brane. The asymptotics of the equatorial geodesics at the AdS horizon also indicated a p-p curvature singularity although an explicit determina- tion was computationaly intractable. There have been other approaches to brane world black holes including numerical studies for off brane metrics and a Hamil- tonian constraint approach to charged black holes [12–18]. In lower dimensions exact studies of brane world black holes [19] involving the AdS C-metric have indicated that the bulk solutions are regular everywhere emphasizing that the bulk singularity in higher dimension is possibly a linearization artifact. However absence of exact bulk metrics in higher dimensions requires a linearized approach and the black string framework is hence physicaly relevant in this context in spite of such a bulk singularity. The brane world constructions must be embedded in an appropriate string theory for consistency, requiring the generalizations of these models to higher dimensions. The generalization of the Randall-Sundrum construction and its variants to higher dimensions with a single space like AdS direction and an ap- propriate codimension one brane is straightforward. Additionaly this may easily be extended to include the full non linear extensions of a Ricci flat metric [20–22]. In higher dimensions also the consistency of such brane world constructions re- quire that gravitational configurations arise from appropriate bulk scenarios. In particular this applies to higher dimensional black holes on the codimension one brane. In this context in an earlier article [23]we had described the N dimensional rotating Myers-Perry [24]black hole on a single (N-1) brane in a (N + 1) dimen- sional RS brane world. The bulk solution in this case was a (N+1) dimensional rotating black string extended in the AdS direction transverse to the (N-1) brane. Analysis of equatorial geodesics again indicated a p-p curvature singularity in the bulk apart from the usual extended singularity on the (N-1) brane. In the recent past there has been remarkable and surprising progress in under- standing higher dimensional black holes. In particular it has been realized that the no hair and the uniqueness theorems are much less restrictive in higher dimen- sions [27]. In four dimensions the no hair theorem characterizes any stationary asymptoticaly flat black hole solution of Einstein-Maxwell system only by their mass, angular momentum and conserved charges whereas the uniqueness theorem forbids event horizons of non spherical toplogies. However the discovery [25] in five dimensions of an asymptoticaly flat stationary black hole solution with a non spherical ring like S2×S1 horizon topology with the possibility of dipole charges, showed that higher dimensional black holes posess remarkably distinctive prop- erties. The static black ring solution [28] was first obtained through the Wick rotation of a neutral solution of an Einstein-Maxwell system [29] although they involved conical singularities. However the stationary solution rotating in the S1 direction was regular everywhere except the usual curvature singularity. For fixed mass the angular momentum of the black ring was bounded below and for a certain range of parameters two black rings and a usual five dimensional rotating Myers-Perry black hole all with the same mass and spin coexist. The charged versions of these black rings were first obtained in the framework of D=5 heterotic supergravity [30] and fully supersymmeric three charged black ring solutions in D=5 followed later from compactifications of black supertubes in D=10 [31, 32]. It was seen that these black rings could also support gauge dipoles independent of the conserved gauge charges entailing an infinite non uniqueness and violating the no hair theorem [33]. As emphasized earlier, for consistency of the brane world scenario it is im- perative that gravitational configurations like black holes on the brane should arise from appropriate bulk solutions. In this context it is but natural to inves- tigate possible bulk configurations in a higher dimensional brane world scenario which would describe five dimensional black rings on the brane. This is espe- cialy relevant for the neutral rotating black rings as they are Ricci flat and hence satisfy the criteria for embedding in higher dimensional Randall-Sundrum brane worlds. Naturaly the absence of exact solutions in higher dimensions require the usual linearized framework to analyse this question. The black string approach is especialy relevant in this context to highlight the physical aspects of such an embedding although it suffers from singular pathologies which are possibly lin- earization artifacts. In this article we address this issue and show that it is possible to consistently embed the five dimensional black ring solution on a single four brane in a (5 + 1) dimensional Randall-Sundrum brane world. Following the black string approach we consider a six dimensional bulk rotating black string extension of the five dimensional black ring. This bulk configuration intercepts the four brane in a five dimensional rotating black ring. In what follows after a brief review of neutral rotating black rings, we obtain their geodesic equations in the plane of the ring analogous to the equatorial plane of black holes with spherical topologies. We further investigate the asymptotic behaviour of both the null and the timelike geodesics in this plane to elucidate the restricted causal structure of the black ring space time. In section three we consider a bulk rotating black string extension of a five dimensional neutral rotating black ring in a six dimensional RS brane world with a single four brane. The bulk black string intercepts the four brane in a five dimensional black ring with the usual spacelike curvature singularity on the brane. Additionaly a curvature singularity also appears at the AdS horizon far away from the four brane. Following the description of a black ring as a boosted black string with periodic identification in a certain limit, the bulk solution may be described as a boosted black two brane with the same periodic identification. We then construct the six dimensional bulk geodesics in the plane of rotation of the ring and show that their projections on the four brane reproduces the usual five dimensional black ring geodesics in the same plane. To study of the nature of the pathological singularity at the AdS horizon we further investigate the late time asymptotics of these geodesics. It is shown that the curvature remains finite along unbound geodesics which reach the AdS horizon. We also discuss the possibility of the bulk solution to pinch off before reaching the AdS horizon due to the usual instabilities and comment on the possible stable solution in the light of the analysis outlined in [9] and [10]. In the last section we provide a summary of our analysis and results and also discuss certain future open issues in this area. 2 The Rotating Neutral Black Ring . In this section we first briefly review the neutral rotating black ring and eluci- date the nature of the adapted coordinate system . We then construct the black ring geodesics restricted to the plane of rotation of the ring which is analogous to the equatorial geodesics in solutions with a spherical topology. Furthermore we analyse the geodesic equations to study the nature of the radial orbits for this plane and their asymptotics. The static neutral black ring was originally discovered through a Wick rotation of certain Kaluza Klein C metrics decribing neutral bubbles [29]. These involved conical singularities and consequent deficit angles leading to either cosmic string defects joining these singularities or deficit membranes. However an analytic continuation led to the original neutral rotat- ing black ring solution which was a five dimensional asymptoticaly flat black hole with a ringlike S2×S1 horizon topology, regular everywhere except at a spacelik e curvature singularity. The original solution was further refined through appro- priate factorizable choice of certain functions appearing in the metric [26,30–32]. The rotating black ring in equlibrium was parametrized by a dimensionless re- duced angular momentum j = 27π which was bounded from below for a fixed mass. It could be shown that in the range 27 ≤ j2 < 1 there existed one Myers- Perry black hole with spherical topology and two black rings with identical mass and angular momenta, in direct violation of the black hole uniqueness theorem. 2.1 Black Ring Metric The metric of the neutral rotating five dimensional black ring in a specific adpated coordinate system which is obtained from the foliation of space-time in terms of the equipotentials of certain 1-form and 2-form gauge potentials is , [32] ds2 = − F (y) F (x) dt− C R 1 + y F (y) (x− y)2 F (x) −G(y) F (y) dψ2 − dy F (x) , (1) where the functions F (ξ) = 1 + λξ, G(ξ) = (1− ξ2)(1 + νξ) , (2) λ(λ− ν) 1 + λ . (3) Here R is a length scale which may be interpreted as the radius of the ring in some limit [32] and the two dimensionless parameters λ and ν which are related to the shape and the rotation velocity of the ring lie in the range 0 < ν ≤ λ < 1 (4) . The range of the spatial co-ordinates (x, y) are required to be, − 1 ≤ x ≤ +1 , −∞ ≤ y ≤ −1 . (5) respectively. The constant y hypersurfaces are nested deformed solid toroids with topology S2 × S1, whereas the coordinate x is like a direction cosine, x = +1 points to the interior of the ring and x = −1 points to the region outside the ring. The solution is a stationary axisymmetric solution with rotation in the ψ direction, and admits t, φ, and ψ Killing isometries. In order to avoid conical singularities at the fixed points x = −1 and y = −1 of the Killing isometries ∂φ and ∂ψ the co-ordinates ψ and φ require to be identified with the equal periods ∆ψ = ∆φ = 4π F (−1) |G′(−1)| . (6) Furthermore the requirement that the orbits of the isometry ∂φ shows no deficit angles at x = +1 lead to the condition 1 + ν2 The co-ordinates (x, φ) parametrize a two-sphere S2, the co-ordinate ψ parametrizes a circle S1 and the solution describes a black ring having a regular horizon of topology S1 × S2 and rotating in the S1 plane. However the horizon geometry is not a simple product of S2 and S1 as the two sphere S2 is deformed there and the deformation grows away from the horizon. The metric reduces to a conventional five dimensional Myers-Perry black hole with rotation in a single plane if, instead of (7), we consider the limit, R → 0, (λ, ν) → 1 and the parameters , a2 = 2R2 (1− ν)2 , (8) are held constant. In this case the co-ordinates (x, φ, ψ) characterises a three- sphere S3 which is a regular horizon of a five dimensional Myers-Perry black hole. The ergosphere and the event horizon of the black ring are located at y = −1/λ and y = −1/ν respectively. At y = −∞ there is a spacelike curvature singularity inside the horizon. Asymptotic infinity is reached as (x, y) → −1. The ADM mass and angular momentum are given as λ(λ− ν)(1 + λ) (1− ν)2 . (10) The curvature squared for the black ring spacetime is computed to be, RµνρσR µνρσ = 6ν2(1 + ν2)2Q(x, y) R4(1 + ν2 + 2νx)6 (x− y)4, (11) where Q(x, y) is a poynomial of degree six in x and y. Hence there is a spacelike curvature singularity at y = −∞ inside the event horizon. In terms of the Myers- Perry co-ordinates (t, r, θ, ψ, φ) the difference (x−y) goes like 1/r2 at large r, i.e. towards spatial infinity, so that the curvature squared goes as RµνρσR µνρσ ∼ as obtained in the case of five dimensional Myers-Perry black hole. The rotating black ring in the limit of large radius R may be described after appropriate coordinate redfinitions as a Schwarzschild black string boosted and periodically identified along the translation invariant direction with a period 2πR [30, 32, 33]. The black string metric is given as ds2 = dw2 − (1− )dt2 + (1− )−1dr2 + r2dΩ2 , (13) where the horizon is at r = r0 and w is the translation invariant direction. The parameter ν = r0/R is seen to correspond to the thickness of the ring or the ratio of the radius of the S2 at the horizon and the ring radius R . The ratio λ/ν then measures the speed of rotation of the ring in the S1 direction and the coordinate ψ = w/R corresponds to a redefined translation invariant direction of the black string which is periodically identified as w = w + 2πR. The speed of rotation is related to the local boost velocity given by 1− (ν/λ) and reduces to 1− (ν2/2) for the black ring space time to exclude any conical singularities. [30] 2.2 Black Ring Geodesics The first order geodesic equations may be derived using the canonical framework [34] from the Lagrangian gµν ẋ µẋν , (14) here µ, ν = 0...4 and the covariant components of the metric tensor are as defined in the previous section and ẋµ = dxµ/dρ with the affine parameter ρ = τ/m [36]for time like geodesics, τ being the proper time andm the mass of the particle. Consequently, for both time like and null geodesics the momenta are pµ = ẋµ. The covariant momenta may be directly obtained from the Lagrangian and are given as pµ = gµν ẋ µ. The norm of the conjugate momenta is then given as, gµνpµpν = −ǫm2 (15) where gµν are the contravariant components of the black ring metric and ǫ = (0, 1) for null and time like geodesics respectively. The black ring spacetime admits three Killing isometries generated by the vector fields ∂t, ∂ψ, and ∂φ corresponding to time translation and the two rotation isometries in the coordinates φ, ψ. These isometries provide three conserved conjugate momenta, pt = −E, pψ = Ψ, pφ = Φ. We consider the geodesics restricted to the plane of rotation of the black ring, outside the ring, i.e, x = −1. It is analogous to an equatorial plane in the spherical case in the sense that it is reflection symmetric and hence geodesics in it with zero initial velocity in the transverse x direction will continue to remain in the plane. The plane x = −1 being a fixed point of the ∂φ isometry, the gφφ component of the metric tensor goes to zero smoothly there. The geodesic equations of motion in the equatorial plane for the t and φ directions are obtained directly from the conserved conjugate momenta. These turn out to be as follows: 1 + λy (1− λ)2 (1 + y)4 (1 + λy)G(y) E − C(1 + y) R(1− λ)(G(y) Ψ (16) CR(1 + y)3 (1− λ)G(y) (1 + y)2(1 + λy) R2(1− λ)G(y) Ψ (17) The form of the y equation for geodesic motion in the equatorial plane is obtained directly from eqn. (15) to be, )2 + gyy gttE2 − 2gtψEΨ+ gψψΨ2 + ǫm2 = 0, (18) where gyy = 1/gyy, g tt = gψψ/D, g ψψ = gtt/D, g tψ = −gtψ/D and D = gttgψψ − Thus, the y equation may be expressed as ẏ2 = − (1 + y) (1− λ)2R2 C2(1 + y)3 + (1− λ)2(1 + νy)(1− y) F (y) 2C(1 + y)2 (1 + λy)(1 + y) − ǫ(1− λ)(1 + νy)(1− y)m2 where ǫ = (0, 1) for null and timelike geodesics respectively. It should be noted that the co-efficient of E2 in the r.h.s of the above equation remains finite and smooth at the ergosphere, y = −1/λ, even though the function F (y) in the denominator vanishes. The eqn (19) should be compared with that appearing in [35] for the null geodesics in the plane of the ring, where a a certain normalization of the metric components have been chosen at asymptotic infinity. The y co-ordinate ranges over the plane of rotation of the ring from the curvature singularity to asymptotic infinity and the above equation is analogous to particle motion in a central potential ẏ2 + Veff(y;E,Ψ) = 0 (20) Towards asymptotic infinity, (x, y) → −1, the effective potential for time like geodesics tends to Veff(y;E,Ψ) → − 2(1− ν) R2(1 + λ) η3(E2 −m2), (21) where η tends to 0 towards asymptotic infinity and is given by η = −(1 + y). Unbound time like geodesics can exist only when E2 −m2 > 0 in which case the effective potential Veff is negative at large distances and approaches zero at asymptotic infinity (x, y = −1). For the case E2 < m2 only bound geodesics exist, in the sense that such geodesics do not reach upto asymptotic infinity. Stable bound orbits are bound orbits which do not end up in the singularity. It is common knowledge that stable bound orbits do not occur in a higher dimensional central potential, even in the case of Newtonian gravity. Thus it is expected that such orbits must be excluded from higher dimensional black hole space times. This was explicitly shown for the equatorial geodesics of a five dimensional Myers- Perry black hole in [36]. This conclusion is expected to also hold for the class of geodesics restricted to the plane of rotation of the ring being considered here. Their existence is indicated by the presence of stable circular orbits. For circular orbits, we have the condition Veff(y = yc) = 0 , ∂Veff (y) = 0 (22) where y = yc is the ‘radius’ of the circular orbit.The condition for stability of the circular orbit is ∂2Veff > 0. (23) −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 Figure 1: Plot of black ring effective potential for ν = 0.46, L = 4.40145 and three different values of E as indicated in the box. Motion is allowed only in the region where Veff < 0. The constants m = R = 1. It is apparent that there are no stable bound orbits. E = 2.0 is close to having an unstable circular orbit, whereas for E = 2.02 there are no inaccessible regions. Since E > 1 all the three curves exhibit unbounded orbits. The case for E < 1 shows an exactly similar behaviour as regards the bound orbits. We get two simulataneous biquadratic equations in E and Ψ from Eq(22) which can be solved in terms of the radius yc for a black ring of specific ν. These values of Ec and Ψc can be then substituted into (23) to obtain a function of yc for a specific black ring [36]. It is difficult to interpret the analytic expressions for Ec,Ψc and that of Eq.(23) in terms of yc. However, numerical plots have been obtained in Fig. 1 for the effective potential Veff(y) against y which clearly shows that stable bound orbits are ruled out both for E2 > m2 and E2 < m2 . 3 Brane World Black Ring In this section we very briefly outline the construction of the Randall-Sundrum braneworld with a single (N-1)-brane in (N+1) dimensions with a single AdS direction transverse to the brane. We then consider the specific case of the five dimensional neutral rotating black ring on a four brane in a (5+1) dimensional Randall-Sundrum braneworld with a single AdS direction transverse to the brane hypersurface. We propose that the appropriate bulk description is provided by a six dimensional rotating black string extension of the five dimensional rotating black ring. The intercept of the bulk solution on the four brane is a five dimen- sional black ring with the usual curvature singularity on the brane hypersurface although an additional bulk singularity also appears at the AdS horizon. We also compute the six dimensional bulk geodesics restricted to the plane of rotation of the black ring. The projection of these bulk geodesics on the four brane reduces to the appropriate class of black ring geodesics on the four brane hypersurface. The y orbits for the bulk solution which reach the AdS horizon are then analyzed using the geodesic equation to elucidate the natuer of the bulk singularity at the AdS horizon. It is seen that the curvature remains finite at the AdS horizon along the unbounded indicating the presence of a mild p-p curvature singularity. 3.1 Black Ring in a RS Brane World The bulk metric for single brane RS brane world in (N +1) dimensions, with one transverse AdS direction to the (N-1) brane is as follows; [19, 21] ds2 = gmndx mdxn = [gµνdx µdxν + dz2]. (24) Here µ, ν = 0 . . . (N − 1) and m,n = 0 . . . (N) and l is the AdS length scale. The transverse coordinate z = 0,∞ are the conformal infinity and the AdS horizon respectively. The actual RS braneworld geometry is obtained by removing the small z region at z = z0 and glueing a mirror copy of the large z geometry at the location of the (N-1) brane which ensures Z2 reflection symmetry. The resulting topology for the double brane RS scenario is essentialy RN × S1 and in the single brane variant considered here the S1 direction is essentialy decompactified with the second regulator brane being at z = ∞. The discontinuity of the extrinsic curvature at the z = z0 surface corresponds to a thin distributional source of stress-energy. From the Israel junctions conditions this may be interpreted as a relativistic (N-1) brane (smooth domain wall) with a corresponding tension [19,21]. The orginal RS model sliced the AdS space-time both at z = 0 and z = l and inserted two (N-1) branes with Z2 reflection symmetry at both hypersurfaces. The Israel junction conditions then required a negative tension for the brane at z = l. The variant considered here may be obtained from the original RS model by allowing the negative tension brane to approach the AdS horizon at z = ∞ . Although we focus here only on the single brane RS model for convenience, our construction may be generalized to the original RS model with double branes in a straightforward manner. The Einstein equations in (N+1) dimensions with a negative cosmological constant continue to be satisfied for any metric gµν which is Ricci flat. The curvature of the modified metric now satisfies RpqrsR pqrs = 2N(N + 1) RµνλκR µνλκ (25) where (p, q) runs over (N +1) dimensions and (µ, ν) over the N dimensions of the brane world volume. The perturbations of the (N+1) dimensional metric around a Ricci flat background are now normalizable modes peaked at the location of the (N-1) brane. Having provided this brief introduction to the single brane RS model in (N+1) dimensions we now specialize to N=5 and consider the bulk description of a five dimensional neutral rotating black ring on the four brane in a six dimensional RS braneworld. To this end we consider a bulk six dimensional black string extension of the five dimensional rotating neutral black ring in the bulk. The black ring being a Ricci flat space-time the bulk black string extension automaticaly satisfies the Einstein equation [21] For a reflection symmetric four brane hypersurface fixed at z = z0 we may introduce the co-ordinate w = z−z0. The bulk metric on either side of the domain wall may now be expressed as ds2 = (z0 + |w|)2 dw2 − F (y) F (x) dt− CR 1 + y F (y) (x− y)2 F (x) −G(y) F (y) dψ2 − dy F (x) where −∞ < w <∞ and the domain wall is located at w = 0. The induced metric on the four brane at z = z0 may be recast into the black ring form by suitably rescaling the coordinates and the parameters. The ADM mass and angular momentum as measured on the brane, scaled by the conformal warp factor, are then given as M , J∗ = J. (27) where M,J are the bulk parameters. The curvature squared for the bulk black string is computed to be; RjklmR jklm = 6(1 + ν2)2ν2Q(x, y) R4(1 + ν2 + 2νx)6 z4(x− y)4 Following Eq(12), towards spatial infinity on the brane the curvature squared behaves as RjklmR jklm ∼ . (29) The curvature invariant diverges at the spacelike singularity on the brane at y = −∞. Additionaly, it is also seen to diverge at the AdS horizon z = ∞ for finite r. As mentioned earlier, such a singularity seems to be a artifact of the linearized approximation. In order to further investigate this issue we need to study the geodesics and their behaviour at the AdS horizon. As mentioned earlier the neutral rotating black ring maybe described in a cer- tain limit as a Schwarzschild black string boosted in the translationaly invariant direction and identified periodicaly. In the braneworld construction that we have developed, this reduces to a six dimensional bulk black two brane boosted along the extended direction on the four brane and identified periodically. In the 5+1 dimensional brane world Eq. (13) generalizes to; ds2 = dz2 + dw2 − (1− r0 )dt2 + (1− r0 )−1dr2 + r2dΩ2 Here u is the translation invariant direction of the black string along the brane hypersurface and z describes the transverse direction. Apart from the conformal factor the coordinate z is a spectator dimension and hence we have a six dimen- sional bulk Schwarzschild black two brane boosted along a translation invariant direction w and periodicaly identified as w ∼ w+2πR. This bulk black two brane in the limit of large boost velocity and a large periodicity R intercepts the four brane in a fast spinning thin five dimensional neutral black ring of large radius R with the usual curvature singularity on the brane. This is obvious as the boost does not involve the transverse z direction and the limit of large radius and high boost velocity are z independent. So in this limit after periodic identification the event horizon has S2×S1×R topology extended in the bulk and periodic in the coordinate w on the four brane. 3.2 The Brane World Geodesics. The geodesic equations for the the bulk spacetime may be obtained as earlier from the Lagrangian L = 1 = gjkẋ j ẋk (31) where gjk are the covariant components of the 5+1 dimensional metric as in eqn. (24) and j, k = 0 . . . 5. Also ẋ = dx/dρ and on time like geodesics the affine parameter ρ = τ/m. Accordingly we have pj = ẋj , pj = gjkẋ k and gjkpjpk = −ǫm2 (32) where ǫ = 0, 1 for null and time like geodesics respectively. The z equation for geodesic motion is obtained from the Lagrangian as . (33) The solution for null geodesics is either z =constant or z = − z1l . (34) For timelike geodesics the solution is z = −z1cosec(ρm/l). (35) Herem is the particle mass for timelike geodesics and we should set z1/m=constant for the null geodesics in this case. The null case z =constant is simply a null geodesic of the five dimensional rotating black ring. We are interested in the other solutions which reach the location of the bulk singularity at the AdS hori- zon z = ∞ for ρ→ 0−. The bulk spacetime has three killing isometries ∂t, ∂ψ, and ∂φ leading to the corresponding conserved momenta pt = −E, pψ = Ψ and pφ = Φ for geodesic motion. Once again we consider only those geodesics in the bulk which, on the 4-brane, are restricted to the plane of rotation of the black ring , i.e in the x = −1 plane. The gφφ component of the 5+1 dimensional metric goes to zero on the plane of rotation so that E and Ψ are the conserved quantities for such geodesics. The geodesic equations for the t and ψ co-ordinates in the plane of rotation of the black ring are given as z2(1− λ) l2(1 + λy) (1− λ)2 (1 + y)4 (1 + λy)G(y) E − z 2C(1 + y)3 l2R(1− λ)(G(y) z2CR(1 + y)3 l2(1− λ)G(y) z2(1 + y)2(1 + λy) l2R2(1− λ)G(y) The y equation of motion for time like and null geodesics in the bulk which reach the AdS horizon is given by + gttE2 − gtψEΨ+ gψψΨ2 = 0. (37) Here the contravariant components of the metric in the equation are essentialy the black ring metric without the bulk conformal factor. The bulk timelike or null geodesics when projected onto the brane reduce to the time like black ring geodesics restricted to the plane of rotation of the ring. The projection to the four brane hypersurface is effected by scaling out the z dependence of the geodesics. First, new parameters γ = z2/m2ρ for null geodesics and γ = (−z2 /lm)cot(mρ/l) for time like geodesics are introduced. We define the rescaled co-ordinates and parameters x = lx̃/z1, y = lỹ/z1, t = lt̃/z1 , R = l2R̃/z2 , λ = z1λ̃/l, ν = z1ν̃/l. The integrals of motion are also rescaled as E = lẼ/z1,Ψ = l 3Ψ̃/z3 The geodesic equation for the y coordinate in the rescaled quantities may then be written as, + Veff(ỹ; Ẽ, Ψ̃) = 0 (38) where Veff is the same effective potential as given in eqn. (20). This is pre- cisely the equation in y for a time like geodesic in the plane of rotation of a five dimensional rotating black ring with an ADM mass M̃ and angular momentum M , J̃ = J (39) and thus existing on the four brane hypersurface located at z = z0 = l 2/z1. The parameter γ now serves as the proper time along the time like geodesic. In order to ascertain the nature of the singularity at the AdS horizon (z = ∞) we need to study the behaviour of the bulk geodesics near the AdS horizon, i.e as ρ → 0−. This is equivalent to γ → ∞, so we need to investigate the late time behaviour of the five dimensional time like geodesics on the four-brane. The geodesics ending into the black ring singularity will take a finite amount of proper time to do so. For infinite proper time the geodesics can either reach up to the asymptotic infinity on the four brane(x̃, ỹ = −z1/l) or remain at a finite distance from the black ring horizon. The geodesics that reach asymptotic infinity on the brane have late time behaviour r̃ ∼ γ Ẽ2 −m2, (40) where r̃2 = − 1 z1/l + ỹ . (41) The co-ordinate r̃ is the radial direction on the brane and it is the same as the radial Myers-Perry coordinate for the black ring in the asymptotic limit modulo certain constants in the plane of rotation of the black ring. It is expected that stable bound orbits do not exist in the case of the five dimensional black rings. So, only unbound geodesics may reach the AdS horizon at z → ∞. Along such orbits the curvature squared, Eq.(28), remains finite, thus indicating the presence of a p-p curvature singularity at the AdS horizon. To explicitly illustrate this, it is necessary to obtain the curvature components in an orthonormal frame parallely propagated on a timelike geodesic to the AdS horizon. Although its simple to demonstrate this in the case of the Schwarzschild black hole in a braneworld for more complicated metrics and higher dimensions the explicit determination of this frame involves several coupled PDE and renders this analysis computationaly intractable. Although we have to emphasize that such frames exist the choice is highly non unique and a specific suitable such frame is complicated to establish even for four dimensional Kerr black holes in a braneworld [11]. 4 Summary and Discussions. To summarize we have described a five dimensional neutral rotating black ring on a four brane in a six dimensional Randall-Sundrum braneworld. As mentioned earlier this has been motivated by the fact that for consistency the usual grav- itational configurations on the brane, in particular black holes must arise from some higher dimensional bulk solutions. The five dimensional black ring being the first asymptoticaly flat solution with a non spherical horizon topology is an interesting configuration to study from a bulk brane world perspective. Espe- cialy as it explicitly violates the no hair and the uniqueness theorem. Due to the absence of suitable exact bulk metrics in D > 4 a linearized framework around a fixed solution is necessary for the analysis of the black ring in a brane world. In this context the bulk black string approach of Chamblin et. al. [7] is especialy relevant to elucidate the physical issues although the pathology of a singularity at the AdS horizon persists. However, absence of such a singularity in lower dimen- sional brane worlds where exact metrics are available shows the bulk singularity to be a linearization artifact. To this end we have considered a bulk six dimensional black string extension of a five dimensional rotating neutral black ring in a 5+1 dimensional Randall- Sundrum braneworld. This choice is consistent with the usual reflection symmet- ric junction conditions on the four brane in such warped compactification models. The bulk black string rotates in the four brane world volume and the induced five dimensional metric on the four brane describes a neutral rotating black ring. This reproduces the usual spacelike curvature singularity of the black ring on the four brane hypersurface. Additionaly a singularity also appears in the bulk at the AdS horizon. After elucidating the geodesics of the rotating black ring restricted to the plane of rotation we have obtained both the timelike and the null geodesics for the black string in the six dimensional bulk. We have further shown that the restricted bulk geodesics projected on the four brane by scaling away the AdS direction exactly match the corresponding class of five dimensional black ring geodesics. The effective potential has been analysed numericaly and we have shown that stable bound geodesics do n ot exist as is expected in D > 4. It has been further shown that the curvature invariant remains finite along un- bounded geodesics which reach the AdS horizon. This clearly indicates that the bulk curvature singularity at the AdS horizon is possibly a p-p curvature sin- gularity although an explicit illustration using parallely propogated orthonormal frames is computationaly intractable. It is mentioned earlier that a fast spinning thin neutral rotating black ring may be described as a black string boosted along the translationaly invariant direction and identified periodically in some limit. We have shown that from the bulk perspective this description involves naturaly a black two brane in the six dimensional bulk orthogonal to the four brane hypersurface. To obtain the black ring on the four brane the black two brane must be boosted along a translationaly invaraint direction longitudinal to the four brane and identified periodicaly along this direction. Due to the direct equivalence of the two metrics it is obvious that the usual matching of the geodesics on the bulk and the brane will continue to hold in this limit . In the black ring limit the event horzion in the bulk would constitute a base S2×S1 on the five dimensional brane hypersurface and a trivial R fibration into the bulk. The issue of stability of the bulk black string configuration is contentious and remains unresolved for axialy symmetric stationary solutions. For AdS solutions one conclusion is that the prefered phase will be an accumulation of a sequence of lower dimensional black holes with the horzion pinched off at some scale. However for the usual Schwarzschild black string this conclusion has been contested where it has been shown that a more likely scenario is an evolution to a translationaly non invariant stable solution [10]. But this although plausible has not yet been generalized explicitly to axialy symmetric solutions. It has been argued that the bulk solution should pinch off due to the instabilities before reaching the singu- larity at the AdS horzion [7, 9]. However this issue is far from being completely settled. It is possible that the pathology at the AdS horizon is a linearization artifact especialy given that lower dimensional exact bulk solutions are regular everywhere. There are several open issues for future studies. Charged rotating black ring solutions have been obtained in the context of string theory through the O(d, d) transformations. These have been further generalized to rotating black rings with dipole charges. In the brane world scenario, bulk configurations which reduce to charged black holes have been investigated. It could be shown in this case that the black hole on the brane developed a tidal charge due to the extra dimensions apart from the usual conserved gauge charge [15]. It would be an interesting exercise to study the brane world formulation of the dipole black rings in this context. Very recently it has been shown that in higher diemnsions it is possible to have stable configurations involving combinations of black rings and black holes. These have been christened black saturn and are remarkably novel solutions of higher dimensional general relativity [37]. Naturally it would be interesting to investigate these configurations from a brane world perspective. It is generaly expected that more such solutions would be possible in the context of higher dimensions. 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0704.0999

Generic character sheaves on disconnected groups and character values

arXiv:0704.0999v1 [math.RT] 7 Apr 2007 GENERIC CHARACTER SHEAVES ON DISCONNECTED GROUPS AND CHARACTER VALUES G. Lusztig Introduction The theory of character sheaves [L3] on a reductive group G over an alge- braically closed field and the theory of irreducible characters of G over a finite field are two parallel theories; the first one is geometric (involving intersection cohomology complexes on G), the second one involves functions on the group of rational points of G. In the case where G is connected, a bridge between the two theories was constructed in [L1] and strengthened in [L2], [S]. In this paper we begin the construction of the analogous bridge in the general case, extending the method of [L1]. Here we restrict ourselves to character sheaves which are ”generic” (in particular their support is a full connected component of G) and show how such character sheaves are related to characters of representations (see Theorem 1.2). Contents 1. Statement of the Theorem. 2. Constructing representations of GF . 3. Proof of Theorem 1.2. 1. Statement of the Theorem 1.1. Let k be an algebraic closure of a finite field Fq. Let G be a reductive algebraic group over k with identity component G0 such that G/G0 is cyclic, generated by a fixed connected component D. We assume that G has a fixed Fq-rational structure with Frobenius map F : G −→ G such that F (D) = D. Let l be a prime number invertible in k; let Q̄l be an algebraic closure of the l-adic numbers. All group representations are assumed to be finite dimensional over Q̄l. We say ”local system” instead of ”Q̄l-local system”. Let B be the variety of Borel subgroups of G0. Now F : G −→ G induces a morphism B −→ B denoted again by F . We fix B∗ ∈ B and a maximal torus T of B∗ such that F (B∗) = B∗, F (T ) = T . Let U∗ be the unipotent radical of B∗. Let Supported in part by the National Science Foundation. Typeset by AMS-TEX http://arxiv.org/abs/0704.0999v1 2 G. LUSZTIG NB∗ (resp. NT ) be the normalizer of B∗ (resp. T ) in G. Let T̃ = NT ∩NB∗, a closed F -stable subgroup of G with identity component T . Let T̃D = T̃ ∩D. Let N = NT ∩ G0. Let W = N /T be the Weyl group. Let D : T −→ T , D : W −→ W be the automorphisms induced by Ad(d) : N −→ N where d is any element of T̃D. Now F : N −→ N induces an automorphism ofW denoted again by F . For w ∈ W let [w] be the inverse image of w under the obvious map N −→ W and let w be the automorphism Ad(x) : T −→ T for any x ∈ [w]. For w ∈ W let Ow be the G 0-orbit in B × B (G0 acting by simultaneous conjugation on both factors) that contains (B∗, xB∗x−1) for some/any x ∈ [w]. Define the ”length function” l : W −→ N by l(w) = dimOw − dimB. For any y ∈ G 0 we define k(y) ∈ N by y ∈ U∗k(y)U∗. For y ∈ G0, τ ∈ T̃ we have k(τyτ−1) = τk(y)τ−1 and F (k(y)) = k(F (y)). For x ∈ G0 we define Fx : G −→ G by Fx(g) = xF (g)x this is the Frobenius map for an Fq-rational structure on G. (Indeed if y ∈ G such that x = y−1F (y), then Ad(y) : G −→ G carries Fx to F .) If w ∈W satisfies D(w) = w and x ∈ [w] then T, T̃ are Fx-stable; thus Fx is the Frobenius map for an Fq-rational structure on T̃ whose group of rational points is T̃ Fx . Since T̃FxD is the set of rational points of T̃D (a homogeneous T -space under left translation) for the rational structure defined by Fx : T̃D −→ T̃D, we have T̃ D 6= ∅. Let Z∅ = {(B0, g) ∈ B ×D; gB0g −1 = B0}. Let d ∈ T̃D. We set Ż∅,d = {(h0U ∗, g) ∈ (G0/U∗)×D; h−10 gh0d −1 ∈ B∗}. Define a∅ : Ż∅,d −→ Z∅ by (h0U ∗, g) 7→ (h0B ∗h−10 , g). Now a∅ is a principal T - bundle where T acts (freely) on Ż∅,d by t0 : (h0U ∗, g) 7→ (h0t 0 , g). Define p∅ : Z∅ −→ D by (B0, g) 7→ g. We define b∅ : Ż∅,d −→ T by (h0U ∗, g) 7→ k(h−10 gh0d Note that b∅ commutes with the T -actions where T acts on T by (a) t0 : t 7→ t0tD(t Let L be a local system of rank 1 on T such that (i) L⊗n ∼= Q̄l for some n ≥ 1 invertible in k; (ii) D∗L ∼= L; From (i),(ii) we see (using [L3, 28.2(a)]) that L is equivariant for the T -action (a) on T . Hence b∗ L is a T -equivariant local system on Ż∅,d. Since a∅ is a principal T -bundle there is a well defined local system L̃∅ on Z∅ such that a L̃∅ = b Note that the isomorphism class of L̃∅ is independent of the choice of d. Assume in addition that: (iii) {w ∈W ;D(w) = w,w∗L ∼= L} = {1}. We show: (b) p∅!L̃∅ is an irreducible intersection cohomology complex on D. We identify Z∅ with the variety X = {(g, xB ∗) ∈ G ×G0/B∗; x−1gx ∈ NB∗} (as in [L3, I, 5.4] with P = B∗, L = T, S = T̃D) by (g, xB ∗) ↔ (xB∗x−1, g). Then L̃∅ becomes the local system Ē on X defined as in [L3, I, 5.6] in terms of the local system E = j∗L on T̃D where j : T̃D −→ T is y 7→ d −1y. (Note that E is equivariant GENERIC CHARACTER SHEAVES ON DISCONNECTED GROUPS AND CHARACTER VALUES3 for the conjugation action of T on T̃D.) In our case we have Ē = IC(X, Ē) since X is smooth. Hence from [L3, I, 5.7] we see that p∅!Ē is an intersection cohomology complex on D corresponding to a semisimple local system on an open dense subset of D which, by the results in [L3, II, 7.10], is irreducible if and only if the following condition is satisfied: if w ∈W,x ∈ [w] satisfy Ad(x)(T̃D) = T̃D and Ad(x) ∗E ∼= E , then w = 1. This is clearly equivalent to condition (iii). This proves (b). From (b) and the definitions we see that p∅!L̃∅[dimD] is a character sheaf on D in the sense of [L3, VI]. A character sheaf on D of this form is said to be generic. We can state the following result. Theorem 1.2. Let A be a generic character sheaf on D such that F ∗A ∼= A where F : D −→ D is the restriction of F : G −→ G. Let ψ : F ∗A −→ A be an isomorphism. Define χψ : D F −→ Q̄l by g 7→ i∈Z(−1) itr(ψ,Hig(A)) where H the i-th cohomology sheaf and Hig is its stalk at g. There exists a G F -module V and a scalar λ ∈ Q̄∗l such that χψ(g) = λtr(g, V ) for all g ∈ D The proof is given in §3. We now make some preliminary observations. In the setup of 1.1 we have A = p∅!L̃∅[dimD] where L satisfies 1.1(i),(ii),(iii) and F ∗(p∅!L̃∅) ∼= p∅!L̃∅. Hence we have p∅!F̃ ∗L∅ ∼= p∅!L̃∅. By a computation in [L3, IV, 21.18] we deduce that there exists w′ ∈ W such that D(w′) = w′, w′∗F ∗L ∼= L. Setting w = F (w′) we see that (a) D(w) = w, F ∗w∗L ∼= L. 1.3. Let w = (w1, w2, . . . , wr) be a sequence in W . Let lw = l(w1)+ l(w2)+ · · ·+ l(wr). Let Zw = {(B0, B1, . . . , Br, g) ∈ B r+1×D; gB0g −1 = Br, (Bi−1, Bi) ∈ Owi(i ∈ [1, r])}. This agrees with the definition in 1.1 when r = 0, that is w = ∅. Let d ∈ T̃D. We define Żw,d as in 1.1 when r = 0 and by Żw,d = {(h0U ∗, h1B ∗, . . . , hr−1B ∗, hrU ∗, g) ∈ (G0/U∗)× (G0/B∗)× . . .× (G0/B∗)× (G0/U∗)×D; k(h−1i−1hi) ∈ [wi](i ∈ [1, r]), h r gh0d −1 ∈ U∗}; when r ≥ 1. Define aw : Żw,d −→ Zw as in 1.1 when r = 0 and by ∗, h1B ∗, . . . , hr−1B ∗, hrU ∗, g) 7→ ∗h−10 , h1B ∗h−11 , . . . , hr−1B ∗hr−1, hrB ∗h−1r , g), when r ≥ 1. Note that aw is a principal T -bundle where T acts (freely) on Żw,d as in 1.1 when r = 0 and by t0 : (h0U ∗, h1B ∗, . . . , hr−1B ∗, hrU ∗, g) 7→ ∗, h1B ∗, . . . , hr−1B ∗, hrdt −1U∗, g) 4 G. LUSZTIG when r ≥ 1. Define pw : Zw −→ D by (B0, B1, . . . , Br, g) 7→ g. In the remainder of this subsection we assume that w1w2 . . . wr = 1; this holds automatically when r = 0. We define bw : Żw,d −→ T as in 1.1 when r = 0 and by ∗, h1B ∗, . . . , hr−1B ∗, hrU ∗, g) 7→ k(h−10 h1)k(h 1 h2) . . . k(h r−1hr) when r ≥ 1. Note that bw commutes with the T -actions where T acts on T as in 1.1(a). Let L be a local system of rank 1 on T such that 1.1(i),(ii) hold. As in 1.1, L is equivariant for the T -action 1.1(a) on T . Hence b∗ L is a T -equivariant local system on Żw,d. Since aw is a principal T -bundle there is a well defined local system L̃w on Zw such that a L̃w = b Lemma 1.4. Assume that w1w2 . . . wr = 1 and that L (as in 1.3) satisfies (i) α̌∗L 6∼= Q̄l for any coroot α̌ : k ∗ −→ T . Then pw!L̃w[lw](lw/2) ∼= p∅!L̃∅. (Note that lw is even.) Assume first that for some i ∈ [1, r] we have wi = w i where w i in W satisfy l(w′iw i ) = l(w i) + l(w i ). Let ′ = (w1, w2, . . . , wi−1, w i , wi+1, . . . , wn). The map (B0, B1, . . . , Br+1, g) 7→ (B0, B1, Bi−1, Bi+1, . . . , Br+1, g) defines an iso- morphism Zw′ −→ Zw compatible with the maps pw′ , pw and with the local systems L̃w′ , L̃w. Since lw′ = lw we have (a) pw!L̃w[lw](lw/2) ∼= pw′!L̃w′ [lw′ ](lw′/2). Using (a) repeatedly we can assume that l(wi) = 1 for all i ∈ [1, r]. We will prove the result in this case by induction on r. Note that r is even. When r = 0 the result is obvious. We now assume that r ≥ 2. Since w1w2 . . . wr = 1, we can find j ∈ [1, r − 1] such that l(w1w2 . . . wj) = j, l(w1w2 . . . wj+1) = j − 1. We can find a sequence w′ = (w′1, w 2, . . . , w r) in W such that l(w i) = 1 for all i ∈ [1, r], 2 . . .w j = w1w2 . . . wj , w j = w j+1, w i = wi for i ∈ [j + 1, r]. Let u = (w1w2 . . . wj , wj+1, . . . , wr) = (w 2 . . . w j , w j+1, . . . , w Using (a) repeatedly we see that pw!L̃w[lw](lw/2) ∼= pu!L̃u[lu](lu/2) ∼= pw′!L̃w′ [lw′ ](lw′/2). Replacing w by w′ we see that we may assume in addition that wj = wj+1 for some j ∈ [1, r − 1]. We have a partition Zw = Z ∪ Z ′′ where Z ′ (resp. ) is defined by the condition Bj−1 = Bj+1 (resp. Bj−1 6= Bj+1). Let w (w1, w, . . . , wj−1, wj+2, . . . , wr), w ′′ = (w1, w, . . . , wj−1, wj+1, . . . , wr). Define c : −→ Zw′ by (B0, B1, . . . , Br, g) 7→ (B0, B1, . . . , Bj−1, Bj+2, . . . , Br, g). GENERIC CHARACTER SHEAVES ON DISCONNECTED GROUPS AND CHARACTER VALUES5 This is an affine line bundle and L̃w|Z′ = c∗L̃w′ . Let p be the restriction of pw to Z ′ . We have p′ = pw′c. Since the induction hypothesis applies to w ′ we have w!(L̃w|Z′w)[lw](lw/2) = pw′!c!c ∗L̃w′ [lw](lw/2) = pw′!L̃w′ [−2](−1)[lw](lw/2) = pw′!L̃w′ [lw′ ](lw′/2) = p∅!L̃∅.(b) Define e : Z ′′ −→ Zw′′ by (B0, B1, . . . , Br, g) 7→ (B0, B1, . . . , Bj−1, Bj+1, . . . , Br, g). Let p′′ be the restriction of pw to Z . We have p′′ = pw′′e. We show that w!(L̃w|Z′′w) = 0. It is enough to show that (c) pw′′!e!(L̃w|Z′′ ) = 0. Hence it is enough to show that e!(L̃w|Z′′ ) = 0. It is also enough to show that, if E is a fibre of e, then Hic(E, L̃w|E) = 0 for any i. As in the proof of [L3, VI, 28.10] we may identify E = k∗ in such a way that L̃w|E becomes α̌ ∗(L) for some coroot α̌ : k∗ −→ T . We then use that Hic(k ∗, α̌∗L) = 0 which follows from α̌∗L 6∼= Q̄l. Using (c) and the exact triangle (pw′′!e!(L̃w|Z′′ ), pw!L̃w, p w!(L̃w|Z′w)) we see that pw!L̃w[lw](lw/2) = p w!(L̃w|Z′w)[lw])(lw/2) = p∅!L̃∅ (the last equality follows from (b)). The lemma is proved. Lemma 1.5. Assume that L (as in 1.3) satisfies 1.1(iii). Then L satisfies 1.4(i). Let RL be the set of roots α : T −→ k ∗ such that the corresponding coroot α̌ satisfies α̌∗L ∼= Q̄l. Let WL be the subgroup of W generated by the reflections with respect to the various α ∈ RL. Since D ∗L ∼= L we have D(WL) = WL. Assume that 1.4(i) does not hold. Then RL 6= ∅ and WL 6= {1}. By [DL, 5.17] the fixed point set of D : WL −→ WL is 6= {1}. Let w ∈ WL − {1} be such that D(d)w = w. Since w ∈WL we have w ∗L ∼= L (see [L3, VI, 28.3(b)]). Thus 1.1(iii) does not hold. The lemma is proved. 2. Constructing representations of GF 2.1. In this section we construct some representations of GF using the method of [DL]. See [M],[DM] for other results in this direction. Let L be a local system of rank 1 on T such that 1.1(i) holds. For any t ∈ T let Lt be the stalk of L at t. Assume that we are given w ∈W and x ∈ [w] such that 6 G. LUSZTIG (i) F ∗xL ∼= L; (Fx : T −→ T as in 1.1). Let φ : F xL −→ L be the unique isomorphism of local systems on T which induces the identity map on L1. For t ∈ T , φ induces an isomorphism LFx(t) −→ Lt. When t ∈ T Fx this is an automorphism of the 1-dimensional vector space Lt given by multiplication by θ(t) ∈ Q̄ l . It is well known that t 7→ θ(t) is a group homomorphism TFx −→ Q̄∗l . Following [DL] we define Y = {hU∗ ∈ G0/U∗; h−1F (h) ∈ U∗xU∗}. For (g, t) ∈ G0F × TFx we define eg,t : Y −→ Y by hU ∗ 7→ ght−1U∗. Note that (g, t) 7→ eg,t is an action of G 0F × TFx on Y . Hence G0F × TFx acts on Hic(Y ) := H c(Y, Q̄l) by (g, τ) 7→ e g−1,τ−1 . We set Hic(Y )θ = {ξ ∈ H c(Y ); e 1,t−1ξ = θ(t) −1ξ for all t ∈ TFx}; this is a G0F × TFx-stable subspace of Hic(Y ). For g ∈ G0F we define ǫg : H c(Y )θ −→ H c(Y )θ by ǫg(ξ) = e g−1,1 . This makes Hic(Y )θ into a G 0F -module. We can find an integer r ≥ 1 such that F r(x) = x, xF (x) . . . F r−1(x) = 1. Indeed we first find an integer r1 ≥ 1 such that F r1(x) = x and then we find an integer r2 ≥ 1 such that (xF (x) . . .F r1−1(x))r2 = 1. Then r = r1r2 has the required properties. Then hU∗ 7→ F r(h)U∗ is a well defined map Y −→ Y denoted again by F r. Also, F r = F rx : G −→ G. (We have F rx (g) = (xF (x) . . . F r−1(x))F r(g)(xF (x) . . .F r−1(x))−1 = F r(g).) Hence F r acts trivially on TFx . We see that F r : Y −→ Y commutes with eg,t : Y −→ Y for any (g, t) ∈ G0F × TFx . Hence (F r)∗ : Hic(Y ) −→ H c(Y ) leaves stable the subspace Hic(Y )θ. Note that: for any i, all eigenvalues of (F r)∗ : Hic(Y ) −→ H c(Y ) are of the form root of 1 times qnr/2 where n ∈ Z. (See [L1, 6.1(e)] and the references there.) Replacing r by an integer multiple we may therefore assume that r satisfies in addition the following condition: (a) for any i, all eigenvalues of (F r)∗ : Hic(Y ) −→ H c(Y ) are of the form q where n ∈ Z. 2.2. We preserve the setup of 2.1 and assume in addition that L satisfies 1.4(i). Let i0 = 2dimU ∗ − l(w). Note that (a) Hic(Y )θ = 0 for i 6= i0; if i = i0 then all eigenvalues of (F r)∗ : Hic(Y )θ −→ Hic(Y )θ are of the form q ir/2. For the first statement in (a) see [DL, 9.9] and the remarks in the proof of [L1, 8.15]. The second statement in (a) is deduced from 2.1(a) as in the proof of [L1, 6.6(c)]. GENERIC CHARACTER SHEAVES ON DISCONNECTED GROUPS AND CHARACTER VALUES7 2.3. We preserve the setup of 2.1 and assume in addition that L satisfies 1.1(ii) and that w ∈W satisfies D(w) = w. From the definitions we see that D : T −→ T commutes with Fx : T −→ T hence D restricts to an automorphism of T Fx and (a) θ(D(t)) = θ(t) for any t ∈ TFx . We show: (b) there exists a homomorphism θ̃ : T̃Fx −→ Q̄∗l such that θ̃|TFx = θ. Let d ∈ T̃FxD . Let n = |G/G 0| = |T̃Fx/TFx |. Then t0 := d n ∈ TFx . Let c ∈ Q̄∗l be such that cn = θ(t0). For any t ∈ T Fx and j ∈ Z we set θ̃(djt) = cjθ(t). This is well defined: if djt = dj t′ with j, j′ ∈ Z, t, t′ ∈ TFx then j′ = j + nj0, j0 ∈ Z and t ′ = t 0 t so that θ(t ′) = cnj0θ(t) and cjθ(t) = cj θ(t′). We show that if j, j′ ∈ Z, t, t′ ∈ TFx then θ̃(djtdj t′) = θ̃(djt)θ̃(dj t′) that is cj+j θ(D−j (t)t′) = cjθ(t)cj θ(t′); this follows from (a). This proves (b). Let Γ = {(g, τ) ∈ GF×T̃Fx ; gτ−1 ∈ G0}, a subgroup of GF×T̃Fx . For (g, τ) ∈ Γ we define eg,τ : Y −→ Y by hU ∗ 7→ ghτ−1U∗. To see that this is well defined we assume that h ∈ G0 satisfies h−1F (h) ∈ U∗xU∗ and (g, τ) ∈ Γ; we compute (ghτ−1)−1F (ghτ−1) = τh−1g−1gF (h)F (τ−1) = τh−1F (h)F (τ−1) ∈ τU∗xU∗F (τ−1) = U∗τxF (τ−1)U∗ = U∗xU∗, since τxF (τ−1) = x (that is Fx(τ) = τ). Note that (g, τ) 7→ eg,τ is an action of Γ on Y (extending the action of G0F × TFx). Hence Γ acts on Hic(Y ) by (g, τ) 7→ e∗ g−1,τ−1 . Note that Hic(Y )θ is a Γ-stable subspace of H c(Y ). This follows from the identity eg−1,τ−1e1,t−1 = e1,τ−1t−1τeg−1,τ−1 for g ∈ GF , τ ∈ T̃Fx , t ∈ TFx together with the identity θ(t) = θ(τ−1tτ) which is a consequence of (a). For g ∈ GF we define ǫg : H c(Y )θ −→ H c(Y )θ by ǫg(ξ) = θ̃(τ)e g−1,τ−1ξ for any ξ ∈ Hic(Y )θ and any τ ∈ T̃ Fx such that gτ−1 ∈ G0. Assume that τ ′ ∈ T̃Fx is another element such that gτ ′−1 ∈ G0. Then τ ′ = τt with t ∈ TFx and θ̃(τ ′)e∗g−1,τ ′−1ξ = θ̃(τ)θ(t)e g−1,τ−1e 1,t−1ξ = θ̃(τ)e g−1,τ−1ξ so that ǫg is well defined. For g, g ′ in GF we choose τ, τ ′ in T̃Fx such that gτ−1 ∈ G0, g′τ ′−1 ∈ G0; we have ǫgǫg′ξ = θ̃(τ ′)θ̃(τ)e∗g−1,τ−1e g′−1,τ ′−1ξ = θ̃(ττ ′)e∗(gg′)−1,(ττ ′)−1ξ = ǫgg′ξ. We see that 8 G. LUSZTIG g 7→ ǫg defines a G F -module structure on Hic(Y )θ extending the G 0F -module structure in 2.1. (Note that this extension depends on the choice of θ̃.) We show: (c) If (g, τ) ∈ Γ then F reg,τ : Y −→ Y is the Frobenius map of an Fq-rational structure on Y . Since eg,t is a part of a Γ-action, it has finite order. Since F r = F rx : G −→ G (see 2.1), we see that F r : Y −→ Y commutes with eg,τ : Y −→ Y . Hence (c) holds. 2.4. We preserve the setup of 2.3 and assume in addition that L satisfies 1.3(i). Let i0 = 2dimU ∗ − l(w). Using 2.2(a), 2.3(c) and Grothendieck’s trace formula we see that for (g, d) ∈ Γ we have (−1)l(w)θ̃(d)qi0r/2tr(ǫg, H c (Y )θ) = θ̃(d) (−1)itr((F r)∗ǫg, H c(Y )θ) = (−1)itr((F r)∗e∗g−1,d−1 , H c(Y )θ) (−1)i|TFx |−1 t∈TFx tr((F r)∗e∗g−1,d−1e 1,t−1 , H c(Y ))θ(t) = |TFx |−1 t∈TFx (−1)itr((F r)∗e∗g−1,(dt)−1 , H c(Y ))θ(t) = |TFx |−1 t∈TFx g−1,(dt)−1 |θ(t) = |TFx |−1 t∈TFx |{hU∗ ∈ (G0/U∗); h−1F (h) ∈ U∗xU∗, h−1g−1F r(h)dt ∈ U∗}|θ(t). 3. Proof of Theorem 1.2 3.1. Let A, ψ, χψ be as in 1.2. Let L, w be as in the end of 1.2. Let x ∈ [w]. From 1.2(a) we see that 2.1(i) holds. Let r ≥ 1 be as in 2.1. Let w = (w, F (w), . . . , F r−1(w)). By the choice of r we have wF (w) . . . F r−1(w) = 1. Define a morphism F̃ : Zw −→ Zw by F̃ (B0, B1, . . . , Br, g) = (F (g −1Br−1g), F (B0), F (B1), . . . , F (Br−1), F (g)). We show: (a) Let g ∈ DF and let F̃g : p (g) −→ p−1 (g) be the restriction of F̃ : Zw −→ Zw. Then F̃g is the Frobenius map of an Fq-rational structure on p It is enough to note that the map Br+1 −→ Br+1 given by (B0, B1, . . . , Br) 7→ (F (g −1Br−1g), F (B0), F (B1), . . . , F (Br−1)) GENERIC CHARACTER SHEAVES ON DISCONNECTED GROUPS AND CHARACTER VALUES9 is the composition of the map F ′ : (B0, B1, . . . , Br) 7→ (F (B0), F (B1), . . . , F (Br)) (the Frobenius map of an Fq-rational structure on B r+1) with the automorphism (B0, B1, . . . , Br) 7→ (g −1Br−1g, B0, B1, . . . , Br−1) of Br+1 which commutes with F ′ and has finite order (since g has finite order in Let d ∈ T̃FxD . Define a morphism F̃ ′ : Żw,d −→ Żw,d by F̃ ′(h0U ∗, h1B ∗, . . . , hr−1B ∗, hrU ∗, g) = (h′0U ∗, h′1B ∗, . . . , h′r−1B ∗, h′rU ∗, F (g)) where h′0 = F (g −1hr−1k(h r−1hr))x −1d, h′r = F (hr−1k(h r−1hr)x h′i = F (hi−1) for i ∈ [1, r − 1]. This is well defined since (F (hr−1k(h r−1hr)x −1)−1F (g)F (g−1hr−1k(h r−1hr))x −1)dd−1 = 1. We show that the T -action on Żw,d (see 1.3) satisfies F̃ ′(t0x̃) = Fx(t0)F̃ ′(x̃) for t0 ∈ T, x̃ ∈ Żw,d. Let (hi) be as above. We must show: F (g−1hr−1k(h r−1hrdt −1))x−1d = F (g−1hr−1k(h r−1hr))x −1dxF (t−10 )x F (hr−1k(h r−1hrdt −1)x−1 = F (hr−1k(h r−1hr)x −1dxF (t0) −1x−1d−1, which follow from F (d) = x−1dx. Note that (b) awF̃ ′ = F̃ aw : Żw,d −→ Zw. We show: (c) |a−1 (y)F̃ | = |TFx | for any y ∈ ZF̃ Since a−1 (y) is a homogeneous T -space this follows from Lang’s theorem applied to (T, Fx). We have (d) pwF̃ = Fpw : Zw −→ D. 3.2. We show: (a) bwF̃ ′ = Fxbw : Żw,d −→ T . Let (h0, h1, . . . , hr, g) ∈ (G 0)r+1 ×D be such that ∗, h1B ∗, . . . , hr−1B ∗, hrU ∗, g) ∈ Żw,d. 10 G. LUSZTIG Let (h′1, h 2, . . . , h r) be as in 3.1. We set µ = k(h−10 h1)k(h 1 h2) . . . k(h r−1hr) ∈ T, µ′ = k(h−10 h1)k(h 1 h2) . . . k(h r−2hr−1) ∈ B ∗F r−1(x)−1B∗ µ̃ = k(h′0 −1h′1)k(h −1h′2) . . . k(h −1h′r) ∈ T so that µ = µ′k(h−1r−1hr) and µ̃ = k(d−1xF (k(h−1r−1hr) −1h−1r−1gh0)) × k(F (h−10 h1)) . . . k(F (h r−3hr−2))k(F (h r−2hr−1k(h r−1hr))x = d−1xF (k(h−1r−1hr) −1)F (d)k(F (d−1)F (h−1r−1gh0))F (µ ′)F (k(h−1r−1hr))x = d−1xF (d)F (µ)x−1 = xF (µ)x−1 = Fx(µ), as required. 3.3. Let φ : F ∗xL −→ L, θ : TFx −→ Q̄∗l be as in 2.1. We shall denote by ? the various isomorphisms induced by φ such as: (a) F̃ ′∗b∗ L = b∗ F ∗xL −→ b∗ L (see 3.2(a)), (b) F̃ ′∗a∗ −→ a∗ L̃w (coming from (a)), (c) a∗ F̃ ∗L̃w −→ a∗ L̃w (see (b) and 3.1(b)), (d) F̃ ∗L̃w −→ L̃w (coming from (c)), (e) pw!F̃ −→ pw!L̃w (coming from (d)), (f) F ∗pw!L̃w −→ pw!L̃w (coming from (e) and 3.1(d)). (g) F ∗(pw!L̃w[lw]) −→ pw!L̃w[lw] (coming from (f)). 3.4. For any g ∈ DF we compute (−1)itr(?,Hig(pw!L̃w)) = (−1)itr(?, Hic(p (g), L̃w)) (g);F̃ (y)=y tr(?, (L̃w)y) where Hi is the i-th cohomology sheaf. (The last two sums are equal by the Grothendieck trace formula applied in the context of 3.1(a).) Using 3.1(c) we see that the last sum equals |TFx |−1 (g))F̃ tr(?, (a∗ L̃w)ỹ) = |T Fx |−1 (g))F̃ tr(?, (b∗ Lw)ỹ) = |TFx |−1 (g))F̃ tr(?, (Lw)bw(ỹ)). GENERIC CHARACTER SHEAVES ON DISCONNECTED GROUPS AND CHARACTER VALUES11 Now a−1 (g))F̃ can be identified with the set of all ∗, h1B ∗, . . . , hr−1B ∗, hrU ∗) ∈ (G0/U∗)×(G0/B∗)× . . .×(G0/B∗)×(G0/U∗) such that (a) k(h−1i−1hi) ∈ F i−1(x)T for i ∈ [1, r], (b) h−1r gh0d −1 ∈ U∗, (c) h0U ∗ = F (g−1hr−1k(h r−1hr))x −1dU∗, (d) hiB ∗ = F (hi−1)B ∗ for i ∈ [1, r − 1]. (We then have automatically hrU ∗ = F (hr−1k(h r−1hr)x −1U∗.) If h0U ∗ is given, then (d) determines successively h2B ∗, . . . hr−1B ∗ in a unique way and (b) deter- mines hrU ∗ in a unique way. We see that the equations (a)-(d) are equivalent to the following equations for h0U h−10 F (h0) ∈ B ∗xB∗, F r−1(h0) −1gh0d −1 ∈ B∗F r−1(x)B∗, F r(h0) −1gh0d −1U∗ = k(F r(h0) −1gF (h0)F (d −1))x−1U∗ (if r ≥ 2) and h−10 gh0d −1 ∈ B∗xB∗, F (h0) −1gh0d −1U∗ = k(F (h0) −1gF (h0)F (d −1))x−1U∗ (if r = 1). In both cases these equations are equivalent to (e) h−10 F (h0) ∈ U ∗txF (t)−1U∗, F r(h0) −1gh0d −1 ∈ F r(t)U∗ for some t ∈ T . We then have F r−1(h0) −1gh0d −1 ∈ U∗F r−1(t)F r−1(x)U∗. For ∗, t as in (e) we compute k(h−10 F (h0))k(F (h0) −1F 2(h0)) . . . k(F r−2(h0) −1F r−1(h0))k(F r−1(h0) −1gh0d = (txF (t)−1)(F (t)F (x)F 2(t−1)) . . . (F r−2(t)F r−2(x)F r−1(t)−1)(F r−1(t)F r−1(x)) = txF (x) . . . F r−1(x) = t. By 3.2(a) the result of the last computation is necessarily in TFx . Thus Fx(t) = t. Hence F r(t) = t and the equations (e) become (f) h−10 F (h0) ∈ U ∗xU∗, F r(h0) −1gh0d −1 ∈ TFxU∗. We see that (−1)itr(?,Hig(pw!L̃w)) = |T Fx |−1 t∈TFx at = |T Fx |−1 t′∈TFx where at = |{hU ∗ ∈ (G0/U∗); h−1F (h) ∈ U∗xU∗, dh−1g−1F r(h)t ∈ U∗}|θ(t), 12 G. LUSZTIG a′t′ = |{hU ∗ ∈ (G0/U∗); h−1F (h) ∈ U∗xU∗, h−1g−1F r(h)dt′ ∈ U∗}|θ(dt′d−1). Comparing with the last formula in 2.4 and using θ(dt′d−1) = θ(t′) for t′ ∈ TFx we obtain (with i0 as in 2.4): (−1)itr(?,Hig(pw!L̃w)) = (−1) l(w)θ̃(d)qi0r/2tr(ǫg, H c (Y )θ). Let us choose an isomorphism pw!L̃w[lw] ∼= p∅!L̃∅. (This exists by 1.4; note that 1.4(i) holds by 1.5.) Via this isomorphism, the isomorphism 3.3(g) corresponds to an isomorphism F ∗(p∅!L̃∅) −→ p∅!L̃∅ that is to an isomorphism ψ ′ : F ∗A −→ A so that ∑ (−1)itr(?,Hig(pw!L̃w)) = (−1)itr(ψ′,Hig(A)) for any g ∈ DF . (We use that lw is even.) Since A is irreducible, we must have ψ = λ′ψ′ for some λ′ ∈ Q̄∗l . It follows that (−1)itr(ψ,Hig(A)) = λ ′(−1)l(w)θ̃(d)qi0r/2tr(ǫg, H c (Y )θ) for any g ∈ DF . Thus Theorem 1.2 holds with V being the GF -module Hi0c (Y )θ, which is irreducible (even as a G0F -module) if G0 has connected centre, but is not necessarily irreducible in general. References [DL] P.Deligne and G.Lusztig, Representations of reductive groups over finite fields, Ann.Math. 103 (1976), 103-161. [DM] F.Digne and J.Michel, Groupes réductifs non connexes, Ann.Sci. École Norm.Sup. 27 (1994), 345-406. [L1] G.Lusztig, Green functions and character sheaves, Ann.Math. 131 (1990), 355-408. [L2] G. Lusztig, Remarks on computing irreducible characters, J.Amer.Math.Soc. 5 (1992), 971-986. [L3] G.Lusztig, Character sheaves on disconnected groups,I, Represent. Th. (electronic) 7 (2003), 374-403; II 8 (2004), 72-124; III 8 (2004), 125-144; IV 8 (2004), 145-178; Er- rata 8 (2004), 179-179; V 8 (2004), 346-376; VI 8 (2004), 377-413; VII 9 (2005), 209-266; VIII 10 (2006), 314-352; IX 10 (2006), 353-379. [M] G.Malle, Generalized Deligne-Lusztig characters, J.Algebra 159 (1993), 64-97. [S] T.Shoji, Character sheaves and almost characters of reductive groups, Adv.in Math. 111 (1995), 244-313; II 111 (1995), 314-354. Department of Mathematics, M.I.T., Cambridge, MA 02139

0704.1000

Measurement of D0-D0bar mixing in D0->Ks pi+ pi- decays

Measurement of D0-D 0 mixing in D0 → K0 − decays L. M. Zhang,37 Z. P. Zhang,37 I. Adachi,7 H. Aihara,45 V. Aulchenko,1 T. Aushev,18, 13 A. M. Bakich,40 V. Balagura,13 E. Barberio,21 A. Bay,18 K. Belous,12 U. Bitenc,14 A. Bondar,1 A. Bozek,27 M. Bračko,20,14 J. Brodzicka,7 T. E. Browder,6 P. Chang,26 Y. Chao,26 A. Chen,24 K.-F. Chen,26 W. T. Chen,24 B. G. Cheon,5 C.-C. Chiang,26 I.-S. Cho,50 Y. Choi,39 Y. K. Choi,39 J. Dalseno,21 M. Danilov,13 M. Dash,49 A. Drutskoy,3 S. Eidelman,1 D. Epifanov,1 S. Fratina,14 N. Gabyshev,1 G. Gokhroo,41 B. Golob,19, 14 H. Ha,16 J. Haba,7 T. Hara,32 N. C. Hastings,45 K. Hayasaka,22 H. Hayashii,23 M. Hazumi,7 D. Heffernan,32 T. Hokuue,22 Y. Hoshi,43 W.-S. Hou,26 Y. B. Hsiung,26 H. J. Hyun,17 T. Iijima,22 K. Ikado,22 K. Inami,22 A. Ishikawa,45 H. Ishino,46 R. Itoh,7 M. Iwasaki,45 Y. Iwasaki,7 N. J. Joshi,41 D. H. Kah,17 H. Kaji,22 S. Kajiwara,32 J. H. Kang,50 H. Kawai,2 T. Kawasaki,29 H. Kichimi,7 H. J. Kim,17 H. O. Kim,39 S. K. Kim,38 Y. J. Kim,4 K. Kinoshita,3 S. Korpar,20, 14 P. Križan,19, 14 P. Krokovny,7 R. Kumar,33 C. C. Kuo,24 A. Kuzmin,1 Y.-J. Kwon,50 J. S. Lee,39 M. J. Lee,38 S. E. Lee,38 T. Lesiak,27 J. Li,6 A. Limosani,21 S.-W. Lin,26 Y. Liu,4 D. Liventsev,13 T. Matsumoto,47 A. Matyja,27 S. McOnie,40 T. Medvedeva,13 W. Mitaroff,11 H. Miyake,32 H. Miyata,29 Y. Miyazaki,22 R. Mizuk,13 Y. Nagasaka,8 I. Nakamura,7 E. Nakano,31 M. Nakao,7 Z. Natkaniec,27 S. Nishida,7 O. Nitoh,48 S. Ogawa,42 T. Ohshima,22 S. Okuno,15 S. L. Olsen,6 Y. Onuki,35 W. Ostrowicz,27 H. Ozaki,7 P. Pakhlov,13 G. Pakhlova,13 C. W. Park,39 H. Park,17 L. S. Peak,40 R. Pestotnik,14 L. E. Piilonen,49 A. Poluektov,1 H. Sahoo,6 Y. Sakai,7 O. Schneider,18 J. Schümann,7 C. Schwanda,11 A. J. Schwartz,3 R. Seidl,9, 35 K. Senyo,22 M. E. Sevior,21 M. Shapkin,12 H. Shibuya,42 S. Shinomiya,32 J.-G. Shiu,26 B. Shwartz,1 J. B. Singh,33 A. Sokolov,12 A. Somov,3 N. Soni,33 S. Stanič,30 M. Starič,14 H. Stoeck,40 K. Sumisawa,7 T. Sumiyoshi,47 S. Suzuki,36 O. Tajima,7 F. Takasaki,7 K. Tamai,7 N. Tamura,29 M. Tanaka,7 G. N. Taylor,21 Y. Teramoto,31 X. C. Tian,34 I. Tikhomirov,13 T. Tsuboyama,7 S. Uehara,7 K. Ueno,26 T. Uglov,13 Y. Unno,5 S. Uno,7 P. Urquijo,21 Y. Usov,1 G. Varner,6 K. Vervink,18 S. Villa,18 A. Vinokurova,1 C. H. Wang,25 M.-Z. Wang,26 P. Wang,10 Y. Watanabe,15 E. Won,16 B. D. Yabsley,40 A. Yamaguchi,44 Y. Yamashita,28 M. Yamauchi,7 C. Z. Yuan,10 C. C. Zhang,10 V. Zhilich,1 and A. Zupanc14 (The Belle Collaboration) 1Budker Institute of Nuclear Physics, Novosibirsk 2Chiba University, Chiba 3University of Cincinnati, Cincinnati, Ohio 45221 4The Graduate University for Advanced Studies, Hayama 5Hanyang University, Seoul 6University of Hawaii, Honolulu, Hawaii 96822 7High Energy Accelerator Research Organization (KEK), Tsukuba 8Hiroshima Institute of Technology, Hiroshima 9University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 10Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 11Institute of High Energy Physics, Vienna 12Institute of High Energy Physics, Protvino 13Institute for Theoretical and Experimental Physics, Moscow 14J. Stefan Institute, Ljubljana 15Kanagawa University, Yokohama 16Korea University, Seoul 17Kyungpook National University, Taegu 18Swiss Federal Institute of Technology of Lausanne, EPFL, Lausanne 19University of Ljubljana, Ljubljana 20University of Maribor, Maribor 21University of Melbourne, School of Physics, Victoria 3010 22Nagoya University, Nagoya 23Nara Women’s University, Nara 24National Central University, Chung-li 25National United University, Miao Li 26Department of Physics, National Taiwan University, Taipei 27H. Niewodniczanski Institute of Nuclear Physics, Krakow 28Nippon Dental University, Niigata 29Niigata University, Niigata 30University of Nova Gorica, Nova Gorica 31Osaka City University, Osaka 32Osaka University, Osaka Typeset by REVTEX http://arxiv.org/abs/0704.1000v2 33Panjab University, Chandigarh 34Peking University, Beijing 35RIKEN BNL Research Center, Upton, New York 11973 36Saga University, Saga 37University of Science and Technology of China, Hefei 38Seoul National University, Seoul 39Sungkyunkwan University, Suwon 40University of Sydney, Sydney, New South Wales 41Tata Institute of Fundamental Research, Mumbai 42Toho University, Funabashi 43Tohoku Gakuin University, Tagajo 44Tohoku University, Sendai 45Department of Physics, University of Tokyo, Tokyo 46Tokyo Institute of Technology, Tokyo 47Tokyo Metropolitan University, Tokyo 48Tokyo University of Agriculture and Technology, Tokyo 49Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 50Yonsei University, Seoul We report a measurement of D0-D 0 mixing in D0 → K0S π +π− decays using a time-dependent Dalitz plot analysis. We first assume CP conservation and subsequently allow for CP violation. The results are based on 540 fb−1 of data accumulated with the Belle detector at the KEKB e+e− collider. Assuming negligible CP violation, we measure the mixing parameters x = (0.80± 0.29+0.09 +0.10−0.07 −0.14)% and y = (0.33 ± 0.24+0.08 +0.06−0.12 −0.08)%, where the errors are statistical, experimental systematic, and systematic due to the Dalitz decay model, respectively. Allowing for CP violation, we obtain the CPV parameters |q/p| = 0.86+0.30 +0.06−0.29 −0.03 ± 0.08 and arg(q/p) = (−14 +16+5+2 −18−3−4) PACS numbers: 13.25.Ft, 11.30.Er, 12.15.Ff Mixing in the D0-D 0 system is predicted to be very small in the Standard Model (SM) [1] and, unlike in K0, B0, and B0s systems, has eluded experimental observa- tion. Recently, evidence for this phenomenon has been found in D0 → K+K−/π+π− [2] and D0 → K+π− [3] decays. It is important to measure D0-D 0 mixing in other decay modes and to search for CP -violating (CPV ) effects in order to determine whether physics contribu- tions outside the SM are present. Here we study the self-conjugate decay D0→K0S π+π−. The time-dependent probability of flavor eigenstates D0 and D 0 to mix to each other is governed by the lifetime τD0 = 1/Γ, and the mixing parameters x = (m1 − m2)/Γ and y = (Γ1 − Γ2)/2Γ. The parameters m1,m2 (Γ1,Γ2) are the masses (decay widths) of the mass eigenstates |D1,2〉 = p|D0〉±q|D 0〉, and Γ = (Γ1+Γ2)/2. The parameters p and q are complex coefficients sat- isfying |p|2 + |q|2 = 1. Various D0 decay modes have been exploited to measure or constrain x and y [4]. For D0→K0S π+π− decays, the time dependence of the Dalitz plot distribution allows one to measure x and y directly. This method was developed by CLEO [5] using 9.0 fb−1 of data; here we extend this method to a data sample 60 times larger. The decay amplitude at time t of an initially produced |D0〉 or |D 0〉 can be expressed as ,m2+, t) = A(m2−,m2+) e1(t) + e2(t) ,m2+) e1(t)− e2(t) ,m2+, t) = A(m2−,m2+) e1(t) + e2(t) ,m2+) e1(t)− e2(t) , (1) where A and A are the amplitudes for |D0〉 and |D 0〉 decays as functions of the invariant-masses-squared vari- ables m2 ≡ m2(K0S π±). The time dependence is con- tained in the terms e1,2(t) = exp[−i(m1,2 − iΓ1,2/2)t]. Upon squaring M and M, one obtains decay rates con- taining terms exp(−Γt) cos(xΓt), exp(−Γt) sin(xΓt), and exp[−(1± y)Γt]. We parameterize the K0Sπ +π− Dalitz distribution fol- lowing Ref. [6]. The overall amplitude as a function of m2+ and m is expressed as a sum of quasi-two-body am- plitudes (subscript r) and a constant non-resonant term (subscript NR): ,m2+) = iφrAr(m2−,m2+) + aNReiφNR , (2) ,m2+) = iφ̄rAr(m2+,m2−) + āNReiφ̄NR . (3) The functions Ar are products of Blatt-Weisskopf form factors and relativistic Breit-Wigner functions [7]. The data were recorded by the Belle detector at the KEKB asymmetric-energy e+e− collider [8]. The Belle detector [9] includes a silicon vertex detector (SVD), a central drift chamber (CDC), an array of aerogel thresh- old Cherenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF), and an elec- tromagnetic calorimeter. We reconstruct D0 candidates via the decay chain D∗+ → π+s D0, D0 → K0Sπ+π− [10]. Here, πs denotes a low-momentum pion, the charge of which tags the fla- vor of the neutral D at production. The K0S candidates are reconstructed in the π+π− final state; we require that the pion candidates form a common vertex sepa- rated from the interaction region and have an invariant mass within ±10 MeV/c2 of m . We reconstruct D0 candidates by combining the K0S candidate with two op- positely charged tracks assigned as pions. These tracks are required to have at least two SVD hits in both r-φ and z coordinates. A D∗+ candidate is reconstructed by com- bining the D0 candidate with a low momentum charged track (the π+s candidate); the resulting D ∗+ momentum in the e+e− center-of-mass (CM) frame is required to be larger than 2.5 GeV/c in order to eliminate BB events and suppress combinatorial background. The charged pion tracks are refitted to originate from a common vertex, which represents the decay point of the D0. The D∗+ vertex is taken to be the intersection of the D0 momentum vector with the e+e− interaction region. The D0 proper decay time is calculated from the projection of the vector joining the two vertices (~L) onto the momentum vector: t = ~L · (~p/p)(m /p). The uncertainty in t (σt) is calculated event-by-event, and we require σt < 1 ps (for selected events, 〈σt〉 ∼ 0.2 ps). The signal and background yields are determined from a two-dimensional fit to the variables m and Q ≡ −mπ)·c2. The variableQ is the kinetic energy released in the decay and equals only 5.9 MeV for D∗+→π+s D0 decays. We parameterize the signal shape by a triple-Gaussian function for m , and the sum of a bifurcated Student t distribution and a Gaussian function for Q. The backgrounds are classified into two types: random πs background, in which a random πs is combined with a true D0 decay, and combinatorial background. The shape of the m distribution for the random πs background is fixed to be the same as that used for the signal. Other background distributions are obtained from Monte Carlo (MC) simulation. We perform a two-dimensional fit to the measured m distributions in a wide range 1.81 GeV/c2 < m 1.92 GeV/c2 and 0<Q< 20 MeV. We define a smaller signal region |m − mD0 | < 15 MeV/c2 and |Q − 5.9 MeV| < 1.0 MeV, corresponding to 3σ intervals in these variables. In this region we find 534410±830 signal events and background fractions of 1% and 4% for the random πs and combinatorial backgrounds, respectively. and Q distributions are shown in Fig. 1 along with projections of the fit result. mKsππ (GeV) 1.825 1.85 1.875 1.9 Signal Random π Combinatorial Q (MeV) 0 5 10 15 20 FIG. 1: The distribution of (a) m with 0 < Q < 20 MeV; (b) Q with 1.81 GeV/c2 < m < 1.92 GeV/c2. Superim- posed on the data (points with error bars) are projections of the m -Q fit. For the events selected in the signal region we perform an unbinned likelihood fit to the Dalitz plot variables and m2+, and the decay time t. For D 0 decays, the likelihood function is fj(mK0 ππ,i, Qi)Pj(m2−,i,m2+,i, ti) , (4) where j = {sig, rnd, cmb} denotes the signal or back- ground components, and the index i runs over D0 candi- dates. The event weights fj are functions of mK0 Q and are obtained from the m -Q fit mentioned above. The probability density function (PDF) Psig(m2−,m2+, t) equals |M(m2−,m2+, t)|2 convolved with the detector response. Resolution effects in two- particle invariant masses are significant only for m2ππ. The latter, and variation of the efficiency across the Dalitz plot, are taken into account using the method described in Ref. [6]. The resolution in decay time t is accounted for by convolving Psig with a resolution function consisting of a sum of three Gaussians with a common mean and widths σk = Sk · σt,i (k = 1− 3). The scale factors Sk and the common mean are free parameters in the fit. The random πs background contains real D 0 and D 0 decays; in this case the charge of the πs is un- correlated with the flavor of the neutral D. Thus the Prnd PDF is taken to be (1 − fw)|M(m2−,m2+, t)|2 + fw|M(m2−,m2+, t)|2, convolved with the same resolution function as that used for the signal, where fw is the wrong-tag fraction. We measure fw = 0.452±0.005 from fitting events in the Q sideband 3 MeV< |Q−5.9 MeV| < 14.1 MeV. For the combinatorial background, Pcmb is the prod- uct of Dalitz-plot and decay time PDFs. The latter is parameterized as the sum of a delta function and an exponential function convolved with a Gaussian resolu- tion function. The timing and Dalitz PDF parameters are obtained from fitting events in the mass sideband 30 MeV/c2< |m −mD0 | < 55 MeV/c2. The likelihood function for D 0 decays, L, has the same form as L, with M and M (appearing in Psig and Prnd) interchanged. To determine x and y, we maximize the sum lnL+lnL. Table I lists the results from two separate fits. In the first fit we assume CP is conserved, i.e., a = ā, φ = φ̄, and p/q = 1. We fit all events in the signal region, where the free parameters are x, y, τ the timing resolution parameters of the signal, and the Dalitz plot resonance parameters ar(NR) and φr(NR). The fit gives τ = (409.9 ± 1.0) fs, which is consistent with the world average [11]. The results for ar and φr for the 18 quasi-two-body resonances used (following the same model as in Ref. [6]) and the NR contribution are listed in Table II. The Dalitz plot and its projections, along with projections of the fit result, are shown in Fig. 2. We estimate the goodness-of-fit of the Dalitz plot through a two-dimensional χ2 test [6] and obtain χ2/ndf = 2.1 for 3653−40 degrees of freedom (ndf). We find that the main features of the Dalitz plot are well reproduced, with some significant but numerically small discrepancies at peaks and dips of the distribution in the very high m2 region. The decay-time distribution for all events, and the ratio of decay-time distribution for events in theK∗(892)+ and K∗(892)− regions, are shown in Fig. 3. TABLE I: Fit results and 95% C.L. intervals for x and y, including systematic uncertainties. The errors are statisti- cal, experimental systematic, and decay-model systematic, respectively. For the CPV -allowed case, there is another so- lution as described in the text. Fit case Parameter Fit result 95% C.L. interval No x(%) 0.80 ± 0.29 +0.09+0.10−0.07−0.14 (0.0, 1.6) CPV y(%) 0.33 ± 0.24 +0.08+0.06−0.12−0.08 (−0.34, 0.96) CPV x(%) 0.81 ± 0.30 +0.10+0.09−0.07−0.16 |x| <1.6 y(%) 0.37 ± 0.25 +0.07+0.07−0.13−0.08 |y| <1.04 |q/p| 0.86+0.30 +0.06−0.29 −0.03 ± 0.08 - arg(q/p)(◦) −14+16+5+2−18−3−4 - For the second fit, we allow for CPV . This introduces the additional free parameters |p/q|, arg(p/q), ār(NR) and φ̄r(NR). The fit gives two solutions: if {x, y, arg(p/q)} is a solution, then {−x, −y, arg(p/q)+π} is an equally good solution. From the fit to data, we find that the Dalitz plot parameters are consistent for the D0 and D 0 samples; hence we observe no evidence for direct CPV . Results for |p/q| and arg(p/q), parameterizing CPV in mixing and interference between mixed and unmixed amplitudes, respectively, are also found to be consistent with CP con- servation. If we fit the data assuming no direct CPV , the values for x and y are essentially the same as those for the TABLE II: Fit results for Dalitz plot parameters. The errors are statistical only. Resonance Amplitude Phase (◦) Fit fraction K∗(892)− 1.629 ± 0.006 134.3 ± 0.3 0.6227 K∗0 (1430) − 2.12 ± 0.02 −0.9± 0.8 0.0724 K∗2 (1430) − 0.87 ± 0.02 −47.3 ± 1.2 0.0133 K∗(1410)− 0.65 ± 0.03 111± 4 0.0048 K∗(1680)− 0.60 ± 0.25 147± 29 0.0002 K∗(892)+ 0.152 ± 0.003 −37.5 ± 1.3 0.0054 K∗0 (1430) + 0.541 ± 0.019 91.8 ± 2.1 0.0047 K∗2 (1430) + 0.276 ± 0.013 −106± 3 0.0013 K∗(1410)+ 0.33 ± 0.02 −102± 4 0.0013 K∗(1680)+ 0.73 ± 0.16 103± 11 0.0004 ρ(770) 1 (fixed) 0 (fixed) 0.2111 ω(782) 0.0380 ± 0.0007 115.1 ± 1.1 0.0063 f0(980) 0.380 ± 0.004 −147.1 ± 1.1 0.0452 f0(1370) 1.46 ± 0.05 98.6 ± 1.8 0.0162 f2(1270) 1.43 ± 0.02 −13.6 ± 1.2 0.0180 ρ(1450) 0.72 ± 0.04 41± 7 0.0024 σ1 1.39 ± 0.02 −146.6 ± 0.9 0.0914 σ2 0.267 ± 0.013 −157± 3 0.0088 NR 2.36 ± 0.07 155± 2 0.0615 1 2 3 2 (GeV2/c4) 10000 1 2 3 2 (GeV2/c4) 20000 40000 1 2 3 2 (GeV2/c4) 10000 0 0.5 1 1.5 2 2 (GeV2/c4) FIG. 2: Dalitz plot distribution and the projections for data (points with error bars) and the fit result (curve). Here, m2± corresponds to m2(K0Sπ ±) for D0 decays and to m2(K0Sπ for D 0 decays. CP -conservation case, and the values for the CPV pa- rameters are further constrained: |q/p| = 0.95+0.22 −0.20 and arg(q/p) = (−2+10 ◦. A check with independent fits to theD0 andD 0 tagged samples gives consistent results for x (y): 0.58%±0.41% (0.45%±0.33%) and 1.04%±0.41% (0.21%± 0.34%), respectively. We consider systematic uncertainties arising from both experimental sources and from the D0→K0S π+π− decay Proper time (fs) -2000 0 2000 4000 FIG. 3: (a) The decay-time distribution for events in the Dalitz plot fit region for data (points with error bars), and the fit projection for the CP -conservation fit (curve). The hatched area represents the combinatorial background contri- bution. (b) Ratio of decay-time distributions for events in the K∗(892)+ and K∗(892)− regions. model. We estimate these uncertainties by varying rel- evant parameters by their ±1σ errors and interpreting the change in x and y as the systematic uncertainty due to that source. The main sources of experimental uncer- tainty are the modeling of the background, the efficiency, and the event selection criteria. We vary the background normalization and timing parameters within their uncer- tainties, and we also set fw equal to its expected value of 0.5 or alternatively let it float. To investigate possi- ble correlations between the Dalitz plot (m2+,m ) dis- tribution and the t distribution of combinatorial back- ground, the Dalitz plot distribution is obtained for three bins of decay time; these PDFs are then used accord- ing to the reconstructed t of individual events. We also try a uniform efficiency function, and we apply a “best- candidate” selection to check the effect of the small frac- tion of multiple-candidate events. We add all variations in x and y in quadrature to obtain the overall experimen- tal systematic error. The systematic error due to our choice of D0 → +π− decay model is evaluated as follows. We vary the masses and widths of the intermediate res- onances by their known uncertainties [11], and we also try fits with Blatt-Weisskopf form factors set to unity and with no q2 dependence in the Breit-Wigner widths. We perform a series of fits successively exclud- ing intermediate resonances that give small contributions (ρ(1450), K∗(1680)+), and we also exclude the NR con- tribution. We account for uncertainty in modeling of the S-wave ππ component by using K-matrix formal- ism [12]. We include an uncertainty due to the effect of around 10-20% bias in the amplitudes for theK∗(1410)±, K∗0 (1430) + andK∗2 (1430) + intermediate states, which we observe in MC studies. Adding all variations in quadra- ture gives the final results listed in Table I. We obtain a 95% C.L. contour in the (x, y) plane by finding the locus of points where −2 lnL increases by 5.99 units with respect to the minimum value (i.e., −2∆ lnL=5.99). All fit variables other than x and y are allowed to vary to obtain best-fit values at each point on the contour. To include systematic uncertainty, we rescale each point on the contour by a factor 1 + r2, where r2 is a weighted average of the ratios of systematic to statistical errors for x and y, where the weights de- pend on the position on the contour. Both the statistical- only and overall contours for both the CPV -allowed and the CP -conservation case are shown in Fig. 4. We note that for the CPV -allowed case, the reflection of these contours through the origin (0, 0) are also allowed re- gions. Projecting the overall contour onto the x, y axes gives the 95% C.L. intervals listed in Table I. After the systematics-rescaling procedure, the no-mixing point (0,0) has a value −2∆ lnL = 7.3; this corresponds to a C.L. of 2.6%. We have confirmed this value by generat- ing and fitting an ensemble of MC fast-simulated exper- iments. x (%) no CPV (stat. only) no CPV CPV (stat. only) -1 0 1 2 FIG. 4: 95% C.L. contours for (x, y): dotted (solid) corre- sponds to statistical (statistical and systematic) contour for no CPV , and dash-dotted (dashed) corresponds to statisti- cal (statistical and systematic) contour for the CPV -allowed case. The point is the best-fit result for no CPV . In summary, we have measured the D0-D 0 mixing parameters x and y using a Dalitz plot analysis of D0 → K0S π+π− decays. Assuming negligible CP vi- olation, we measure x = (0.80 ± 0.29+0.09+0.10 −0.07−0.14)% and y = (0.33 ± 0.24+0.08+0.06 −0.12−0.08)%, where the errors are sta- tistical, experimental systematic, and decay-model sys- tematic, respectively. Our results disfavor the no-mixing point x = y = 0 with a significance of 2.2σ, while the one dimensional significance for x > 0 is 2.4σ. We have also searched for CPV ; we see no evidence for this and constrain the CPV parameters |q/p| and arg(q/p). We thank the KEKB group for excellent operation of the accelerator, the KEK cryogenics group for efficient solenoid operations, and the KEK computer group and the NII for valuable computing and Super-SINET net- work support. We acknowledge support from MEXT and JSPS (Japan); ARC and DEST (Australia); NSFC and KIP of CAS (China); DST (India); MOEHRD, KOSEF and KRF (Korea); KBN (Poland); MES and RFAAE (Russia); ARRS (Slovenia); SNSF (Switzerland); NSC and MOE (Taiwan); and DOE (USA). [1] A. F. Falk et al., Phys. Rev. D 65, 054034 (2002); I. I. Bigi, N. Uraltsev, Nucl. Phys. B 592, 92 (2001); A. F. Falk et al., Phys. Rev. D 69, 114021 (2004); A. A. Petrov, Int. J. Mod. Phys. A21, 5686 (2006). [2] M. Starič et al. (Belle Collaboration), Phys. Rev. Lett. 98, 211803 (2007). [3] B. Aubert et al. (BaBar Collaboration), Phys. Rev. Lett. 98, 211802 (2007). [4] For a review see: D. M. Asner, D0- D 0 Mixing, in Ref. [11]. [5] D. M. Asner et al. (CLEO Collaboration), Phys. Rev. D 72, 012001 (2005) and arXiv: hep-ex/0503045v3. [6] A. Poluektov et al. (Belle Collaboration), Phys. Rev. D 73, 112009 (2006). [7] S. Kopp et al. (CLEO Collaboration), Phys. Rev. D 63, 092001 (2001). [8] S. Kurokawa, E. Kikutani et al., Nucl. Instrum. Methods Phys. Res. Sect. A 499, 1 (2003), and other papers in this volume. [9] A. Abashian et al. (Belle Collaboration), Nucl. In- strum. Methods Phys. Res. Sect. A 479, 117 (2002); Z. Natkaniec et al. (Belle SVD2 Group), Nucl. Instrum. Methods Phys. Res. Sect. A 560, 1 (2006). [10] Charge conjugate decays are implied unless explicitly stated otherwise. [11] W.-M. Yao et al. (Particle Data Group), J. Phys. G 33, 1 (2006). [12] J. M. Link et al. (FOCUS Collaboration), Phys. Lett. B 585, 200 (2004); B. Aubert et al. (BaBar Collaboration), arXiv: hep-ex/0507101. http://arxiv.org/abs/hep-ex/0503045 http://arxiv.org/abs/hep-ex/0507101

0704.1001

Tautological relations in Hodge field theory

TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY A. LOSEV, S. SHADRIN, AND I. SHNEIBERG Abstract. We propose a Hodge field theory construction that captures algebraic properties of the reduction of Zwiebach invari- ants to Gromov-Witten invariants. It generalizes the Barannikov- Kontsevich construction to the case of higher genera correlators with gravitational descendants. We prove the main theorem stating that algebraically defined Hodge field theory correlators satisfy all tautological relations. From this perspective the statement that Barannikov-Kontsevich construction provides a solution of the WDVV equation looks as the simplest particular case of our theorem. Also it generalizes the particular cases of other low-genera tautological relations proven in our earlier works; we replace the old technical proofs by a novel conceptual proof. Contents 1. Introduction 1 2. Gromov-Witten theory 5 3. Zwiebach invariants 9 4. Construction of correlators in Hodge field theory 12 5. String, dilaton, and tautological relations 17 6. Vanishing of the BV structure 21 7. Main Lemma 24 8. Proof of Theorem 3 27 References 33 1. Introduction In this paper we present an attempt to formalize what may be called a string field theory (SFT) for (closed) topological strings with Hodge property. From the very first days of string theory it was considered as a kind of generalization of the perturbative expansion of the quantum field theory in the (functional) integral representation. The space of graphs with g loops with metrics on edges (Schwinger proper times) was gen- eralized to moduli space of Riemann surfaces. Indeed, the latter space really looks like a principle U(1)n bundle over the former space near the http://arxiv.org/abs/0704.1001v1 2 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG points of maximal degeneracy (i.e., where maximal number of handles are pinched). A natural question is whether there are special string theories that degenerate exactly to quantum field theories (may be, of the special kind). Would it happen such theories should enjoy both finiteness of string theory and (functional) integral description of quantum field theory. One of the first attempts to construct a theory of this type was done by Zwiebach in [37]. He divided the moduli space into two re- gions: the internal piece and the boundary. He observed that surfaces representing the boundary region may be constructed from those rep- resenting the internal piece by gluing them with the help of cylinders (with flat metric). Therefore, he proposed to take integrals over the moduli spaces in two steps: first, to take an integral over the internal pieces, such that this would produce vertices, and then take an integral along metrics on cylinders, that would exactly reproduce integral along the Schwinger parameters on graphs in QFT prescription. In this approach, he came with the infinite number of vertices of different internal genera and with different number of external legs. However, he observed that such vertices satisfy quadratic relations that where a quantum version of some infinity-structure. At that time com- munity of theoretical physicists seemed not to be impressed by the Lagrangian with infinite number of (almost uncomputable1) vertices. The next attempt was done by Witten [36]. He assumed that in topological string theories there may be a limit in the space of two- dimensional theories such that the measure of integration goes to the vicinity of the points of maximal degeneration. In the type B theories such limit seems to be the large volume limit of the target space; this motivated Witten’s Chern-Simons-like representation for the topologi- cal string theory. This approach was further developed by Bershadsky, Cecotti, Ooguri, and Vafa in [3]. We note that the tropical limit of Gromov-Witten theory [26] (type A topological strings) seems to real- ize the same QFT degeneration of string theory. Indeed, the tropical limit of a Riemann surface mapped to a toric variety is represented by the graph mapped to the moment map domain. In the development of topological string theory it became clear that the proper object is not just a measure on the moduli space of complex structures of Riemann surfaces, but rather a differential form on this space. In original formulation these differential forms were assigned to the tensor algebra of cohomology of some complex; such objects are called Gromov-Witten invariants. We say that Gromov-Witten invariants are QFT-like if the differential forms of non-zero degree have support only in a vicinity of the points of maximal degeneration. 1 Note, that computation of an integral over a subspace with a boundary is harder than that one over a compact space. TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 3 We generalized the definition of Gromov-Witten invariants in [22] by lifting it from the cohomology of a complex to the full complex. Such generalization involved enlargement of the moduli space from Deligne- Mumford space to Kimura-Stasheff-Voronov space [15], and we called it Zwiebach invariants (in fact, some pieces of this construction appeared earlier in [37] and [8]). The complex of states involved in the definition of Zwiebach invariants is a bicomplex due to the action of the second differential. The second differential represents the substitution of a special vector field corresponding to the constant rotation of the phase of the local coordinate at a marked point into differential forms on the Kimura-Stasheff-Voronov space. Once we have some Zwiebach invariants, it is possible to produce new Zwiebach invariants by contraction of an acyclic Hodge sub-bicomplex. In fact, it is one of the main properties of Zwiebach invariants. Con- sider a sub-bicomplex, where these two differentials act freely. We call it Hodge contractible bicomplex. The operation of contraction of a Hodge contractible bicomplex turns Zwiebach invariants into induced Zwiebach invariants on the coset with respect to contactible bicomplex. Induced Zwiebach invariants are differential forms whose support is a union of the support of the initial Zwiebach invariants and small neigh- bourhoods of the points of maximal degeneration. This procedure is a generalization from intervals to cylinders of the procedure of induction of L∞-structures, see e. g. [30, 28]. This way we can obtain QFT-like Gromov-Witten invariants. We just should start with Zwiebach invariants that have (in some suit- able sense) no support inside the Kimura-Stasheff-Voronov spaces. In fact, it is even enough to consider a weaker condition, motivated by applications. That is, usually people consider the integrals of Gromov- Witten invarians only over the tautological classes in the moduli space of curves. So, we call a set of Zwiebach invariants vertex-like if the integral over the Kimura-Stasheff-Voronov spaces of any their non- zero component multipled by the pullback (from the Deligne-Mumford space) of any tautological class vanishes. Consider vertex-like Zwiebach invariants. Assume we contract a Hodge contractible bicomplex down to cohomology. We obtain differ- ential forms on the Deligne-Mumford space, such that the integral of the product of any such form of non-zero degree with any tautological class vanishes the interior of the moduli space. Integrals of such forms over the moduli spaces turn out to be sums over graphs (corresponding to degenerate Riemann surfaces). They resemble Feynman diagramms, and generation function for the integral over moduli spaces resemble diagrammatic expansion of perturbative quantum field theory. In this paper, we don’t construct examples of vertex-like Zwiebach invariants (we are going to do this explicitly in a future publication, as well as the corresponding theory for the spaces introduced in [20, 21]). 4 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG Rather we conjecture that they exist and study the consequences of this assumption. We call the emerging construction the Hodge field theory, and now we will explain it in some detail. First of all, degree zero parts of vertex-like Zwiebach invariants in- duce the structure of homotopy cyclic Hodge algebra on the target complex [22]. We remind that a cyclic Hodge algebra is just a Hodge dGBV-algebra with one additional axiom (1/12-axiom, see below). In fact, this structure is interesting by itself, without any reference to Zwiebach invariants. It has first appeared in the paper of Barannikov and Kontsevich [1]; it captures the properties of polyvector fields on Calabi-Yau manifolds. More examples of dGBV algebras are studied in [25] and [23]. It is possible to understand the structure of dGBV- algebra as a natural generalization of the algebraic structure studied in [19]. In the Hodge field theory construction we consider only a particular case, where we obtain axioms of of a cyclic Hodge algebra itself, not up to homotopy. We are aware of the fact that demanding existance of vertex-like Zwiebach invariants simultaneously with vanishing homo- topy piece of cyclic Hodge algebra conditions may be too restrictive, and while considering only those relations that lead to axioms of cyclic Hodge algebra may be too weak, however we proceed. In the Hodge field theory construction we define graph expressions for the analogues of Gromov-Witten invariants multiplied by tautological classes using only cyclic Hodge algebra data. We call them Hodge field theory correlators. The corresponding action of the Hodge field theory is written down explicitely in Section 6.2. Our main result is the proof that the Hodge field theory correla- tors satisfy all universal equations that follow from relations among tautological classes in cohomology. The first result of this kind is due to Barannikov and Kontsevich. They have noticed that there is a solution of the WDVV equation that is associated to a dGBV-algebra (this solution is the critical value of the BCOV action [3], see [1, Appendix] and [22, Appendix]). Later, we reproved this in [22]. Then, in [22, 31, 32, 33] we proved some other low-genera universal equations. Here we generalize all these result and put all calculations done before in a proper framework. In particular, the main problem for us was to define a graph expres- sion in tensors of a cyclic Hodge algebra that corresponds to the full Gromov-Witten potential with descendants. The first steps were done in [31, 32], where we introduced the definition of descendants at one point in Hodge field theory (mostly for combinatorial reasons). But then we observed that it is a part of a natural definition of potential with descendants in cyclic Hodge algebras that appears as a special case of degeneration of vertex-like Zwiebach invariants multiplied by tautological classes. TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 5 In this paper, we present and study this construction. We prove in a completely algebraic way that Hodge field theory correlators satisfy the same equations as a Gromov-Witten potential: string, dilaton, and the whole system of PDEs coming from tautological relations in the cohomology of the moduli space of curves (see also [33] for some pre- liminary results). In what follows we will not only present the proof but also will do our best relating algebraic definitions and statements on Hodge field theory to analoguous constructions and theorems in the theory of Zwiebach invariants. 1.1. Organization of the paper. In Section 2 we remind all neces- sary facts about the axiomatic Gromov-Witten theory. In Section 3 we define Zwiebach invariants and explain the motivation to consider the sums over graphs in cyclic Hodge algebras. In Section 4 we de- fine cyclic Hodge algebras and the corresponding descendant potential. In Section 5 we state the main properties of the descendant potential in cyclic Hodge algebras, and the rest of the paper is devoted to the proofs. 1.2. Acknowledgments. A.L. was supported by the Russian Fed- eral Agency of Atomic Energy and by the grants INTAS-03-51-6346, NSh-8065.2006.2, NWO-RFBR-047.011.2004.026 (RFBR-05-02-89000- NWO-a), and RFBR-07-02-01161-a. S.S. was supported by the grant SNSF-200021-115907/1. S.S. is grateful to the participants of the Moduli Spaces program at the Mittag- Leffler Institute (Djursholm, Sweden) for the fruitful discussions of the preliminary versions of the results of this paper. The remarks of C. Faber, O. Tommasi, and D. Zvonkine were especially helpful. I.S. was supported by the grant RFBR-06-01-00037. 2. Gromov-Witten theory In this section we remind what is Gromov-Witten theory and explain its basic properties that we are going to reproduce in Hodge field theory construction. 2.1. Gromov-Witten invariants. Let us fix a finite dimensional vec- tor space H0 over C together with the choice of a homogeneous basis H0 = 〈e1, . . . , es〉 and a non-degenerate scalar product ηij = (·, ·) on it. Let e1 be a distinguished even element of the basis. Consider the moduli spaces of curves Mg,n. On each Mg,n we take a differential form Ωg,n of mixed degree with values in H 0 . The whole system of forms {Ωg,n} is called Gromov-Witten invariants, if it satisfies the axioms [18, 24]: (1) There are two actions of the symmetric group Sn on Ωg,n. First, we can relabel the marked points on curves in Mg,n; second, we can interchange the factors in the tensor product H⊗n0 . We 6 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG require that Ωg,n is equivariant with respect to these two actions of Sn. In other words, one can think that each copy of H0 in the tensor product is assigned to a specific marked point on curves in Mg,n. (2) The forms must be closed, dΩg,n = 0. (3) Consider the mapping π : Mg,n+1 → Mg,n forgetting the last marked point. Then the correspondence between Ωg,n and Ωg,n+1 is given by the formula (1) π∗Ωg,n = (Ωg,n+1, e1) . The meaning of the right hand side is the following. We want to turn a H⊗n+10 -valued form into a H 0 -valued one. So, we take the copy of H0 corresponding to the last marked point and contract it with the vector e1 using the scalar product. (4) Consider an irreducible boundary divisor inMg,n, whose generic point is represented by a two-component curve. It is the image of a natural mapping σ : Mg1,n1+1 ×Mg1,n2+1 → Mg,n, where g = g1 + g1 and n = n1 + n2. We require that (2) σ∗Ωg,n = Ωg1,n1+1 ∧ Ωg2,n2+1, η Here on the right hand side we contract with a scalar product the two copies of H0 that correspond to the node. In the same way, consider the divisor of genus g − 1 curves with one self-intersection. It is the image of a natural mapping σ : Mg−1,n+2 → Mg,n. In this case, we require that (3) σ∗Ωg,n = Ωg−1,n+2, η As before, we contract two copies of H0 corresponding to the node. (5) We also assume that (Ω0,3, e1 ⊗ ei ⊗ ej) = (ei, ej) = ηij . 2.2. Gromov-Witten potential. Let us associate to each ei the set of formal variables Tn,i, n = 0, 1, 2 . . . . By Fg denote the formal power series in these variables defined as (4) Fg := a1,...,an≥0 ψaii , ejTai,j The first sum is taken over n ≥ 3 for g = 0, n ≥ 1 for g = 1, and n ≥ 0 for g ≥ 2. On the right hand side, we contract each copy of H0 with the factor of the tensor product associated to the same marked point. The formal power series F := exp( g≥0 ~ g−1Fg) is called Gromov- Witten potential associated to the system of Gromov-Witten invariants {Ωg,n}. The coefficients of Fg, g ≥ 0, are called correlators and denoted TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 7 (5) 〈τa1,i1 . . . τan,in〉g := Vectors ei1 , . . . , ein are called primary fields. The main properties of GW potentials come from geometry of the moduli space of curves. First, one can prove that coefficients of F satisfy string and dilaton equations: 〈τ0,1 τaj ,ij〉g = 〈τaj−1,ij k 6=j τak ,ik〉g;(6) 〈τ1,1 τaj ,ij〉g = (2g − 2 + n)〈 τaj ,ij〉g.(7) The string equation is a corollary of the fact that π∗ψj = ψj − Dj ; here π : Mg,n+1 → Mg,n is the projection forgetting the last marked point andDj is the divisor inMg,n+1 whose generic point is represented by a two-component curve with one node such that one component has genus 0 and contains exactly two marked points, the i-th and the (n+ 1)-th ones. It is assumed that j=1 aj > 0. The dilaton equation is a corollary of the fact that, in the same notations, π∗ψn+1 = 2g−2+n. Of course, we assume that 2g−2+n > 0. Second, any relation in the cohomology of Mg,n among natural ψ-κ- strata gives a relation for the correlators. Let us explain this in more detail. 2.3. Tautological relations. 2.3.1. Stable dual graphs. The moduli space of curves Mg,n [12] has a natural stratification by the topological type of stable curves. We can combine natural strata with ψ-classes at marked points and at nodes and κ-classes on the moduli spaces of irreducible components. These objects are called ψ-κ-strata. A convenient way to describe a ψ-κ-stratum in Mg,n is the language of stable dual graphs. Take a generic curve in the stratum. To each irreducible component we associate a vertex marked by its genus. To each node we associate an edge connecting the corresponding vertices (or a loop, if it is a double point of an irreducible curve). If there is a marked point on a component, then we add a leaf at the corresponding vertex, and we label leaves in the same way as marked points. If we multiply a stratum by some ψ-classes, then we just mark the corre- sponding leaves or half-edges (in the case when we add ψ-classes at nodes) by the corresponding powers of ψ. Also we mark each vertex by the κ-class associated to it. 8 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG Let us remark that by κ-classes on Mg,n we mean the classes (8) κk1,...,kl := π∗ where π : Mg,n+l → Mg,n is the projections forgetting the last l marked points. It is just another additive basis in the ring generated by the ordinary κ-classes (κk, k ≥ 1, in our notations). The basic properties of these classes are stated in [6]. 2.3.2. Integrals over ψ-κ-strata. Using the properties of GW invariants, one can express the integral of Ωg,n over a ψ-κ-stratum S in terms of correlators. Consider a special case, when S is represented by a two-vertex graph with no ψ- and κ-classes. Then, according to axiom 4, the integral of Ωg,n is the product of integrals of Ωg1,n1+1 and Ωg2,n2+1 over the moduli spaces corresponding to the vertices, contracted by the scalar product: Ωg,n, Mg1,n1+1 Ωg1,n1+1, eij ⊗ ei′ Mg2,n2+1 Ωg2,n2+1, eij ⊗ ei′′ Here we assume that the genus of one component of a generic curve in S is g1 and n1 marked points with labels j ∈ J1, |J1| = n1, are on this component. The other component has genus g2 and n2 marked points with labels j ∈ J2, |J2| = n2, lie on it. Of course, g1 + g2 = g, n1 + n2 = n. Now consider a special case, when S is represented by a one-vertex graph with ψ- and κ-classes. Let us assign a vector in the basis of H0 to each leaf (to each marked point). Then, according to axiom 3 the integral j κb1,...,bk , is equal to Mg,n+k Ωg,n+k n+j , eij ⊗ e Combining these two special cases one can obtain an expression in correlators that corresponds to an arbitrary ψ-κ-stratum. TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 9 2.3.3. Relations for correlators. Suppose that we have a linear combi- nation L of ψ-κ-strata that is equal to 0 in the cohomology of Mg,n (a tautological relation). Since dΩg,n = 0, the integral of Ωg,n, j=1 eij over L is equal to zero, for an arbitrary choice of primary fields. This gives an equation for correlators. Usually, one consider also the pull-backs of L to Mg,n+n′, n ′ ≥ 0, multiplied by arbitrary monomials of ψ-classes. Of course, they are also represented as vanishing linear combinations of ψ-κ-strata. This gives a system of PDEs for the formal power series Fg, g ≥ 0. For the detailed description of the correspondence between tautological relations and universal PDEs for GW potentials see, e. g., [10] or [6]. There are 8 basic tautological relations known at the moment: WDVV, Getzler, Belorousski-Pandharipande, and topological recursion rela- tions in M0,4, M1,1, M2,1, M2,2, M3,1 [10, 9, 2, 16]. 3. Zwiebach invariants In Gromov-Witten theory (and also in topological string theory) the Gromov-Witten invariants is usually a structure on the cohomology of a target manifold (the space H0) of on the cohomology of a com- plex of some other gometric origin. We have introduced the notion of Zwiebach invariants in [22] in order to formalize in a convenient way what physicists mean by topological conformal quantum field theory at the level of a complex rather than at the level of the cohomology. The very general principles of homological algebra imply that al- gebraic stuctures on the cohomology are often induced by some fun- damental structures on a full complex (the standard example is the induction of the infinity-structures from differential graded algebraic structures). Such induction usually can be represented as a sum over trees with vertices corresponding to fundamental operations and edges corresponding to the homotopy that contracts the complex to its co- homology. We would like to stress that Gromov-Witten invariants also can be considered as an induced structure on the cohomology of a complex. In this case, the fundamental structure on the whole complex is deter- mined by Zwiebach invariants. We are able to associate some structure on a bicomplex with a special compactification of the moduli space curves (Kimura-Stasheff-Voronov compactification). So, complexes are replaced by bicomplexes, where the second differential reflects the rotation of attached cylinders (or circles). This is an appearance of the string nature of the problem. As an induced structure we indeed obtain a Gromov-Witten-type theory that, under some additional assumptions, can be presented in terms of a sum over graphs. Below we explain the whole construction, following [22] and with some additional details. 10 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG 3.1. Kimura-Stasheff-Voronov spaces. We remind the construc- tion of the Kimura-Stasheff-Voronov compactification Kg,n of the mod- uli space of curves of genus g with n marked point. It is a real blow-up of Mg,n; we just remember the relative angles at double points. We can also choose an angle of the tangent vector at each marked point; this way we get the principal U(1)n-bundle over Kg,n. We denote the total space of this bundle by Sg,n. There are also the standard mappings between different spaces Sg,n. First, one can consider the projection π : Sg,n+1 → Sg,n forgetting the last marked point. Suppose that under the projection we have to con- tract a sphere that contains the points xi, xn+1, and a node. Denote the natural coordinates on the circles corresponding to xi and a node on a curve in Sg,n+1 by φi and θ. Let φ̃i be a coordinate on the circle corresponding to xi in Sg,n. Then φ̃i = φi + θ under the projection π. In the same way, if we contract a sphere that contains two nodes and xn+1, then θ̃ = θ1 + θ2, where θ1 and θ2 are the coordinates on the circles corresponding to the two nodes of a curve in Sg,n+1 and θ̃ is a coordinate on the circle at the resulting node in Sg,n In the same way, when we consider the mappings σ : Sg1,n1+1 × Sg1,n2+1 → Sg,n representing the natural boundary components of Sg,n, we have θ = φn1+1 + φn2+1, where φn1+1 and φn2+1 are the coordinates on the circles corresponding the points that are glued by σ into the node and θ is the coordinate on the circle at the node. For the map- ping σ : Sg−1,n+2 → Sg,n we also have θ = φn+1 + φn+2 with the same notations. 3.2. Zwiebach invariants. Let us fix a Hodge bicomplex H with two differentials denoted by Q and G− and with an even scalar product η = (·, ·) invariant under the differentials: (12) (Qv,w) = ±(v,Qw), (G−v, w) = ±(v,G−w). The Hodge property means that (13) H = H0 ⊕ 〈eα, Qeα, G−eα, QG−eα〉, where QH0 = G−H0 = 0 and H0 is orthogonal to H4. Below we consider the action of Q and G− on H ⊗n. We denote by Q(k) and G − the action of Q and G− respectively on the k-th component of the tensor product. On each Sg,n we take a differential form Cg,n of the mixed degree with values in H⊗n0 . The whole system of forms {Cg,n} is called Zwiebach invariants, if it satisfies the axioms: (1) Ωg,n is Sn-equivariant. (2) (d+Q)Ωg,n = 0, Q = TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 11 (3) (G − + ık)Cg,n = 0 for all 1 ≤ k ≤ n (we denote by ık the substitution of the vector field generating the action on Sg,n of the k-th copy of U(1)); Cg,n is invariant under the action of U(1)n; (4) π∗Cg,n = (Cg,n+1, e1), where π : Sg,n+1 → Sg,n is the mapping forgetting the last marked point. (5) σ∗Cg,n = (Cg1,n1+1 ∧ Cg2,n2+1, η −1), where σ : Sg1,n1+1×Sg1,n2+1 → Sg,n represents the boundary component. In the same way, σ∗Cg,n = (Cg−1,n+2, η −1) for the mapping σ : Sg−1,n+2 → Sg,n. (6) (C0,3, e1 ⊗ vα ⊗ vβ) = ((Id+ dφ2G−)vα, (Id+ dφ3G−)vβ), φ2 and φ3 are the coordinates on the circles at the corresponding points. Zwiebach invariants on the bicomplex with zero differentials deter- mine Gromov-Witten invariants. Indeed, in this case the factorization property implies that {Cg,n} is lifted from the blowdown of Kimura- Stasheff-Voronov spaces, i.e. it is determined by a set of forms on Deligne-Mumford spaces. Then it is easy to check that this system of forms satisfied all axioms of Gromov-Witten invariants. 3.3. Induced Zwiebach invariants. Induced Zwiebach invariants are obtained by the contraction of H4. We denote by G+ the contraction operator. This means that G+H0 = 0, Π = {Q,G+} is the projection to H4 along H0, {G+, G−} = 0, and (G+v, w) = ±(v,G+w). We construct an induced Zwiebach form C indg,n on a homotopy equiv- alent modification S̃g,n of the space Sg,n. At each boundary component γ we glue the cylinder γ× [0,+∞] such that γ in Sg,n is identified with γ × {0} in the cylinder. So, we have the mappings σ̃ : S̃g1,n1+1×S̃g1,n2+1× [0,+∞] → S̃g,n and σ̃ : S̃g−1,n+2 × [0,+∞] → S̃g,n representing the boundary components with glued cylinders. We take a form Cg,n, restrict it to H 0 , and extend it to the glued cylinder by the rule that (14) σ̃∗C indg,n = C indg1,n1+1 ∧ C g2,n2+1 , [e−tΠ−dt·G+ ] in the first case and (15) σ̃∗C indg,n = C indg−1,n+2, [e −tΠ−dt·G+ ] in the second case, where [e−tΠ−dt·G+ ] is the bivector obtained from the operator e−tΠ−dt·G+ , t is the coordinate on [0,+∞]. This determines C indg,n completely. Now it is a straightforward calculation to check that the forms C indg,n are (d+Q)-closed and satisfy the factorization property when restricted to the strata γ × {+∞}. 3.4. Induced Gromov-Witten theory. The induced Zwiebach in- variants determine Gromov-Witten invariants. The correlators of the corresponding Gromov-Witten potential are given by the integrals over 12 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG the fundamental cycles of K̃g,n (we just forget the circles at marked points in S̃g,n) of the forms C ψaii . In fact, the fundamental class of K̃g,n is represented as a sum over all irreducible boundary strata in Mg,n. Indeed, a boundary stratum γ in Mg,n has real codimension equal to the doubled number of the nodes of its generic curve. But then we add in K̃g,n a real two-dimensional cylinder for each node. A simple explicit calculation allows to ex- press the integral over the component of the fundamental cycle of K̃g,n corresponding to γ. It splits into the integrals of the initial Zwiebach invariants (multiplied by ψ-classes) over the moduli spaces correspond- ing to the irreducible components of curves in γ; they are contracted with the bivectors [G−G+] (obtained from the operator G−G+ via the scalar product) corresponding to the nodes according to the topology of curves in γ. So, we represent the correlators of the induced Gromov-Witten the- ory as sums over graphs. Then one can observe that C0,3 determines a multiplication on H . Topology of the spaces S0,4 and S1,1 implies that the whole algebraic structure that we obtain on H is the structure of cyclic Hodge algebra up to Q-homotopy, see [22]. Let us assume that the initial system of Zwiebach invariants is simple enough, i. e., it induces the explicit structure of cyclic Hodge algebra on H and only the integrals of the zero-degree parts of the initial Zwiebach invariants (multiplied by ψ-classes) are non-vanishing on fundamental cycles. In this case, the induced Gromov-Witten potential can be described in very simple algebraic terms. It is the motivation of the definition of the Hodge field theory construction given in the next section. 4. Construction of correlators in Hodge field theory In this section, we describe in a very formal algebraic way the sum over graphs obtained as an expression for the Gromov-Witten potential induced from Zwiebach invariants in the previous section. 4.1. Cyclic Hodge algebras. In this section, we recall the definition of cyclic Hodge dGBV-algebras [22, 31, 32, 24] (cyclic Hodge algebras, for short). A supercommutative associative C-algebra H with unit is called cyclic Hodge algebra, if there are two odd linear operators Q,G− : H → H and an even linear function : H → C called integral. They must satisfy the following axioms: (1) (H,Q,G−) is a bicomplex: (16) Q2 = G2− = QG− +G−Q = 0; (2) H = H0 ⊕ H4, where QH0 = G−H0 = 0 and H4 is repre- sented as a direct sum of subspaces of dimension 4 generated TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 13 by eα, Qeα, G−eα, QG−eα for some vectors e ∈ H4, i. e. (17) H = H0 ⊕ 〈eα, Qeα, G−eα, QG−eα〉 (Hodge decomposition); (3) Q is an operator of the first order, it satisfies the Leibniz rule: (18) Q(ab) = Q(a)b+ (−1)ãaQ(b) (here and below we denote by ã the parity of a ∈ H); (4) G− is an operator of the second order, it satisfies the 7-term relation: G−(abc) = G−(ab)c+ (−1) b̃(ã+1)bG−(ac) + (−1) ãaG−(bc)(19) −G−(a)bc− (−1) ãaG−(b)c− (−1) ã+b̃abG−(c). (5) G− satisfies the property called 1/12-axiom: (20) str(G− ◦ a·) = (1/12)str(G−(a)·) (here a· and G−(a)· are the operators of multiplication by a and G−(a) respectively, str means supertrace). Define an operator G+ : H → H related to the particular choice of Hodge decomposition. We put G+H0 = 0, and on each subspace 〈eα, Qeα, G−eα, QG−eα〉 we define G+ as G+eα = G+G−eα = 0,(21) G+Qeα = eα, G+QG−eα = G−eα. We see that [G−, G+] = 0; Π4 = [Q,G+] is the projection to H4 along H0; Π0 = Id− Π4 is the projection to H0 along H4. Consider the integral : H → C. We require that Q(a)b = (−1)ã+1 aQ(b),(22) G−(a)b = (−1) aG−(b), G+(a)b = (−1) aG+(b). These properties imply that G−G+(a)b = aG−G+(b), Π4(a)b = aΠ4(b), and Π0(a)b = aΠ0(b). We can define a scalar product on H as (a, b) = ab. We suppose that this scalar product is non-degenerate. Using the scalar product we may turn any operator A : H → H into the bivector that we denote by [A]. 14 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG 4.2. Tensor expressions in terms of graphs. Here we explain a way to encode some tensor expressions over an arbitrary vector space in terms of graphs. Consider an arbitrary graph (we allow graphs to have leaves and we require vertices to be at least of degree 3, the definition of graph that we use can be found in [24]). We associate a symmetric n-form to each internal vertex of degree n, a symmetric bivector to each egde, and a vector to each leaf. Then we can substitute the tensor product of all vectors in leaves and bivectors in edges into the product of n-forms in vertices, distributing the components of tensors in the same way as the corresponding edges and leaves are attached to vertices in the graph. This way we get a number. Let us study an example: v ⊗ v w ⊗ w We assign a 5-form x to the left vertex of this graph and a 3-form y to the right vertex. Then the number that we get from this graph is x(a, b, c, v, w) · y(v, w, d). Note that vectors, bivectors and n-forms used in this construction can depend on some variables. Then what we get is not a number, but a function. 4.3. Usage of graphs in cyclic Hodge algebras. Consider a cyclic Hodge algebra H . There are some standard tensors over H , which we associate to elements of graphs below. Here we introduce the notations for these tensors. We always assign the form (24) (a1, . . . , an) 7→ a1 · · · · · an to a vertex of degree n. There is a collection of bivectors that will be assigned below to edges: [G−G+], [Π0], [Id], [QG+], [G+Q], [G+], and [G−]. In pictures, edges with these bivectors will be denoted by , , , QG+ , G+Q , respectively. Note that an empty edge corresponding to the bivector [Id] can usually be contracted (if it is not a loop). The vectors that we will put at leaves depend on some variables. Let {e1, . . . , es} be a homogeneous basis of H0. In particular, we assume that e1 is the unit of H . To each vector ei we associate formal variables Tn,i, n ≥ 0, of the same parity as ei. Then we will put at a leaf one of the vectors En = i=1 eiTn,i, n ≥ 0, and we will mark such leaf by the TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 15 number n. In our picture, an empty leaf is the same as the leaf marked by 0. 4.3.1. Remark. There is a subtlety related to the fact that H is a Z2- graded space. In order to give an honest definition we must do the following. Suppose we consider a graph of genus g. We can choose g edges in such a way that the graph being cut at these edge turns into a tree. To each of these edges we have already assigned a bivector [A] for some operator A : H → H . Now we have to put the bivector [JA] instead of the bivector [A], where J is an operator defined by the formula J : a 7→ (−1)ãa. In particular, consider the following graph (this is also an example to the notations given above): An empty loop corresponds to the bivector [Id]. An empty leaf corre- sponds to the vector E0. A trivalent vertex corresponds to the 3-form given by the formula (a, b, c) 7→ If we ignore this remark, then what we get is just the trace of the operator a 7→ E0 · a. But using this remark we get the supertrace of this operator. In fact, this subtlety will play no role in this paper. It affects only some signs in calculations and all these signs will be hidden in lemmas shared from [22, 31]. So, one can just ignore this remark. 4.4. Correlators. We are going to define the potential using correla- tors. Let (27) 〈τk1(V1) . . . τkn(Vn)〉g be the sum over graphs of genus g with n leaves marked by τki(Vi), i = 1, . . . , n, where V1, . . . , Vn are vectors in H , and τki are just formal symbols. The index of each internal vertex of these graphs is ≥ 3; we associate to it the symmetric form (24). There are two possible types of edges: edges marked by [G−G+] (thick black dots in pictures, “heavy edges” in the text) and edges marked by [Id] (empty edges). Since an empty edge connecting two different vertices can be contracted, we assume that all empty edges are loops. Consider a vertex of such graph. Let us describe all possible half- edges adjusted to this vertex. There are 2g, g ≥ 0, half-edges coming from g empty loops; m half-edges coming from heavy edges of graph, and l leaves marked τka1 (Va1), . . . , τkal (Val). Then we say that the type of this vertex is (g,m; ka1, . . . , kal). We denote the type of a vertex v by (g(v), m(v); ka1(v), . . . , kal(v)(v)). Consider a graph Γ in the sum determining the correlator (28) 〈τk1(V1) . . . τkn(Vn)〉g 16 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG We associate to Γ a number: we contract according to the graph struc- ture all tensors corresponding to its vertices, edges, and leaves (for leaves, we take vectors V1, . . . , Vn). Let us denote this number by T (Γ). Also we weight each graph by a coefficient which is the product of two combinatorial constants. The first factor is equal to (29) V (Γ) = v∈V ert(Γ) 2 g(v)g(v)! |aut(Γ)| Here |aut(Γ)| is the order of the automorphism group of the labeled graph Γ, V ert(Γ) is the set of internal vertices of Γ. In other words, we can label each vertex v by g(v), delete all empty loops, and then we get a graph with the order of the automorphism group equal to 1/V (Γ). The second factor is equal to (30) P (Γ) = v∈V ert(Γ) Mg(v),m(v)+l(v) a1(v) 1 . . . ψ al(v)(v) The integrals used in this formula can be calculated with the help of the Witten-Kontsevich theorem [35, 17, 29, 27, 13, 14, 4]. So, the whole contribution of the graph Γ to the correlator is equal to V (Γ)P (Γ)T (Γ). One can check that the non-trivial contribution to the correlator 〈τk1(V1) . . . τkn(Vn)〉g is given only by graphs that have exactly 3g − 3 + n− i=1 ki heavy edges. The geometric meaning here is very clear. The number T (Γ) comes from the integral of the induced Gromov-Witten invariants of degree zero, while the coefficient V (Γ)P (Γ) is exactly the combinatorial in- terpretation of the intersection number of ψk11 . . . ψ n with the stratum whose dual graph is obtained from Γ by the procedure described after the definition of V (Γ). 4.5. Potential. We fix a cyclic Hodge algebra and consider the formal power series F = F(Tn,i) defined as (31) F = exp g−1Fg = exp a1,...,an∈Z≥0 〈τa1(Ea1) . . . τan(Ean)〉g Abusing notations, we allow to mark the leaves by τa(Ea), Ea, or a; all this variants are possible and denote the same. 4.6. Trivial example. For example, consider the trivial cyclic Hodge algebra: H = H0 = 〈e1〉, Q = G− = 0, e1 = 1. Then Ea = e1 · ta, and the correlator 〈τa1(Ea1) . . . τan(Ean)〉g consists just of one graph with TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 17 one vertex, g empty loops, and n leaves marked by a1, . . . , an. The explicit value of the coefficient of this graph is, by definition, (32) 〈τa1 . . . τan〉g := ψa11 . . . ψ So, in the case of trivial cyclic Hodge algebra we obtain exactly the Gromov-Witten potential of the point (i. e., Ωg,n, e ≡ 1)) that we denote below by F pt. 4.6.1. Remark about notations. Abusing notation, we use the same symbol 〈 〉g for the correlators in GW theory and in Hodge field the- ory. We hope that it does not lead to a confusion. For instance, 〈τa1(Ea1) . . . τan(Ean)〉g in the trivial example above is the correlator of the trivial Hodge field theory, while 〈τa1 . . . τan〉g is the correlator of the trivial GW theory. 5. String, dilaton, and tautological relations In this section, we prove that the potential (31) satisfies the same string and dilaton equations as GW potentials. 5.1. String equation. Theorem 1. If j=1 aj > 0, we have: (33) 〈τ0(e1) τaj (eij )〉g = 〈τaj−1(eij ) k 6=j τak(eik)〉g; Proof. Consider a graph Γ contributing to the correlator on the left hand side of the string equation. The special leaf that we are going to remove is marked by τ0(e1) and is attached to a vertex v of genus gv (i. e., with gv attached light loops) with lv more attached leaves labeled by indices in Iv, |Iv| = lv, and mv attached half-edges coming from heavy edges and loops. Let us remove the leaf τ0(e1) and change the label of one of the leaves attached to the same vertex from τaj (eij ) to τaj−1(eij ). This way we obtain a graph Γj contributing to the j-th summand of the right hand side of (33). We take the sum of these graphs over j ∈ Iv. Of course, we skip the summands where aj = 0. Note that this sum is not empty (if Γ gives a non-zero contribution to the left hand side of (33)). Indeed, if it is empty, this means that aj = 0 for all j ∈ Iv. Therefore, since we expect that the contribution to P (Γ) of the vertex v on the left hand side is nonzero, it follows that gv = 0 and mv + lv = 2. So, there are three possible local pictures: τ0(ej) τ0(e1) , and τ0(ej1) τ0(ej2) τ0(e1) . 18 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG The first picture can be replaced with the bivector [G−G+G−G+], which is equal to zero. Therefore, T (Γ) is also equal to zero. In the second case, we also get 0 since G−G+(e1ej) = G−G+(E0) = 0. The third picture is possible only when it is the whole graph, and this is in contradiction with the assumption that j=1 aj > 0. Note also that T (Γ) = T (Γj) and V (Γ) = V (Γj) for all j. Indeed, we have just removed the leaf with the unit of the algebra, so this can’t change anything in the contraction of tensors. Therefore, T (Γ) = T (Γj). Also both the leaf τ0(e1) and the vertex v are the fixed points of any automorphism of Γ. The same is for the vertex corresponding to v in Γj . Therefore, the automorphism groups are isomorphic for both graphs. Since we make no changes for empty loops, it follows that V (Γ) = V (Γj). Let us prove that P (Γ) = P (Γj). Indeed, the vertices of Γ and Γj are in a natural one-to-one correspondence. Moreover, the lo- cal pictures for all of them except for v and its image in Γj are the same. Therefore, the corresponding intersection numbers contributing to P (Γ) and P (Γj) are the same. The unique difference appeares when we take the intersection numbers corresponding to v and its images in Γj, j ∈ Iv. But then we can apply the string equation (6) of the GW theory of the point (32), and we see that Mgv,kv+lv+1 Mgv,kv+lv k 6=j (here o : Iv → {1, . . . , l} is an arbitrary on-to-one mapping). This implies that P (Γ) = P (Γj). So, we have V (Γ)P (Γ)T (Γ) = V (Γj)P (Γj)T (Γj). In order to complete the proof of (33), we should just notice that when we write down this expression for all graphs contributing to the left hand side of (33), we use each graph contributing to the right hand side of (33) exactly once. 5.2. Dilaton equation. Theorem 2. If 2g − 2 + n > 0, we have: (36) 〈τ1(e1) τaj (eij )〉g = (2g − 2 + n)〈 τaj (eij )〉g; Proof. Consider a graph Γ contributing to the correlator on the left hand side of (36). The special leaf that we are going to remove is marked by τ1(e1) and is attached to a vertex v of genus gv (i. e., with gv attached light loops) with lv more attached leaves labeled by indices in Iv, |Iv| = lv, and mv attached half-edges coming from heavy edges and loops. TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 19 Let us remove the leaf τ1(e1). We obtain a graph Γ ′ contributing to the right hand side of (33). Let us prove this. Indeed, if we remove a leaf and don’t get a proper graph, it follows that we have a trivalent vertex. Since the contribution of this vertex to P (Γ) should be non- zero, it follows that the unique possible local picture is (37) τ1(e1) . But this picture is the whole graph, and it is in contradiction with the condition 2g − 2 + n > 0. The same argument as in the proof of the string equation shows that T (Γ) = T (Γ′) and V (Γ) = V (Γ′). Also, the contribution to P (Γ) and P (Γ′) of all vertices except for the changed one is the same. The change of the intersection number corresponding to the vertex v is captured by the dilaton equation (6) of the trivial GW theory (32): Mgv,kv+lv+1 = (2gv−2+kv+ lv) Mgv,kv+lv (again, o : Iv → {1, . . . , l} is an arbitrary on-to-one mapping). This implies that P (Γ) = (2gv − 2 + kv + lv)P (Γ ′), and, therefore, (39) V (Γ)P (Γ)T (Γ) = (2gv − 2 + kv + lv)V (Γ ′)P (Γ′)T (Γ′). Let us write down the last equation for all graphs Γ contributing to the left hand side (36). Observe that any graph Γ′ contributing to the right hand side occurs |V ert(Γ′)| times, since the leaf τ1(e1) could be attached to any its vertex. Therefore, any graph Γ′ contributing to the right hand side of (36) appears in these equations with the coefficient v∈V ert(Γ′) (2gv − 2 + kv + lv) = 2g − 2 + n. This completes the proof. 5.3. Tautological equations. As we have explained in Section 2.3.3, any linear relation L among ψ-κ-strata in the cohomology of the mod- uli space of curves gives rise to a family of universal relations for the correlators of a Gromov-Witten theory. Theorem 3 (Main Theorem). The system of universal relations com- ing from a tautological relation in the cohomology of the moduli space of curves holds for the correlators 〈 j=1 τaj (eij )〉g of cyclic Hodge algebra. Note that some special cases of this theorem were proved in [22, 31, 32]. Our argument below is a natural generalization of the technique introduced in these papers. Also we are able now to give an explana- tion why we have managed to perform all our calculations there, see Remark 8.6.1. 20 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG Let us give here a brief account of the proof of this theorem. First, the definition of correlators of the Hodge field theory can be extended to the intersection with an arbitrary tautological class α of degree K in the space Mg,n, not only the a monomial in ψ-classes. In that case, we do the following. Again, we consider the sum over all graphs Γ with 3g − 3 + n − K heavy edges, and the number T (Γ) is defined as above. Instead of the coefficient V (Γ)P (Γ) we use the intersection number of α with the stratum whose dual graph is obtained from Γ by the procedure described right after the definition of V (Γ). Namely, a vertex with g loops is replaced by a vertex marked by g. This definition is very natural from the point of view of Zwiebach invariants. However, we know from Gromov-Witten theory that this extension of the notion of correlator is unnecessary. Indeed, all integrals with arbitrary tautological classes can be expressed in terms of the integrals with only ψ-classes via some universal formulas. The main question is whether these universal formulas also work in Hodge field theory. Actually, the main result that more or less immediately proves the theorem is the positive answer to this question. 5.3.1. Organization of the proof. The rest of the paper is devoted to the proof of Main Theorem, and here we would like to overview it here. In Section 6, we study the structure of graphs that can appear in formulas for the correlators of Hodge field theory. We prove that if T (Γ) 6= 0 and there is at least one heavy edge in Γ, then all vertices have genus ≤ 1, i. e., there is at most one empty loop at any vertex. This basically means that in calculations we’ll have to deal only with genera 0 and 1. Also this allows us to write down the action of a Hodge field theory. In Section 7, we prove the main technical result (Main Lemma). In- formally, it states that Q = −G−ψ when we apply these two operators to the correlators of a Hodge field theory. In order to prove it, we look at a small piece (consisting just of one heavy edge and one or two vertices that are attached to it) in one of the graphs of a correlator. Of course, in the correlator we can vary this small piece in an arbitrary way, such that the rest of the graph remains the same. So, when we consider the sum of all these small pieces, it is also a correlator of the Hodge field theory. Thus we reduce the proof to a special case of the whole statement. But since the genus of a vertex is ≤ 1, it appear now to be a low-genera statement that can be done by a straightforward calculation. In Section 8, we present the proof of the Main Theorem. Consider a ψ-κ-stratum α whose stable dual graph has k ≥ 1 edges. There is a universal expression of the integral over α coming from Gromov-Witten theory. It includes k entries of the scalar product restricted to H0. In terms of graphs, it means that we are to introduce new edges with the TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 21 bivector [Π0] on them, and there are k such edges in our expression. A direct corollary of the Main Lemma is that we can always replace [Π0] by [Id]− ψ[G−G+]− [G−G+]ψ. In Sections 8.2, 8.3, and 8.4, we show that when we replace [Π0] by [Id] − ψ[G−G+] − [G−G+]ψ at all edges corresponding to the scalar product restricted to H0, we obtain a new expression for the integral over α that again contains only heavy edges and empty loops, as any ordinary correlator. The main problem now is to understand the com- binatorial coefficient of a graph Γ obtained this way. Since we have a sum over graphs with heavy edges and empty loops, it is natural to identify again these graphs with the corresponding strata in the moduli space of curves. Then we can calculate the intersection index of the stratum corresponding to a graph Γ and the initial class α. Roughly speaking, the main thing that we have to do is to decide about each node in α (represented initially by [Π0]), whether we have this node in stratum corresponding to Γ. If yes, then we have an excessive intersection (so, we must put −ψ on one of the half-edges of the corresponding edge), and we keep this edge in Γ (so, we replace [Π0] with −ψ[G−G+] or −[G−G+]ψ). If no, then we don’t have this edge in Γ, so we contract [Π0], i. e., replace it with [Id]. So, the procedure that we used to get rid of the scalar product is the same as the procedure of the intersection of α with strata of the complementary dimension. This means (Section 8.5) that the universal formula coming from Gromov-Witten theory is equivalent to the natu- ral formula for the “correlator with α” coming from Zwiebach theory. The tautological relation is a sum of classes equal to zero. So, while the universal formula coming from Gromov-Witten theory gives (in the case of a vanishing class) a non-trivial expression in correlators, the nat- ural formula coming from Zwiebach theory gives identically zero. This proves our theorem, see Section 8.6. 6. Vanishing of the BV structure In this section, we recall several useful lemmas shared in [34, 31]. In particular, these lemmas give some strong restrictions on graphs that can give a non-zero contirbution to the correlators defined above. 6.1. Lemmas. Lemma 1. [34, 31] The following vectors and bivectors are equal to zero: , G− . Also let us remind another lemma in [31] that is very useful in cal- culations. 22 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG Lemma 2. [31] For any vectors V0, V1, . . . , Vk, k ≥ 2, ...A1 A2 Ak ...G−A1 A2 Ak + · · ·+ ...A1 Ak−1 G−Ak ,(42) ...A1 A2 Ak ...G−A1 A2 Ak + · · ·+ ...A1 Ak−1 G−Ak .(43) Both lemmas are just simple corollaries of the axioms of cyclic Hodge algebra. 6.2. Structure of graphs. Consider a graph studied in Section 4.4. It can have leaves, empty and heavy loops, and heavy edges. Consider a vertex of such graph. Let us assume that there are A empty loops, B heavy loops, C heavy edges going to the other vertices of the graph, and D leaves attached to this vertex: (44) D C This picture can be considered as an C +D form. Let us denote it by Φ(A,B,C,D). Lemma 3. If A ≥ 2 and B + C ≥ 1, then Φ(A,B,C,D) = 0. In other words, if there are at least two empty loops at a vertex, then there should not be any heavy loops or edges attached to this vertex. Otherwise the contribution of the whole graph vanishes. This implies Corollary 1. In the definition of correlators one should consider only graphs of one of the following two types: (1) One-vertex graphs with no heavy edges (loops). (2) Arbitrary graphs with at most one empty loop at each vertex. The contribution of all other graphs vanishes. This corollary dramatically simplifies all our calculations with graphs given below. Also we can write down now the action of the Hodge field theory. TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 23 Let F 0g (v0, v1, v2, . . . ), vi ∈ H ⊗ C[[{Tn,i}]] be the “dimension zero” part of the potential of the Hodge field theory, namely, (45) F 0g := a1+···+an=3g−3+n 〈τa1(va1) . . . τan(van)〉g. The first sum is taken over n ≥ 0 such that 2g − 2 + n > 0. So, it is exactly the generating function for the vertices of our graph expressions. Then the action of the Hodge field theory is equal to (46) A(v) := F 00 (E0 +G−v, E1, E2, . . . ) + ~F 1 (E0 +G−v, E1, E2, . . . ) gF 0g (E0, E1, E2, . . . )− Qv ·G−v. If we put Tn,i = 0 for n ≥ 1, then we immediately obtain the BCOV- type action discussed in [1, Appendix] and [22, Appendix]. The similar actions were also studied in [5] and [7]. 6.3. Proof of Lemma 3. We consider the form Φ(A,B,C,D) and we assume that A ≥ 2. First, let us study the case when C ≥ 1. In this case, our C+D-form can be represented as a contraction via the bivector [Id] of two forms, Φ(A−2, B, C−1, D+1) and Φ(2, 0, 1, 1). Let us prove that the last one is equal to zero. Indeed, this two-form can be represented as Φ̃(·, G+·), where the two-form Φ̃ is represented by the picture According to Lemma 1, Φ̃ = 0. Therefore, Φ(2, 0, 1, 1) = 0 and the whole form Φ(A,B,C,D) is also equal to zero. Now consider the case when B ≥ 1. In this case, our C + D-form can be represented as a contraction via the bivector [Id] of two forms, Φ(A − 2, B − 1, C,D + 1) and Φ(2, 1, 0, 1). Let us prove that the last one is equal to zero. Indeed, (48) Φ(2, 1, 0, 1) = = Here the first equality is definition of Φ(2, 1, 0, 1), the second one is just an equivalent redrawing, the third equality is application of Lemma 2, the fourth one is again an equivalent redrawing. 24 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG The last picture contains the bivector (47) which is equal to zero according to Lemma 1. Therefore, the whole picture is equal to zero, and Φ(2, 1, 0, 1) = 0. So, the whole form Φ(A,B,C,D) is equal to zero also in this case. This proves the lemma. 7. Main Lemma 7.1. Statement. The main technical tool that we use in the proof of Theorem 3 is the lemma that we prove in this section. Lemma 4 (Main Lemma). For any v1, . . . , vn ∈ H, a1, . . . , an ≥ 0, 〈τa1(v1) . . . τai−1(vi−1)τai(Q(vi))τai+1(vi+1) . . . τan(vn)〉g = 〈τa1(v1) . . . τai−1(vi−1)τai+1(G−(vi))τai+1(vi+1) . . . τan(vn)〉g. A simple corollary of this lemma is the following: Lemma 5. For any w ∈ H, v1, . . . , vn ∈ H0, (50) 〈τa0(Qw)τa1(v1) . . . τan(vn)〉g + 〈τa0+1(G−w)τa1(v1) . . . τan(vn)〉g = 0. In other words, we can state informally that Q+ ψG− = 0. 7.2. Special cases. The proof of the lemma can be reduced to a small number of special cases. We consider correlators whose graphs have only one heavy edge. 7.2.1. The first case is the following: Let i=1 ai = n− 4. We prove that for any v1, . . . , vn ∈ H , 〈τa1(v1) . . . τai−1(vi−1)τai(Q(vi))τai+1(vi+1) . . . τan(vn)〉0 = 〈τa1(v1) . . . τai−1(vi−1)τai+1(G−(vi))τai+1(vi+1) . . . τan(vn)〉0. First, we see that according to the definition of the correlator, the left hand side of Equation (51) is the sum over graphs with two vertices and with [G−G+] on the unique edge that connects the vertices. For each I ⊔J = {1, . . . , n} we can consider the corresponding distribution of leaves between the vertices (to be precise, let us assume that 1 ∈ I). Then the coefficient of such graph is 〈τ0 i∈I τai〉0〈τ0 j∈J τaj〉0, and we take the sum over all possible positions of Q at the leaves. Using the Leibniz rule for Q and the property that [Q,G−G+] = −G−, we see that this sum is equal to the sum over graphs with two vertices and with [−G−] on the unique edge that connects the vertices. TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 25 For each I ⊔J = {1, . . . , n}, |I|, |J | ≥ 2, we consider the corresponding distribution of leaves between the vertices. Then the coefficient of such graph is still 〈τ0 i∈I τai〉0〈τ0 j∈J τaj〉0, and the underlying tensor expression can be written (after we multiply the whole sum by −1) as i∈I\{1} vi ·G−( Let us recall that the 7-term relation for G− implies that vj) = i,j∈J, i<j G−(vivj) k∈J\{i,j} vk(53) − (|J | − 2) G−(vj) i∈J\{i,j} Using this, we can rewrite the whole sum over graphs as 1<i<j k 6=1,i,j vk ·G−(vivj) I′⊔J ′⊔{i,j}={2,...,n} 〈τa1τ0 τak〉0〈τaiτajτ0 k∈J ′ τak〉0 i 6=1 j 6=1,i vj ·G−(vi) I′⊔J ′⊔{i}={2,...,n} (|J ′| − 1) 〈τa1τ0 τaj〉0〈τaiτ0 j∈J ′ τaj〉0 Using that I′⊔J ′⊔{i,j}={2,...,n} 〈τa1τ0 τak〉0〈τaiτajτ0 k∈J ′ τak〉0 = 〈τa1+1 k 6=1 τak〉0, Equation (53), and the fact that (56) (n− 3)〈τa1+1 j 6=1 τaj 〉0 I′⊔J ′⊔{i}={2,...,n} (|J ′| − 1) 〈τa1τ0 τaj〉0〈τaiτ0 j∈J ′ τaj〉0 = 〈τai+1 j 6=i τaj〉0, 26 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG we can rewrite Expression (54) as G−(vi), j 6=i 〈τai+1 j 6=i τaj 〉0. The last formula coincides by definition with the right hand side of Equation (51) multiplied by −1. This proves the first special case. 7.2.2. The second case is in genus 1. Let i=1 ai = n− 1. We prove that for any v1, . . . , vn ∈ H , 〈τa1(v1) . . . τai−1(vi−1)τai(Q(vi))τai+1(vi+1) . . . τan(vn)〉1 = 〈τa1(v1) . . . τai−1(vi−1)τai+1(G−(vi))τai+1(vi+1) . . . τan(vn)〉1. According to the definition of the correlator, the left hand side of Equation (58) is the sum over graphs of two possible types. The first type include graphs with two vertices and two edges. The first edge is heavy and connects the vertices; the second edge is an empty loop attached to the first vertex. For each I ⊔ J = {1, . . . , n}, |J | ≥ 2, we can consider the corresponding distribution of leaves between the vertices (we assume that leaves with indices in I are at the first edge). Then the coefficient of such graph is 〈τ0 i∈I τai〉1〈τ0 j∈J τaj 〉0. The second type include graphs with one vertex and one heavy loop. All leaves are attached to this vertex, and the coefficient of such graph is 〈τ 20 i=1 τai〉0. For both types of graphs, we take the sum over all possible positions of Q at the leaves. Using the Leibniz rule for Q and the property that [Q,G−G+] = −G−, we get the same graphs as before, but there is no Q, and instead of [G−G+] we have −[G−] on the corresponding edge. Using Lemma 2, we move −G− in graphs of the first type to the leaves marked by indices in J . Using the 1/12-axiom and Lemma 2, we move −G− in graphs of the second type to all leaves. This way we get graphs of the same type in both cases. We get graphs with one vertex, one empty loop attached to it, all leaves are also attached to this vertex, and there is −G− on one of the leaves. One can easily check that the coefficient of the graphs with −G− at the i-th leaf is equal to I⊔J⊔{i}={1,...,n} τak〉1〈τ0τai τak〉0 + 〈τ 20 τak〉0 = 〈τai+1 k 6=i τak〉1 TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 27 It is exactly the unique graph contributing to the i-th summand of the right hand side of Equation (58), and the coefficient is right. This proves that special case. 7.2.3. Consider g ≥ 2. Let i=1 ai = 3g + n − 4. In this case, the statement that for any v1, . . . , vn ∈ H , 〈τa1(v1) . . . τai−1(vi−1)τai(Q(vi))τai+1(vi+1) . . . τan(vn)〉g = 〈τa1(v1) . . . τai−1(vi−1)τai+1(G−(vi))τai+1(vi+1) . . . τan(vn)〉g, is immediately reduced to 0 = 0; it is a simple corollary of Lemma 1. 7.3. Proof of Main Lemma. Consider the left hand side of Equa- tion (49). As usual, using the Leibniz rule for Q and the property that [Q,G−G+] = −G−, we can remove all Q, but then we must change one of [G−G+] on edges to −[G−]. Let us cut out the peaces of graphs that includes this edges with −[G−], all empty loops, leaves and halves of heavy edges attached to the ends of this special edge. Since we consider the sum over all possible graphs contributing to correlators, these small pieces can be gathered into groups according to the type of the rest of the initial graph. Each group forms exactly one of the special cases studied above. So, we know that −G− should jump either to one of the leaves or to one of the heavy edges attached to the ends of its edge. In the first case, we get exactly the graphs in the right hand side of Equation (49); in the second case, we get zero. One can easily check that we get the right coefficients for the graphs in the right hand side of Equation (49). This proves the lemma. 8. Proof of Theorem 3 8.1. Equivalence of expression in graphs. Consider the expres- sion in correlators corresponding to a ψ-κ-stratum as it is described in Section 2.3.2. To each vertex of the corresponding stable dual graph we assign the sum of graphs that forms correlator in the sense of Sec- tion 4.4. The leaves of these graphs corresponding to the edges of the stable dual graph (nodes) are connected in these pictures by edges with [Π0] (the restriction of the scalar product toH0). We call the edges with [Π0] “white edges” and mark them in pictures by thick white points, see (25). The axioms of cyclic Hodge algebra imply a system of linear equa- tions for the graphs of this type. In particular, it has appeared that playing with this linear equations we can always get rid of white edges in the sum of pictures corresponding to a stable dual graph, see [22, 31, 32]. However, previously it was just an experimental fact. Now we can show how it works in general. 28 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG The numerous examples of the correspondence between stable dual graphs and graphs expressions in cyclic Hodge algebras and also of the linear relations implied by the axioms of cyclic Hodge algebra are given in [22, 31, 32]. Below, we explain how one can represent the expression in correlators corresponding to a ψ-κ-stratum in terms of graphs with only empty and heavy edges and with no white edges. The unique tool that we need is Lemmas 4 and 5 proved above. 8.2. The simplest example. Consider a stable dual graph with two vertices and one edge connecting them: g1 g2 ψa1 . . . ψan1 ψb1 . . . ψbn2 κu1,...,ul1 κv1,...,vl2 ψa0 ψb0 The corresponding expression in correlators is (62) 〈τa0(ej1) τai(e•) τui(e1)〉g1· ηj1j2 · 〈τb0(ej2) τbi(e•) τvi(e1)〉g1 (here we denote by e• an arbitrary choice of ei ∈ H0). It is convenient for us to rewrite this expression as (63) 〈τa0(xα1) τai(e•) τui(e1)〉g1· α1α2 · 〈τb0(xα2) τbi(e•) τvi(e1)〉g1, TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 29 where {xα} is the basis of the whole H . Using the fact that Π0 = Id−QG+ −G+Q and applying Lemma 5, we obtain (64) 〈τa0(xα1) τai(e•) τui(e1)〉g1· α1α2 · 〈τb0(xα2) τbi(e•) τvi(e1)〉g1 = 〈τa0(xα1) τai(e•) τui(e1)〉g1· [Id]α1α2 · 〈τb0(xα2) τbi(e•) τvi(e1)〉g1 − 〈τa0+1(xα1) τai(e•) τui(e1)〉g1· [G−G+] α1α2 · 〈τb0(xα2) τbi(e•) τvi(e1)〉g1 − 〈τa0(xα1) τai(e•) τui(e1)〉g1· [G−G+] α1α2 · 〈τb0+1(xα2) τbi(e•) τvi(e1)〉g1 In all three summands of the right hand side we still have two correla- tors, whose leaves corresponding to the nodes are connected by some special edges. But now the connecting edge is either marked by [Id] (an empty edge) or by [G−G+] (an ordinary heavy edge). So, this way we get rid of the white edge in this case. Informally, in terms of pictures, we can describe Equation (64) as When we put ψ, we mean that we add one more ψ-class at the node at the corresponding branch of the curve. Dashed circles denote corre- lators. 8.3. Example with two nodes. Now we consider an example of stra- tum, whose generic point is represented by a three-component curve. Again, we allow arbitrary ψ-classes at marked points and two branches at nodes. 30 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG We perform the same calculation as above, but now we explain it in terms of informal pictures from the very beginning. So, the first step is the same as above: =(66) Then we apply Lemma 4 to each of the summands in the right hand side: =(67) We take the sum of these three expressions, and we see that all pictures where we have edges with [G−] and [G+] are cancelled. So, we get an TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 31 expression for the sum of graphs representing the initial stratum in terms of graphs with only empty and heavy edges. 8.4. General case. The general argument is exactly the same as in the second example. In fact, this gives a procedure how to write an expression in graphs with only empty and heavy edges (and no white edges) starting from a stable dual graph. Let us describe this proce- dure. Take a stable dual graph corresponding to a ψ-κ-stratum of dimen- sion k in Mg,n. First, we are to decorate it a little bit. For each edge, we either leave it untouched, or substitute it with an arrow (in two possible ways). At the pointing end of the arrow, we increase the number of ψ-classes by 1. Each of these graphs we weight with the inversed order of its automorphism group (automorphisms must pre- serve all decorations) multiplied by (−1)arr, where arr is the number of arrows. Consider a decorated dual graph. To each its vertex we associate the corresponding correlator of cyclic Hodge algebra (we add new leaves in order to represent κ-classes). Then we connect the leaves corresponding to the nodes either by empty edges (if the corresponding edge of the decorated graph is untouched) or by heavy edges (if the corresponding edge of the dual graph is decorated by an arrow). It is obvious that the number of heavy edges in the final graphs is equal to k. 8.5. Coefficients. We can simplify the resulting graphs obtained in the previous subsection. First, we can contract empty edges (as much as it is possible; it is forbidden to contract loops). Second, we can remove leaves added for the needs of κ-classes. Indeed, each such leaf is equipped with a unit of H , so it doesn’t affect the contraction of tensors corresponding to a graph. Moreover, when we remove all leaves corresponding to κ-classes, we still have graph with at least trivalent vertices. Otherwise, this graph is equal to zero, c. f. arguments in the proofs of string and dilaton equations. So, we obtain final graphs that have the same number of heavy edges as the dimension of the initial ψ-κ-stratum, the same number of leaves as the initial dual graph, and some number of empty loops, at most one at each vertex. The exceptional case is when k = 0; in this case we obtain only one graph, with one vertex, n leaves, and g empty loops. In the first case, let us turn a graph like this into a stable dual graph. Just replace its vertices with no empty loops by vertices of genus zero, vertices with empty loops by vertices of genus one, heavy edges are edges, and leaves are leaves. There are no ψ- or κ- classes. It is obvious that the codimension of the stratum corresponding to this dual graph is k. Indeed, in this case it is just the number of nodes. 32 A. LOSEV, S. SHADRIN, AND I. SHNEIBERG So, to each ψ-κ-stratum X of dimension k > 0 we associate a linear combination ciYi of strata of codimension k with no ψ- or κ-classes, whose curves have irreducible components of genus 0 and 1 only. Proposition 1. We have ci = X · Yi. Proof. We prove it two steps. First, consider a one-vertex stable dual graph with no edges (just a correlator). In this case, the intersection numberX ·Yi is just by definition ci = V (Γi)P (Γi), where Γi is the cyclic Hodge algebra graph that turns into Yi via the procedure described above. Then, consider a stable dual graph with one edge. It is the intersec- tion of the one-vertex stable dual graph with an irreducible component of the boundary. For a given Yi, this component of the boundary either intersects it transversaly, or we have an excessive intersection. In the first case, the corresponding node is not represented in Yi. This means that in Γi it should be an empty edge. In the second case, this node is one of the nodes of Yi, so it should be a heavy edge of Γi. Also, it is an excessive intersection, so we are to add the sum of ψ-classes with the negative sign at the marked points (half-edges) corresponding to the node, see [11, Appendix]. Exactly the same argument works for an arbitrary number of nodes, we just extend it by induction. � In the case of k = 0, we get just one final graph with coefficient c. Proposition 2. If k = 0, the coefficient of the final graph is equal to the number of points in the initial ψ-κ-stratum. Proof. If k = 0, this means that each of the vertices of the initial stable dual graph also has dimension 0, and the corresponding correlator of cyclic Hodge algebra is represented by one one-vertex graph with no heavy edges. Also this means that each edge of the initial stable dual graph is replaced in the algorithm above by an empty edge. So, we can think that we just work with the correlators of the Gromov-Witten theory of the point. In this case Proposition becomes obvious. � Now we deduce Theorem 3 from these propositions. 8.6. Proof of Theorem 3. Consider the system of subalgebras (70) RH∗1 (Mg,n) ⊂ RH ∗(Mg,n) of the cohomological tautological algebras of Mg,n generated by strata with no ψ- or κ-classes and with irreducible curves of genus 0 and 1 only. Let L be a linear combination of ψ-κ-strata of dimension k in Mg,n. Then the expression in correlators of cyclic Hodge algebras correspond- ing to L is equaivalent to a sum of some graphs with coefficients equal to the intersection of L with classes in RHk1 (Mg,n). TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 33 So, if the class of L is equal to zero, then the corresponding equation (and also the whole system of equations that we described in Sec- tion 2.3.3) for correlators of cyclic Hodge algebra is valid. Theorem is proved. 8.6.1. Remark. Evidently, RH∗1 (Mg,n) is a module over RH ∗(Mg,n). Also it is obvious that RH∗1 (Mg,n) is closed under pull-backs and push- forwards via the forgetful morphisms. This explains why it was enough to make only one check in the simplest case in order to get the system of equations in [22, 32] (cf. an argument in the last section in [22]). 8.7. An interpretation of Propositions 1 and 2. From the point of view of the theory of Zwiebach invariants, both propositions look very natural. Indeed, we try to give a graph expression for the integral of an induced Gromov-Witten form multiplied by a tautological class X . Since we know that we are able to integrate only degree zero parts of induced Gromov-Witten invariants, we should just take the sum over all graphs that correspond to the strata of complimentary dimension in RH∗1 (Mg,n). The coefficients are to be the intersection numbers of these strata with X . 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[32] S. Shadrin, I. Shneiberg, Belorousski-Pandharipande relation in dGBV alge- bras, J. Geom. Phys. 57 (2007), no. 2, 597–615. http://arxiv.org/abs/math/0601760 http://arxiv.org/abs/math/0508384 http://arxiv.org/abs/hep-th/0610326 http://arxiv.org/abs/math/0101147 http://arxiv.org/abs/math/0611912 http://arxiv.org/abs/math/0507106 TAUTOLOGICAL RELATIONS IN HODGE FIELD THEORY 35 [33] I. Shneiberg, Topological recursion relations in M2,2, to appear in Funct. Anal. Appl. (2007). [34] U. Tillmann, Vanishing of the Batalin-Vilkovisky algebra structure for TCFTs, Comm. Math. Phys. 205 (1999), no. 2, 283-286. [35] E. Witten, Two dimensional gravity and intersection theory on moduli space. Surveys in Differential Geometry, vol. 1 (1991), 243–310. [36] E. Witten, Chern-Simons gauge theory as a string theory, The Floer memorial volume, 637–678, Progr. Math. 133, Birkhaeuser, Basel, 1995. [37] B. Zwiebach, Closed string field theory: quantum action and the Batalin- Vilkovisky master equation, Nuclear Phys. B 390 (1993), no. 1, 33–152. Institute for Theoretical and Experimental Physics, Bolshaya Che- remushkinskaya 25, Moscow, 117218, Russia. E-mail address : [email protected] Department of Mathematics, University of Zurich, Winterthurer- strasse 190, CH-8057 Zurich, Switzerland. E-mail address : [email protected] Department of Algebra, Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, GSP, Moscow, 119899, Rus- E-mail address : [email protected] 1. Introduction 2. Gromov-Witten theory 3. Zwiebach invariants 4. Construction of correlators in Hodge field theory 5. String, dilaton, and tautological relations 6. Vanishing of the BV structure 7. Main Lemma 8. Proof of Theorem 3 References

0704.1002

SBF: multi-wavelength data and models

**FULL TITLE** ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION** **NAMES OF EDITORS** SBF: multi-wavelength data and models M. Cantiello1,2, G. Raimondo2, J.P. Blakeslee1, E. Brocato2, M. Capaccioli3 Abstract. Recent applications have proved that the Surface Brightness Fluc- tuations (SBF) technique is a reliable distance indicator in a wide range of dis- tances, and a promising tool to analyze the physical and chemical properties of unresolved stellar systems, in terms of their metallicity and age. We present the preliminary results of a project aimed at studying the evolutionary properties and distance of the stellar populations in external galaxies based on the SBF method. On the observational side, we have succeeded in detecting I-band SBF gra- dients in six bright ellipticals imaged with the ACS, for these same objects we are now presenting also B-band SBF data. These B-band data are the first fluctuations magnitude measurements for galaxies beyond 10 Mpc. To analyze the properties of stellar populations from the data, accurate SBF models are essential. As a part of this project, we have evaluated SBF magnitudes from Simple Stellar Population (SSP) models specifically optimized for the purpose. A wide range of chemical compositions and ages, as well as different choices of the photometric system have been investigated. All mod- els are available at the Teramo-Stellar Populations Tools web site: www.oa- teramo.inaf.it/SPoT. We have measured B- and I-band SBF magnitudes for 6 elliptical galax- ies observed with the ACS camera on board of HST: NGC1407, NGC3258, NGC3268, NGC4696, NGC5322 and NGC5557. Concerning I-band images, their high S/N ratio allowed us to obtain SBF measurements in different regions of the galaxies – 5 concentric annuli (Cantiello et al. 2005). On the contrary, the B-band images have low S/N (∼1), and SBF amplitudes can be measured only in one single annulus. The reliability of these B-band measurements has been verified via numerical simulations, by using a procedure which is able to reproduce realistic images of elliptical galaxies, including the stellar SBF signal. The general lack of B-band SBF data hampered up to now a detailed com- parison with models, our observational data represent the first sample of B- and I-band SBF measurements for a fair sample of distant galaxies. Figure 1 (left panels) shows the comparison of absolute SBF magnitudes versus (B-I)0 color data with SSP models from the Teramo Stellar Populations Tools group (SPoT models, Raimondo et al. 2005). SBF and color data appear generally well reproduced by means of standard SSP models in the M̄I vs. (B-I)0 panel. However, there is a considerable mismatch between SBF models and data for Dep. of Physics and Astronomy, Washington State University, Pullman, WA 99164 INAF-Oss. Astronomico di Teramo, Via M. Maggini, 64100, Teramo, Italy INAF-Oss. Astronomico di Capodimonte, Vicolo Moiariello 16, 80131, Napoli, Italy http://arxiv.org/abs/0704.1002v1 2 Cantiello M. et al. some objects in the M̄B vs. (B-I)0 panel. Such disagreement does not depend on the distance modulus adopted to estimate the absolute SBF magnitudes, in fact the same mismatch is present also in the distance-free SBF-color vs. color (B-I)0 (Cantiello et al. 2006, ApJ submitted). In addition, adopting other standard SSP models from literature (e.g. Blakeslee et al. 2001) or also non-standard SSP models (e.g. alpha enhanced models) the disagreement is not removed. Figure 1. Left panels: SBF absolute magnitudes vs. the (B-I)0 color derived from HST data (full dots). Full squares mark the only two other galaxies with literature data. SSP models are from the SPoT group for the labeled chemical compositions and 2 Gyr ≤ t ≤ 14 Gyr (symbols of increasing size mark older ages). Right panels: same data as left panels but compared to CSP models. One possible solution seems to be the use of Composite Stellar Populations (CSP). In Figure 1 (right panels) we compare the Blakeslee et al. (2001) CSP models with the present data. These CSP models are obtained combining SSP models in such a way to mimic, at least approximately, the evolution of an elliptical galaxy. With these models the disagreement between SBF data & models disappears, as it is completely accounted for by CSP with a fraction of old and metal-poor (t∼ 14 Gyr, [Fe/H]∼-1.3) stars as high as 8%, combined with a dominant contribution from an old and metal rich stellar component. In conclusion, our data seem to show that while the integrated properties of some galaxies might be well interpreted within the scenario of classical SSP models, there are few objects whose observational properties can only be inter- preted by means of more complex stellar populations systems. In this view, SBF and SBF colors, coupled with classical photometric data appear to be a very in- teresting tool to understand the properties of the unresolved stellar systems in distant galaxies. References Blakeslee, J.P., Vazdekis, A., Ajhar, E. 2001, MNRAS, 320, 193 Cantiello, M. et al. 2005, ApJ, 634, 239 Raimondo, G., et al. 2005, AJ, 130, 2625

0704.1003

On the choice of coarse variables for dynamics

Microsoft Word - coarse_variables_preprint.doc On the choice of coarse variables for dynamics Amit Acharya Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA 15213 Ph. – (412) 268 4566, Fax – (412) 268 7813, email- [email protected] Abstract Two ideas for the choice of an adequate set of coarse variables allowing approximate autonomous dynamics for practical applications are presented. The coarse variables are meant to represent averaged behavior of a fine-scale autonomous dynamics. Keywords: Coarse variables, autonomous dynamics, delay-reconstruction To appear in International Journal for Multiscale Computational Engineering 1. Introduction The goal of this paper is to lay out some ideas for the further development of an invariant- manifold-theory inspired computational approach to the problem of coarse-graining an autonomous system of ODE (fine system). Coarse variables are introduced as either functions of the fine state or time-averages of functions of the fine state. The objective is to come up with a closed theory of evolution for the coarse variables. In our past work [1, 2, 3] that we have called the method of Parametrized Locally Invariant Manifolds (PLIM), we have shown that this goal can be achieved in the context of hard nonlinear problems involving dissipation and/or oscillatory response. Strictly speaking, the achievement is, however, only partial in the sense that one requires knowledge of fine initial conditions to ensure a correct coarse-grained response even though the developed coarse equations are posed purely in terms of the coarse variables. This realization is intimately tied to the understanding of the emergence of memory effects in coarse response of an autonomous fine theory, a feature that can also be interpreted as stochastic effects in coarse response. In this paper, we examine two methods for the selection of a small number of coarse variables designed to allow for an autonomous coarse response, thus allowing unambiguous initialization of the coarse theory with information only on the coarse state. PLIM is an algorithm for developing approximate, but micro-dynamics-consistent, equations of evolution for user-defined coarse variables. The broad idea here is to calculate parametrizations of an appropriate collection of locally invariant manifolds of the fine dynamics a priori, with the coordinates being the coarse variables (observables). With a database devised to store this information, it becomes possible to define a closed dynamics for the coarse variables. PLIM as well as the Mori-Zwanzig Projection Operator Technique imply that if coarse variables are chosen ‘arbitrarily’, then more variables will, in general, be required to have an autonomous coarse theory that can be initialized unambiguously. Here, we propose two possible strategies for making a choice of such coarse variables. Connections of our work with, and a detailed review of, other multiscale strategies are provided in [1,2,3]. In particular, we note the work of Kevrekidis and co-workers [10,11,12] where the emphasis is not on deriving the form of the coarse equations at all but nevertheless make consistent predictions of coarse evolution based on a carefully crafted strategy for utilizing short bursts of microscopic simulations. The main point of the paper is how to start from a fine dynamics with some idea of what time- averaged coarse variables one might be interested in and proceed to augment this problem in a well-defined manner so that coarse response can indeed be computed. In a rough sense, it is specifically designed to deal with problems where the ‘as-received’ fine scale problem does not readily have an obvious slow macroscopic dynamics associated with it. The procedure tries to augment the problem definition so that an appropriate macroscopic dynamics becomes associated with the augmented microscopic problem. This work is mathematically formal. 2. Background The autonomous fine dynamics is defined as ( ) ( )( ) ( )0 . t H f t (1) f is an N -dimensional vector of fine degrees of freedom and H is a generally nonlinear function of fine states, denoted as the vector field of the fine dynamical system. Equation (1) 2 represents the specification of initial conditions. N can be large in principle, and the function H rapidly oscillating. Let Λ be a user-specified function of the fine states producing vectors with m components whose time averages over intervals of period τ can be measured in principle and are of physical interest. These time averages are considered as some of the coarse variables of interest. Let us also define the remaining list of ‘instantaneous’ coarse variables p with n components through the relationship ( )p fΠ= , (2) where Π is user-defined. Often, such variables may be required to incorporate external driving influences (e.g. loadings) on an assembly whose time-averaged behavior needs to be explored. Given the fixed time interval τ characterizing the resolution of coarse measurements in time, a coarse trajectory corresponding to each fine trajectory ( )f ⋅ is defined as the following pair of functions of time: ( ) ( )( ) ( ) ( )( ) c t f s ds p t f t ∫ (3) Roughly speaking, it is a closed statement of evolution for the pair ( ),c p that we seek. The statement is unambiguous only after we specify what sort of initial conditions we may want to prescribe. For the purpose of this section we assume that fine initial conditions are known with certainty. Then, the goal is to develop a closed evolution equation for ( ),c p , i.e. an equation that can be used for evolving ( ),c p without concurrently evolving (1), corresponding to fine trajectories out of a prescribed set of fine initial conditions. Clearly, ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) t f t f t t f t H f t Λ τ Λ ⎡ ⎤= + −⎢ ⎥⎣ ⎦ . (4) If we now introduce a forward trajectory ( )ff ⋅ corresponding to a trajectory ( )f ⋅ as ( ) ( ):ff t f t τ= + , (5) then ( ) ( )( )f f t H f t = . (6) Also, given an initial state f∗ we denote by f∗∗ the state defined as the solution of (1) evaluated at time τ . With these definitions in hand, we augment the fine dynamics (1) to ( ) ( )( ) ( ) ( )( ) t H f t t H f t (7) and apply invariant manifold techniques to (7). In detail, on an m n+ - dimensional coarse phase space whose generic element we denote as ( ),c p , we seek functions fG and G that satisfy the first-order, quasilinear partial differential equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 to I I lm n N f fk k K I f fk l K k l K II lm n N fk k K I fk l K k l K G G G H G H G c p f G G G H G H G c p f = = = = = = ⎫⎪∂ ∂⎛ ⎞ ∂ ⎪⎡ ⎤⎟⎜ − + = ⎪⎟⎜ ⎢ ⎥⎟ ⎪⎜ ⎣ ⎦⎝ ⎠∂ ∂ ∂ ⎪⎪ =⎬ ⎪∂⎛ ⎞∂ ∂ ⎪⎡ ⎤⎟⎜ ⎪− + =⎟⎜ ⎪⎢ ⎥⎟⎜ ⎣ ⎦⎝ ⎠ ⎪∂ ∂ ∂ ⎪⎭ (8) at least locally in ( ),c p -space. Assuming that we have such a pair of functions over the domain containing the point (9) defined from (1) and (3) which, moreover, satisfies the conditions fG c p f G c p f ∗ ∗ ∗∗ ∗ ∗ ∗ (10) it is easy to see that a local-in-time fine trajectory defined by ( ) ( ) ( )( ) ( ) ( ) ( )( ) f ft G c t p t t G c t p t (11) through the coarse local trajectory satisfying ( )( ) ( )( ) ( )( ) ( )( ) G c p G c p G c p H G c p ⎡ ⎤= −⎢ ⎥⎣ ⎦ ∑ (12) is the solution of (7) (locally). A solution pair ( ),fG G of (8) represents a parametrization of a locally invariant manifold of the dynamics (7). By a locally invariant manifold we mean a set of points in phase space such that the vector field of (7) is tangent to the set at all points. Thus, a trajectory of (7) exits a locally invariant manifold only through the boundary of the manifold. Also, note that if ( ),fG G and ( ),fG G are two solutions to (8) and (10) on an identical local domain in ( ),c p -space containing ( ),c p∗ ∗ and ( ) ( )( ),c p⋅ ⋅ and ( ) ( )( ),c p⋅ ⋅ are the corresponding coarse trajectories defined as solutions to (12), then local uniqueness of solutions to (7) implies ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) , : : , , : : , . f f f fG c t p t t t G c t p t G c t p t t t G c t p t = = = = = = (13) Thus, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ; locally in time, 0 0 ; 0 0 dc dc dp dp t t t t dt dt dt dt c c p p (14) from (12), and assuming ( ),c p and ( ),c p are continuous, ( ) ( ), ,c p c p≡ , locally. Hence, given any pair of mappings ,fG G satisfying (8) and (10) on a domain containing ( ),c p∗ ∗ , we consider (12) as the consistent, closed theory for the evolution of the coarse variables c . Obstruction to the construction of solutions to (8) is explored in [1], providing one reason for seeking multiple local solutions as implemented in [2]. This paper seeks to determine coarse functions such that multiple local solutions are not required, as far as possible. Notice that if equations (4)1 , (5), (7) are viewed as a system, then this system has a singular perturbation structure for τ large. We are interested in the evolution of the ‘ c ’ variables which are coarse/macroscopic variables. Section 3 briefly outlines a concrete and systematic procedure of how to append this system with more memory variables (in mechanics parlance, internal variables) so as to obtain an unambiguously initializable coarse dynamics. As explained in [1,2,3], an arbitrary choice of coarse variables (12) will not, in general, result in unique evolution of the coarse state out of a specified coarse initial condition. As well, conservation properties of the fine system are not expected to be preserved in the behavior of the time averages of fine variables. Thus, fixing attention on a fixed set of arbitrarily chosen coarse variables would imply what would seem like stochastic coarse response with dissipation. This is also the content of the main result of the Mori-Zwanzig projection Operator Technique within which Langevin dynamics can be derived. In this work, we propose a different strategy – we would like to start with a particular physically motivated set of coarse variables, but then would like to augment this set with more appropriately chosen variables so that the augmented set displays autonomous response. These extra variables effectively are memory (delay) variables corresponding to the original set of coarse variables. In the next section, we outline a procedure for the selection of such extra variables and then the use of the PLIM methodology to set up the autonomous coarse response. 3. Variables for autonomous coarse response: The Delay Reconstruction Technique The main conceptual ingredient of this technique is Takens’s embedding theorem [6]: Theorem: Let M be a compact manifold of dimension m . For pairs ( ), yϕ , : M Mϕ → a smooth diffeomorphism and :y M R→ (reals) a smooth function, it is a generic property that the map ( ) y M RϕΦ +→ , defined by ( ) ( ) ( ) ( ) ( )( ) , , , , y x y x y x y xϕΦ ϕ ϕ= is an embedding; by ‘smooth’ we mean at least 2C . Practically, Takens’s theorems suggest, and were motivated by the idea, that a single measurable signal (the function y ) of a complicated, possibly high-dimensional, dynamics (the mapping ϕ ), can in principle reveal all qualitative features of the underlying dynamics through the study of the delay-reconstruction map. Thus the delay reconstruction technique has been used by workers to make statements about qualitative features of dynamics. Of particular interest is the work in [4] and [5] where an algorithm for a systematic unfolding of delay-reconstructed trajectories is introduced, corresponding to an autonomous original dynamics. Essentially, starting from a one-component delay reconstruction of a trajectory of the original dynamics, more delay components are added till the point where the trajectory in delay-reconstruction space has no self-intersections. The number of components required is then declared to be representative of the number 2m of the theorem. Ding et al. [7] show that provided the function y satisfies certain smoothness assumptions, the correlation dimension of the delay-reconstructed signal with progressively more components hits a plateau when it becomes just greater than the correlation dimension of the attractor of the original dynamics. On the other hand, physical intuition suggests that any kind of measuring device acts as a filter that cannot measure variations below its resolution. Thus if y were to represent a moving time average, then it could not possibly reveal all features of the original dynamics. In a deep physical sense, were this not the case, macroscopic physics would not be possible – time-averaged signals display much gentler and lower dimensional dynamics. It is this idea that we would now like to pursue. Suggestive practical examples of this feature of dynamics can be found in [4]. Consider Axiom A diffeomorphisms for which we have ergodicity on attractors. Consider an observable of the form ( ) ( ) y x x ′ = ∑ , (15) where we assume that ϕ takes values in k for some positive integer k . Due to ergodicity, this map has a constant value almost everywhere on a set of non-vanishing Lebesgue measure in k containing the attractor. Thus the set of points generated by the dynamics corresponding to this observable along almost all trajectories on the attractor has dimension 0 , whereas the original attractor has some finite, possibly large, dimension. If y′ (or each of its component functions) satisfied the smoothness hypotheses of Takens’s theorem, then this would be a contradiction, as can be seen by utilizing a theorem of Eckmann and Ruelle [8] related to the determination of correlation dimension. Of course, it does not, since the value of y′ at a fixed point of ϕ is the fixed point itself whereas for an evaluation at a slight perturbation off the fixed point, the value is the ergodic average implying that the observable is, in all likelihood, discontinuous, let alone smooth. If we step back from an infinite sum as in (15) and perform finite sums with large N for realistic dynamics where the map ϕ may not have enough smoothness to be a diffeomorphism (e.g. the interatomic force for the Lennard-Jones potential contains odd powers of square roots of the interatomic distance), we do not expect the above argument to change drastically, in the sense of a discontinuous function being approached as a limit of smooth functions. Therefore, time-averaged observables, reflecting real measurements, maybe expected to display lower- dimensional dynamics. Moreover, such variables may be very useful for coarse behavior. Thus the idea with regard to definition of an autonomous coarse dynamics is to choose a set of physically motivated, time-averaged coarse variables corresponding to the original high- dimensional dynamics. An aperiodic, dense(on the attractor) original trajectory is then delay- reconstructed in terms of these variables plus added delay components, up to the point where the reconstructed trajectory has no self-intersections. The original set of coarse variables plus their delay counterparts form the augmented set of coarse variables that form an autonomous coarse dynamics. One now uses PLIM with this set of coarse variables to establish the coarse dynamics. The procedure outlined in Section 2 in dealing with one set of delay variables ( )ff is easily generalized to deal with multiple (sets of) delay variables with different delays. If indeed the delay-reconstructed trajectory has no self-intersection and the embedding dimension is small compared to the dimension of the attractor or its enveloping inertial manifold for the original dynamics, then this implies that we have a one-to-one mapping between sets of different dimension. In this connection, the work of Sauer et al. [9] should be noted, especially their filtered delay embedding theorem. Again there are strong hypotheses involved, and, interestingly, their Self-Intersection theorem does not put a lower-bound on the dimension of the self-intersection set, thus leaving a ray of hope with regard to the existence of a one-one map. Of course, it should be noted that one–to-one maps between sets of different dimensions can be continuous (in the small-to-large dimension direction) but nowhere differentiable, but this may be acceptable in the PLIM approximation methodology in approaching such functions as limits of piecewise-smooth continuous functions. As an example of the application of these ideas, one may consider the Frenkel-Kontorova model of a chain of atoms interacting through linear springs as well as with a nonlinear substrate potential. The chain may be assumed fixed at one end and a load applied at the other. Of interest is the time averaged stress strain (end-load-displacement) curve of this 1-d assembly. PLIM is applied to this problem setup in [2]. However, strictly speaking (and as mentioned in [2]), the problem there is solved only for coarse evolution corresponding to the fine trajectory starting from the stress free initial state. With the developments suggested in this paper, it may be hoped that the two average stress and strain coarse variables can be systematically augmented with more memory variables such that coarse evolution based on the developed theory is provably representative (at least in a formal sense) of coarse evolution corresponding to a large class of fine trajectories. Remark: We note here that in the conventional applications of the delay-reconstruction technique, non-generic or non-smooth observables leading to dimension change, e.g. Broomhead et al. [13] and Pecora and Carroll [14], are considered anomalous and to be avoided. Of particular interest to this work, Broomhead et al. [13] explicitly construct an example involving a nonrecursive filter, the inverse all-pole filter, that reduces dimension under delay-reconstruction of a particular signal. For the question of coarse-graining/averaging, however, it seems that it is precisely such non-generic and/or non-smooth functions that should be relevant. Indeed, it makes sense that a set of coarse observables executing autonomous macroscopic dynamics is special and cannot be chosen as any generic, smooth function(s). 4. Variables for autonomous coarse response: Adapted projections In this proposed approach to find coarse variables that evolve autonomously, we consider the following argument: let Π be a scalar function on the fine phase space, representing the definition of the sought-for coarse variable. Let f be a fine trajectory. Defining the coarse trajectory corresponding to f by c , we have ( ) ( )( ) ( ) ( )( ) ( )( ): c t f t c t f t H f t = ⇒ = . (16) Now, in general, given Π , many fine states will correspond to a single coarse state. Under the circumstances, one way to ensure an autonomous coarse response is to ensure that on any set of fine states where the evaluation of Π agrees, so should the right-hand-side of the c evolution. Mathematically, we have the following: we are interested in determining a function Π that has the following property. Let c be arbitrarily fixed. Then define ( ){ }: :cW f f cΠ= = . (17) Now require that Π satisfy ( ) ( ) cf H f Af on cW , (18) where cA is a constant depending on c but independent of f . Equations (17) and (18) together imply that a level set of ( )Π ⋅ should also be a level set of the function ( ) ( )H . One way to require this is to demand that ( ) H f f f ⎛ ⎞∂ ∂ ∂ ⎟⎜= ⎟⎜ ⎟⎜ ⎟⎜∂ ∂ ∂⎝ ⎠ (19) where λ is an arbitrary scalar. Choosing it to be a derivative of an arbitrary function of a single variable, i.e. of the form λ ϕ Π=∂ ∂ , we have the necessary condition that ( ) 0H ⎛ ⎞∂ ∂ ⎟⎜ − ⎟=⎜ ⎟⎜ ⎟⎜∂ ∂⎝ ⎠ . (20) Thus, if we now require Π to satisfy the first-order linear PDE on the fine phase space ( ) H , (21) it can be show by reversing the above arguments that ( )c cϕ= (22) would be the correct autonomous, coarse evolution equation for the coarse variable defined by Π obtained as a solution to (21), for arbitrary choices of ϕ in (21). Thus an entire class of coarse variables can be defined based on the choice of ϕ , which is a somewhat surprising result. It may be hoped that this class contains physically meaningful coarse variable definitions. More importantly, it perhaps suggests that the choice of appropriate coarse variables cannot be left completely unconstrained, and their definition requires physical guidance. Equation (21) is a linear first order PDE for Π . Characteristic curves for the PDE are solutions to the original set of fine system of ODE. They do not intersect (and meet only at fixed points) because of the autonomous nature and smoothness of the fine evolution. Thus shocks do not exist, and it appears reasonable to expect to numerically approximate the PDE without any further conditions. 5. Acknowledgments: It is a pleasure to acknowledge discussions with Luc Tartar and Noel Walkington. In particular, LT suggested the local characterization (19) and NW pointed out the backward-in-time uniqueness for smooth ODE. 6. References [1] Acharya, A., Parametrized invariant manifolds: A recipe for multiscale modeling? Computer Methods in Applied Mechanics and Engineering, 194, 3067-3089, 2005. [2] Acharya, A. and Sawant, A., On a computational approach for the approximate dynamics of averaged variables in nonlinear ODE systems: toward the derivation of constitutive laws of the rate type, Journal of the Mechanics and Physics of Solids, 54, 2183-2213. [3] Sawant, A., Acharya, A., Model reduction via Parametrized Locally Invariant manifolds: Some Examples, Computer Methods in Applied Mechanics and Engineering, 195, 6287-6311, 2005. [4] Abarbanel, H. D. I., Analysis of observed chaotic data, Institute for Nonlinear Science Series, Springer-Verlag. [5] Kennel, M. B., Abarbanel, H. D. I. False neighbors and false strands: a reliable minimum embedding dimension algorithm, Physical review E, 66, 026209, 2002. [6] Takens, F., Detecting strange attractors in turbulence, In: Dynamical Systems and Turbulence, Lecture Notes in Mathematics, 898, Warwick 1980, Ed. D. A. Rand, L. S. Young, 366-381, 1980. [7] Ding, M., Grebogi, C., Ott, E., Sauer, T., Yorke, J. A. Estimating correlation dimension from a chaotic time series: when does plateau onset occur? Physica D, 69, 404-424, 1993. [8] Eckmann J.-P., Ruelle, D. Ergodic theory of chaos and strange attractors, Reviews of Modern Physics, 57, 3, 617-656, 1985. [9] Sauer, T., Yorke, J. A., Casdagli, M. Embedology, Journal of Statistical Physics, 65, 3/4, 579-616, 1991. [10] Gear, C. W., Kevrekidis, I. G., Theodoropoulos, C., Coarse Integration/Bifurcation Analysis via Microscopic Simulators: micro-Galerkin methods, , Comp. Chem. Engng., 26, 941-963, 2002. [11] Kevrekidis, I. G., C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg and K. Theodoropoulos, Equation-free coarse-grained multiscale computation: enabling microscopic simulators to perform system-level tasks, Comm. Math. Sciences, 1(4), 715-762, 2003. [12] Kevrekidis, I. G., C. William Gear, G. Hummer, Equation-free: the computer-assisted analysis of complex, multiscale systems, A.I.Ch.E Journal, 50(7), 1346-1354, 2004. [13] Broomhead, D. S., Huke, J. P., Muldoon, M. R. Linear filters and non-linear systems, Journal of the Royal Statistical Society, Ser. B, 54(2), 373-382, 1992. [14] Pecora, L. M., Carroll, T. L. Discontinuous and non-differentiable functions and dimension increase induced by filtering chaotic data, Chaos, 6(3), 432-439, 1996.

0704.1004

Simulating CCD images of elliptical galaxies

**FULL TITLE** ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION** **NAMES OF EDITORS** Simulating CCD images of elliptical galaxies M. Cantiello1,2, G. Raimondo2, E. Brocato2, J.P. Blakeslee1, M. Capaccioli3 Abstract. We introduce a procedure developed by the “Teramo Stellar Pop- ulations Tools” group (Teramo-SPoT), specifically optimized to obtain realistic simulations of CCD images of elliptical galaxies. Particular attention is devoted to include the Surface Brightness Fluctua- tion (SBF) signal observed in ellipticals and to simulate the Globular Cluster (GC) system in the galaxy, and the distribution of background galaxies present in real CCD frames. In addition to the physical properties of the simulated objects - galaxy distance and brightness profile, luminosity function of GC and background galaxies, etc. - the tool presented allows the user to set some of the main instrumental properties - FoV, zero point magnitude, exposure time, etc. The light coming from distant galaxies includes a specific SBF signal, essen- tially correlated with the properties of the host stellar system (Tonry & Schnei- der 1988). The existence of these luminosity fluctuations is due to the statistical correlation between adjacent galaxy regions (pixels). Since its introduction, the SBF method has been used as a reliable distance indicator for elliptical galaxies (e.g. Tonry et al. 2001) and, more recently, as a tracer of stellar population properties (e.g. Cantiello et al. 2003, 2005; Raimondo et al. 2005, R05). In order to derive SBF magnitudes from CCD images of elliptical galaxies, very high quality CCD data are required. We have developed a tool to simulate CCD images of elliptical galaxies including the SBF signal and other properties of the galaxy - surface brightness profile, distance, color profiles, contamination of background galaxies, etc. Due to its statistical nature, a reliable simulation of the SBF signal needs: i) to accurately reproduce the details of the statistics governing the stellar SBF sig- nal, and ii) to take into account the presence of any other source of fluctuations. To include SBF signal in the simulations we use the Teramo-SPoT Single-burst Stellar Populations (SSP) models (R05, visit also the SPoT website www.oa- teramo.inaf.it/SPoT). These models are provided by computing a number Nsim of independent SSP simulations for a large range of ages and chemical composi- tions. The latter property of SPoT SSP models is at the base of our simulations of realistic galaxies: we start with a galaxy having an analytic Sersic r1/n profile, then, for each pixel [i,j] at the radius r∗ we substitute the analytic magnitude profile µth(r∗)[i,j] with the integrated magnitudes µsim as evaluated in one of the Nsim independent SSP simulations, having assumed 〈µsim〉 = µth(r∗). In this Dep. of Physics and Astronomy, Washington State University, Pullman, WA 99164 INAF-Oss. Astronomico di Teramo, Via M. Maggini, 64100, Teramo, Italy INAF-Oss. Astronomico di Capodimonte, Vicolo Moiariello 16, 80131, Napoli, Italy http://arxiv.org/abs/0704.1004v1 2 Cantiello et al. way the poissonian correlation between adjacent pixels is introduced, preserving the galaxy brightness profile. To be realistic, the simulation must also include the presence of GC and background galaxies - which, in addition, can strongly affect the fluctuations signal derived from the CCD. These sources are indeed included into our simu- lations according to their typical luminosity functions, i.e., the total luminosity function is assumed to be the sum of a power law for galaxies, and a gaus- sian distribution for the GC component. The characteristic parameters of these functions can be arbitrarily set by the user. Once the luminosity functions are randomly populated all the background galaxies are randomly distributed in the frame, while the GC spatial distribution is additionally convolved with an inverse power law centered on the galaxy. Finally, a uniform sky value is included, and the detector noise is added according to the readout-noise and gain values of the selected instrument. After the galaxy profile - including SBF -, the GC system, the background galaxies, and the detector noise properties have been properly chosen, the simu- lation can be carried out. The panels of Figure 1 show the frames associated to some of the steps described above, the final frame simulated, and the luminosity functions of GC and background galaxies. Figure 1. The first three panels of the figure (left to right) show the profile of the galaxy simulated, the frame of GC and background galaxies, and the sum of the previous two frames plus sky and noise, respectively. ACS camera properties are adopted for the instrumental characteristics. The last plot shows the luminosity functions adopted of the GC (short dashed line) and the background galaxies (long dashed line), and their sum (solid line). The capabilities of the procedure here briefly described makes it useful to simulate astronomical data for a wide range of applications. As a specific case we mention the use of this tool to simulate realistic runs at defined telescopes with the aim of measuring SBF. For example, we have applied this technique to simulate ISAAC@VLT Ks-band, and ACS@HST F814W-band (reported Figure 1) images of given ellipticals in order to evaluate the proper exposure times required to reach a defined S/N ratio for objects at different distances. References Cantiello, M., Raimondo, G., Brocato, E., Capaccioli, M. 2003, AJ, 611, 670 Cantiello, M., et al. 2005, ApJ, 634, 239 Raimondo, G., et al. 2005, AJ, 130, 2625 Simulating images of elliptical galaxies 3 Tonry, J. & Schneider, D.P. 1988, AJ, 96, 807 Tonry, J. et al. 2001, AJ, 546, 681

0704.1005

Constructions of Kahler-Einstein metrics with negative scalar curvature

CONSTRUCTIONS OF KÄHLER-EINSTEIN METRICS WITH NEGATIVE SCALAR CURVATURE Jian Song1 Ben Weinkove2 Johns Hopkins University Harvard University Department of Mathematics Department of Mathematics Baltimore, MD 21218 Cambridge, MA 02138 Abstract. We show that on Kähler manifolds with negative first Chern class, the sequence of algebraic metrics introduced by H. Tsuji converges uniformly to the Kähler-Einstein metric. For algebraic surfaces of general type and orbifolds with isolated singularities, we prove a convergence result for a modified version of Tsuji’s iterative construction. 1. Introduction AKähler manifold with negative first Chern class admits a unique Kähler- Einstein metric. This was shown by Yau in his seminal paper on the Calabi conjecture [Y1], and also independently by Aubin [A]. Yau later posed the question of whether, in general, Kähler-Einstein metrics could be obtained as a limit of algebraic metrics induced from embeddings into projective space [Y2]. Donaldson [D1] showed that if a polarized variety (X,L) with discrete automorphism group admits a constant scalar curvature Kähler metric, then there is indeed a sequence of algebraic ‘balanced metrics’ [Zh] converging to it. Donaldson’s proof makes use of the Tian-Yau-Zelditch expansion of the Bergman kernel [Ti], [Ze] (see also [C]) and Lu’s [L] computation of the sec- ond coefficient. Very recently, Donaldson [D2] has described how numerical approximations to these balanced metrics could be used to compute, to a good accuracy, explicit Kähler-Einstein metrics on certain varieties. Tsuji [Ts] has considered a different way of producing Kähler-Einstein metrics by algebraic approximations. He introduced a new iterative proce- dure on varieties of general type with the aim of describing (possibly sin- gular) Kähler-Einstein metrics. In the case when the first Chern class is The first-named author is on leave for the semester and is visiting MSRI, Berkeley, CA as a postdoctoral fellow. He is supported in part by National Science Foundation grant DMS 0604805. Part of this research was carried out while the second-named author was a short-term visitor at MSRI in January 2007. He is supported in part by National Science Foundation grant DMS 0504285. http://arxiv.org/abs/0704.1005v1 negative, Tsuji proved that his iteration converges, in a certain weak sense, to the Kähler-Einstein metric. In this paper, we give a uniform convergence result and describe how his procedure may be modified to obtain results in the case of algebraic surfaces of general type, and on orbifolds with isolated singuarities. We now describe Tsuji’s iteration. Let X be a compact Kähler mani- fold of complex dimension n with ample canonical bundle KX . Let ωKE =√ (gKE)ijdz i ∧ dzj be the Kähler-Einstein metric satisfying Ric(ωKE) = − ∂∂ logωnKE = −ωKE ∈ c1(X). Fix a Hermitian metric hKE on KX by setting hKE = (det gKE) Let m0 ≥ 1 be an integer such that Km0X is base point free and let hm0 be any Hermitian metric on Km0X . Define a sequence of Hermitian metrics hm on K X for m > m0 inductively as follows. Suppose that hm is a given Hermitian metric on KmX . To define hm+1, first define an inner product 〈 , 〉Tm+1 on the space of sections of Km+1X by 〈s, t〉Tm+1 = hm ⊗ s⊗ t, for s, t ∈ H0(X,Km+1X ), (1.1) where hm⊗s⊗t is regarded as a volume form onX. Now let (σ m+1, . . . , σ (Nm+1) m+1 ), for Nm+1+1 = dimH 0(X,Km+1X ), be an orthonormal basis of H 0(X,Km+1X ) with respect to this inner product. Then define the Hermitian metric hm+1 on Km+1X by hm+1 = (m+ n+ 1)! (m+ 1)! m+1 ⊗ σ Observe that this metric is independent of the choice of orthonormal basis. We have the following theorem on the convergence of the metrics hm, strengthening the result given in [Ts]. Theorem 1 Let X be a compact Kähler manifold with ample canonical bundle. Let hKE, ωKE and the sequence of Hermitian metrics hm be as above. There exists a constant C depending only on X and hm0 such that for m ≥ m0, C logm m hKE ≤ h1/mm ≤ e C logm m hKE. (1.2) Hence h m converges uniformly on X to hKE as m→ ∞. We now describe a modification of Tsuji’s iteration. Let β be a continu- ous function on a variety X with 0 ≤ β ≤ 1. Here we allow X to have mild singularities. We also remove the assumption that KX be ample. We only require that there exists an m0 ≥ 1 such that Km0X is base point free. Let hm0 be a Hermitian metric on K X . Given 0 < ε ≤ 1, we inductively define a sequence of Hermitian metrics hm,ε = hm,ε(β, hm0) on KX as follows. As- suming that hm,ε is given, define an inner product 〈 , 〉Tm+1,ε on the space of sections of Km+1X by 〈s, t〉Tm+1,ε = βεhm,ε ⊗ s⊗ t, for s, t ∈ H0(X,Km+1X ). Then define the Hermitian metric hm+1,ε on K hm+1,ε = (m+ n+ 1)! (m+ 1)! m+1,ε ⊗ σ m+1,ε where (σ m+1,ε, . . . , σ (Nm+1) m+1,ε ) is an orthonormal basis of H 0(X,Km+1X ) with respect to the inner product 〈 , 〉Tm+1,ε . We call hm,ε the modified Tsuji iteration. It depends on ε and β. First, we consider the case when X is an algebraic surface of general type. Let E = iEi be the sum of the nonsingular rational curves Ei of self intersection −1 (or (−1)-curves, for short) on X. Let τ : X → Xmin be a holomorphic map blowing down these curves onX, so that Xmin is a minimal surface of general type. Now if h is any Hermitian metric on KXmin, then Ω = h−1(z1, . . . , zn)( −1/2π)ndz1∧dz1∧ · · ·∧dzn∧dzn can be regarded as a volume form on Xmin. Using coordinates w i on X, there is a holomorphic section S−1 of the line bundle [E ] associated to E and vanishing of order one on E satisfying (τ∗h)−1 ⊗ |S−1|2 (w1, . . . , wn) dw1 ∧ dw1 ∧ · · · ∧ dwn ∧ dwn = τ∗Ω, for any such h. Xmin has no (−1)-curves, but may have (−2)-curves. Let f : Xmin → Xcan be its canonical map. Xcan is an surface with ample canonical bundle KXcan and, at worst, isolated orbifold singularities. By the orbifold version of the results of [Y1], [A] (see [K], for example), there exists an orbifold Kähler- Einstein metric ωKE on Xcan, with corresponding Hermitian metric hKE on KXcan . Define a Kähler metric on Xmin by ωmin = f ∗ωKE and a Hermitian metric on KXmin by hmin = f ∗hKE. Note that ωmin and hmin are not smooth in general, although hmin is continuous on Xmin (see [TZ]). Let C = be the sum of the (−2)-curves on Xmin and let S−2 be a holomorphic section of the line bundle [C], vanishing of order one on C. Fix a smooth Hermitian metric hC on [C], and assume that supXmin |S−2| = 1. Let β be the smooth function on X defined by β = τ∗|S−2|2hC , and let h∞ = τ ∗hmin. For this β and some initial Hermitian metric hm0 on K X , let hm,ε = hm,ε(β, hm0) be the sequence of Hermitian metrics in the modified Tsuji iteration as described above. Then we have the following result. Theorem 2 Let X be an algebraic surface of general type. With hm,ε = hm,ε(β, hm0) and S−1 as described above, for every sequence εj → 0, lim sup h1/mm,εj → h∞ ⊗ |S−1| −2, as j → ∞, almost everywhere on X. Note that since KX = τ ∗KXmin + [E ], we can regard h∞ ⊗ |S−1|−2 as a singular Hermitian metric on KX . We now consider the case whenX is an orbifold with isolated singularities with ample canonical bundle. Theorem 3 Let (X,ωKE) be a Kähler-Einstein orbifold with KX ample and only isolated singularities at points p1, . . . , pk. Let β be a continuous function on X satisfying 0 ≤ β ≤ 1 and β(x) = 0 if and only if x = pi for some i. Then, with hm,ε = hm,ε(β, hm0) as above, for every sequence εj → 0, lim sup h1/mm,εj → hKE, as j → ∞, almost everywhere on X. We end the introduction with a couple of remarks. 1. Tsuji’s iteration has some similarities to Donaldson’s TK -iteration (see [D2], section 2.2.2) which in the case of KX ample should yield a ‘canonically balanced metric’ using sections of a fixed power k of the canonical line bundle. As k → ∞, the limit of these metrics is expected to be the Kähler-Einstein metric. On the other hand, Tsuji’s method is a single iterative process. 2. Donaldson has suggested that the dynamical systems introduced in [D2] are likely to be discrete approximations to the Kähler-Ricci and Calabi flows. It would be interesting to know whether Tsuji’s iteration could be viewed in a similar light. The outline of the paper is as follows. Our main technique is the peak section method of [Ti]. This is described in section 2, and extended to orbifolds with isolated singularities (for related results on the Szegö kernel for orbifolds, see [S], [DLM], [P]). In sections 3 and 4 we prove the main theorems. 2. Peak sections Now suppose that (X,ω) be a compact Kähler manifold of complex di- mension n with ω ∈ c1(L) for an ample line bundle L. Fix a Hermitian metric h on L satisfying − ∂∂ log h = ω. Write 〈·, ·〉L2(hm) and ‖ · ‖L2(hm) for the L2 inner product and norm on H0(X,Lm) with respect to the Her- mitian metric hm and volume form 1 ωn. We use the following lemma from [Ti]. Lemma 2.1 There exists m1 > 0 depending only on X, L and h such that for every x0 ∈ X and m ≥ m1 there is a global holomorphic section sm,x0 of Lm satisfying the following. (i) |sm,x0 |2hm(x0) = 1 and ‖sm,x0‖2L2(hm) = (m+ n)! (1 + O(m−1)), where f(m) = O(m−1) means that |f(m)|m ≤ A for a constant A depending only on X, L and h. (ii) For t any holomorphic section of Lm which vanishes at x0, ∣〈sm,x0 , t〉L2(hm) ∣ ≤ B ‖sm,x0‖L2(hm)‖t‖L2(hm), with a constant B depending only on X, L and h. Proof We outline here the proof of part (i), since we will explicitly refer to this method in the orbifold case. For (ii) we refer the reader to [Ti]. Pick a normal coordinate chart (U, (z1, z2, . . . , zn)) for g centered at the point x0. Let η : [0,∞) → [0, 1] be a cut-off function satisfying η(t) = 1 for t ≤ 1 η(t) = 0 for t ≥ 1 and −4 ≤ η′(t) ≤ 0, |η′′(t)| ≤ 8. Define a weight function ψ, which for m sufficiently large is supported in U , by setting ψ(z) = (n+ 2)η am|z|2 am|z|2 , (2.1) for z ∈ U , and ψ ≡ 0 outside U , where am = m/(logm)2. A short calculation shows that √ ∂∂ψ(z) ≥ −C(n+ 2)amω, for |z| > 0. (2.2) Now let ψi be a decreasing sequence of smooth functions on X such that ψ = lim ψi and ∂∂ψi ≥ −C(n+ 2)amω, (2.3) where the constant C may be different from the one in (2.2). It follows that for sufficiently large m, ∂∂ψi − ∂∂ log hm +Ric(ω) ≥ ω. (2.4) Choose a local holomorphic section v of L so that |v|2h(x0) = 1 and (∂|v|2h)(x0) = 0. Let w be the smooth local section of Lm defined by w = ∂ am|z|2 We apply the L2 estimate of Hörmander [H] (cf. Proposition 1.1 of [Ti]) to the (1,0) form w: there exists a smooth global section u of Lm satisfying ∂u = w and |u|2hme−ψ |w|2hme−ψ Here we are using the fact that the weight function ψ can be approximated by ψi satisfying (2.4). Observe that w vanishes identically outside the region 2/am ≤ |z|2 ≤ 4/am, and that in U , |w|2hme−ψ ≤ Cam|vm|2hm. It follows that |u|2hme−ψ (logm)2 2/am≤|z|2≤4/am |vm|2hm From our choice of coordinates, |v|2h(z) = 1− |z|2 +O(|z|3), and so locally |vm|2hm )m(√−1 dz1 ∧ dz1 ∧ · · · ∧ dzn ∧ dzn. Hence |u|2hm (logm)2 ≤ C(logm)2n−2m− logm/2−n. (2.5) Now set sm,x0 = η am|z|2 vm − u. Since |u|2hme−ψ < ∞, we have u(z) = O(|z|2) by the definition of ψ. Hence |sm,x0 |2hm(x0) = 1. Calculate |sm,x0 |2hm am|z|2 |vm|2hm |u|2hm + 2Re am|z|2 vm, u . (2.6) From (2.5) the last two terms are O(m−q) for any q. For the first term, observe that |z|2≤2/am |vm|2hm am|z|2 |vm|2hm |z|2≤4/am |vm|2hm (2.7) We will make use of the following elementary lemma: Lemma 2.2 Fix b > 0 and q > 0. Then |z|2≤b/am (1−|z|2)mdz1∧dz1∧· · ·∧dzn∧dzn = (m+ n)! +O(m−q), where the term O(m−q) depends only on b, q and n. Then for any fixed b > 0 and for m sufficiently large, |z|2≤b/am |vm|2hm (m+ n)! (1 + O(m−1)). From (2.6) and (2.7) we obtain |sm,x0 |2hm (m+ n)! (1 + O(m−1)), as required. � Assume now that (X,ω) is a Kähler orbifold of complex dimension n with isolated orbifold singularities at points p1, . . . , pk. Then for each x ∈ {p1, . . . , pk} there is an open neighborhood Vx ⊂ X containing x, a finite subgroup Gx ∈ U(n), a Gx-invariant open neighborhood Ṽx of the origin in n and a projection map πx : Ṽx → Ṽx/Gx ∼= Vx with πx(0) = x. Let h be an orbifold Hermitian metric on an orbifold line bundle L with − ∂∂ log h = ω. We show the following. Lemma 2.3 There exists m1 > 0 depending only on X, L and h such that for each m ≥ m1 the following holds. Let x0 ∈ X. Then (i) If x0 satisfies (d(x0, pi)) 2 ≤ 1 , for some i ∈ {1, . . . , k}, where d( , ) is the distance function on X with respect to ω, then there is a global holomorphic section sm,x0 of L m satisfying |sm,x0 |2hm(x0) = 1 (m+ n)! ‖sm,x0‖2L2(hm) = |Gpi | +O(m−1). (ii) If x0 satisfies < (d(x0, pi)) , for some i ∈ {1, . . . , k}, where am = m/(logm) 2, then there is a global holomorphic section sm,x0 of L m satisfying |sm,x0 |2hm(x0) = 1 and 1 + O(m−1) ≤ (m+ n)! ‖sm,x0‖2L2(hm) ≤ C2 1 + O(m−1) for positive constants C1 and C2 depending only on |Gpi |. (iii) If x0 satisfies (d(x0, pi)) , for every i ∈ {1, . . . , k}, then there exists a global holomorphic section sm,x0 of L m satisfying |sm,x0 |2hm(x0) = 1 and (m+ n)! ‖sm,x0‖2L2(hm) = 1 + O(m Proof We will choose m1 ≫ 1 later in the proof, depending only on X, L and h. Let m ≥ m1. For (i), assume first that x0 ∈ X − {p1, . . . , pk}. We may assume without loss of generality that (d(x0, p1)) 2 ≤ 1 and d(x0, pi) ≥ c > 0 for i = 2, . . . , k for some uniform c. Dropping the subscript p1, we have a uniformizing coordinate system π : Ṽ → Ṽ /G ∼= V centered at p1 ∈ V , so that at 0 ∈ Ṽ , the metric g is the identity. The metric is G-invariant and smooth in Ṽ and has vanishing first derivatives at the origin. Set |G| = l. Since the singularity is isolated, the only fixed point of the action is 0 ∈ Ṽ , and so Ṽ − {0} is a l-fold cover of V − {p1}. The preimage of x0 under the map π consists of l distinct points which we will write as x̃1, . . . , x̃l ∈ Ṽ ⊂ Cn. We may assume that 0 < |x̃1|2 = |x̃2|2 = · · · = |x̃l|2 < Let η and ψ be the functions defined earlier in the smooth case, and let ψ̃ be a weight function on U given by ψ̃(z) = ψ(z − x̃i). Observe that ψ̃ is G-invariant, since ψ(z) is a function of |z|2 only. Hence ψ̃ can be regarded as a smooth function on X in the orbifold sense. Note also that ψ̃ is non-positive everywhere. We have −1∂∂ψ̃(z) ≥ −C(n+ 2)amω(z), for z ∈ Ṽ − {x1, . . . , xl}. Hence for sufficiently large m, with ψ̃j approximating ψ̃ as before, −1∂∂ψ̃j − ∂∂ log hm +Ric(ω) ≥ Let v be a local orbifold holomorphic section of L. We may assume without loss of generality that |v|2h(p1) = 1. Pulling back to Ṽ we have (∂|v|2h)(0) = 0. Let w be the smooth local section of Lm defined in Ṽ by am|z − x̃i|2 Notice that w is G-invariant. If m1 is sufficiently large then |x̃i − x̃j|2 ≤ for all i, j and it follows that w vanishes identically in the regions |z − x̃i|2 ≤ |z − x̃i|2 ≥ for all i. In addition, w vanishes in the regions |z|2 ≤ 1 |z|2 ≥ 16 It follows that w descends to a smooth global orbifold section of Lm and ψ̃ is uniformly bounded whenever w is not identically zero. Hence in V , |w|2hme−ψ̃ ≤ Cam|vm|2hm . |w|2hme−ψ̃ and we can apply the orbifold version of Hörmander’s estimates to obtain a smooth global orbifold section u of Lm satisfying ∂u = w and |u|2hme−ψ̃ |w|2hme−ψ̃ (logm)2 1/4am≤|z|2≤16/am |vm|2hm We can write |v|2h(z) = 1 − |z|2 + O(|z|3) and it follows that, by a similar argument as in the smooth case, |u|2hm ≤ C(logm)2n−2m− logm/2−n. Now set sm,x0(z) = am|z − x̃i|2 vm(z)− u(z), so that sm,x0 is a global holomorphic orbifold section of L m. Notice that since |u|2hme−ψ̃ it follows from the definition of ψ̃ that u(z − x̃i) = O(|z − x̃i|2) for each i. Hence u(x0) = 0 and since |sm,x0 |2hm(x0) = |vm|2hm(x0), we have 1 ≥ |sm,x0 |2hm(x0) ≥ (1− )m = em log(1−2/m 2) = 1−O(m−q), (2.8) for any q. Calculate, remembering that Ṽ −{0} is an l-fold cover of V −{p1}, |sm,x0 |2hm am|z − x̃i|2 |vm|2hm am|z − x̃i|2 vm, u |u|2hm . (2.9) The last two terms are O(m−q) for any q. For the first term, observe that |z|2≤1/4am |vm|2hm am|z − x̃i|2 |vm|2hm |z|2≤16/am |vm|2hm . (2.10) From Lemma 2.2 we obtain |sm,x0 |2hm l(m+ n)! (1 + O(m−1)). Then from (2.8) we obtain the required section by rescaling. The case when x0 is one of the singular points pi is easier, since we can take the weight function to be ψ, which is of course G-invariant. The proof follows as in the smooth case, except that a factor of l arises when estimating the integral of |sm,x0 |2hm . We now consider case (ii). We divide this into two parts: < (d(x0, p1)) ≤ (d(x0, p1))2 < for a constant A≫ 0 depending on l to be determined later. For (a), we can use almost the same argument as in (i). The only differ- ence is that (2.8) becomes 1 ≥ |sm,x0 |2hm(x0) ≥ c > 0, for a constant c depending on A. The required estimate follows after scaling sm,x0 . For (b) we argue as follows. Using the notation above, we work in the coordinate patch Ṽ and consider the same weight function ψ̃. Let v1 be a local holomorphic section of π∗L over Ṽ with the property that |v1|2h(z) = 1− |z − x̃1|2 +O(|z − x̃1|3). Observe that v1 is not G-invariant. Writing the elements of G as γ1, . . . , γl with γ1 the identity element, we set vi = γ i v1, so that |vi|2h(z) = 1− |z − x̃i|2 +O(|z − x̃i|3). Now define a local G-invariant section ŵ of Lm over Ṽ by ŵ(z) = am|z − x̃i|2 vmi (z). We have ≤ |x̃i − x̃j |2 ≤ , (2.11) and by a similar argument as in case (i), |ŵ|2hme−ψ̃ Hence we can obtain a smooth global orbifold section û of Lm satisfying ∂û = ŵ and |û|2ω ≤ C(logm)2n−2m− logm/2−n. Let sm,x0 be the global holomorphic orbifold section of L m given by sm,x0(z) = am|z − x̃i|2 vmi (z)− û(z). Notice that û(x0) = 0. For i 6= j we have |vmi |2hm(x̃j) ≤ e−A/8, using (2.11), and if A is sufficiently large depending only on l it follows that 1 ≥ |sm,x0 |2hm(x0) ≥ c > 0, (2.12) for c depending only on l. Now |sm,x0 |2hm am|z − x̃i|2 am|z − x̃i|2 vmi , û |û|2hm . (2.13) As before, the last two terms are O(m−q) for any q, and (m+ n!) (1 + O(m−1)) ≤ 1 am|z − x̃i|2 l(m+ n!) (1 + O(m−1)), (2.14) for a constant c′ > 0 depending only on l. Combining (2.12), (2.13) and (2.14) completes part (b) of (ii). For case (iii) we can avoid the singularities using the same argument as in the smooth case. � Remark 2.1 The result of Lemma 2.3.(ii) is clearly not sharp. It would be interesting to know what the optimal estimates are in this case. 3. Convergence of Tsuji’s iteration In this section we give a proof of Theorem 1. We begin with a simple and well-known observation, which we will be useful later. Let X be any set, and let H be a finite dimensional vector subspace of the vector space of functions from X to C. Suppose that H is equipped with an inner product 〈 , 〉H . For any orthonormal basis (v0, . . . , vN ) of H, define a function ρ : X → R by ρ(x) = i=0 |vi|2(x). Note that the function ρ is independent of the choice of orthonormal basis. Now fix x ∈ X . Then it is possible to choose an orthonormal basis (v0, . . . , vN ) such that vi(x) = 0 for i = 1, 2, . . . , N . The observation is that ρ(x) = sup |v|2(x) ∣ v ∈ H, ‖v‖H = 1 = |v0|2(x). (3.1) We now turn to the proof of Theorem 1. Notice that, in addition to a Hermitian metric hm on K X for each m ≥ m0, we have defined by (1.1) an inner product 〈 , 〉Tm on H0(X,KmX ) for each m ≥ m0 + 1. Also, from the Hermitian metric hKE on KX we have the L 2 inner product on H0(X,KmX ) given by hmKE and ωnKE. We will denote this inner product simply by 〈·, ·〉KE. We will use Lemma 2.1 on the existence of peak sections to prove the following: Lemma 3.1 Let m1 be the constant of Lemma 2.1 for L = KX , h = hKE. Assume m1 ≥ m0. There exists A depending only on X such that for all m ≥ m1, hm1KE ≤ hm ≤ sup hm1KE (3.2) Proof In the course of this proof, the constant A may change from line to line. We will prove the upper bound on hm first. We use induction. Obviously, the inequality holds for m = m1. Let Cm = sup and assume that hm ≤ CmhmKE. Notice that for any section t of H 0(X,Km+1X ), ‖t‖2Tm+1 ≤ Cm ‖t‖ Fix a point x0 ∈ X. Let sm+1,x0 ∈ H0(X,Km+1X ) be a peak section as constructed in Lemma 2.1. Then calculate, from the definition of hm+1, (m+ 1)! (m+ n+ 1)! hm+1(x0) ≤ ‖sm+1,x0‖ |sm+1,x0 | hm+1KE (x0) ≤ Cm ‖sm+1,x0‖ hm+1KE (x0) (m+ 1)! (m+ n+ 1)! 1 + O hm+1KE (x0), and it follows that hm+1 ≤ sup We turn now to the lower bound for hm. Again we use induction and assume that hm ≥ DmhmKE for Dm = inf Fix x0 inX. Let (σ m+1, . . . , σ (Nm+1) m+1 ) be an orthonormal basis of H 0(X,Km+1X ) with respect to the inner product 〈 , 〉Tm+1 . We may assume that m+1(x0) = 0 for i = 1, . . . , Nm+1. Observe that 1 = ‖σ(0)m+1‖2Tm+1 ≥ Dm‖σ m+1‖2KE. Then (m+ 1)! (m+ n+ 1)! hm+1(x0) = h KE (x0) |σ(0)m+1|2hm+1 ≥ Dmhm+1KE (x0) ‖σ(0)m+1‖2KE |σ(0)m+1|2hm+1 . (3.3) Now let (τ m+1, . . . , τ (Nm+1) m+1 ) be an orthonormal basis of H 0(X,Km+1X ) with respect to the inner product 〈 , 〉KE. As before we may assume that m+1(x0) = 0 for i = 1, . . . , Nm+1. Then it follows that if t is any section of H0(X,Km+1X ), we have ‖t‖2KE ≤ |τ (0)m+1| (x0). (3.4) Hence (m+ 1)! (m+ n+ 1)! hm+1(x0) ≥ Dmhm+1KE (x0) |τ (0)m+1|2hm+1 . (3.5) We consider again the peak section sm+1,x0 . Define real numbers a0, . . . , aNm+1 sm+1,x0 = Then, using the second part of Lemma 2.1, a2i = sm+1,x0 , ‖sm+1,x0‖KE and so a2i ≤ A ‖sm+1,x0‖2KE (m+ 1)2 Now notice that a20 = ‖sm+1,x0‖2KE − ≥ ‖sm+1,x0‖2KE 1−A 1 (m+ 1)2 Now since |sm+1,x0 |2hm+1 (x0) = 1, we have |τ m+1|2hm+1 (x0) = 1/a 0. Then from (3.5) we have hm+1(x0) ≥ Dmhm+1KE (x0) and the required lower bound follows. � From this, we can prove Theorem 1. Proof of Theorem 1 Raise (3.2) to the power 1/m. For the upper bound of Theorem 1, observe that for a constant C depending only on A. The lower bound follows similarly. 4. The modified iteration We give a proof of Theorem 2. We omit the proof of Theorem 3, since it is simpler and follows along the same lines. We first prove a convergence result on the minimal surface Xmin of general type. Consider a Hermitian metric hm0 on K and write β = |S−2|2hC , for |S−2| as in the introduction. Recall that hmin is the Hermitian metric on KXmin given by f ∗hKE, for hKE the Hermitian metric on KXcan corresponding to the Kähler-Einstein metric ωKE. Consider the sequence of metrics hm,ε = hm,ε(β, hm0) on K Theorem 4.1 For every sequence εj → 0, lim sup h1/mm,εj → hmin, as j → ∞, almost everywhere on Xmin. To prove this, we will need two lemmas. Lemma 4.1 There exist m1 > 0 and A depending only on Xmin, β and ε such that for all m ≥ m1, βεhm,ε ≤ sup hm1,ε hmmin . (4.1) Proof We will use induction, for m1 to be determined later. Assume the inequality holds for hm. Let Cm = sup hm1,ε hm1min Then βεhm,ε ≤ Cmhmmin. It follows that ‖t‖2Tm+1,ε ≤ Cm‖t‖ for t any global section of Km+1Xmin . Since the inequality for hm+1,ε obviously holds at points on C, it is sufficient to prove it at a fixed point y0 ∈ Xmin − C. Write x0 = f(y0) ∈ Xcan. By Lemma 2.3 there is a global holomorphic section sm+1,x0 of K satisfying |sm+1,x0 |2hm+1 (x0) = 1 and βε(y0) |sm+1,x0 |2hm+1 (m+ 1)! (m+ n+ 1)! 1 + O(m−1) as long as m1 is chosen to be sufficiently large. Then (m+ 1)! (m+ n+ 1)! βε(y0)hm+1,ε(y0) ≤ βε(y0) ‖f∗(sm+1,x0)‖ Tm+1,ε |f∗(sm+1,x0)| hm+1min (y0) ≤ Cmβε(y0) |sm+1,x0 |2hm+1 hm+1min (y0) (m+ 1)! (m+ n+ 1)! 1 + O hm+1min (y0), and the lemma follows. � For the lower bound of hm,ε, we use a modification of a lemma of Tsuji [Ts]: Lemma 4.2 There exist constants m2 and B depending only on Xmin such that for all m ≥ m2 and 0 < ε ≤ 1, βεh−1/mm,ε ≤ V m−m2+1 βεh−1/m2m2,ε )m2/m (4.2) for V = ωnmin Proof Set Lm,ε = βεh−1/mm,ε (m+ n)! (Nm + 1), for Nm + 1 = dimH 0(Xmin,K ). Then we claim that Lm,ε ≤ c1/mm L (m−1)/m m−1,ε . (4.3) Given (4.3), we can finish the proof of the lemma as follows. First, by Riemann-Roch, there exist constants m2 and B such that for m ≥ m2, cm ≤ V From (4.3), arguing by induction, we have Lm,ε ≤ (cmcm−1 · · · cm2) Lm2/mm2,ε , and the inequality (4.2) follows immediately. It remains to show (4.3). Using Hölder’s inequality, Lm,ε = βεh−1/mm,ε h 1/(m−1) m−1,ε h −1/(m−1) m−1,ε βε(h−1/mm,ε h 1/(m−1) m−1,ε ) −1/(m−1) m−1,ε −1/(m−1) m−1,ε βεh−1m,εhm−1,ε (m−1)/m m−1,ε (m+ n)! σ(i)m,ε ⊗ σ m,ε ⊗ hm−1,ε (m−1)/m m−1,ε = c1/mm L (m−1)/m m−1,ε , and this completes the proof of the lemma. � We can now use these lemmas to prove a convergence result for the metrics hm,ε. Proof of Theorem 4.1 From Lemma 4.1 we have h−1/mm,ε ≥ E(m, ε)h−1min on Xmin − C, where E(m, ε) → 1 as m→ ∞. From Lemma 4.2, we have βεh−1/mm,ε ≤ V F (m, ε), where F (m, ε) → 1 as m→ ∞. Writing hε = lim supm→∞ h m,ε , we have h−1ε − h−1min h−1min h−1min = lim inf βε(h−1/mm,ε − E(m, ε)h−1min) ≤ lim inf βεh−1/mm,ε − βεh−1min ≤ V − βεh−1min → 0, as ε→ 0. Theorem 4.1 follows. � Finally, we complete the proof of Theorem 2. Proof of Theorem 2 Using the notation given in the introduction, there is an isomorphism Θ : H0(Xmin,K ) → H0(X,KmX ) given by Θ(s) = τ∗s⊗ Sm−1. Then, given an inner product Tm,ε on H0(X,KmX ), we can define an inner product T̂m,ε on H 0(Xmin,K ) by 〈s, t〉 T̂m,ε = 〈Θ(s),Θ(t)〉Tm,ε . Then given an initial Hermitian metric hm0 on K X we can obtain an inner product 〈·, ·〉 T̂m0+1,ε on Km0+1Xmin and hence a Hermitian metric ĥm0+1,ε . Applying the modified Tsuji iteration as in the case of Theorem 4.1, we obtain a sequence of Hermitian metrics ĥm,ε form ≥ m0+1 onKmXmin. From the definition of S−1, one can check that hm,ε = τ ∗ĥm,ε ⊗ |S−1|−2m. Indeed, assuming inductively that hm,ε = τ ∗ĥm,ε ⊗ |S−1|−2m, denote by 〈·, ·, 〉′ T̂m+1,ε the inner product induced by ĥm,ε on H 0(Xmin,K ). We need to show that 〈s, t〉′ T̂m+1,ε = 〈s, t〉 T̂m+1,ε for s, t ∈ H0(Xmin,Km+1Xmin). But 〈s, t〉′ T̂m+1,ε ĥm,ε ⊗ s⊗ t hm,ε ⊗ |S−1|2m ⊗ (τ∗s)⊗ (τ∗t)⊗ |S−1|2 = 〈Θ(s),Θ(t)〉Tm+1,ε = 〈s, t〉 T̂m+1,ε Now by Theorem 4.1, we see that for any sequence εj → 0, we have lim sup ĥ1/mm,εj → hmin, almost everywhere. Theorem 2 follows immediately. � References [A] Aubin, T. Équations du type Monge-Ampère sur les variétés kählériennes compactes. Bull. Sci. Math. (2) 102 (1978), no. 1, 63–95. [C] Catlin, D. The Bergman kernel and a theorem of Tian, Analysis and geometry in several complex variables (Katata, 1997), 1–23, Trends Math., Birkhäuser Boston, Boston, MA, 1999. [DLM] Dai, X., Liu, K. and Ma, X. On the asymptotic expansion of Bergman Kernel, J. Differential Geom. 72 (2006), no. 1, 1–41. [D1] Donaldson, S.K. Scalar curvature and projective embeddings. I., J. Dif- ferential Geom. 59 (2001), no. 3, 479–522. [D2] Donaldson, S.K. Some numerical results in complex differential geom- etry, preprint, math.DG/0512625. [H] Hörmander, L. An introduction to complex analysis in several variables, Second revised edition. North-Holland Mathematical Library, Vol. 7. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. [K] Kobayashi, R., Einstein-Kähler V -metrics on open Satake V -surfaces with isolated quotient singularities. Math. Ann. 272 (1985), no. 3, 385– [L] Lu, Z. On the lower order terms of the asymptotic expansion of Tian- Yau-Zelditch, Amer. J. Math. 122 (2000), no. 2, 235–273. [P] Paoletti, R. Szegö kernels and finite group actions, Trans. Amer. Math. Soc. 356 (2004), no. 8, 3069–3076. [S] Song, J. The Szegö kernel on an orbifold circle bundle, preprint, math.DG/0405071. [Ti] Tian, G. On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), 99–130. [TZ] Tian, G. and Zhang, Z. On the Kähler-Ricci flow on projective man- ifolds of general type, Chinese Ann. Math. Ser. B 27 (2006), no. 2, 179–192. [Ts] Tsuji, H. Dynamical construction of Kähler-Einstein metrics, preprint, math.AG/06066262. [Y1] Yau, S.-T. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Comm. Pure Appl. Math. 31 (1978), no.3, 339–411. [Y2] Yau, S.-T. Open problems in geometry, Proc. Symposia Pure Math. 54 (1993), 1–28 http://arxiv.org/abs/math/0512625 http://arxiv.org/abs/math/0405071 http://arxiv.org/abs/math/0606626 [Ze] Zelditch, S. Szegö kernels and a theorem of Tian, Internat. Math. Res. Notices (1998), no. 6, 317–331. [Zh] Zhang, S. Heights and reductions of semi-stable varieties, Compositio Math. 104 (1996), 77–105.

0704.1006

A Denjoy Theorem for commuting circle diffeomorphisms with mixed Holder derivatives

arXiv:0704.1006v1 [math.DS] 8 Apr 2007 A Denjoy Theorem for commuting circle diffeomorphisms with mixed Hölder derivatives Victor Kleptsyn & Andrés Navas Abstract. We prove that if d ≥ 2 is an integer number and fk, k ∈ {1, . . . , d}, are C 1+τk commuting circle diffeomorphisms with τk ∈]0, 1[ and τ1 + · · · + τd > 1, then the fk’s are simultaneously (topologically) conjugate to rotations provided that their rotation numbers are independent over the rationals. Keywords: Denjoy Theorem, centralizers, Hölder derivative. Introduction Starting from the seminal works by Poincaré [13] and Denjoy [3], a deep theory for the dy- namics of circle diffeomorphisms has been developed by many authors [1, 7, 8, 17], and most of the fundamental related problems have been already solved. Quite surprisingly, the case of several commuting diffeomorphisms is rater special, as it was pointed out for the first time by Moser [9] in relation to the problem of the smoothness for the simultaneous conjugacy to rotations. Roughly speaking, in this case it should be enough to assume a joint Diophantine condition on the rotation numbers which does not imply a Diophantine condition for any of them (see the recent work [5] for the solution of the C∞ case of Moser’s problem). A similar phenomenon concerns the classical Denjoy Theorem. Indeed, in [4] it was proved that if d ≥ 2 is an integer number and τ > 1/d, then the elements f1, . . . , fd of any family of C1+τ commuting circle diffeomorphisms are simultaneously (topologically) conjugate to rotations provided that their rotation numbers are independent over the rationals (that is, no non trivial linear combination of them with rational coefficients equals a rational number). In other words, the classical (and nearly optimal) C2 hypothesis for Denjoy Theorem can be weakened in the case of several commuting diffeomorphisms. The first and main result of this work is a generalization of this fact to the case of different regularities. Theorem A. Let d ≥ 2 be an integer number and τ1, . . . , τd be real numbers in ]0, 1[ such that τ1+ · · ·+ τd > 1. If fk, k ∈ {1, . . . , d}, are respectively C 1+τk circle diffeomorphisms which have ro- tation numbers independent over the rationals and which do commute, then they are simultaneously (topologically) conjugate to rotations. Since the probabilistic arguments of [4] cannot be applied to the case of different regularities, the preceding result is much more than a straightforward generalization of Theorem A of [4]. Indeed, for the proof here we use a key new argument which is somehow more deterministic. Theorem A is (almost) optimal (in the Hölder scale), in the sense that if one decreases slightly the regularity assumptions then it is no longer true. The following result relies on classical constructions by Bohl [2], Denjoy [3], Herman [7], and Pixton [12], and its proof consists on an easy extension of the construction given by Tsuboi in [16]. http://arxiv.org/abs/0704.1006v1 Denjoy Theorem for commuting diffeomorphisms 2 Theorem B. Let d ≥ 2 be an integer number and τ1, . . . , τd be real numbers in ]0, 1[ such that τ1 + · · ·+ τd < 1. If ρ1, . . . , ρd are elements in R/Z which are independent over the rationals, then there exist C1+τk circle diffeomorphisms fk, k ∈ {1, . . . , d}, having rotation numbers ρk, which do commute, and such that none of them is topologically conjugate to a rotation. It is well known that the techniques developed for Denjoy Theory can be applied to the study of group actions on the interval. In this direction we should point out that the methods of this paper also allow to extend (in a straightforward way) the so called “Generalized Kopell Lemma” and the “Denjoy-Szekeres Type Theorem” (Theorems B and C of [4] respectively) for Abelian groups of interval diffeomorphisms under analogous hypothesis of different regularities. Furthermore, the construction of counter-examples for both of them when these hypothesis do not hold can be also extended to this context. We leave the verification of all of this to the reader. Acknowledgments. It is a pleasure to thank Bassam Fayad and Sergey Voronin for their encour- agements, as well as the Independent University of Moscow for the hospitality during the conference “Laminations and Group Actions in Dynamics” held in February 2007. The first author was sup- ported by the Swiss National Science Foundation. This work was also funded by the RFBR grants 7-01-00017-a and CNRS-L−a 05-01-02801, and by the CONICYT grant 7060237. 1 A general principle revisited As it is well known since the classical works by Denjoy, Schwartz and Sacksteder [3, 14, 15], if I is a wandering interval1 for the dynamics of a finitely generated semigroup Γ of C1+lip diffeomorphisms of the closed interval or the circle (on which we will always consider the normalized length), one can control the distortion of the elements of Γ over (a slightly larger interval than) I in terms of the sum of the lengths of the images of I along the corresponding sequence of compositions and a uniform Lipschitz constant for the derivatives of the (finitely many) generators of Γ. If τ belongs to ]0, 1[ and Γ consists of C1+τ diffeomorphisms, the same is true provided that the sum of the τ -powers of the lengths of the corresponding images of I is finite (this last condition does not follow from the disjointness of these intervals !): see for instance [4], Lemma 2.2. It is not difficult to prove a similar statement for the case of different regularities, and this is precisely the content of the following lemma. However, to the difference of [4], here we will deal with finite sequences of compositions by a technical reason which will be clear at the end of the next section. Lemma 1.1. Let Γ be a semigroup of (orientation preserving) diffeomorphisms of the circle or the closed interval which is generated by finitely many elements gk, k ∈ {1, . . . , l}, which are respectively of class C1+τk , where τk∈]0, 1]. Let Ck denote the τk-Hölder constant of the function log(g k), and let C = max{C1, . . . , Cl} and τ = max{τ1, . . . , τl}. Given n0 ∈ N, for each n ≤ n0 let us chose kn ∈ {1, . . . , l}, and for a fixed interval I let S > 0 be a constant such that ∣gkn · · · gk1(I) τkn+1 . (1) If n ≤ n0 is such that gkn · · · gk1(I) does not intersect I but is contained in the L-neighborhood of I, where L := |I|/2 exp(2τCS), then gkn · · · gk1 has a hyperbolic fixed point. 1We say that an interval is wandering if its images by different elements of the underlying semigroup are disjoint. Denjoy Theorem for commuting diffeomorphisms 3 Proof. Let J = [a, b] be the (closed) 2L-neighborhood of I, and let I ′ (resp. I ′′) the connected component of J \I to the right (resp. to the left) of I. We will prove by induction on j∈{0, . . . , n0} that the following two conditions are satisfied: (i)j |gkj · · · gk1(I ′)| ≤ |gkj · · · gk1(I)|, (ii)j sup{x,y}⊂I∪I′ (gkj ···gk1) (gkj ···gk1) ≤ exp(2τ CS). Condition (ii)0 is trivially satisfied, whereas condition (i)0 is satisfied since |I ′| = 2L ≤ |I|. Assume that (i)i and (ii)i hold for each i ∈ {0, . . . , j − 1}. Then for every x, y in I ∪ I ′ we have (gkj · · · gk1) (gkj · · · gk1) ∣ log(g′ki+1(gki · · · gk1(x))) − log(g (gki · · · gk1(y))) Cki+1 ∣gki · · · gk1(x)− gki · · · gk1(y) τki+1 |gki · · · gk1(I)|+ |gki · · · gk1(I )τki+1 ≤ C 2τ |gki · · · gk1(I)| τki+1 ≤ C 2τS. This shows (ii)j. To verify (i)j first note that there must exist x ∈ I and y ∈ I ′ such that |gkj · · · gk1(I)| = |I| · (gkj · · · gk1) ′(x) and |gkj · · · gk1(I ′)| = |I ′| · (gkj · · · gk1) ′(y). Therefore, by (ii)j , |gkj · · · gk1(I |gkj · · · gk1(I)| (gkj · · · gk1) (gkj · · · gk1) |I ′| ≤ exp(2τCS) |I ′| which proves (i)j . Obviously, similar arguments show that (i)j and (ii)j also hold for every j ∈ {0, . . . , n0} when we replace I ′ by I ′′. Now for simplicity let us denote hj = gkj · · · gk1 . Assume that hn(I) is contained in the L- neighborhood of the interval I (see Figure 1). Then property (i)n gives hn(J) ⊂ J , and this already implies that hn has a fixed point x in J . (The reader will see that the existence of this fixed point together with the fact that hn 6= id is the only information that we will retain for the proof of Theorem A.) To conclude we would like to show that the fixed point x is hyperbolic. To do this just note that, if hn(I) does not intersect I, then there exists y ∈ I such that h′n(y) = |hn(I)| Therefore, by (ii)n, h′n(x) ≤ h n(y) exp(2 τCS) ≤ L exp(2τCS) and this finishes the proof. � Denjoy Theorem for commuting diffeomorphisms 4 ....................... ............................... ................................................................................................................................................................................. ................................................................. ........... ................................................................................................................................................................................................................................................................................................................................................................. ...................................... .............................. ......................... ...................... .................... ................... . ............................ ............. .......... .......... hn hn ........................................................... ......... Figure 1 hn(I)I hyperbolic fixed point 2 Proof of Theorem A Recall the following well known argument (see for instance [6], Proposition 6.17, or [11], Lemma 4.1.4). If f1, . . . , fd are commuting circle homeomorphisms, then there is a common invariant probability measure µ on S1. Moreover, if the rotation number of at least one of them is irrational, then there is no finite orbit for the group action, and the measure µ has no atom. Therefore, the distribution function Fµ : S 1 → R/Z, Fµ(x) := µ([0, x[), gives a (simultaneous) semiconjugacy between the maps f1, . . . , fd and the rotations corresponding to their rotation numbers. Thus, for the proof of Theorem A we have to show that this semiconju- gacy is in fact a conjugacy, and our strategy for proving this (under the hypothesis of the Theorem) is the classical one and goes back to Schwartz [15]. Indeed, in the contrary case the support of µ would be a (minimal) invariant Cantor set, and the connected components of its complement would correspond to the maximal wandering open intervals. Fixing one of these intervals, say I, we will search for a sequence of compositions hn = fkn · · · fk1 satisfying the hypothesis of Lemma 1.1. This will allow us to conclude that some hn has a (hyperbolic) fixed point, thus implying that its rotation number is equal to zero. However, this is in contradiction to the fact that the rotation numbers of the fk’s are independent over the rationals (it is easy to verify that the rotation number restricted to any group of circle homeomorphisms which preserves a probability measure on S1 is a group homomorphism: see again [6] or [11]). In order to ensure the existence of the sequence (hn) the main idea of [4] was to endow the space of all (infinite) sequences of compositions with a natural probability measure, and then to prove that the “generic ones” satisfy many nice properties as for instance the convergence of the sum (1) as n0 goes to infinity. It seems that such a probabilistic argument cannot be applied to the case of different regularities, and we will need to introduce a new argument which is somehow more deterministic, since it gives partial information on the sequence that we find. For simplicity we will first deal with the case d=2. 2.1 The case d = 2 Although not explicitly stated in [4], the main probabilistic argument for the proof of the Generalized Denjoy Theorem therein is not a dynamical issue, but it is just a statement concerning the finiteness of the sum of the τ -powers of some positive real numbers. To be more concrete (at least in the case d = 2 and when τ > 1/2), if (ℓi,j) is a double-indexed sequence of positive numbers with finite total sum (where i and j are non negative integers), then with respect to some natural probability distribution on the space of infinite paths (i(n), j(n))n≥0 satisfying i(0) = j(0) = 0, i(n+1) ≥ i(n), j(n+1) ≥ j(n) and i(n+1)+ j(n+1) = 1+ i(n)+ j(n), one has almost everywhere Denjoy Theorem for commuting diffeomorphisms 5 the convergence of the sum ℓτi(n),j(n). The first goal of this section is to prove the existence of paths sharing a similar property in the case of different exponents τ1, τ2 in ]0, 1[ (with τ1 + τ2 > 1). A substantial difference here is that we will construct our sequence by concatenating infinitely many finite paths, and each one of these paths will be chosen among finitely many ones. To do this we begin with the following elementary lemma. Lemma 2.1. Let ℓi,j be positive real numbers, where i ∈ {1, . . . ,m} and j ∈ {1, . . . , n}. Assume that the total sum of the ℓi.j’s is less than or equal to 1. If τ belongs to ]0, 1[, then there exists k ∈ {1, . . . , n} such that ℓτi,k ≤ Proof. We will show that the mean value of the function k 7→ i=1 ℓ i,k is less than or equal to m1−τ/nτ , from where the claim of the lemma follows immediately. To do this first note that, by Hölder’s inequality, for each fixed k ∈ {1, . . . , n} one has ℓτi,k = (ℓτi,k) i=1, (1) ∥(ℓτi,k) · ‖(1)mi=1‖1/(1−τ) = m1−τ . Thus, by using Hölder’s inequality again one obtains ℓτi,k , (1) · ‖(1) k=1‖1/(1−τ) which finishes the proof. � Now we explain the main idea of our construction. Let us assume that the total sum of the double-indexed sequence of positive numbers ℓi,j is ≤ 1, and suppose that the numbers τ1∈]0, 1[ and τ2∈]0, 1[ such that τ1+τ2 > 1 are fixed. Denoting by [[a, b]] the set of integers between a and b (with a and b included when they are in Z), let us consider any sequence of rectangles Rm ⊂ N0×N0 such that R0 = {(0, 0)}, R2m+1 = [[im, im+1]]× [[jm, jm+2]] and R2m+2 = [[im, im+2]]× [[jm+1, jm+2]], where (im)m≥1 and (jm)m≥1 are strictly increasing sequences of non negative integers numbers satisfying i0 = i1 = 0 and j0 = j1 = 0 (see Figure 2). Denoting by Xm and Ym respectively the number of points on the horizontal and vertical sides of each Rm, a direct application of Lemma 2.1 gives us, for ε := 1− τ1 − τ2 > 0 and each m ≥ 0: Denjoy Theorem for commuting diffeomorphisms 6 – an integer r(2m+ 1) ∈ [[im, im+1]] such that r(2m+1),j Y 1−τ22m+1 Y τ12m+1 · Y −ε2m+1, – an integer r(2m+ 2) ∈ [[jm+1, jm+2]] such that i,r(2m+2) Y τ12m+2 Y τ12m+2 ·X−ε2m+2. Starting from the origin and following the corresponding horizontal and vertical lines, we find an infinite path (i(n), j(n))n≥0 satisfying i(0) = j(0) = 0, i(n + 1) ≥ i(n), j(n+ 1) ≥ j(n), i(n+ 1) + j(n + 1) = 1 + i(n) + j(n), and such that the sum τα(n) i(n),j(n) is bounded by · Y −ε2m+1 + ·X−ε2m+2 , (3) where α(n) := 1 if |i(n + 1)− i(n)| = 1 and α(n) := 2 if |j(n + 1)− j(n)| = 1. Figure 2 i0 = i1 i2 i3 i4 i5 j0 =j1 ••••• ••••••••••••••••••••••• ••••••••••••••••••••••••••••• R5 R6 R7 R8 Denjoy Theorem for commuting diffeomorphisms 7 Now let us consider any choice such that im = [4 mτ1 ] and jm = [4 mτ2 ] for m large enough. Writing am ≃ bm when (am) and (bm) are sequences of positive numbers such that (am/bm) remains bounded and away from zero, for such a choice we have Xm ≃ 2 mτ1 and Ym ≃ 2 mτ2 . Thus, (2mτ1)τ2 (2mτ2)τ1 and therefore there exists C > 0 such that, for each m ≥ 0, Y τ1m This implies that the sum in (3) is bounded by S := C  = C 4τ2ε − 1 4τ1ε − 1 , (4) and so the value of the sum (2) is finite (and also bounded by S). We can now proceed to the proof of Theorem A in the case d=2. Assume by contradiction that fk, k∈{1, 2}, are respectively C 1+τk commuting circle diffeomorphisms which are not simultaneously conjugate to rotations and which have rotation numbers independent over the rationals. Let I be a connected component of the complement of the invariant minimal Cantor set for the group action, and let ℓi,j = |f 2 (I)|. We obviously have i,j ℓi,j ≤ 1, and so we can apply all our previous discussion to this sequence. In particular, there exists an infinite path (i(n), j(n)) starting at the origin and such that the sum τα(n) i(n),j(n) is bounded by the number S > 0 defined by (4). If for n ≥ 1 we let kn = α(n − 1) ∈ {1, 2}, then we obtain a sequence of compositions hn = fkn · · · fk1 such that the preceding sum coincides term by term with |fkn · · · fk1(I)| τkn+1 . Thus, in order to apply Lemma 1.1 to get a contradiction, we just need to verify that, for some n ≥ 1, the hypothesis that hn(I) = fkn · · · fk1(I) is contained in the L-neighborhood of I is satisfied (where L := |I|/2 exp(2τCS), τ := max{τ1, τ2}, and C := max{C1, . . . , Cd}, with Ck being the τk- Hölder constant for the function log(f ′k)). To to this first note that, if we collapse all the connected components of the complement of the minimal invariant Cantor set, then we obtain a topological circle Ŝ1 on which the original diffeomor- phisms induce naturally minimal homeomorphisms f̂1 and f̂2 which are simultaneously conjugate to rotations. Moreover, the L-neighborhood of I becomes a non degenerate interval Û ; thus, there exists N ∈ N such that the intervals f̂−11 (Û), . . . , f̂ 1 (Û), as well as f̂ 2 (Û), . . . , f̂ 2 (Û), cover the circle Ŝ1. This easily implies that for any image I0 of I by some element of the semigroup generated by f1 and f2 there exists k and k ′ in {1, . . . , N} such that fk1 (I0) and f 2 (I0) are contained in the L-neighborhood of I. Now it is easy to see that, for the sequence of compositions that we found, for every N̄ ∈ N there exists some integer r ∈ N such that kr = kr+1 = . . . = kr+N̄ . For N̄ = N this obviously implies that at least one of the intervals hr+1(I), . . . , hr+N (I) is contained in the L-neighborhood of I, thus finishing the proof. Denjoy Theorem for commuting diffeomorphisms 8 Figure 3 ••••••••• ••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• R′2 R R′4 R R′6 R x′0 x y′0=y y′2=y y′4=y y′6=y We would like to close this section by giving a different type of choice for the sequence of rectangles which is simpler to describe and for which the preceding arguments are also valable. (For simplicity, we will use a similar construction to deal with the case d > 2, altough the preceding one still applies). This sequence (R′m)m≥0 is of the form [[0, x m]] × [[0, y m]], where (x m) and (y are non decreasing sequences of positive integer numbers such that x′0 = y 0 = 0, x m > x m−1 and y′m = y m−1 if m is odd, and x m = x m−1 and y m > y m−1 if m is even. If (ℓi,j) is a double-indexed sequence of positive real numbers with total sum ≤ 1, we chose these integer numbers in such a way that x′2m+1 = x 2m+2 = [4 mτ1 ] and y′2m = y 2m+1 = [4 mτ2 ] for m large enough. As before, inside the rectangle Rm there is a “good” vertical (resp. horizontal) segment of line Lm for m even (resp. odd). Therefore, for each M0 ∈ N we can concatenate these segments between Lm−1 ∩ Lm and Lm ∩ Lm+1 at the m th step for m < M0, and between LM0−1 ∩ LM0 and the point of LM0 on the boundary of RM0 at the last step (see Figure 3). In this way we obtain a path (starting at the origin) of finite length n(M0)− 1 for which the sum n(M0)−1 τα(n) i(n),j(n) is bounded by some number S > 0 which is independent of M0. Now let fk, k ∈{1, 2}, be two commuting circle diffeomorphisms of class C 1+τk which are not simultaneously conjugate to rotations. Fix again one of the maximal wandering open intervals for the dynamics, say I, and let ℓi,j = |f 2 (I)|. (Note that i,j ℓi,j ≤ 1.) The method above gives us a family of finite paths, and each of these paths determines uniquely a sequence of compositions. Remark however that there is a little difference here, since we allow the use of the inverses of f1 and f2. Therefore, in order to apply Lemma 1.1, we will need to consider now {f1, f 1 , f2, f 2 } as being our system of generators, and therefore we put τ = max{τ1, τ2} and C = max{C1, C2, C where Ci (resp. C i) is a τi-Hölder constant for the function log(f i) (resp. log((f ′)). As in the previous proof, we need to verify that, for some M0 ∈ N, there exists a non trivial element in the sequence of compositions (hn) associated to its corresponding finite path which sends I inside the L-neighborhood of itself, where L := |I|/2 exp(2τCS). As before, for proving this it suffices to show that for every N there exists r ∈ N such that one has hr+i+1 = f1hr+i for each i ∈ {0, . . . , N − 1}, or hr+i+1 = f2hr+i for each i ∈ {0, . . . , N − 1}. However, this last property is always satisfied if Denjoy Theorem for commuting diffeomorphisms 9 M0 is big enough so that the number of points with integer coordinates in the line segment LM0 contained in RM0 \ RM0−1 is greater than N . Note that it is in this last argument where we use the fact that we keep only finite sequences of compositions, altough our method combined with a diagonal type argument easily shows the existence of an infinite sequence for which the sum (2) converges. 2.2 The general case In the case d = 2, the “good” paths leading to the sequence of compositions which allows to apply Lemma 1.1 were obtained by concatenating horizontal and vertical lines. When d > 2 we will need to concatenate lines in several (namely d) directions, and the geometrical difficulty for doing this is evident: in dimension bigger than 2, two lines in different directions do not necessarily intersect. To overcome this difficulty we will use the fact that, at each step (i.e. inside each rectangle), there is not only one finite path which is good, but this is the case for a “large proportion” of finite paths. We first reformulate Lemma 2.1 in this direction. Lemma 2.2. Let ℓi,j be positive real numbers, where i ∈ {1, . . . ,m} and j ∈ {1, . . . , n}. Assume that the total sum of the ℓi.j’s is less than or equal to 1. If τ belongs to ]0, 1[ and A > 1, then for a proportion of indexes k ∈ {1, . . . , n} greater than or equal to (1− 1/A) we have ℓτi,k ≤ A Proof. As in the proof of Lemma 2.1, the mean value of the function ℓτi,k (5) is less than or equal to m1−τ/nτ . The claim of the lemma then follows as a direct application of Chebychev’s inequality: the proportion of points for which the value of (5) is greater than this mean value times A cannot exceed 1/A. � Now let (ℓi1,...,id) be a multi-indexed sequence of positive real numbers having total sum ≤ 1, and let τ1, . . . , τd be real numbers in ]0, 1[. Starting with R0 = [[0, 0]] d, let us consider a sequence (Rm)m≥0 of rectangles of the form Rm = [[0, x1,m]]× · · · × [[0, xd,m]] satisfying xk,m ≥ xk,m−1 for each k ∈ {1, . . . , d}, with strict inequality if and only if k ≡ m (mod d). For each m ≥ 1 denote by s(m) ∈ {1, . . . , d} the residue class (mod d) of m, and denote by Fm the face [[0, x1,m]]× · · · × [[0, xs(m)−1,m]]× {0} × [[0, xs(m)+1,m]]× · · · × [[0, xd,m]] of Rm. For each (i1, . . . , is(m)−1, 0, is(m)+1, . . . , id) belonging to this face Fm we consider the sum xs(m),m τs(m) i1,...,is(m)−1,j,is(m)+1,...,id By Lemma 2.2, if Am > 1 then the proportion of points in Fm for which this sum is bounded by (1 + xs(m),m) 1−τs(m) j 6=s(m) (1 + xj,m) τs(m) = Am · 1−τs(m) s(m),m j 6=s(m) τs(m) is at least equal to (1− 1/Am), where Xj,m := 1+xj,m. In order to concatenate the corresponding lines we will use the following elementary lemma. Denjoy Theorem for commuting diffeomorphisms 10 ................. .................. s(m)-direction .................. .................. ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ...... ............ ............ ............ ............ ............ ............ ............ ............ ....... ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ...... ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ...... .................. ...................... Fm+1• s(m+ 1)-direction ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ...... ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ...... ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ..... ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ..... .................. .................. Figure 4 (i1, . . . , is(m+1)−2, 0, 0, is(m+1)+1, . . . , id) admissible in Cm (i1, . . . , is(m+1)−1, 0, 0, is(m+1)+2, . . . , id) admissible in Cm+1 Lemma 2.3. Let us chose inside each rectangle (Rm)m≥1 a set L(m) of (complete) lines in the corresponding s(m)-direction whose proportion (with respect to all the lines in that direction inside (Rm)) is at least (1 − 1/Am). If M0∈N is such that m=1 1/Am<1, then there exists a sequence of lines Lm ∈ L(m), m ∈ {0, . . . ,M0}, such that Lm+1 intersects Lm for every m < M0. Proof. Let us denote by Cm the (d− 2)-dimensional face of Rm given by [[0, x1,m]]× · · · × [[0, xs(m)−1,m]]× {0} × {0} × [[0, xs(m)+2,m]]× · · · × [[0, xd,m]]. Call a point (i1, . . . , is(m)−1, 0, 0, is(m)+2 , . . . , id) ∈ Cm admissible if there exists a sequence of lines Li∈L(i), i∈{0, . . . ,m}, such that Li intersects Li+1 for every i∈{0, . . . ,m− 1}, and such that Lm projects in the s(m)-direction into a point (i1, . . . , is(m)−1, 0, is(m)+1, is(m)+2, . . . , id) ∈ Fm for some is(m)+1∈ [[0, xs(m)+1,m+1]]. We will show that the proportion of admissible points in CM0 is greater than or equal to P := 1− Am > 0. To prove this, for each m ≥ 0 let us denote by Pm the proportion of admissible points in Cm. Since R0 reduces to the origin, it suffices to show that, for all m ≥ 0, Pm+1 ≥ Pm − To prove this inequality first note that each line Lm+1 ∈ L(m + 1) determines uniquely a point (i1, . . . , is(m+1)−1, 0, is(m+1)+1, . . . , id)∈Fm+1. The projection into Cm of this line then corresponds to the point (i1, . . . , is(m+1)−2, 0, 0, is(m+1)+1 , . . . , id). If this is an admissible point of Cm then we can concatenate the line Lm+1 to the sequence of lines corresponding to it (see Figure 4). Now the proportion of lines in L(m + 1) being at least Denjoy Theorem for commuting diffeomorphisms 11 1− 1/Am+1, the proportion of those lines which project on Cm into an admissible point is at least equal to − (1− Pm) = Pm − By projecting in the (s(m+1)+1)-direction, this obviously implies that the proportion of admissible points in Cm+1 is also greater than or equal to Pm − 1/Am+1, thus finishing the proof. � Observe that a sequence of lines Lm as above determines a finite path (starting at the origin) of points (x1(n), . . . , xd(n)) having non negative integer coordinates such that the distance between two consecutive ones is equal to 1. Moreover, if we denote by n(M0) the length of this path plus 1, the corresponding sum n(N0)−1 τα(n) x1(n),...,xd(n) is bounded by (1 + xs(m),m) 1−τs(m) i 6=s(m) (1 + xi,m) τs(m) 1−τs(m) s(m),m j 6=s(m) τs(m) , (7) where α(n) equals the unique index in {1, . . . , d} for which |xα(n)(n+ 1)− xα(n)(n)| = 1. Now let us define Am=2 εmτs(m)/2A, where A is a large enough constant so that m≥0 1/Am<1, and let us consider any choice of the xk,m’s so that Xk,m ≃ 2 mτk . For such a choice we have j 6=k = X−εk,m · j 6=k ≃ 2−εmτk · j 6=k (2mτk )τj (2mτj )τk = 2−εmτk , (8) where ε := 1 − τ1 − · · · − τd > 0. Therefore, for each M0 ∈ N the preceding lemma provides us a sequence of lines Lm, m ∈ {0, . . . ,M0}, such that Lm+1 intersects Lm for each m < M0, and such that the corresponding expression (7) is bounded from above by 2εmτs(m)/2A · j 6=k ≤ AC ′ 2−εmτs(m)/2 ≤ AC ′ 2−εmτ ′/2 =: S < ∞, (9) where τ ′ := min{τ1, . . . , τd} and C ′ is a constant (independent of M0) giving an upper bound for the quotient between the left and the right hand expressions in (8). With all this information in mind we can proceed to the proof of Theorem A in the case d > 2 in the very same way as in the (second proof for the) case d = 2. Indeed, assume that fk, k ∈ {1, . . . , d}, are circle diffeomorphisms as in the statement of the theorem which are not conjugate to rotations, and let I be a maximal open wandering interval for the dynamics (i.e. a connected component of the complement of the minimal invariant Cantor set). Clearly, we can apply all our previous discussion to the multi-indexed sequence (ℓi1,...,id) defined by ℓi1,...,id = |f 1 · · · f (I)|. In particular, for each M0 ∈ N we can find a finite path so that the sum (6) is bounded by the number S > 0 defined by (9) (which is independent of M0). Each such a path induces canonically a finite sequence of compositions by the fk’s and their inverses. Therefore, in order to apply Lemma 1.1 to get a contradiction, we need to verify that some of such sequences contains a (non trivial) element hn which sends I into its L-neighborhood for L := |I|/2 exp(2τCS), where τ := max{τ1, . . . , τd} and Denjoy Theorem for commuting diffeomorphisms 12 C := max{C1, . . . , Cd, C 1, . . . , C d}, with Ck (resp. C k) being the τk-Hölder constant of the function log(f ′k) (resp. log((f )′). To ensure this last property let U be the L-neighborhood of I, and let N ∈ N be such that, given any wandering interval, among the first N iterates of f1, as well as for f2, . . . , fd, at least one of them sends this interval inside U . If we take M0 large enough so that the number of points with integer coordinates in LM0 which are contained in RM0 \RM0−1 exceeds N , then one can easily see that the associated sequence of compositions contains the desired element hn. This finishes the proof of Theorem A. 3 Proof of Theorem B The strategy for the proof of Theorem B is well known. We prescribe the rotation numbers ρ1, . . . , ρd (which are supposed to be independent over the rationals), we fix a point p ∈ S 1, and for each (i1, . . . , id) ∈ Z d we replace the point Ri1ρ1 · · ·R (p) by an interval Ii1,...,id of length ℓi1,...,id in such a way that the total sum of the ℓi1,...,id ’s is finite. Doing this we obtain a new circle on which the rotations Rρk induce nice homeomorphisms if we extend them apropiately to the intervals Ii1,...,id (outside these intervals the induced homeomorphisms are canonically defined). More precisely, as it is well explained in [4, 7, 10, 16], if there exists a constant C ′ > 0 so that for all (i1, . . . , id) ∈ Z and all k ∈ {1, . . . , d} one has ℓi1,...,1+ik,...,id ℓi1,...,ik,...,id i1,...,ik,...,id ≤ C ′, (10) then one can perform the extension to the intervals Ii1,...,id in such a way the resulting maps fk, k∈{1, . . . , d}, are respectively C1+τk diffeomorphisms and commute, and moreover their derivatives are identically equal to 1 on the invariant minimal Cantor set.2 Indeed, one possible extension is given by fk(x) = (ϕIi1,...,ik,...,id )−1 ◦ϕIi1,...,1+ik,...,id (x), where x belongs to the interior of the interval Ii1...,ik,...,id. Here, ϕI:]a, b[→ R denotes the map ϕI(x) = It turns out that a good choice for the lengths is ℓi1,...,id = 1 + |i1|1/τ1 + · · · |id| Indeed, on the one hand, if we decompose the sum of the ℓi1,...,id ’s according to the biggest |ij | we obtain (i1,...,id)∈Z ℓi1,...,id ≤ 1 + |ij | 1/τj ≤ |ik| for all j ∈ {1, . . . , d} |ik| ≥ 1 1 + |i1|1/τ1 + · · · |id| 2Condition (10) is also necessary under these requirements. Indeed, there must exist a point in Ii1,...,ik,...,id for which the derivative of the corresponding map fk equals ℓi1,...,1+ik,...,id/ℓi1,...,ik,...,id . Since the derivative of fk at the end points of Ii1,...,ik,...,id is assumed to be equal to 1, condition (10) holds for C ′ being the τk-Hölder constant of the derivative of fk. Denjoy Theorem for commuting diffeomorphisms 13 and therefore, for some constant C > 0, this sum is bounded by card{(i1, . . . , id) : |ij | 1/τj ≤ n1/τk for all j∈{1, . . . , d}, ik = n} 1 + n1/τk ≤ 1 + C n1/τk j 6=k nτj/τk = 1 + C j 6=k τj)/τk n1/τk = 1 + C n(1−τk−ε)/τk n1/τk = 1 + C n1+ε/τk where ε := 1− (τ1 + · · ·+ τd). (Remark that, since ε > 0, the last infinite sum converges.) On the other hand, the left hand expression in (10) is equal to F (i1, . . . , id) := |1 + ik| 1/τk − |ik| 1 + |i1|1/τ1 + · · ·+ |1 + ik| 1/τk + · · ·+ |id| 1 + |i1| 1/τ1 + · · · + |ik| 1/τk + · · ·+ |id| In order to obtain an upper bound for this expression first note that, if ik ≥ 0, then F (i1, . . . , ik, . . . , id) ≤ F (i1, . . . ,−1− ik, . . . , id). Therefore, we can restrict to the case where ik < 0. For this case, denoting B = 1 + j 6=k |ij | and a = |ik| we have F (i1, . . . , id) = a1/τk − (a− 1)1/τk B + (a− 1)1/τk B + a1/τk a1/τk − (a− 1)1/τk B + (a− 1)1/τk )1−τk B + a1/τk B + (a− 1)1/τk Both factors in the last expression are decreasing in B. Thus, since B ≥ 1, F (i1, . . . , id) ≤ a1/τk − (a− 1)1/τk 1 + (a− 1)1/τk )1−τk 1 + a1/τk 1 + (a− 1)1/τk Now note that a ≥ 1. For a = 1 the right hand expression above equals 2τk . If a > 1 then the Mean Value Theorem gives the estimate a1/τk − (a − 1)1/τk ≤ a /τk, and therefore the preceding expression is bounded from above by ((a− 1)1/τk )1−τk a1/τk (a− 1)1/τk · 2 = 21/τk We have then shown that for any (i1, . . . , id) ∈ Z d one has F (i1, . . . , id) ≤ 21/τk . In other words, if τ ′ = min{τ1, . . . , τd} then inequality (10) holds for each (i1, . . . , id) ∈ Z d and every k ∈ {1, . . . , d} for the constant C ′ = 21/τ /τ ′, and this finishes the proof of Theorem B. Denjoy Theorem for commuting diffeomorphisms 14 References [1] Arnol’d, V. Small denominators I. Mapping the circle onto itself. Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 21-86. [2] Bohl, P. Uber die hinsichtlich der unabhängigen variabeln periodische differential gleichung erster ord- nung. Acta Math. 40 (1916), 321-336. [3] Denjoy, A. Sur les courbes définies par des équations différentielles à la surface du tore. J. Math. Pures et Appl. 11 (1932), 333-375. [4] Deroin, B., Kleptsyn, V. & Navas, A. Sur la dynamique unidimensionnelle en régularité intermédiaire. To appear in Acta Math. [5] Fayad, B. & Khanin, K. Smooth linearisation of commuting circle diffeomorphisms. Preprint (2006). [6] Ghys, É. Groups acting on the circle. L’Enseig. Math. 47 (2001), 329-407. [7] Herman, M. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. de l’IHÉS 49 (1979), 5-234. [8] Katznelson, Y., Ornstein, D. The differentiability of the conjugation of certain diffeomorphisms of the circle. Erg. Theory and Dynam. Systems 9 (1989), 643-680. [9] Moser, J. On commuting circle mappings and simultaneous Diophantine approximations. Math. Z. 205 (1990), 105-121. [10] Navas, A. Growth of groups and diffeomorphisms of the interval. Preprint (2006). [11] Navas, A. Grupos de difeomorfismos del ćırculo. Monograf́ıas del IMCA, Lima, Perú (2006). [12] Pixton, D. Nonsmoothable, unstable group actions. Trans. of the AMS 229 (1977), 259-268. [13] Poincaré, H. Mémoire sur les courbes définies par une équation différentielle. Journal de Mathématiques 7 (1881), 375-422, and 8 (1882), 251-296. [14] Sacksteder, R. Foliations and pseudogroups. Amer. J. Math. 87 (1965), 79-102. [15] Schwartz, A. A generalization of Poincaré-Bendixon theorem to closed two dimensional manifolds. Amer. J. Math. 85 (1963), 453-458. [16] Tsuboi, T. Homological and dynamical study on certain groups of Lipschitz homeomorphisms of the circle. J. Math. Soc. Japan 47 (1995), 1-30. [17] Yoccoz, J. C. Centralisateurs et conjugaison différentiable des difféomorphismes du cercle. Petits diviseurs en dimension 1. Astérisque 231 (1995), 89-242. Victor Kleptsyn Université de Genève, 2-4 rue du Lièvre, Case postale 64, 1211 Genève 4, Suisse ([email protected]) Andrés Navas Universidad de Santiago de Chile, Alameda 3363, Santiago, Chile ([email protected])

0704.1007

Transient Dynamics of Sparsely Connected Hopfield Neural Networks with Arbitrary Degree Distributions

Transient Dynamics of Sparsely Connected Hopfield Neural Networks with Arbitrary Degree Distributions Pan Zhang and Yong Chen ∗ Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China Abstract Using probabilistic approach, the transient dynamics of sparsely connected Hopfield neural networks is studied for arbitrary degree distributions. A recursive scheme is developed to determine the time evolution of overlap parameters. As illustrative examples, the explicit calculations of dynamics for networks with binomial, power- law, and uniform degree distribution are performed. The results are good agreement with the extensive numerical simulations. It indicates that with the same average degree, there is a gradual improvement of network performance with increasing sharpness of its degree distribution, and the most efficient degree distribution for global storage of patterns is the delta function. Key words: neural networks, complex networks, degree distribution, probability theory PACS: 87.10.+e, 89.75.Fb, 87.18.Sn, 02.50.-r As a tractable toy model of associative memories and can also be viewed as an extension of the Ising model, Hopfield neural networks [1] received lots of attention in recent two decades. Equilibrium properties of fully-connected Hopfield neural network have been well studied using spin-glass theory, es- pecially the replica method [2,3]. Dynamics is also studied using generating functional method [4] and signal-to-noise analysis [5,6,7] . Given the huge number of neurons, there is only small number of intercon- nections in human brain cortex (∼ 1011 neurons and ∼ 1014 synapses). In order to simulate a biological genuine model rather than the fully-connected networks, various random diluted models were studied, including extremely diluted model [8,9], finite diluted model [10,11], and finite connection model ⋆ Physica A 387, 1009(2008) ∗ Corresponding author. Email address: [email protected] Preprint submitted to Elsevier 19 November 2018 http://arxiv.org/abs/0704.1007v2 [12,13]. But neural connectivity is suggested to be far more complex than fully random graph, e.g. the networks of c.elegans and cat’s cortical neural were re- ported to be small-world and scale-free, respectively [14,15]. To go one step closer to more biological realistic model, many numerical studies are carried out, focusing on how the topology, the degree distribution, and clustering co- efficient of a network topology affect the computational performance of the Hopfield model [16,17,18,19]. With the same average connection, random net- work was reported to be more efficient for storage and retrieval of patterns than either small-world network or regular network [17]. Torres et al. reported that the capacity of storage is higher for neural network with scale-free topol- ogy than for highly random diluted Hopfield networks [18]. However, to our best knowledge, there are no any theoretical results of either dynamics or statics yet. The goal of this paper is to analytically study the dynamics of Hopfield model for a sparsely connected topology whose degree distribution is not restricted to a specific distribution (e.g. Poisson) but can take arbitrary forms. Another question investigated in this paper is how the degree distribution of connection topology influences the network performance, especially whether there exists an optimal degree distribution given a fixed number of nodes and connections. Let us consider a system of N spins or neurons, the state of the spins takes si (t) = ±1 and updates synchronously with the following probability, Prob[si (t + 1) |hi (t)] = eβsi(t+1)hi(t) 2 cosh (βhi (t)) , (1) where β is the inverse temperature and the local field of neuron i is defined hi (t) = Jijsj (t) . (2) We store q = αN random patterns ξµ = (ξ 1 , . . . , ξ N) in networks, where α is called the loading ratio. The couplings are given by the Hebb rule, Jij = j , (3) where Cij is the adjacency matrix (Cij = 1 if j is connected to i, Cij = 0 otherwise). In contrast to spin glasses or many other physical systems, the interactions between biological neurons are not symmetric: neuron i may in- fluence neuron j even if neuron j has no influence on neuron i. So in our model, Cij and Cji are chosen independently. Degree of spin i, ki = j=1Cij, denotes the number of spins that are connected to i. We consider the case that neurons are sparsely connected, it means that N → ∞, ki → ∞ but ki/N → 0. For example, we can take ki = O(lnN). And in this paper, the degrees of neurons are set as an arbitrary distribution p (ki = k). We use g (·) to express the transfer function, si (t+ 1) = g (hi(t)) . (4) Without loss of generality, let us consider the case to retrieve ξ1. We define m (t) as the overlap parameter between network state s (t) and the first pattern ξ1 as m (t) = ξ1i si (t) . (5) Then the local field at time t can be represented by hi (t) = j 6=i j sj (t) + j 6=i j sj (t) , (6) where the first term is the signal from ξ1 and the second one is crosstalk noise from other patterns. Our aim is to determine the form of the local field in the thermodynamic limit N → ∞. We apply the law of large numbers to the signal term and find that it converges to ξ1i m (t) in the thermodynamic limit. To show this point intuitively, we can simply replace the signal term by its average, j 6=i j sj (t) =ξ j sj (t) . (7) This formula is exact in the thermodynamic limit because the whole system is assumed to be self-averaging. Since Cij and ξ 1 are independent of each other, we can write the average of product as the product of average, j sj (t) = ξ1i 〈Cij〉 ξ1j sj (t) . (8) Using definition of ki together with Eq. (5), we have 〈Cij〉 = ki/N and m (t) = ξ1j sj (t) . So we have following formula, j 6=i j sj (t) = ξ m (t) . (9) Taking a closer look at the second term in Eq. (6), if all the terms in the sum (with regard to µ) are independent, we are able to apply the central limit theorem to it. As pointed out in Ref. [8], two conditions are essential for the independence of terms in the sum: first is that almost all feedback loops are eliminated, and the second is that with probability 1, any two neurons have different clusters of ancestors, i.e. they will remain independent because they receive inputs from two trees which have no neurons in common. In our model, because of the sparsely connected architecture together with the high asymmetry of synaptic connections, two conditions are both satisfied. Thus the second term in Eq. (6) converges to a zero-mean Gaussian form N (q−1)ki where (q−1)ki is the variance of Gaussian noise. Then the local field of neuron i can be expressed by hi (t) = ξ m (t) +N (q − 1) ki . (10) Note that similar treatment of local field can also be found in [7]. Then the average state of neuron i was given formally by 〈si (t + 1)〉 = dz (2π) −z2/2 m (t) + (q − 1) ki , (11) where 〈〉ξ1 stands for averaging over distribution of ξ i , and P (ξ) = [δ(ξ + 1) + δ(ξ − 1)] /2. When self-averaging is assumed, the average of neuron state in the next time can be obtained by taking average over all N neurons, 〈s (t+ 1)〉 = dz (2π) −z2/2 m (t) + (q − 1) ki .(12) Using the concept of degree distribution, we only need to take average over the degree distributions as 〈s (t+ 1)〉 = dkp (k) dz (2π) −z2/2 m (t) + (q − 1) k .(13) The overlap parameters are obtained in the similar way, m (t+ 1) = dkp (k) dz (2π) −z2/2 m (t) + (q − 1) k .(14) When focusing on the most interested case of zero temperature (β → ∞), transfer function g (·) is replaced by sgn (·). From Eq. (14) one gets m (t+ 1)= dkp (k) m(t)+ (q−1)k dz (2π) −z2/2 m(t)+ (q−1)k dz (2π) −z2/2 m(t)+ (q−1)k dz (2π) −z2/2 m(t)+ (q−1)k dz (2π) −z2/2 Then, the last equation can be further simplified to m (t+ 1) = dkp (k) erf m (t) (q − 1) /k  , (16) where erf (u) = −x2/2 dx. (17) This finishes the Signal-to-Noise derivation of overlap parameter at zero tem- perature. As long as the degree distribution of network is determined, using Eq. (14-17), one can calculate temporal evolution of overlap parameters up to an arbitrary time step. Using auxiliary thermal fields γ (t) to express the stochastic dynamics [7], it is easy to extend the method to arbitrary temperatures by averaging the zero temperature results over the auxiliary fields. s (t+ 1) = g (h (t) + γ (t) /β) , (18) and the probability density of γ (t) is given by p (γ (t)) = 1− tanh2 (γ (t)) . (19) 0 2 4 6 8 10 in theory in simulation Fig. 1. Time evolution of overlap parameters for Hopfield network with delta func- tion degree distribution. Initial overlaps range from 1.0 to 0.1 (top to bottom). N is 50000. Each neuron has 100 degrees and 20 patterns are stored in networks. For illustrative examples, we apply our theory to networks with some specific degree distributions and numerical simulations are performed to verify the theoretical results. In all of our numerical simulations, we set N = 5 × 104 and the average degree k̄ = 100, varying only the arrangement of connections. Each neuron is connected on average to 0.2% of the other neurons compared to ∼ 0.1% in the mouse cortex [20]. The first numerical experiment is the delta function p (k) = δ k − k̄ , (20) which means that every neuron has exactly k̄ connections. In practice, the con- nection topology is generated by randomizing a regular lattice which average degree is k̄. Time evolutions of overlap parameters from theory and numerical simulations are plotted in Fig. 1. The second degree distribution is binomial distribution which comes from a Erdös-Renyi random graph [21] (see the left panel of Fig. 2) p (k) = CkN . (21) The temporal evolution of overlap parameters are presented in the right panel 0 2 4 6 8 10 in theory in simulation 0 50 100 150 200 Degree Fig. 2. Left panel: the normalized binomial degree distribution of networks. Right panel: the temporal evolution of overlap parameters for Hopfield network with de- gree distribution shown in the left panel (Erdös-Renyi random graph). Initial over- laps range from 1.0 to 0.1 (top to bottom). N = 50000, k̄ = 100, and 20 patterns are stored in networks. 0 2 4 6 8 10 in theory in simulation 54.59815 148.41316 403.42879 9.11882E-4 0.00248 0.00674 0.01832 0.04979 0.13534 0.36788 Degree Fig. 3. Left panel: the normalized power-law degree distribution (log-log scale). Right panel: the time evolution of overlap parameters for Hopfield network with degree distribution plotted in the left panel. Initial overlaps range from 1.0 to 0.1 (top to bottom). N = 50000, k̄ = 100, and 20 patterns are stored in networks. of Fig. 2. The third one is power-law distribution (see the left panel of Fig. 3) p (k) = k̄2k−3, (22) 0 5 10 15 20 1.0 A 18.0 18.5 19.0 Fig. 4. Theoretical comparison of time evolutions of overlap parameters in networks with the same average degree but different degree distributions. Degree distribution of A is the delta function, B is binomial, and C is power-law. The inset shows in detail that performance of network with delta function degree distribution is slightly better than that with binomial distribution (Erdös-Renyi random graph). N = 50000, k̄ = 100, and 20 patterns are stored in networks. which is of great importance because it may comes from preferential attach- ment in the growth process of neurons [22]. The right panel of Fig. 3 shows the temporal evolutions of overlap parameters. It is obvious that the theoretical results from our scheme are consistent with the simulations for the above degree distributions. Emerging naturally from the above statements, which form of degree distribution is the best one? To investigate how degree distributions influence the performance of networks, we theoretically compare time evolutions of overlap parameters with delta function (A), binomial (B), and power-law degree distribution (C) in Fig. 4. The inset shows the details near stationary states. It indicates that network with delta degree distribution performs slightly better than that with binomial distribution. The most rapid degradation in overlap occurs in network with power-law distribution. This behavior can be interpreted as follows. Using Eq. (16), it is easy to find that an individual neuron with fewer degrees suffers more perturbations from crosstalk noise. In the case of power-law degree dis- tribution, degrees are not uniformly distributed in networks and there are too many neurons with small number of connections, which leads to negative per- formance of the entire networks. However, note that despite the disadvantages of power-law distribution, hubs (subset of networks which has higher degrees) 0 50 100 150 200 Degree 0 5 10 15 20 17 18 19 Fig. 5. Theoretical comparison of time evolutions of overlap parameters for Hopfield networks for uniform degree distributions with different width. Degree distribution of A is delta function. Left panel: the uniform degree distributions with various width, 50 (B), 100 (C), 150 (D), and 200 (E). The inset in right panel shows in detail that the performance of A is slightly better than that of B. N = 50000, k̄ = 100, and 55 patterns are stored in networks. in networks may be useful for partial storage [17]. In addition, for verifying the above statements, we construct the special cases that the degree distributions of networks is uniform with various widthes, 50 (B), 100 (C), 150 (D), 200 (E), and the delta function A which width is 0 (see the left panel of Fig. 5). The temporal evolutions of overlaps are plotted in the right panel of Fig. 5, and the inset shows the detailed information near stationary states. It was found that the much more widespread degree distribution tends to induce worse performance of networks. In summary, the transient dynamics of sparsely connected Hopfield model with arbitrary degree distributions is studied in this paper. It was found that the delta function degree distribution is optimal in terms of network performance, and there is a gradual improvement for network performance with increasing sharpness of its degree distribution. We would like to emphasize that the model investigated in this paper is a simple relaxation to real network topology, by neglecting loops in it. But Ref. [23] suggested that the feedback loops together with their correlations exist in networks and play important role in network dynamics even in the case of sparsely connected systems. It would be interesting to investigate Hopfield model with real complicated topology influenced both by degree distributions and loops (feedbacks and correlations). Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant No. 10305005 and by the Special Fund for Doctor Programs at Lanzhou University. One of us (PZ) thanks Dong Liu for useful suggestions. References [1] Hopfield, J. J. (1982). Neural Networks and Physical Systems with Emergent Collective Computational Abilities. Proc. Nat. Acad. Sci. USA, 79(8), 2554- 2558. [2] Amit, D. J., Gutfreund, H., & Sompolinsky, H. (1985). Spin-glass models of neural networks. Phys. Rev. A, 32(2), 1007-1018. [3] Amit, D. J., Gutfreund H., & Sompolinsky, H. (1985). Storing Infinite Numbers of Patterns in a Spin-Glass Model of Neural Networks. Phys. Rev. Lett., 55, 1530-1533. [4] Coolen, A. C. C. (2000). Statistical Mechanics of Recurrent Neural Networks II. arXiv:cond-mat/0006011. [5] Amari, S., & Maginu, K. (1988). Statistical neurodynamics of associative memory. Neural Networks, 1, 63-73. [6] Okada, M. (1995). A Hierarchy of Macrodynamical Equations for Associative Memory. Neural Networks, 8, 833-838. [7] Bolle, D., Blanco, J. B., & Verbeiren T. (2004). The signal-to-noise analysis of the Little-Hopfield model revisited. J. Phys. A, 37, 1951-1969. [8] Derrida, B., Gardner E., & Zippelius, A. (1987). An exactly solvable asymmetric neural network model. Europhys. Lett., 4, 167-173. [9] Patrick, A. E., & Zagrebnov, V. A. (1990). Parallel dynamics for an extremely diluted neural network. J. Phys. A, 23, L1323-L1337. [10] Theumann, W. K. (2003). Mean-field dynamics of sequence processing neural networks with finite connectivity. Physica A, 328, 1-12. [11] Zhang, P., & Chen, Y. (2007). Statistical neurodynamics for sequence processing neural networks with finite dilution. Lect. Note Comput. Sci., 4491, 1144-1152. [12] Wemmenhove, B., & Coolen, A. C. C. (2003). Finite connectivity attractor neural networks. J. Phys. A, 36, 9617-9633. [13] Castillo, I. P., & Skantzos, N. S. (2004). The Little-Hopfield model on a Random Graph. J. Phys. A, 37, 9087-9099. http://arxiv.org/abs/cond-mat/0006011 [14] Stephan, K.E., Kamper, L., Bozkurt, A., Burns, G.A.P.C., Young, M.P., & Koter, R. (2001). Advanced database methodology for the collation of connectivity data on the Macaque brain (CoCoMac). Phil. Trans. R. Soc. Lond. B Biol. Sci. 356, 1159-1186; Cherniak, C. (1994). Component placement optimization in the brain. J. Neurosci. 14, 2418-2427; Scannell, J. W., Burns, G. A. P. C., Hilgetag, C. C., ONeill, M. A., & Young, M. P. (1999). The Connectional Organization of the Cortico-thalamic System of the Cat. Cerebral Cortex 9, 277-299. [15] Watts D. J., & Strogatz, S. H. (1998). Collective dynamics of ’small-world’ networks. Nature(London), 393, 440-442. [16] Simard, D., Nadeau L., & Krögerar, H. (2005). Fastest learning in small world neural networks. Phys. lett. A 336(11), 8-15; Li, C. & Chen, G. (2003). Stability of a neural network model with small-world connections. Phys. Rev. E 68, 052901-4; Davey, N. & Adams, R. (2004). High capacity associative memories and connection constraints. Connection Science 16(1), 47-65; Davey N., Christianson, B. & Adams, R. (2004). High capacity associative memories and small world networks. Neural Networks Proceedings 1, 182; Stauffer, D., Aharony A., da, da Fontoura Costa L., & Adler, J. (2003). Efficient Hopfield pattern recognition on a scale-free neural network. Euro. Phys. J. B 32, 395-399. [17] McGraw, P. N., & Menzinger, M. (2003). Topology and computational performance of attractor neural networks. Phys. Rev. E, 68, 047102-047105. [18] Torres, J. J., Munoz, M. A., Marro, J., & Garrido, P. L. (2003). Influence of topology on the performance of a neural network, arXiv:cond-mat/0310205. [19] Kim, B. J. (2004). Performance of networks of artificial neurons: The role of clustering. Phys. Rev. E, 69, 045101-045104. [20] Braitenberg, V. & Schüz, A. (1998). Cortex: Statistics and Geometry of Neruonal Connectivity. Springer-Verlag, Berlin. [21] Erdös, P., & Renyi, A. (1959). On random graphs. Publ. Math. (debrecen) 6, 290-297. [22] Barabási, A. L., & Albert, R. (1999). Emergence of scaling in random networks, Science, 286, 509-512; Barabási, A. L., Albert, R., & Jeong, H. (1999). Mean- field theory for scale-free random networks. Physica A, 272, 173-187. [23] Zhang, P., & Chen Y. (2007). Topology and dynamics of attractor neural networks: the role of loopiness, arxiv:cond-mat/0703405. http://arxiv.org/abs/cond-mat/0310205 http://arxiv.org/abs/cond-mat/0703405 References

0704.1009

Lectures on derived and triangulated categories

LECTURES ON DERIVED AND TRIANGULATED CATEGORIES BEHRANG NOOHI These are the notes of three lectures given in the International Workshop on Noncommutative Geometry held in I.P.M., Tehran, Iran, September 11-22. The first lecture is an introduction to the basic notions of abelian category theory, with a view toward their algebraic geometric incarnations (as categories of modules over rings or sheaves of modules over schemes). In the second lecture, we motivate the importance of chain complexes and work out some of their basic properties. The emphasis here is on the notion of cone of a chain map, which will consequently lead to the notion of an exact triangle of chain complexes, a generalization of the cohomology long exact sequence. We then discuss the homotopy category and the derived category of an abelian category, and highlight their main properties. As a way of formalizing the properties of the cone construction, we arrive at the notion of a triangulated category. This is the topic of the third lecture. Af- ter presenting the main examples of triangulated categories (i.e., various homo- topy/derived categories associated to an abelian category), we discuss the prob- lem of constructing abelian categories from a given triangulated category using t-structures. A word on style. In writing these notes, we have tried to follow a lecture style rather than an article style. This means that, we have tried to be very concise, keeping the explanations to a minimum, but not less (hopefully). The reader may find here and there certain remarks written in small fonts; these are meant to be side notes that can be skipped without affecting the flow of the material. With a few insignificant exceptions, the topics are arranged in linear order. A word on references. The references given in these lecture notes are mostly sugges- tions for further reading and are not meant for giving credit or to suggest originality. Acknowledgement. I would like to thank Masoud Khalkhali, Matilde Marcolli, and Mehrdad Shahshahani for organizing the excellent workshop, and for inviting me to participate and lecture in it. I am indebted to Masoud for his continual help and support. At various stages of preparing these lectures, I have benefited from discussions with Snigdhayan Mahanta . I thank Elisenda Feliu for reading the notes and making useful comments and, especially for creating the beautiful diagrams that I have used in Lecture 2! http://arXiv.org/abs/0704.1009v1 2 BEHRANG NOOHI Contents Lecture 1: Abelian categories 3 1. Products and coproducts in categories 3 2. Abelian categories 4 3. Categories of sheaves 7 4. Abelian category of quasi-coherent sheaves on a scheme 9 5. Morita equivalence of rings 10 6. Appendix: injective and projective objects in abelian categories 11 Lecture 2: Chain complexes 13 1. Why chain complexes? 13 2. Chain complexes 14 3. Constructions on chain complexes 15 4. Basic properties of cofiber sequences 17 5. Derived categories 20 6. Variations on the theme of derived categories 21 7. Derived functors 23 Lecture 3: Triangulated categories 24 1. Triangulated categories 24 2. Cohomological functors 27 3. Abelian categories inside triangulated categories; t-structures 28 4. Producing new abelian categories 30 5. Appendix I: topological triangulated categories 31 6. Appendix II: different illustrations of TR4 31 References 32 LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 3 Lecture 1: Abelian categories Overview. In this lecture, we introduce additive and abelian categories, and dis- cuss their most basic properties. We then concentrate on the examples of abelian categories that we are interested in. The most fundamental example is the cate- gory of modules over a ring. The next main class of examples consists of various categories of sheaves of modules over a space; a special type of these examples, and a very important one, is the category of quasi-coherent sheaves on a scheme. The idea here is that one can often recover a space from an appropriate category of sheaves on it. For example, we can recover a scheme from the category of quasi- coherent sheaves on it. This point of view allows us to think of an abelian category as “a certain category of sheaves on a certain hypothetical space”. One might also attempt to extract the “ring of functions” of this hypothetical space. Under some conditions this is possible, but the ring obtained is not unique. This leads to the notion of Morita equivalence of rings. The slogan is that the “ring of functions” on a noncommutative space is well defined only up to Morita equivalence. 1. Products and coproducts in categories References: [HiSt], [We], [Fr], [Ma], [GeMa]. Let C be a category and {Xi}i∈I a set of objects in C. The product i∈I Xi is an object in C, together with a collection of morphisms i∈I Xi → Xi, satisfying the following universal property: Given any collection of morphisms fi : Y → Xi, i∈I Xi The coproduct i∈I Xi is an object in C, together with a collection of morphisms ιi : Xi → i∈I Xi, satisfying the following universal property: Given any collection of morphisms gi : Xi → Y , i∈I Xi Y Xigi Remark 1.1. Products and coproducts may or may not exist, but if they do they are unique up to canonical isomorphism. Example 1.2. 4 BEHRANG NOOHI 1. C = Sets: = disjoint union, =cartesian product. 2. C = Groups: = free product, =cartesian product. 3. C = UnitalCommRings: finite = ⊗Z, =cartesian product. 4. C = Fields: = does not exist, =does not exist. 5. C = R-Mod=left R-modules, R a ring: finite finite = ⊕, the usual direct sum of modules. Exercise. Show that in R-Mod there is a natural morphism i∈I Xi → i∈I Xi, and give an example where this is not an isomorphism. 2. Abelian categories References: [HiSt], [We], [Fr], [Ma], [GeMa]. We discuss three types of categories: Ab-categories, additive categories, and abelian categories. Each type of category has more structure/properties than the previous one. AnAb-category is a category C with the following extra structure: each HomC(X,Y ) is endowed with the structure of an abelian group. We require that composition is linear: u(f + g) = uf + ug, (f + g)v = fv + gv. An additive functor between Ab-categories is a functor that induces group ho- momorphisms on Hom-sets. An additive category is a special type of Ab-category. More precisely, an additive category is an Ab-category with the following properties: ◮ There exists a zero object 0 such that ∀X, HomC(0, X) = {0} = HomC(X, 0). ◮ Finite sums exist. (Equivalently, finite products exist; see Proposition 2.1.) Before defining abelian categories, we discuss some basic facts aboutAb-categories. Proposition 2.1. Let C be an Ab-category with a zero object, and let {Xi}i∈I be a finite set of objects in C. Then Xi exists if an only if Xi exists. In this case, Xi and Xi are naturally isomorphic. Proof. Exercise. (Or See [HiSt].) � LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 5 Notation. Thanks to Proposition 2.1, it makes sense to use the same symbol ⊕ for both product and coproduct (of finitely many objects) in an additive category. Exercise. In Example 1.2, which ones can be made into additive categories? A morphism f : B → C in an additive category is called a monomorphism if −→ C, f ◦ g = 0 ⇒ g = 0. A kernel of a morphism f : B → C in an additive category is a morphism i : A→ B such that f ◦ i = 0, and that for every g : X → B with f ◦ g = 0 Proposition 2.2. If i is a kernel for some morphism, then i is a monomorphism. The converse is not always true. We can also make the dual definitions. A morphism f : B → C in an additive category is called an epimorphism if −→ D, h ◦ f = 0 ⇒ h = 0. A cokernel of a morphism f : B → C in an additive category is a morphism p : C → D such that p ◦ f = 0, and such that for every h : B → Y with h ◦ f = 0 Proposition 2.3. If p is a cokernel for some morphism, then p is an epimorphism. The converse is not always true. Remark 2.4. Kernels and cokernels may or may not exist, but if they do they are unique up to a canonical isomorphism. Example 2.5. 1. In A = R-Mod kernels and cokernels always exist. 6 BEHRANG NOOHI 2. Let A be the category of finitely generated free Z-modules. Then kernels and cokernels always exist. (Exercise. What is the cokernel of Z −→ Z?) 3. Let A be the category of C-vector spaces of even dimension. Then kernels and cokernels do not always exist in A. (Give an example.) The same thing is true if A is the category of infinite dimensional vector spaces. An abelian category is an additive category A with the following properties: ◮ Kernels and cokernels always exist in A. ◮ Every monomorphism is a kernel and every epimorphism is a cokernel. Main example. For every ring R, the additive category R-Mod is abelian. Remark 2.6. Note that an Ab-category is a category with an extra structure. How- ever, an additive category is just an Ab-category which satisfies some property (but no additional structure). An abelian category is an additive category which satisfies some more properties. Exercise. In Example 2.5 show that (1) is abelian, but (2) and (3) are not. In (2) the map Z −→ Z is an epimorphism that is not the cokernel of any morphism, because it is also a monomorphism! Proposition 2.7. Let f : B → C be a morphism in an abelian category. Let i : ker(f) → B be its kernel and p : C → coker(f) its cokernel. Then there is a natural isomorphism coker(i) ∼−→ ker(p) fitting in the following commutative diagram: ker(f) coker(f) coker(i) ker(p) Corollary 2.8. Every morphism f : B → C in an abelian category has a unique (up to a unique isomorphism) factorization fmono Corollary 2.9. In an abelian category, mono + epi ⇔ iso. The object I (together with the two morphisms fepi and fmono) in the above corol- lary is called the image of f and is denoted by im(f). The morphism fmono factors through every monomorphism into C through which f factors. Dually, fepi factors through every epimorphism originating from B through which f factors. Either of these properties characterizes the image. LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 7 In an abelian category, a sequence 0 −→ A −→ C −→ 0 is called exact if f is a monomorphism, g is an epimorphism, and im(f) = ker(g). An additive functor between abelian categories is called exact if it takes exact sequences to exact sequences. We saw that R-modules form an abelian category for every ring R. In fact, every small abelian category is contained in some R-Mod. (A category is called small if its objects form a set.) Theorem 2.10 (Freyd-Mitchell Embedding Theorem [Fr, Mi1]). Let A be a small abelian category. Then there exists a unital ring R and an exact fully faithful functor A → R-Mod. 3. Categories of sheaves References: [Ha1], [KaSch], [Iv], [GeMa]. The main classes of examples of abelian categories are categories of sheaves over spaces. We give a quick review of sheaves and describe the abelian category structure on them. Let C be an arbitrary category (base) and A another category (values). A presheaf on C with values in A is a functor F : Cop → A. A morphism f : F → G of presheaves is a natural transformation of functors. The category of presheaves is denoted by PreSh(C,A). Typical example. Let X be a topological space, and let C = OpenX be the category whose objects are open sets of X and whose morphisms are inclusions. Let A = Ab, the category of abelian groups. A presheaf F of abelian groups on X consists of: ⊲ A collection of abelian groups F(U), ∀ U open; ⊲ “Restriction” homomorphisms F(U) → F(V ), ∀ V ⊆ U. Restriction homomorphisms should respect triple inclusions W ⊆ V ⊆ U and be equal to the identity for U ⊆ U . Example 3.1. 1. (Pre)sheaf of continuous functions on a topological space X. The assign- U 7→ OcontX (U) = {continuous functions on U} 8 BEHRANG NOOHI is a presheaf on X . The restriction maps are simply restriction of func- tions. In fact, this is a presheaf of rings because restriction maps are ring homomorphisms. 2. Constant presheaf. Let A be an abelian group. The assignment U 7→ A is a presheaf of groups. The restriction maps are the identity maps. Variations. There are many variations on these examples. For instance, in (1) one can take the (pre)sheaf of Cr functions on a Cr-manifold, or (pre)sheaf of holomorphic functions on a complex manifold, and so on. These are called structure sheaves. Idea: Structure sheaves encode all the information about the structure in question (e.g, Cr, analytic, holomorphic, etc.). So, for instance, a complex manifold X can be recovered from its underlying topological space Xtop and the sheaf OholoX . That is, we can think of the pair (Xtop, OholoX ) as a complex manifold. Exercise. Formulate the notion of a holomorphic map of complex manifolds purely in terms of the pair (Xtop,OholoX ). Proposition 3.2. Let C be an arbitrary category, and A an abelian category. Then PreSh(C,A) is an abelian category. The kernel and cokernel of a morphism f : F → G of presheaves are given by ker(f) : U 7→ ker −→ G(U) coker(f) : U 7→ coker −→ G(U) A presheaf F, say of abelian groups, rings etc., on X is called a sheaf if for every open U ⊆ X and every open cover {Uα} of U the sequence F(Uα) −→ F(Uα ∩ Uβ) (fα) 7→ (fα|Uα∩Uβ − fβ|Uα∩Uβ ) is exact. Example 3.3. Structure sheaves (e.g., Example 3.1.1) are sheaves! More generally, every vector bundle E → X gives rise to a sheaf of vector spaces via the assignment U 7→ E(U), where E(U) stands for the space of sections of E over U . Is a constant presheaf (Example 3.1.2) a sheaf? For an abelian category A, we denote the full subcategory of PreSh(X,A) whose objects are sheaves by Sh(X,A). Proposition 3.4. The category Sh(X,A) is abelian. Remark 3.5. LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 9 1. Monomorphisms in Sh(X,A) are the same as the ones in PreSh(X,A), but epimorphisms are different: f : F → G is an epimorphism if for every open U and every a ∈ G(U), ∃ {Uα}, open cover of U, such that: ∀α, a|Uα is in the image of f(Uα) : F(Uα) → G(Uα). 2. Kernels in Sh(X,A) are defined in the same way as kernels in PreSh(X,A), but cokernels are defined differently: if f : F → G is a morphism of sheaves, the cokernel of f is the sheaf associated to the presheaf U 7→ coker −→ G(U) We remark that there is a general procedure for producing a sheaf Fsh out of a presheaf F. This is called sheafification. The sheaf Fsh is called the sheaf associated to F and it comes with a natural morphism of presheaves i : F → Fsh which is universal among morphisms F → G to sheaves G. More details on this can be found in [Ha1]. 4. Abelian category of quasi-coherent sheaves on a scheme References: [Ha1], [GeMa]. We give a super quick review of schemes. We then look at the category of quasi-coherent sheaves on a scheme. The affine scheme (SpecR,OR) associated to a commutative unital ring R is a topological space SpecR, the space of prime ideals in R, together with a sheaf of rings OR, the structure sheaf, on it. Recall that, in the topology of SpecR (called the Zariski topology) a closed set is the set V (I) of all prime ideals containing a given ideal I. The sheaf OR is uniquely determined by the fact that, for every f ∈ R, the ring of sections of OR over the open set Uf := SpecR − V (f) is the localization R(f), that is, OR(Uf ) = R(f). In particular, OR(SpecR) = R. (Note: the open sets Uf form a basis for the Zariski topology on SpecR.) A scheme X consists of a pair (XZar,OX) where X is a topological space and OX is a sheaf of rings on XZar. We require that X can be covered by open sets U such that each (U,OU ) is isomorphic to an affine scheme. (Here, OU is the restriction of OX to U .) Remark 4.1. This definition is modeled on Example 3.1.1; in particular, read the paragraph after the example. 10 BEHRANG NOOHI A sheaf of modules over a scheme X is a sheaf F of abelian groups on XZar such that, for every open U ⊆ XZar, F(U) is endowed with an OX(U)-module structure (and restriction maps respect the module structure). A sheaf of modules F over X is called quasi-coherent if for every inclusion of the form SpecS = V ⊆ U = SpecR of open sets in XZar we have F(V ) ∼= S ⊗R F(U). Proposition 4.2. The category OX-Mod of OX-modules on a scheme X is an abelian category. The full subcategory Quasi-CohX is also an abelian category. Example 4.3. To an R-module M there is associated a quasi-coherent sheaf M̃ on X = SpecR which is characterized by the property that M̃(Uf ) = M(f), for every f ∈ R. In fact, every quasi-coherent sheaf on SpecR is of this form. More precisely, we have an equivalence of categories ([Ha1], Corollary II.5.5) Quasi-CohSpecR ∼= R-Mod. The category Quasi-CohX is a natural abelian category associated with a scheme X . This allows us to do homological algebra on schemes (e.g., sheaf coho- mology). The following reconstruction theorem states that, indeed, Quasi-CohX captures all the information about X . Proposition 4.4 (Gabriel-Rosenberg Reconstruction Theorem [Ro]). A scheme X can be reconstructed, up to isomorphism, from the abelian category Quasi-CohX . More generally, Rosenberg [Ro], building on the work of Gabriel, associates to an abelian category A a topological space Spec A together with a sheaf of rings OA on it. In the case where A = Quasi-CohX , the pair (Spec A,OA) is naturally isomorphic to (XZar,OX). The above theorem is a starting point in non-commutative algebraic geometry. It means that one can think of the abelian category Quasi-CohX itself as a space. 5. Morita equivalence of rings References: [We], [GeMa]. We saw that, by the Gabriel-Rosenberg Reconstruction Theorem, we can regard the abelian category Quasi-CohX as being “the same” as the scheme X itself. The Gabriel-Rosenberg Reconstruction Theorem has a more classical precursor. Theorem 5.1 (Gabriel [Ga, Fr]). Let R be a unital ring (not necessarily com- mutative). Then R-Mod has a small projective generator (e.g., R itself), and is closed under arbitrary coproducts. Conversely, let A be an abelian category with a small projective generator P which is closed under arbitrary coproducts. Let R = (EndP )op. Then A ∼= R-Mod. LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 11 Remark 5.2. This, however, is not exactly a reconstruction theorem: the projective gen- erator P is never unique, so we obtain various rings S = EndP such that A ∼= S-Mod. Nevertheless, all such rings are regarded as giving the same “noncommutative scheme”. So, from the point of view of noncommutative geometry they are the same. By the Gabriel-Rosenberg Reconstruction Theorem, if such R and S are both commutative, then they are necessarily isomorphic. Two rings R and S are called Morita equivalent if R-Mod ∼= S-Mod. Example 5.3. For any ring R, the ring S = Mn(R) of n × n matrices over R is Morita equivalent to R. (Proof. In the above theorem take P = R⊕n, or use the next theorem.) The following characterization of Morita equivalence is important. The proof is not hard. Theorem 5.4. Let R and S be rings. Then the following are equivalent: i. The categories R-Mod and S-Mod are equivalent. ii. There is an S−R bimodule M such that the functor M⊗R− : R-Mod → S- Mod is an equivalence of categories. iii. There is a finitely generated projective generator P for R-Mod such that S ∼= EndP . 6. Appendix: injective and projective objects in abelian categories We will need to deal with injective and projective objects in the next lecture, so we briefly recall their definition. An object P in an abelian category is called projective if it has the following lifting property: Equivalently, P is projective if the functor HomA(P,−) : A → Ab takes exact se- quences in A to exact sequences of abelian groups. An object I in an abelian category is called injective if it has the following exten- sion property: 12 BEHRANG NOOHI Equivalently, I is injective if the functor HomA(−, I) : A op → Ab takes exact se- quences in A to exact sequences of abelian groups. We say that A has enough projectives (respectively, enough injectives), if for every object A there exists an epimorphism P → A where P is projective (respectively, a monomorphism A→ I where I is injective). In a category with enough projectives (respectively, enough injectives) one can always find projective resolutions (respectively, injective resolutions) for objects. The category R-Mod has enough injectives and enough projectives. The categories PreSh(C,Ab), PreSh(C, R-Mod), Sh(C,Ab), Sh(C, R-Mod), and Quasi-CohX have enough injectives, but in general they do not have enough projectives. Exercise. Show that the abelian category of finite abelian groups has no injective or projective object other than 0. Show that in the category of vector spaces every object is both injective and projective. LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 13 Lecture 2: Chain complexes Overview. Various cohomology theories in mathematics are constructed from chain complexes. However, taking the cohomology of a chain complex kills a lot of information contained in that chain complex. So, it is desirable to elevate the cohomological constructions to the chain complex level. In this lecture, we introduce the necessary machinery for doing so. The main tool here is the mapping cone construction, which should be thought of as a homotopy cokernel construction. We discuss in some detail the basic properties of the cone construction; we will see, in particular, how it allows us to construct long exact sequences on the chain complex level, generalizing the usual cohomology long exact sequences. In practice, one is only interested in chain complexes up to quasi-isomorphism. This leads to the notion of the derived category of chain complexes. The cone construction is well adapted to the derived category. Derived categories provide the correct setting for manipulating chain complexes; for instance, they allow us to construct derived functors on the chain complex level. 1. Why chain complexes? Chain complexes arise naturally in many areas of mathematics. There are two main sources for (co)chain complexes: chains on spaces and resolutions. We give an example for each. Chains on spaces. Let X be a topological space. There are various ways to associate a chain complex to X . For example, singular chains, cellular chains (if X is a CW complex), simplicial chains (if X is triangulated), de Rham complex (if X is differentiable), and so on. These complexes encode a great deal of topological information about X in terms of algebra (e.g., homology and cohomology). Observe the following two facts: • Such constructions usually give rise to a chain complex of free modules. • Various chain complexes associated to a given space are chain homotopy equivalent (hence, give rise to the same homology/cohomology). Resolutions. Let us explain this with an example. Let R be a ring, and M and N R-modules. Recall how we compute Tori(M,N). First, we choose a free resolution · · · → P−2 → P−1 → P 0 ︸ ︷︷ ︸ Then, we define Tori(M,N) = H −i(P • ⊗N). Observe the following two facts: 14 BEHRANG NOOHI • The complex P • is a complex of free modules. • Any two resolutions P • and Q• of M are chain homotopy equivalent (hence Tor is well-defined). Conclusion. In both examples above, we replaced our object with a chain complex of free modules that was unique up to chain homotopy. We could then extract infor- mation about our object by doing algebraic manipulations (e.g., taking homology) on this complex. Therefore, the real object of interest is the (chain homotopy class of) a chain complex (of, say, free modules). Of course, instead of working with the complex itself, one could choose to work with its (co)homology, but one loses some information this way. Remark 1.1. Complexes of projective modules (e.g, projective resolutions) work equally well as complexes of free modules. Sometimes, we are in a dual situation where complexes of injective modules are more appropriate (e.g., in computing Ext groups). 2. Chain complexes References: [We], [GeMa], [Ha2], [HiSt], [KaSch], [Iv], and any book on algebraic topology. We quickly recall a few definitions. We prefer to work with cohomological index- ing, so we work with cochain complexes. A cochain complex C in an abelian category A is a sequence of objects in A · · · −→ Cn−1 −→ Cn −→ Cn−1 −→ · · · , d2 = 0. Such a sequence is in general indexed by Z, but in many applications one works with complexes that are bounded below, bounded above, or bounded on both sides. A chain map f : B → C between two cochain complexes is a sequence of fn : Bn → Cn of morphisms such that the following diagram commutes · · · · · · · · · · · · A null homotopy for a chain map f : B → C is a sequence sn : Bn → Cn−1 such fn = sn+1 ◦ dn + dn−1 ◦ sn, ∀n LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 15 We say two chain maps f, g : B → C are chain homotopic if f − g is null ho- motopic. We say f : B → C is a chain homotopy equivalence if there exists g : C → B such that f ◦ g and g ◦ f are chain homotopy equivalent to the corre- sponding identity maps. To any chain complex C one can associate the following objects in A: Zn(C) = ker(Cn −→ Cn+1) Bn(C) = im(Cn−1 −→ Cn) Hn(C) = Zn(C)/Bn(C) Exercise. Show that a chain map f : B → C induces morphisms on each of the above objects. In particular, we have induced morphisms Hn(f) : Hn(B) → Hn(C) for all n. Show that if f and g are chain homotopic, then they induce the same map on cohomology. In particular, a chain homotopy equivalence induces isomorphisms on cohomologies. A chain map f : B → C is called a quasi-isomorphism if it induces isomorphisms on all cohomologies. (So every chain homotopy equivalence is a quasi-isomorphism.) Exercise. When is a chain complex C quasi-isomorphic to the zero complex? (Hint: exact.) When is C chain homotopy equivalent to the zero complex? (Hint: split exact.) A short exact sequence of chain complexes is a sequence 0 → A→ B → C → 0 of chain maps which is exact at every n ∈ Z. Such an exact sequence gives rise to a long exact sequence in cohomology · · · (A) → H (B) → H (A) → H (B) → H → · · · A short exact sequence as above is called pointwise split (or semi-split) if every epimorphism Bn → Cn admits a section, or equivalently, every monomorphism An → Bn is a direct summand. Exercise. Does pointwise split imply that the maps ∂ in the above long exact sequence are zero? Is the converse true? (Answer: both implications are false.) 3. Constructions on chain complexes References: [We], [GeMa], [Ha2], [HiSt], [KaSch], [Iv]. In the following table we list a few constructions that are of great importance in algebraic topology. The left column presents the topological construction, and the right column is the cochain counterpart. The right column is obtained from the left column as follows: imagine that the spaces are triangulated and translate the topological constructions in terms of the simplices; this gives the chain complex picture. The cochain complex picture is ob- tained by the appropriate change in indexing (from homological to cohomological). 16 BEHRANG NOOHI The reader is strongly encouraged to do this as an exercise. It is extremely important to pay attention to the orientation of the simplices, and keep track of the signs accordingly. Spaces Cochains f : X → Y continuous map f : B → C cochain map Cyl(f) = (X × I) ∪f YX Y Mapping cylinderCyl(f)n := Bn ⊕ Bn+1 ⊕ CnBn ⊕ Bn+1 ⊕ Cn d−→ Bn+1 ⊕ Bn+2 ⊕ Cn+1(b′, b, c) 7→ `dB(b′) − b,−dB(b), f(b) + dC(c)´ dB − idB 0 0 −dB 0 0 f dC Cone(f) = Cyl(f)/XX Y Mapping coneCone(f) = Cyl(f)/B Cone(f)n := Bn+1 ⊕ Cn Bn+1 ⊕ Cn −→ Bn+2 ⊕ Cn+1 (b, c) 7→ − dB(b), f(b) + dC(c) Cone(X) = Cone(idX) = X × I/X × {0}X ConeCone(B) := Cone(idB)Cone(B)n = Bn+1 ⊕ Bn Bn+1 ⊕ Bn −→ Bn+2 ⊕ Bn+1 (b, b′) 7→ − d(b), b + d(b′) Σ(X) = Cone(X)/X = X × I/X × {0, 1}X Shift (Suspension)B[1] := Cone(B)/BB[1]n := Bn+1 B[1]n −→ B[1]n+1 b 7→ −d(b) LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 17 Some history. These constructions were originally used by topologists (e.g., Puppe [Pu]) as a unified way of treating various cohomology theories for topological spaces (the so- called generalized cohomology theories [Ad]), and soon they became a standard tool in stable homotopy theory. Around the same time, Grothendieck and Verdier developed the same machinery for cochain complexes, to be used in Grothendieck’s formulation of duality theory and the Riemann-Roch theorem for schemes [Ha2]. This led to the invention of derived categories and triangulated categories [Ve]. As advocated by Grothendieck, it has become more and more apparent over time that working in derived categories is a better alternative to working with cohomology. Nowadays people consider even richer structures such as dg-categories [BoKa, Ke3], A∞-categories [Ke4], (stable) model categories ([Ho], Section 7), etc. The idea is that, the higher one goes in this hierarchy, the more higher- order cohomological information (e.g., Massey products, Steenrod operations, or other cohomology operations) is retained. In these lectures we will not go beyond derived categories. An important fact. Taking the mapping cone of f : X → Y is somewhat like taking cokernels: we are essentially killing the image of f , but softly (i.e., we make it contractible). That is why Cone(f) is sometimes called the homotopy cofiber or homotopy cokernel of f ; see [May1], §8.4, especially, the first lemma on page 58. Proposition 3.1. In the following diagram of cochain complexes, if g and h are chain equivalences, then so is the induced map on the mapping cones Cone(f) C′ Cone(f A similar statement is true for topological spaces. Remark 3.2. The above proposition is still true if we use quasi-isomorphisms instead of chain homotopy equivalences. Similarly, the topological version remains valid if we use weak equivalences (i.e., maps that induce isomorphisms on all homotopy groups) instead of homotopy equivalences. Exercise. Give an example (in both the topological and chain complex settings) to show that the above proposition is not true if we use the strict cofiber Y/f(X) instead of Cone(f). (Therefore, since we are interested in chain homotopy equiv- alence classes, or quasi-isomorphism classes, of cochain complexes, Cone(f) is a better-behaved notion than Y/f(X).) Exercise. Formulate a universal property for the mapping cone construction. 4. Basic properties of cofiber sequences References: [We], [GeMa], [KaSch], [Ha2], [Ve]. The sequence −→ C −→ Cone(f) 18 BEHRANG NOOHI (or any sequence quasi-isomorphic to such a sequence) is called a cofiber sequence. The same definition can be made with topological spaces. We list the basic prop- erties of cofiber sequences. 1. Exact sequence, basic form. An important property of cofiber sequences is that they give rise to long exact sequences. Here is a baby version which is very easy to prove and is left as an exercise. Proposition 4.1. Let B → C → Cone(f) be a cofiber sequence. Then the sequence Hn(B) → Hn(C) → Hn(Cone(f)) is exact for every n. Remark 4.2. This is of course part of a long exact sequence · · · → Hn(B) → Hn(C) → Hn(Cone(f)) → Hn+1(B) → Hn+1(C) → Hn+1(Cone(f)) → · · · that we will discuss shortly. If we use the fact that a short exact sequence of cochain complexes gives rise to a long exact sequence of cohomology groups, this is easily proven by showing that the cofiber sequence B → C → Cone(f) is chain homotopy equivalent to the short exact sequence B → Cyl(f) → Cone(f). The equivalence C ∼ Cyl(f) is given by C −→ Cyl(f) c 7→ (0, 0, c) Cyl(f) −→ C Bn ⊕Bn+1 ⊕ Cn −→ Cn (b′, b, c) 7→ f(b′) + c. Remark 4.3. We have a similar statement for a cofiber sequence X → Y → Cone(f) of topological spaces. The corresponding long exact sequence in (co)homology should then be interpreted as the long exact sequence of relative (co)homology groups. More specifically, if f is an inclusion, we have H∗(Cone(f)) ∼= H∗(Y, X) and H (Cone(f)) ∼= H (Y, X), where the right hand sides are relative (co)homology groups. 2. A short exact sequence is a cofiber sequence. We pointed out in Re- mark 4.2 that a cofiber sequence is chain equivalent (hence quasi-isomorphic) to a pointwise split short exact sequence. The converse is also true. Proposition 4.4. Consider a short exact sequence of cochain complexes 0 → B → C → C/B → 0. Then the natural chain map ϕ : Cone(f) → C/B defined by Bn+1 ⊕ Cn → Cn/Bn (b, c) 7→ c̄ is a quasi-isomorphism. If the sequence is pointwise split, with s : Cn/Bn → Cn a choice of splitting, then ϕ is a chain equivalence. The inverse is given by ψ : C/B → Cone(f) LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 19 Cn/Bn → Bn+1 ⊕ Cn sd(x) − ds(x), s(x) Exercise. Let f : B →֒ C be an inclusion ofR-modules, viewed as cochain complexes concentrated in degree 0. Show that ϕ : Cone(f) → C/B defined above is a chain equivalence if and only if f is split. 3. Iterate of a cofiber sequence. Consider a cofiber sequence −→ Cone(f). Observe that g : C → Cone(f) is a pointwise split inclusion. This implies that the mapping cone of g is naturally chain equivalent to Cone(f)/C = B[1]. More precisely, we have the following commutative diagram: Cone(f) Cone(f) Cone(g). ch. eqϕ Note that we have natural chain equivalences Cone(∂) ∼ Cone(h) ∼ coker(h) = C[1]. Indeed, a careful chasing through the above chain equivalences (for which we have given explicit formulas) shows that the sequence Cone(f) −→ B[1] −f [1] −→ C[1] is a cofiber sequence. We can now iterate this process forever and produce a long sequence · · · ∂[−1] −→ Cone(f) −→ B[1] −f [1] −→ C[1] −g[1] −→ Cone(f)[1] −∂[1] −→ · · · (⋆) in which every three consecutive terms form a cofiber sequence. Remark 4.5. A similar long exact sequence can be constructed for a map f : X → Y of topological spaces, and it is called a Puppe sequence (or cofiber sequence [May1] §8.4). A Puppe sequence, however, only extends to the right. The reason for this is that the suspension functor on the category of topological spaces is not invertible (as opposed to the shift functor for cochain complexes). The Puppe sequence can be used, among other things, to give a natural construction of the long exact (co)homology sequence of a pair. 4. Long exact sequence of a cofiber sequence. Applying Proposition 4.1 to the above long exact sequence of chain complexes, and observing that Hi(B[n]) = Hi+n(B), we obtain a long exact sequence of cohomology groups · · · → Hn(B) → Hn(C) → Hn(Cone(f)) → Hn+1(B) → Hn+1(C) → Hn+1(Cone(f)) → · · · . 20 BEHRANG NOOHI Moral. By exploiting the notion of cone of a map in a systematic way, we can elevate many basic cohomological constructions to the level of chain complexes. For this to work conveniently, one needs to invert chain equivalences so that one can treat such morphisms as isomorphisms. Indeed, for various reasons, inverting only chain equivalences is usually not enough. For example, recall that one of our motivations for working with chain complexes was that we wanted to be able to replace an objectM ∈ A with a better behaved resolution (projective or injective) of it. Since the resolution map is only a quasi-isomorphism and not a chain equivalence in general, it is only after inverting quasi-isomorphisms that we can regard an M ∈ A and its resolution as the “same”. 5. Derived categories References: [We], [GeMa], [Ha2], [KaSch], [Iv], [Ve], [Ke1], [Ke2]. Let A be an abelian category. The homotopy category K(A) is the category obtained by inverting all chain equivalences in the category Ch(A) of chain complexes. More precisely, there is a natural functor Ch(A) → K(A) which has the following universal property: If a functor F : Ch(A) → C sends chain equivalences to isomorphisms, then Ch(A) If in the above definition we replace ‘chain equivalence’ by ‘quasi-isomorphism’, we arrive at the definition of the derived category D(A). We list the basic properties of K(A) and D(A). 1. Explicit construction of K(A). We can, alternatively, define the homotopy category K(A) to be the category whose objects are the ones of Ch(A) and whose morphisms are defined by HomK(A)(B,C) := HomCh(A)(B,C)/N where N stands for the group of null homotopic maps. (Exercise. Show that this category satisfies the required universal property.) This, in particular, implies that K(A) is an additive category. LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 21 2. Cohomology functors. The cohomology functors Hn : Ch(A) → A factor through K(A) and D(A): Ch(A) 3. Explicit construction of D(A). The objects of D(A) can again be taken to be the ones of Ch(A). Description of morphisms in D(A) is, however, slightly more involved than the case of K(A). It requires the notion of calculus of fractions in a category, which we will not get into for the sake of brevity. Just to give an idea how this works, first we observe that D(A) can, equiva- lently, be defined by inverting quasi-isomorphisms in K(A). The class of quasi- isomorphisms in K(A) satisfies the axioms of the calculus of fractions ([GeMa], Definition III.2.6), and this allows one to give an explicit description of morphisms in D(A) (see [GeMa], Section III.2, [Ha2], Section I.3, or [We], Section 10.3). A consequence of this is that, for every morphism f : B → C in D(A), one can find morphisms s, t, g and h in K(A) such that s and t are quasi-isomorphisms and the following diagrams commute in D(A): Also, for any two morphisms f, f ′ : B → C, now in K(A), which become equal in D(A), there are s and t as above such that f ◦ s = f ′ ◦ s and t ◦ f = t ◦ f ′. 4. Cofiber sequences. There is a notion of cofiber sequence in both K(A) and D(A). Every morphism f : B → C fits in a sequence ⋆ as in page 19 in which every three consecutive terms form a cofiber sequence. 6. Variations on the theme of derived categories References: [We], [GeMa], [Ha2], [KaSch], [Iv], [Ke1], [Ke2]. To an abelian category A we can associate various types of homotopy or derived categories by imposing certain boundedness conditions on the chain complexes in question. The bounded below derived category D+(A) is the category obtained from in- verting the quasi-isomorphisms in the category of bounded below chain complexes 22 BEHRANG NOOHI Ch+(A). Equivalently, D+(A) is obtained from inverting the quasi-isomorphisms in the homotopy category K+(A) of Ch+(A). In the same way one can define bounded derived categories D−(A) and Db(A), where b stands for bounded on two sides. Remark 6.1. One can use the calculus of fractions in K+(A) (respectively, K−(A), Kb(A)) to give a description of morphisms in D+(A) (respectively, D−(A), Db(A)). It follows that each of these bounded derived categories can be identified with a full subcategory of D(A). In particular, Db(A) = D+(A) ∩ D−(A). We will say more on this in the next lecture. There is an alternative way of computing morphisms in bounded derived cate- gories using injective or projective resolutions. It is summarized in the following theorems. Theorem 6.2 ([GeMa], § III.5.21, [We], Theorem 10.4.8). Let I+(A) ⊂ K+(A) be the full subcategory consisting of complexes whose terms are all injective. Then the composition I+(A) K+(A) D+(A) is fully faithful. It is an equivalence if A has enough injectives. The same thing is true if we replace + by − and injective by projective: P−(A) K−(A) D−(A) . The (first part of the) above theorem says that, if I and J are bounded below complexes of injective objects, then we have HomD(A)(J, I) = HomK(A)(J, I). (This is actually true if J is an arbitrary bounded below complex; see [We], Corollary 10.4.7). In particular, if I and J are quasi-isomorphic, then they are chain equiva- lent. This is good news, because we have seen that computing Hom is much easier in K(A). Now if A has enough injectives, it can be shown that for every bounded C, there exists a bounded below complex of injectives I, and a quasi-isomorphism C ∼−→ I. This is called an injective resolution for C. So, by virtue of the above fact, injective resolutions can be used to compute morphisms in derived categories: HomD(A)(B,C) = HomK(A)(B, I). Exercise. Set Homi D(A)(B,C) := Hom(B,C[i]). Let B and C be objects in A, viewed as complexes concentrated in degree 0. Assume A has enough injectives. Show that D(A)(B,C) ∼= Ext i(B,C). LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 23 Use the second part of the theorem to give a way of computing Ext groups using a projective resolution for C, if they exist. Compute Homi K(A)(B,C) and compare it with Homi D(A)(B,C). Exercise. Give an example of a non-zero morphism in D(A), with A your favorite abelian category, which induces zero maps on all cohomologies. 7. Derived functors References: [We], [GeMa], [Ha2], [KaSch], [Iv], [Ke1], [Ke2]. To keep the lecture short, we will skip the very important topic of derived func- tors and confine ourselves to an example: the derived tensor ⊗. The reader is encouraged to consult the given references for the general discussion of derived functors, especially the all important derived hom RHom. Let A = R-Mod. The idea is that we want to have a notion of tensor product for chain complexes which is well defined on D(A). The usual tensor product of chain complexes (Definition [We], §2.7.1) does not pass to derived categories. This is because tensor product is not exact. More precisely, if A → A′ is a quasi-isomorphism, then A ⊗ B → A′ ⊗ B may no longer be. (For a counterexample, let A be a short exact sequence, A′ = 0, and B a complex concentrated in degree 0.) The usual tensor product of chain complexes DOES pass to homotopy categories. This is because a null homotopy s for a chain map A → A′ gives rise to a null homotopy s⊗B for A⊗B → A′ ⊗B (exercise). In particular, taking all complexes to be (bounded-above) complexes of projective modules, we get a well-defined tensor product − ⊗ − : P−(A) × P−(A) → P−(A). Since R-Mod has enough projectives, we obtain, via the equivalence P−(A) ∼= D−(A) of Theorem 6.2, the desired tensor product on the bounded above derived category: ⊗− : D−(A) × D−(A) → D−(A). More explicitly, A ⊗B is defined to be P ⊗Q, where P → A and Q→ B are certain chosen projective resolutions. (In fact, it is enough to resolve only one of A or B.) Exercise. Show that if M,N ∈ A are viewed as complexes concentrated in degree 0, then H−i(M ⊗N) ∼= Tori(M,N). 24 BEHRANG NOOHI Lecture 3: Triangulated categories Overview. The main properties of the cone construction for chain complexes can be formalized into a set of axioms. This leads to the notion of a triangulated category, the main topic of this lecture. A triangulated category is an additive category in which there is an abstract notion of mapping cone. The cohomology functors on chain complexes can also be studied at an abstract level in a triangulated category. The main example of a triangulated category is the derived category D(A) of chain complexes in an abelian category A. Using a t-structure on a triangulated category, we can produce an abelian category, called the heart of the t-structure. For example, there is a standard t-structure on D(A) whose heart is A. The t- structure is not unique, and by varying it we can produce new abelian categories. We show how using a torsion theory on an abelian category A we can produce a new t-structure on D(A). The heart of this t-structure is then a new abelian category B. For example, this method has been used in noncommutative geometry to “deform” the abelian category A of coherent sheaves on a torus X . The new abelian category B can then be thought of as the category of coherent sheaves on a “noncommutative deformation” of X , a noncommutative torus. 1. Triangulated categories References: [We], [GeMa], [Ha2], [Ne], [BeBeDe], [KaSch], [Iv], [Ve], [Ke1], [Ke2]. We formalize the main properties of the mapping cone construction and define triangulated categories. Let T be an additive category equipped with an auto-equivalence X 7→ X [1] called shift (or translation). By a triangle in T we mean a sequence X → Y → Z → X [1]. We sometimes write this as A triangulation on T is a collection of triangles, called exact (or distinguished) triangles, satisfying the following axioms: TR1. a) X −→ X −→ 0 −→ X [1] is an exact triangle. b) Any morphism f : X → Y is part of an exact triangle. c) Any triangle isomorphic to an exact triangle is exact. TR2. X → X [1] exact ⇔ Y → X [1] −f [1] → Y [1] exact. TR3. In the diagram LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 25 X Y Z X [1] X ′[1] if the horizontal rows are exact, and the left square commutes, then the dotted arrow can be filled (not necessarily uniquely) to make the diagram commute. TR4 (short imprecise version). Given f : X → Y and g : Y → Z, we have the following commutative diagram in which every pair of same color arrows is part of an exact triangle U V W Idea: “(Z/X)/(Y/X) = Z/Y )”. See Appendix II for a more precise state- ment. A triangulated category is an additive category T equipped with an auto- equivalence X 7→ X [1] and a triangulation. A triangle functor (or a exact functor)1 is a functor between triangulated cate- gories which commutes with the translation functors (up to natural transformation) and sends exact triangles to exact triangles. Remark 1.1. A triangulated category can be thought of as a category in which there are well-behaved notions of homotopy kernel and homotopy cokernel2 (hence also various other types of homotopy limits and colimits). Some immediate consequences. Let T be a triangulated category, and let X → Y → Z → X [1] be an exact triangle in T. Then, the following are true: 0. The opposite category Top is naturally triangulated. 1. Any length three portion of the sequence · · · −→ Z[−1] −h[−1] −→ X[1] −f [1] −→ Y [1] −g[1] −→ Z[1] −h[1] −→ · · · 1This is not a very good terminology, because there is also a notion of t-exact functor with respect to a t-structure. 2Indeed not quite well-behaved (read, functorial), essentially due to the fact that the morphism whose existence is required in TR3 may not be unique. This turns out to be problematic. A way to remedy this is to work with DG categories; see [BoKa]. 26 BEHRANG NOOHI is an exact triangle. 2. In the above exact sequence, any two consecutive morphisms compose to zero. Proof. Enough to check g ◦ f = 0. Apply TR1.a and TR3 to X [1] X [1] 3. Let T be any object. Then, Hom(T,X) → Hom(T, Y ) → Hom(T, Z) is exact. Therefore, the long sequence of (1) gives rise to a long exact sequence of abelian groups. The same thing is true for Hom(X,T ) → Hom(Y, T ) → Hom(Z, T ). Remark 1.2. In fact, TR3 and half of TR2 follow from the rest of the axioms. For this, see [May2]. Also see [Ne] for another formulation of TR4; especially, Remark 1.3.15 and Proposition 1.4.6. The main example. The proof of the following theorem can be found in [GeMa] or [KaSch]. (We have sketched some of the ideas in Lecture 2.) Theorem 1.3. Let A be an abelian category. Then K(A) and D(A) are triangulated categories. In both cases, the distinguished triangles are the ones obtained from cofiber sequences. Equivalently, the distinguished triangles are the ones obtained from the pointwise split short exact sequences. (To see the equivalence, note that one implication follows from Proposition 4.4 of Lecture 2. The other follows from the fact that the cofiber sequence B → C → Cone(f) is chain homotopy equivalent to the pointwise split short exact sequence B → Cyl(f) → Cone(f); see Remark 4.2 of Lecture 2.) In the case of D(A) we get the same triangulation if we take all short exact sequences. This also follows from Proposition 4.4 of Lecture 2. The following proposition allows us to produce more triangulated categories from the above basic ones. Proposition 1.4. Let A be an abelian category, and let C be a full additive sub- category of Ch(A). Let K ⊂ K(A) be the corresponding quotient category and D the localization of K with respect to quasi-isomorphisms. Assume C is closed under translation, quasi-isomorphisms, and forming mapping cones. Then K and D are triangulated categories and we have fully faithful triangle functors K → K(A) LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 27 D → D(A). Proof. The proof in the case of K is straightforward. The case of D → D(A) follows easily from the existence of calculus of fractions on K(A). (Our discussion of calculus of fractions in Lecture 2 is enough for this.) � Exercise. Show that D−(A) can, equivalently, be defined by inverting the quasi- isomorphisms in the category C of chain complexes with bounded above cohomology. The similar statement is true in the case of D+(A) and Db(A). (Hint: construct a right inverse to the inclusion ι : Ch (A) →֒ C by choosing an appropriate truncation of each complex in C; see page 29 to learn how to truncate. Show that this becomes an actual inverse to ι when we pass to derived categories.) Corollary 1.5. We have the following fully faithful triangle functors: −(A),K+(A),Kb(A) →֒ K(A), −(A),D+(A),Db(A) →֒ D(A). Proof. This is an immediate corollary of the previous proposition. (We also need to use the previous exercise.) � Important examples to keep in mind: 1. A = R-Mod. 2. A = sheaves of R-modules over a topological space. 3. A = presheaves of R-modules over a topological space. 4. A = sheaves of OX -modules on a scheme X . 5. A = quasi-coherent sheaves on a scheme X . 2. Cohomological functors References: [We], [GeMa], [BeBeDe] [Ha2], [KaSch], [Ve]. Let T be a triangulated category and A an abelian category. An additive functor H : T → A is called a cohomological functor if for every exact triangle X → Y → Z → X [1], H(X) → H(Y ) → H(Z) is exact in A. If we set Hn(X) := H(X [n]), we obtain the following long exact sequence: · · · −→ Hn−1(Z) → Hn(X) → Hn(Y ) → Hn(Z) → Hn+1(X) → Hn+1(Y ) → Hn+1(Z) → · · · Example 2.1. 28 BEHRANG NOOHI 1. Let T be any of K∗(A) or D∗(A), ∗ = ∅,−,+, b. Then the functorH : T → A, X 7→ H0(X) is a cohomological functor. 2. For any T , Hom(T,−) : T → A is a cohomological functor. Similarly, Hom(−, T ) : T → Aop is a cohomological functor. Exercise. Show that (1) is a special case of (2). 3. Abelian categories inside triangulated categories; t-structures References: [GeMa], [KaSch], [BeBeDe]. The triangulated categories associated to an abelian category A contain A as a full subcategory. More precisely, Proposition 3.1. Let T be any of K∗(A) or D∗(A), ∗ = ∅,−,+, b. Then the functor A → T defined by A 7→ · · · → 0 → A→ 0 → · · · (A is sitting in degree zero) is fully faithful. Observe that this is not the only abelian subcategory of T. For example, we have Z many copies of A in T (simply take shifts of the above embedding). We may also have abelian categories in T that are not isomorphic to A. As we saw above, the triangulated structure of T is not sufficient to recover the abelian subcategory A. What extra structure do we need on T in order to reconstruct A? A t-structure on a triangulated category T is a pair (T≤0,T≥0) of saturated (i.e., closed under isomorphism) full subcategories such that: t1. If X ∈ T≤0, Y ∈ T≥1, then Hom(X,Y ) = 0. t2. T≤0 ⊆ T≤1, and T≥1 ⊆ T≥0. t3. For every X ∈ T, there is an exact triangle A→ X → B → A[1] such that A ∈ T≤0 and B ∈ T≥1. Notation: T≤n = T≤0[−n] and T≥n = T≥0[−n]. Main example. Let T = D(A). Set ≤0 = {X ∈ D(A) | Hi(X) = 0, ∀i > 0}, ≥0 = {X ∈ D(A) | Hi(X) = 0, ∀i < 0}. The proof that this is a t-structure is not hard. The axioms (t1) and (t2) are straightforward. To verify (t3), we define truncation functors τ≤0 : T → T LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 29 τ≥1 : T → T τ≤0(X) := · · · → X −2 → X−1 → ker d→ 0 → 0 → · · · τ≥1(X) := X/τ≤0(X) = · · · → 0 → 0 → X0/ kerd→ X 1 → X2 → · · · . It is easy to see that τ≤0X → X → τ≥1X → τ≤0X [1] is exact (Lecture 2, Proposition 4.4). So, we can take A = τ≤0X and B = τ≥1X . Remark 3.2. This t-structure induces t-structures on each of D−,+,b(A). Exercise. Show that this does not give a t-structure on K(A). (Hint: show that (t1) fails by taking X = Y to be an exact complex that is not contractible, e.g., a non-split short exact sequence.) The truncation functors discussed above can indeed be defined for any t-structure. Proposition 3.3 ([BeBeDe], §1.3). Let T be a triangulated category equipped with a t-structure (T≤0,T≥0). i. The inclusion T≤n →֒ T admits a right adjoint τ≤n : T → T ≤n, and the inclusion T≥n →֒ T admits a left adjoint τ≥n : T → T ii. For every X ∈ T and every n ∈ Z, there is a unique d that makes the triangle τ≤nX −→ X −→ τ≥n+1X −→ τ≤nX [1] exact. Remark 3.4. An aisle in a triangulated category is a full saturated subcategory U that is closed under extensions, U[1] ⊆ U, and such that U → T admits a right adjoint. For example, U = T≤0 is an aisle. Conversely, any aisle U is equal to T≤0 for a unique t-structure (T≤0, T≥0); see [KeVo]. Proposition 3.5 ([BeBeDe], §1.3). Let a ≤ b. Then τ≥a and τ≤b commute, in the sense that τ≤bX X τ≥aX τ≥aτ≤bX τ≤bτ≥aX is commutative. Furthermore, ϕ is necessarily an isomorphism. We set τ[a,b] := τ≥aτ≤bX . We call T ≤0∩T≥0 the heart (or core) of the t-structure. Theorem 3.6 ([BeBeDe], §1.3). The heart C := T≤0 ∩ T≥0 is an abelian category. The functor H0 := τ[0,0] : T → C is a cohomological functor. Example 3.7. The heart of the standard t-structure on D(A) is A, and the functor τ[0,0] is the usual H 30 BEHRANG NOOHI Remark 3.8. In the above theorem, T may not be equivalent to any of D∅,−,+,b(C). The moral of the story is that, by varying t-structures inside a triangulated category (e.g., D(A)), we can produce new abelian categories. 4. Producing new abelian categories Reference: [HaReSm]. One can use t-structures to produce new abelian categories out of given ones. This technique appeared for the first time in [BeBeDe] where they alter the t- structures on various derived categories of sheaves on a space to produce new abelian categories of perverse sheaves (see loc. cit. §2 and also Theorem 1.4.10). Another way to produce t-structures is via torsion theories. For an application of this in noncommutative geometry see [Po]. A torsion theory on an abelian category A is a pair (T,F) of full additive subcat- egories of A such that: • For every T ∈ T and F ∈ F, we have Hom(T, F ) = 0. • For every A ∈ A, there is a (necessarily unique) exact sequence 0 → T → A→ F → 0, T ∈ T, F ∈ F. Example 4.1. Let A = Ab be abelian categories. Take T to be the torsion abelian groups and F the torsion-free abelian groups. Exercise. Prove that for a torsion theory (T,F) we have T⊥ = F and ⊥F = T, that F = {F ∈ A | ∀T ∈ T, Hom(T, F ) = 0}, T = {T ∈ A | ∀F ∈ F, Hom(T, F ) = 0}. Out of a given torsion theory (T,F) on an abelian category A we can construct a new abelian category B in which the roles of T and F are interchanged. We construct B as follows. Consider the following t-structure on D = D(A): ≤0 = {B ∈ D≤0 | H0(B) ∈ T}, ≥0 = {B ∈ D≥−1 | H−1(B) ∈ F}. This is easily seen to be a t-structure, so, by Theorem 3.6, B := D≤0 ∩ D≥0 is an abelian category. More precisely, B is equivalent to the full subcategory of D(A) consisting of complexes d : A−1 → A0 such that ker d ∈ F and cokerd ∈ T. Remark 4.2. Let T′ and F′ be essential images of F[1] and T in B. Then (T′,F′) is a torsion theory for B. (However, if we repeat the same process as above for this torsion theory, we may not get back A.) LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 31 5. Appendix I: topological triangulated categories References: ([Ho], Section 7), [Ma], ([We], Section 10.9). In our examples, we considered only triangulated categories that come from algebra (e.g, derived categories). There are certain triangulated categories that arise from topology (e.g, stable homotopy categories) that we have not discussed in these notes but which are worthy of attention. In line with our topological-vs-chain- complex comparison picture of Lecture 2, we say a few words about the homotopy category of spectra. Spectra. Recall from Lecture 2 that the cone construction (which is the main input in the construction of derived categories) was motivated by a topological construc- tion on spaces. One may wonder then whether one could repeat the construction of the derived categories in the topological setting. The problem that arises here is that the suspension functor X 7→ SX is not an auto-equivalence of the category of topological spaces (while its chain complex counterpart, the shift functor C 7→ C[1], is). To remedy this, one can formally “invert” the suspension functor. This gives rise to the category of Spanier-Whitehead spectra ([Ma], Chapter 1) from which we can construct a triangulated category by inverting weak equivalences, in the same way we obtained the derived category by inverting quasi-isomorphisms. What made topologists unhappy about the category of Spanier-Whitehead spec- tra is that it is not closed under arbitrary colimits. 3 Nowadays there are bet- ter categories of spectra that do not have this deficiency. They go under various names such as spectra, Ω-spectra, symmetric spectra, etc. All these categories come equipped with a notion of weak equivalence, and inverting the weak equivalences in any of these categories gives rise to the same triangulated category, the homotopy category of spectra. Understanding the homotopy category of spectra is an area of active research that is called stable homotopy theory. To see how triangulated categories emerge in this context consult the given references. 6. Appendix II: different illustrations of TR4 One way of illustrating TR4 is via the following “braid” diagram: Z W U [1] . . . Y V Y [1] . . . W [−1] U X [1] Z[1] The axiom TR4 can be stated as saying that, given the three solid strands of braids, the dotted strand can be filled so that the diagram commutes. It is understood that each of the four long sequences is exact (i.e., obtained from a distinguished triangle). Note that we are requiring the existence of only three 3Actually neither are the bounded derived categories! 32 BEHRANG NOOHI consecutive dotted arrows; the rest of the sequence is uniquely determined by these three. Another depiction of the axiom TR4 is via the following octahedron: The axiom requires that the dotted arrows could be filled. It is understood that the four mixed-color triangles are commutative, and the four unicolor triangles are exact. References [Ad] J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1995. [BeBeDe] A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), 5–171, Astérisque, 100, Soc. Math. France, Paris, 1982 [BoKa] A. I. Bondal, M. M. Kapranov, Framed triangulated categories, (Russian) Mat. Sb. 181 (1990), no. 5, 669–683; translation in Math. USSR-Sb. 70 (1991), no. 1, 93–107. [Fr] P. Freyd, Abelian categories, Harper, 1966. [Ga] P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448. [GeMa] S. I. Gelfand, Y. I. Manin, Methods of homological algebra, Springer-Verlag, Berlin, 1996. [Gr] A. Grothendieck, Sur quelques points d’algébre homolgiques, Tôhoku Math. J. 9 (1957), 119–221. [HaReSm] D. Happel, I. Reiten, S. O. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 575. [Ha1] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer-Verlag, New York–Heidelberg, 1977. [Ha2] R. Hartshorne, Residues and duality, Lecture Notes in Math. 20, Springer-Verlag, 1966. [HiSt] P. Hilton, U. Stammbach, A course in homological algebra, Graduate Texts in Mathematics 4, Springer-Verlag, New York, 1997. [Ho] M. Hovey, Model categories, Mathematical Surveys and Monographs 63, American Mathe- matical Society, Providence, RI, 1999. [Iv] B. Iversen, Cohomology of sheaves, Universitext, Springer-Verlag, Berlin, 1986. [Bernhard Keller’s Webpage] http://www.math.jussieu.fr/%7Ekeller/publ/index.html [Ke1] B. Keller, Introduction to abelian and derived categories, in Representations of reductive groups, edited by R. W. Carter and M. Geck, Cambridge University Press 1998, 41-62 (avail- able on Keller’s webpage). [Ke2] B. Keller, Derived categories and their uses, in Handbook of algebra, edited by M. Hazewinkel, Elsevier 1996 (available on Keller’s webpage). [Ke3] B. Keller, On differential graded categories, preprint (available on Keller’s webpage). [Ke4] B. Keller, Introduction to A∞-algebras and modules, Homology, Homotopy, and Applica- tions 3 (2001), 1–35 (available on Keller’s webpage). [KeVo] B. Keller, D. Vossieck, Aisles in derived categories, Bull. Soc. Math. Belg. Ser. A 40 (1988), no. 2, 239-253. [KaSch] M. Kashiwara, P. Schapira, Sheaves on Manifolds, Grundlehren der Mathematichen Wis- senschaften 292, Springer-Verlag, 1990. LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 33 [Li] J. Lipman, Notes on derived categories and derived functors, available at http://www.math.purdue.edu/∼lipman. [Ma] S. Mac Lane, Categories for the working mathematician, Second edition. Graduate Texts in Mathematics 5, Springer-Verlag, New York, 1998. [Ma] H. R. Margolis, Spectra and the Steenrod algebra North-Holland Mathematical Library 29, North-Holland Publishing Co., Amsterdam, 1983. [May1] J. P. May A concise course in algebraic topology, Chicago Lectures in Mathematics, Uni- versity of Chicago Press, Chicago, IL, 1999. [May2] J. P. May, The axioms for triangulated categories, preprint. [Mi1] B. Mitchel, The full embedding theorem, Am. J. Math. 86 (1964), 619–637. [Mi2] B. Mitchel, Rings with several objects, Adv. in Math. 8 (1972), 1–161. [Ne] A. Neeman, Triangulated categories, Annals of Mathematics Studies, 148. Princeton Univer- sity Press, Princeton, NJ, 2001. [Po] A. Polishchuk, Noncommutative two-tori with real multiplication as noncommutative projec- tive varieties, J. Geom. Phys. 50, no. 1-4, 162-187. [Pu] D. Puppe, On the formal structure of stable homotopy theory, in Colloq. Alg. Topology, Aarhus University (1962), 65-71. [Ro] A. Rosenberg, The spectrum of abelian categories and reconstructions of schemes, in Rings, Hopf Algebras, and Brauer groups, 257–274, Lectures Notes in Pure and Appl. Math. 197, Marcel Dekker, New York, 1998. [Ve] J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239, 1996. [We] C. A. Weibel, Introduction to homological algebra, Cambridge Studies in Advanced Mathe- matics 38, Cambridge University Press, 1994. E-mail address: [email protected] Mathematics Department, Florida State University, 208 Love Building, Tallahassee, FL 32306-4510, U.S.A. Lecture 1: Abelian categories 1. Products and coproducts in categories 2. Abelian categories 3. Categories of sheaves 4. Abelian category of quasi-coherent sheaves on a scheme 5. Morita equivalence of rings 6. Appendix: injective and projective objects in abelian categories Lecture 2: Chain complexes 1. Why chain complexes? 2. Chain complexes 3. Constructions on chain complexes 4. Basic properties of cofiber sequences 5. Derived categories 6. Variations on the theme of derived categories 7. Derived functors Lecture 3: Triangulated categories 1. Triangulated categories 2. Cohomological functors 3. Abelian categories inside triangulated categories; t-structures 4. Producing new abelian categories 5. Appendix I: topological triangulated categories 6. Appendix II: different illustrations of TR4 References

0704.1010

Group actions on algebraic stacks via butterflies

GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES BEHRANG NOOHI Abstract. We introduce an explicit method for studying actions of a group stack G on an algebraic stack X. As an example, we study in detail the case where X = P(n0, · · · , nr) is a weighted projective stack over an arbitrary base S. To this end, we give an explicit description of the group stack of automorphisms of P(n0, · · · , nr), the weighted projective general linear 2-group PGL(n0, · · · , nr). As an application, we use a result of Colliot-Thélène to show that for every linear algebraic group G over an arbitrary base field k (assumed to be reductive if char(k) > 0) such that Pic(G) = 0, every action of G on P(n0, · · · , nr) lifts to a linear action of G on Ar+1. 1. Introduction The aim of this work is to propose a concrete method for studying group actions on algebraic stacks. Of course, in its full generality this problem could already be very difficult in the case of schemes. The case of stacks has yet an additional layer of difficulty due to the fact that stacks have two types of symmetries: 1- symmetries (i.e., self-equivalences) and 2-symmetries (i.e., 2-morphisms between self-equivalences). Studying actions of a group stack G on a stack X can be divided into two sub- problems. One, which is of geometric nature, is to understand the two types of sym- metries alluded to above; these can be packaged in a group stack AutX. The other, which is of homotopy theoretic nature, is to get a hold of morphisms G → AutX. Here, a morphism G → AutX means a weak monoidal functor; two morphisms f, g : G→ AutX that are related by a monoidal transformation ϕ : f ⇒ g should be regarded as giving rise to the “same” action. Therefore, to study actions of G on X one needs to understand the group stack AutX, the morphisms G→ AutX, and also the transformations between such mor- phisms. Our proposed method, uses techniques from 2-group theory to tackle these problems. It consists of two steps: 1) finding suitable crossed module models for AutX and G; 2) using butterflies [No3, AlNo1] to give a geometric description of morphisms G → AutX and monoidal transformations between them. Finding a ‘suitable’ crossed module model for AutX may not always be easy, but we can go about it by choosing a suitable ‘symmetric enough’ atlas X → X. This can be used to find an approximation of AutX (Proposition 6.2), and if we are lucky (e.g., when X = P(n0, · · · , nr)) it gives us the whole AutX. 2 BEHRANG NOOHI Once crossed module models for G and AutX are found, the butterfly method reduces the action problem to standard problems about group homomorphisms and group extensions, which can be tackled using techniques from group theory. Organization of the paper Sections §3–§5 are devoted to setting up the basic homotopy theory of 2-group actions and using butterflies to formulate our strategy for studying actions. To illustrate our method, in the subsequent sections we apply these ideas to study group actions on weighted projective stacks. In §6 we define weighted projective general linear 2-groups PGL(n0, n1, · · · , nr) and prove (see Theorem 6.3) that they model AutP(n0, · · · , nr); we prove this over any base scheme S, generalizing the case S = SpecC proved in [BeNo]: Theorem 1.1. Let AutPS(n0, n1, · · · , nr) be the group stack of automorphisms of the weighted projective stack PS(n0, n1, · · · , nr) relative to an arbitrary base scheme S. Then, there is a natural equivalence of group stacks PGLS(n0, n1, · · · , nr)→ AutPS(n0, n1, · · · , nr). Here, PGLS(n0, n1, · · · , nr) stands for the group stack associated to the crossed module PGLS(n0, · · · , nr) (see §6 for the defnition). We analyze the structure of PGL(n0, n1, · · · , nr) in detail in §7. In Theorem 7.7 we make explicit the structure of PGL(n0, · · · , nr) = [Gm → G] by writing G as a semidirect product of a reductive part (product of general linear groups) and a unipotent part (successive semidirect product of linear affine groups). In light of the two step approach discussed above, Theorems 6.3 and 7.7 enable us to study actions of group schemes (or group stacks, for that matter) on weighted projective stacks in an explicit manner. This is discussed in §9. We classify actions of a group scheme G on PS(n0, n1, · · · , nr) in terms of certain central extensions of G by the multiplicative group Gm. We also describe the stack structure of the corresponding quotient (2-)stacks. As a consequence, we obtain the following (see Theorem 9.1). Theorem 1.2. Let k be a field and G a connected linear algebraic group over k, assumed to be reductive if char(k) > 0. Let X = P(n0, n1, · · · , nr) be a wighted projective stack over k. Suppose that Pic(G) = 0. Then, every action of G on X lifts to a linear action of G on Ar+1. In a forthcoming paper, we use the results of this paper (more specifically, The- orem 6.3), together with the results of [No2], to give a complete classification, and explicit construction, of twisted forms of weighted projective stacks; these are the weighted analogues of Brauer-Severi varieties. Contents 1. Introduction 1 2. Notation and terminology 3 3. Review of 2-groups and crossed modules 3 4. 2-groups over a site and group stacks 4 4.1. Presheaves of weak 2-groups over a site 5 4.2. Group stacks over C 5 4.3. Equivalences of group stacks 6 GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 3 5. Actions of group stacks 7 5.1. Formulation in terms of crossed modules and butterflies 8 6. Weighted projective general linear 2-groups PGL(n0, n1, ..., nr) 9 6.1. Automorphism 2-group of a quotient stack 9 6.2. Weighted projective general linear 2-groups 12 7. Structure of PGL(n0, n1, ..., nr) 13 8. Some examples 18 9. 2-group actions on weighted projective stacks 20 9.1. Description of the quotient 2-stack 21 References 22 2. Notation and terminology Our notation for 2-groups and crossed modules is that of [No1] and [No3], to which the reader is referred to for more on 2-group theory relevant to this work. In particular, we use mathfrak letters G, H for 2-groups or crossed modules. By a weak 2-group we mean a strict monoidal category G with weak inverses (Definition 3.1). If the inverses are also strict, we call G a strict 2-group. By a stack we mean a presheaf of groupoids (and not a category fibered in groupoids) over a Grothendieck site that satisfies the decent condition. We use mathcal letters X, Y,... for stacks. Given a presheaf of groupoids X over a site, its stackification is denoted by Xa. We use the same notation for the sheafification of a presheaf of sets (or groups). Them-dimensional general linear group scheme over SpecR is denoted by GL(m,R). When R = Z, this is abbreviated to GL(m). The corresponding projectivized gen- eral linear group scheme is denoted by PGL(m); this notation does not conflict with the notation PGL(n0, n1, · · · , nr) for a weighted projective general linear 2-group (§6) because in the latter case we always assume r ≥ 1. 3. Review of 2-groups and crossed modules A strict 2-group is a group object in the category of groupoids. Equivalently, a strict 2-group is a strict monoidal groupoid G in which every object has a strict inverse; that is, multiplication by an object induces an isomorphism from G onto itself. A morphism of 2-groups is, by definition, a strict monoidal functor. The weak 2-groups we will encounter in this paper are less weak than the ones discussed in [No1, No3]. We hope that this change in terminology is not too con- fusing for the reader. Definition 3.1. A weak 2-group is a strict monoidal groupoid G in which multi- plication by an object induces an equivalence of categories from G to itself. By a morphism of weak 2-groups we mean a strict monoidal functor. By a weak mor- phism we mean a weak monoidal functor. (We will not encounter weak morphisms until later sections.) The set of isomorphism classes of objects in a 2-group G is denoted by π0G; this is a group. The automorphism group of the identity object 1 ∈ ObG is denoted by π1G; this is an abelian group. 4 BEHRANG NOOHI Weak 2-groups and strict monoidal functors between them form a category W2Gp which contains the category 2Gp of strict 2-groups as a full subcategory.1 Morphisms in W2Gp induce group homomorphisms on π0 and π1. In other words, we have functors π0, π1 : W2Gp → Gp; the functor π1 indeed lands in the full subcategory of abelian groups. A morphism between weak 2-groups is called an equivalence if the induced homomorphisms on π0 and π1 are isomorphisms. Note that an equivalence may not have an inverse. The following lemma is straightforward. Lemma 3.2. Let f : H→ G be a morphism of weak 2-groups. Then f , viewed as a morphism of underlying groupoids, is fully faithful if and only if π0f : π0H→ π0G is injective and π1f : π1H→ π1G is an isomorphism. It is an equivalence of groupoids if and only if both π0f and π1f are isomorphisms. A crossed module G = [∂ : G1 → G0] consists of a pair of groups G0 and G1, a group homomorphism ∂ : G1 → G0, and a (right) action of G0 on G1, denoted −a. This action lifts the conjugation action of G0 on the image of ∂ and descends the conjugation action of G1 on itself. In other words, the following axioms are satisfied: • ∀β ∈ G1,∀a ∈ G0, ∂(βa) = a−1∂(β)a; • ∀α, β ∈ G1, β∂(α) = α−1βα. It is easy to see that the kernel of ∂ is a central (in particular abelian) subgroup of G1; we denote this abelian group by π1G. The image of ∂ is always a normal subgroup of G0; we denote the cokernel of ∂ by π0G. A morphism of crossed modules is a pair of group homomorphisms which commute with the ∂ maps and respect the actions. Such a morphism induces group homomorphisms on π0 and Crossed modules and morphisms between them form a category, which we denote by XMod. We have functors π0, π1 : XMod → Gp; the functor π1 indeed lands in the full subcategory of abelian groups. A morphism in XMod is said to be an equivalence if it induces isomorphisms on π0 and π1. Note that an equivalence may not have an inverse. There is a well-known natural equivalence of categories 2Gp ∼= XMod; see [No1], §3.3. This equivalence respects the functors π0 and π1. This way, we can think of a crossed module as a strict 2-group, and vice versa. For this reason, we may sometimes abuse terminology and use the term (strict) 2-group for an object which is actually a crossed module; we hope that this will not cause any confusion. Note that W2Gp contains 2Gp as a full subcategory. 4. 2-groups over a site and group stacks First a few words on terminology. For us a stack is presheaf of groupoids over a Grothendieck site (and not a category fibered in groupoids) that satisfies the descent condition. This may be somewhat unusual for algebraic geometers who are used to categories fibered in groupoids, but it makes the exposition simpler. Of course, it is standard that this point of view is equivalent to the one via categories fibered in groupoids. Just to recall how this equivalence works, to any category fibered in groupoids X one can associate a presheaf X of groupoids over C which is defined 1Both W2Gp and 2Gp are 2-categories but we will ignore the 2-morphisms for the time being and only look at the underlying 1-category. GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 5 as follows. By definition, X is the presheaf that assigns to an object U ∈ C the groupoids X(U) := Hom(U,X), where U stands for the presheaf of sets represented by U and Hom is computed in the category of stacks over C. Conversely, to any presheaf of groupoids one associates a category fibered in groupoids defined via the Grothendieck construction. For more on this we refer the reader to [Ho], especially §5.2. 4.1. Presheaves of weak 2-groups over a site. Let C be a Grothendieck site. Let W2GpC be the category of presheaves of weak 2-groups over C; that is, the category of contravariant functors from C to W2Gp. We define 2GpC and XModC analogously. There is a natural equivalence of categories 2GpC ' XModC. In particular, we can think of a presheaf of crossed modules as a presheaf of (strict) 2-groups. Note that W2GpC contains 2GpC as a full subcategory. Let X be a presheaf of groupoids over C. To X we associate a presheaf of weak 2- groups AutX ∈W2GpC which parametrizes auto-equivalences of X. By definition, AutX is the functor that associates to an object U in C the weak 2-group of self- equivalences of XU , where XU is the restriction of X to the comma category CU . (The ‘comma category’, or the ‘over category’, CU is the category of objects in C over U .) Notice that in the case where X is a stack, AutX, viewed as a presheaf of groupoids, is also a stack. Indeed, AutX is almost a group object in the category of stacks over C. To be more precise, AutX is a group stack in the sense of Definition 4.1 below. Let G ∈ W2GpC be a presheaf of weak 2-groups on C. We define π 0 G to be the presheaf U 7→ π0 , and π0G to be the sheaf associated to π Similarly, π 1 G is defined to be the presheaf U 7→ π1 , and π1G to be the sheaf associated to π We define π0G and π1G for a presheaf of crossed modules G ∈ XModC in a similar manner. The equivalence of categories between 2GpC and XModC respects 0 , π0, π 1 and π1. Lemma 3.2 remains valid in this setting if instead of π0 and π1 we use π 0 and π 4.2. Group stacks over C. We recall the definition of a group stack from [Bre]. We modify Breen’s definition by assuming that our group stacks are strictly as- sociative and have strict units. This is all we will need because the group stack AutX of self-equivalences of a stack X (indeed, any presheaf of groupoids X) has this property, and that is all we are concerned with in this paper. Definition 4.1 ([Bre], page 19). Let C be a Grothendieck site. By a group stack over C we mean a stack G that is a strict monoid object in the category of stacks over C and for which weak inverses exist. By a morphism of group stacks we mean a strict monoidal functor. That is, a morphism of stacks that strictly respects the monoidal structures. By a weak morphism we mean a weak monoidal functor. The condition on existence of weak inverses means that for every U ∈ ObC and every object a in the groupoid G(U), multiplication by a induces an equivalence of categories from G(U) to itself (or equivalently, an equivalence of stacks from XU to itself). This condition is equivalent to saying that, for every U ∈ ObC, X(U) is a weak 2-group. More concisely, it is equivalent to (pr,mult) −→ G× G being an equivalence of stacks. 6 BEHRANG NOOHI Remark 4.2. It is well known that a weak group stack can always be strictified to a strict one. So, theoretically speaking, the strictness of monoidal structure in Definition 4.1 is not restrictive. However, given fixed (strict) group stacks G and H, strict morphisms H → G are not adequate. We will see in the subsequent sections that when studying group actions on stacks we can not avoid weak morphisms. In this section, however, we will only discuss strict morphisms. Let grStC be the category of group stacks and strict morphisms between them (Definition 4.1); this is naturally a full subcategory of W2GpC. There are natural functors W2GpC → grStC and XModC → grStC. The former is the stackification functor that sends a presheaf of groupoids to its as- sociated stack; note that since the stackification functor preserves products, we can carry over the monoidal structure from a presheaf of groupoids to its stackification. The latter functor is obtained from the former by precomposing with the natural fully faithful functor XModC → W2GpC (see the beginning of §4.1). Given a presheaf of crossed modules [∂ : G1 → G0], the associated group stack has as un- derlying stack the quotient stack [G0/G1], where G1 acts on G0 by multiplication on the right (via ∂). Definition 4.3. Let X be a presheaf of groupoids over C. We define πpreX to be the presheaf that sends an object U in C to the set of isomorphism classes in X(U). We denote the sheaf associated to πpreX by πX. For a global section e of X, we define AutX(e) to be sheaf associated to the presheaf that sends an object U in C to the group of automorphisms, in the groupoid X(U), of the object eU ; note that when X is a stack this presheaf is already a sheaf and no sheafification is needed. Note that when G is the underlying presheaf of groupoids of a presheaf of weak 2-groups G ∈W2GpC, then π0G = πG and π1G = AutG(e), where e is the identity section of G. 4.3. Equivalences of group stacks. There are two ways of defining the notion of equivalence between group stacks. One way is to regard them as stacks and use the usual notion of equivalence of stacks. The other is to regard them as presheaves of weak 2-groups and use π0 and π1 (see §4.1). The next lemma shows that these two definitions agree. Lemma 4.4. Let G and H be group stacks, and let f : H → G be a morphism of group stacks. Then, the following are equivalent: (i) f is an equivalence of stacks; (ii) The induced maps π0f : π 0 H → π 0 G and π1f : π 1 H → π 1 G are isomorphisms of presheaves of groups; (iii) The induced maps π0f : π0H → π0G and π1f : π1H → π1G are isomor- phisms of sheaves of groups. Proof. The only non-trivial implication is (iii)⇒ (ii). In the proof we will use the following standard fact from closed model category theory. Theorem ([Hi], Theorem 3.2.13). Let M be a closed model cate- gory, L a localizaing class of morphisms in M, and ML the localized model category. Let X and Y be fibrant objects (i.e., L-local ob- jects) in ML, and let f : Y→ X be a morphism in M that is a weak GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 7 equivalence in the localized model structure ML (that is, f is an L-local weak equivalence). Then, f is a weak equivalence in M. We will apply the above theorem with M being the model structure on the category GpdC of presheaves of groupoids on C in which weak equivalences are morphisms that induce isomorphisms (of presheaves of groups) on π 0 and π and fibrations are objectwise. We take L to be the class of hypercovers. The weak equivalences in the localized model structure will then be the ones inducing isomorphism (of sheaves of groups) on π0 and π1. The main reference for this is [Ho]. Let us now prove (iii)⇒ (ii). It is shown in [Ho] that G and H are L-local objects (see §5.2 and §7.3 of [ibid.]). By hypothesis, f induces isomorphisms (of sheaves) on π0 and π1, so it is a weak equivalence in the localized model structure. Therefore, since G and H are L-local, f is already a weak equivalence in the non-localized model structure. This exactly means that π0f : π 0 H→ π 0 G and π1f : π 1 H→ π are isomorphisms of presheaves. � Lemma 4.5. Let X be a presheaf of groupoids over C and ϕ : X→ Xa its stackifi- cation. Then, we have the following (see Definition 4.3 for notation): (i) The induced morphism πX→ π(Xa) is an isomorphism of sheaves of sets; (ii) For every global section e of X, the natural map AutX(e)→ AutXa(e) is an isomorphism of sheaves of groups. Proof. This is a simple sheaf theory exercise. We include the proof of (i). Proof of (ii) is similar. First we prove that πϕ : πX → π(Xa) is injective. Let U ∈ ObC, and let x, y be element in πX(U) such that πϕ(x) = πϕ(y). We have to show that x = y. By passing to a cover of U , we may assume x and y lift to objects x̄ and ȳ in X(U). We will show that there is an open cover of U over which x̄ and ȳ become isomorphic. Since ϕ(x̄) and ϕ(ȳ) become equal in π(Xa), there is a cover {Ui} of U such that there is an isomorphism αi : ϕ(x̄|Ui) ∼−→ ϕ(ȳ|Ui) in the groupoid Xa(Ui), for every i. By replacing {Ui} with a finer cover, we may assume that αi come from X(Ui). (More precisely, αi = ϕ(βi), where βi is a morphism in the groupoid X(Ui).) This implies that, for every i, x̄|Ui and ȳ|Ui are isomorphic as objects of the groupoid X(Ui). This is exactly what we wanted to prove. Having proved the injectivity, to prove the surjectivity it is enough to show that every object x in π(Xa)(U) is in the image of ϕ, possibly after replacing U by an open cover. By choosing an appropriate cover, we may assume x lifts to Xa(U). Since Xa is the stackification of X, we may assume, after refining our cover, that x is in the image of X(U)→ Xa(U). The claim is now immediate. � Lemma 4.6. Let G = [G1 → G0] be a presheaf of crossed modules, and let G = [G0/G1] be the corresponding group stack. Then, we have natural isomorphisms of sheaves of groups πiG ∼−→ πiG, i = 1, 2. Proof. Apply Lemma 4.5. � 5. Actions of group stacks In this section we present an interpretation of an action of a group stack on a stack in terms of butterflies. We begin with the definition of an action. 8 BEHRANG NOOHI Definition 5.1. Let X be a stack and G a group stack. By an action of G on X we mean a weak morphism f : G→ AutX. We say two actions f and f ′ are equivalent if there is a monoidal transformation ϕ : f → f ′. In the case where G is a group (over the base site), it is easy to see that our definition of action is equivalent to Definitions 1.3.(i) of [Ro]. A monoidal trans- formation ϕ : f → f ′ between two such actions is the same as the structure of a morphism of G-groupoids (in the sense of Definitions 1.3.(ii) of [Ro]) on the identity map idX : X→ X, where the source and the target are endowed with the G-groupoid structures coming from the actions f and f ′, respectively. 5.1. Formulation in terms of crossed modules and butterflies. Butterflies were introduced in [No3] as a convenient way of encoding weak morphisms between 2-groups (rather, crossed modules representing the 2-groups). The theory was fur- ther extended in [AlNo1] to the relative case (over a Grothendieck site). We will use this theory to translate problems about 2-group actions on stacks to certain group extension problems. We begin by recalling the definition of a butterfly (see [No3], Definition 8.1 and [AlNo1], §4.1.3). Definition 5.2. Let G = [ϕ : G1 → G0] and H = [ψ : H1 → H0] be crossed modules. By a butterfly from H to G we mean a commutative diagram of groups H0 G0 in which both diagonal sequences are complexes, and the NE-SW sequence, that is, G1 → E → H0, is short exact. We require that ρ and σ satisfy the following compatibility with actions. For every x ∈ E, α ∈ G1, and β ∈ H1, ι(αρ(x)) = x−1ι(α)x, κ(βσ(x)) = x−1κ(β)x. A morphism between two butterflies (E, ρ, σ, ι, κ) and (E′, ρ′, σ′, ι′, κ′) is a mor- phism f : E → E′ commuting with all four maps (it is easy to see that such an f is necessarily an isomorphism). We define B(H,G) to be the groupoid of butterflies from H to G. This definition is justified by the following result (see [No3] and [AlNo1]). Theorem 5.3. Let G = [G1 → G0] and H = [H1 → H0] be crossed modules of sheaves of groups over the cite C. Let G = [G0/G1] and H = [H0/H1] be the corresponding quotient group stacks. Then, there is a natural equivalence of groupoids B(H,G) ∼= Homweak(H,G). Here, the right hand side stands for the groupoid whose objects are weak morphisms of group stacks and whose morphisms are monoidal transformations. The above result can be interpreted as follows. A butterfly as in the theorem gives rise to a canonical zigzag in XModC ∼←− E→ G, GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 9 where E = [H1 ×G1 κ·ι−→ E]. After passing to the associated stacks, it gives rise to a zigzag in grStC ∼←− E→ G, which after inverting the left map (as a weak morphism), results in a weak morphism H→ G. It follows from this description of a butterfly that π0 and π1 are functorial with respect to butterflies. Furthermore, the equivalence of Theorem 5.3 respects π0 and π1 (see Lemma 4.6). When [G1 → G0] is a crossed module model for the group stack AutX of auto- equivalences of a stack X, then it follows from Theorem 5.3 that an action of H = [H0/H1] on X is the same thing as an isomorphism class of a butterfly as in Definition 5.2. In other words, to give an action of H on X, we need to find an extension E of H0 by G1, together with group homomorphisms κ : H1 → E and ρ : E → G0 satisfying the conditions of Definition 5.2. This summarizes our strategy for studying group actions on stacks. To show its usefulness, in the subsequent sections we will apply this method to the case where X = PS(n0, n1, · · · , nr) is a weighted projective stack over a base scheme S. 6. Weighted projective general linear 2-groups PGL(n0, n1, ..., nr) In this section we introduce weighted projective general linear 2-group schemes and prove that they model self-equivalences of weighted projective stacks (Theorem 6.3). We begin by some general observations about automorphism 2-groups of quotient stacks. From now on, we assume that C = SchS is the big site of schemes over a base scheme S, endowed with a subcanonical topology (say, étale, Zariski, fppf, fpqc, etc.). 6.1. Automorphism 2-group of a quotient stack. We define a crossed module in S-schemes [∂ : G1 → G0] to be a pair of S-group schemes G0 and G1, an S-group scheme homomorphism ∂ : G1 → G0, and a (right) action of G0 on G1 satisfying the axioms of a crossed module. These are precisely the representable objects in XModSchS ; in other words, a crossed module in schemes [∂ : G1 → G0] gives rise to a presheaf of crossed modules U 7→ [∂(U) : G1(U)→ G0(U)]. We often abuse terminology and call a crossed module in schemes over S simply a strict 2-group scheme over S. The following two propositions generalize Lemma 8.2 of [BeNo]. Proposition 6.1. Let S be a base scheme. Let A be an abelian affine group scheme over S acting on a S-scheme X, and let X = [X/A] be the quotient stack. Let G be those automorphisms of X which commute with the A action; this is a sheaf of groups on SchS. We have the following: (i) With the trivial action of G on A, the natural map ϕ : A → G becomes a crossed modules in SchS-schemes. (ii) Let G be the group stack associated to [ϕ : A→ G]. Then, there is a natural morphism of group stacks G→ AutX. Furthermore, this morphism induces an isomorphism of sheaves of groups π1G ∼−→ π1(AutX). 10 BEHRANG NOOHI Proof. Part (i) is straightforward, because ϕ maps A to the center of G. Let G denote the presheaf of 2-groups associated to [ϕ : A→ G]. To prove part (ii), it is enough to construct a morphism of presheaves of 2-groups G → AutX and show that it has the required properties. Stackification of this map gives us the desired map (Lemma 4.6). Let us construct the morphism G→ AutX. We give the effect of this morphism on the sections over S. Since everything commutes with base change, the same construction works for every U → S in the site SchS and gives rise to the desired morphism. of presheaves. To define G(S)→ AutX(S), recall the explicit description of the S-points of the quotient stack [X/A]: Ob[X/A](S) = (T, α) | T an A-torsor over S α : T → X an A-map Mor[X/H](S)((T, α), (T )) = {f : T → T ′ an A-torsor map s.t. α′ ◦ f = α} Any element of g ∈ G(S) induces an automorphism of X relative to S (keep the same torsor T and compose α with the action of g on X). Also, for any element a ∈ A(S), there is a natural 2-isomorphism from the identity automorphism of X to the automorphism induced by ϕ(a) ∈ G(S) (which is by definition the same as the action of a). It is given by the multiplication action of a−1 on the torsor T (remember that A is abelian) which makes the following triangle commute Interpreted in the language of 2-groups, this gives a morphism of 2-groups G(S)→ AutX(S). To prove that G→ AutX induces an isomorphism on π1, we show that, for every U → S in the site SchS , the morphism of 2-groups G(U) → AutX(U) induces an isomorphism on π1. Again, we may assume that U = S. We know that the group of 2-isomorphisms from the identity automorphism of X to itself is naturally isomorphic to the group of global sections of the inertia stack of X. In the case X = [X/A], this is naturally isomorphic to the group of elements of A(S) which act trivially on X. Note that this group is naturally isomorphic to π1G(S). Therefore, the map G(S)→ AutX(S) induces an isomorphism on π1. � Proposition 6.2. Notation being as in Proposition 6.1, assume that X is a proper Deligne-Mumford stack over S, and that X → S has geometrically connected and reduced fibers. Also, assume that A fits in an extension 0→ A0 → A→ A/A0 → 0 where A/A0 is finite over S and A0 has geometrically connected fibers (this is auto- matics, for example, in the case where A is smooth and the number of its geometric connected components is a locally constant function on S). Then, G → AutX is fully faithful (as a morphism of presheaves of groupoids). In particular, the induced map π0G→ π0(AutX) of sheaves of groups is injective. GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 11 Proof. As in Proposition 6.1, let G denote the presheaf of 2-groups associated to [ϕ : A → G]. We need to show that, for every U → S in the site SchS , G(U) → AutX(U) is fully faithful; since AutX is a stack, it would then follow that the stackified morphism G→ AutX is also fully faithful. We may assume that U = S. By Proposition 6.1.(ii) and Lemma 3.2, it is enough to prove that if the action of g ∈ G(S) on X is 2-isomorphic to the identity, then g is of the form ϕ(a), for some a ∈ A(S). Let us fix such a 2-isomorphism. The effect of this 2-isomorphism on the A-torsor on X corresponding the point X → [X/A], viewed as an object in the groupoid [X/A](X) of X-points of [X/A], is given by an A-torsor map F : A ×S X → A ×S X which makes the following A-equivariant triangle commute: A×S X µ // X A×S X Here, A×S X is the trivial A-torsor on X and µ is the action of A on X. Precomposing F with the canonical section X → A×S X (corresponding to the identity element of A) and then projecting onto the first factor, we obtain a map f : X → A relative to S. The proposition follows from the following. Claim. The map f is constant, in the sense that it factors through an S-point a : S → A of A. Furthermore, the effect of a on X (induced from the action of A on X) is the same as the effect of g on X. Let us prove the claim. It follows from the commutativity of the above diagram that, for any point x in X, the effect of g on x is the same as the effect of f(x) on x.2 In other words, f(x)g−1 leaves x fixed. Applying this to ax instead of x, and using the fact that a and f(x)g−1 commute, we find that f(ax)g−1 also leaves x fixed, for every a ∈ A. This implies that, for any point x of X, and any a ∈ A, the element r(a, x) := f(ax)f(x)−1 leaves x fixed. Therefore, the map ρ : A ×S X → A ×S X, ρ(a, x) := (r(a, x), x) factors through the stabilizer group scheme τ : Σ→ X. Thus, we have a commutative triangle A×S X Now, consider the short exact sequence 0→ A0 → A→ A/A0 → 0, where A0 is a group scheme over S with geometrically connected fibers and A/A0 is finite over S. Since τ : Σ→ X has discrete fibers (because X is Deligne-Mumford) the restriction of ρ to A0 ×S X factors through the identity section. Hence, for every a ∈ A0 and x ∈ X (over the same point in S), r(a, x) = f(ax)f(x)−1 is the identity element of A. This implies that f : X → A is A0-equivariant (for the trivial action of A0 on A). So, we obtain an induced map λ : [X/A0]→ A (relative to S). Since [X/A0] is finite over [X/A], and [X/A] is proper over S, the structure map 2When we say a “point” of X we mean a scheme T over S and a morphism T → X relative to 12 BEHRANG NOOHI π : [X/A0]→ S is proper. From our assumptions we have that π has geometrically connected and reduced fibers. Base change then implies that π∗O[X/A0] = OS . Since A is affine over S, it follows that λ is constant, i.e., factors through a section a : S → A. Since f : X → A factors through λ, it also factors through a. By construction, the effect of a on X is the same as the effect of g on X, which is what we wanted to prove. � 6.2. Weighted projective general linear 2-groups. Since the construction of the weighted projective stacks, and also of the weighted projective general linear 2-group schemes, commutes with base change, we can work over Z. We begin with some notation. We denote the multiplicative group scheme over SpecZ by Gm,Z, or simply Gm. The affine (r + 1)-space over a base scheme S is denoted by Ar+1S ; when the base scheme is SpecR it is denoted by Ar+1R , and when the base scheme is SpecZ simply by Ar+1. Since r will be fixed throughout this section, we will usually denote Ar+1S − {0} by US . We will abbreviate USpecR and USpecZ to UR and U, respectively. We fix a Grothendieck topology on SchS that is not coarser than Zariski. Let n0, n1, · · · , nr be a sequence of positive integers, and consider the weight (n0, n1, · · · , nr) action of Gm on U = Ar+1−{0}. (That is, for every scheme T , an el- ement t ∈ Gm(T ) acts on UT by multiplication by (tn0 , tn1 , · · · , tnr ).) The quotient stack of this action is called the weighted projective stack of weight (n0, n1, · · · , nr) and is denoted by PZ(n0, n1, · · · , nr), or simply by P(n0, n1, · · · , nr). The weighted projective general linear 2-group scheme PGL(n0, n1, · · · , nr) is defined to be the 2-group scheme associated to the crossed module [∂ : Gm → Gn0,n1,··· ,nr ], where Gn0,n1,··· ,nr is the group scheme, over Z, of all Gm-equivariant (for the above weighted action) automorphisms of U. More precisely, the T -points of Gn0,n1,··· ,nr are automorphisms f : UT → UT that commute with the Gm-action. The homomorphism ∂ : Gm → Gn0,n1,··· ,nr is the one induced from the Gm-action itself. We take the action of Gn0,n1,··· ,nr on Gm to be trivial. The associated group stack is denoted by PGL(n0, n1, · · · , nr), and is called the projective general linear group stack of weight (n0, n1, · · · , nr). The following theorem says that a weighted projective general linear 2-group scheme is a model for the group stack of self-equivalences of the corresponding weighted projective stack. A special case of this theorem (namely, the case where the base scheme is C) was proved in ([BeNo], Theorem 8.1). We briefly sketch how the proof in [ibid.] can be modified to cover the general case. Theorem 6.3. Let AutP(n0, n1, · · · , nr) be the group stack of automorphisms of the weighted projective stack P(n0, n1, · · · , nr). Then, the natural map PGL(n0, n1, · · · , nr)→ AutP(n0, n1, · · · , nr) is an equivalence of group stacks. In particular, we have isomorphisms of sheaves of groups π0AutP(n0, n1, · · · , nr) ∼= π0PGL(n0, n1, · · · , nr) ∼= π0 PGL(n0, n1, · · · , nr), π1AutP(n0, n1, · · · , nr) ∼= π1PGL(n0, n1, · · · , nr) ∼= π1 PGL(n0, n1, · · · , nr) ∼= µd, GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 13 where d = gcd(n0, n1, · · · , nr) and µd stands for the multiplicative group scheme of dth roots of unity. In order to prove our main result (Theorem 6.3) we need the following result about line bundles on weighted projective stacks. For more details on this, the reader is referred to [No4]. More general results about Picard stacks of algebraic stacks can be found in [Bro]. Proposition 6.4. Let P = PS(n0, n1, · · · , nr), where S = SpecR is the spectrum of a local ring. Then every line bundle on P is of the form O(d) for some d ∈ Z. Proof. In the proof we use stack versions of Grothendieck’s base change and semi- continuity results ([Ha], III. Theorem 12.11). We will assume that R is Noetherian. In the case where R is a field, the assertion is easy to prove using the fact that the Picard group of P is isomorphic to the Weil divisor class group. To prove the general case, let x be the closed point of S = SpecR. Let L be a line bundle on P. After twisting with on appropriate O(d), we may assume Lx ∼= O. We will show that L is trivial. We have H1(Px,Lx) = H 1(Px,Ox) = 0. Hence, by semicontinuity, H1(Py,Ly) = 0 for every point y of S. Base change implies that R1f∗(L) = 0, and that R 0f∗(L) = f∗(L) is locally free (necessarily of rank 1). Therefore, f∗(L) is free of rank 1 and, by base change, H 0(Py,Ly) is 1-dimensional as a k(y)-vector space, for every y in S. In fact, this is true for every tensor power L⊗n, n ∈ Z. So, Ly is trivial for every y in S. (Note that, when k is a field, dimkH 0(Pk(n0, n1, · · · , nr),O(d)) is equal to the number of solutions of the equation a1n0 + a2n1 + · · ·+ arnr = d in non-negative integers ai.) Now let s be a generating section of f∗(L) ∼= R. It follows that f∗(s) is a generating section of L. So L is trivial. � Proof of Theorem 6.3. We apply Propositions 6.1 and 6.2 with S = SpecZ, X = Ar+1 − {0}, and H = Gm. This implies that PGL(n0, n1, · · · , nr)→ AutP(n0, n1, · · · , nr) is a fully faithful morphism of stacks. That is, for every scheme U , the morphism of groupoids PGL(n0, n1, · · · , nr)(U)→ AutP(n0, n1, · · · , nr)(U) is fully faithful. All that is left to show is that it is essentially surjective. Since PGL(n0, n1, · · · , nr) and AutP(n0, n1, · · · , nr) are both stacks, it is enough to prove this for U = SpecR, where R is a local ring. In this case, we know by Proposition 6.4 that PicP(n0, n1, · · · , nr) ∼= Z. We can now proceed exactly as in ([BeNo], Theorem 8.1). The isomorphisms stated at the end of the theorem follow from Lemma 4.4 and Lemma 4.6. � 7. Structure of PGL(n0, n1, ..., nr) In this section we give detailed information about the structure of the group Gn0,n1,··· ,nr . We show that, as a group scheme over an arbitrary base, it splits as a semi-direct product of a reductive group scheme and a unipotent group scheme. The reductive part is a product of a copies of the general linear groups. The unipotent part is a successive semi-direct product of vector groups; see Theorem 7.7. 14 BEHRANG NOOHI Throughout this section, the action of Gm on U = Ar+1−{0} means the weight (n0, n1, · · · , nr) action. To shorten the notation, we denote the group Gn0,n1,··· ,nr by G. The rank m general linear group scheme over SpecR is denoted by GL(m,R). When R = Z, this is abbreviated to GL(m). We always assume r ≥ 1. The corresponding projectivized group scheme is denoted by PGL(m); this notation does not conflict with the notation PGL(n0, n1, · · · , nr) for a weighted projective general linear 2-group as in the latter case we have at least two variables. We begin with a simple lemma. Lemma 7.1. Let R be an arbitrary ring, and let f be a global section of the structure sheaf of UR = Ar+1R − {0}, r ≥ 1. Then f extends uniquely to a global section of Ar+1R . Proof. Let Ui = SpecR[x0, · · · , xr, x−1i ] and consider the covering UR = ∪ i=1Ui. We show that the restriction fi := f |Ui is a polynomial for every i. To see this, observe that, except possibly for xi, all variables occur with positive powers in fi. To show that xi also occurs with a positive power, pick some j 6= i and use the fact that xi occurs with a positive power in fj |Ui∩Uj = fi|Ui∩Uj . Therefore, for every i, fi actually lies in R[x0, · · · , xr, x−1i ]. Since fj |Ui = fi|Uj , it is obvious that all fi are actually the same and provide the desired extension of f to UR. � From now on, we will use a slightly different notation with indices. Namely, we assume that the weights are m1 < m2 < · · · < mt, with each mi appearing exactly ri ≥ 1 times in the weight sequence (so in the previous notation we would have r + 1 = r1 + · · · + rt). We denote the corresponding projective general linear 2- group by PGL(m1 : r1,m2 : r2, · · · ,mt : rt). We use the coordinates xij , 1 ≤ i ≤ t, 1 ≤ j ≤ ri, for Ar+1. We think of xij as a variable of degree mi. We will usually abbreviate the sequence xi1, · · · , xiri to x i. Similarly, a sequence F i1, · · · , F iri of polynomials is abbreviated to Fi. Let R be a ring. The following proposition tells us how a Gm,R-equivariant automorphisms of UR looks like. Proposition 7.2. Let F : UR → UR be a Gm-equivariant map. Then F is of the form (Fi)1≤i≤t, where for every i, each component F j ∈ R[x j ; 1 ≤ i ≤ t, 1 ≤ j ≤ ri] of Fi is a weighted homogeneous polynomial of weight mi. Proof. The fact that components of F are polynomial follows from Lemma 7.1. The statement about homogeneity of F ij is a simple exercise in polynomial algebra and is left to the reader. � In the above proposition, each F ij can be written in the form F j = L j + P where Lij is linear in the variables x 1, · · · , xiri , and P j is a homogeneous polynomial of degree mi in variables x b with a < i. Let LF := (L i)1≤i≤t be the linear part of F . It is again a Gm-equivariant endomorphism of U. Proposition 7.3. Let F be as in the Proposition 7.2. The assignment F 7→ LF respects composition of endomorphisms. In particular, if F is an automorphism, then so is LF . Proof. This follows from direct calculation, or, alternatively, by using the fact that LF is simply the derivative of F at the origin. � GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 15 Corollary 7.4. There is a natural split homomorphism φ : G→ GL(r1)×GL(r2)× · · · ×GL(rt). Next we give some information about the structure of the kernel U of φ. It consists of endomorphisms F = (F ij )i,j , where F j has the form F ij = x j + P Here, P ij is a homogeneous polynomial of degree mi in variables x b with a < i. Indeed, it is easily seen that, for an arbitrary choice of the polynomials P ij , the resulting endomorphism F is automatically invertible. So, to give such an F ∈ U is equivalent to giving an arbitrary collection of polynomials {P ij}1≤i≤t,1≤j≤ri such that each P ij is a homogeneous polynomial of degree mi in variables x b with a < i. So, from now on we switch the notation and denote such an element of U by (P ij )i,j . Proposition 7.5. For each 1 ≤ a ≤ t, let Ua ⊆ U be the set of those endomor- phisms F = (P ij )i,j for which P j = 0 whenever i 6= a. Let Ka denote the set of monomials of degree ma in variables x j, i < a, and let ka be the cardinality of Ka. (In other words, ka is the number of solutions of the equation zi,j = ma in non-negative integers zi,j.) Then we have the following: (i) Ua is a subgroup of U and is canonically isomorphic to the vector group scheme Ara ⊗ AKa ∼= Araka . (Note: U1 is trivial.) (ii) If a < b, then Ua normalizes Ub. (iii) The groups Ua, 1 ≤ i ≤ t, generate U and we have Ua ∩ Ub = {1} if a 6= b. Proof of (i). The action of (P ij )i,j ∈ Ua on A r+1 is given by (x1, · · · ,xa, · · · ,xt) 7−→ (x1, · · · ,xa + Pa, · · · ,xt). So, if AKa stands for the vector group scheme on the basis Ka, there is a canonical isomorphism Ua ∼= AKa ∼= Ara ⊗ AKa . Proof of (ii). Let G = (Qij)i,j be an element in Ua and F = (P j )i,j an element in Ub. By (i), the inverse of G is G −1 = (−Qij)i,j . Let us analyze the effect of the composite G ◦ F ◦G−1 on Ar+1: (x1, · · · ,xa, · · · ,xb, · · · ,xt) G 7−→ (x1, · · · ,xa −Qa, · · · ,xb, · · · ,xt) F7−→ (x1, · · · ,xa −Qa, · · · ,xb + Rb, · · · ,xt) G7−→ (x1, · · · ,xa, · · · ,xb + Rb, · · · ,xt). Here the polynomial Rbk, 1 ≤ k ≤ rb, is obtained from P k by substituting the variables xaj with the polynomial x Proof of (iii). Easy. � 16 BEHRANG NOOHI Part (ii) implies that each Ua acts by conjugation on each of Ua+1, Ua+2,· · · , 3 To fix the notation, in what follows we let the conjugate of an automorphism f by an automorphism g to be g ◦ f ◦ g−1. Notation. Let {Ua}ta=1 be a family of subgroups of a group U which satisfies the following properties: 1) Each Ua normalizes every Ub with a < b; 2) No two distinct Ua intersect; 3) The Ua generate U . In this case, we say that U is a successive semi-direct product of the Ua, and use the notation U ∼= Ut o · · ·o U2 o U1. The following is an immediate corollary of Proposition 7.5. Corollary 7.6. There is a natural decomposition of U as a semi-direct product U ∼= Ut o · · ·o U2 o U1, where Ua ∼= Araka is the group introduced in Proposition 7.5. (Note that U1 is trivial.) In the next theorem we use the notation Am for two things. One that has already appeared is the affine group scheme of dimension m. When there is a group scheme G involved, we also use the notation Am for the trivial representation of G on Am. Theorem 7.7. There is a natural decomposition of G as a semi-direct product G ∼= Ut o · · ·o U2 o U1 o GL(r1)× · · · ×GL(rt) where Ua ∼= Araka and ka is as in Proposition 7.5. (Note that U1 is trivial.) Furthermore, for every 1 ≤ a ≤ t, the action of GL(ra) leaves each Ub invariant. We also have the following: (i) When a > b the induced action of GL(ra) on Ub is trivial. (ii) When a = b the induced action of GL(ra) on Ua is naturally isomorphic to the representation ρ ⊗ AKa , where ρ is the standard representation of GL(ra) and Ka is as in Proposition 7.5. (Recall that Ua is canonically isomorphic to Ara ⊗ AKa .) (iii) When a < b the action of GL(ra) on Ub is naturally isomorphic to the representation ⊕ 0≤l≤bmb Arbdl ⊗ ρ̂⊗l. Here ρ̂ stands for the inverse transpose of ρ, and dl is the number of mono- mials of degree mb in variables x j, i < b, i 6= a; so dl also depends on a and b. (In other words, dl is the number of solutions of the equation i 6=a zi,j = mb − lma in non-negative integers zi,j.) Proof. Let g ∈ GL(ra) and F ∈ Ub. As in the proof of Proposition 7.5.i, we analyze the effect of the composite g ◦ F ◦ g−1 on Ar+1. The element g ∈ GL(ra) acts on Ar+1 as follows: it leaves every component xji invariant if i 6= a and on the coordinates xa1 , · · · , xara it acts linearly (like the action of an ra × ra matrix on a column vector). 3All group actions in this section are assumed to be on the left. GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 17 Proof of (i). The effect of g ∈ GL(ra) only involves the variables xa1 , · · · , xara and does not see any other variable, whereas the effect of F ∈ Ub only involves the variables xij , i ≤ b. Since b < a, these two are independent of each other. That is, F and g commute. Proof of (ii). Assume F = (P ij )i,j ; so P j = 0 if i 6= a. The effect of g ◦ F ◦ g −1 can be described as follows: (x1, · · · ,xa, · · · ,xt) g 7−→ (x1, · · · ,ya, · · · ,xt) F7−→ (x1, · · · ,ya + Pa, · · · ,xt) g7−→ (x1, · · · ,xa + Qa, · · · ,xt). Here, yaj is the linear combination of x 1 , · · · , xara , the coefficients being the en- tries of the jth row of the matrix g−1. Similarly, Qaj is the linear combination of P a1 , · · · , P ara , coefficients being the entries of the j th row of the matrix g. Proof of (iii). Assume F = (P ij )i,j ; so P j = 0 if i 6= b. Let y a be as in (ii). The effect of g ◦ F ◦ g−1 can be described as follows: (x1, · · · ,xa, · · · ,xb, · · · ,xt) g 7−→ (x1, · · · ,ya, · · · ,xb, · · · ,xt) F7−→ (x1, · · · ,ya, · · · ,xb + Rb, · · · ,xt) g7−→ (x1, · · · ,xa, · · · ,xb + Rb, · · · ,xt). Here the polynomials Rbk, 1 ≤ k ≤ rb, are obtained from P k by substituting the variable xaj with y Let λ be the representation of GL(ra) on the space V of homogenous polynomials of degree mb which acts as follows: it takes a polynomial P ∈ V and substitutes the variables xaj , 1 ≤ j ≤ ra, with y j . From the description above, we see that the representation of GL(ra) on Ub is a direct sum of rb copies of λ. We will show that 0≤l≤bmb Adl ⊗ ρ̂⊗l. To obtain the above decomposition, simply note that a polynomial in V can be uniquely written in the form ∑ 0≤l≤bmb SlTl, where Tl is a homogenous polynomial of degree lma in variables x 1 , · · · , xara , and Sl is a homogenous polynomial of degree mb− lma in the rest of the variables. The action of GL(ra) leaves Sl intact and acts on Tl by the l th power of the inverse transpose of the standard representation. � The actions of various pieces in the above semi-direct product decomposition, though explicit, are tedious to write down, except for small values of t. We give some examples in §8. Let us denote Ut o · · ·o U2 o U1 by U and define the crossed module PGL(n0, n1, · · · , nr)red := [∂ : Gm → GL(r1)× · · · ×GL(rt)], where the kth factor of ∂(λ) is the size rk scalar matrix λ mk . Theorem 7.7 then implies that PGL(n0, n1, · · · , nr) ∼= U o PGL(n0, n1, · · · , nr)red. 18 BEHRANG NOOHI We think of U as the unipotent radical and PGL(n0, n1, · · · , nr)red as the reductive part of PGL(n0, n1, · · · , nr). Remark 7.8. It is perhaps useful to put the above result in the general context of algebraic group theory. Recall that every algebraic group G over a field fits in a short exact sequence 1→ U → G→ Gred → 1, where U is the unipotent radical of G and Gred is reductive. The sequence is not split in general. In our case, the group scheme Gn0,n1,··· ,nr admits such a short sequence over an arbitrary base and, furthermore, the sequence is split. The general theory of unipotent groups tells us that any unipotent group over a perfect field admits a filtration whose graded pieces are vector groups. This filtration splits, but only in the category of schemes (i.e., the splitting maps may not be group homomorphisms). In our case, however, the group scheme U admits such a filtration over an arbitrary base. Furthermore, the filtration is split group theoretically. 8. Some examples In this section we look at some explicit examples of weighted projective general linear 2-groups. Example 8.1. Weight sequence m < n,m - n. In this case we have t = 2, and r1 = r2 = 1 and k1 = 0. So G ∼= Gm ×Gm. Example 8.2. Weight sequence m < n,m | n. In this case we have t = 2, r1 = r2 = 1, and k1 = 1. So we have G ∼= Ao (Gm ×Gm). The action of an element (λ1, λ2) ∈ Gm ×Gm on an element a ∈ A is given by (λ1, λ2) · a = λ2λ More explicitly, an element in G is map of the form (x, y) 7→ (λ1x, λ2y + ax Note the similarity with the group of 2× 2 lower-triangular matrices. Example 8.3. Weight sequence n = m. We obviously have G ∼= GL(2). Example 8.4. Weight sequence 1, 2, 3. First we determine U . A typical element in U is of the form (x, y, z) 7→ (x, y + ax2, z + bx3 + cxy). We have U2 = A and U3 = A2. The action of an element a ∈ U2 on an element (b, c) ∈ U3 is given by (b− ac, c). That is, a acts on U3 = A2 by the matrix( So, U ∼= A⊕2 oA. Finally, we have G ∼= U o (Gm)3 = A⊕2 oAo (Gm)3, where the action of an element (λ1, λ2, λ3) ∈ (Gm)3 on an element (a, b, c) ∈ U is given by (λ−21 λ2a, λ 1 λ3b, λ 2 λ3c). GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 19 Example 8.5. Weight sequence 1, 2, 4. An element in U has the general form (x, y, z) 7→ (x, y + ax2, z + bx4 + cx2y + dy2). We have U2 = A and U3 = A3. The action of an element a ∈ U2 on an element (b, c, d) ∈ U3 is given by the matrix  1 −a a20 1 −2a 0 0 1 So, U ∼= A⊕3 oA. Finally, we have G ∼= U o (Gm)3 = A⊕3 oAo (Gm)3, where the action of an element (λ1, λ2, λ3) ∈ (Gm)3 on an element (a, b, c, d) ∈ U is given by (λ−21 λ2a, λ 1 λ3b, λ 2 λ3c, λ 2 λ3d). Next we look at PGL(m1 : r1,m2 : r2, · · · ,mt : rt). Recall that, as a crossed module, this is given by [∂ : Gm → G], where ∂ is the obvious map coming from the action of Gm on Ar+1, and the action of G on Gm is the trivial one. Observe that the map ∂ factors though the component GL(r1)× · · · ×GL(rt) of G. So, let us define L to be the cokernel of the following map: r1︷ ︸︸ ︷ m1 , . . . , λ ,·····, rt︷ ︸︸ ︷ mt , . . . , λ ) // GL(r1)× · · · ×GL(rt). From Theorem 7.7 we immediately obtain the following. Proposition 8.6. Let L be the group defined in the previous paragraph, and let k = gcd(m1, · · · ,mt). We have natural isomorphisms of group schemes π0 PGL(m1 : r1,m2 : r2, · · · ,mt : rt) ∼= Ut o · · ·o U2 o U1 o L, π1 PGL(m1 : r1,m2 : r2, · · · ,mt : rt) ∼= µk. Our final result is that, if all weights are distinct (that is, ri = 1), then the corresponding projective general linear 2-group is split. Proposition 8.7. Let {m1, · · · ,mt} be distinct positive integers, and consider the projective general linear 2-group PGL(m1,m2, · · · ,mt). Then, the projection map G → π0 PGL(m1,m2, · · · ,mt) splits. In particular, PGL(m1,m2, · · · ,mt) is split. That is, it is completely classified by its homotopy group schemes: π0 PGL(m1, · · · ,mt) ∼= Ut o · · ·o U2 o U1 o (Gm)t−1, π1 PGL(m1, · · · ,mt) ∼= µk. Proof. By Theorem 7.7 and Proposition 8.6 we know that G ∼= Ut o · · · o U2 o U1 o (Gm)t and π0 PGL(m1,m2, · · · ,mt) ∼= Ut o · · ·oU2 oU1 oL, where L is the cokernel of the map α : Gm (λm1 ,··· ,λmt ) // (Gm)t. So it is enough to show that the image of µ is a direct factor. Note that if we divide all the mi by their greatest common divisor, the image of α does not change. 20 BEHRANG NOOHI So, we may assume gcd(m1, · · · ,mt) = 1. Let M be a t × t integer matrix whose determinant is 1 and whose first column is (m1, · · · ,mt). The matrix M gives rise to an isomorphism µ : (Gm)t → (Gm)t whose restriction to the subgroup Gm × {1}t−1 ∼= Gm is naturally identified with α. The subgroup µ({1} × (Gm)t−1) ⊂ (Gm)t is the desired complement of the image of α. � Corollary 8.8. Let m, n be distinct positive integers, and let k = gcd(m,n). Then PGL(m,n) is a split 2-group. That is, it is classified by its homotopy groups: π0 PGL(m,n) ∼= Gm, if m < n,m - n AoGm, if m < n,m | n π1 PGL(m,n) ∼= µk. (In the case m | n, the action of Gm on A in the cross product A o Gm is simply the multiplication action.) Proof. Everything is clear, except perhaps a clarification is in order regarding the parenthesized statement. Observe that the Gm appearing in the cross product AoGm is indeed the cokernel of the map α : Gm (λm,λn) // (Gm)2, which is naturally identified with the subgroup {1} ×Gm ⊂ (Gm)2. Therefore, by the formula of Example 8.2, the action of an element λ ∈ Gm on an element a ∈ A is given by λa. � Finally, for the sake of completeness, we include the following. Proposition 8.9. The 2-group PGL(k, k, · · · , k), k appearing t times, is given by the following crossed module: (λk,··· ,λk)// GL(t)]. We have π0 PGL(k, · · · , k) ∼= PGL(t) and π1 PGL(k, · · · , k) ∼= µk. In particular, PGL(1, 1, · · · , 1), 1 appearing t times, is equivalent to the group scheme PGL(t). 9. 2-group actions on weighted projective stacks In this section we combine the method developed in §3–§5 with the results about the structure of AutP(n0, n1, · · · , nr) to study 2-group actions on a weighted pro- jective stack P(n0, n1, · · · , nr). The goal is to illustrate how one can classify 2-group actions using butterflies and how to describe the corresponding quotient 2-stacks. Below, all group scheme are assumed to be flat and of finite presentation over a fixed base S. Let H be a group stack and [ψ : H1 → H0] a presentation for it as a crossed module in schemes. By Theorem 5.3, to give an action of H on P(n0, n1, · · · , nr) is equivalent to giving a butterfly diagram }} ρ && H0 Gn0,n1,··· ,nr GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 21 In other words, to give an action of H on X, we need to find a central extension 1→ Gm → E → H0 → 1 of H0 by Gm, together with • a lift κ : H1 → E of ψ to E such that κ(βσ(x)) = x−1κ(β)x, for every β ∈ H1 and x ∈ E; • an extension of the weighted action of Gm to a linear action of E on Ar+1 which is trivial on the image of κ. The following result is more or less immediate from the above description of a group action. Theorem 9.1. Let k be a field and H a connected linear algebraic group over k, assumed to be reductive if char k > 0. Let X = P(n0, n1, · · · , nr) be a wighted projective stack over k. Suppose that Pic(H) = 0. Then, every action of H on X lifts to an action of H on Ar+1 via a homomorphism H → Gn0,n1,··· ,nr . Proof. An action of H on X lifts to Ar+1 if and only if the corresponding butterfly is equivalent to a strict one. By ([AlNo1], Proposition 4.5.3) this is equivalent to the central extension 1→ Gm → E → H → 1 being split. By ([C-T], Corollary 5.7) such central extensions are classified by Pic(H), which in our case is assumed to be trivial. Any choice of a splitting amounts to a lift of the action of H to Ar+1. � This result is essentially saying that the obstruction to lifting the H-action from P(n0, n1, · · · , nr) to Ar+1 lies in Pic(H). 9.1. Description of the quotient 2-stack. Given an action of a group stack H on the weighted projective stack P(n0, n1, · · · , nr), we can use the associated butterfly to get information about the quotient 2-stack [P(n0, n1, · · · , nr)/H]. First, notice that [κ : H1 → E] is also a crossed module in schemes. Denote the associated group stack by H′. It acts on P(n0, n1, · · · , nr) via ρ. We have [P(n0, n1, · · · , nr)/H] ∼= [(Ar+1 − {0})/H′]. Note that the right hand side is the quotient stack of the action of a group stack on an honest scheme, namely, Ar+1 − {0}. It is easy to describe its 2-stack structure by looking at cohomologies of the NW-SE sequence κ−→ E ρ−→ Gn0,n1,··· ,nr of the butterfly. Set [P(n0, n1, · · · , nr)/H]1 := [(Ar+1 − {0})/ cokerκ], [P(n0, n1, · · · , nr)/H]0 := [(Ar+1 − {0})/ im ρ]. (Here, cokerκ and im ρ are the sheaf theoretic cokernel and image of the corre- sponding maps which we assume are representable.) Then, [P(n0, n1, · · · , nr)/H]1 is the best approximation of [P(n0, n1, · · · , nr)/H] by a 1-stack, in the sense that it is obtained by killing off the 2-automorphisms of the 2-stack [P(n0, n1, · · · , nr)/H]. More precisely, there is a natural map [P(n0, n1, · · · , nr)/H]→ [P(n0, n1, · · · , nr)/H]1 22 BEHRANG NOOHI making the former a 2-gerbe over the latter for the 2-group [kerκ→ 1]. Similarly, [P(n0, n1, · · · , nr)/H]0 is an orbifold (i.e., a Deligne-Mumford stack which is gener- ically a scheme) obtained by quotienting out the generic 1-automorphisms. More precisely, there is a natural map [P(n0, n1, · · · , nr)/H]1 → [P(n0, n1, · · · , nr)/H]0 making the former a gerbe over the latter for the group ker ρ/ imκ (namely, the middle cohomology of the NW-SE sequence). It follows that the quotient 2-stack [P(n0, n1, · · · , nr)/H] is equivalent to a 1- stack if and only if κ is injective; it is an orbifold if and only if the NW-SE sequence is left exact. Example 9.2. Suppose that H is an honest group scheme and denote it by H. Then, to give an action of H on P(n0, n1, · · · , nr) is equivalent to giving a central extension 1→ Gm → E → H → 1 of H by Gm, together with a linear action of E on Ar+1 extending the weighted action of Gm. We have [P(n0, n1, · · · , nr)/H] ∼= [(Ar+1 − {0})/E], which is an honest 1-stack. Example 9.3. Suppose that H is the group stack associated to [A → 1], where A is an abelian group scheme. We rename H to A[1]. Then, to give an action of A[1] on P(n0, n1, · · · , nr) is equivalent to giving a character κ : A → µd ⊂ Gm, where d is the greatest common divisor of (n0, n1, · · · , nr). The quotient 2-stack [P(n0, n1, · · · , nr)/A[1]] is a 1-stack if and only if κ is injective. Assume this to be the case and identify A with the corresponding subgroup of µd. Then, roughly speaking, the quotient stack [P(n0, n1, · · · , nr)/A[1]] is obtained by killing the A in µd ⊆ Ix at every inertia group Ix of P(n0, n1, · · · , nr). (Note that the generic inertia group of P(n0, n1, · · · , nr) is µd.) For example, if the base is an algebraically closed field of characteristic prime to d, then [P(n0, n1, · · · , nr)/A[1]] ∼= P( , · · · , where a is the order of A. References [AlNo1] E. Aldrovandi, B. Noohi, Butterflies I: morphisms of 2-group stacks, Adv. Math. 221 (2009), no. 3, 687–773. [BeNo] K. Behrend, B. Noohi, Uniformization of Deligne-Mumford analytic curves, J. reine angew. Math. 599 (2006), 111–153. [Bre] L. Breen, On the classification of 2-gerbes and 2-stacks, Astérisque No. 225, 1994. [Bro] S. Brochard, Foncteur de Picard d’un champ algébrique, Math. Ann. 343 (2009), no. 3, 541–602. [C-T] J-L. Colliot-Thélène, Résolutions flasques des groupes linéaires connexes, J. reine angew. Math. 618 (2008), 77–133. [Ha] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. [Hi] P. S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Mono- graphs, 99, American Mathematical Society, Providence, RI, 2003. [Ho] Sh. Hollander, A homotopy theory for stacks, math.AT/0110247. GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 23 [No1] B. Noohi, Notes on 2-groupoids, 2-groups and crossed modules, Homotopy, Homology, and Applications, 9 (2007), no. 1, 75–106. [No2] , Group cohomology with coefficients in a crossed module, J. Inst. Math. Jussieu, 10 (2011), no. 2, 359–404. [No3] , On weak maps between 2-groups, preprint, arXiv:math/0506313v3 [math.CT]. [No4] , Picard stack of a weighted projective stack, preprint available at http://www.maths.qmul.ac.uk/∼noohi/research.html. [Ro] M. Romagny, Group actions on stacks and applications, Michigan Math. J. 53, Issue 1 (2005), 209–236. 1. Introduction 2. Notation and terminology 3. Review of 2-groups and crossed modules 4. 2-groups over a site and group stacks 4.1. Presheaves of weak 2-groups over a site 4.2. Group stacks over C 4.3. Equivalences of group stacks 5. Actions of group stacks 5.1. Formulation in terms of crossed modules and butterflies 6. Weighted projective general linear 2-groups PGL(n0,n1,...,nr) 6.1. Automorphism 2-group of a quotient stack 6.2. Weighted projective general linear 2-groups 7. Structure of PGL(n0,n1,...,nr) 8. Some examples 9. 2-group actions on weighted projective stacks 9.1. Description of the quotient 2-stack References

0704.1011

Modal Extraction in Spatially Extended Systems

Modal Extraction in Spatially Extended Systems Kapilanjan Krishan Department of Physics and Astronomy, University of California - Irvine, Irvine, California 92697 Andreas Handel Department of Biology, Emory University, Atlanta, Georgia 30322 Roman O. Grigoriev and Michael F. Schatz Center for Nonlinear Science and School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332 (Dated: August 10, 2021) We describe a practical procedure for extracting the spatial structure and the growth rates of slow eigenmodes of a spatially extended system, using a unique experimental capability both to impose and to perturb desired initial states. The procedure is used to construct experimentally the spectrum of linear modes near the secondary instability boundary in Rayleigh-Bénard convection. This technique suggests an approach to experimental characterization of more complex dynamical states such as periodic orbits or spatiotemporal chaos. PACS numbers: Numerous nonlinear nonequilibrium systems in nature and in technology exhibit complex behavior in both space and time ; understanding and characterizing such behav- ior (spatiotemporal chaos) is a key unsolved problem in nonlinear science [1]. Many such systems are modelled by partial differential equations; hence, in principle, their dynamics takes place in an infinite dimensional phase space. However, dissipation often acts to confine these systems’ asymptotic behavior to finite-dimensional sub- spaces known as invariant manifolds [2]. Knowledge of the invariant manifolds provides a wealth of dynamical information; thus, devising methodologies to determine invariant manifolds from experimental data would signif- icantly advance understanding of spatiotemporal chaos. In this Letter, we describe experiments in Rayleigh- Bénard convection where several slow eigenmodes and their growth rates associated with instability of roll states are extracted quantitatively. Rayleigh-Bénard convec- tion (RBC) serves well as a model spatially extended sys- tem; in particular, the spiral defect chaos (SDC) state in RBC is considered an outstanding example of spatiotem- poral chaos. In SDC the spatial structure is primarily composed of curved but locally parallel rolls, punctuated by defects (Fig. 1) [3, 4]. The recurrent formation and drift of defects in SDC is believed to play a key role in driving spatiotemporal chaos; moreover, many aspects of defect nucleation in SDC are related to defect formation observed at the onset of instability in patterns of straight, parallel rolls in RBC [10]. We obtain experimentally a low-dimensional description of the modes responsible for the nucleation of one important class of defects (disloca- tions), by first imposing reproducibly a linearly stable, straight roll state (stable fixed point) near instability on- set. This state is subsequently subjected to a set of dis- tinct, well-controlled perturbations, each of which initi- FIG. 1: Shadowgraph visualization reveals spontaneous de- fect nucleation in the spiral defect chaos state of Rayleigh- Benard convection. Two convection rolls are compressed to- gether (higher contrast region in left image). (b.) A short time later (right image), one of the rolls pinches off and two dislocations form. ates a relaxational trajectory from the disturbed state to the (same) fixed point. An ensemble of such trajectories is used to construct a suitable basis for describing the em- bedding space by means of a modified Karhunen-Loeve decomposition. The dynamical evolution of small distur- bances is then characterized by computing both finite- time Lyapunov exponents and the spatial structure of the associated eigenmodes (a similar approach was car- ried out numerically by Egolf et al. [5]). This capability is an important step toward developing a systematic way of characterizing and, perhaps, controlling, spatiotempo- rally chaotic states like SDC where localized “pivotal” events like defect formation play a central role in driving complex behavior. The convection experiments are performed with gaseous CO2 at a pressure of 3.2 MPa. A 0.697±0.06 mm-thick gas layer is contained in a 27 mm square cell, which is confined laterally by filter paper. The layer http://arxiv.org/abs/0704.1011v1 FIG. 2: Experimental images illustrate the flow response to two different perturbations applied, in turn, to the same state of straight convection rolls. Each image represents the dif- ference between the perturbed and unperturbed convection states and therefore, each image highlights the effect of a given perturbation on the flow. In the two cases shown, the localized perturbation is applied directly on a region of either downflow (left image) or upflow (right image). In all cases, the disturbance created by the perturbation decays away and the flow returns to the original unperturbed state. is bounded on top by a sapphire window and on the bottom by a sheet of 1 mm-thick glass neutral den- sity filter(NDF). The neutral density filter is bonded to a heated metal plate with heat sink compound. The temperature of the sapphire window held constant at 21.3 ◦C by water cooling. The temperature difference between the top and bottom plates ∆T is held fixed at 5.50 ± 0.01 ◦C by computer control of a thin film heater attached to the bottom metal plate. These condi- tions correspond to a dimensionless bifurcation parame- ter ǫ=(∆T−∆Tc)/∆Tc = 0.41, where ∆Tc is the temper- ature difference at the onset of convection. The vertical thermal diffusion time, computed to be 2.1 s at onset, represents the characteristic timescale for the system. We use laser heating to alter the convective patterns that occur spontaneously. A focused beam from an Ar- ion laser is directed through the sapphire window at a spot on the NDF. Absorption of the laser light by the NDF increases the local temperature of the bottom boundary and hence that of the gas, thereby inducing locally a convective upflow. The convection pattern may be modifed, either locally or globally, by rastering the hot spot created by the laser beam. The beam is steered using two galvanometric mirrors rotating about axes or- thogonal to each other under computer control. The in- tensity of the beam is modulated using an acousto-optic modulator. This technique of optical actuation is used to impose convection patterns with desired properties, to perturb these convection patterns and to change the boundary conditions. Similar approaches for manipulat- ing convective flows were explored earlier using a high intensity lamp and masks [11] in RBC and a rastered infrared laser in Bénard-Marangoni convection [12]. The experiments begin by using laser heating to im- pose a well-specified basic state of stable straight rolls. The basic state is typically arranged to be near the on- set of instability by imposing a sufficiently large pattern wavenumber such that at fixed ǫ the system’s parame- ters are near the skew-varicose stability boundary [10]. In this regime, the modes responsible for the instability are weakly damped and, therefore, can be easily excited. The linear stability of the basic state is probed by ap- plying brief pulses of spatially localized laser heating. For stable patterns, all small disturbances eventually relax. To excite all modes governing the disturbance evolution, we apply a set of localized perturbations consistent with symmetries of the (ideal) straight roll pattern – continu- ous translation symmetry in the direction along the rolls and discrete translation symmetry in the perpendicular direction plus the reflection symmetry in both directions. Therefore, localized perturbations applied across half a wavelength of the pattern form a ”basis” for all such per- turbations – any other localized perturbation at a differ- ent spatial location is related by symmetry. Localized perturbations are produced in the experiment by aiming the laser beam to create a “hot spot” whose location is stepped from the center of a (cold) downflow region to the center of an adjacent (hot) upflow region in differ- ent experimental runs. The perturbations typically last approximately 5 s and have a lateral extent of approxi- mately 0.1 mm, which is less than 10 % of the pattern wavelength. The evolution of the perturbed convective flow is mon- itored by shadowgraph visualization. A digital camera with a low-pass filter (to filter out the reflections from the Ar-ion laser) is used to capture a sequence of 256× 256 pixel images recorded with 12 bits of intensity resolution at a rate of 41 images per second. A background im- age of the unperturbed flow is subtracted from each data image; such sequences of difference images comprise the time series representing the evolution of the perturbation (Fig 2). The total power for each (difference) image in a time series is obtained from 2-D spatial Fourier transforms. The resulting time series of total power shows a strong transient excursion (corresponding to the initial response of the convective flow to a localized perturbation by laser heating) followed by exponential decay as the system re- laxes back to the stable state of straight convection rolls. We restrict further analysis to the region of exponential decay, which typically represents about 3.5 seconds of data for each applied perturbation. The dimensionality of the raw data is too high to permit direct analysis, so each difference image is first windowed (to avoid aliasing effects) and Fourier filtered by discarding the Fourier modes outside a 31 × 31 win- dow centered at the zero frequency. The discarded high- frequency modes are strongly damped and contain less than 1% of the total power. The basis of 312 Fourier modes still contains redundant information, so we fur- KL mode 1 KL mode 2 KL mode 3 KL mode 4 FIG. 3: The first four Karhunen-Loeve eigenvectors are shown for a perturbed roll state near the skew-varicose boundary of Rayleigh-Bénard convection. The eigenvectors are ordered by their eigenvalues (largest to smallest), which are propotional to the amount of power contained in the corresponding eigen- vector. ther reduce the dimensionality of the embedding space by projecting the disturbance trajectories onto the “opti- mal” basis constructed using a variation of the Karhunen- Loeve (KL) decomposition [6, 7]. The correlation matrix C is computed using the Fourier filtered time series xs(t), (xs(t)− 〈xs(t)〉t)(x s(t)− 〈xs(t)〉t) †, (1) where the index s labels different initial conditions and the origin of time t = 0 corresponds to the time when the perturbation applied by the laser is within the linear neighbourhood of the statioary state. The angle brack- ets with the subscript t indicate a time average. The eigenvectors of C are the KL basis vectors. It is worth noting that the average performed to compute C rep- resents an ensemble average over different initial condi- tions (obtained by applying different perturbations); this is distinctly different from the standard implementation of KL decomposition where statistical time averages are typically employed. The spatial structures of the first four KL vectors are shown in Fig. 3. We find that the first 24 basis vectors capture over 90% of the total power, so an embedding space spanned by these vectors represents well the relax- ational dynamics about the straight roll pattern. In our convection experiments, the KL eigenvectors show two distinct length scales. The first two dominant vectors KL basis vector 1 FIG. 4: A two-dimensional projection of the experimental time series (symbols) and the least squares fits (continuous curves). The time series have been shifted such that the fixed point is at the origin. are spatially localized, while the remaining vectors are spatially extended. This is consistent with earlier work as suggested in [4]. More quantitative information can be obtained by find- ing the eigenmodes of the system, excited by the pertur- bation, and their growth rates. These can be extracted from a nonlinear least squares fit with the cost function i,s,t i (t)− i (∞) + , (2) where xsi (t) is a projection of the perturbation at time t in the time series s onto the ith KL basis vector. In the fit k and λk are the kth eigenmode and its growth rate and Ask is the initial amplitude of the kth eigenmode excited in the experimental time series s. The fixed points xs(∞) are chosen to be different for the differing time series in the ensemble to account for a slow drift in the parameters and we assume that only n eigenmodes are excited. The results for an ensemble of time series correspond- ing to seven point perturbations applied across a wave- length of the pattern with n = 6 are shown in Figs. 4-5. (With seven different initial conditions we cannot hope to distinguish more than seven different modes). In par- ticular, Fig. 4 shows the projection of the experimental time series and the least squares fit on the plane spanned by the first two KL basis vectors. Such extraction of the linear manifold in experiments on spatially extended systems without the knowledge of the dynamical equa- tions of the system aids in the application of techniques that are well developed for low dimensional systems. The manifolds of fixed points and periodic orbits are of par- ticular interest in chaotic systems. The extracted growth rates λk are shown in Fig. 5. Not surprisingly, since the pattern is stable the growth rates are negative. The leading eigenmode (see Fig. 6) is spatially extended and shows a diagonal structure charac- 1 2 3 4 5 6 Mode number FIG. 5: The growth rates of the six dominant eigenmodes and the error bars extracted from the least squares fit. The growth rates have been non-dimensionalized by the vertical thermal diffusion time. teristic of the skew-varicose instability in an unbounded system. This is also expected as the pattern is near the skew-varicose instability boundary. The second eigen- mode is spatially localized and has no analog in spatially unbounded systems. The subsequent modes are again spatially delocalized and likely correspond to the Gold- stone modes of the unbounded system (e.g., overall trans- lation of the pattern) which are made weakly stable due to confinement by the lateral boundaries of the convec- tion cell. If the system is brought across the stability boundary, one of the modes is expected to become unstable (with- out significant change in its spatial structure), thereby determining further (nonlinear) evolution of the system towards a state with a pair of dislocation defects. We would also expect the spatially localized eigenmodes (like the second one in Fig. 6) to preserve their structure if the base state is smoothly distorted (as it would be, e.g., in the SDC state shown in Fig. 1), indicating the same type of a spatially localized instability. Our further ex- perimental studies will aim to confirm these expectations. Defects represent a type of “coherent structure” in spiral defect chaos. Previous efforts have used coher- ent structures to characterize spatiotemporally chaotic extended systems in both models [7] and experiments [8]; the use of coherent structures to parametrize the in- variant manifold was pioneered by Holmes et al. [6] in the context of turbulence. In practice coherent struc- tures are usually extracted using the Karhunen-Loéve (or proper orthogonal) decomposition of time series of system states, which picks out the statistically impor- tant patterns. This prior work has met with only limited success – indeed, it is unclear whether statistically im- portant patterns are dynamically important. An alterna- tive approach has been proposed by Christiansen et al. Eigenmode 1 Eigenmode 2 Eigenmode 3 Eigenmode 4 FIG. 6: Four dominant eigenmodes extracted from the least squares fit. [9], who suggested instead to use the recurrent patterns corresponding to the low-period unstable periodic orbits (UPO) of the system, which are dynamically more impor- tant. Our work sets the stage for attempting the more ambitious task of extraction of UPOs and their stability properties from experimental data. Summing up, we have developed an experimental tech- nique which allows extraction of quantitative information describing the dynamics and stability of a pattern form- ing system near a fixed point. This technique should be applicable to a broad class of patterns, including unstable fixed points, periodic orbits and segments of chaotic tra- jectories. Moreover, we expect that a similar approach could be applied to other pattern forming systems, con- vective or otherwise, as long as a method of spatially distibuted actuation of their state can be devised. [1] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851-1112 (1993) [2] P. Manneville, Dissipative structures and weak turbulence (Academic Press, 1993). [3] S. W. Morris, E. Bodenschatz, D. S. Cannel and G. Ahlers, Phys. Rev. Lett. 71, 2026-2029 (1993) [4] D. A. Egolf, E. V. Melnikov and E. Bodenschatz, Phys. Rev. Lett. 80, 3228-3231 (1998). [5] D. A. Egolf, I. V. Melnikov, W. Pesch and R. E. Ecke, Nature 404, 733-736 (2000) [6] P. J. Holmes, J. L. Lumley, and G. Berkooz, Turbu- lence, coherent structures, dynamical systems and sym- metry (Cambridge University Press, 1996). [7] L. Sirovich, Physica A 37, 126 (1989) [8] F. Qin, E. E. Wolf and H. C. Chang, Phys. Rev. Lett. 72(10), 1459 (1994) [9] F. Christiansen, P. Cvitanovic and V. Putkaradze, Non- linearity 10, 55-70 (1997). [10] F. H. Busse, J. Math. Phys. 46, 140 (1967) [11] M. M. Chen and J. A. Whitehead, J. Fluid Mech. 31, 1 (1968); F. H. Busse and J. A. Whitehead, J. Fluid Mech. 47, 305 (1971). [12] D. Semwogerere and M. F. Schatz, Phys. Rev. Lett. 88, 054501(2002)

0704.1012

Unstable and Stable Galaxy Models

UNSTABLE AND STABLE GALAXY MODELS YAN GUO AND ZHIWU LIN Abstract. To determine the stability and instability of a given steady galaxy configuration is one of the fundamental problems in the Vlasov theory for galaxy dynamics. In this article, we study the stability of isotropic spherical symmetric galaxy models f0(E), for which the distribution function f0 depends on the particle energy E only. In the first part of the article, we derive the first sufficient criterion for linear instability of f0(E) : f0(E) is linearly unstable if the second-order operator A0 ≡ −∆+ 4π (E){I −P}dv has a negative direction, where P is the projection onto the function space {g(E,L)}, L being the angular momentum [see the explicit formulae (27) and (26)]. In the second part of the article, we prove that for the important King model, the corresponding A0 is positive definite. Such a positivity leads to the nonlinear stability of the King model under all spherically symmetric pertur- bations. 1. Introduction A galaxy is an ensemble of billions of stars, which interact by the gravitational field which they create collectively. For galaxies, the collisional relaxation time is much longer than the age of the universe ([8]). The collisions can therefore be ignored and the galactic dynamics is well described by the Vlasov - Poisson system (collisionless Boltzmann equation) (1) ∂tf + v · ∇xf −∇xU · ∇vf = 0, ∆U = 4π f(t, x, v)dv, where (x, v) ∈ R3 × R3, f(t, x, v) is the distribution function and Uf (t, x) is its gravitational potential. The Vlasov-Poisson system can also be used to describe the dynamics of globular clusters over their period of orbital revolutions ([11]). One of the central questions in such galactic problems, which has attracted considerable attention in the astrophysics literature, of [7], [8], [11], [31] and the references there, is to determine dynamical stability of steady galaxy models. Stability study can be used to test a proposed configuration as a model for a real stellar system. On the other hand, instabilities of steady galaxy models can be used to explain some of the striking irregularities of galaxies, such as spiral arms as arising from the instability of an initially featureless galaxy disk ([7]), ([32]). In this article, we consider stability of spherical galaxies, which are the simplest elliptical galaxy models. Though most elliptical galaxies are known to be non- spherical, the study of instability and dynamical evolution of spherical galaxies could be useful to understand more complicated and practical galaxy models . By Jeans’s Theorem, a steady spherical galaxy is of the form f0(x, v) ≡ f0(E,L2), http://arxiv.org/abs/0704.1012v2 2 YAN GUO AND ZHIWU LIN where the particle energy and total momentum are |v|2 + U0(x), L2 = |x× v|2 , and U0(x) = U0 (|x|) satisfies the self-consistent Poisson equation. The isotropic models take the form f0(x, v) ≡ f0(E). The cases when f ′0(E) < 0 has been widely studied and these models are known to be linearly stable to both radial ([9]) and non-radial perturbations ([2]). The well-known Casimir-Energy functional (as a Liapunov functional) (2) H(f) ≡ Q(f) + |v|2f − 1 |∇xUf |2, is constant along the time evolution. If f ′0(E) < 0, we can choose the Casimir function Q0 such that Q′0(f0(E)) ≡ −E for all E. By a Taylor expansion of H(f)−H(f0), it follows that formally the first variation at f0 is zero, that is, H(1)(f0(E)) = 0 (on the support of f0(E)), and the second order variation of H at f0 is (3) H(2)f0 [g] ≡ {f0>0} −f ′0(E) dxdv − 1 |∇xUg|2dx where Q′′(f0) = , g = f − f0 and ∆Ug = gdv. In the 1960s, Antonov ([1], [2]) proved that (4) H(2)f0 [Dh] = ∫ ∫ |Dh|2 |f ′0(E)| dxdv − |∇ψh|2 dx is positive definite for a large class of monotone models. Here D = v · ∇x −∇xU0 · ∇v, h(x, v) is odd in v and −∆ψ = Dhdv. He showed that such a positivity is equivalent to the linear stability of f0(E). In [9], Doremus, Baumann and Feix proved the radial stability of any monotone spherical models. Their proof was further clarified and simplified in [10], [37], [22], and more recently in [33], [21]. In particular, this implies that any monotone isotropic models are at least linearly stable. Unfortunately, despite its importance and a lot of research (e.g., [20], [5], [6], [13]), to our knowledge, no rigorous and explicit instability criterion of non-monotone models has been derived. When f ′0(E) changes sign, functional H is indefinite and it gives no stability information, although it seems to suggest that these mod- els are not energy minimizers under symplectic perturbations. In this paper, we first obtain the following instability criterion for general spherical galaxies. For any function g with compact support within the support of f0(E), we define the |f ′0(E)| −weighted L2 R3 ×R3 space L2|f ′0| with the norm ‖·‖|f ′0| as (5) ||h||2|f ′ |f ′0(E)|h2dxdv. UNSTABLE AND STABLE GALAXY MODELS 3 Theorem 1.1. Assume that f0(E) has a compact support in x and v, and f bounded. For φ ∈ H1, define the quadratic form (6) (A0φ, φ) = |∇φ|2dx + 4π f ′0(E) (φ− Pφ) dxdv, where P is the projector of L2|f ′0| kerD = and more explicitly Pφ is given by (18) for radial functions and (26) for general functions. If there exists φ0 ∈ H1 such that (7) (A0φ0, φ0) < 0, then there exists λ0 > 0 and φ ∈ H2, f (x, v) given by (14), such that eλ0t[f, φ] is a growing mode to the Vlasov-Poisson system (1) linearized around [f0(E), Uf0 ] . A similar instability criterion can be obtained for symmetry preserving pertur- bations of anisotropic spherical models f0 , see Remark 2. We note that the term Pφ in the instability criterion is highly non-local and this reflects the collective nature of stellar instability. The proof of Theorem 1.1 is by extending an approach developed in [25] for 1D Vlasov-Poisson, which has recently been generalized to Vlasov-Maxwell systems ([26], [28]). There are two elements in this approach. One is to formulate a family of dispersion operators Aλ for the potential, depending on a positive parameter λ. The existence of a purely growing mode is reduced to find a parameter λ0 such that the Aλ0 has a kernel. The key observation is that these dispersion operators are self-adjoint due to the reversibility of the particle trajectories. Then a continuation argument is applied to find the parameter λ0 corresponding to a growing mode, by comparing the spectra of Aλ for very small and large values of λ. There are two new complications in the stellar case. First, the essential spectrum of Aλ is [0,+∞) and thus we need to make sure that the continuation does not end in the essential spectrum.This is achieved by using some compactness property due to the compact support of the stellar model. Secondly, it is more tricky to find the limit of Aλ when λ tends to zero. For that, we need an ergodic lemma (Lemma 2.4) and use the integrable nature of the particle dynamics in a central field to derive an expression for the projection Pφ appeared in the limit. In the second part of the article, we further study the nonlinear (dynamical) stability of the normalized King model: (8) f0 = [e E0−E − 1]+ motivated by the study of the operator A0. The famous King model describes isothermal galaxies and the core of most globular clusters [24]. Such a model provides a canonical form for many galaxy models widely used in astronomy. Even though f ′0 < 0 for the King model, it is important to realize that, because of the Hamiltonian nature of the Vlasov-Poisson system (1), linear stability fails to imply nonlinear stability (even in the finite dimensional case). The Liapunov functional is usually required to prove nonlinear stability. In the Casimir-energy functional (2), it is natural to expect that the positivity of such a quadratic formH(2)f0 [g] should imply stability for f0(E). However, there are at least two serious mathematical difficulties. First of all, it is very challenging to use the positivity of H(2)f0 [g] to control higher 4 YAN GUO AND ZHIWU LIN order remainder in H(f)−H(f0) to conclude stability [38]. For example, one of the remainder terms is f3 whose L2 norm is difficult to be bounded by a power of the stability norm. The non-smooth nature of f0(E) also causes trouble here. Second of all, even if one can succeed in controlling the nonlinearity, the positivity of H is only valid for certain perturbation of the form g = Dh [22]. It is not clear at all if any arbitrary, general perturbation can be reduced to the form Dh. To overcome these two difficulties, a direct variational approach was initiated by Wolansky [39], then further developed systematically by Guo and Rein in [14], [15], [17], [18], [19]. Their method avoids entirely the delicate analysis of the second order variation H(2)f0 in (3), which has led to first rigorous nonlinear stability proof for a large class of f0(E). The high point of such a program is the nonlinear stability proof for every polytrope [18] f0(E) = (E0 − E)k+. Their basic idea is to construct galaxy models by solving a variational problem of minimizing the energy under some constraints of Casimir invariants. A concentration-compactness argument is used to show the convergence of the minimizing sequence. All the models constructed in this way are automatically stable. Unfortunately, despite its success, the King model can not be studied by such a variational approach. The Casimir function for a normalized King model is (9) Q0(f) = (1 + f) ln(1 + f)− 1− f, which has very slow growth for f → ∞. As a result, the direct variational method fails. Recently, Guo and Rein [21] proved nonlinear radial stability among a class of measure-preserving perturbations Sf0 ≡ f(t, r, vr, L) ≥ 0 : Q(f, L) = Q(f0, L), for Q ∈ C∞c and Q(0, L) ≡ 0. The basic idea is to observe that for perturbations in the class Sf0 , one can write g = f − f0 as Dh = {h,E}. Therefore, H(2)f0 [g] = H [Dh], for which the positivity was proved in [22] for radial perturbations. To avoid the difficulty of controlling the remainder term by H(2)f0 [g], an indirect contradiction argument was used in [21]. As our second main result of this article, we establish nonlinear stability of King’s model for general perturbations with spherical symmetry: Theorem 1.2. The King’s model f0 = [e E0−E − 1]+ is nonlinearly stable under spherically symmetric perturbations in the following sense: given any ε > 0 there exists ε1 > 0 such that for any compact supported initial data f(0) ∈ C1c with spherical symmetry, if d (f (0) , f0) < ε1 then 0≤t<∞ d (f (t) , f0) < ε, where the distance functional d (f, f0) is defined by (35). For the proof, we extended the approach in [27] for the 1 1 D Vlasov-Maxwell model. To prove nonlinear stability, we study the Taylor expansion ofH(f)−H(f0). Two difficulties as mentioned before are: to prove the positivity of the quadratic form and to control the remainder. We use two ideas introduced in [27]. The first idea is to use any finite number of Casimir functional Qi f, L2 as constraints. The difference from [21] is that we do not impose Qi f, L2 f0, L in the perturbation class, but expand the invariance equationQi f (t) , L2 f0, L UNSTABLE AND STABLE GALAXY MODELS 5 f (0) , L2 f0, L to the first order. In this way, we get a constraint for g = f −f0 in the form that the coefficient of its projection to ∂1Qi f0, L is small. Putting these constraints together, we deduce that a finite dimensional projection of g to the space spanned by f0, L is small. To control the remainder term, we use a duality argument. Noting that it is much easier to control the potential φ, we use a Legendre transformation to reduce the nonlinear term in g to a new one in φ only. The key observation is that the constraints on g in the projection form are nicely suited to the Legendre transformation and yields a non-local nonlinear term in φ only with the projections kept. By performing a Taylor expansion of this non-local nonlinear term in φ, the quadratic form becomes a truncated version of (A0φ, φ) defined by (6), whose positivity can be shown to be equivalent to that of Antonov functional. The the remainder term now is only in terms of φ and can be easily controlled by the quadratic form. The new complication in the stellar case is that the steady distribution f0 (E) is non-smooth and compactly supported. Therefore, we split the perturbation g into inner and outer parts, according to the support of f0. For the inner part, we use the above constrainted duality argument and the outer part is estimated separately. 2. An Instability Criterion We consider a steady distribution f0 (x, v) = f0(E) has a bounded support in x and v and f ′0 is bounded, where the particle energy E = 1 |v|2 + U0(x). The steady gravitational potential U0(x) satisfies a nonlinear Poisson equation ∆U0 = 4π f0dv. The linearized Vlasov-Poisson system is (11) ∂tf + v · ∇xf −∇xU0 · ∇vf = ∇xφ · ∇vf0, ∆φ = 4π f(t, x, v)dv. A growing mode solution (eλtf(x, v), eλtφ(x)) to (1) with λ > 0 satisfies (12) λf + v · ∇xf −∇xU0 · ∇vf = f ′0v · ∇xφ. We define [X(s;x, v), V (s;x, v)] as the trajectory of dX(s;x,v) = V (s;x, v) dV (s;x,v) = −∇xU0 such that X(0;x, v) = x, and V (0;x, v) = v. Notice that the particle energy E is constant along the trajectory. Integrating along such a trajectory for −∞ ≤ s ≤ 0, we have f(x, v) = eλsf ′0(E)V (s;x, v) · ∇xφ(X(s;x, v))ds(14) = f ′0(E)φ(x) − f ′0(E) λeλsφ(X(s;x, v))ds. 6 YAN GUO AND ZHIWU LIN Plugging it back into the Poisson equation, we obtain an equation for φ −∆φ+ [4π f ′0(E)dv]φ − 4π f ′0(E) λeλsφ(X(s;x, v))dsdv = 0. We therefore define the operator Aλ as Aλφ ≡ −∆φ+ [4π f ′0(E)dv]φ − 4π f ′0(E) λeλsφ(X(s;x, v))dsdv. Lemma 2.1. Assume that f0(E) has a bounded support in x and v and f bounded. For any λ > 0, the operator Aλ : H 2 → L2 is self-adjoint with the essential spectrum [0,+∞) . Proof. We denote Kλφ = −4π[ f ′0(E)dv]φ + 4π f ′0(E) λeλsφ(X(s;x, v))dsdv. Recall that f0 (x, v) = f0(E) has a compact support ⊂ S ⊂ R3x × R3v. We may assume S = Sx×Sv, both balls in R3. Let χ = χ (|x|) be a smooth cut-off function for the spatial support of f0 in the physical space Sx; that is, χ ≡ 1 on the spatial support of f0 and has compact support inside Sx. Let Mχ be the operator of multiplication by χ. Then Kλ = KλMχ =MχKλ =MχKλMχ. Indeed, f ′0 (x, v) = f 0 (X(s;x, v), V (s;x, v)) because of the invariance of E under the flow. So (Kλφ) (x) = −4π[ f ′0(E)dv]φ + 4π f ′0(E) λeλsφ(X(s;x, v))dsdv(15) = −4π[ f ′0(E)dv]φ + 4π ∫ ∫ 0 λeλs (f ′0(E)φ) (X(s;x, v))dsdv = (MχKλMχφ)(x). First we claim that ‖Kλ‖L2→L2 ≤ 8π |f ′0(E)| dv Indeed, the L2 norm for the first term in Kλ is easily bounded by 4π f ′0(E)dv For the second term, we have for any ψ ∈ L2, 4πλeλsf ′0(E)φ(X(s;x, v))dsdvψ(x)dx| |f ′0(E)|φ2(X(s;x, v))dvdx |f ′0(E)|ψ2(x)dvdx |f ′0(E)|φ2(x)dvdx |f ′0(E)|ψ2(x)dvdx |f ′0(E)|φ2(x)dvdx |f ′0(E)|ψ2(x)dvdx |f ′0(E)| dv ‖φ‖2 ‖ψ‖2 . UNSTABLE AND STABLE GALAXY MODELS 7 Moreover, we have that Kλ is symmetric Indeed, for fixed s, by making a change of variable (y, w) → (X(s;x, v), V (s;x, v)), so that (z, v) = (X(−s; y, w), V (−s; y, w)), we deduce that 4πf ′0(E) λeλsφ(X(s;x, v))dsdvψ(x)dx 4πf ′0(E)φ(y)ψ(X(−s; y, w))dydwds 4πf ′0(E) λeλsψ(X(−s; y,−w))φ(y)dydwds 4πf ′0(E) λeλsψ(X(s;x, v))φ(x)dvdxds. Here we have used the fact [X(s; y, w), V (s; y, w)] = [X(−s; y,−w),−V (s; y,−w)] in the last line. Hence (Kλφ, ψ) = (φ,Kλψ). Since Kλ = KλMχ and Mχ is compact from H 2 into L2 space with support in Sx, so Kλ is relatively compact with respect to −∆. Thus by Kato-Relich and Weyl’s Theorems, Aλ : H 2 → L2 is self-adjoint and σess(Aλ) = σess(−∆). � Lemma 2.2. Assume that f ′0(E) has a bounded support in x and v and f bounded. Let k(λ) = inf φ∈D(Aλ),||φ||2=1 (φ,Aλφ), then k(λ) is a continuous function of λ when λ > 0. Moreover, there exists 0 < Λ <∞ such that for λ > Λ (17) k(λ) ≥ 0. Proof. Fix λ0 > 0, φ ∈ D(Aλ), and ||φ||2 = 1. Then k(λ0) ≤ (φ,Aλ0φ) ≤ (φ,Aλφ) + |(φ,Aλ0φ)− (φ,Aλφ)| ≤ (φ,Aλφ) + 4π |f ′0(E)| [λeλs − λ0eλ0s]φ(X(s;x, v))φ(x)dsdvdx ≤ (φ,Aλφ) + 4π |f ′0(E)| [λ̃|s|eλ̃s + eλ̃s]dλ̃φ(X(s;x, v))φ(x)dsdvdx ≤ (φ,Aλφ) + C [λ̃|s|eλ̃s + eλ̃s]dλ̃ds ≤ (φ,Aλφ) + C| lnλ− lnλ0|. We therefore deduce that by taking the infimum over all φ, k(λ0) ≤ k(λ) + C| lnλ− lnλ0|. Same argument also yields k(λ) ≤ k(λ0) + C| lnλ − lnλ0|.Thus |k(λ0)− k(λ)| ≤ C| lnλ− lnλ0| and k(λ) is continuous for λ > 0. To prove (17), by (14), we recall from Sobolev’s inequality in R3 8 YAN GUO AND ZHIWU LIN |(Kλφ, ψ)| = 4πf ′0(E)e λs∇φ(X(s;x, v))V (s)dsdvψ(x)dx |ψ|2|f ′0(E)|dvdx |∇φ(X (s))|2|f ′0(E)||V (s) |2dxdv]1/2ds |ψ|2|f ′0(E)|dvdx )1/2 ∫ ∫ v2|∇φ(x)|2|f ′0(E)|dxdv]1/2ds ||ψ||6||∇φ||2 ≤ ||∇ψ||2||∇φ||2, since f0 has compact support. Therefore, (Aλφ, φ) = ||∇φ||2 − (Kλφ, φ) ≥ (1 − )||∇φ||2 ≥ 0 for λ large. � We now compute limλ→0+Aλ. We first consider the case when the test function φ is spherically symmetric. Lemma 2.3. For spherically symmetric function φ(x) = φ (|x|) , we have (Aλφ, φ) = (A0φ, φ) ≡ |∇φ|2dx+ 4π f ′0(E)dvφ − 32π3 minU0 f ′0(E) ∫ r2(E,L) r1(E,L) 2(E−U0−L2/2r2) ∫ r2(E,L) r1(E,L) 2(E−U0−L2/2r2) |∇φ|2 + 32π3 f ′0(E) ∫ r2(E,L) r1(E,L) (φ− φ̄)2 drdEdL√ 2(E − U0 − L2/2r2) Proof. Given the steady state f0(E), U0(|x|) and any radial function φ (|x|) . To find the limit of (Aλφ, φ) = |∇φ|2dx+ 4π f ′0(E)dvφ 2dx(19) f ′0(E) λeλsφ(X(s;x, v))ds φ (x) dxdv, we study the following (20) lim λeλsφ(X(s;x, v))ds. Note that we only need to study (20) for points (x, v) with E = 1 |v|2+U0| (x|) < E0 and L = |x× v| > 0, because in the third integral of (19) f ′0(E) has support in UNSTABLE AND STABLE GALAXY MODELS 9 {E < E0} and the set {L = 0} has a zero measure. We recall the linearized Vlasov- Poisson system in the r, vr, L coordinates takes the form ∂tf + vr∂rf + − ∂rU0 ∂vrf = ∂rUf∂vrf0, ∂rrUf + ∂rUf = 4π For the corresponding linearized system, for points (x, v) with E < E0 and L > 0, the trajectory of (X(s;x, v), V (s;x, v)) in the coordinate (r, E, L) is a periodic motion described by the ODE (see [8]) dr(s) = vr(s), dvr(s) = −U ′0(r) + with the period T (E,L) = 2 ∫ r2(E,L) r1(E,L) 2(E − U0 − L2/2r2) where 0 < r1(E,L) ≤ r2(E,L) < +∞ are zeros of E − U0 − L2/2r2.So by Lin’s lemma in [[25]], λeλsφ(X(s;x, v))ds = φ(X(s;x, v))ds. Since φ(X(s;x, v) = φ(r(s)), a change of variable from s→ r(s) leads to φ(X(s;x, v))ds = 2 φ(r)dr 2(E − U0 − L2/2r2) For any function g(r, E, L), we define its trajectory average as ḡ(E,L) ≡ ∫ r2(E,L) r1(E,L) g(r,E,L)dr√ 2(E−U0−L2/2r2) ∫ r2(E,L) r1(E,L) 2(E−U0−L2/2r2) λeλsφ(X(s;x, v))ds = 2 φ(r)dr 2(E − U0 − L2/2r2) /T (E,L) = φ̄ (E,L) 10 YAN GUO AND ZHIWU LIN and the integrand in third term of (19) converges pointwise to f ′0(E)φ̄φ. Thus by the dominated convergence theorem, we have (Aλφ, φ) = |∇φ|2dx+ 4π f ′0(E)φ 2dxdv − 4π f ′0(E)φ̄φ dxdv |∇φ|2dx+ 4π f ′0(E)φ 2dxdv − 32π3 minU0 f ′0(E) ∫ r2(E,L) r1(E,L) φ̄ (E,L)φ (r) drdEdL 2(E − U0 − L/2r2) |∇φ|2dx+ 4π f ′0(E)φ 2dxdv − 32π3 minU0 f ′0(E) ∫ r2(E,L) r1(E,L) 2(E−U0−L/2r2) ∫ r2(E,L) r1(E,L) 2(E−U0−L/2r2) |∇φ|2 + 32π3 f ′0(E) ∫ r2(E,L) r1(E,L) (φ− φ̄)2 drdEdL 2(E − U0 − L/2r2) This finishes the proof of the lemma. � To compute limλ→0+(Aλφ, φ) for more general test function φ, we use the fol- lowing ergodic lemma which is a direct generalization of the result in [26]. Lemma 2.4. Consider the solution (P (s; p, q) , Q (s; p, q)) to be the solution of a Hamiltonian system Ṗ = ∂qH (P,Q) Q̇ = −∂pH (P,Q) with (P (0) , Q (0)) = (p, q) ∈ Rn ×Rn. Denote Qλm = λeλsm (P (s) , Q (s)) ds. Then for anym (p, q) ∈ L2 (Rn ×Rn), we have Qλm→ Pm strongly in L2 (Rn ×Rn). Here P is the projection operator of L2 (Rn ×Rn) to the kernel of the transport operator D = ∂qH∂p − ∂pH∂q and Pm is the phase space average of m in the set traced by the trajectory. Proof. Denote U (s) : L2 (Rn ×Rn) → L2 (Rn ×Rn) to be the unitary semigroup U (s)m = m (P (s) , Q (s)). By Stone Theorem ([40]), U (s) is generated by iR = D, where R = −iD is self-adjoint and U (s) = eiαsdMα where Mα;α ∈ R1 is spectral measure of R. So λeλsm(P (s), Q(s))ds = eiαsdMαm ds = λ+ iα dMαm. UNSTABLE AND STABLE GALAXY MODELS 11 On the other hand, the projection is P = M{0} = ξdMα where ξ(α) = 0 for α 6= 0 and ξ(0) = 1. Therefore λeλsm(P (s), Q(s))ds − Pm λ+ iα − ξ(α) d‖Mαm‖2L2 by orthogonality of the spectral projections. By the dominated convergence theo- rem this expression tends to 0 as λ→ 0+, as we wished to prove. The explaination of Pm as the phase space average of m is in our remark below. � Remark 1. Since λeλsds = 1, the function (x, v) = λeλsm (P (s), Q(s)) ds is a weighted time average of the observable m along the particle trajectory. By the same proof of Lemma 2.4, we have (22) lim m (P (s), Q(s)) ds = Pm. But from the standard ergodic theory ([3]) of Hamiltonian systems, the limit of the above time average in (22) equals the phase space average of m in the set traced by the trajectory. Thus Pm has the meaning of the phase space average of m and Lemma 2.4 states that the limit of the weighted time average (21) yields the same phase space average. In particular, if the particle motion is ergodic in the invariant set SI determined by the invariants E1, · · · , Ik, and if dσI denotes the induced measure of Rn ×Rn on SI , then (23) Pm = σI (SI) m (p, q) dσI (p, q) . For integral systems, using action angle variables (J1, · · · , Jn;ϕ1, · · · , ϕn) we have (Pm) (J1, · · · , Jn) = (2π)−n · · · m (J1, · · · , Jn, ϕ1, · · · , ϕn) dϕ1, · · · dϕn for the generic case with independent frequencies (see [4]). Recall the weighted L2 space L2|f ′0| in (5). Then U (s) : L2|f ′0| → L2|f ′0| defined by U (s)m = m (X(s;x, v), V (s;x, v)) is an unitary group, where (X(s;x, v), V (s;x, v)) is the particle trajectory (13). The generator of U (s) is D = v ·∂x−∇xU0 ·∇v and R = −iD is self-adjoint by Stone Theorem. By the same proof, Lemma 2.4 is still valid in L2|f ′0| . In particular, for any φ (x) ∈ L2 we have λeλsφ(X(s;x, v))ds → Pφ in L2|f ′0| , where P is the projector of L2|f ′0| to kerD. Now we derive an explicit formula for the above limit Pφ. Note that as in the proof of lemma 2.3, we only need to derive the formula of Pφ for points (x, v) with E < E0 and L > 0. Since U0 (x) = U0 (r), the particle motion (13) in such a center field is integrable and has been well studied (see e.g. [8], [4]). For particles with 12 YAN GUO AND ZHIWU LIN energy E < E0 < 0, L > 0 and momentum ~L = x× v, the particle orbit is a rosette in the annulus AE,L = {r1(E,L) ≤ r ≤ r2(E,L)} = E − U0 − L2/2r2 ≥ 0 lying on the orbital plane perpendicular to ~L. So we can consider the particle motion to be planar. For such case, the action-angle variables are as follows (see e.g. [30]): the actions variables are T (E,L) , Jθ = L, where T (E,L) = 2 ∫ r2(E,L) r1(E,L) 2(E − U0 − L2/2r2) is the radial period, the angle variable ϕr is determined by dϕr = T (E,L) 2(E − U0 − L2/2r2) and ϕθ = θ −∆θ where d (∆θ) = Lr−2 − Ωθ 2(E − U0 − L2/2r2) Ωθ (E,L) = T (E,L) ∫ r2(E,L) r1(E,L) 2(E − U0 − L2/2r2) is the average angular velocity. For any function φ (x) ∈ H2 , we denote φ~L (r, θ) to be the restriction of φ in the orbital plane perpendicular to ~L. Then by (24), for the generic case when the radial and angular frequencies are independent, we have E, ~L = (2π) φ~Ldϕθdϕr(26) πT (E,L) ∫ r2(E,L) r1(E,L) φ~L (r, θ) dθdr 2(E − U0 − L2/2r2) In particular, for a spherically symmetric function φ = φ (r), we recover (27) (Pφ) (E,L) = 2 T (E,L) ∫ r2(E,L) r1(E,L) φ(r)dr 2(E − U0 − L2/2r2) We thus conclude the following Lemma 2.5. Assume that f0(E) has a bounded support in x and v and f bounded. For any φ ∈ H1 , we have (Aλφ, φ) = (A0φ, φ) |∇φ|2dx+ 4π f ′0(E)dvφ 2dx− 4π f ′0(E) (Pφ) |∇φ|2dx+ 4π f ′0(E) (φ− Pφ) UNSTABLE AND STABLE GALAXY MODELS 13 where P is the projector of L2|f ′0| to kerD and more explicitly Pφ is given by (26). The limiting operator A0 is (29) A0φ = −∆φ+ [4π f ′0(E)dv]φ − 4π f ′0(E)Pφdv. Now we give the proof of the instability criterion. Proof of Theorem 1.1. We define λ∗ = sup k(λ)<0 By Lemmas 2.1 and 2.5, we deduce that −∞ < λ∗ ≤ Λ <∞. Therefore, by the continuity of k(λ), we have k(λ∗) = 0. Hence, there exists an increasing sequence of λn < λn+1 < λ∗ so that λn → λ∗, kn ≡ k(λn) < 0, and kn → k(λ∗) = 0. Therefore, kn are negative eigenvalues. By Lemma 2.2, we get a sequence φn ∈ H2 such that (30) Aλnφn = knφn with kn < 0, kn → 0 and λn → λ0 > 0, as n → ∞. Recall χ the cutoff function of the support of f0(E) such that χ ≡ 1 for f0(E) > 0. We claim that χφn is a nonzero function for any n. Suppose otherwise, χφn ≡ 0, then from the equation (30) we have (−∆− kn)φn = 0 which implies that φn = 0, a contradiction.Thus we can normalize φn by ‖χφn‖2 = 1. Taking inner product of (30) with φn and integrating by parts, we have ‖▽φn‖22 ≤ −4π f ′0(E)φ n dvdx+ 4πf ′0(E) λnsφn(X(s;x, v))dsφn (x) dx = −4π f ′0(E) (χφn) 4πf ′0(E) λns (χφn) (X(s;x, v))ds (χφn) (x) dx f ′0(E)dv ‖χφn‖22 . Here in the second equality above, we use the fact χ = 1 on the support of f ′0(E) (f0(E)) and that (χφn) (X(s;x, v)) = φn(X(s;x, v)χ due to the invariance of the support under the trajectory flow, as in (15). In the last inequality, we use the same estimate as in (16). Thus, ||φn||L6 ≤ C sup ‖▽φn‖2 < C for some constant C′ independent of n. Then there exists φ ∈ L6 and ∇φ ∈ L2 such that φn → φ weakly in L6, and ∇φn → ∇φ weakly in L2. 14 YAN GUO AND ZHIWU LIN This implies that χφn → χφ strongly in L2. Therefore ‖χφ‖2 = 1 and thus φ 6= 0. It is easy to show that φ is a weak solution of Aλ0φ = 0 or (31) −∆φ = −[4π f ′0(E)dv]φ + 4πf λ0sφ(X(s;x, v))dsdv = ρ. We have that ρdx = −4π f ′0(E)φ (x) dxdv + 4πf ′0(E)φ(X(s;x, v))dxdvds = −4π f ′0(E)φ (x) dxdv + 4πf ′0(E)φ(x)dxdvds = 0 and by (31) ρ has compact support in Sx, the x−support of f0(E). Therefore from the formula φ (x) = |x−y| dy, we have φ (x) = ρ (y) |x− y| ρ (y) |x− y| ρ (y) dy = O |x|−2 for x large, and thus φ ∈ L2. By elliptic regularity, φ ∈ H2. We define f (x, v) by (14), then f ∈ L∞ with the compact support in S. Now we show that eλ0t[f, φ] is a weak solution to the linearized Vlasov-Poisson system. Since φ satisfies the Poisson equation (31), we only need to show that f satisfies the linearized Vlasov equation (12) weakly. For that, we take any g ∈ C1c 3 × R3 , and R3×R3 (Dg) fdxdv R3×R3 (Dg) (f ′0(E)φ(x)) dxdv − R3×R3 (Dg) f ′0(E) λ0sφ(X(s;x, v))dsdxdv = I + II. Since D is skew-adjoint, the first term is I = − R3×R3 gD (f ′0(E)φ) dxdv = − R3×R3 f ′0(E)gDφdxdv. UNSTABLE AND STABLE GALAXY MODELS 15 For the second term, II = − R3×R3 f ′0(E) Dg(x, v) φ (X(s;x, v)) dxdvds R3×R3 f ′0(E) (Dg) (X(−s), V (−s))φ (x) dxdvds R3×R3 f ′0(E) g (X(−s), V (−s)) ds φ (x) dxdv R3×R3 f ′0(E) λ0g (x, v)− λ0sg (X(−s), V (−s)) ds φ (x) dxdv R3×R3 f ′0(E)λ0φ (x)− f ′0(E) λ0sφ (X(s), V (s)) ds g (x, v) dxdv R3×R3 f ′0(E)φ (x)− f ′0(E) λ0sφ (X(s), V (s)) ds g dxdv = .λ0 R3×R3 fgdxdv. Thus we have R3×R3 (Dg) fdxdv = R3×R3 (λ0f − f ′0(E)Dφ) gdxdv which implies that f is a weak solution to the linearized Vlasov equation λ0f +Df = f 0 (E) v · ∇xφ. Remark 2. Consider an anisotropic spherical galaxy with f0 (x, v) = f0 For a radial symmetric growing mode eλt (φ, f) with φ = φ (|x|) and f = f |x| , E, L2 The linearized Vlasov equation (11) becomes λf + v · ∇xf −∇xU0 · ∇vf = ∇xφ · ∇vf0 = ∇xφ · |x× v|2 = φ′ (|x|) x v + 2 [(x× v)× x] v · ∇xφ, which is of the same form as in the isotropic case (20). So by the same proof of The- orem 1.1, we also get an instability criterion for radial perturbations of anisotropic galaxy, in terms of the quadratic form (18) with f ′0(E) being replaced by 3. Nonlinear Stability of the King’s Model In the second half of the article, we investigate the nonlinear stability of the King model (8). We first establish: Lemma 3.1. Consider spherical models f0 = f0 (E) with f 0 < 0. The operator A0 : H r → L2r A0φ = −∆φ+ [4π f ′0dv]φ − 4π f ′0Pφdv 16 YAN GUO AND ZHIWU LIN is positive, where H2r and L r are spherically symmetric subspaces of H 2 and L2, and the projection Pφ is defined by (27). Moreover, for φ ∈ H2r we have (32) (A0φ, φ) ≥ ε |∇φ|22 + |φ| for some constant ε > 0. Proof. Define k0 = inf (A0φ, φ) / (φ, φ) .We want to show that k0 > 0. First, by using the compact embedding of H2r →֒ L2r it is easy to show that the minimum can be obtained and k0 is the lowest eigenvalue. Let A0φ0 = k0φ0 with φ0 ∈ H2r and ‖φ0‖2 = 1. The fact that k0 ≥ 0 follows immediately from Theorem 1.1 and the nonexistence of radial modes ([9], [22]) for monotone spherical models. The proof of k0 > 0 is more delicate. For that, we relate the quadratic form (A0φ, φ) to the Antonov functional (4). We define D = v ·∂x−∇xU0 ·∇v to be the generator of the unitary group U (s):L |f ′0| → L2,r|f ′0| defined by U (s)m = m (X(s;x, v), V (s;x, v)) . Here L |f ′0| is the spherically symmetric subspace of L2|f ′0| , which is preserved under the flow mapping U (s). By the definition of Pφ, we have φ0 − Pφ0 ⊥ kerD. By Stone theorem iD is self-adjoint and in particular D is closed. Therefore by the closed range theorem ([40]), we have (kerD) = R (D) , where R (D) is the range of D. So there exists h ∈ L2,r|f ′0| such that Dh = φ0−Pφ0. Moreover, since φ0−Pφ0 is even in v and the operator D reverses the parity in v, the function h is odd in v. Define f− = f ′0h. We have k0 = (A0φ0, φ0) = |∇φ0|2 dx + 4π f ′0 (φ0 − Pφ0) |∇φ0|2 dx− 8π |f ′0| (φ0 − Pφ0)φ0dxdv |f ′0| (φ0 − Pφ0) |Df−|2 |f ′0| dxdv + 2 Df−dvdx + |∇φ0|2 dx |Df−|2 |f ′0| dxdv + |∇φ0|2 dx |Df−|2 |f ′0| dxdv + |∇φ0|2 − 2∇φ0 · ∇φ− |Df−|2 |f ′0| dxdv − 1 where ∆φ− = 4π Df−dv.Notice that the last expression above is the Antonov functional 4πH (f−, f−). Since f− is spherical symmetric and odd in v,we have H (f−, f−) > 0 by the proof in [22] which was further clarified in [33] and [21]. Therefore we get k0 > 0 as desired and (A0φ, φ) ≥ k0 |φ|22. UNSTABLE AND STABLE GALAXY MODELS 17 To get the estimate (32), we rewrite (A0φ, φ) = ε |∇φ|2 dx + 4π f ′0 (φ− Pφ) + (1− ε) (A0φ, φ) |∇φ|2 dx− 4πε ‖φ− Pφ‖2L2 + (1− ε) k0 |φ|22 |∇φ|2 dx− 8πε ‖φ‖2L2 + (1− ε) k0 |φ|22 (since ‖P‖L2 |∇φ|2 dx+ ((1− ε) k0 − Cε) |φ|22 ≥ ε |∇φ|2 dx+ |φ|22 if ε is small enough. � Next, we will approximate the kerD by a finite dimensional approximation. Let {ξi(E,L) = αi(E)βi(L)}∞i=1 be a smooth orthogonal basis for the subspace kerD = {g(E,L)} ⊂ L2,r|f ′0| .Define the finite-dimensional projection operator PN : L2,r|f ′0| |f ′0| (33) PNh ≡ (h, ξi)|f ′0|ξi and the operator AN : H2r → L2r by ANφ = −∆φ+ [4π f ′0dv]φ− 4π f ′0PNφdv. Lemma 3.2. There exists K, δ0 > 0 such that when N > K we have ANφ, φ ≥ δ0 |∇φ|22 for any φ ∈ H2r . Proof. First we have AN → A0 strongly in L2. In deed, for any φ ∈ H2r , ∥ANφ−A0φ 4πf ′0 (PNφ− Pφ) dv ≤ C ‖PNφ− Pφ‖L2 as N → ∞.We claim that for N sufficiently large, the lowest eigenvalue of AN is at least k0/2 where k0 > 0 is the lowest eigenvalue of A0. Suppose otherwise, then there exists a sequence {λn} and {φn} ⊂ H2r with λn < k0/2, ‖φn‖2 = 1 and Anφn = λnφn. This implies that ∆φn is uniformly bounded in L 2, by elliptic estimate we have ‖φn‖H2 ≤ C for some constant C independent of n. Therefore there exists φ0 ∈ H2r such that φn → φ0 weakly in H2r . By the compact embedding of H2r →֒ L2r, we have φn → φ0 strongly in L2r and ‖φ0‖2 = 1. The strong convergence of Anφ0 → A0φ0 implies that Anφn → A0φ0 weakly in L2. Let λn → λ0 ≤ k0/2, then we have A0φ0 = λ0φ0, a contradiction. Therefore we have ANφ, φ ≥ k0/2 |φ|22 for φ ∈ H2r , when N is large enough. The estimate (34) is by the same proof of (32) in Lemma 3.1. � 18 YAN GUO AND ZHIWU LIN Recalling (8) with f0 = [e E0−E−1]+ and Q0(f) = (f+1) ln(f+1)−f, we further define functionals (related to the finite dimensional approximation of kerD) as Ai(f) ≡ αi(− ln(s+ 1) + E0)ds, Qi(f, L) ≡ Ai(f)βi(L), for 1 ≤ i ≤ N. for 1 ≤ i ≤ N. Clearly, ∂1Qi(f0, L) = αi(− ln(f0 + 1) + E0)βi(L) = αi(E)βi(L) = ξi(E,L), where {ξi(E,L)}Ni=1 are used to define PN in Lemma 3.2. Define the Casimir functional (E0 < 0 ) I(f) = [Q0(f) + |v|2f − E0f ]dxdv − |∇φ|2dx which is invariant of the nonlinear Vlasov-Poisson system. We introduce additional N invariants Ji(f, L) ≡ Qi(f, L)dxdv. for 1 ≤ i ≤ N . We define Ω to be the support of f0(E). We first consider I(f)− I(f0) = [Q0(f)−Q0(f0) + |v|2(f − f0)− E0(f − f0)]dxdv ∇U0 · ∇(U − U0)− |∇(U − U0)|2dx [Q0(f)−Q0(f0) + (E − E0)(f − f0)]dxdv − |∇(U − U0)|2dx. We define g = f − f0, φ = U − U0 gin ≡ (f − f0)1Ω, gout ≡ (f − f0)1Ωc , ∆φin ≡ gin, ∆φout ≡ gout . And we define the distance function for nonlinear stability as d(f, f0) ≡ [Q0(gin + f0)−Q0(f0) + (E − E0)gin]dxdv |∇φin|2dx Q0(gout)dxdv + (E − E0)goutdxdv = din + |∇φin|2dx + dout, for which each term is non-negative. We therefore split: I(f)− I(f0) [Q0(f0 + gin)−Q0(f0) + (E − E0)gin]dxdv − |∇φin|2dx Q0(gout)dxdv + (E − E0)goutdxdv − |∇φout|2dx− ∇φout · ∇φindx = Iin + Iout . UNSTABLE AND STABLE GALAXY MODELS 19 In the estimates below, we use C,C′, C′′ to denote general constants depending only on f0 and quantities like ‖f (t)‖Lp (p ∈ [1,+∞]) which equals ‖f (0)‖Lp and therefore always under control. We first estimate ‖∇φout‖22 to be of higher order of d, which also implies that ∇φout · ∇φindx is of higher order of d. Lemma 3.3. For ε > 0 sufficiently small, we have |∇φout|2dx ≤ C εd(f, f0) + [d(f, f0)] Proof. In fact, since |∇φout|2dx ≤ C|| gout dv||2L6/5 ≤ C|| gout 1E0≤E≤E0+εdv||2L6/5 + C|| gout 1E>E0+εdv||2L6/5 . The first term is bounded by g2out dv] 1E0≤E≤E0+εdv] 3/5dx g2out dvdx]× 1E0≤E≤E0+εdv] 3/2dx ≤ Cε[ g2out dvdx] ≤ Cε[ g2out dvdx] ≤ Cεd(f, f0). In the above estimates, we use that Q0(gout)dvdx ≥ c g2out dvdx and 1E0≤E≤E0+εdv ≤ Cε, which can be checked by an explicit computation when ε > 0 is sufficiently small such that E0 + ε ≤ 0. On the other hand, by the standard estimates (see [12, P. 120-121]) gout 1E>E0+εdv||2L6/5 gout 1E>E0+εdxdv |v|2gout 1E>E0+εdxdv (E − E0)gout 1E>E0+εdxdv (E − E0)gout 1E>E0+εdxdv + 2 sup |U0| gout 1E>E0+εdxdv 2 sup |U0| d5/3. 20 YAN GUO AND ZHIWU LIN By Lemma 3.3, we have ∇φout · ∇φindx ≤ ‖∇φout‖2 ‖∇φin‖2 ε1/3d(f, f0) + [d(f, f0)] and therefore for ε sufficiently small, (36) Iout ≥ dout − C ε1/3d(f, f0) + [d(f, f0)] 4/3 + [d(f, f0)] To estimate Iin, we split it into three parts: [Q0(f0 + gin)−Q0(f0) + (E − E0)gin + φingin]dxdv + |∇φin|2dx (1− τ) [Q0(f0 + gin)−Q0(f0) + (E − E0)gin + (I − PN )φingin]dxdv + |∇φin|2dx + (1− τ) PNφingindxdv = I1in + I in + I where ∆φin = 4π gin dv. We estimate each term in the following lemmas. Lemma 3.4. (38) I1 din − Cτ |∇φin|2dx. Proof. In fact, since the integration region Ω is finite, we have I1in =τ [Q0(f0 + gin)−Q0(f0) + (E − E0)gin + φingin]dxdv + |∇φin|2dx [Q0(f0 + gin)−Q0(f0) + (E − E0)gin]dxdv − Cτ ||φin||L6 ||gin||L6/5 [Q0(f0 + gin)−Q0(f0) + (E − E0)gin]dxdv − C′τ ||∇φin||L2 ||gin||2 din − C′′τ ||∇φin||22, since din = [Q0(f0 + gin)−Q0(f0) + (E − E0)gin]dxdv ≥ C||gin||22. To estimate I2in, we need the following pointwise duality lemma from elementary calculus. Lemma 3.5. For any c, and any h, we have gc,f0 (h) = Q0(h+ f0)−Q0(f0)−Q′0(f0)h− ch ≥ (f0 + 1)(1 + c− ec). Proof. Direct computation yields that the minimizer fc of gc,f0 (h) satisfies the Euler-Lagrange equation ln (fc + f0 + 1)− ln (f0 + 1)− c = 0, UNSTABLE AND STABLE GALAXY MODELS 21 fc = (f0 + 1) (e c − 1) . Thus by using the Euler-Lagrange equation, we deduce min gc,f0 (h) = gc,d (fc) = (fc + f0 + 1) ln(1 + fc + f0) − (f0 + 1) ln(1 + f0)− [1 + ln(f0 + 1)]fc − cfc = (fc + f0 + 1)[ln(1 + fc + f0)− ln(f0 + 1)− c] + fc ln(1 + f0) + c(f0 + 1)− [1 + ln(f0 + 1)]fc = (f0 + 1)(1 + c− ec). Lemma 3.6. (39) I2 (1− τ) δ0 |∇φin|2dx− CeC Proof. Recall (37). By using Lemma 3.5 for c = − (φin − PNφin) and using the Taylor expansion, we have I2in = (1 − τ) [Q0(f0 + gin)−Q0(f0) + (E − E0)gin + (φin − PNφin) fin]dxdv (1− τ) |∇φin|2dx (1− τ) |∇φin|2dx+ (1− τ) (f0 + 1)1Ω(1 + φin − PNφin − eφin−PNφin)dxdv ≥ 1− τ |∇φin|2dx− 4π |f ′0 (E)| (φin − PNφin) − Ce|φin−PNφin|∞ |f ′0 (E)| |φin − PNφin| dxdv (Note (f0(E) + 1)1Ω = |f ′0(E)|) ≥ (1− τ) δ0 |∇φin|2dx− Ce|φin−PNφin|∞ |f ′0 (E)| |φin − PNφin| dxdv. In the last line, we have used Lemma 3.2. To estimate the last term above and conclude our lemma, it suffices to show |φin − PNφin|∞ ≤ CNd This follows from the facts that for the fixed N smooth functions ξi, we have |PNφin|∞ = (φin, ξi)|f ′0|ξi ≤ CN |φin|∞ , and since φ is spherically symmetric, |φin| (r) = u2ρin (u) du+ uρin (u)du R |ρin|2 ≤ C ′′ ‖gin‖2 ≤ CNd where ρin = gindv and R is the support radius of ρin. � 22 YAN GUO AND ZHIWU LIN We now estimate the term PNφinfindxdv, for which we use the additional invariants. Lemma 3.7. For any ε > 0, we have ∣ ≤ C(d1/2(0) + ε1/2d1/2 + 1 d)d1/2. Proof. By the definition of I3in in (37), it suffices to estimate (gin, ξi). We expand Ji(f, L)− Ji(f0, L) = Ji(f0 + gin, L)− Ji(f0, L) + Ji(gout, L) = (gin , ξi) +O(d) + Ji(gout, L). Notice that |Ji(gout, L)| ≤ C||gout||L1 ≤ C||1{E0≤E≤E0+ε}gout||L1 + C||1{E≥E0+ε}gout||L1 ≤ ε1/2||gout||L2 + ||1{E≥E0+ε}(E − E0)gout||L1 ≤ C[ε 1/2d1/2 + It thus follows that |(gin , ξi)| ≤ |Ji(f(0), L)− Ji(f0, L)|+ C[ε1/2d1/2 + ≤ C[d1/2(0) + ε1/2d1/2 + 1 Therefore ∣I3in ∣ = (1− τ) PNφingin dxdv (φin, ξi)|f ′0|ξi gin dxdv (φin, ξi)|f ′0| |(ξi, gin)| ≤ C′ |φin|∞ |(ξi, gin)| ≤ Cd1/2[d1/2(0) + ε1/2d1/2 + Now we prove the nonlinear stability of King model. Proof of Theorem 1.2. The global existence of classical solutions of 3D Vlasov- Poisson system was shown in [34] for compactly supported initial data f (0) ∈ C1c . Let the unique global solution be (f (t) , φ (t)). Let d (t) = d(f (t) , f0). Combining estimates (36), (38), (39) and (40), we have I(f (0))− I(f0) = I(f (t))− I(f0) ≥ dout + din + (1− τ) δ0 |∇φin|2dx ε1/3d (t) + d (t) d (t) − CeC ′d(t) d (t) − Cd (t)1/2 [d1/2(0) + ε1/2d (t)1/2 + d (t)]. UNSTABLE AND STABLE GALAXY MODELS 23 Thus by choosing ε and τ sufficiently small, there exists δ′ > 0 such that I(f (0))− I(f0) ≥ δ′d(t)− C d (t) + d (t) + d (t) − CeC ′d(t) d (t) − Cd (t)1/2 d1/2(0). It is easy to show that I(f (0)) − I(f0) ≤ C′′d (0). Define the functions y1 (x) = δ′x2 − CeC′xx3 − C x8/3 + x10/3 + x3 and y2 (x) = Cd (0) x + C′′d (0). Then above estimates implies that y1 d (t) d (t) . The function y1 is in- creasing in (0, x0) where x0 is the first maximum point. So if d (0) is sufficiently small, the line y = y2 (x) intersects the curve y = y1 (x) at points x1, x2, · · · , with x1 (d (0)) < x0 < x2 (d (0)) < · · · . Thus the inequality y1 (x) ≤ y2 (x) is valid in disjoint intervals [0, x1 (d (0))] and [x2 (d (0)) , x3 (d (0))], · · · . Because d (t) is continuous, we have that d (t) < x1 (d (0)) for all t < ∞, provided we choose d (0) < x0. Since x1 (d (0)) → 0 as d (0) → 0, we deduce the nonlinear stability in terms of the distance functional d (t) Acknowledgements This research is supported partly by NSF grants DMS-0603815 and DMS-0505460. We thank the referees for comments and corrections. References [1] Antonov, V. A. Remarks on the problem of stability in stellar dynamics. Soviet Astr, AJ., 4, 859-867 (1961). [2] Antonov, V. A., Solution of the problem of stability of stellar system Emden’s density law and the spherical distribution of velocities, Vestnik Leningradskogo Universiteta, Leningrad University, 1962. [3] Arnold, V. I., Avez, A., Ergodic problems of classical mechanics, W. A. Benjamin, Inc., New York-Amsterdam 1968. [4] Arnold, V. I., Mathematical methods of classical mechanics, Springer-Verlag, New York- Heidelberg, 1978. [5] Barnes, J.; Hut, P.; Goodman, J., Dynamical instabilities in spherical stellar systems, Astro- physical Journal, vol. 300, p. 112-131, 1986. [6] Bartholomew, P., On the theory of stability of galaxies, Monthly Notices of the Royal Astro- nomical Society, Vol. 151, p. 333 (1971). [7] Bertin, Giuseppe, Dynamics of Galaxies, Cambridge University Press, 2000. [8] Binney, J., Tremaine, S., Galactic Dynamics. Princeton University Press, 1987. [9] Doremus, J. P.; Baumann, G.; Feix, M. R., Stability of a Self Gravitating System with Phase Space Density Function of Energy and Angular Momentum, Astronomy and Astrophysics, Vol. 29, p. 401 (1973). [10] Gillon, D.; Cantus, M.; Doremus, J. P.; Baumann, G., Stability of self-gravitating spherical systems in which phase space density is a function of energy and angular momentum, for spherical perturbations, Astronomy and Astrophysics, vol. 50, no. 3, p. 467-470, 1976. [11] Fridman, A., Polyachenko, V., Physics of Gravitating System Vol I and II, Springer-Verlag, 1984. [12] Glassey, Robert T., The Cauchy problem in kinetic theory, SIAM, Philadelphia, PA, 1996. [13] Goodman, Jeremy, An instability test for nonrotating galaxies, Astrophysical Journal, vol. 329, p. 612-617, 1988. [14] Guo, Y., Variational method for stable polytropic galaxies, Arch. Rational Mech. Anal., 147, 225-243, 1999. 24 YAN GUO AND ZHIWU LIN [15] Guo, Y., On generalized Antonov stablility criterion for polytropic steady states, Contem. Math., 263, 85-107, 1999. [16] Guo, Y., Rein, G., Stable steady states in stellar dynamics, Arch. Rational Mech. Anal., 147, no. 3, 225-243, (1999). [17] Guo, Y., Rein, G., Existence and stability of Camm type steady states in galactic dynamics, Indiana U. Math. J., 48, 1237-1255, 1999. [18] Guo, Y., Rein, G., Isotropic steady states in stellar dynamics, Commun. Math. Phys., 219, 2001. [19] Guo, Y., Rein, G., Isotropic steady states in stellar dynamics revisited., Los Alamos Preprint, 2002. [20] Henon, M., Numerical Experiments on the Stability of Spherical Stellar Systems, Astronomy and Astrophysics, Vol. 24, p. 229 (1973). [21] Guo, Y., Rein, G., Stability of the King Model and Symmetric Measure-Preserving Pertur- bations, Preprint. [22] Kandrup, H.; Signet, J. F.; A simple proof of dynamical stability for a class of spherical clusters. The Astrophys. J. 298, p. 27-33.(1985) [23] Kandrup, Henry E., A stability criterion for any collisionless stellar equilibrium and some concrete applications thereof, Astrophysical Journal, vol. 370, p. 312-317, 1991. [24] King, Ivan R., The structure of star clusters. III. Some simple dynamical models, Astronom- ical Journal, Vol. 71, p. 64 (1966). [25] Lin, Zhiwu, Instability of periodicBG waves, Math. Res. Letts., 8, 521-534(2001). [26] Lin, Zhiwu and Strauss, Walter, Linear stability and instability of relativistic Vlasov-Maxwell systems, to appear in Comm. Pure Appl. Math. [27] Lin, Zhiwu and Strauss, Walter, Nonlinear stability and instability of relativistic Vlasov- Maxwell systems, to appear in Comm. Pure Appl. Math. [28] Lin, Zhiwu and Strauss, Walter, A sharp stability criterion for the Vlasov-Maxwell systems, submitted. [29] Lynden-Bell, D., The Hartree-Fock exchange operator and the stability of galaxies, Monthly Notices of the Royal Astronomical Society, Vol. 144, p.189, 1969. [30] Lynden-Bell, D. Lectures on stellar dynamics. Galactic dynamics and N-body simulations (Thessaloniki, 1993), 3–31, Lecture Notes in Phys., 433, Springer, Berlin, 1994. [31] Merritt, David, Elliptical Galaxy Dynamics, The Publications of the Astronomical Society of the Pacific, Volume 111, Issue 756, pp. 129-168. [32] Palmer, P. L., Stability of collisionless stellar systems: mechanisms for the dynamical struc- ture of galaxies, Kluwer Academic Publishers, 1994. [33] Perez, Jerome and Aly, Jean-Jacques, Stability of spherical stellar systems - I. Analytical results, Monthly Notices of the Royal Astronomical Society, Volume 280, Issue 3, pp. 689- 699, 1996. [34] Pfaffelmoser, K, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations 95 (1992), no. 2, 281–303. [35] Rein, G.: Collisionless Kinetic Equations from Astrophysics - The Vlasov-Poisson system. Preprint 2005. [36] Schaeffer, Jack, Steady states in galactic dynamics, Arch. Ration. Mech. Anal. 172 (2004), no. 1, 1–19. [37] Sygnet, J. F.; des Forets, G.; Lachieze-Rey, M.; Pellat, R., Stability of gravitational systems and gravothermal catastrophe in astrophysics, Astrophysical Journal, vol. 276, p. 737-745, 1984. [38] Wan, Y-H., On onlinear stability of isotropic models in stellar dynamics, Arch. Rational. Mech. Anal., 147, (1999) 245-268. [39] Wolansky, G., On nonlinear stability of polytropic galaxies. Ann. Inst. Henri Poincare. (1999), 16, 15-48. [40] Yosida, Kôsaku, Functional analysis, Sixth edition. Grundlehren der Mathematischen Wis- senschaften, 123. Springer-Verlag, 1980. UNSTABLE AND STABLE GALAXY MODELS 25 Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, RI 02912, USA E-mail address: [email protected] Mathematics Department, University of Missouri, Columbia, MO 65211 USA E-mail address: [email protected] 1. Introduction 2. An Instability Criterion 3. Nonlinear Stability of the King's Model References

0704.1013

Flops connect minimal models

7 Flops connect minimal models Yujiro Kawamata October 30, 2018 Abstract A result by Birkar-Cascini-Hacon-McKernan together with the bound- edness of length of extremal rays implies that different minimal models can be connected by a sequence of flops. A flop of a pair (X,B) is a flip of a pair (X,B′) which is crepant for KX + B where B ′ is a suitably chosen different boundary. We prove the following: Theorem 1. Let f : (X,B) → S and f ′ : (X ′, B′) → S be projective morphisms from Q-factorial terminal pairs of varieties and Q-divisors such that KX+B and KX′ +B ′ are relatively nef over S. Assume that there exists a birational map α : X 99K X ′ such that α∗B = B ′, where the lower asterisk denotes the strict transform. Then α is decomposed into a sequence of flops. More precisely, there exist an effective Q-divisor D on X such that (X,B +D) is klt and a factorization of the birational map α X = X0 99K X1 99K · · · 99K Xt = X which satisfy the following conditions: (1) αi : Xi−1 → Xi (1 ≤ i ≤ t) is a flip for the pair (Xi, Bi +Di) over S, where Bi and Di are strict transforms of B and D, respectively. (2) αi is crepant for KXi−1 + Bi−1 in the sense that the pull-backs of KXi−1 +Bi−1 and KXi +Bi coincide on a common log resolution. We remark that the boundary B need not be assumed to be big as in [1] Corollary 1.1.3. For example, a birational map between Calabi-Yau man- ifolds can be decomposed into a sequence of flops. The number of marked http://arxiv.org/abs/0704.1013v1 minimal models which are birationally equivalent to a fixed pair is finite if B is big ([1] Corollary 1.1.5), but it is not the case in general (cf. [4]), where a marked minimal model is a pair consisting of a minimal model and a fixed birational map to it. If we relax the condition for the pairs to being klt, then we should allow crepant blowings up besides flops. The theorem was already proved in the case dimX = 3 and B = 0; first in [2] assuming the abundance which was proved afterwards, and later in [5] without assumption. Proof. It is well-known that α is an isomorphism in codimension 1 because (X,B) and (X ′, B′) are terminal and KX + B and KX′ + B ′ are relatively nef (cf. [2]). We recall the proof for reader’s convenience. Let µ : V → X and µ′ : V → X ′ be common log resolutions. We write KV = µ ∗(KX +B)− µ B + E = (µ′)∗(KX′ +B ′)− (µ′ )−1B′ + E ′ where E and E ′ are effective divisors whose supports coincide with the excep- tional loci of µ and µ′, respectively, because (X,B) and (X ′, B′) are terminal. Assume that there is a prime divisor on V which is contracted by µ but not by µ′. Then it is an irreducible component of E but not of E ′. We set F = min{E,E ′}, Ē = E − F and Ē ′ = E ′ − F . By the Hodge index theo- rem, there exists a curve C on V which is contracted by µ and is contained in Supp(Ē) but not in Supp(µ−1 B + Ē ′) and such that (Ē · C) < 0. Since B ≥ (µ′ )−1B′, we have ((µ′)∗(KX′ +B ′) + µ−1 B − (µ′ )−1B′ + Ē ′) · C) ≥ 0. But this is a contradiction to (µ∗(KX +B) + Ē) · C) < 0. The case where there is a prime divisor on V which is contracted by µ′ but not by µ is treated similarly. Let L′ be an effective f ′-ample divisor on X ′, and L its strict transform on X . There exists a small positive number l such that (X,B + lL) is klt. If KX + B + lL is f -nef over S, then α becomes a morphism by the base point free theorem, hence an isomorphism since X is Q-factorial. Therefore we may assume that KX +B + l ′L is not f -nef over S for any 0 < l′ ≤ l. Let H be an effective divisor on X such that (X,B + lL+ tH) is klt and KX + B + lL + tH is f -nef for some positive number t. We shall run the MMP for the pair (X,B + l′L) over S with scaling of H for some l′. Since α is an isomorphism in codimension 1, there are only flips in this MMP. The following lemma shows that we can choose extremal rays such that the flips are crepant with respect to KX +B. Let k be a positive integer such that k(KX +B) is a Cartier divisor. We set e = 1 2k dimX+1 Lemma 2. (1) There exists an extremal ray R for (X,B + lL) over S such that ((KX +B) ·R) = 0. (2) Let t0 = min{t ∈ R | ((KX +B + lL+ tH) · R) ≥ 0 for all extremal rays R for (X,B + lL) over S s.t. ((KX +B) ·R) = 0}. Then KX +B + elL+ et0H is f -nef, and there exists an extremal ray R for (X,B+elL) over S such that ((KX+B+elL+et0H)·R) = ((KX+B)·R) = 0. Proof. (1) Since KX +B + elL is not nef, there exists an extrenal ray R for (X,B+ elL) over S. Then R is also an extremal ray for (X,B+ lL) because (X,B) is f -nef. Since the pair (X,B+ lL) is klt, R is generated by a rational curve C, which is mapped to a point on S, such that 0 > ((KX +B + lL) · C) ≥ −2 dimX by [3]. We claim that ((KX +B) ·C) = 0. Indeed we have otherwise ((KX +B) · C) ≥ 1/k, hence ((KX +B + elL) · C) 2k dimX + 1 ((KX +B + lL) · C) + 2k dimX 2k dimX + 1 ((KX +B) · C) 2k dimX + 1 (−2 dimX + 2dimX) = 0 a contradiction. (2) If KX + B + elL + et0H is not f -nef, then there exists an extremal ray R for (X,B + elL + et0H) over S. Then R is also an extremal ray for (X,B+ lL+t0H) because (X,B) is f -nef. Since the pair (X,B+ lL+t0H) is klt, R is generated by a rational curve C such that ((KX+B+lL+t0H)·C) ≥ −2 dimX by [3]. Then we have ((KX +B + elL+ et0H) · C) 2k dimX + 1 ((KX +B + lL+ t0H) · C) + 2k dimX 2k dimX + 1 ((KX +B) · C) 2k dimX + 1 (−2 dimX + 2dimX) = 0 a contradiction. Therefore KX +B + elL+ et0H is f -nef. Since B + lL is f -big, the number of extremal rays for (X,B + lL) over S is finite. Hence there exists such an R that ((KX +B + lL+ t0H) · R) = ((KX +B) · R) = 0. We note that we can deduce (1) from only the finiteness of extremal rays, but not (2). The point is that the number e stays independent of t0 during the MMP. We run the MMP for (X,B+elL) with scaling ofH . We take an extremal ray R such that ((KX +B+ elL+ et0H) ·R) = ((KX +B) ·R) = 0. The flip exists by [1] Corollary 1.4.1. Since (lL+ t0H) ·R) = 0, the pair (X,B+ lL+ t0H) remains to be klt after the flip. We also note that k(KX+B) remains to be a Cartier divisor after the flip by the base point free theorem. Therefore we can continue the process. By the termination theorem of directed flips ([1] Corollary 1.4.2), we complete our proof. References [1] Caucher Birkar, Paolo Cascini, Christopher D. Hacon, James McK- ernan. Existence of minimal models for varieties of log general type. math.AG/0610203. [2] Kawamata, Yujiro. Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces. Ann. of Math. 127 (1988), 93–163. [3] Kawamata, Yujiro. On the length of an extremal rational curve. In- vent. Math. 105 (1991), 609–611. [4] Kawamata, Yujiro. On the cone of divisors of Calabi-Yau fiber spaces. Internat. J. Math. 8 (1997), 665–687. http://arxiv.org/abs/math/0610203 [5] Kollár, János. Flops. Nagoya Math. J. 113(1989), 15–36. Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo, 153-8914, Japan [email protected]

0704.1014

A product formula for volumes of varieties

7 A product formula for volumes of varieties Yujiro Kawamata October 28, 2018 The volume v(X) of a smooth projective variety X is defined by v(X) = lim sup dimH0(X,mKX) md/d! where d = dimX . This is a birational invariant. Theorem 0.1. Let f : X → Y be a surjective morphism of smooth projective varieties with connected fibers. Assume that both Y and the general fiber F of f are varieties of general type. Then v(Y ) v(F ) where dX = dimX, dY = dimY and dF = dimF . Proof. Let H be an ample divisor on Y . There exists a positive integer m0 such that m0KY −H is effective. Let ǫ be a positive integer. By Fujita’s approximation theorem ([1]), after replacing a birational model of X , there exists a positive integer m1 and ample divisors L on F such thatm1KF−L is effective and v( L) > v(F )−ǫ. By Viehweg’s weak positivity theorem ([2]), there exists a positive integer k such that Sk(f OX(m1KX/Y )⊗OY (H)) is generically generated by global sections for a positive integer k. k is a function on H and m1. We have rank Im(SmSk(f OX(m1KX/Y )) → f∗OX(km1mKX/Y )) ≥ dimH0(F, kmL) ≥ (v(F )− 2ǫ) (km1m) http://arxiv.org/abs/0704.1014v1 for sufficiently large m. dimH0(X, km1mKX) ≥ dimH0(Y, k(m1 −m0)mKY )× (v(F )− 2ǫ) (km1m) ≥ (v(Y )− ǫ) (k(m1 −m0)m) (v(F )− 2ǫ) (km1m) ≥ (v(Y )− 2ǫ)(v(F )− 2ǫ) (km1m) dY !dF ! if we take m1 large compared with m0 such that (v(Y )− ǫ) (v(Y )− 2ǫ) m1 −m0 )dY . Remark 0.2. If X = Y × F , then we have an equality in the formula. We expect that the equality implies the isotriviality of the family. References [1] Fujita, Takao. Approximating Zariski decomposition of big line bundles. Kodai Math. J. 17 (1994), no. 1, 1–3. [2] Viehweg, Eckart. Weak positivity and the additivity of the Kodaira di- mension for certain fibre spaces. Algebraic varieties and analytic vari- eties (Tokyo, 1981), 329–353, Adv. Stud. Pure Math., 1, North-Holland, Amsterdam, 1983. Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo, 153-8914, Japan [email protected]

0704.1015

An Alternative Topological Field Theory of Generalized Complex Geometry

arXiv:0704.1015v3 [hep-th] 26 Aug 2007 YITP-04-15 An Alternative Topological Field Theory of Generalized Complex Geometry Noriaki IKEDA1∗ and Tatsuya TOKUNAGA2 † 1Department of Mathematical Sciences, Ritsumeikan University Kusatsu, Shiga 525-8577, Japan 2Yukawa Institute for Theoretical Physics, Kyoto University Kyoto 606-8502, Japan October 27, 2018 Abstract We propose a new topological field theory on generalized complex geometry in two dimension using AKSZ formulation. Zucchini’s model is A model in the case that the generalized complex structure depends on only a symplectic structure. Our new model is B model in the case that the generalized complex structure depends on only a complex structure. ∗E-mail address: [email protected] †E-mail address: [email protected] http://arxiv.org/abs/0704.1015v3 1 Introduction In [1][2], Zucchini has constructed a two dimensional topological sigma model on generalized complex geometry [3] [4] [5] by the AKSZ formulation [6] (also see [7]), which is a general geometrical framework to construct a topological sigma model by the Batalin-Vilkovisky formalism [8]. Also, there are many recent papers [9]-[32] on this topic. Zucchini’s model is a generalization of the Poisson sigma model and is similar to A model in [6]. However B model looks different from the Zucchini model because B model has more fields than the Zucchini model has. In this paper, we propose an alternative realization of generalized complex geometry by a topological field theory by the AKSZ formulation. Our model is similar to B model, not A model in the sense of AKSZ, as a worldsheet action of a topological sigma model with superifields on a supermaifold. Our model is the first candidate which naturally includes B model and may be related to a topological string theory on generalized Calabi-Yau geometry [23] [24]. First we construct a three dimensional topological field theory of generalized complex geometry with a nontrivial 3-form H , which has Zucchini’s model as a boundary action. This topological field theory is a reconstruction by the AKSZ formulation of the model proposed in the paper [33]. Next after a dimensional reduction, we derive a topological field theory of generalized complex geometry in two dimensions from three dimensions. We can see that this model has a generalized complex structure as a consistency condition of a topological BV action. If the generalized complex structure is a complex structure, our model has one parameter marginal deformation of the model without changing a complex structure, and reduces to B model in a limit of the deformation. If the generalized complex structure is a symplectic structure, our model becomes a new 2D topological sigma model with a symplectic structure. The paper is organized as follows. In section 2, the AKSZ actions of A model, B model and the Zucchini model are reviewed. In section 3, three dimensional topological field the- ory of generalized complex geometry is rederived in the AKSZ formulation. In section 4, we derive a two dimensional topological field theory of generalized complex geometry and check its properties. In section 5, our model is reduced in two special ways. Section 6 includes conclusion and discussion. In appendix A, a generalized complex structure is briefly summa- rized. In appendix B, the AKSZ formulation of the Batalin-Vilkovisky formalism in general n dimensions is reviewed. 2 A Model, B Model and Zucchini Model In this section, we review the AKSZ formulation of topological sigma models such as A model, B model and the Zucchini model. 2.1 A Model and B Model A model and B model are defined on the graded bundle T ∗[1]M ⊕ (T [1]M ⊕ T ∗[0]M) . (1) Here E = TM , n = 2 and p ≥ 1 in the general graded bundles (100). Local coordinates are written by superfields on this bundle: (φi,B1i,A1 i,B0,i). φ i is a map φi : ΠTΣ → M , and B1i is a basis of sections of ΠT ∗Σ ⊗ φ∗(T ∗[1]M). A1i is a basis of sections of ΠT ∗Σ ⊗ φ∗(T [1]M), and B0i is a basis of sections of ΠT ∗Σ ⊗ φ∗(T ∗[0]M). The antibracket on this bundle (1) is (F,G) ≡ F ∂B1,i ∂B1,i ∂B0,i ∂B0,i G (2) from (102). The A model action with a symplectic form Qij in [34] is SAQ = Qij(φ)dφ idφj , (3) where d is a superderivative d = θµ∂µ. where the integration ΠTΣ means the integration on the supermanifold, ΠTΣ d 2θd2σ. This action is consistent if and only if the 2-formQ = 1 Qijdφ satisfies the symplectic condition dMQ = 0, namely ∂kQij + ∂iQjk + ∂jQki = 0. (4) This model is rewritten by the AKSZ formulation on the graded bundle T ∗[1]M⊕(T [1]M ⊕ T ∗[0]M). We introduce Ai1, B0i and B1i as auxiliary fields, and rewrite the action using the first order formalism. The action in AKSZ formulation is SAQ = B1idφ i −B0idAi1 −B1iAi1 + Qij(φ)A . (5) We can check that (SAQ, SAQ) = 0 if and only if the 2-formQ satisfies the symplectic condition Also, A model action with a Poisson bivector P ij is SAP = B1idφ i −B0idAi1 + P ij(φ)B1iB1j , (6) which is called the Poisson sigma model [35][36]. The consistency condition (SAP , SAP ) = 0 is satisfied if and only if P ij is a Poisson bivector field i.e. P il∂lP jk + P jl∂lP ki + P kl∂lP ij = 0. (7) B model with a complex structure J ij is B1idφ i −B0idAi1 + J ij(φ)B1iA ∂J ik (φ)B0iA 1, (8) which is a covariant form of B model action in [6], but is different from the action in [37]. We can check that the consistency condition (SB, SB) = 0 is satisfied if and only if J j satisfies the integrability condition for the complex structure J li∂lJ j − J lj∂lJki − Jkl∂iJ lj + Jkl∂jJ li = 0. (9) 2.2 Zucchini Model In [1], Zucchini has proposed a topological sigma model with a generalized complex structure on a two dimensional worldsheet Σ. Although he called this model ”the Hitchin sigma model”, here we call it the Zucchini model. First we consider H = 0 case. The action of the Zucchini’s model is B1idφ P ij(φ)B1iB1j + Qij(φ)dφ idφj + J ij(φ)B1idφ j . (10) The master equation (SZ , SZ) = 0 is satisfied if P , Q and J satisfy the conditions for a generalized complex structure (73), (74) and (75). We can see that the Batalin-Vilkovisky structure of this model defines a generalized complex structure on a target manifold M . If J ij = 0 in the action (10), the action reduces to the summation of two realizations of A model such that (3) + (6). However, if P ij = Qij = 0, the action (10) does not reduce to the B model action (8). So we can not easily see whether the Zucchini model can be related to B model. Also, we can consider b-transformation property of this model [1]. The b-transformation is defined by (77), (83) and = φi, B̂1i = B1i + bijdφ j . (11) The b-transformation produces the b field term such as ŜZ = SZ − bijdφ idφj. (12) This suggests that the Zucchini action with H 6= 0 should have a Wess-Zumino term SZH = B1idφ P ijB1iB1j + Qijdφ idφj + J ijB1idφ Hijkdφ idφjdφk,(13) where X is a three dimensional worldvolume such that Σ = ∂X is a two dimensional boundary of X . 3 3D Topological Field Theory with Generalized Com- plex Structures from 2D Zucchini Model In this section, we review a three dimensional topological field theory with a generalized complex structure from the Zucchini model in two dimensions. Here this topological field theory is redefined by the AKSZ formulation, which was not explicitly written in [33]. 3.1 H = 0 case Let X be a three dimensional worldvolume with a coordinate (σM) for M = 1, 2, 3, and Σ = ∂X be a two dimensional boundary of X . First we consider H = 0 case. By using the Stokes theorem, we can see the action (10) as B1idφ P ijB1iB1j + Qijdφ idφj + J ijB1idφ dB1idφ ∂P ij dφkB1iB1j + P ijdB1iB1j + dφkdφidφj ∂J ij dφkB1idφ j + J ijdB1idφ j , (14) where d is a three dimensional derivative d = θM∂M . φ i and B1i can be extended to those on X such that φi : ΠTX → M and B1i is a basis of sections of ΠT ∗X ⊗ φ∗(T ∗[1]M). We introduce a superfield Ai1 with total degree one, which is a basis of a section of ΠT ∗(T [1]M) such that Ai1 = dφ i, and a superfield B2i with total degree two, which is a basis of a section of ΠT ∗X ⊗ φ∗(T ∗[2]M) such that B2i = −dB1i. Moreover, we introduce two Lagrange multiplier fields Y 2i and Z 1 in order to realize two equations such as A 1 = dφ and B2i = −dB1i by the equations of motion. The superfield Y 2i with total degree two is a section of ΠT ∗X ⊗ φ∗(T ∗[2]M), and the superfield Zi1 with total degree one is a section of ΠT ∗X ⊗ φ∗(T [1]M). The 3D action (14) is equivalent to −B2iAi1 + ∂P ij 1B1iB1j − P ijB2iB1j + ∂J ij 1B1iA 1 − J ijB2iA 1 + (A 1 − dφ i)Y 2i + (B2i + dB1i)Z 1. (15) We define Y ′2i = Y 2i − 12B2i and Z 1 = Z 1 − 12A 1. The action (15) is rewritten as SZ = Sa + Sb + total derivative ; −Y ′2idφi + dB1iZ ′i1 + Y ′2iAi1 +B2iZ ′i1 , B2idφ B1idA 1 − J ijB2iA 1 − P ijB2iB1j + 1B1k + ∂P jk 1B1jB1k. (16) where Sa is independent of a generalized complex structure. Sb can be written as 〈0 +B2, d(φ+ 0)〉+ 〈A1 +B1, d(A1 +B1)〉 −〈0 +B2,J (A1 +B1)〉 − 〈A1 +B1,Ai1 (A1 +B1)〉+ total derivative,(17) which is analogical with the B model action (8). The antibracket (P-structure) on X , which is induced from the antibracket (2) on Σ, for i, B2,i, A1 i and B1,i is given by the antibracket (102) in n = 3. In order to define the antibrackets for Y ′2i and Z 1 , we introduce two antibracket conjugate fields X i, which are maps from ΠTX to M , and V 1i, which are sections of ΠT ∗X ⊗ φ∗(T ∗[1]M). The model is defined on the graded bundle of the direct product of T ∗[2]M ⊕ (T [1]M ⊕ T ∗[1]M) and (T [0]M⊕T ∗[2]M)⊕ (T [1]M ⊕ T ∗[1]M). The second bundle is represented by auxiliary fields. The antibracket is (F,G) ≡ F ∂B2,i ∂B2,i ∂B1,i ∂B1,i ∂Y ′2,i ∂Y ′2,i ∂Z1′i ∂V 1,i ∂V 1,i ∂Z1′i G. (18) We can check that SZ satisfies the master equation (SZ , SZ) = 0 if J , P and Q are components of the generalized complex structure (72). We can take the proper boundary conditions Σ = ∂X ; //|∂X = 0,B2i//|∂X = 0,Y ′2i//|∂X = 0,Z1 //|∂X = 0, (19) such that the total derivative terms on the master equation (SZ , SZ) vanish. Here // means that we take the components which are tangent to the boundary ∂X . Also, because (Sa, Sa) = (Sa, Sb) = 0, Sb satisfies the master equation (Sb, Sb) = 0 Aijk = Bijk = Cijk = 0, ∂iDjkl + (ijkl cyclic) = 0, (20) where Aijk, Bijk, Cijk and Djkl are defined in Appendix A. Therefore, we can see Sb as a three dimensional AKSZ action with generalized complex structure. We discuss why the condition is not Djkl = 0 but ∂iDjkl + (ijkl cyclic) = 0 in subsection 3.3. We call Sb three dimensional generalized complex sigma model. We consider 3D b-transformation property from the 2D b-transformations (11) and the conditions Ai1 = dφ and B2i = −dB1i. 3D b-transformations are = φi, = Ai1, B̂1i = B1i + bijA B̂2i = B2i − d(bijAj1), 2i = Y bijdA −J lk 1 − P lk 1B1k + d(J kbliA 1) + d(P lkbliB1k), = Z ′i1 . (21) We can see that 3D action (16) is invariant under the b-transformation such that ŜZ = SZ . (22) 3.2 H 6= 0 case I :Action induced from the Zucchini model In the similar way, we can consider the case of a twisted generalized complex structure with H 6= 0. From the Zucchini model with H 6= 0 (13), a three dimensional action is derived as SZH = Sa + SHb + total derivative ; −Y ′2idφ i + dB1iZ 1 + Y i +B2iZ SHb = B2idφ B1idA 1 − J ijB2iA 1 − P ijB2iB1j + Hijk + 1B1k + ∂P jk 1B1jB1k. (23) This action (23) satisfies the master equation (SZH, SZH) = 0, if J , P , Q and H are compo- nents of a twisted generalized complex structure (87). However, this action is not b-invariant under the b-transformation (21), (77) and (83). The action (23) transforms under the b- transformation as ŜZH = SZH − 1 = SZH − (dMb)[ijk]A 1, (24) which has been expected from b-transformation property (12) in the two dimensional model. Since H is closed, from the Poincaré Lemma, we can locally write H with a 2-form q on M such as Hijk = . (25) 1 term in (23) can be absorbed to Q by a local b-transformation qij = bij in the action (23), and we obtain just the H = 0 action (16). In other words, the H terms in (23) are consistent up to H-exact terms as a global theory, and this model is meaningful only as a cohomology class in H3(M). It is a gerbe gauge transformation dependence [1]. If we set Qij = J j = 0 in (13), we obtain the AKSZ formulation of the WZ-Poisson sigma model [38]: SWZP = B1idφ P ijB1iB1j + Hjkldφ idφjdφk. (26) From (23), the 3D topological sigma model equivalent to (26) is SWZP = Sa + SWZPb ; −Y ′2idφ i + dB1iZ 1 + Y 1 +B2iZ SWZPb = B2idφ dB1iA 1 − P ijB2iB1j + HijkA ∂P jk 1B1jB1k. (27) 3.3 H 6= 0 case II : b-invariant action We can construct a b-invariant action with H 6= 0 in three dimensional manifold X . We introduce other H terms. SI = Sa + SIb ; −Y ′2idφ i + dB1iZ 1 + Y 1 +B2iZ SIb = B2idφ B1idA 1 − J ijB2iA 1 − P ijB2iB1j + J liHjkl + −P klHijl − 1B1k + ∂P jk 1B1jB1k. (28) SI satisfies the master equation (SI , SI) = 0 under the antibracket (18) if and only if J , P , Q and H are components of a twisted generalized complex structure. Namely, the master equation (SI , SI) = 0 gives AHijk = BHijk = CHijk = 0, ∂iDHjkl + (ijkl cyclic) = 0, (29) where AHijk, BHijk, CHijk and DHjkl are defined in Appendix A. The integrability condition is not DHijk = 0 but ∂iDHjkl + (ijkl cyclic) = 0 because the action SI is b-transformation invariant, H ijk has b-transformation ambiguity by (83), and H is defined as a cohomology class in H3(M) in a twisted generalized complex structure. Since SIa does not depend on a twisted generalized complex structure, (SIb, SIb) = 0 is satisfied under the condition (29). We can introduce the coupling constants by redefining Y ′2i and Z1 ′i to g1Y 2i and g2Z 1 . If we take the limits that g1 → 0 and g2 → 0, then SI → SIb and a twisted generalized complex structure does not change. We call this model SIb a three dimensional twisted generalized complex sigma model. We can change the b-transformation so that the action SI is invariant, though the action (28) is not invariant under the original b-transformation (21). The b-transformations for B2i and Y ′2i are changed to B̂2i = B2i − 2i = Y 1 − bijdZ 1, (30) and b-transformations for the other fields are the same as (21). Then we can check ŜI = SI after short calculation. 4 2D Topological Field Theory of Generalized Complex Geometry In this section, we propose a new two dimensional topological field theory of generalized complex geometry using the 3D topological field theory. First, only a part of the 3D BV formalism action is dimensionally reducted to in two dimension, and next this is modified in the 2D BV formalism such that the master equations determine just generalized complex structures. One important reason to have to take this unusual way is that generally, master equations of BV formalisms are not kept by a dimensional reduction. 4.1 H = 0 First we consider the H = 0 case. We consider a dimensional reduction, which can keep a generalized complex structure, from a three dimensional worldvolume X to a two dimensional manifold Σ′. X is compactified to Σ′ × S1. Then ΠTX is compactified to ΠTΣ′ × ΠTS1. It should be noticed that Σ′ is generally a different manifold from Σ. Here, we take X = Σ×R+, where Σ has a local coordinate (σ1, σ2) and R+ = [0,∞) has a local coordinate (σ3). The second component (σ2) is compactified such that Σ′ = L×R+, whose local coordinate is (σ1, σ3), where L is a manifold in one dimension. We formulate the dimensional reduction from a general three dimensional manifold X to a general two dimensional manifold Σ′. Here we ignore Kaluza-Klein modes and consider only massless sectors, because we will see that the consistent BV action can be constructed in two di- mension even if these KK modes are omitted. It is not our purpose that we derive the two dimensional model which is completely equivalent to the 3D topological field theory. The target graded bundle for the three dimensional model, T ∗[2]M ⊕ (T [1]M ⊕ T ∗[1]M), re- duces to the graded bundle for the two dimensional model, (T ∗[1]M ⊕ (T [−1]M ⊕ T ∗[2]M))⊕ ((T [0]M ⊕ T ∗[1]M)⊕ (T [1]M ⊕ T ∗[0]M)). Under the dimensional reduction (σ1, σ2, σ3) → (σ1, σ3), the fields are reduced as follows. i(σ1, σ2, σ3) = φ̃ (σ1, σ3) + θ2φ̃−1 (σ1, σ3), 1, σ2, σ3) = Ã1 (σ1, σ3) + θ2α̃0 i(σ1, σ3), B1i(σ 1, σ2, σ3) = B̃1i(σ 1, σ3) + θ2β̃0i(σ 1, σ3), B2i(σ 1, σ2, σ3) = B̃2i(σ 1, σ3) + θ2β̃1i(σ 1, σ3), (31) where φ̃−1 has the total degree −1, φ̃ , α̃0 i and β̃0i have the total degree 0, Ã1 , B̃1i and β̃1i have the total degree 1, and B̃2i has the total degree 2. All these superfields do not depend on θ2. The antibracket induced from three dimensions is (F,G) ≡ F ∂β̃1i ∂β̃1i ∂φ̃−1 ∂B̃2i ∂B̃2i ∂φ̃−1 ∂β̃0i ∂β̃0i ∂B̃1i ∂B̃1i G. (32) We take a three dimensional AKSZ action Sb (16) with a generalized complex structure. The existence of the negative total degree superfield φ̃−1 complexifies the dimensional reduction in the AKSZ formulation. Generally in [39], it is known that even if we substitute (31) to (16), we do not obtain the correct AKSZ action in two dimensions, and we need more φ̃−1 terms. In order to derive the correct AKSZ action, first we should consider the dimensional reduction via the non-BV formalism. The superfields are expanded by the ghost numbers to i = φ(0)i + φ(−1)i + φ(−2)i + φ(−3)i, B1i = B 1,i +B 1,i +B 1,i +B 1,i , i = A 1 + A 1 + A (−1)i 1 + A (−2)i B2,i = B 2,i +B 2,i +B 2,i +B 2,i , (33) where φ(−1)i ≡ θMφ(−1)iM , etc. After setting all the antifield with negative ghost numbers to zero, the following non-BV action is 2i dφ (0)i + 1i dA 1 − J ijB 1 − P ijB ∂φ(0)i (φ(0)i)A ∂φ(0)i ∂φ(0)j (φ(0)i)A ∂P jk ∂φ(0)i (φ(0)i)A 1k . (34) Since by the dimensional reduction, the fields reduce to φ(0)i(σ1, σ2, σ3) = φ̃ (σ1, σ2), 1, σ2, σ3) = Ã1 (σ1, σ3) + θ2α̃0 (0)i(σ1, σ3), 1i (σ 1, σ2, σ3) = B̃1 1, σ3) + θ2β̃0 1, σ3), 2i (σ 1, σ2, σ3) = B̃2 1, σ3) + θ2β̃1 1, σ3), (35) the action (34) reduces to i dφ̃ + B̃1 i dα̃0 (0)i + Ã1 − J ijÃ1 i + P ijB̃1 i β̃1  α̃0 (0)k + − ∂J  β̃0  Ã1  α̃0 (0)j − ∂P  Ã1 ∂P jk  B̃1 j B̃1 J ijα̃0 (0)j + P ijβ̃0 i , (36) up to total derivative terms. Therefore the action S R of a 2D topological field theory is R = S 0 + S i dφ̃ + B̃1 i dα̃0 (0)i + Ã1 −J ijÃ1 i + P ijB̃1 i β̃1  α̃0 (0)k + − ∂J  β̃0  Ã1  α̃0 (0)j − ∂P  Ã1 ∂P jk  B̃1 j B̃1 J ijα̃0 (0)j + P ijβ̃0 i . (37) Next we formulate the action SR by the AKSZ formulation. We define SR = S0 + S1 where S0 and S1 are AKSZ actions for S 0 and S 1 , respectively. S0 is easily derived after substituting (31) to (16); β̃1idφ̃ − B̃2idφ̃−1 + B̃1idα̃0 i + Ã1 dβ̃0i up to total derivative terms. The condition (S1, S1) = 0 comes from (38) and (SR, SR) = 0. We introduce an negative total degree, which is defined as one for φ̃−1, and zero for the other fields. We can expand S1 for the negative total degree such as S1 = p=0 S 1 , where i1 · · · φ̃−1 ipL[p]i1···ip(φ̃, Ã1 , α̃0, B̃1i, β̃0i, B̃2i, β̃1i) (39) are the negative total degree p terms. Therefore SR = S0 + 1 . (40) Here we write the first two actions S 1 and S 1 with the negative total degree zero and one by substituting (31) to (16), −J ijÃ1 β̃1i + P B̃1iβ̃1j  α̃0  β̃0k  Ã1  α̃0 j − ∂P  Ã1 B̃1k + ∂P jk  B̃1jB̃1k J ijα̃0 j + P ijβ̃0j B̃2i, (41) ∂J ij B̃2iÃ1 ∂P ij B̃2iB̃1j − ∂2Qjk  Ã1 B̃1k − ∂2P jk B̃1jB̃1k . (42) 1 for p > 1 are recursively derived from the master equation (S1, S1) = p=0{(S1, S1)}[p] = 0. It should be noticed that since a target space M has finite dimensions, S 1 is nonzero for only a finite number of p. This action is a special case of a nonlinear gauge theory with 2-forms (a generalization of the Poisson sigma model) analyzed in the paper [39][40][41]. 4.2 H 6= 0 Here we consider H 6= 0 case. A 2D topological field theory of twisted generalized complex geometry is derived in a similar way in subsection 4.1 from H-terms in section 3.2: SR = S0 + 1 ; (43) −J ijÃ1 β̃1i + P B̃1iβ̃1j 3Hijk +  α̃0  β̃0k  Ã1  α̃0 j − ∂P  Ã1 B̃1k + ∂P jk  B̃1jB̃1k J ijα̃0 j + P ijβ̃0j B̃2i, (44) ∂J ij B̃2iÃ1 ∂P ij B̃2iB̃1j − Hijk +  Ã1  Ã1 B̃1k − ∂2P jk B̃1jB̃1k , (45) and S 1 for p > 1 are recursively derived from (SR, SR). Also, from b-invariant H-terms in section 3.3, SR = S0 + 1 ; (46) −J ijÃ1 β̃1i + P B̃1iβ̃1j 3J liHjkl +  α̃0 −P klHjkl −  β̃0k  Ã1 P klHjkl +  α̃0 j − ∂P  Ã1 B̃1k + ∂P jk  B̃1jB̃1k J ijα̃0 j + P ijβ̃0j B̃2i, (47) ∂J ij B̃2iÃ1 ∂P ij B̃2iB̃1j − JmiHjkm +  Ã1 −P kmHjkm −  Ã1 B̃1k − ∂2P jk B̃1jB̃1k , (48) and S 1 for p > 1 are recursively derived from (SR, SR). 5 Two Special Reductions to Complex Geometry and Symplectic Geometry In this section, we consider two special reductions related to complex geometry and of sym- plectic geometry. 5.1 Complex geometry First we consider our model in complex geometry, which is the case that P = Q = H = 0 in the action (40). We redefine superfields as , φ̃−1 = λφ̃′ , α̃0 i = λα̃′ B̃1i = λB̃ , β̃0i = −β̃′0i, B̃2i = λB̃ , β̃1i = , (49) where λ is a constant. After this redefinition, the action (40) is SR = S0 + 1 , (50) i − Ã′ + B̃′ , (51) J ijβ̃′1ià − J ijα̃′0  , (52) λφ̃−1 ∂J ij ∂2Jkj , (53) and S 1 has at least the higher order of λ than λ p because φ̃−1 = λφ̃′ . We can take the limit λ −→ 0 with preserving the complex structure. S [p]1 for p > 0 reduces to zero, and the 2D action is SRJ = β̃1idφ̃ − Ã1 dβ̃0i + J jβ̃1iÃ1 ∂J ik β̃0iÃ1 . (54) This action is nothing but the B model action (8) up to a total derivative and the all over factor 1 , which depends on only J ij. The master equation (SbJ , SbJ) = 0 impose the condition that J ij is a complex structure. We make a comment about the difference between the action (50) with a finite λ and the B model action (54) with λ → 0. Following the well-known method in [6], we can see that the topological string theory has to be deformed by the other terms in (50) than in the B model. In the calculation of [6], we may locally take the complex structure as a constant, and the kinetic terms (51) and two terms in (52) without the derivatives of J ij are only different parts from those in the B model. Here it should be noted that although these deformed parts may seem to decouple to the B model part, the interactions between them can come from the non-constant metric. These deformed parts couple to only the metric on the bosonic space of φ̃ , which is independent of φ̃−1 , because there is no metric with fermionic indices on the fermionic space of φ̃−1 . So these deformed parts can be seen as a topological theory with only B field-like couplings on the fermionic space of φ̃−1 . Physically, we may assume that there is no topological information along fermionic directions, although this situation with no metric is special. Therefore in this assumption, we can see that our action (50) is equivalent to topological string theory, called topological B model. As a future work, it would be interesting to check this equivalence more carefully. 5.2 Symplectic geometry Next we consider our model in symplectic geometry, which is the case that J = H = 0 in the action (40). We redefine superfields as , φ̃−1 = µφ̃′ = µÃ′ , α̃0 i = α̃′ B̃1i = , β̃0i = −µβ̃′0i, B̃2i = µB̃ , β̃1i = , (55) where µ is a constant. After this redefinition, the 2D action (40) reduces to SR = S0 + + B̃′ − Ã′ P ijB̃′ ∂P jk  (57)  α̃′ j − 1 ∂P jk + P ijβ̃′ µφ̃−1 ∂P ij B̃2iB̃1j − ∂2Qjk ∂2P jk ,(58) and S 1 is at least the higher order of λ than λ p because φ̃−1 = µφ̃′ After taking the limit µ −→ 0 with preserving the symplectic structure, S [p]1 for p > 0 reduces to zero, and the 2D action is SRP = β̃1idφ̃ + B̃1idα̃0 i + P ijB̃1iβ̃1j + ∂P jk B̃1jB̃1k. (59) The BV condition (SbP , SbP ) = 0 is satisfied if and only if P ij is a Poisson structure (the inverse of a symplectic structure) (7). It should be noticed that although this action (59) depends on only a symplectic structure P ij, this action is a different realization of the Poisson structure from the A model (6), because we can also check that this model is not equivalent to topological string theory following the similar way as in [6]. 6 Conclusions and Discussion We have constructed a topological field theory with a generalized complex structure in three dimensions and two dimensions using the AKSZ formulation. Our model reduces to B model in a limit if the generalized complex structure is only a complex structure, although Zucchini model reduces to A model in the limit that the generalized complex structure is only a symplectic structure. It would be interesting to check that the Zucchini model and our model are equivalent to a topological string theory with a generalized complex structure [23][24], which is constructed from the twisted N = (2, 2) supersymmetric sigma model with a non-trivial B field. Appendix A. Generalized Complex Structure In this appendix A, we summarize a generalized complex structure, based on description of section 3 in [11] and section 2 in [1]. Let M be a manifold of even dimension d with a local coordinate {φi}. We consider the vector bundle TM ⊕ T ∗M . We denote a section as X + ξ ∈ C∞(TM ⊕ T ∗M) where X ∈ C∞(TM) and ξ ∈ C∞(T ∗M). TM ⊕ T ∗M is equipped with a natural indefinite metric of signature (d, d) defined by 〈X + ξ, Y + η〉 = 1 (iXη + iY ξ), (60) for X + ξ, Y + η ∈ C∞(TM ⊕ T ∗M), where iV is an interior product with a vector field V . In the Cartesian coordinate (∂/∂φi, dφi), The metric is written as follows: , (61) We define a Courant bracket on TM ⊕ T ∗M as follows: [X + ξ, Y + η] = [X, Y ] + LXη −LY ξ − dM(iXη − iY ξ), (62) with X + ξ, Y + η ∈ C∞(TM ⊕T ∗M), where LV denotes Lie derivation with respect a vector field V and dM is the exterior differential of M . This bracket is antisymmetric but do not satisfy the Jacobi identity. We may consider a so called Dorfman bracket as follows: (X + ξ) ◦ (Y + η) = [X, Y ] + LXη − iY dξ, (63) which satisfies the Jacobi identity but is not antisymmetric. Antisymmetrization of a Dorfman bracket coincides with a Courant bracket. A generalized almost complex structure J is a section of C∞(End(TM ⊕ T ∗M)), which is an isometry of the metric 〈 , 〉, J ∗IJ = I, and satisfies J 2 = −1. (64) A b-transformation is an isometry defined by exp(b)(X + ξ) = X + ξ + iXb, (65) where b ∈ C∞(∧2T ∗M) is a 2–form. A Courant bracket is covariant under the b-transformation [exp(b)(X + ξ), exp(b)(Y + η)] = exp(b)[X + ξ, Y + η], (66) if the 2–form b is closed. The b-transform of J is defined by Ĵ = exp(−b)J exp(b). (67) J has the ± −1 eigenbundles because J 2 = −1, In order to divide TM ⊕ T ∗M to each eigenbundle, we need complexification of TM ⊕ T ∗M , (TM ⊕ T ∗M)⊗ C. The projectors on the eigenbundles are defined by −1J ). (68) The generalized almost complex structure J is integrable if Π∓[Π±(X + ξ),Π±(Y + η)] = 0, (69) for any X + ξ, Y + η ∈ C∞(TM ⊕ T ∗M), where the bracket is the Courant bracket. Then J is called a generalized complex structure. Integrability is equivalent to the single statement N(X + ξ, Y + η) = 0, (70) for all X + ξ, Y + η ∈ C∞(TM ⊕ T ∗M), where N is the generalized Nijenhuis tensor defined N(X + ξ, Y + η) = [X + ξ, Y + η]− [J (X + ξ),J (Y + η)] + J [J (X + ξ), Y + η] +J [X + ξ,J (Y + η)]. (71) The b-transform Ĵ of a generalized complex structure J is a generalized complex structure if the 2–form b is closed. We decompose a generalized almost complex structure J in coordinate form as follows , (72) where J,K ∈ C∞(TM ⊗ T ∗M), P ∈ C∞(∧2TM), Q ∈ C∞(∧2T ∗M). Then the conditions J ∗IJ = I, and J 2 = −1 derive i = −J ij J ikJ j + P ikQkj + δ j = 0, J ikP kj + J jkP ki = 0, j +QjkJ i = 0, (73) where P ij + P ji = 0, Qij +Qji = 0. (74) The integrability condition (69) is equivalent to the following condition Aijk = Bijk = Cijk = Dijk = 0, (75) where Aijk = P il∂lP jk + P jl∂lP ki + P kl∂lP ij, Bijk = J li∂lP jk + P jl(∂iJkl − ∂lJki) + P kl∂lJ j i − J j l∂iP lk, Cijk = J li∂lJkj − J lj∂lJki − Jkl∂iJ lj + Jkl∂jJ li +P kl(∂lQij + ∂iQjl + ∂jQli), Dijk = J li(∂lQjk + ∂kQlj) + J lj(∂lQki + ∂iQlk) +J lk(∂lQij + ∂jQli)−Qjl∂iJ lk −Qkl∂jJ li −Qil∂kJ lj . (76) Here ∂i is a differentiation with respect to φ i. The b–transform is Ĵ ij = J j − P ikbkj, P̂ ij = P ij, Q̂ij = Qij + bikJ j − bjkJki + P klbkiblj . (77) where bij + bji = 0. The usual complex structures J is embedded in generalized complex structures as the special form 0 −tJ . (78) Indeed, one can check this form satisfies conditions, (73) and (75) if and only if J is a complex structure. Similarly, the usual symplectic structures Q is obtained as the special form of generalized complex structures 0 −Q−1 . (79) This satisfies (73) and (75) if and only if Q is a symplectic structure, i. e. it is closed. Other exotic examples exist. There exists manifolds which cannot support any complex or symplectic structure, but admit generalized complex structures. The Courant bracket on TM ⊕ T ∗M can be modified by a closed 3–form. Let H ∈ C∞(∧3T ∗M) be a closed 3–form. We define the H twisted Courant brackets by [X + ξ, Y + η]H = [X + ξ, Y + η] + iX iYH, (80) where X + ξ, Y + η ∈ C∞(TM ⊕ T ∗M). Under the b-transform with b a closed 2–form, [exp(b)(X + ξ), exp(b)(Y + η)] = exp(b)[X + ξ, Y + η], (81) holds with the brackets [ , ] replaced by [ , ]H . For a non closed b, one has [exp(b)(X + ξ), exp(b)(Y + η)]H−dM b = exp(b)[X + ξ, Y + η]H . (82) So, the b-transformation shifts H by the exact 3–form dMb: Ĥ = H − dMb. (83) One can define an H twisted generalized Nijenhuis tensor NH as follows N(X + ξ, Y + η) = [X + ξ, Y + η]H − [J (X + ξ),J (Y + η)]H + J [J (X + ξ), Y + η]H +J [X + ξ,J (Y + η)]H , (84) by using the brackets [ , ]H instead of [ , ]. A generalized almost complex structure J is H integrable if NH(X + ξ, Y + η) = 0, (85) for all X + ξ, Y + η ∈ C∞(TM ⊕ T ∗M). Then we call J an twisted generalized complex structure. The H integrability conditions is as follows: AHijk = BHijk = CHijk = DHijk = 0, (86) where AHijk = Aijk, BHijk = Bijk + P jlP kmHilm CHijk = Cijk − J liP kmHjlm + J ljP kmHilm, DHijk = Dijk −Hijk + J liJmjHklm + J ljJmkHilm + J lkJmiHjlm. (87) Appendix B. AKSZ Formulation of Batalin-Vilkovisky Formalism In the appendix B, we review the AKSZ formulation in any dimension [42]. In order to construct and analyze topological field theories systematically, it is useful to use Batalin- Vilkovisky formalism. The geometric structure of the AKSZ formulation is called Batalin- Vilkovisky Structures. B-1. Batalin-Vilkovisky Structures on Graded Vector Bundles Let M be a smooth manifold in d dimensions. If we consider We define a supermanifold ΠT ∗M . Mathematically, ΠT ∗M , whose bosonic part is M , is defined as a cotangent bundle with reversed parity of the fiber. That is, a base manifold M has a Grassman even coordinate and the fiber of ΠT ∗M has a Grassman odd coordinate. We introduce a grading called total degrees, which is denoted |F | for a function F . The coordinates of the base manifold have grade zero and the coordinates of the fiber have grade one. Similarly, we can define ΠTM for a tangent bundle TM . ΠTM is also called a supermanifold. We must consider more general assignments for the degree of the fibers of T ∗M or TM . For an integer p, we define T ∗[p]M , which is called a graded cotangent bundle. T ∗[p]M is a cotangent bundle, whose fiber has the degree p. This degree is also called the total degree. A coordinate of the bass manifold have the total degree zero and a coordinate of the fiber have the total degree p. If p is odd, the fiber is Grassman odd, and if p is even, the fiber is Grassman even. We define a graded tangent bundle T [p]M in the same way. We consider a vector bundle E. A graded vector bundle E[p] is defined in the similar way. E[p] is a vector bundle whose fiber has a shifted degree by p. Note that only the degree of fiber is shifted, and the degree of base space is not shifted. We consider a Poisson manifold N with a Poisson bracket {∗, ∗}. If we shift the total degree, we can construct a graded manifold (a graded cotangent bundle or a graded vector bundle) Ñ from N . Then a Poisson structure {∗, ∗} shifts to a graded Poisson structure by grading of Ñ . The graded Poisson bracket is called an antibracket and denoted by (∗, ∗). (∗, ∗) is graded symmetric and satisfies the graded Leibniz rule and the graded Jacobi identity with respect to grading of the manifold. The antibracket (∗, ∗) with the total degree −n + 1 satisfies the following identities: (F,G) = −(−1)(|F |+1−n)(|G|+1−n)(G,F ), (F,GH) = (F,G)H + (−1)(|F |+1−n)|G|G(F,H), (FG,H) = F (G,H) + (−1)|G|(|H|+1−n)(F,H)G, (−1)(|F |+1−n)(|H|+1−n)(F, (G,H)) + cyclic permutations = 0, (88) where F,G and H are functions on Ñ , and |F |, |G| and |H| are total degrees of the functions, respectively. The graded Poisson structure is also called P-structure. If n = 1, the antibracket is equivalent to the Schouten bracket. For higher n, the antibracket is equivalent to the Loday bracket [43] with the degree −n + 1. Typical examples of Poisson manifold N are a cotangent bundle T ∗M and a vector bundle E ⊕ E∗. First we consider a cotangent bundle T ∗M . Since T ∗M has a natural symplectic structure, we can define a Poisson bracket induced from the symplectic structure. If we take a local coordinate φi on M and a local coordinate Bi of the fiber, we can define a Poisson bracket as follows: {F,G} ≡ F G, (89) where F and G are functions on T ∗M , and ∂ /∂ϕ and ∂ /∂ϕ are the right and left differen- tiations with respect to ϕ, respectively. Here we shift the degree of fiber by p, i.e. the space T ∗[p]M . Then a Poisson structure shifts to a graded Poisson structure. The corresponding graded Poisson bracket is called antibracket, (∗, ∗). Let φi be a local coordinate of M and Bn−1,i a basis of the fiber of T ∗[p]M . The antibracket (∗, ∗) on a cotangent bundle T ∗[p]M is expressed as: (F,G) ≡ F ∂Bp,i ∂Bp,i G. (90) The total degree of the antibracket (∗, ∗) is −p. This antibracket satisfies the property (88) for −p = −n + 1. Next, we consider a vector bundle E ⊕ E∗. There is a natural Poisson structure on the fiber of E ⊕ E∗ induced from a paring of E and E∗. If we take a local coordinate Aa on the fiber of E and Ba on the fiber of E ∗, we can define {F,G} ≡ F G, (91) where F and G are functions on E ⊕ E∗. We shift the degrees of fibers of E and E∗ like E[p]⊕E∗[q], where p and q are positive integers. The Poisson structure changes to a graded Poisson structure (∗, ∗). Let Apa be a basis of the fiber of E[p] and Bq,a a basis of the fiber of E∗[q]. The antibracket is represented as (F,G) ≡ F ∂Bq,a G− (−1)pqF ∂Bq,a G. (92) The total degree of the antibracket (∗, ∗) is −p − q. This antibracket satisfies the property (88) for −p− q = −n + 1. We define a Q-structure. A Q-structure is a function S on a graded manifold Ñ which satisfies the classical master equation (S, S) = 0. S is called a Batalin-Vilkovisky action. We require that S satisfy the compatibility condition S(F,G) = (SF ,G) + (−1)|F |+1(F, SG), (93) where F andG are arbitrary functions. (S, F ) = δF generates an infinitesimal transformation, which is a BRST transformation, which coincides with the gauge transformation of the theory. The AKSZ formulation of the Batalin-Vilkovisky formalism is defined as a P-structure and a Q-structure on a graded manifold. B-2. Batalin-Vilkovisky Structures of Topological Sigma Models In this subsection, we explain Batalin-Vilkovisky structures of topological sigma models. Let X be a base manifold in n dimensions, with or without boundary, and M be a target manifold in d dimensions. We denote φ a smooth map from X to M . We consider a supermanifold ΠTX , whose bosonic part is X . ΠTX is defined as a tangent bundle with reversed parity of the fiber. We take a local coordinate of ΠTX , (σµ, θµ), where σµ is a coordinate on the base space and θµ is a super coordinate on the fiber and µ = 1, 2, · · · , n. We extend a smooth function φ to a function on the supermanifold φ : ΠTX → M . φ is called a superfield and an element of ΠT ∗X ⊗M . We introduce a new non-negative integer grading on ΠT ∗X . A coordinate σµ on a base manifold has zero and a coordinate θµ on the fiber has one. This grading is called the form degree. We denote degF the form degree of the function F . The total degree defined in the previous section is a grading with respect to M , on the other hand The form degree is a grading with respect to X . We define a ghost number ghF such that ghF = |F | − degF . W assign the ghost numbers of σµ and θµ zero. Thus σµ has the total degree zero and θµ has total degree one. We consider a P-structure on T ∗[p]M . We take p = n−1 to construct a Batalin-Vilkovisky structure in a topological sigma model on a general n dimensional worldvolume. We consider T ∗[n− 1]M for an n-dimensional base manifold X . Let a superfield φi be local a coordinate of ΠT ∗X ⊗M , where i, j, k, · · · are indices of the local coordinate on M . Let a superfield Bn−1,i be a basis of sections of ΠT ∗X ⊗φ∗(T ∗[n− 1]M). Expansions to component fields of the superfields are the following: i = φ(0)i + θµ1φ(−1)iµ1 + θµ1θµ2φ(−2)iµ1µ2 + · · ·+ θµ1 · · · θµnφ(−n)iµ1···µn , (94) Bn−1,i = B (n−1) n−1,i + · · ·+ (n− 1)! θµ1 · · · θµn−1B(0)µ1···µn−1n−1,i + θµ1 · · · θµnB(−1)µ1···µnn−1,i, where (p) is the ghost number of the component field. From (90) in the previous subsection, we define an antibracket (∗, ∗) on a cotangent bundle T ∗[n− 1]M as (F,G) ≡ F ∂Bn−1,i ∂Bn−1,i G, (95) where F and G are functions of φi and Bn−1,i. The total degree of the antibracket is −n+1. If F and G are functionals of φi and Bn−1,i, we understand an antibracket is defined as (F,G) ≡ ∂Bn−1,i ∂Bn−1,i G, (96) where the integration ΠTX means the integration on the supermanifold, ΠTX d nθdnσ. Through this article, we always understand an antibracket on two functionals in a similar manner and abbreviate this notation. Next we consider a P-structure on E ⊕ E∗. In a topological sigma model in n dimension worldvolume, we assign the total degree of p and q such that p+ q = n− 1. The total graded bundle is E[p] ⊕ E∗[n − p − 1], where −n + 1 ≤ p ≤ n − 1, p 6= 0. Let Apap be a basis of sections of ΠT ∗X ⊗φ∗(E[p]) and Bn−p−1,ap a basis of the fiber of ΠT ∗X ⊗φ∗(E∗[n− p− 1]). Expansions to component fields of the superfields are ap = A(p)app + θ µ1A(p−1)apµ1p + · · ·++ (p− 1)! θµ1 · · · θµ(p−1)A(0)apµ1···µ(p−1)p + · · ·+ 1 θµ1 · · · θµn , A(−n+p)apµ1···µnp (97) Bn−p−1,ap = B (n−p−1) n−p−1,ap + θµ1B (n−p−2) µ1n−p−1,ap + · · ·+ 1 (n− p− 1)! θµ1 · · · θµ(n−p−1)B(0)µ1···µ(n−p−1)n−p−1,ap + · · ·+ 1 θµ1 · · · θµnB(−p−1)µ1···µnn−p−1,ap, From (92), we define the antibracket as (F,G) ≡ F ∂Apap ∂Bn−p−1,ap G− (−1)npF ∂Bn−p−1,ap ∂Apap G. (98) We need to consider various grading assignments for E ⊕ E∗, because each assignment induces different Batalin-Vilkovisky structures. In order to consider all independent assign- ments, we define the following bundle. 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0704.1017

Magnetic susceptibility and specific heat of the spin-1/2 Heisenberg model on the kagome lattice and experimental data on ZnCu3(OH)6Cl2

EPJ manuscript No. (will be inserted by the editor) Magnetic susceptibility and specific heat of the spin-1 Heisenberg model on the kagome lattice and experimental data on ZnCu3(OH)6Cl2. Grégoire Misguich1 and Philippe Sindzingre2 1 Service de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France 2 Laboratoire de Physique Théorique de la Matière condensée, Univ. P. et M. Curie, 75252 Paris Cedex, France the date of receipt and acceptance should be inserted later Abstract. We compute the magnetic susceptibility and specific heat of the spin- 1 Heisenberg model on the kagome lattice with high-temperature expansions and exact diagonalizations. We compare the results with the experimental data on ZnCu3(OH)6Cl2 obtained by Helton et al. [Phys. Rev. Lett. 98, 107204 (2007)]. Down to kBT/J ' 0.2, our calculations reproduce accurately the experimental susceptibility, with an exchange interaction J ' 190 K and a contribution of 3.7% of weakly interacting impurity spins. The comparison between our calculations of the specific heat and the experiments indicate that the low- temperature entropy (below ∼20 K) is smaller in ZnCu3(OH)6Cl2 than in the kagome Heisenberg model, a likely signature of other interactions in the system. PACS. 75.50.Ee Antiferromagnetics – 75.10.Jm Quantized spin models – 75.40.Cx Static properties 1 Introduction After many years of theoretical investigations, the nature of the ground-state of the spin- 1 Heisenberg model on the kagome lattice is still not known. Although all numerical studies have concluded to the absence of long-range mag- netic (Néel) order [1,2,3,4,5,6,7,8], many basic questions such as the existence of spontaneously broken symmetries, or the existence of a finite gap to magnetic excitations re- main open. In fact, many different states of matter have been proposed for the kagome Heisenberg antiferromag- net: Z2 gapped topological liquids [9,10], valence-bond crystals [11,12,13,14], critical spin liquids with gapless spinons [15,12]. Recently, a promising spin- 1 antiferromagnetic insula- tor with an ideal kagome geometry, ZnCu3(OH)6Cl2, has been synthesized and studied for its magnetic properties [16,17,18,19,20]. Because these studies did not detect any kind ordering (nor spin freezing) down to 50mK, it could represent one of the first and most remarkable realization of a 2D quantum spin liquid [21,22]. To extract some information about the low-energy physics of the kagome Heisenberg model from the experiments on ZnCu3(OH)6Cl2, it is important to first analyze in a quan- titative way the possible role of magnetic defects (and other “perturbations”to this model) in this compound. In this paper we compare the experimental data for the magnetic susceptibility χ and specific heat cv obtained by Correspondence to: [email protected] Helton et al.[16] with calculations for the spin- 1 Heisen- berg model on the kagome lattice based on exact diago- nalization (ED) data (partial spectrum of a 36-site cluster and full spectrum for 24-site and 18-sites clusters) and high-temperature series expansion [23,24]. Down to tem- perature kBT/J = 0.2, the experimental susceptibility χexp(T ) can be very well fitted by that of the kagome lat- tice Heisenberg model with J ' 190 K plus a contribution of about 4% of impurity spins with weak mutual inter- actions (modeled by a ferromagnetic Curie-Weiss temper- ature of ' −6 K), likely mostly due to antisite disorder (Cu substituted to Zn on sites between the kagome planes) [20,25]. The low temperature specific heat is dominated by impurities (and other perturbations) below 2 K and by phonons above 15 K. In the intermediate range, the calculated specific heat appears to be larger than in the experiment. We comment on this feature at the end of the paper. 2 Uniform static susceptibility The spin- 1 Heisenberg model reads: H = J 〈i,j〉 Si · Sj − gµBH Szi (1) where the sum runs over pairs of nearest neighbor sites on the kagome lattice and H is an external magnetic field. 2 G. Misguich and P. Sindzingre: Magnetic susceptibility and specific heat of the spin- 1 Heisenberg model ... To fix the notations, we define the (zero-field) uniform susceptibility per site χ(T ) as χ(T ) = gµB 〈Szi 〉 where N is the total number of spins. The high-temperature expansion of χ has been com- puted up to order O (included) by Elstner and Young [23]: χ(T ) = χth(t = kBT/J) χth(t) = (1/4)t −1 − (1/4)t−2 + · · · (2) where C0 = 0.25(gµB)2 is the Curie constant and t the re- duced temperature. The truncated series χn(t) = i=0 cit at order n = 14 and n = 15 agree with a relative error smaller than 10−2 for t > 1. Down to this temperature, they already provide good approximations to χth(t). The convergence of high-T series can be improved using Padé approximants (PA). In the present case the PA provide a reliable estimate of χth(t) at least down to t ∼ 0.5.1 One representative PA (numerator of degree 8 and denomina- tor of degree 7) is displayed Fig. 1. On a small enough system, it is possible to obtain by ED the full spectrum (2N energy levels for N spins). We have done so for two 18-site and 24-site kagome clus- ters (with periodic boundary conditions). Then thermo- dynamic quantities can be computed exactly as a func- tion of T . For bigger systems, where one can still compute some eigenstates by ED, one may use the approximate method described in reference [26] to compute thermody- namic quantities. footnote In this method, one constructs the density of states in each symmetry sector from the exact low and high energy states obtained from ED and approximating in between the unknown part of the spec- trum with a smooth density of states. This smooth part is constructed so that the first moments of the density of states (Tr[Hn]), in each symmetry sector of the finite clus- ter, are exact up to n = 5. This Ansatz guaranties that the thermodynamics becomes exact at low T (when thermal excitations only involve the eigenstates computed exactly) as well as at high T (when a re-summed high-temperature expansion up to T−5 is valid). The results for the susceptibility χ(t) are shown in Fig. 1. Above t = 0.2, the relative difference between the (exact) N = 18 and N = 24 curves is smaller than 0.5%. We therefore make the (rather safe) assumption that our finite-size results are good approximations to the thermo- dynamic limit down to tmin ' 0.2. This represents a small gain over the coupled-cluster expansion of reference [27], which is valid above t ∼ 0.3. The χ(t) obtained with the approximate method for N = 36 sites also agrees (with a relative error smaller than 2%) with the N = 24 results down to t ' 0.2. Slightly be- low, the 36-site susceptibility is still increasing and might be a better approximation to the infinite-size limit than the 24-site curve. Still, it is not possible decide at which t finite-size (and/or errors due to the approximation in the 1 Comparing the PA and the exact curves for 18 and 24-site clusters shows that the PA is in fact correct down to T ∼ 0.4J , see Fig. 1. density of states) will become too important. Safely, we only use the theoretical results (noted χth) above t ' 0.2 to fit the experimental data χexp for the susceptibility. We fit χexp to a sum of contributions from the kagome spins χs and the impurities χimp in the following way: χexp(T ) = χs(kBT/J) + χimp(T ) (3) with χimp(T ) = T + θimp where J is the (unknown) magnetic exchange in between the spins in the kagome planes, x an impurity concentra- tion, C0 = 0.25(gµB)2/kB the Curie constant and θimp the Curie-Weiss temperature of the system of impurities. Eq. 4 is the leading term in a high-temperature expan- sion for the system of impurities, which provides a sim- plified picture of their interactions. To be applicable, T should therefore be large compared to θimp. We assume that the system of impurities does not perturb the the kagome spins. We optimized numerically the parameters so that χs fits the theoretical results χth for t ≥ 0.2. As can be seen on Fig. 1, an almost perfect agreement can be obtained be- tween χs = J4C0 (χexp(T )−χimp(T )) (squares) and the the- oretical estimates for the kagome susceptibility χth with the following parameters: C0 = 0.504 K cm3/(mol of Cu) (equivalent to a gyromagnetic factor g = 2.32), J = 190.4 K, x = 0.03655 and θimp = −6.1 K. We note that the value of J is in rough agreement with the values reported in reference [16] (17meV'200 K) and [27] (170 K). We also check a posteriori that the lowest temperature of the fit (t = 0.2 ' 38 K) is much bigger than |θimp|, so that a Curie-Weiss approximation for the impurities is justified. 6 K is also approximatively the transition temperature re- ported in ZnxCu4−x(OH)6Cl3 for x < 0.6 (replacing some Zn by Cu between the kagome planes) [17], so this energy scale may correspond to some couplings for spins located between the planes, where magnetic impurities could sit. We eventually notice that down to t = 0.15 (that is below the lowest temperature used for the fit), χexp − χimp con- tinues to increase and to follow the N = 36 curve. This suggests that the maximum of the kagome susceptibility could indeed be below t = 0.15. Rigol and Singh [27] analyzed the same experimen- tal data with another high-temperature method (coupled cluster expansion) and obtained a somewhat different con- clusion. They argued that Dzyaloshinskii-Moriya (DM) in- teractions provide a better description of the low-temperature increase of the susceptibility than impurities. Although we agree that DM interactions are certainly present in ZnCu3(OH)6Cl2 and that they should affect the physics of the system (at least at low-temperatures), it is also clear that a few percent of impurities must be present too and should have a visible effect on the susceptibility, even at rather high temperatures. According to reference [27], free impurities cannot explain the sharp increase of χexp below 60 K. In our analysis, this issue is solved by al- lowing for a small ferromagnetic Curie-Weiss temperature G. Misguich and P. Sindzingre: Magnetic susceptibility and specific heat of the spin- 1 Heisenberg model ... 3 Fig. 1. (color online) Magnetic susceptibility per spin as a function of temperature. Dashed (magenta) curve : experimen- tal data (Helton et al.), multiplied by J/(4C0) as a function of kBT/J with J ' 190 K and C0 = 0.504 K cm3/(mol of Cu). Black squares: χs = (χexp(T ) − χimp(T )), obtained from the experimental data χexp by subtracting the contribu- tion χimp of a concentration x = 0.03655 of impurity spins with a Curie-Weiss temperature θimp = −6.1 K. Red (resp. cyan) curve : Exact χth for a N = 24 (resp. 18) site kagome cluster with periodic boundary conditions. Blue curve: Results for N = 36 spins obtained with the (approximate) method of reference [26]. Green curve: Padé approximant from the high- temperature expansion at order t−15. The Padé approximant is not converged below t ' 0.4 whereas the finite-size curves are practically converged to the thermodynamic limit down to t ' 0.2. Below t = 0.2J , the later curves are only indicative. θimp ' −6 K for the impurities.2 One can indeed see that, once χimp has been subtracted, the experimental results show a saturation of χ around 20 K, in rough agreement with the measurements of reference [19]. The location of the maximum we obtain is however quite sensitive to the value x of the impurity concentration. 3 Specific heat The experimental data for the specific heat cv(T ) are only available at very low temperature where the size effects on the ED results are large. We therefore also applied the high-temperature entropy method [28]. It combines three pieces of information about the system: 1) The high 2 The Curie-Weiss temperature is an average of the differ- ent exchange constants. However, due to the complexity of the interactions between impurities (disorder), the ferromagnetic sign of θimp does not necessarily imply that they behave ferro- magnetically at low temperatures. temperature series expansion of cv(T ), up to T−17 [23, 24]. 2) The ground-state energy per site e0 of the Hamil- tonian. Here we use the following estimate e0 = 2〈0|Si · Sj |0〉 = −0.44 [24]. 3) The exponent α describing the low- temperature behavior of the specific heat: cv(t→ 0) ∼ tα. The method then provides a set of cv(t) curves (for dif- ferent Padé approximants) which all satisfy exactly the following properties: i) cv(T → 0) ∼ Tα, ii) cv(T →∞) ∼ the series expansion, iii) cv(T )dT = −Ne0, and iv)∫∞ cv(T )/TdT = NkB ln(2). When the value of e0 and α are both correct, one usually gets a large number of very similar curves but if either is incorrect, only a few and scat- tered curves will be obtained (more details in Refs. [28, 24]). In the case of the kagome antiferromagnet, the value of α is not known and the entropy method gives a reason- able convergence for α = 1 and α = 2 [24]. Motivated by the experimental observation [16] of a low-temperature cv with an exponent smaller than one, we also include here a calculation with α = 0.5. The results for some valid3 PA from orders 14 to 17 are displayed in Fig. 2 together with the exact result obtained by ED of the full spectrum of a 24-site kagome cluster. For t > 0.3, these results are practically exact. In the two scenarios α = 1 and 2 there is a significant dispersion of the results for t < 0.3 from one PA to another [24].4 In both case, cv show a low-t peak or shoulder as found from ED of finite-size systems. The choice α = 0.5 leads to a smoother cv(t) and improves significantly the convergence. It is however not clear if this improved con- vergence for small values of α (0.5 . α . 1) indicates that α is actually smaller than one or an artifact of the present entropy method, which might be “slower” to stabilize a low-t peak (as with α = 2, see Fig. 2), than a smooth curve (as with α = 0.5). In any case, this is clearly re- lated to the unusually large low-temperature entropy of the kagome system. Fig. 2 also displays the experimental results (black squares) obtained by Helton et al. [16]. The only parame- ter here is the exchange constant, taken to be J ' 190 K from the fits of the susceptibility data. Above 15 K, the phonon contribution is dominant and we have to focus on the lower temperatures to analyze the magnetic contribu- tion. Below 15 K the order of magnitude agrees with our calculation but there is no quantitative agreement. Several “perturbations” such as weakly interacting magnetic im- purities or magnetic anisotropies should indeed contribute to the specific heat at such low temperatures. Results for the integrated entropy S(T ) = cv(x)/xdx are plotted in Fig. 3, where one sees that the choice of the 3 By the entropy method, the entropy s(e) is obtained as the power α/(α + 1) of a rational fraction (PA) of the energy per site e. The specific heat curve is then obtained parametrically through T (e) = 1/s′(e) and cv(e) = −s′(e)2/s′′(e). Only the PA which satisfy s(e) > 0, s′(e) > 0 and s′′(e) < 0 in the range ]e0, 0[ are physically “valid”. 4 This is due to the finite order in the high-temperature expansion. Still, for a given value of α, we believe that this method gives a qualitatively correct picture for cv(T ), even at low T. 4 G. Misguich and P. Sindzingre: Magnetic susceptibility and specific heat of the spin- 1 Heisenberg model ... Fig. 2. (color online) Specific heat cv vs temperature T . Black squares: experimental data from Helton et al. [16], as- suming J ' 190 K. Red triangles: exact specific heat of a 24- site kagome cluster. Green (full), blue (dashed) and magenta (dotted) curves: cv calculated by the entropy method for three different values of the low-temperature exponent (α = 2, 1 and 0.5). All valid PA from order 14 to 17 with numerator and de- nominator of degrees ≥ 3 are shown. For a given α and at each temperature, their dispersion provides a rough estimate of the error bars. Fig. 3. (color online) Entropy S vs temperature. Square, tri- angles and circles: experimental data from Helton et al. [16] with magnetic field B= 0, 5 and 14 Teslas, plotted as a func- tion of kBT/J with J = 190 K. Green (full), blue (dashed) and magenta (dotted) curves: entropy calculated by the en- tropy method for α = 2, α = 1 and α = 0.5 (same PA as in Fig. 2). low-T exponent α of cv has practically no influence on the theoretical S(t = kBT/J) above t ' 0.06 and that, around this temperature, the experimental value is significantly lower than in our calculations. Of course, subtracting the contribution of the phonons (hard to estimate quantita- tively) and from the impurities would make the discrep- ancy even larger. Concerning the impurities, one sees in Fig. 3 that an applied magnetic field of 5 and 14 Teslas is enough to significantly reduce S(t) for t . 0.06. Such fields are low in comparison to J but of the order of the estimated coupling between the impurities. The difference between the curves at 0 and 5 (or 14) Teslas may thus provide a rough estimate of the contribution of the impu- rities. Note that S(t) at 5 Teslas become closer to the S(t) computed by the entropy method with α = 2. The experi- mental value α < 1 could be due to the impurities and the actual value of α for the kagome spins could be larger than 1. In any case, at t = 0.06(' 11 K), the experimental en- tropy (' 0.2 ln 2) is thus at least 7% of ln 2 below the the- oretical estimates (0.27 ln 2). This seems a rather robust indication that some additional interactions play some role in this energy range, by freezing some degrees of freedom of the spins in the kagome planes and pushing the corre- sponding entropy to higher energies. We may mention in particular DM interactions [27], non-magnetic impurities in the kagome planes (“dilution”) [20] and interactions between impurities and the kagome spins. We conclude that 15∼20 K is a minimal temperature for a kagome lat- tice Heisenberg model description of ZnCu3(OH)6Cl2 to be valid. Acknowledgments We are grateful to C. Lhuillier for many discussions and comments about the manuscript. We also thank P. Mendels and F. Bert for useful discussions. GM also thanks Y. S. Lee and J. Helton for discussions and for providing their data as well as P.A. Lee, Y. Ran, T. Senthil, X.-G. Wen for discussions on related topics. Note added : After the first submission of this manuscript, two preprints [20,25] (neutron scattering) confirmed the importance of magnetic impurities (from 6% to 10%) in ZnCu3(OH)6Cl2. The smaller value found here could be due to our simplified model to fit the magnetic suscepti- bility. References 1. V. Elser, Phys. Rev. Lett. 62, 2405 (1989) 2. C. Zeng and V. Elser, Phys. Rev. B 42, 8436 (1990) 3. J. T. Chalker and J. F. G. Eastmond, Phys. Rev. B 46, 14201 (1992) 4. R. R. P. Singh and D. A. Huse, Phys. Rev. Lett. 68 1766, (1992) 5. P. W. Leung and V. Elser, Phys. Rev. B 47, 5459 (1993) 6. T. Nakamura and S. Miyashita, Phys. Rev. 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0704.1018

Environmental dielectric screening effect on exciton transition energies in single-walled carbon nanotubes

C:/Documents and Settings/yohno/デスクトップ/岩崎論文/revtex4/manuscript070408.dvi Environmental dielectric screening e�ect on exciton transition energies in single�walled carbon nanotubes Yutaka Ohno��� � Shinya Iwasaki�� Yoichi Murakami�� Shigeru Kishimoto�� Shigeo Maruyama�� and Takashi Mizutani�� y �Department of Quantum Engineering� Nagoya University� Furo�cho� Chikusa�ku� Nagoya ��������� Japan �Department of Mechanical Engineering� The University of Tokyo� ��� Hongo� Bunkyo�ku� Tokyo �������� Japan �Dated� April �� ����� Abstract Environmental dielectric screening e ects on exciton transition energies in single walled carbon nanotubes �SWNTs� have been studied quantitatively in the range of dielectric constants from ��� to � by immersing SWNTs bridged over trenches in various organic solvents by means of photoluminescence and the excitation spectroscopies� With increasing environmental dielectric constant ��env�� both E�� and E�� exhibited a redshift by several tens meV and a tendency to saturate at a �env � � without an indication of signi�cant �n�m� dependence� The redshifts can be explained by dielectric screening of the repulsive electron electron interaction� The �env dependence of E�� and E�� can be expressed by a simple empirical equation with a power law in �env� Eii � ii � A� env� We also immersed a sample in sodium dodecyl sulfate �SDS� solution to investigate the e ects of wrapping SWNTs with surfactant� The resultant E�� and E��� which agree well with Weisman�s data �Nano Lett� �� �� � ���� ��� are close to those of �env of �� However� in addition to the shift due to dielectric screening� another shift was observed so that the ��n � m� family patterns spread more widely� similar to that of the uniaxial stress induced shift� I� INTRODUCTION Optical spectroscopy of single�walled carbon nanotubes �SWNTs� has received increasing attention� not only for assigning the chiral vector� �n�m�� of the SWNTs��� but also to study the physics of the one�dimensional excitons���� In paticular� photoluminescence �PL� and the excitation spectroscopies are the most widely used for characterizing SWNTs with resonant Raman scattering spectroscopy� Two�photon absorption spectroscopy� and Rayleigh scat� tering spectroscopy combined with TEM or SEM observation are also powerful techniques to investigate excitonic states of SWNTs and the bundle e�ect� Recently� the environmental e�ect is one of the topics most investigated for its e�ect on the optical properties of SWNTs� ��� It is known that the optical transition energies vary� depending on the kind of surrounding surfactant used to individualize SWNTs� ��� Lefebvre et al� have reported the optical transition energies of SWNTs bridging between two micro� pillars fabricated on a Si wafer show a blueshift as compared to the SDS�wrapped SWNTs��� We have also compared air�suspended SWNTs grown on a quartz substrate with a grating structure to SDS�wrapped SWNTs� and have shown that the energy di�erences between air�suspended and SDS�wrapped SWNTs depend on �n�m�� especially on chiral angle and on type of SWNTs� type�I � n m mod � � �� or type�II � n m mod � � ���� These energy variations due to environmental conditions have been thought to be produced by the di�erence in the dielectric constant of the surrounding materials� The energy of the many�body Coulomb interactions between carriers depends on the environmental dielectric constant ��env�� because the electric force lines contributing to the Coulomb interactions pass through the matrix� as well as the inside of the SWNT����� Note that the environmental e�ect is caused not only by the dielectric screening of Coulomb interactions� but also by chemical����� and mechanical conditions��� Finnie et al� have reported that gas adsorption a�ects the emission energy of SWNTs��� Investigations into the environmental e�ect have not been comprehensive� despite its importance in understanding the optical properties of SWNTs� Quantitative and separate investigations are necessary to understand the contributions of various environmental factors� In this study� we have focused on the dielectric screening e�ect� which has been quantitatively investigated by immersing the SWNTs grown over trenches into various liquids with di�erent �env� from �� to ��� by means of photoluminescence and the excitation spectroscopies� II� EXPERIMENTAL For the study on environmental e�ects� the SWNTs grown between two micro�pillars���� or over a trench����� are suitable because the environmental conditions can be controlled intentionally� The samples used in this work are SWNTs grown on a quartz substrate with periodic trenches� as shown in Fig� �� Both the period and depth of the trenches are �m� The trenches were formed by photolithography� Al�metal evaporation and lift�o�� and subsequent reactive�ion etching of the quartz with the Al mask� The SWNTs were grown by alcohol catalytic chemical vapor deposition�� at ����C for �� s� after spin coating of a water solution of Co acetate� By optimizing the growth condition� isolated bridging SWNTs can be formed over the trenches� The density of the SWNTs was ��� �m�� along the direction of the trench� Such low�density� isolated SWNTs were necessary to observe luminescence when the sample was immersed in liquids� On the other hand� in case of a sample with a relatively high�density of SWNTs as � �m���� PL intensity was degraded by immersing in liquids� even though strong PL was obtained in air� This is probably because the SWNTs form bundles with neighbors upon immersion in liquids� in the case of a sample with a high�density of SWNTs� The sample was mounted in a vessel with a quartz window� and immersed in various organic solvents with �env from �� to �� �see Table I�� We also immersed a sample in ��wt� D�O solution of sodium�dodecyl�sulfate �SDS� to investigate the e�ect of wrapping with surfactant molecules� PL and the excitation spectra were measured using a home� made facility consisting of a tunable� continuous�wave Ti�Sapphire laser �������� nm�� a monochromator with a focal length of � cm� and a liquid�N��cooled InGaAs photomultiplier tube ���������� nm�� The excitation wavelength was monitored by a laser wavelength meter� The diameter of the laser spot on the sample surface was �� mm� so that an ensemble of many SWNTs was detected� III� RESULTS AND DISCUSSION Figure shows PL maps of SWNTs �a���g� in various organic solvents with di�erent �env and �h� in SDS solution� By immersing the sample in the organic solvents� the emission and excitation peaks� which respectively correspond to E�� and E��� showed redshifts and spectral broadening� In the PL map measured in SDS solution��� the peak positions agree well with those of Weisman�s empirical Kataura plot for SDS�wrapped SWNTs represented by crosses�� The E���E�� plots in air ��env � ����� hexane ��� �� chloroform ������ and SDS solution are shown in Fig� � � Both of E�� and E�� showed redshifts with increasing �env� without signi�cant �n�m� dependence� The amounts of the redshifts are ���� meV for E�� and ���� meV for E��� The redshift with increasing �env is consistent with the theoretical work by Ando��� The optical transition energy in SWNTs is given by a summation of the bandgap and the exciton binding energies� When the �env increases� the Coulomb interaction is enhanced and the exciton binding energy decreases� This leads to a blueshift of the optical transition energy in the SWNT� It should be noted that the bandgap is renormalized by the electron�electron repulsion interaction� and consequently the change in the �env a�ects not only the exciton binding energy� but also the bandgap� According to theoretical work������ the repulsive electron�electron energy is larger in magnitude than the exciton binding energy� Therefore� if we consider that both energies show a similar dependence on �env� the decrease in the electron�electron repulsion energy exceeds the decrease in the exciton binding energy when �env increases� This results in a redshift in optical transition energies� The redshifts observed in the present experiment are attributed to the decrease in the repulsion energy of the electron�electron interaction with an increase in the �env� Note that even though the redshift with increasing �env is consistent with the theoretical studies� the amount of the redshift is much smaller than the calculations� This is probably because in the present experiments� the �env only outside of the SWNTs was varied� whereas the theoretical studies used a dielectric constant for the whole system� E�� and E�� are plotted as a function of �env in Fig� �� The energy shifts show a tendency to saturate at �env � �� The energy variations can be �tted to an empirical expression with a power law in � Eii � E env ��� where E�ii corresponds to a transition energy when �env is in�nity� A corresponds to the maximum value of the energy change by � � and � is a �tting parameter� A and � are �� meV and �� for E��� and meV and ��� for E��� respectively� on average� Perebeinos et al� have reported power law scaling for excitonic binding energy by dielectric constant� where the scaling factor � is estimated to be � ��� in the range of � � ��� Although the power law expression of Eq� � is quite similar to the Perebeinos�s scaling law for exciton binding energy� the present empirical expression is attributed to the scaling of electron�electron repulsion energy by �env� rather than exciton �electron�hole� binding energy as described above� Quite recently� such a power�law�like�downshift in optical transition energy with �env has been obtained by theoretical calculations based on a tight�binding model��� We have previously pointed out that the E�� and E�� varies depending on �n�m�� in particular on chiral angle and on the type of SWNTs �type�I or type�II�� comparing those of SDS�wrapped SWNTs to those of air�suspended SWNTs� The same behavior occurred by immersing a sample in SDS solution� as shown in Fig� �h�� Most E�� and E�� of SDS� wrapped SWNTs show a redshift� except for E�� of near�zigzag type�II SWNTs� which show a blueshift as compared to those in air� This behavior is inconsistent with the results of the �env dependence as described above� in which �n�m� dependence is not signi�cant� Comparing the PL maps in SDS solution to those in organic solvents as shown in Fig� �� the equivalent �env of SDS�wrapped SWNTs would be � � In addition to the shift due to the dielectric screening� the peak positions of SDS�wrapped SWNTs shift so as the � n family patterns spread more widely� This suggest that wrapping with SDS has another e�ect in addition to the dielectric screening e�ect� The behavior of the additional shifts is similar to the shift induced by uniaxial stress� reported by Arnold et al��� At present� it still remains an issue whether such uniaxial strain is induced in the SWNTs by wrapping with surfactants or not� A charging e�ect due to SDS� which is an anionic surfactant� should also be considered� Further study is necessary to understand the e�ects of surfactants on optical transition energies in SWNTs� Finally� we note the broadening of the PL spectra in liquids� The representative spectra are shown in Fig� �� The peaks show a broadening� in addition to the redshift with increasing �env� The linewidth increased from � meV in air to �� meV in acetonitrile for � ��� SWNTs� This linewidth broadening is probably attributed to inhomogeneous broadening due to local �uctuations of �env on the nano�scale dimension� The dielectric constants we used in Table I are macroscopic values� On such small dimensions as exciton diameters of a few nm��� the size of molecules of the organic solvents would be considerable so that the local �env would �uctuate depending on the number and orientation of the organic molecules� This would result in inhomogeneous broadening of the PL spectrum� IV� SUMMARY In summary� dielectric screening e�ects due to the environment around SWNTs on exciton transition energies in the SWNTs have been studied quantitatively by means of PL and the excitation spectroscopies� We varied the �env from ��� to �� by immersing the samples with SWNTs bridging over trenches in various organic solvents with di�erent �envs� With increasing �env� both E�� and E�� showed a redshift by ���� meV for E�� and ���� meV for E��� and a tendency to saturate at �env � �� without a signi�cant �n�m� dependence� The redshift can be explained by dielectric screening of repulsive electron�electron energy� The �env dependence of E�� and E�� were expressed by a simple empirical equation with a power law in �env� The equivalent �env of SDS�wrapped SWNTs was estimated to be � � It was suggested that the e�ect of wrapping SWNTs with SDS was not only a dielectric screening e�ect� but also another e�ect which caused an energy shift like a uniaxial�stress�induced shift� � Also at PRESTO� Japan Science and Technology Agency� � � Honcho� Kawaguchi� Saitama ����� Japan� Electronic address� yohno�nuee�nagoya�u�ac�jp y Also at Institute for Advanced Research� Nagoya University� Furo cho� Chikusa ku� Nagoya� ����� Japan � H� Kataura� Y� Kumazawa� Y� Maniwa� I� Umezu� S� Suzuki� Y� Ohtsuka� and Y� Achiba� Synthetic Metals ���� ���� ������� � S� M� Bachilo� M� S� Strano� C� Kittrell� R� H� Hauge� R� E� Smalley� and R� B� Weisman� Science ���� � �� ������� � R� B� Weisman and S� M� Bachilo� Nano Lett� �� �� � ���� �� � Y� Miyauchi� S� Chiashi� Y� Murakami� Y� Hayashida� S� Maruyama� Chem� Phys� Lett� ���� ��� ������� � H� Htoon� M� J� O�Connell� P� J� Cox� S� K� Doorn� and V� I� Klimov� Phys� Rev� Lett� ��� ������ ������� � F� Wang� G� Dukovic� L� E� Brus� T� F� Heinz� Science ���� � � ������� F� Wang� M� Y� Sfeir� L� Huang� X� M� H� Huang� Y� Wu� J� Kim� J� Hone� S� O�Brien� L� E� Brus� and T� F� Heinz� Phys� Rev� Lett � � ������ ������� � P� T� Araujo� S� K� Doorn� S� Kilina� S� Tretiak� E� Einarsson� S� Maruyama� H� Chacham� M� A� Pimenta� and A� Jorio� Phys� Rev� Lett� ��� ������ ������� V� C� Moore� M� S� Strano� E� H� Haroz� R� H� Hauge� R� E� Smalley� J� Schmidt� and Y� Talmon� Nano Lett� �� � �� ���� �� �� S� G� Chou� H� B� Ribeiro� E� B� Barros� A� P� Santos� D� Nezich� Ge� G� Samsonidze� C� Fantini� M� A� Pimenta� A� Jorio� F� Plentz Filho� M� S� Dresselhaus� G� Fresselhaus� R� Saito� M� Zheng� G� B� Onoa� E� D� Semke� A� K� Swan� M� S� �Unl�u� and B� B� Goldberg� Chem� Phys� Lett� ���� ��� ������� �� C� Fantini� A� Jorio� M� Souza� M� S� Strano� M� S� Dresselhaus� and M� A� Pimenta� Phys� Rev� Lett� ��� ������ ������� �� J� Lefebvre� J� M� Fraser� Y� Homma� and P� Finnie� Appl� Phys� A ��� ���� ������� �� T� Hertel� A� Hagen� V� Talalaev� K� Arnold� F� Hennrich� M� Kappes� S� Rosenthal� J� McBride� H� Ulbricht� and E� Flahaut� Nano Lett� � ��� ������� �� P� Finnie� Y� Homma� and J� Lefebvre� Phys� Rev� Lett� ��� ������ ������� �� Y� Ohno� S� Iwasaki� Y� Murakami� S� Kishimoto� S� Maruyama� and T� Mizutani� Phys� Rev� B ��� � ���� ������� �� T� Ando J� Phys� Soc� Japan ��� ��� ������� � V� Perebeinos� J� Terso � and Ph� Avouris� Phys� Rev� Lett� ��� ������ ������� �� K� Arnold� S� Lebedkin� O� Kiowski� F� Hennrich� and M� M� Kappes� Nano Lett� �� � �� ������� J� Lefebvre� Y� Homma� and P� Finnie� Phys� Rev� Lett� ��� ������ ���� �� �� Y� Ohno� S� Kishimoto� and T� Mizutani� Nanotechnology ��� ��� ������� �� Y� Murakami� Y� Miyauchi� S� Chiashi� and S� Maruyama� Chem� Phys� Lett� ���� � ���� �� �� In order to form micelle structure� the sample was kept in SDS solution for one night before the PL measurement� �� Y� Miyauchi� R� Saito� K� Saito� Y� Ohno� S� Iwasaki� T� Mizutani� J� Jiang� S� Maruyama �will be published elsewhere� �� C� D� Spataru� S� Ismail Beigi� L� X� Benedict� and S� G� Louie� Phys� Rev� Lett� ��� ������ ������� TABLE I� �env of air and various liquids used in this study� solvent �env air ��� hexane ��� chloroform ��� ethyl acetate ��� dichloromethane ��� acetone �� acetonitrile � FIG� �� Plan view and bird view �inset� SEM images of SWNTs bridging over trenches on a quartz substrate� Pt thin �lm was deposited on the sample in order to avoid charge up of the insulating substrate and to observe individual SWNTs� By optimizing growth condition� individual SWNTs can be obtained� 0.8 0.9 1.0 1.1 0.9 1.0 1.1 0.9 1.0 1.1 0.9 1.0 1.1 (a) air (b) hexane (c) chloroform (d) ethyl acetate (e) dichloromethane (f) acetone (g) acetonitrile (h) SDS FIG� �� PL contour maps for SWNTs� �a� in air ��env � ����� �b� hexane ��env � ����� �c� chloroform ��env � ����� �d� ethyl acetate ��env � ����� �e� dichloromethane ��env � ����� �f� acetone ��env � ������ and �g� acetonitrile ��env � ����� and �h� SDS�D�O solution� The crosses and open circles represent peak positions in air and in the liquid� In �h�� stars represent the peak positions of wrapped SWNTs reported by Weisman et al�� The PL intensity color schemes are linear� but di erent scaling factors were used for each maps� The emission and excitation maxima correspond to E�� and E��� respectively� FIG� � E�� versus E�� plots in air ��env � ����� hexane ��env � ����� chloroform ��env � ����� The stars represent the peak positions of SDS wrapped SWNTs�� Both E�� and E�� show redshifts with increasing �env with a small �n�m� dependence� The peak positions of SDS wrapped SWNTs deviate from the line of dielectric screening e ect� FIG� �� �env dependences of E�� and E�� of various �n�m� SWNTs� The solid lines are �tting curves given by an empirical expression with power law in � FIG� �� PL spectra in various liquids� With increasing � � the emission spectra show linewidth broadening in addition to redshift�

0704.1019

Cyclic cohomology of certain nuclear Fr\'echet and DF algebras

CYCLIC COHOMOLOGY OF CERTAIN NUCLEAR FRÉCHET AND DF ALGEBRAS ZINAIDA A. LYKOVA Abstract. We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain ⊗̂-algebras. We use well-developed homolog- ical techniques together with some niceties of the theory of locally convex spaces to generalize the results known in the case of Banach algebras and their inverse limits to wider classes of topological algebras. To this end we show that, for a con- tinuous morphism ϕ : X → Y of complexes of complete nuclear DF -spaces, the isomorphism of cohomology groups Hn(ϕ) : Hn(X ) → Hn(Y) is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective ⊗̂-algebras: the tensor algebra E⊗̂F generated by the duality (E,F, 〈·, ·〉) for nuclear Fréchet spaces E and F or for nuclear DF -spaces E and F ; nuclear biprojective Köthe algebras λ(P ) which are Fréchet spaces or DF -spaces; the algebra of distributions E∗(G) on a compact Lie group G. 2000 Mathematics Subject Classification: Primary 19D55, 22E41, 16E40, 46H40. 1. Introduction Cyclic cohomology groups of topological algebras play an essential role in non- commutative geometry [2]. There has been a number of papers addressing the calculation of cyclic-type continuous homology and cohomology groups of some Ba- nach, C∗- and topological algebras; see, e.g., [2, 8, 14, 17, 18, 20, 32]. However, it remains difficult to describe these groups explicitly for many topological algebras. To compute the continuous Hochschild and cyclic cohomology groups of Fréchet algebras one has to deal with complexes of complete DF -spaces. Here, in addition to presenting known homological techniques we also supply technical enhancements that permit the necessary generalization of results known in the case of Banach algebras and their inverse limits to wider classes of topological algebras notably to those that occur in noncommutative geometry. The category of Banach spaces has the useful property that it is closed under passage to dual spaces. Fréchet spaces do not have this property: the strong dual Date: 22 August 2007. Key words and phrases. Cyclic cohomology, Hochschild cohomology, nuclear DF -spaces, locally convex algebras, nuclear Fréchet algebra. I am indebted to the Isaac Newton Institute for Mathematical Sciences at Cambridge for hos- pitality and for generous financial support from the programme on Noncommutative Geometry while this work was carried out. http://arxiv.org/abs/0704.1019v2 2 Z. A. Lykova of a Fréchet space is a complete DF -space. DF -spaces have the awkward feature that their closed subspaces need not be DF -spaces. However, closed subspaces of complete nuclear DF -spaces are again DF -spaces [21, Proposition 5.1.7]. In Section 3 we use the strongest known results on the open mapping theorem to give sufficient conditions on topological spaces E and F to imply that any continuous linear operator T from E∗ onto F ∗ is open. This allows us to prove the following results. In Lemma 3.6 we show that, for a continuous morphism ϕ : X → Y of complexes of complete nuclear DF -spaces, the isomorphism of cohomology groups Hn(ϕ) : Hn(X )→ Hn(Y) is automatically topological. We use this fact to describe explicitly up to topological isomorphism the continuous Hochschild and cyclic cohomology groups of nuclear ⊗̂-algebras A which are Fréchet spaces or DF -spaces and have trivial Hochschild homology HHn(A) for all n ≥ 1 (Theorem 5.3). In Proposition 4.4, under the same condition on HHn(A), we give explicit formulae, up to isomorphism of linear spaces, for continuous cyclic-type homology of A in a more general category of underlying spaces. In Theorem 6.8 the continuous cyclic-type homology and cohomology groups are described up to topological isomorphism for the following classes of biprojective ⊗̂- algebras: the tensor algebra E⊗̂F generated by the duality (E, F, 〈·, ·〉) for nuclear Fréchet spaces or for nuclear complete DF -spaces E and F ; nuclear biprojective Fréchet Köthe algebras λ(P ); nuclear biprojective Köthe algebras λ(P )∗ which are DF -spaces; the algebra of distributions E∗(G) and the algebra of smooth functions E(G) on a compact Lie group G. 2. Definitions and notation We recall some notation and terminology used in homology and in the theory of topological algebras. Homological theory can be found in any relevant textbook, for instance, Loday [16] for the pure algebraic case and Helemskii [7] for the continuous case. Throughout the paper ⊗̂ is the projective tensor product of complete locally convex spaces, by X⊗̂n we mean the n-fold projective tensor power X⊗̂ . . . ⊗̂X of X , and id denotes the identity operator. We use the notation Ban, Fr and LCS for the categories whose objects are Banach spaces, Fréchet spaces and complete Hausdorff locally convex spaces respectively, and whose morphisms in all cases are continuous linear operators. For topological homology theory it is important to find a suitable category for the underlying spaces of the algebras and modules. In [7] Helemskii constructed homology theory for the following categories Φ of underlying spaces, for which he used the notation (Φ, ⊗̂). Definition 2.1. ([7, Section II.5]) A suitable category for underlying spaces of the algebras and modules is an arbitrary complete subcategory Φ of LCS having the following properties: (i) if Φ contains a space, it also contains all those spaces topologically isomorphic to it; Cyclic cohomology of nuclear Fréchet and DF algebras 3 (ii) if Φ contains a space, it also contains any of its closed subspaces and the completion of any its Hausdorff quotient spaces; (iii) Φ contains the direct sum and the projective tensor product of any pair of its spaces; (iv) Φ contains C. Besides Ban, Fr and LCS important examples of suitable categories Φ are the categories of complete nuclear spaces [31, Proposition 50.1], nuclear Fréchet spaces and complete nuclear DF -spaces. As to the above properties for the category of complete nuclear DF -spaces, recall the following results. By [12, Theorem 15.6.2], if E and F are complete DF -spaces, then E⊗̂F is a complete DF -space. By [21, Proposition 5.1.7], a closed linear subspace of a complete nuclear DF -space is also a complete nuclear DF -space. By [21, Proposition 5.1.8], each quotient space of a complete nuclear DF -space by a closed linear subspace is also a complete nuclear DF -space. By definition a ⊗̂-algebra is a complete Hausdorff locally convex algebra with jointly continuous multiplication. A left ⊗̂-module X over a ⊗̂-algebra A is a com- plete Hausdorff locally convex spaceX together with the structure of a leftA-module such that the map A×X → X , (a, x) 7→ a ·x is jointly continuous. For a ⊗̂-algebra A, ⊗̂A is the projective tensor product of left and right A-⊗̂-modules (see [6], [7, II.4.1]). The category of left A-⊗̂-modules is denoted by A-mod and the category of A-⊗̂-bimodules is denoted by A-mod-A. Let K be one of the above categories. A chain complex X∼ in the category K is a sequence of Xn ∈ K and morphisms dn · · · ← Xn ← Xn+1 ← Xn+2 ← . . . such that dn ◦ dn+1 = 0 for every n. The homology groups of X∼ are defined by Hn(X∼) = Ker dn−1/Im dn. A continuous morphism of chain complexes ψ∼ : X∼ → P∼ induces a continuous linear operator Hn(ψ∼) : Hn(X∼)→ Hn(P∼) [9, Definition 0.4.22]. If E is a topological vector space E∗ denotes its dual space of continuous linear functionals. Throughout the paper, E∗ will always be equipped with the strong topology unless otherwise stated. The strong topology is defined on E∗ by taking as a basis of neighbourhoods of 0 the family of polars V 0 of all bounded subsets V of E; see [31, II.19.2]. For any ⊗̂-algebra A, not necessarily unital, A+ is the ⊗̂-algebra obtained by adjoining an identity to A. For a ⊗̂-algebra A, the algebra Ae = A+⊗̂A called the enveloping algebra of A, where A + is the opposite algebra of A+ with multiplication a · b = ba. A complex of A-⊗̂-modules and their morphisms is called admissible if it splits as a complex in LCS [7, III.1.11]. A module Y ∈ A-mod is called flat if for any admissible complex X of right A-⊗̂-modules the complex X⊗̂AY is exact. A module Y ∈ A-mod-A is called flat if for any admissible complex X of A-⊗̂-bimodules the 4 Z. A. Lykova complex X⊗̂AeY is exact. For Y,X ∈ A-mod-A, we shall denote by Tor n (X, Y ) the nth homology of the complex X⊗̂AeP, where 0← Y ← P is a projective resolution of Y in A-mod-A, [7, Definition III.4.23]. It is well known that the strong dual of a Fréchet space is a complete DF -space and that nuclear Fréchet spaces and complete nuclear DF -spaces are reflexive [21, Theorem 4.4.12]. Moreover, the correspondence E ↔ E∗ establishes a one-to-one relation between nuclear Fréchet spaces and complete nuclear DF -spaces [21, The- orem 4.4.13]. DF -spaces were introduced by A. Grothendieck in [5]. Further we shall need the following technical result which extends a result of Johnson for the Banach case [11, Corollary 1.3]. Proposition 2.2. Let (X , d) be a chain complex of (a) Fréchet spaces and continuous linear operators, or (b) complete nuclear DF -spaces and continuous linear operators, and let N ∈ N. Then the following statements are equivalent: (i) Hn(X , d) = {0} for all n ≥ N and HN−1(X , d) is Hausdorff; (ii) Hn(X ∗, d∗) = {0} for all n ≥ N. Proof. Recall that Hn(X , d) = Ker dn−1/Im dn and H n(X ∗, d∗) = Ker d∗n/Im d Let L be the closure of Im dN−1 in XN−1. Consider the following commutative diagram 0 ← L ←− XN ←− XN+1 ←− . . . ↓ i ւ dN−1 in which i is the natural inclusion and j is a corestriction of dN−1. The dual com- mutative diagram is the following 0 → L∗ −→ X∗N −→ X∗N+1 −→ . . . ↑ i∗ ր d∗N−1 X∗N−1 It is clear that HN−1(X , d) is Hausdorff if and only if j is surjective. Since i is injective, condition (i) is equivalent to the exactness of diagram (1). On the other hand, by the Hahn-Banach theorem, i∗ is surjective. Thus condition (ii) is equivalent to the exactness of diagram (2). In the case of Fréchet spaces, by [18, Lemma 2.3], the exactness of the complex (1) is equivalent to the exactness of the complex (2). In the case of complete nuclear DF -spaces, by [21, Proposition 5.1.7], L is the strong dual of a nuclear Fréchet space. By [21, Theorem 4.4.12], complete nuclear DF -spaces are reflexive, and therefore the complex (1) is the dual of the complex (2) of nuclear Fréchet spaces and continuous linear operators. By [18, Lemma 2.3], the exactness of the complex (1) is equivalent to the exactness of the complex (2). The proposition is proved. � Cyclic cohomology of nuclear Fréchet and DF algebras 5 3. The open mapping theorem in complete nuclear DF -spaces It is known that there exist closed linear subspaces of DF -spaces that are not DF -spaces. For nuclear spaces, however, we have the following. Lemma 3.1. [21, Proposition 5.1.7] Each closed linear subspace F of the strong dual E∗ of a nuclear Fréchet space E is also the strong dual of a nuclear Fréchet space. In a locally convex space a subset is called a barrel if it is absolutely convex, absorbent and closed. Every locally convex space has a neighbourhood base con- sisting of barrels. A locally convex space is called a barrelled space if every barrel is a neighbourhood [26]. By [26, Theorem IV.1.2], every Fréchet space is barrelled. By [26, Corollary IV.3.1], a Hausdorff locally convex space is reflexive if and only if it is barrelled and every bounded set is contained in a weakly compact set. Thus the strong dual of a nuclear Fréchet space is barrelled. For a generalization of the open mapping theorem to locally convex spaces, V. Pták introduced the notion of B-completeness in [24]. A subspace Q of E∗ is said to be almost closed if, for each neighbourhood U of 0 in E, Q∩U0 is closed in the relative weak* topology σ(E∗, E) on U0. A locally convex space E is said to be B-complete or fully complete if each almost closed subspace of E∗ is closed in the weak* topology σ(E∗, E). Theorem 3.2. [24]. Let E be a B-complete locally convex space and F be a barrelled locally convex space. Then a continuous linear operator f of E onto F is open. Recall [10, Theorem 4.1.1] that a locally convex space E is B-complete if and only if each linear continuous and almost open mapping f of E onto any locally convex space F is open. By [10, Proposition 4.1.3], every Fréchet space is B-complete. Theorem 3.3. Let E be a semi-reflexive metrizable barrelled space, F be a Hausdorff reflexive locally convex space and let E∗ and F ∗ be the strong duals of E and F respectively. Then a continuous linear operator T of E∗ onto F ∗ is open. Proof. By [10, Theorem 6.5.10] and by [10, Corollary 6.2.1], the strong dual E∗ of a semi-reflexive metrizable barrelled space E is B-complete. By [26, Corollary IV.3.2], if a Hausdorff locally convex space is reflexive, so is its dual under the strong topology. By [26, Corollary IV.3.1], a Hausdorff reflexive locally convex space is barrelled. Hence F ∗ is a barrelled locally convex space. Therefore, by Theorem 3.2, T is open. � Corollary 3.4. Let E and F be nuclear Fréchet spaces and let E∗ and F ∗ be the strong duals of E and F respectively. Then a continuous linear operator T of E∗ onto F ∗ is open. For a continuous morphism of chain complexes ψ∼ : X∼ → P∼ in Fr, a surjective map Hn(ϕ) : Hn(X ) → Hn(Y) is automatically open, see [7, Lemma 0.5.9]. To get the corresponding result for dual complexes of Fréchet spaces one has to assume nuclearity. 6 Z. A. Lykova Lemma 3.5. Let (X , dX ) and (Y , dY) be chain complexes of nuclear Fréchet spaces and continuous linear operators and let (X ∗, d∗X ) and (Y ∗, d∗Y) be their strong dual complexes. Let ϕ : X ∗ → Y∗ be a continuous morphism of complexes. Suppose that ϕ∗ = H n(ϕ) : Hn(X ∗, d∗X )→ H n(Y∗, d∗Y) is surjective. Then ϕ∗ is open. Proof. Let σY∗ : Ker (d Y)n → H n(Y∗, d∗Y) be the quotient map. Consider the map ψ : Ker (d∗X )n ⊕ Y n−1 → Ker (d Y)n ⊂ Y given by (x, y) 7→ ϕn(x) + (d Y)n−1(y). By Lemma 3.1, Ker (d∗X )n and Ker (d Y)n are the strong duals of nuclear Fréchet spaces and hence are barrelled. By [10, Theorem 6.5.10] and [10, Corollary 6.2.1], the strong dual of a semi-reflexive metrizable barrelled space is B-complete. Thus Ker (d∗X )n, Y n−1 and Ker (d∗X )n ⊕ Y ∼= [(Ker (d∗X )n) ∗ ⊕ Yn−1] are B-complete. By assumption ϕ∗ maps H n(X ∗, d∗X ) onto H n(Y∗, d∗Y), which im- plies that ψ is a surjective linear continuous operator from the B-complete locally convex space Ker (d∗X )n ⊕ Y n−1 to the barrelled locally convex space Ker (d Therefore, by Theorem 3.2, ψ is open. Consider the diagram Ker (d∗X )n ⊕ Y → Ker (d∗X )n → Hn(X ∗, d∗X ) ↓ ψ ↓ ϕ∗ Ker (d∗Y)n −→ Hn(Y∗, d∗Y) in which j is a projection onto a direct summand and σX ∗ and σY∗ are the natural quotient maps. Obviously this diagram is commutative. Note that the projection j and quotient maps σX ∗ , σY∗ are open. As ψ is also an open map, so is σY∗ ◦ ψ = ϕ∗ ◦ σX ∗ ◦ j. Since σX ∗ ◦ j is continuous, ϕ∗ is open. � Corollary 3.6. Let (X , dX ) and (Y , dY) be cochain complexes of complete nuclear DF -spaces and continuous linear operators, and let ϕ : X → Y be a continuous morphism of complexes. Suppose that ϕ∗ = H n(ϕ) : Hn(X , dX ) → H n(Y , dY) is surjective. Then ϕ∗ is open. Proof. By [21, Theorem 4.4.13], (X , dX ) and (Y , dY) are strong duals of chain com- plexes (X ∗, d∗X ) and (Y ∗, d∗Y) of nuclear Fréchet spaces and continuous operators. The result follows from Lemma 3.5. � 4. Cyclic and Hochschild cohomology of some ⊗̂-algebras One can consult the books by Loday [16] or Connes [2] on cyclic-type homological theory. Let A be a ⊗̂-algebra and let X be an A-⊗̂-bimodule. We assume here that the category of underlying spaces Φ has the properties from Definition 2.1. Let us recall the definition of the standard homological chain complex C∼(A, X). For n ≥ 0, let Cyclic cohomology of nuclear Fréchet and DF algebras 7 Cn(A, X) denote the projective tensor product X⊗̂A . The elements of Cn(A, X) are called n-chains. Let the differential dn : Cn+1 → Cn be given by dn(x⊗ a1 ⊗ . . .⊗ an+1) = x · a1 ⊗ . . .⊗ an+1+ (−1)k(x⊗ a1 ⊗ . . .⊗ akak+1 ⊗ . . .⊗ an+1) + (−1) n+1(an+1 · x⊗ a1 ⊗ . . .⊗ an) with d−1 the null map. The homology groups of this complex Hn(C∼(A, X)) are called the continuous Hochschild homology groups of A with coefficients in X and denoted by Hn(A, X) [7, Definition II.5.28]. We also consider the cohomology groups Hn((C∼(A, X)) ∗) of the dual complex (C∼(A, X)) ∗ with the strong dual topology. For Banach algebras A, Hn((C∼(A, X)) ∗) is topologically isomorphic to the Hochschild cohomologyHn(A, X∗) of A with coefficients in the dual A-bimodule X∗ [7, Definition I.3.2 and Proposition II.5.27]. The weak bidimension of a Fréchet algebra A is dbwA = inf{n : H n+1(C∼(A, X) ∗) = {0} for all Fréchet A−bimodules X}. The continuous bar and ‘naive’ Hochschild homology of a ⊗̂-algebra A are defined respectively as Hbar∗ (A) = H∗(C(A), b ′) and Hnaive∗ (A) = H∗(C(A), b), where Cn(A) = A ⊗̂(n+1), and the differentials b, b′ are given by b′(a0 ⊗ · · · ⊗ an) = (−1)i(a0 ⊗ · · · ⊗ aiai+1 ⊗ · · · ⊗ an) and b(a0 ⊗ · · · ⊗ an) = b ′(a0 ⊗ · · · ⊗ an) + (−1) n(ana0 ⊗ · · · ⊗ an−1). Note that Hnaive∗ (A) is just another way of writing H∗(A,A), the continuous homol- ogy of A with coefficients in A, as described in [7, 11]. There is a powerful method based on mixed complexes for the study of the cyclic- type homology groups; see papers by C. Kassel [13], J. Cuntz and D. Quillen [4] and J. Cuntz [3]. We shall present this method for the category LCS of locally convex spaces and continuous linear operators; see [1] for the category of Fréchet spaces. A mixed complex (M, b, B) in the category LCS is a familyM = {Mn}n≥0 of locally convex spaces Mn equipped with continuous linear operators bn : Mn → Mn−1 and Bn : Mn → Mn+1, which satisfy the identities b 2 = bB + Bb = B2 = 0. We assume that in degree zero the differential b is identically equal to zero. We arrange the 8 Z. A. Lykova mixed complex (M, b, B) in the double complex . . . . . . . . . . . . b ↓ b ↓ b ↓ b ↓ b ↓ There are three types of homology theory that can be naturally associated with a mixed complex. The Hochschild homology Hb∗(M) of (M, b, B) is the homology of the chain complex (M, b), that is, Hbn(M) = Hn(M, b) = Ker {bn :Mn →Mn−1}/Im {bn+1 :Mn+1 →Mn}. To define the cyclic homology of (M, b, B), let us denote by BcM the total complex of the above double complex, that is, · · · → (BcM)n → (BcM)n−1 → · · · → (BcM)0 → 0, where the spaces (BcM)0 =M0, . . . , (BcM)2k−1 =M1 ⊕M3 ⊕ · · · ⊕M2k−1 (BcM)2k =M0 ⊕M2 ⊕ · · · ⊕M2k are equipped with the product topology, and the continuous linear operators b+B are defined by (b+B)(y0, . . . , y2k) = (by2 +By0, . . . , by2k +By2k−2) (b+B)(y1, . . . , y2k+1) = (by1, . . . , by2k+1 +By2k−1). The cyclic homology of (M, b, B) is defined to be H∗(BcM, b+B). It is denoted by Hc∗(M, b, B). The periodic cyclic homology of (M, b, B) is defined in terms of the complex · · · → (BpM)ev → (BpM)odd → (BpM)ev → (BpM)odd → . . . , where even/odd chains are elements of the product spaces (BpM)ev = M2n and (BpM)odd = M2n+1, respectively. The spaces (BpM)ev/odd are locally convex spaces with respect to the product topology [15, Section 18.3.(5)]. The continuous differential b+B is defined as an obvious extension of the above. The periodic cyclic homology of (M, b, B) is Hpν (M, b, B) = Hν(BpM, b+B), where ν ∈ Z/2Z. There are also three types of cyclic cohomology theory associated with the mixed complex, obtained when one replaces the chain complex of locally convex spaces Cyclic cohomology of nuclear Fréchet and DF algebras 9 by its dual complex of strong dual spaces. For example, the cyclic cohomology associated with the mixed complex (M, b, B) is defined to be the cohomology of the dual complex ((BcM) ∗, b∗ + B∗) of strong dual spaces and dual operators; it is denoted by H∗c (M ∗, b∗, B∗). Consider the mixed complex (Ω̄A+, b̃, B̃), where Ω̄ nA+ = A ⊗̂(n+1) ⊕A⊗̂n and b 1− λ 0 −b′ ; B̃ = where λ(a1⊗· · ·⊗an) = (−1) n−1(an⊗a1⊗· · ·⊗an−1) and N = id+λ+· · ·+λ n−1 [16, 1.4.5]. The continuous Hochschild homology of A, the continuous cyclic homology of A and the continuous periodic cyclic homology of A are defined by HH∗(A) = H ∗(Ω̄A+, b̃, B̃), HC∗(A) = H ∗(Ω̄A+, b̃, B̃) and HP∗(A) = H ∗ (Ω̄A+, b̃, B̃) where Hb∗, H ∗ and H ∗ are Hochschild homology, cyclic homology and periodic cyclic homology of the mixed complex (Ω̄A+, b̃, B̃) in LCS, see [17]. There is also a cyclic cohomology theory associated with a complete locally convex algebra A, obtained when one replaces the chain complexes of A by their dual complexes of strong dual spaces. Lemma 4.1. (i) Let A be a [nuclear] Fréchet algebra. Then the following complexes (C(A), b), (Ω̄A+, b̃), (BcΩ̄A+, b̃+ B̃) and (BpΩ̄A+, b̃+ B̃) are complexes of [nuclear] Fréchet spaces and continuous linear operators. (ii) Let A be a [nuclear] ⊗̂-algebra which is a DF -space. Then the following com- plexes (C(A), b), (Ω̄A+, b̃), and (BcΩ̄A+, b̃+ B̃) are complexes of [nuclear] complete DF -spaces and continuous linear operators, and (BpΩ̄A+, b̃ + B̃) is a complex of [nuclear] complete locally convex spaces and continuous linear operators, but it is not a DF -space in general. Proof. It is well known that Fréchet spaces are closed under countable cartesian products and projective tensor product [31]; nuclear locally convex spaces are closed under cartesian products, countable direct sums and projective tensor product [12, Corollary 21.2.3]; complete DF -spaces are closed under countable direct sums, pro- jective tensor product, but not under infinite cartesian products [12, Theorem 12.4.8 and Theorem 15.6.2]. � Propositions 4.2 and 4.3 below are proved by the author in [17, 18] and show the equivalence between the continuous cyclic (co)homology of A and the contin- uous periodic cyclic (co)homology of A when A has trivial continuous Hochschild (co)homology HHn(A) for all n ≥ N for some integer N . Here we add in these statements certain topological conditions on the algebra which allow us to show that isomorphisms of (co)homology groups are automatically topological. Proposition 4.2. [17, Proposition 3.2] Let A be a complete locally convex algebra. Then, for any even integer N , say N = 2K, and the following assertions, we have (i)N ⇒ (ii)N ⇒ (iii)N ⇒ (ii)N+1 and (ii)N ⇒ (iv)N : 10 Z. A. Lykova (i)N H naive n (A) = {0} for all n ≥ N and H n (A) = {0} for all n ≥ N − 1; (ii)N HHn(A) = {0} for all n ≥ N ; (iii)N for all k ≥ K, up to isomorphism of linear spaces, HC2k(A) = HCN(A) and HC2k+1(A) = HCN−1(A); (iv)N up to isomorphism of linear spaces, HP0(A) = HCN(A) and HP1(A) = HCN−1(A). For Fréchet algebras the isomorphisms in (iii)N and (iv)N are automatically topolog- ical. For a nuclear ⊗̂-algebra A which is a DF -space the isomorphisms in (iii)N are automatically topological. Proof. A proof of the statement is given in [17, Proposition 3.2]. Here we add a part on the automatic continuity of the isomorphisms. In view of the proofs of [17, Propositions 2.1 and 3.2] it is easy to see that isomorphisms of homology in (iii)N and (iv)N are induced by continuous morphisms of complexes. Note that by Lemma 4.1, for a Fréchet algebra A, the following complexes (BcΩ̄A+, b̃ + B̃) and (BpΩ̄A+, b̃ + B̃) are complexes of Fréchet spaces and continuous linear operators. Thus, for Fréchet algebras, by [7, Lemma 0.5.9], isomorphisms of homology groups are topological. By Lemma 4.1, for a nuclear ⊗̂-algebra A which is a DF -space, the following com- plex (BcΩ̄A+, b̃ + B̃) is a complex of nuclear complete DF -spaces and continuous linear operators. By Corollary 3.6, for complete nuclear DF -spaces the isomor- phisms for homology groups in (iii)N are also topological. � Proposition 4.3. [18, Proposition 3.1] Let A be a complete locally convex algebra. Then, for any even integer N , say N = 2K, and the following assertions, we have (i)N ⇒ (ii)N ⇒ (iii)N ⇒ (ii)N+1 and (ii)N ⇒ (iv)N : (i)N H naive(A) = {0} for all n ≥ N and H bar(A) = {0} for all n ≥ N − 1; (ii)N for all n ≥ N, HH n(A) = {0}; (iii)N for all k ≥ K, up to isomorphism of linear spaces, HC 2k(A) = HCN(A) and HC2k+1(A) = HCN−1(A); (iv)N up to isomorphism of linear spaces, HP 0(A) = HCN(A) and HP 1(A) = HCN−1(A). For nuclear Fréchet algebras the isomorphisms in (iii)N and (iv)N are topological isomorphisms. For a nuclear ⊗̂-algebra A which is a DF -space the isomorphisms in (iii)N are topological isomorphisms. Proof. We need to add to the proof of [18, Proposition 3.1] the following part on automatic continuity. In view of the proof of [18, Proposition 3.1] it is easy to see that the isomorphisms of cohomology groups in (iii)N and (iv)N are induced by continuous morphisms of complexes. For nuclear Fréchet algebras, by Lemma 4.1, the complexes ((BcΩ̄A+) ∗, b̃∗ + B̃∗) and ((BpΩ̄A+) ∗, b̃∗ + B̃∗) are complexes of strong duals of nuclear Fréchet spaces. By Lemma 3.5, the isomorphisms of cohomology groups in (iii)N and (iv)N are topological. Cyclic cohomology of nuclear Fréchet and DF algebras 11 For a nuclear ⊗̂-algebra A which is a DF -space, by Lemma 4.1 and by [21, Theorem 4.4.13], the chain complex (BcΩ̄A+, b̃+ B̃) is the strong dual of a complex of nuclear Fréchet spaces. By [21, Theorem 4.4.12], complete nuclear DF -spaces and nuclear Fréchet spaces are reflexive. Therefore ((BcΩ̄A+) ∗, b̃∗ + B̃∗) is a complex of nuclear Fréchet spaces. Thus, by [7, Lemma 0.5.9], the isomorphisms of cohomology groups in (iii)N are topological. � The space of continuous traces on a topological algebra A is denoted by Atr, that Atr = {f ∈ A∗ : f(ab) = f(ba) for all a, b ∈ A}. The closure in A of the linear span of elements of the form {ab− ba : a, b ∈ A} is denoted by [A,A]. Recall that b0 : A⊗̂A → A is uniquely determined by a ⊗ b 7→ ab− ba. Proposition 4.4. Let A be in Φ and be a ⊗̂-algebra. (i) Suppose that the continuous cohomology groups Hnaiven (A) = {0} for all n ≥ 1 and Hbarn (A) = {0} for all n ≥ 0. Then, up to isomorphism of linear spaces, HHn(A) = {0} for all n ≥ 1 and HH0(A) = A/Im b0; HC2ℓ(A) = A/Im b0 and HC2ℓ+1(A) = {0} for all ℓ ≥ 0; HP0(A) = A/Im b0 and HP1(A) = {0}. (ii) Suppose that the continuous cohomology groups Hnnaive(A) = {0} for all n ≥ 1 and Hnbar(A) = {0} for all n ≥ 0. Then, up to isomorphism of linear spaces, HHn(A) = {0} for all n ≥ 1 and HH0(A) = Atr; HC2ℓ(A) = Atr and HC2ℓ+1(A) = {0} for all ℓ ≥ 0; HP0(A) = Atr and HP1(A) = {0}. Proof. (i). One can see that Hbarn (A) = {0} for all n ≥ 0 implies that HHn(A) = H naive n (A) for all n ≥ 0, see [17, Section 3]. Note that by definition of the ‘naive’ Hochschild homology of A, Hnaive0 (A) = A/Im b0. Therefore, HHn(A) = {0} for all n ≥ 1 and HH0(A) = A/Im b0. From the exactness of the long Connes-Tsygan sequence of continuous homology it follows that HC0(A) = H naive 0 (A) = A/Im b0 and HC1(A) = {0}. The rest of Statement (i) follows from Proposition 4.2. (ii) It is known that Hnbar(A) = {0} for all n ≥ 0, implies HH n(A) = Hnnaive(A) for all n ≥ 0. By definition of the ‘naive’ Hochschild cohomology of A, H0naive(A) = Atr. Thus HHn(A) = {0} for all n ≥ 1 and HH0(A) = Atr. From the exactness of the long Connes-Tsygan sequence of continuous cohomology it follows that HC0(A) = H0naive(A) = A tr and HC1(A) = {0}. The rest of Statement (ii) follows from Proposition 4.3. � 12 Z. A. Lykova 5. Cyclic-type cohomology of biflat ⊗̂-algebras Recall that a ⊗̂-algebra A is said to be biflat if it is flat in the category of A-⊗̂- bimodules [7, Def. 7.2.5]. A ⊗̂-algebra A is said to be biprojective if it is projective in the category of A-⊗̂-bimodules [7, Def. 4.5.1]. By [7, Proposition 4.5.6], a ⊗̂- algebra A is biprojective if and only if there exists an A-⊗̂-bimodule morphism ρA : A → A⊗̂A such that πA ◦ ρA = idA, where πA is the canonical morphism πA : A⊗̂A → A, a1 ⊗ a2 7→ a1a2. It can be proved that any biprojective ⊗̂-algebra is biflat and A = A2 = Im πA [7, Proposition 4.5.4]. Here A2 is the closure of the linear span of the set {a1 · a2 : a1, a2 ∈ A} in A. A ⊗̂-algebra A is said to be contractible if A+ is is projective in the category of A-⊗̂-bimodules. A ⊗̂-algebra A is contractible if and only if A is biprojective and has an identity [7, Def. 4.5.8]. For biflat Banach algebras A, Helemskii proved A = A2 = Im πA [7, Proposition 7.2.6] and gave the description of the cyclic homology HC∗ and cohomology HC ∗ groups of A in [8]. Later the author generalized Helemskii’s result to inverse limits of biflat Banach algebras [17, Theorem 6.2] and to locally convex strict inductive limits of amenable Banach algebras [18, Corollary 4.9]. Proposition 5.1. Let A be in Φ and be a biflat ⊗̂-algebra such that A = Im πA; in particular, let A ∈ Φ be a biprojective ⊗̂-algebra. Then (i) Hnaiven (A) = {0} for all n ≥ 1, H naive 0 (A) = A/Im b0 and H n (A) = {0} for all n ≥ 0; (ii) for the homology groups HH∗, HC∗ and HP∗ of A we have the isomorphisms of linear spaces (5). If, furthermore, A is a Fréchet space or A is a nuclear DF -space, then Hnaive0 (A) = A/[A,A] is Hausdorff, and, for a biflat A, A = A2 implies that A = Im πA. Proof. By [7, Theorem 3.4.25], up to topological isomorphism, the homology groups Hnaiven (A) = Hn(A,A) = Tor n (A,A+) for all n ≥ 0. Since A is biflat, by [7, Proposition 7.1.2], Hnaiven (A) = {0} for all n ≥ 1. By [7, Theorem 3.4.26], up to topological isomorphism, the homology groups Hbarn (A) = Hn+1(A,C) = Tor n+1(C,C) for all n ≥ 0, where C is the trivial A-bimodule. Note that, for the trivial A- bimodule C, there is a flat resolution 0← C← A+ ← A← 0 in the category of left or right A-⊗̂-modules. By [7, Theorem 3.4.28], Hbarn (A) = TorAn+1(C,C) = {0} for all n ≥ 1. By assumption, A = Im πA, hence H 0 (A) = A/Im πA = {0}. Thus the conditions of Proposition 4.4 (i) are satisfied. In the categories of Fréchet spaces and complete nuclear DF -spaces, the open mapping theorem holds – see Corollary 3.4 for DF -spaces. Thus, by [7, Proposi- tions 3.3.5 and 7.1.2], up to topological isomorphism, Hnaive0 (A) = Tor 0 (A,A+) is Cyclic cohomology of nuclear Fréchet and DF algebras 13 Hausdorff. Since A is biflat, by [7, Proposition 7.1.2], TorA0 (C,A) is also Hausdorff. By [7, Proposition 3.4.27], A2 = Im πA. � A ⊗̂-algebra A is amenable if A+ is a flat A-⊗̂-bimodule. For a Fréchet algebra A amenability is equivalent to the following: for all Fréchet A-bimodules X , H0(A, X) is Hausdorff and Hn(A, X) = {0} for all n ≥ 1. Recall that an amenable Banach algebra A is biflat and has a bounded approximate identity [7, Theorem VII.2.20]. Lemma 5.2. Let A be an amenable ⊗̂-algebra which is a Fréchet space or a nu- clear DF -space. Then Hnaiven (A) = {0} for all n ≥ 1, H naive 0 (A) = A/[A,A] and Hbarn (A) = {0} for all n ≥ 0. Proof. In the categories of Fréchet spaces and complete nuclear DF -spaces, the open mapping theorem holds. Therefore, by [7, Theorem III.4.25 and Proposition 7.1.2], up to topological isomorphism, for the trivial A-bimodule C, Hbarn (A) = Hn+1(A,C) = Tor n+1(C,A+) = {0} for all n ≥ 0; Hnaiven (A) = Hn(A,A) = Tor n (A,A+) = {0} for all n ≥ 1 and Hnaive0 (A) = Tor 0 (A,A+) is Hausdorff, that is, H naive 0 (A) = A/[A,A]. � Theorem 5.3. Let A be a ⊗̂-algebra which is a Fréchet space or a nuclear DF - space. Suppose that the continuous homology groups Hnaiven (A) = {0} for all n ≥ 1, Hnaive0 (A) is Hausdorff and H n (A) = {0} for all n ≥ 0. In particular, asssume that A is a biflat algebra such that A = A2 or A is amenable. Then (i) up to topological isomorphism, HHn(A) = {0} for all n ≥ 1 and HH0(A) = A/[A,A]; HC2ℓ(A) = A/[A,A] and HC2ℓ+1(A) = {0} for all ℓ ≥ 0; (ii) up to topological isomorphism for Fréchet algebras and up to isomorphism of linear spaces for nuclear DF -algebras, (8) HP0(A) = A/[A,A] and HP1(A) = {0}; (iii) Hnnaive(A) = {0} for all n ≥ 1; Hnbar(A) = {0}; for all n ≥ 0; (iv) up to topological isomorphism for nuclear Fréchet algebras and nuclear DF - algebras and up to isomorphism of linear spaces for Fréchet algebras, HHn(A) = {0} for all n ≥ 1 and HH0(A) = Atr; HC2ℓ(A) = Atr and HC2ℓ+1(A) = {0} for all ℓ ≥ 0; (v) up to topological isomorphism for nuclear Fréchet algebras and up to isomorphism of linear spaces for Fréchet algebras and for nuclear DF -algebras, (11) HP0(A) = Atr and HP1(A) = {0}. 14 Z. A. Lykova Proof. In view of Proposition 5.1 and Lemma 5.2, a biflat algebra A such that A = A2 and an amenable A satisfy the conditions of the theorem. By Proposition 2.2, firstly, Hbarn (A) = {0} for all n ≥ 0 if and only if H bar(A) = {0} for all n ≥ 0; and, secondly, Hnnaive(A) = {0} for all n ≥ 1 if and only if Hnaiven (A) = {0} for all n ≥ 1 and H naive 0 (A) is Hausdorff. By Proposition 4.4, we have isomorphisms of linear spaces in (i) – (v). In Propo- sitions 4.2 and 4.2 we show also when the above isomorphisms are automatically topological. � Remark 5.4. Recall that, for a biflat Banach algebra A, dbwA ≤ 2 [27, Theorem 6]. By [14, Theorem 5.2], for a Banach algebra A of a finite weak bidimension dbwA, we have isomorphisms between the entire cyclic cohomology and the periodic cyclic cohomology of A, HE0(A) = HP 0(A) = Atr and HE1(A) = HP 1(A) = {0}. The entire cyclic cohomology HEk(A) of A for k = 0, 1 are defined in [2, IV.7]. In [25, Theorem 6.1] M. Puschnigg extended M. Khalkhali’s result on the isomorphism HEk(A) = HP k(A) for k = 0, 1 from Banach algebras to some Fréchet algebras. The following statement shows that the above theorems give the explicit descrip- tion of cyclic type homology and cohomology of the projective tensor product of two biprojective ⊗̂-algebras. Proposition 5.5. Let B and C be biprojective ⊗̂-algebras. Then the projective tensor product A = B⊗̂C is a biprojective ⊗̂-algebra. Proof. Since B is biprojective, there is a morphism of B-⊗̂-bimodules ρB : B → B⊗̂B such that πB ◦ ρB = idB. A similar statement is valid for C. Let i be the topological isomorphism i : (B⊗̂B)⊗̂(C⊗̂C)→ (B⊗̂C)⊗̂(B⊗̂C) given by (b1⊗ b2)⊗ (c1⊗ c2) 7→ (b1⊗ c1)⊗ (b2⊗ c2). Note that πB⊗̂C = (πB⊗̂πC)◦ i It is routine to check that ρB⊗̂C : B⊗̂C → (B⊗̂C)⊗̂(B⊗̂C) defined by ρB⊗̂C = i◦(ρB⊗ρC) is a morphism of B⊗̂C-⊗̂-bimodules and πB⊗̂C◦ρB⊗̂C = idB⊗̂C. � Remark 5.6. For amenable Banach algebras B and C, B. E. Johnson showed that the Banach algebra A = B⊗̂C is amenable [11]. By [19, Proposition 5.4], for a biflat Banach algebra A, each closed two-sided ideal I with bounded approximate identity is amenable and the quotient algebra A/I is biflat. Thus the explicit description of cyclic type homology and cohomology of such I and A/I is also given in Theorem 5.3. One can find a number of examples of biflat and simplicially trivial Banach and C∗- algebras in [17, Example 4.6, 4.9]. Cyclic cohomology of nuclear Fréchet and DF algebras 15 6. Applications to the cyclic-type cohomology of biprojective ⊗̂-algebras In this section we present examples of nuclear biprojective ⊗̂-algebras which are Fréchet spaces or DF -spaces and the continuous cyclic-type homology and cohomol- ogy of these algebras. Example 6.1. Let G be a compact Lie group and let E(G) be the nuclear Fréchet algebra of smooth functions on G with the convolution product. It was shown by Yu.V. Selivanov that A = E(G) is biprojective [29]. Let E∗(G) be the strong dual to E(G), so that E∗(G) is a complete nuclear DF - space. This is a ⊗̂-algebra with respect to convolution multiplication: for f, g ∈ E∗(G) and x ∈ E(G), < f ∗ g, x >=< f, y >, where y ∈ E(G) is defined by y(s) =< g, xs >, s ∈ G and xs(t) = x(s −1t), t ∈ G. J.L. Taylor proved that the algebra of distributions E∗(G) on a compact Lie group G is contractible [30]. Example 6.2. Let (E, F ) be a pair of complete Hausdorff locally convex spaces endowed with a jointly continuous bilinear form 〈·, ·〉 : E ×F → C that is not iden- tically zero. The space A = E⊗̂F is a ⊗̂-algebra with respect to the multiplication defined by (x1 ⊗ x2)(y1 ⊗ y2) = 〈x2, y1〉x1 ⊗ y2, xi ∈ E, yi ∈ F. Yu.V. Selivanov proved that this algebra is biprojective and usually non unital [28, 29]. More exactly, if A = E⊗̂F has a left or right identity, then E or F respectively is finite-dimensional. If the form 〈·, ·〉 is nondegenerate, then A = E⊗̂F is called the tensor algebra generated by the duality (E, F, 〈·, ·〉). In particular, if E is a Banach space with the approximation property, then the algebra A = E⊗̂E∗ is isomorphic to the algebra N (E) of nuclear operators on E [7, II.2.5]. 6.1. Köthe sequence algebras. The following results on Köthe algebras can be found in A. Yu. Pirkovskii’s papers [22, 23]. A set P of nonnegative real-valued sequences p = (pi)i∈N is called a Köthe set if the following axioms are satisfied: (P1) for every i ∈ N there is p ∈ P such that pi > 0; (P2) for every p, q ∈ P there is r ∈ P such that max{pi, qi} ≤ ri for all i ∈ N. Suppose, in addition, the following condition is satisfied: (P3) for every p ∈ P there exist q ∈ P and a constant C > 0 such that pi ≤ Cq for all i ∈ N. For any Köthe set P which satisfies (P3), the Köthe space λ(P ) = {x = (xn) ∈ C N : ‖x‖p = |xn|pn <∞ for all p ∈ P} is a complete locally convex space with the topology determined by the family of seminorms {‖x‖p : p ∈ P} and a ⊗̂-algebra with pointwise multiplication. The ⊗̂-algebras λ(P ) are called Köthe algebras. By [21] and [6], for a Köthe set, λ(P ) is nuclear if and only if (P4) for every p ∈ P there exist q ∈ P and ξ ∈ ℓ1 such that pi ≤ ξiqi for all i ∈ N. 16 Z. A. Lykova By [22, Theorem 3.5], λ(P ) is biprojective if and only if (P5) for every p ∈ P there exist q ∈ P and a constant M > 0 such that p2i ≤Mqi for all i ∈ N. The algebra λ(P ) is unital if and only if n pn <∞ for every p ∈ P . Example 6.3. Fix a real number 1 ≤ R ≤ ∞ and a nondecreasing sequence α = (αi) of positive numbers with limi→∞ αi =∞. The power series space ΛR(α) = {x = (xn) ∈ C N : ‖x‖r = |xn|r αn <∞ for all 0 < r < R} is a Fréchet Köthe algebra with pointwise multiplication. The topology of ΛR(α) is determined by a countable family of seminorms {‖x‖rk : k ∈ N} where {rk} is an arbitrary increasing sequence converging to R. By [23, Corollary 3.3], ΛR(α) is biprojective if and only if R = 1 or R =∞. By the Grothendieck-Pietsch criterion, ΛR(α) is nuclear if and only if for limn 0 for R <∞ and limn <∞ for R =∞, see [22, Example 3.4]. The algebra ΛR((n)) is topologically isomorphic to the algebra of functions holo- morphic on the open disc of radius R, endowed with Hadamard product, that is, with “co-ordinatewise” product of the Taylor expansions of holomorphic functions. Example 6.4. The algebra H(C) ∼= Λ∞((n)) of entire functions, endowed with the Hadamard product, is a biprojective nuclear Fréchet algebra [23]. Example 6.5. The algebra H(D1) ∼= Λ1((n)) of functions holomorphic on the open unit disc, endowed with the Hadamard product, is a biprojective nuclear Fréchet algebra. Moreover it is contractible, since the function z 7→ (1− z)−1 is an identity for H(D1) [23]. For any Köthe space λ(P ) the dual space λ(P )∗ can be canonically identified with {(yn) ∈ C N : ∃p ∈ P and C > 0 such that |yn| ≤ Cpn for all n ∈ N}. It is shown in [23] that, for a biprojective Köthe algebra λ(P ), λ(P )∗ is a sequence algebra with pointwise multiplication. The algebra λ(P )∗ is unital if and only if there exists p ∈ P such that inf i pi > 0. Example 6.6. The nuclear Fréchet algebra of rapidly decreasing sequences s = {x = (xn) ∈ C N : ‖x‖k = |xn|n k <∞ for all k ∈ N} is a biprojective Köthe algebra [22]. The algebra s is topologically isomorphic to Λ∞(α) with αn = logn [23]. The nuclear Köthe ⊗̂-algebra s ∗ of sequences of poly- nomial growth is contractible [30]. Example 6.7. [23, Section 4.2] Let P be a Köthe set such that pi ≥ 1 for all p ∈ P and all n ∈ N. Then the formula 〈a, b〉 = i aibi defines a jointly continuous, nondegenerate bilinear form on λ(P ) × λ(P ). Thus M(P ) = λ(P )⊗̂λ(P ) can be considered as the tensor algebra generated by the duality (λ(P ), λ(P ), 〈·, ·〉), and so is biprojective. There is a canonical isomorphism between M(P ) and the algebra Cyclic cohomology of nuclear Fréchet and DF algebras 17 λ(P × P ) of N×N complex matrices (aij)(ij)∈N×N satisfying the condition ‖a‖p = i,j |aij|pipj <∞ for all p ∈ P with the usual matrix multiplication. In particular, for P = {(nk)n∈N : k = 0, 1, . . . }, we obtain the biprojective nuclear Fréchet algebra ℜ = s⊗̂s of “smooth compact operators” consisting of N×N com- plex matrices (aij) with rapidly decreasing matrix entries. Here s is from Example Theorem 6.8. Let A be a ⊗̂-algebra belonging to one of the following classes: (i) A = E(G) or A = E∗(G) for a compact Lie group G; (ii) A = E⊗̂F , the tensor algebra generated by the duality (E, F, 〈·, ·〉) for nuclear Fréchet spaces E and F (e.g., ℜ = s⊗̂s) or for nuclear complete DF -spaces E and (iii) Fréchet Köthe algebras A = λ(P ) such that the Köthe set P satisfies (P3), (P4) and (P5); in particular, Λ1(α) such that limn = 0 or Λ∞(α) such that <∞. (e.g., H(D1), s, H(C)). (iv) Köthe algebras A = λ(P )∗ which are the strong duals of λ(P ) from (iii). (v) the projective tensor product A = B⊗̂C of biprojective nuclear ⊗̂-algebras B and C which are Fréchet spaces or DF -spaces; in particular, A = E(G)⊗̂ℜ. Then, up to topological isomorphism, Hnaiven (A) = {0} for all n ≥ 1 and H naive 0 (A) = A/[A,A]; Hbarn (A) = {0} for all n ≥ 0; HHn(A) = {0} for all n ≥ 1 and HH0(A) = A/[A,A]; HC2ℓ(A) = A/[A,A] and HC2ℓ+1(A) = {0} for all ℓ ≥ 0; Hnnaive(A) = {0} for all n ≥ 1; H bar(A) = {0} for all n ≥ 0; HHn(A) = {0} for all n ≥ 1 and HH0(A) = Atr; HC2ℓ(A) = Atr and HC2ℓ+1(A) = {0} for all ℓ ≥ 0; and, up to topological isomorphism for Fréchet algebras and up to isomorphism of linear spaces for DF -algebras, HP0(A) = A/[A,A] and HP1(A) = {0}; HP0(A) = Atr and HP1(A) = {0}. Proof. We have mentioned above that the algebras in (i)-(iii) and (v) are biprojective and nuclear. By [23, Corollary 3.10], for any nuclear biprojective Fréchet Köthe algebra λ(P ), the strong dual λ(P )∗ is a nuclear, biprojective Köthe ⊗̂-algebra which is a DF -space. For nuclear Fréchet algebras and for nuclear DF -algebras, the conditions of Theorem 5.3 are satisfied. Therefore, for the homology and cohomology groups HH and HC of A we have the topological isomorphisms (7) and (10). For the periodic cyclic homology and cohomology groups HP of A, for Fréchet algebras, we have topological isomorphisms and, for nuclear DF -algebras, isomorphisms of linear spaces (8) and (11). It is obvious that, for commutative algebras, Atr = A∗ and A/[A,A] = A. � 18 Z. A. Lykova The cyclic-type homology and cohomology of E(G) for a compact Lie group G were calculated in [20]. References [1] J. Brodzki and Z. A. Lykova: “Excision in cyclic type homology of Fréchet algebras”, Bull. London Math. Soc., Vol. 33, (2001), pp. 283-291. [2] A. Connes: Noncommutative geometry, Academic Press, London, 1994. [3] J. Cuntz: “Cyclic theory and the bivariant Chern-Connes character”, In: Noncommutative geometry,” Lecture Notes in Math., no. 1831, Springer, Berlin, 2004, pp. 73–135. [4] J. Cuntz and D. 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Johnson: Cohomology in Banach algebras, Mem. Amer. Math. Soc., Vol. 127, 1972. [12] H. Jarchow: Locally convex spaces, B.G. Teubner, Stuttgart, 1981. [13] C. Kassel: “Cyclic homology, comodules, and mixed complexes”, J. Algebra, Vol. 107, (1987), pp. 195-216. [14] M. Khalkhali: “Algebraic connections, universal bimodules and entire cyclic cohomology”, Comm. Math. Phys., Vol. 161, (1994), pp. 433–446. [15] G. Köthe: Topological Vector Spaces, I, Die Grundlehren der mathematischen Wissenschaften, 159, Springer Verlag, Berlin, New-York, 1979. [16] J.-L. Loday: Cyclic Homology, Springer Verlag, Berlin, 1992. [17] Z.A. Lykova: “Cyclic cohomology of projective limits of topological algebras”, Proc. Edin- burgh Math. Soc., Vol. 49, (2006), pp. 173–199. [18] Z.A. Lykova: “Cyclic-type cohomology of strict inductive limits of Fréchet algebras”, J. Pure Appl. Algebra, Vol. 205, (2006), pp. 471–497. [19] Z.A. Lykova and M.C. White: “Excision in the cohomology of Banach algebras with coeffi- cients in dual bimodules”, In: E. Albrecht and M. Mathieu (Eds.), Banach Algebras’97, Walter de Gruyter Publishers, Berlin, 1998, pp. 341–361. [20] R. Meyer: “Comparisons between periodic, analytic and local cyclic cohomology”, ArXiv:math.KT/0205276 v2 18 Dec 2003. [21] A. Pietsch: Nuclear Locally Convex Spaces, Springer Verlag, Berlin, 1972. [22] A.Yu. Pirkovskii: “Biprojective topological algebras of homological bidimension 1”, J. Math. Sci., Vol. 111, No. 2, (2001), pp. 3476-3495. [23] A.Yu. Pirkovskii: “Homological bidimension of biprojective topological algebras and nuclear- ity”, Acta Univ. Oulu. Ser Rerum Natur., Vol. 408, (2004), pp. 179–196. [24] V. Pták: “On complete topological linear spaces”, Czech. Math. J., Vol. 78 (3), (1953), pp. 301–364. (Russian with English summary) http://arxiv.org/abs/math/0205276 Cyclic cohomology of nuclear Fréchet and DF algebras 19 [25] M. Puschnigg: “Excision in cyclic homology theories”, Invent. Math., Vol. 143, (2001), pp. 249–323. [26] A.P. Robertson and W. Robertson: Topological vector spaces, Cambridge Univ. Press, 1973. [27] Yu.V. Selivanov: “Cohomology of biflat Banach algebras with coefficients in dual bimodules”, Functional Anal. Appl., Vol. 29, (1995), pp. 289–291. [28] Yu.V. Selivanov: “Biprojective Banach algebras”, Math. USSR Izvestija, Vol. 15, (1980), pp. 387–399. [29] Yu.V. Selivanov: “Biprojective topological algebras”, Unpublished manuscript, 1996. [30] J.L. Taylor: “Homology and cohomology for topological algebras”, Adv. Math., Vol. 9, (1972), pp. 137–182. [31] F. Treves: Topological vector spaces, distributions and kernels, Academic Press, New York, London, 1967. [32] M. Wodzicki: “Vanishing of cyclic homology of stable C∗-algebras”, C. R. Acad. Sci. Paris I , Vol. 307, (1988), pp. 329–334. School of Mathematics and Statistics, University of Newcastle,, Newcastle upon Tyne, NE1 7RU, UK ([email protected]) 1. Introduction 2. Definitions and notation 3. The open mapping theorem in complete nuclear DF-spaces 4. Cyclic and Hochschild cohomology of some -algebras 5. Cyclic-type cohomology of biflat -algebras 6. Applications to the cyclic-type cohomology of biprojective -algebras 6.1. Köthe sequence algebras References

0704.1020

The on-line shortest path problem under partial monitoring

The on-line shortest path problem under partial monitoring András György Tamás Linder Gábor Lugosi György Ottucsák ∗ October 27, 2018 Abstract The on-line shortest path problem is considered under various models of partial monitor- ing. Given a weighted directed acyclic graph whose edge weights can change in an arbitrary (adversarial) way, a decision maker has to choose in each round of a game a path between two distinguished vertices such that the loss of the chosen path (defined as the sum of the weights of its composing edges) be as small as possible. In a setting generalizing the multi-armed bandit problem, after choosing a path, the decision maker learns only the weights of those edges that belong to the chosen path. For this problem, an algorithm is given whose average cumulative loss in n rounds exceeds that of the best path, matched off-line to the entire sequence of the edge weights, by a quantity that is proportional to 1/ n and depends only polynomially on the number of edges of the graph. The algorithm can be implemented with linear complexity in the number of rounds n and in the number of edges. An extension to the so-called label efficient setting is also given, in which the decision maker is informed about the weights of the edges corresponding to the chosen path at a total of m ≪ n time instances. Another extension is shown where the decision maker competes against a time-varying path, a generalization of the problem of tracking the best expert. A version of the multi-armed bandit setting for shortest path is also discussed where the decision maker learns only the total weight of the chosen path but not the weights of the individual edges on the path. Applications to routing in packet switched networks along with simulation results are also presented. Index Terms: On-line learning, shortest path problem, multi-armed bandit problem. ∗A. György is with the Machine Learning Research Group, Computer and Automation Research Institute of the Hungarian Academy of Sciences, Kende u. 11-13, Budapest, Hungary, H-1111 (email: [email protected]). T. Linder is with the Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada K7L 3N6 (email: [email protected]). G. Lugosi is with ICREA and Department of Economics, Universitat Pompeu Fabra, Ramon Trias Fargas 25-27, 08005 Barcelona, Spain (email: [email protected]). Gy. Ottucsák is with Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Magyar Tudósok Körútja 2., Budapest, Hungary, H-1117. He is also with the Machine Learning Research Group, Computer and Automation Research Institute of the Hungarian Academy of Sciences (email: [email protected]). This research was supported in part by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, the Mobile Innovation Center of Hungary, by the Natural Sciences and Engineering Research Council (NSERC) of Canada, by the Spanish Ministry of Science and Technology grant MTM2006-05650, by the PASCAL Network of Excellence under EC grant no. 506778 and by the High Speed Networks Laboratory, Department of Telecommuni- cations and Media Informatics, Budapest University of Technology and Economics. Parts of this paper have been presented at COLT’06. http://arxiv.org/abs/0704.1020v1 1 Introduction In a sequential decision problem, a decision maker (or forecaster) performs a sequence of actions. After each action the decision maker suffers some loss, depending on the response (or state) of the environment, and its goal is to minimize its cumulative loss over a certain period of time. In the setting considered here, no probabilistic assumption is made on how the losses corresponding to different actions are generated. In particular, the losses may depend on the previous actions of the decision maker, whose goal is to perform well relative to a set of reference forecasters (the so-called “experts”) for any possible behavior of the environment. More precisely, the aim of the decision maker is to achieve asymptotically the same average (per round) loss as the best expert. Research into this problem started in the 1950s (see, for example, Blackwell [5] and Hannan [18] for some of the basic results) and gained new life in the 1990s following the work of Vovk [29], Littlestone and Warmuth [24], and Cesa-Bianchi et al. [7]. These results show that for any bounded loss function, if the decision maker has access to the past losses of all experts, then it is possible to construct on-line algorithms that perform, for any possible behavior of the environment, almost as well as the best of N experts. More precisely, the per round cumulative loss of these algorithms is at most as large as that of the best expert plus a quantity proportional to lnN/n for any bounded loss function, where n is the number of rounds in the decision game. The logarithmic dependence on the number of experts makes it possible to obtain meaningful bounds even if the pool of experts is very large. In certain situations the decision maker has only limited knowledge about the losses of all possible actions. For example, it is often natural to assume that the decision maker gets to know only the loss corresponding to the action it has made, and has no information about the loss it would have suffered had it made a different decision. This setup is referred to as the multi-armed bandit problem, and was considered, in the adversarial setting, by Auer et al. [1] who gave an algorithm whose normalized regret (the difference of the algorithm’s average loss and that of the best expert) is upper bounded by a quantity which is proportional to N lnN/n. Note that, compared to the full information case described above where the losses of all possible actions are revealed to the decision maker, there is an extra N factor in the performance bound, which seriously limits the usefulness of the bound if the number of experts is large. Another interesting example for the limited information case is the so-called label efficient decision problem (see Helmbold and Panizza [22]) in which it is too costly to observe the state of the environment, and so the decision maker can query the losses of all possible actions for only a limited number of times. A recent result of Cesa-Bianchi, Lugosi, and Stoltz [9] shows that in this case, if the decision maker can query the losses m times during a period of length n, then it can achieve O( lnN/m) average excess loss relative to the best expert. In many applications the set of experts has a certain structure that may be exploited to construct efficient on-line decision algorithms. The construction of such algorithms has been of great interest in computational learning theory. A partial list of works dealing with this problem includes Herbster and Warmuth [19], Vovk [30], Bousquet and Warmuth [6], Helmbold and Schapire [27], Takimoto and Warmuth [28], Kalai and Vempala [23], György at al. [13, 14, 15]. For a more complete survey, we refer to Cesa-Bianchi and Lugosi [8, Chapter 5]. In this paper we study the on-line shortest path problem, a representative example of structured expert classes that has received attention in the literature for its many applications, including, among others, routing in communication networks; see, e.g., Takimoto andWarmuth [28], Awerbuch et al. [3], or György and Ottucsák [17], and adaptive quantizer design in zero-delay lossy source coding; see, György et al. [13, 14, 16]. In this problem, a weighted directed (acyclic) graph is given whose edge weights can change in an arbitrary manner, and the decision maker has to pick in each round a path between two given vertices, such that the weight of this path (the sum of the weights of its composing edges) be as small as possible. Efficient solutions, with time and space complexity proportional to the number of edges rather than to the number of paths (the latter typically being exponential in the number of edges), have been given in the full information case, where in each round the weights of all the edges are revealed after a path has been chosen; see, for example, Mohri [26], Takimoto and Warmuth [28], Kalai and Vempala [23], and György et al. [15]. In the bandit setting only the weights of the edges or just the sum of the weights of the edges composing the chosen path are revealed to the decision maker. If one applies the general bandit algorithm of Auer et al. [1], the resulting bound will be too large to be of practical use because of its square-root-type dependence on the number of paths N . On the other hand, using the special graph structure in the problem, Awerbuch and Kleinberg [4] and McMahan and Blum [25] managed to get rid of the exponential dependence on the number of edges in the performance bound. They achieved this by extending the exponentially weighted average predictor and the follow-the- perturbed-leader algorithm of Hannan [18] to the generalization of the multi-armed bandit setting for shortest paths, when only the sum of the weights of the edges is available for the algorithm. However, the dependence of the bounds obtained in [4] and [25] on the number of rounds n is significantly worse than the O(1/ n) bound of Auer et al. [1]. Awerbuch and Kleinberg [4] consider the model of “non-oblivious” adversaries for shortest path (i.e., the losses assigned to the edges can depend on the previous actions of the forecaster) and prove an O(n−1/3) bound for the expected per-round regret. McMahan and Blum [25] give a simpler algorithm than in [4] however obtain a bound of the order of O(n−1/4) for the expected regret. In this paper we provide an extension of the bandit algorithm of Auer et al. [1] unifying the advantages of the above approaches, with a performance bound that is polynomial in the number of edges, and converges to zero at the right O(1/ n) rate as the number of rounds increases. We achieve this bound in a model which assumes that the losses of all edges on the path chosen by the forecaster are available separately after making the decision. We also discuss the case (considered by [4] and [25]) in which only the total loss (i.e., the sum of the losses on the chosen path) is known to the decision maker. We exhibit a simple algorithm which achieves an O(n−1/3) per-round regret with high probability against “non-oblivious” adversary. In this case it remains an open problem to find an algorithm whose cumulative loss is polynomial in the number of edges of the graph and decreases as O(n−1/2) with the number of rounds. In Section 2 we formally define the on-line shortest path problem, which is extended to the multi-armed bandit setting in Section 3. Our new algorithm for the shortest path problem in the bandit setting is given in Section 4 together with its performance analysis. The algorithm is extended to solve the shortest path problem in a combined label efficient multi-armed bandit setting in Section 5. Another extension, when the algorithm competes against a time-varying path is studied in Section 6. An algorithm for the “restricted” multi-armed bandit setting (when only the sums of the losses of the edges are available) is given in Section 7. Simulation results are presented in Section 8. 2 The shortest path problem Consider a network represented by a set of vertices connected by edges, and assume that we have to send a stream of packets from a distinguished vertex, called source, to another distinguished vertex, called destination. At each time slot a packet is sent along a chosen route connecting source and destination. Depending on the traffic, each edge in the network may have a different delay, and the total delay the packet suffers on the chosen route is the sum of delays of the edges composing the route. The delays may change from one time slot to the next one in an arbitrary way, and our goal is to find a way of choosing the route in each time slot such that the sum of the total delays over time is not significantly more than that of the best fixed route in the network. This adversarial version of the routing problem is most useful when the delays on the edges can change dynamically, even depending on our previous routing decisions. This is the situation in the case of ad-hoc networks, where the network topology can change rapidly, or in certain secure networks, where the algorithm has to be prepared to handle denial of service attacks, that is, situations where willingly malfunctioning vertices and links increase the delay; see, e.g., Awerbuch et al. [3]. This problem can be cast naturally as a sequential decision problem in which each possible route is represented by an action. However, the number of routes is typically exponentially large in the number of edges, and therefore computationally efficient algorithms are called for. Two solutions of different flavor have been proposed. One of them is based on a follow-the-perturbed- leader forecaster, see Kalai and Vempala [23], while the other is based on an efficient computation of the exponentially weighted average forecaster, see, for example, Takimoto and Warmuth [28]. Both solutions have different advantages and may be generalized in different directions. To formalize the problem, consider a (finite) directed acyclic graph with a set of edges E = {e1, . . . , e|E|} and a set of vertices V . Thus, each edge e ∈ E is an ordered pair of vertices (v1, v2). Let u and v be two distinguished vertices in V . A path from u to v is a sequence of edges e(1), . . . , e(k) such that e(1) = (u, v1), e (j) = (vj−1, vj) for all j = 2, . . . , k − 1, and e(k) = (vk−1, v). Let P = {i1, . . . , iN} denote the set of all such paths. For simplicity, we assume that every edge in E is on some path from u to v and every vertex in V is an endpoint of an edge (see Figure 1 for examples). PSfrag replacements Figure 1: Two examples of directed acyclic graphs for the shortest path problem. (a) (b) In each round t = 1, . . . , n of the decision game, the decision maker chooses a path It among all paths from u to v. Then a loss ℓe,t ∈ [0, 1] is assigned to each edge e ∈ E. We write e ∈ i if the edge e ∈ E belongs to the path i ∈ P, and with a slight abuse of notation the loss of a path i at time slot t is also represented by ℓi,t. Then ℓi,t is given as ℓi,t = and therefore the cumulative loss up to time t of each path i takes the additive form Li,t = ℓi,s = where the inner sum on the right-hand side is the loss accumulated by edge e during the first t rounds of the game. The cumulative loss of the algorithm is L̂t = ℓIs,s = ℓe,s . It is well known that for a general loss sequence, the decision maker must be allowed to use randomization to be able to approximate the performance of the best expert (see, e.g., Cesa-Bianchi and Lugosi [8]). Therefore, the path It is chosen randomly according to some distribution pt over all paths from u to v. We study the normalized regret over n rounds of the game L̂n −min where the minimum is taken over all paths i from u to v. For example, the exponentially weighted average forecaster ([29], [24], [7]), calculated over all possible paths, has regret L̂n −min ln(1/δ) with probability at least 1− δ, where N is the total number of paths from u to v in the graph and K is the length of the longest path. 3 The multi-armed bandit setting In this section we discuss the “bandit” version of the shortest path problem. In this setup, which is more realistic in many applications, the decision maker has only access to the losses corresponding to the paths it has chosen. For example, in the routing problem this means that information is available on the delay of the route the packet is sent on, and not on other routes in the network. We distinguish between two types of bandit problems, both of which are natural generalizations of the simple bandit problem to the shortest path problem. In the first variant, the decision maker has access to the losses of those edges that are on the path it has chosen. That is, after choosing a path It at time t, the value of the loss ℓe,t is revealed to the decision maker if and only if e ∈ It. We study this case and its extensions in Sections 4, 5, and 6. The second variant is a more restricted version in which the loss of the chosen path is observed, but no information is available on the individual losses of the edges belonging to the path. That is, after choosing a path It at time t, only the value of the loss of the path ℓIt,t is revealed to the decision maker. Further on we call this setting as the restricted bandit problem for shortest path. We consider this restricted problem in Section 7. Formally, the on-line shortest path problem in the multi-armed bandit setting is described as follows: at each time instance t = 1, . . . , n, the decision maker picks a path It ∈ P from u to v. Then the environment assigns loss ℓe,t ∈ [0, 1] to each edge e ∈ E, and the decision maker suffers loss ℓIt,t = ℓe,t. In the unrestricted case the losses ℓe,t are revealed for all e ∈ It, while in the restricted case only ℓIt,t is revealed. Note that in both cases ℓe,t may depend on I1, . . . , It−1, the earlier choices of the decision maker. For the basic multi-armed bandit problem, Auer et al. [1] gave an algorithm, based on exponen- tial weighting with a biased estimate of the gains (defined, in our case, as gi,t = K− ℓi,t), combined with uniform exploration. Applying their algorithm to the on-line shortest path problem in the bandit setting results in a performance that can be bounded, for any 0 < δ < 1 and fixed time horizon n, with probability at least 1− δ, by L̂n −min ≤ 11K N ln(N/δ) K lnN (The constants follow from a slightly improved version; see Cesa-Bianchi and Lugosi [8].) However, for the shortest path problem this bound is unacceptably large because, unlike in the full information case, here the dependence on the number of all paths N is not merely logarithmic, while N is typically exponentially large in the size of the graph (as in the two simple examples of Figure 1). In order to achieve a bound that does not grow exponentially with the number of edges of the graph, it is imperative to make use of the dependence structure of the losses of the different actions (i.e., paths). Awerbuch and Kleinberg [4] and McMahan and Blum [25] do this by extending low complexity predictors, such as the follow-the-perturbed-leader forecaster [18], [23] to the restricted bandit setting. However, in both cases the price to pay for the polynomial dependence on the number of edges is a worse dependence on the length n of the game. 4 A bandit algorithm for shortest paths In this section we describe a variant of the bandit algorithm of [1] which achieves the desired performance for the shortest path problem. The new algorithm uses the fact that when the losses of the edges of the chosen path are revealed, then this also provides some information about the losses of each path sharing common edges with the chosen path. For each edge e ∈ E, and t = 1, 2, . . ., introduce the gain ge,t = 1− ℓe,t, and for each path i ∈ P, let the gain be the sum of the gains of the edges on the path, that is, gi,t = ge,t . The conversion from losses to gains is done in order to facilitate the subsequent performance analysis. To simplify the conversion, we assume that each path i ∈ P is of the same length K for some K > 0. Note that although this assumption may seem to be restrictive at the first glance, from each acyclic directed graph (V,E) one can construct a new graph by adding at most (K−2)(|V |−2)+1 vertices and edges (with constant weight zero) to the graph without modifying the weights of the paths such that each path from u to v will be of length K, where K denotes the length of the longest path of the original graph. If the number of edges is quadratic in the number of vertices, the size of the graph is not increased substantially. A main feature of the algorithm below is that the gains are estimated for each edge and not for each path. This modification results in an improved upper bound on the performance with the number of edges in place of the number of paths. Moreover, using dynamic programming as in Takimoto and Warmuth [28], the algorithm can be computed efficiently. Another important ingredient of the algorithm is that one needs to make sure that every edge is sampled sufficiently often. To this end, we introduce a set C of covering paths with the property that for each edge e ∈ E there is a path i ∈ C such that e ∈ i. Observe that one can always find such a covering set of cardinality |C| ≤ |E|. We note that the algorithm of [1] is a special case of the algorithm below: For any multi- armed bandit problem with N experts, one can define a graph with two vertices u and v, and N directed edges from u to v with weights corresponding to the losses of the experts. The solution of the shortest path problem in this case is equivalent to that of the original bandit problem with choosing expert i if the corresponding edge is chosen. For this graph, our algorithm reduces to the original algorithm of [1]. A BANDIT ALGORITHM FOR SHORTEST PATHS Parameters: real numbers β > 0, 0 < η, γ < 1. Initialization: Set we,0 = 1 for each e ∈ E, wi,0 = 1 for each i ∈ P, and W 0 = N . For each round t = 1, 2, . . . (a) Choose a path It at random according to the distribution pt on P, defined by pi,t = (1− γ)wi,t−1 W t−1 if i ∈ C (1− γ)wi,t−1 W t−1 if i 6∈ C. (b) Compute the probability of choosing each edge e as qe,t = i:e∈i pi,t = (1− γ) i:e∈iwi,t−1 W t−1 |{i ∈ C : e ∈ i}| (c) Calculate the estimated gains g′e,t = ge,t+β if e ∈ It otherwise. (d) Compute the updated weights we,t = we,t−1e ηg′e,t wi,t = we,t = wi,t−1e where g′ i,t = e∈i g e,t, and the sum of the total weights of the paths W t = wi,t. The main result of the paper is the following performance bound for the shortest-path bandit algorithm. It states that the per round regret of the algorithm, after n rounds of play, is, roughly, of the order of K |E| lnN/n where |E| is the number of edges of the graph, K is the length of the paths, and N is the total number of paths. Theorem 1 For any δ ∈ (0, 1) and parameters 0 ≤ γ < 1/2, 0 < β ≤ 1, and η > 0 satisfying 2ηK|C| ≤ γ, the performance of the algorithm defined above can be bounded, with probability at least 1− δ, as L̂n −min ≤ Kγ + 2ηK2|C|+ + |E|β. In particular, choosing β = , γ = 2ηK|C|, and η = 4nK2|C| yields for all n ≥ , 4|C| lnN L̂n −min 4K|C| lnN + |E| ln |E| The proof of the theorem is based on the analysis of the original algorithm of [1] with necessary modifications required to transform parts of the argument from paths to edges, and to use the connection between the gains of paths sharing common edges. For the analysis we introduce some notation: Gi,n = gi,t and G i,n = g′i,t for each i ∈ P and Ge,n = ge,t and G e,n = g′e,t for each e ∈ E, and Ĝn = gIt,t. The following lemma, shows that the deviation of the true cumulative gain from the estimated cumulative gain is of the order of n. The proof is a modification of [8, Lemma 6.7]. Lemma 1 For any δ ∈ (0, 1), 0 ≤ β < 1 and e ∈ E we have Ge,n > G e,n + Proof. Fix e ∈ E. For any u > 0 and c > 0, by the Chernoff bound we have P[Ge,n > G e,n + u] ≤ e−cuEec(Ge,n−G e,n) . (1) Letting u = ln(|E|/δ)/β and c = β, we get e−cuEec(Ge,n−G e,n) = e− ln(|E|/δ)Eeβ(Ge,n−G e,n) = Eeβ(Ge,n−G e,n) , so it suffices to prove that Eeβ(Ge,n−G e,n) ≤ 1 for all n. To this end, introduce Zt = e β(Ge,t−G e,t) . Below we show that Et[Zt] ≤ Zt−1 for t ≥ 2 where Et denotes the conditional expectation E[·|I1, . . . , It−1] . Clearly, Zt = Zt−1 exp ge,t − 1{e∈It}ge,t + β Taking conditional expectations, we obtain Et[Zt] = Zt−1Et ge,t − 1{e∈It}ge,t + β = Zt−1e qe,t Et ge,t − 1{e∈It}ge,t ≤ Zt−1e qe,t Et 1 + β ge,t − 1{e∈It}ge,t ge,t − 1{e∈It}ge,t = Zt−1e qe,t Et 1 + β2 ge,t − 1{e∈It}ge,t ≤ Zt−1e qe,t Et 1 + β2 1{e∈It}ge,t ≤ Zt−1e ≤ Zt−1. (4) Here (2) holds since β ≤ 1, ge,t − 1{e∈It} ≤ 1 and ex ≤ 1 + x + x2 for x ≤ 1. (3) follows from 1{e∈It} = ge,t. Finally, (4) holds by the inequality 1 + x ≤ ex. Taking expectations on both sides proves E[Zt] ≤ E[Zt−1]. A similar argument shows that E[Z1] ≤ 1, implying E[Zn] ≤ 1 as desired. ✷ Proof of Theorem 1. As usual in the analysis of exponentially weighted average forecasters, we start with bounding the quantity ln Wn . On the one hand, we have the lower bound i,n − lnN ≥ ηmax G′i,n − lnN . (5) To derive a suitable upper bound, first notice that the condition η ≤ γ 2K|C| implies ηg′ i,t ≤ 1 for all i and t, since ηg′i,t = η g′e,t ≤ η 1 + β ≤ ηK(1 + β)|C| where the second inequality follows because qe,t ≥ γ/|C| for each e ∈ E. Therefore, using the fact that ex ≤ 1 + x+ x2 for all x ≤ 1, for all t = 1, 2, . . . we have W t−1 wi,t−1 W t−1 pi,t − γ|C|1{i∈C} pi,t − γ|C|1{i∈C} 1 + ηg′i,t + η ηg′i,t + η pi,tg i,t + pi,tg i,t (7) where (6) follows form the definition of pi,t, and (7) holds by the inequality ln(1 + x) ≤ x for all x > −1. Next we bound the sums in (7). On the one hand, pi,tg i,t = g′e,t = g′e,t i∈P:e∈i g′e,tqe,t = gIt,t + |E|β. On the other hand, pi,tg i,t = g′e,t pi,tK i∈P:e∈i e,tqe,t qe,tg β + 1{e∈It}ge,t ≤ K(1 + β) g′e,t where the first inequality is due to the inequality between the arithmetic and quadratic mean, and the second one holds because ge,t ≤ 1. Therefore, W t−1 (gIt,t + |E|β) + η2K(1 + β) g′e,t . Summing for t = 1, . . . , n, we obtain Ĝn + n|E|β η2K(1 + β) G′e,n Ĝn + n|E|β η2K(1 + β) |C|max G′i,n (8) where the second inequality follows since e∈E G e,n ≤ i∈C G i,n. Combining the upper bound with the lower bound (5), we obtain Ĝn ≥ (1− γ − ηK(1 + β)|C|)max G′i,n − lnN − n|E|β. (9) Now using Lemma 1 and applying the union bound, for any δ ∈ (0, 1) we have that, with probability at least 1− δ, Ĝn ≥ (1− γ − ηK(1 + β)|C|) Gi,n − − 1− γ lnN − n|E|β , where we used 1− γ − ηK(1 + β)|C| ≥ 0 which follows from the assumptions of the theorem. Since Ĝn = Kn− L̂n and Gi,n = Kn− Li,n for all i ∈ P, we have L̂n ≤ Kn (γ + η(1 + β)K|C|) + (1− γ − η(1 + β)K|C|)min + (1− γ − η(1 + β)K|C|) lnN + n|E|β with probability at least 1− δ. This implies L̂n −min Li,n ≤ Knγ + η(1 + β)nK2|C|+ lnN + n|E|β ≤ Knγ + 2ηnK2|C|+ K + n|E|β with probability at least 1− δ, which is the first statement of the theorem. Setting and γ = 2ηK|C| results in the inequality L̂n −min Li,n ≤ 4ηnK2|C|+ nK|E| ln which holds with probability at least 1 − δ if n ≥ (K/|E|) ln(|E|/δ) (to ensure β ≤ 1). Finally, setting 4nK2|C| yields the last statement of the theorem (n ≥ 4 lnN |C| is required to ensure γ ≤ 1/2). ✷ Next we analyze the computational complexity of the algorithm. The next result shows that the algorithm is feasible as its complexity is linear in the size (number of edges) of the graph. Theorem 2 The proposed algorithm can be implemented efficiently with time complexity O(n|E|) and space complexity O(|E|). Proof. The two complex steps of the algorithm are steps (a) and (b), both of which can be computed, similarly to Takimoto and Warmuth [28], using dynamic programming. To perform these steps efficiently, first we order the vertices of the graph. Since we have an acyclic directed graph, its vertices can be labeled (in O(|E|) time) from 1 to |V | such that u = 1, v = |V |, and if (v1, v2) ∈ E, then v1 < v2. For any pair of vertices u1 < v1 let Pu1,v1 denote the set of paths from u1 to v1, and for any vertex s ∈ V , let Ht(s) = i∈Ps,v Ĥt(s) = i∈Pu,s we,t . Given the edge weights {we,t}, Ht(s) can be computed recursively for s = |V | − 1, . . . , 1, and Ĥt(s) can be computed recursively for s = 2, . . . , |V | in O(|E|) time (letting Ht(v) = Ĥt(u) = 1 by definition). In step (a), first one has to decide with probability γ whether It is generated according to the graph weights, or it is chosen uniformly from C. If It is to be drawn according to the graph weights, it can be shown that its vertices can be chosen one by one such that if the first k vertices of It are v0 = u, v1, . . . , vk−1, then the next vertex of It can be chosen to be any vk > vk−1, satisfying (vk−1, vk) ∈ E, with probability w(vk−1,vk),t−1Ht−1(vk)/Ht−1(vk−1). The other computationally demanding step, namely step (b), can be performed easily by noting that for any edge (v1, v2), q(v1,v2),t = (1− γ) Ĥt−1(v1)w(v1,v2),t−1Ht−1(v2) Ht−1(u) |{i ∈ C : (v1, v2) ∈ i}| as desired. ✷ 5 A combination of the label efficient and bandit settings In this section we investigate a combination of the multi-armed bandit and the label efficient problems. This means that the decision maker only has access to the loss of the chosen path upon request and the total number of requests must be bounded by a constant m. This combination is motivated by some applications, in which feedback information is costly to obtain. In the general label efficient decision problem, after taking an action, the decision maker has the option to query the losses of all possible actions. For this problem, Cesa-Bianchi et al. [9] proved an upper bound on the normalized regret of order O(K ln(4N/δ)/(m)) which holds with probability at least 1− δ. Our model of the label-efficient bandit problem for shortest paths is motivated by an application to a particular packet switched network model. This model, called the cognitive packet network, was introduced by Gelenbe et al. [11, 12]. In these networks a particular type of packets, called smart packets, are used to explore the network (e.g., the delay of the chosen path). These packets do not carry any useful data; they are merely used for exploring the network. The other type of packets are the data packets, which do not collect any information about their paths. The task of the decision maker is to send packets from the source to the destination over routes with minimum average transmission delay (or packet loss). In this scenario, smart packets are used to query the delay (or loss) of the chosen path. However, as these packets do not transport information, there is a tradeoff between the number of queries and the usage of the network. If data packets are on the average α times larger than smart packets (note that typically α ≫ 1) and ǫ is the proportion of time instances when smart packets are used to explore the network, then ǫ/(ǫ+ α(1− ǫ)) is the proportion of the bandwidth sacrificed for well informed routing decisions. We study a combined algorithm which, at each time slot t, queries the loss of the chosen path with probability ǫ (as in the solution of the label efficient problem proposed in [9]), and, similarly to the multi-armed bandit case, computes biased estimates g′ i,t of the true gains gi,t. Just as in the previous section, it is assumed that each path of the graph is of the same length K. The algorithm differs from our bandit algorithm of the previous section only in step (c), which is modified in the spirit of [9]. The modified step is given below: MODIFIED STEP FOR THE LABEL EFFICIENT BANDIT ALGORITHM FOR SHORTEST PATHS (c’) Draw a Bernoulli random variable St with P(St = 1) = ǫ, and compute the estimated gains g′e,t = ge,t+β ǫqe,t St if e ∈ It ǫqe,t St if e /∈ It . The performance of the algorithm is analyzed in the next theorem, which can be viewed as a combination of Theorem 1 in the preceding section and Theorem 2 of [9]. Theorem 3 For any δ ∈ (0, 1), ǫ ∈ (0, 1], parameters η = ǫ lnN 4nK2|C| , γ = 2ηK|C| ≤ 1/2, and n|E|ǫ ≤ 1, and for all K2 ln2(2|E|/δ) |E| lnN |E| ln(2|E|/δ) , 4|C| lnN the performance of the algorithm defined above can be bounded, with probability at least 1− δ, as L̂n −min K|C| lnN + 5 |E| ln 8K ln |E| ln 2N If ǫ is chosen as (m − 2m ln(1/δ))/n then, with probability at least 1 − δ, the total number of queries is bounded by m (see [8, Lemma 6.1]) and the performance bound above is of the order |E| ln(N/δ)/m. Similarly to Theorem 1, we need a lemma which reveals the connection between the true and the estimated cumulative losses. However, here we need a more careful analysis because the “shifting term” ǫqe,t St, is a random variable. Lemma 2 For any 0 < δ < 1, 0 < ǫ ≤ 1, for any n ≥ 1 K2 ln2(2|E|/δ) |E| lnN K ln(2|E|/δ) parameters 2ηK|C| ≤ γ, η = ǫ lnN 4nK2|C| and β = n|E|ǫ ≤ 1 , and e ∈ E, we have Ge,n > G e,n + Proof. Fix e ∈ E. Using (1) with u = 4 and c = , it suffices to prove for all n that ec(Ge,n−G ≤ 1 . Similarly to Lemma 1 we introduce Zt = e c(Ge,t−G e,t) and we show that Z1, . . . , Zn is a super- martingale, that is Et[Zt] ≤ Zt−1 for t ≥ 2 where Et denotes E[·|(I1, S1), . . . , (I t−1, St−1)]. Taking conditional expectations, we obtain Et[Zt] = Zt−1Et ge,t− 1{e∈It} Stge,t+Stβ qe,tǫ ≤ Zt−1Et 1 + c ge,t − 1{e∈It}Stge,t + Stβ qe,tǫ ge,t − 1{e∈It}Stge,t + Stβ qe,tǫ . (10) Since ge,t − 1{e∈It}Stge,t + Stβ qe,tǫ = − β ge,t − 1{e∈It}Stge,t qe,tǫ 1{e∈It}Stge,t qe,tǫ qe,tǫ we get from (10) that Et[Zt] ≤ Zt−1Et 1− cβ qe,tǫ 21{e∈It}Stge,tβ q2e,tǫ 2ge,tStβ qe,tǫ q2e,tǫ ≤ Zt−1 qe,tǫ . (11) Since c = βǫ/4 we have − β + c qe,tǫ = −3β qe,tǫ 4qe,t 4qe,t β3|C| ≤ 0, (13) where (12) follows from qe,t ≥ γ|C| and (13) holds by β2|C| ≤ 1 , and the last inequality is ensured by n ≥ K 2 ln2(2|E|/δ) ǫ|E| lnN , the assumption of the lemma. Combining (11) and (13) we get that Et[Zt] ≤ Zt−1. Taking expectations on both sides of the inequality, we get E[Zt] ≤ E[Zt−1] and since E[Z1] ≤ 1, we obtain E[Zn] ≤ 1 as desired. ✷ Proof of Theorem 3. The proof of the theorem is a generalization of that of Theorem 1, and follows the same lines with some extra technicalities to handle the effects of the modified step (c’). Therefore, in the following we emphasize only the differences. First note that (5) and (7) also hold in this case. Bounding the sums in (7), one obtains pi,tg i,t = (gIt,t + |E|β) and ∑ pi,tg i,t ≤ K(1 + β) g′e,t . Plugging these bounds into (7) and summing for t = 1, . . . , n, we obtain (gIt,t + |E|β ) + η2K(1 + β) (1− γ)ǫ |C|max G′i,n . Combining the upper bound with the lower bound (5), we obtain (gIt,t + |E|β ) ≥ 1−γ− ηK(1 + β)|C| G′i,n− . (14) To relate the left-hand side of the above inequality to the real gain t=1 gIt,t, notice that (gIt,t + |E|β) − (gIt,t + |E|β) is a martingale difference sequence with respect to (I1, S1), (I2, S2), . . .. Now for all t = 1, . . . , n, we have the bound X2t |(I1, S1), . . . , (It−1, St−1) (gIt,t + |E|β)2 ∣∣∣∣(I1, S1), . . . , (I t−1, St−1) ≤ (K + |E|β) = σ2, (15) where (15) holds by n ≥ |E| ln(2|E|/δ) (to ensure β|E| ≤ K). We know that for all t. Now apply Bernstein’s inequality for martingale differences (see Lemma 9 in the Appendix) to obtain Xt > u , (16) where From (16) we get (gIt,t + |E|β) ≥ Ĝn + βn|E|+ u . (17) Now Lemma 2, the union bound, and (17) combined with (14) yield, with probability at least 1− δ, Ĝn ≥ 1− γ − ηK(1 + β)|C| Gi,n − − βn|E| − u since the coefficient of G′i,n is greater than zero by the assumptions of the theorem. Since Ĝn = Kn− L̂n and Gi,n = Kn− Li,n, we have L̂n ≤ 1− γ − K(1 + β)η|C| Li,n +Kn K(1 + β)η|C| 1− γ − K(1 + β)η|C| + βn|E|+ ≤ min Li,n +Kn K(1 + β)η|C| + 5βn|E|+ + u , where we used the fact that K = βn|E|. Substituting the value of β, η and γ, we have L̂n −min Li,n ≤Kn 2Kη|C| 2Kη|C| + 5βn|E|+ u n|C| lnN n|E|K ln(2|E|/δ) K|C| lnN + 5 |E| ln(2|E|/δ) + 8K ln (2/δ) ln (2/δ) as desired. ✷ 6 A bandit algorithm for tracking the shortest path Our goal in this section is to extend the bandit algorithm so that it is able to compete with time- varying paths under small computational complexity. This is a variant of the problem known as tracking the best expert ; see, for example, Herbster and Warmuth [19], Vovk [30], Auer and Warmuth [2], Bousquet and Warmuth [6], Herbster and Warmuth [20]. To describe the loss the decision maker is compared to, consider the following “m-partition” prediction scheme: the sequence of paths is partitioned into m+1 contiguous segments, and on each segment the scheme assigns exactly one of the N paths. Formally, an m-partition Part(n,m, t, i) of the n paths is given by an m-tuple t = (t1, . . . , tm) such that t0 = 1 < t1 < · · · < tm < n+1 = tm+1, and an (m+ 1)-vector i = (i0, . . . , im) where ij ∈ P. At each time instant t, tj ≤ t < tj+1, path ij is used to predict the best path. The cumulative loss of a partition Part(n,m, t, i) is L(Part(n,m, t, i)) = tj+1−1∑ ℓij ,t = tj+1−1∑ ℓe,t. The goal of the decision maker is to perform as well as the best time-varying path (partition), that is, to keep the normalized regret L̂n −min L(Part(n,m, t, i)) as small as possible (with high probability) for all possible outcome sequences. In the “classical” tracking problem there is a relatively small number of “base” experts and the goal of the decision maker is to predict as well as the best “compound” expert (i.e., time-varying expert). However in our case, base experts correspond to all paths of the graph between source and destination whose number is typically exponentially large in the number of edges, and therefore we cannot directly apply the computationally efficient methods for tracking the best expert. György, Linder, and Lugosi [15] develop efficient algorithms for tracking the best expert for certain large and structured classes of base experts, including the shortest path problem. The purpose of the following algorithm is to extend the methods of [15] to the bandit setting when the forecaster only observes the losses of the edges on the chosen path. A BANDIT ALGORITHM FOR TRACKING SHORTEST PATHS Parameters: real numbers β > 0, 0 < η, γ < 1, 0 ≤ α ≤ 1. Initialization: Set we,0 = 1 for each e ∈ E, wi,0 = 1 for each i ∈ P, and W 0 = N . For each round t = 1, 2, . . . (a) Choose a path It according to the distribution pt defined by pi,t = (1− γ)wi,t−1 W t−1 if i ∈ C; (1− γ)wi,t−1 W t−1 if i 6∈ C. (b) Compute the probability of choosing each edge e as qe,t = i:e∈i pi,t = (1− γ) i:e∈iwi,t−1 W t−1 |{i ∈ C : e ∈ i}| (c) Calculate the estimated gains g′e,t = ge,t+β if e ∈ It; otherwise. (d) Compute the updated weights vi,t = wi,t−1e wi,t = (1− α)vi,t + where g′ i,t = e∈i g e,t and W t is the sum of the total weights of the paths, that W t = vi,t = wi,t. The following performance bounds shows that the normalized regret with respect to the best time-varying path which is allowed to switch pathsm times is roughly of the order ofK (m/n)|C| lnN . Theorem 4 For any δ ∈ (0, 1) and parameters 0 ≤ γ < 1/2, α, β ∈ [0, 1], and η > 0 satisfying 2ηK|C| ≤ γ, the performance of the algorithm defined above can be bounded, with probability at least 1− δ, as L̂n −min L(Part(n,m, t, i)) ≤ Kn (γ + η(1 + β)K|C|) + K(m+ 1) |E|(m+ 1) + βn|E|+ 1 αm(1− α)n−m−1 In particular, choosing K(m+ 1) |E|(m+ 1) , γ = 2ηK|C|, α = (m+ 1) lnN +m ln e(n−1) 4nK2|C| we have, for all n ≥ max K(m+1) |E|(m+1) , 4|C|D L̂n −min L(Part(n,m, t, i)) ≤ 2 4K|C|D + |E|(m + 1) ln |E|(m + 1) where D = (m+ 1) lnN +m 1 + ln The proof of the theorem is a combination of that of our Theorem 1 and Theorem 1 of [15]. We will need the following three lemmas. Lemma 3 For any 1 ≤ t ≤ t′ ≤ n and any i ∈ P, vi,t′ wi,t−1 ≥ eηG i,[t,t′](1− α)t′−t where G′ i,[t,t′] τ=t g i,τ . Proof. The proof is a straightforward modification of the one in Herbster and Warmuth [19]. From the definitions of vi,t and wi,t (see step (d) of the algorithm) it is clear that for any τ ≥ 1, wi,τ = (1− α)vi,τ + W τ ≥ (1− α)eηg i,τwi,τ−1 . Applying this equation iteratively for τ = t, t+1, . . . , t′ − 1, and the definition of vi,t (step (d)) for τ = t′, we obtain vi,t′ = wi,t′−1e i,t′ ≥ eηg t′−1∏ (1− α)eηg wi,t−1 i,[t,t′](1− α)t′−twi,t−1 which implies the statement of the lemma. ✷ Lemma 4 For any t ≥ 1 and i, j ∈ P, we have Proof. By the definition of wi,t we have wi,t = (1− α)vi,t + W t ≥ W t ≥ vj,t . This completes the proof of the lemma. ✷ The next lemma is a simple corollary of Lemma 1. Lemma 5 For any δ ∈ (0, 1), 0 ≤ β ≤ 1, t ≥ 1 and e ∈ E we have Ge,t > G e,t + |E|(m+ 1) |E|(m+ 1) Proof of Theorem 4. The theorem is proved the same way as Theorem 1 until (8), that is, Ĝn + n|E|β η2K(1 + β) |C|max G′i,n . (18) Let Part(n,m, t, i) be an arbitrary partition. Then the lower bound is obtained as vim,n (recall that im denotes the path used in the last segment of the partition). Now vim,n can be rewritten in the form of the following telescoping product vim,n = wi0,t0−1 vi0,t1−1 wi0,t0−1 wij ,tj−1 vij−1,tj−1 vij ,tj+1−1 wij ,tj−1 Therefore, applying Lemmas 3 and 4, we have vim,n ≥ wi0,t0−1 )m m∏ (1− α)tj+1−1−tjeηG ij ,[tj ,tj+1−1] ′(Part(n,m,t,i))(1− α)n−m−1. Combining the lower bound with the upper bound (18), we have αm(1− α)n−m−1 ηG′(Part(n,m, t, i)) Ĝn + n|E|β η2K(1+β) |C|maxi∈P G′i,n , where we used the fact that Part(n,m, t, i) is an arbitrary partition. After rearranging and using maxi∈P G i,n ≤ maxt,iG′(Part(n,m, t, i)) we get Ĝn ≥ (1− γ − ηK(1 + β)|C|) max G′(Part(n,m, t, i)) −n|E|β − 1− γ αm(1− α)n−m−1 Now since 1 − γ − ηK(1 + β)|C| ≥ 0, by the assumptions of the theorem and from Lemma 5 with an application of the union bound we obtain that, with probability at least 1− δ, Ĝn ≥ (1− γ − ηK(1 + β)|C|) G(Part(n,m, t, i))− K(m+ 1) |E|(m+ 1) − n|E|β − 1− γ αm(1− α)n−m−1 Since Ĝn = Kn− L̂n and G(Part(n,m, t, i)) = Kn− L(Part(n,m, t, i)), we have L̂n ≤ (1− γ − ηK(1 + β)|C|) min L(Part(n,m, t, i)) +Kn (γ + η(1 + β)K|C|) + (1− γ − η(1 + β)K|C|) K(m+ 1) |E|(m+ 1) + n|E|β αm(1− α)n−m−1 This implies that, with probability at least 1− δ, L̂n −min L(Part(n,m, t, i)) ≤ Kn (γ + η(1 + β)K|C|) + K(m+ 1) |E|(m+ 1) + n|E|β + 1 αm(1− α)n−m−1 . (20) To prove the second statement, let H(p) = −p ln p − (1 − p) ln(1 − p) and D(p ‖ q) = p ln p (1− p) ln 1−p . Optimizing the value of α in the last term of (20) gives αm(1− α)n−m−1 (m+ 1) ln (N) +m ln + (n−m− 1) ln (m+ 1) ln (N) + (n − 1)(Db(α∗ ‖ α) +Hb(α∗)) where α∗ = m . For α = α∗ we obtain αm(1− α)n−m−1 ((m+ 1) ln (N) + (n− 1)(Hb(α∗))) ((m+ 1) ln (N) +m ln((n − 1)/m) +(n−m− 1) ln(1 +m/(n−m− 1))) ((m+ 1) ln (N) +m ln((n − 1)/m) +m) (m+ 1) ln (N) +m ln e(n− 1) where the inequality follows since ln(1 + x) ≤ x for x > 0. Therefore L̂n −min L(Part(n,m, t, i)) ≤ Kn (γ + η(1 + β)K|C|) + K(m+ 1) |E|(m+ 1) + n|E|β + which is the first statement of the theorem. Setting K(m+ 1) |E|(m+ 1) , γ = 2ηK|C|, and η = 4nK2|C| results in the second statement of the theorem, that is, L̂n −min L(Part(n,m, t, i)) 4K|C|D + |E|(m+ 1) ln |E|(m+ 1) Similarly to [15], the proposed algorithm has an alternative version, which is efficiently com- putable: AN ALTERNATIVE BANDIT ALGORITHM FOR TRACKING SHORTEST PATHS For t = 1, choose I1 uniformly from the set P. For t ≥ 2, (a) Draw a Bernoulli random variable Γt with P(Γt = 1) = γ. (b) If Γt = 1, then choose It uniformly from C. (c) If Γt = 0, (c1) choose τt randomly according to the distribution P{τt = t′} = (1−α)t−1Z1,t−1 for t′ = 1 α(1−α)t−t Wt′Zt′,t−1 for t′ = 2, . . . , t where Zt′,t−1 = i∈P e i,[t′,t−1] for t′ = 1, . . . , t− 1, and Zt,t−1 = N ; (c2) given τt = t ′, choose It randomly according to the probabilities P{It = i|τt = t′} = i,[t′,t−1] Zt′,t−1 for t′ = 1, . . . , t− 1 for t′ = t. In a way completely analogous to [15], in this alternative formulation of the algorithm one can compute the probabilities P{It = i|τt = t′} and the normalization factors Zt′,t−1 efficiently. Using the fact that the baseline bandit algorithm for shortest paths has an O(n|E|) time complexity by Theorem 2, it follows from Theorem 3 of [15] that the time complexity of the alternative bandit algorithm for tracking the shortest path is O(n2|E|). 7 An algorithm for the restricted multi-armed bandit problem In this section we consider the situation where the decision maker receives information only about the performance of the whole chosen path, but the individual edge losses are not available. That is, the forecaster has access to the sum ℓIt,t of losses over the chosen path It but not to the losses {ℓe,t}e∈It of the edges belonging to It. This is the problem formulation considered by McMahan and Blum [25] and Awerbuch and Kleinberg [4]. McMahan and Blum provided a relatively simple algorithm whose regret is at most of the order of n−1/4, while Awerbuch and Kleinberg gave a more complex algorithm to achieve O(n−1/3) regret. In this section we combine the strengths of these papers, and propose a simple algorithm with regret at most of the order of n−1/3. Moreover, our bound holds with high probability, while the above-mentioned papers prove bounds for the expected regret only. The proposed algorithm uses ideas very similar to those of McMahan and Blum [25]. The algorithm alternates between choosing a path from a “basis” B to obtain unbiased estimates of the loss (exploration step), and choosing a path according to exponential weighting based on these estimates. A simple way to describe a path i ∈ P is a binary row vector with |E| components which are indexed by the edges of the graph such that, for each e ∈ E, the eth entry of the vector is 1 if e ∈ i and 0 otherwise. With a slight abuse of notation we will also denote by i the binary row vector representing path i. In the previous sections, where the loss of each edge along the chosen path is available to the decision maker, the complexity stemming from the large number of paths was reduced by representing all information in terms of the edges, as the set of edges spans the set of paths. That is, the vector corresponding to a given path can be expressed as the linear combination of the unit vectors associated with the edges (the eth component of the unit vector representing edge e is 1, while the other components are 0). While the losses corresponding to such a spanning set are not observable in the restricted setting of this section, one can choose a subset of P that forms a basis, that is, a collection of b paths which are linearly independent and each path in P can be expressed as a linear combination of the paths in the basis. We denote by B the b × |E| matrix whose rows b1, . . . , bb represent the paths in the basis. Note that b is equal to the maximum number of linearly independent vectors in {i : i ∈ P}, so b ≤ |E|. Let ℓ t denote the (column) vector of the edge losses {ℓe,t}e∈E at time t, and let ℓ (ℓb1,t, . . . , ℓbb,t) T be a b-dimensional column vector whose components are the losses of the paths in the basis B at time t. If α (i,B) , . . . , α (i,B) are the coefficients in the linear combination of the basis paths expressing path i ∈ P, that is, i = j=1 α (i,B) j, then the loss of path i ∈ P at time t is given by ℓi,t = 〈i, ℓ t 〉 = (i,B) 〈bj, ℓ(E)t 〉 = (i,B) bj ,t (21) where 〈·, ·〉 denotes the standard inner product in R|E|. In the algorithm we obtain estimates ℓ̃ bj ,t of the losses of the basis paths and use (21) to estimate the loss of any i ∈ P as ℓ̃i,t = (i,B) bj ,t . (22) It is algorithmically advantageous to calculate the estimated path losses ℓ̃i,t from an intermediate estimate of the individual edge losses. LetB+ denote the the Moore-Penrose inverse ofB defined by B+ = BT (BBT )−1, where BT denotes the transpose of B and BBT is invertible since the rows of B are linearly independent. (Note thatB+ = B−1 if b = |E|). Then letting ℓ̃(B)t = (ℓ̃b1,t, . . . , ℓ̃bb,t) t = B it is easy to see that ℓ̃i,t in (22) can be obtained as ℓ̃i,t = 〈i, ℓ̃ t 〉, or equivalently ℓ̃i,t = ℓ̃e,t. This form of the path losses allows for an efficient implementation of exponential weighting via dynamic programming [28]. To analyze the algorithm we need an upper bound on the magnitude of the coefficients α (i,B) For this, we invoke the definition of a barycentric spanner from [4]: the basis B is called a C- barycentric spanner if |α(i,B) | ≤ C for all i ∈ P and j = 1, . . . , b. Awerbuch and Kleinberg [4] show that a 1-barycentric spanner exists if B is a square matrix (i.e., b = |E|) and give a low-complexity algorithm which finds a C-barycentric spanner for C > 1. We use their technique to show that a 1-barycentric spanner also exists in case of a non-square B, when the basis is chosen to maximize the absolute value of the determinant of BBT . As before, b denotes the maximum number of linearly independent vectors (paths) in P. Lemma 6 For a directed acyclic graph, the set of paths P between two dedicated nodes has a 1- barycentric spanner. Moreover, let B be a b×|E| matrix with rows from P such that det[BBT ] 6= 0. If B−j,i is the matrix obtained from B by replacing its jth row by i ∈ P and ∣∣det B−j,iB ]∣∣ ≤ C2 ∣∣det ]∣∣ (23) for all j = 1, . . . , b and i ∈ P, then B is a C-barycentric spanner. Proof. Let B be a basis of P with rows b1, . . . , bb ∈ P that maximizes |det[BBT ]|. Then, for any path i ∈ P, we have i = j=1 α (i,B) j for some coefficients {α(i,B) }. Now for the matrix B−1,i = [i T , (b2)T , . . . , (bb)T ]T we have ∣∣det B−1,iB ∣∣∣det B−1,ii T ,B−1,i(b 2)T ,B−1,i(b 3)T , . . . ,B−1,i(b ∣∣∣∣∣∣∣ (i,B) B−1,ib ,B−1,i(b 2)T ,B−1,i(b 3)T , . . . ,B−1,i(b ∣∣∣∣∣∣∣ ∣∣∣∣∣∣ (i,B) B−1,i(b j)T ,B−1,i(b 2)T ,B−1,i(b 3)T , . . . ,B−1,i(b ∣∣∣∣∣∣ = |α(i,B) ∣∣det B−1,iB (i,B) )2 ∣∣det where last equality follows by the same argument the penultimate equality was obtained. Repeating the same argument for B−j,i, j = 2, . . . , b we obtain ∣∣det B−j,iB ]∣∣ = (i,B) )2 ∣∣det ]∣∣ . (24) Thus the maximal property of |det[BBT ]| implies |α(i,B) | ≤ 1 for all j = 1, . . . , b. The second statement follows trivially from (23) and (24). ✷ Awerbuch and Kleinberg [4] also present an iterative algorithm to find a C-barycentric spanner if B is a square matrix. Starting from the identity matrix, their algorithm replaces a row of the matrix in each step by maximizing the determinant with respect to the given row. This is done by calling an oracle function, and it is shown that the oracle is called O(b logC b) times. In case B is not a square matrix, the algorithm carries over if we have access to an alternative oracle that can maximize |det[BBT ]|: Starting from an arbitrary basis B we can iteratively replace one row in each step, using the oracle, to maximize the determinant |det[BBT ]| until (23) is satisfied for all j and i. By Lemma 6, this results in a C-barycentric spanner. Similarly to [4], it can be shown that the oracle is called O(b logC b) times for C > 1. For simplicity (to avoid carrying the constant C), assume that we have a 2-barycentric spanner B. Based on the ideas of label efficient prediction, the next algorithm gives a simple solution to the restricted shortest path problem. The algorithm is very similar to that of the algorithm in the label efficient case, but here we cannot estimate the edge losses directly. Therefore, we query the loss of a (random) basis vector from time to time, and create unbiased estimates ℓ̃ bj ,t of the losses of basis paths ℓ bj ,t, which are then transformed into edge-loss estimates. A BANDIT ALGORITHM FOR THE RESTRICTED SHORTEST PATH PROBLEM Parameters: 0 < ǫ, η ≤ 1. Initialization: Set we,0 = 1 for each e ∈ E, wi,0 = 1 for each i ∈ P, W 0 = N . Fix a basis B, which is a 2-barycentric spanner. For each round t = 1, 2, . . . (a) Draw a Bernoulli random variable St such that P(St = 1) = ǫ; (b) If St = 1, then choose the path It uniformly from the basis B. If St = 0, then choose It according to the distribution {pi,t}, defined by pi,t = wi,t−1 W t−1 (c) Calculate the estimated loss of all edges according to t = B where ℓ̃ t = {ℓ̃ e,t }e∈E , and ℓ̃ t = (ℓ̃ , . . . , ℓ̃ ) is the vector of the estimated losses bj ,t = bj ,t1{It=b for j = 1, . . . , b. (d) Compute the updated weights we,t = we,t−1e −ηℓ̃e,t , wi,t = we,t = wi,t−1e e∈i ℓ̃e,t , and the sum of the total weights of the paths W t = wi,t . The performance of the algorithm is analyzed in the next theorem. The proof follows the argu- ment of Cesa-Bianchi et al. [9], but we also have to deal with some additional technical difficulties. Note that in the theorem we do not assume that all paths between u and v have equal length. Theorem 5 Let K denote the length of the longest path in the graph. For any δ ∈ (0, 1), parameters 0 < ǫ ≤ 1 and η > 0 satisfying η ≤ ǫ2, and n ≥ 8b ln 4bN , the performance of the algorithm defined above can be bounded, with probability at least 1− δ, as L̂n −min Li,n ≤ K + nǫ+ 2nǫ ln 4 In particular, choosing and η = ǫ2 we obtain L̂n −min Li,n ≤ 9.1K2b (Kb ln(4bN/δ))1/3 n2/3 . The theorem is proved using the following two lemmas. The first one is an easy consequence of Bernstein’s inequality: Lemma 7 Under the assumptions of Theorem 5, the probability that the algorithm queries the basis more than nǫ+ 2nǫ ln 4 times is at most δ/4. Using the estimated loss of a path i ∈ P given in (22), we can estimate the cumulative loss of i up to time n as L̃i,n = ℓ̃i,t . The next lemma demonstrates the quality of these estimates. Lemma 8 Let 0 < δ < 1 and assume n ≥ 8b ln 4bN . For any i ∈ P, with probability at least 1− δ/4, pi,tℓi,t − pi,tℓ̃i,t ≤ Furthermore, with probability at least 1− δ/(4N), L̃i,n − Li,n ≤ Proof. We may write pi,tℓi,t − pi,tℓ̃i,t = (i,B) bj ,t − ℓ̃bj ,t pi,tα (i,B) bj ,t − ℓ̃bj ,t bj ,t . (25) Note that for any bj, X bj ,t, t = 1, 2, . . . is a martingale difference sequence with respect to (It, St), t = 1, 2, . . . as Etℓ̃b,t = ℓb,t. Also, bj ,t pi,tα (i,B) bj ,t (i,B) )2 K2b bj ,t| ≤ ∣∣∣∣∣ pi,tα (i,B) ∣∣∣∣∣ ∣∣∣ℓbj ,t − ℓ̃bj ,t ∣∣∣ ≤ ∣∣∣α(i,B) where the last inequalities in both cases follow from the fact that B is a 2-barycentric spanner. Then, using Bernstein’s inequality for martingale differences (Lemma 9), we have, for any fixed bj , bj ,t ≥ where we used (26), (27) and the assumption of the lemma on n. The proof of the first statement is finished with an application of the union bound and its combination with (25). For the second statement we use a similar argument, that is, (ℓ̃i,t − ℓi,t) = (i,B) bj ,t − ℓbj ,t) ≤ ∣∣∣α(i,B) ∣∣∣∣∣ bj ,t − ℓbj ,t) ∣∣∣∣∣ ∣∣∣∣∣ bj ,t − ℓbj ,t) ∣∣∣∣∣ . (29) Now applying Lemma 9 for a fixed bj we get bj ,t − ℓbj ,t) ≥ because of Et[(ℓ̃bj ,t − ℓbj ,t)2] ≤ and −K ≤ ℓ̃ bj ,t − ℓbj ,t ≤ K . The proof is completed by applying the union bound to (30) and combining the result with (29). ✷ Proof of Theorem 5. Similarly to earlier proofs, we follow the evolution of the term ln Wn the same way as we obtained (5) and (7), we have ≥ −ηmin L̃i,n − lnN pi,tℓ̃i,t + pi,tℓ̃ Combining these bounds, we obtain L̃i,n − pi,tℓ̃i,t + pi,tℓ̃ −1 + ηKb pi,tℓ̃i,t , because 0 ≤ ℓ̃i,t ≤ 2Kbǫ . Applying the results of Lemma 8 and the union bound, we have, with probability 1− δ/2, Li,n − −1 + ηKb )( n∑ pi,tℓi,t − pi,tℓi,t + . (31) Introduce the sets = {t : 1 ≤ t ≤ n and St = 0} and T n = {t : 1 ≤ t ≤ n and St = 1} of “exploitation” and “exploration” steps, respectively. Then, by the Hoeffding-Azuma inequality [21] we obtain that, with probability at least 1− δ/4, pi,tℓi,t ≥ ℓIt,t − |Tn|K2 Note that for the exploration steps t ∈ T n, as the algorithm plays according to a uniform distribu- tion instead of pi,t, we can only use the trivial lower bound zero on the losses, that is, t∈T n pi,tℓi,t ≥ t∈T n ℓIt,t −K|T n| . The last two inequalities imply pi,tℓi,t ≥ L̂n − |Tn|K2 −K|T n| . (32) Then, by (31), (32) and Lemma 7 we obtain, with probability at least 1− δ, L̂n −min + nǫ+ 2nǫ ln 4 where we used L̂n ≤ Kn and |Tn| ≤ n. Substituting the values of ǫ and η gives L̂n −min Li,n ≤ K2bnǫ+ Knǫ+Knǫ+ Knǫ+ nǫ ≤ 9.1K2bnǫ where we used 2nǫ ln 4 ln 4N = nǫ, and lnN ≤ nǫ (from the assumptions of the theorem). ✷ 8 Simulation results To further investigate our new algorithms, we have conducted some simple simulations. As the main motivation of this work is to improve earlier algorithms in case the number of paths is exponentially large in the number of edges, we tested the algorithms on the small graph shown in Figure 1 (b), which has one of the simplest structures with exponentially many (namely 2|E|/2) paths. The losses on the edges were generated by a sequence of independent and uniform random variables, with values from [0, 1] on the upper edges, and from [0.32, 1] on the lower edges, resulting in a (long-term) optimal path consisting of the upper edges. We ran the tests for n = 10000 steps, with confidence value δ = 0.001. To establish baseline performance, we also tested the EXP3 algorithm of Auer et al. [1] (note that this algorithm does not need edge losses, only the loss of the chosen path). For the version of our bandit algorithm that is informed of the individual edge losses (edge-bandit), we used the simple 2-element cover set of the paths consisting of the upper and lower edges, respectively (other 2-element cover sets give similar performance). For our restricted shortest path algorithm (path-bandit) the basis {uuuuu, uuuul, uuull, uulll, ullll} was used, where u (resp. l) in the kth position denotes the upper (resp. lower) edge connecting vk−1 and vk. In this example the performance of the algorithm appeared to be independent of the actual choice of the basis; however, in general we do not expect this behavior. Two versions of the algorithm of Awerbuch and Kleinberg [4] were also simulated. With its original parameter setting (AwKl), the algorithm did not perform well. However, after optimizing its parameters off-line (AwKl tuned), substantially better performance was achieved. The normalized regret of the above algorithms, averaged over 30 runs, as well as the regret of the fixed paths in the graph are shown in Figure 2. Although all algorithms showed better performance than the bound for the edge-bandit algo- rithm, the latter showed the expected superior performance in the simulations. Furthermore, our algorithm for the restricted shortest path problem outperformed Awerbuch and Kleinberg’s (AwKl) algorithm, while being inferior to its off-line tuned version (AwKl tuned). It must be noted that similar parameter optimization did not improve the performance of our path-bandit algorithm, which showed robust behavior with respect to parameter tuning. 9 Conclusions We considered different versions of the on-line shortest path problem with limited feedback. These problems are motivated by realistic scenarios, such as routing in communication networks, where the vertices do not have all the information about the state of the network. We have addressed the problem in the adversarial setting where the edge losses may vary in an arbitrary way; in particular, they may depend on previous routing decisions of the algorithm. Although this assumption may neglect natural correlation in the loss sequence, it suits applications in mobile ad-hoc networks, where the network topology changes dynamically in time, and also in certain secure networks that has to be able to handle denial of service attacks. Efficient algorithms have been provided for the multi-armed bandit setting and in a combined label efficient multi-armed bandit setting, provided the individual edge losses along the chosen path are revealed to the algorithms. The normalized regrets of the algorithms, compared to the performance of the best fixed path, converge to zero at an O(1/ n) rate as the time horizon n grows to infinity, and increases only polynomially in the number of edges (and vertices) of the graph. Earlier methods for the multi-armed bandit problem either do not have the right O(1/ convergence rate, or their regret increase exponentially in the number of edges for typical graphs. 0 2000 4000 6000 8000 10000 Number of packets edge-bandit path-bandit AwKl tuned bound for edge-bandit Figure 2: Normalized regret of several algorithms for the shortest path problem. The gray dotted lines show the normalized regret of fixed paths in the graph. The algorithm has also been extended so that it can compete with time varying paths, that is, to handle situations when the best path can change from time to time (for consistency, the number of changes must be sublinear in n). In the restricted version of the shortest path problem, where only the losses of the whole paths are revealed, an algorithm with a worse O(n−1/3) normalized regret was provided. This algorithm has comparable performance to that of the best earlier algorithm for this problem [4], however, our algorithm is significantly simpler. Simulation results are also given to assess the practical performance and compare it to the theoretical bounds as well as other competing algorithms. It should be noted that the results are not entirely satisfactory in the restricted version of the problem, as it remains an open question whether the O(1/ n) regret can be achieved without the exponential dependence on the size of the graph. Although we expect that this is the case, we have not been able to construct an algorithm with such a proven performance bound. 10 Appendix Lemma 9 (Bernstein’s inequality for martingale differences [10].) Let X1, . . . ,Xn be a martingale difference sequence such that Xt ∈ [a, b] with probability one (t = 1, . . . , n). Assume that, for all t, X2t |Xt−1, . . . ,X1 ≤ σ2 a.s. Then, for all ǫ > 0, Xt > ǫ 2nσ2+2ǫ(b−a)/3 and therefore 2nσ2 ln δ−1 + 2 ln δ−1(b− a)/3 References [1] P. Auer, N. Cesa-Bianchi, Y. Freund, and R. Schapire. The non-stochastic multi-armed bandit problem. SIAM Journal on Computing, 32(1):48–77, 2002. [2] P. Auer and M. K. Warmuth, “Tracking the best disjunction,” Machine Learning, vol. 32, no. 2, pp. 127–150, 1998. [3] B. Awerbuch, D. Holmer, H. Rubens, and R. Kleinberg. Provably competitive adaptive routing. In Proceedings of IEEE INFOCOM 2005, volume 1, pages 631–641, March 2005. [4] B. Awerbuch and R. D. Kleinberg. Adaptive routing with end-to-end feedback: distributed learning and geometric approaches. In Proceedings of the 36th Annual ACM Symposium on the Theory of Computing, STOC 2004, pages 45–53, Chicago, IL, USA, Jun. 2004. ACM Press. [5] D. Blackwell. An analog of the minimax theorem for vector payoffs. Pacific Journal of Math- ematics, 6:1–8, 1956. [6] O. Bousquet and M. K. Warmuth. Tracking a small set of experts by mixing past posteriors. Journal of Machine Learning Research, 3:363–396, Nov. 2002. [7] N. Cesa-Bianchi, Y. Freund, D. P. Helmbold, D. Haussler, R. Schapire, and M. K. Warmuth. How to use expert advice. Journal of the ACM, 44(3):427–485, 1997. [8] N. Cesa-Bianchi and G. Lugosi. Prediction, Learning, and Games. Cambridge University Press, Cambridge, 2006. [9] N. Cesa-Bianchi, G. Lugosi, and G. Stoltz. Minimizing regret with label efficient prediction. IEEE Trans. Inform. Theory, IT-51:2152–2162, June 2005. [10] D. A. Freedman. On tail probabilities for martingales. Annals of Probability, 3:100–118, Feb. 1975. [11] E. Gelenbe, M. Gellman, R. Lent, P. Liu, and P. Su. Autonomous smart routing for network QoS. In Proceedings of First International Conference on Autonomic Computing, pages 232– 239, New York, May 2004. IEEE Computer Society. [12] E. Gelenbe, R. Lent, and Z. Xhu. Measurement and performance of a cognitive packet network. Journal of Computer Networks, 37:691–701, 2001. [13] A. György, T. Linder, and G. Lugosi. Efficient algorithms and minimax bounds for zero-delay lossy source coding. IEEE Transactions on Signal Processing, 52:2337–2347, Aug. 2004. [14] A. György, T. Linder, and G. Lugosi. A ”follow the perturbed leader”-type algorithm for zero-delay quantization of individual sequences. In Proc. Data Compression Conference, pages 342–351, Snowbird, UT, USA, Mar. 2004. [15] A. György, T. Linder, and G. Lugosi. Tracking the best of many experts. In Proceedings of the 18th Annual Conference on Learning Theory, COLT 2005, pages 204–216, Bertinoro, Italy, Jun. 2005. Springer. [16] A. György, T. Linder, and G. Lugosi. Tracking the best quantizer. In Proceedings of the IEEE International Symposium on Information Theory, pages 1163–1167, Adelaide, Australia, June- July 2005. [17] A. György and Gy. Ottucsák. Adaptive routing using expert advice. The Computer Journal, 49(2):180–189, 2006. [18] J. Hannan. Approximation to Bayes risk in repeated plays. In M. Dresher, A. Tucker, and P. Wolfe, editors, Contributions to the Theory of Games, volume 3, pages 97–139. Princeton University Press, 1957. [19] M. Herbster and M. K. Warmuth. Tracking the best expert. Machine Learning, 32(2):151–178, 1998. [20] M. Herbster and M. K. Warmuth, “Tracking the best linear predictor,” Journal of Machine Learning Research, vol. 1, pp. 281–309, 2001. [21] W. Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58:13–30, 1963. [22] D.P. Helmbold and S. Panizza. Some label efficient learning results. In Proceedings of the 10th Annual Conference on Computational Learning Theory, pages 218–230. ACM Press, 1997. [23] A. Kalai and S Vempala. Efficient algorithms for the online decision problem. In B. Schölkopf and M. Warmuth, editors, Proceedings of the 16th Annual Conference on Learning Theory and the 7th Kernel Workshop, COLT-Kernel 2003, pages 26–40, New York, USA, Aug. 2003. Springer. [24] N. Littlestone and M. K. Warmuth. The weighted majority algorithm. Information and Computation, 108:212–261, 1994. [25] H. B. McMahan and A. Blum. Online geometric optimization in the bandit setting against an adaptive adversary. In Proceedings of the 17th Annual Conference on Learning Theory, COLT 2004, pages 109–123, Banff, Canada, Jul. 2004. Springer. [26] M. Mohri. General algebraic frameworks and algorithms for shortest distance problems. Tech- nical Report 981219-10TM, AT&T Labs Research, 1998. [27] R. E. Schapire and D. P. Helmbold. Predicting nearly as well as the best pruning of a decision tree. Machine Learning, 27:51–68, 1997. [28] E. Takimoto and M. K. Warmuth. Path kernels and multiplicative updates. Journal of Machine Learning Research, 4:773–818, 2003. [29] V. Vovk. Aggregating strategies. In Proceedings of the Third Annual Workshop on Computa- tional Learning Theory, pages 372–383, Rochester, NY, Aug. 1990. Morgan Kaufmann. [30] V. Vovk. Derandomizing stochastic prediction strategies. Machine Learning, 35(3):247–282, Jun. 1999. Introduction The shortest path problem The multi-armed bandit setting A bandit algorithm for shortest paths A combination of the label efficient and bandit settings A bandit algorithm for tracking the shortest path An algorithm for the restricted multi-armed bandit problem Simulation results Conclusions Appendix

0704.1021

Curvature estimates for Weingarten hypersurfaces in Riemannian manifolds

CURVATURE ESTIMATES FOR WEINGARTEN HYPERSURFACES IN RIEMANNIAN MANIFOLDS CLAUS GERHARDT Abstract. We prove curvature estimates for general curvature func- tions. As an application we show the existence of closed, strictly convex hypersurfaces with prescribed curvature F , where the defining cone of F is Γ+. F is only assumed to be monotone, symmetric, homogeneous of degree 1, concave and of class Cm,α, m ≥ 4. Contents 1. Introduction 1 2. Curvature estimates 5 3. Proof of Theorem 1.5 9 References 9 1. Introduction Let N = Nn+1 be a Riemannian manifold, Ω ⊂ N open, connected and precompact, and M ⊂ Ω a closed connected hypersurface with second fun- damental form hij , induced metric gij and principal curvatures κi. M is said to be a Weingarten hypersurface, if, for a given curvature function F , its principal curvatures lie in the convex cone Γ ⊂ Rn in which the curvature function is defined, M is then said to be admissible, and satisfies the equation (1.1) F |M = f where the right-hand side f is a prescribed positive function defined in Ω̄. When proving a priori estimates for solutions of (1.1) the concavity of F plays a central role. As usual we consider F to be defined in a cone Γ as well as on the space of admissible tensors such that (1.2) F (hij) = F (κi). Date: September 2, 2021. 2000 Mathematics Subject Classification. 35J60, 53C21, 53C44, 53C50, 58J05. Key words and phrases. curvature estimates, Weingarten hypersurface, curvature flows. This work has been supported by the Deutsche Forschungsgemeinschaft. http://arxiv.org/abs/0704.1021v2 2 CLAUS GERHARDT Notice that curvature functions are always assumed to be symmetric and if F ∈ Cm,α(Γ ), 2 ≤ m, 0 < α < 1, then F ∈ Cm,α(SΓ ), where SΓ ⊂ T 0,2(M) is the open set of admissible symmetric tensors with respect to the given metric gij . The result is due to Ball, [1], see also [7, Theorem 2.1.8]. The second derivatives of F then satisfy (1.3) F ij,klηijηkl = ∂κi∂κj ηiiηjj + Fi − Fj κi − κj (ηij) 2 ≤ 0 ∀ η ∈ S, where S ⊂ T 0,2(M) is the space of symmetric tensors, if F is concave in Γ , cf. [4, Lemma 1.1]. However, a mere non-positivity of the right-hand side is in general not sufficient to prove a priori estimates for the κi resulting in the fact that only for special curvature functions for which a stronger estimate was known such a priori estimates could be derived and the problem (1.1) solved, if further assumptions are satisfied. Sheng et al. then realized in [9] that the term (1.4) Fi − Fj κi − κj (ηij) was all that is needed to obtain the stronger concavity estimates under certain circumstances. Indeed, if the κi are labelled (1.5) κ1 ≤ · · · ≤ κn, then there holds: 1.1. Lemma. Let F be concave and monotone, and assume κ1 < κn, then (1.6) Fi − Fj κi − κj (ηij) κn − κ1 (Fn − Fi)(ηni) for any symmetric tensor (ηij), where we used coordinates such that gij = δij . Proof. Without loss of generality we may assume that the κi satisfy the strict inequalities (1.7) κ1 < · · · < κn, since these points are dense. The concavity of F implies (1.8) F1 ≥ · · · ≥ Fn, cf. [2, Lemma 2], where (1.9) Fi = the last inequality is the definition of monotonicity. The inequality then follows immediately. � CURVATURE ESTIMATES 3 The right-hand side of inequality (1.6) is exactly the quantity that is needed to balance a bad technical term in the a priori estimate for κn, at least in Riemannian manifolds, as we shall prove. Unfortunately, this doesn’t work in Lorentzian spaces, because of a sign difference in the Gauß equations. The assumptions on the curvature function are very simple. 1.2. Assumption. Let Γ ⊂ Rn be an open, symmetric, convex cone con- taining Γ+ and let F ∈ C m,α(Γ ) ∩ C0(Γ̄ ), m ≥ 4, be symmetric, monotone, homogeneous of degree 1, and concave such that (1.10) F > 0 in Γ (1.11) F |∂Γ = 0. These conditions on the curvature function will suffice. They could have been modified, even relaxed, e.g., by only requiring that logF is concave, but then the condition (1.12) F ijgij ≥ c0 > 0, which automatically holds, if F is concave and homogeneous of degree 1, would have been added, destroying the aesthetic simplicity of Assumption 1.2. Our estimates apply equally well to solutions of an equation as well as to solutions of curvature flows. Since curvature flows encompass equations, let us state the main estimate for curvature flows. Let Ω ⊂ N be precompact and connected, and 0 < f ∈ Cm,α(Ω̄). We consider the curvature flow (1.13) ẋ = −(Φ− f̃)ν x(0) = x0, where Φ is Φ(r) = r and f̃ = f , x0 is the embedding of an initial admissible hypersurface M0 of class C m+2,α such that (1.14) Φ− f̃ ≥ 0 at t = 0, where of course Φ = Φ(F ) = F . We introduce the technical function Φ in the present case only to make a comparison with former results, which all use the notation for the more general flows, easier. We assume that Ω̄ is covered by a Gaussian coordinate system (xα), 0 ≤ 1 ≤ n, such that the metric can be expressed as (1.15) ds̄2 = e2ψ{(dx0)2 + σijdx and Ω̄ is covered by the image of the cylinder (1.16) I × S0 where S0 is a compact Riemannian manifold and I = x 0(Ω̄), x0 is a global coordinate defined in Ω̄ and (xi) are local coordinates of S0. 4 CLAUS GERHARDT Furthermore we assume that M0 and the other flow hypersurfaces can be written as graphs over S0. The flow should exist in a maximal time interval [0, T ∗), stay in Ω, and uniform C1-estimates should already have been established. 1.3. Remark. The assumption on the existence of the Gaussian co- ordinate system and the fact that the hypersurfaces can be written as graphs could be replaced by assuming the existence of a unit vector field η ∈ C2(T 0,1(Ω̄)) and of a constant θ > 0 such that (1.17) 〈η, ν〉 ≥ 2θ uniformly during the flow, since this assumption would imply uniform C1- estimates, which are the requirement that the induced metric can be esti- mated accordingly by controlled metrics from below and above, and because the existence of such a vector field is essential for the curvature estimate. If the flow hypersurfaces are graphs in a Gaussian coordinate system, then such a vector field is given by (1.18) η = (ηα) = e ψ(1, 0, . . . , 0) and the C1-estimates are tantamount to the validity of inequality (1.17). In case N = Rn+1 and starshaped hypersurfaces one could also use the (1.19) 〈x, ν〉, cf. [3, Lemma 3.5]. Then we shall prove: 1.4. Theorem. Under the assumptions stated above the principal curva- tures κi of the flow hypersurfaces are uniformly bounded from above (1.20) κi ≤ c, provided there exists a strictly convex function χ ∈ C2(Ω̄). The constant c only depends on |f |2,Ω, θ, F (1, . . . , 1), the initial data, and the estimates for χ and those of the ambient Riemann curvature tensor in Ω̄. Moreover, the κi will stay in a compact set of Γ . As an application of this estimate our former results on the existence of a strictly convex hypersurface M solving the equation (1.1), [4, 5], which we proved for curvature functions F of class (K), are now valid for curvature functions F satisfying Assumption 1.2 with Γ = Γ+. We are even able to solve the existence problem by using a curvature flow which formerly only worked in case that the sectional curvature of the ambient space was non-positive. CURVATURE ESTIMATES 5 1.5. Theorem. Let F satisfy the assumptions above with Γ = Γ+ and assume that the boundary of Ω has two components (1.21) ∂Ω = M1 ∪ M2, where the Mi are closed, connected strictly convex hypersurfaces of class Cm+2,α, m ≥ 4, which can be written as graphs in a normal Gaussian coordi- nate system covering Ω̄, and where we assume that the normal of M1 points outside of Ω and that of M2 inside. Let 0 < f ∈ C m,α(Ω̄), and assume that M1 is a lower barrier for the pair (F, f) and M2 an upper barrier, then the problem (1.1) has a strictly convex solution M ∈ Cm+2,α provided there exists a strictly convex function χ ∈ C2(Ω̄). The solution is the limit hypersurface of a converging curvature flow. 2. Curvature estimates Let M(t) be the flow hypersurfaces, then their second fundamental form i satisfies the evolution equation, cf. [7, Lemma 2.4.1]: 2.1. Lemma. The mixed tensor h i satisfies the parabolic equation (2.1) i − Φ̇F i;kl = Φ̇F klhrkh i − Φ̇Fhrih rj + (Φ− f̃)hki h − f̃αβx gkj + f̃αν i + Φ̇F kl,rshkl;ih + Φ̈FiF j + 2Φ̇F klR̄αβγδx − Φ̇F klR̄αβγδx rj − Φ̇F klR̄αβγδx + Φ̇F klR̄αβγδν νγxδl h i − Φ̇F R̄αβγδν γxδmg + (Φ− f̃)R̄αβγδν γxδmg + Φ̇F klR̄αβγδ;ǫ{ν mj + ναx Let η be the vector field (1.18), or any vector field satisfying (1.17), and (2.2) ṽ = 〈η, ν〉, then we have: 2.2. Lemma (Evolution of ṽ). The quantity ṽ satisfies the evolution equa- (2.3) ˙̃v − Φ̇F ij ṽij =Φ̇F j ṽ − [(Φ− f̃)− Φ̇F ]ηαβν − 2Φ̇F ijhkjx ηαβ − Φ̇F ijηαβγx − Φ̇F ijR̄αβγδν xδjηǫx − f̃βx k ηαg 6 CLAUS GERHARDT The derivation is elementary, see the proof of the corresponding lemma in the Lorentzian case [7, Lemma 2.4.4]. Notice that ṽ is supposed to satisfy (1.17), hence (2.4) ϕ = − log(ṽ − θ) is well defined and there holds (2.5) ϕ̇− Φ̇F ijϕij = −{ ˙̃v − Φ̇F ij ṽij} ṽ − θ − Φ̇F ijϕiϕj . Finally, let χ be the strictly convex function. Its evolution equation is (2.6) χ̇− Φ̇F ijχij = −[(Φ− f̃)− Φ̇F ]χαν α − Φ̇F ijχαβx ≤ −[(Φ− f̃)− Φ̇F ]χαν α − c0Φ̇F ijgij where c0 > 0 is independent of t. We can now prove Theorem 1.4: Proof of Theorem 1.4. Let ζ and w be respectively defined by ζ = sup{ hijη iηj : ‖η‖ = 1 },(2.7) w = log ζ + ϕ+ λχ,(2.8) where λ > 0 is supposed to be large. We claim that w is bounded, if λ is chosen sufficiently large. Let 0 < T < T ∗, and x0 = x0(t0), with 0 < t0 ≤ T , be a point in M(t0) such that (2.9) sup w < sup{ sup w : 0 < t ≤ T } = w(x0). We then introduce a Riemannian normal coordinate system (ξi) at x0 ∈ M(t0) such that at x0 = x(t0, ξ0) we have (2.10) gij = δij and ζ = h Let η̃ = (η̃i) be the contravariant vector field defined by (2.11) η̃ = (0, . . . , 0, 1), and set (2.12) ζ̃ = hij η̃ gij η̃ ζ̃ is well defined in neighbourhood of (t0, ξ0). Now, define w̃ by replacing ζ by ζ̃ in (2.8); then, w̃ assumes its maximum at (t0, ξ0). Moreover, at (t0, ξ0) we have (2.13) ζ = ḣnn, and the spatial derivatives do also coincide; in short, at (t0, ξ0) ζ̃ satisfies the same differential equation (2.1) as hnn. For the sake of greater clarity, let us therefore treat hnn like a scalar and pretend that w is defined by (2.14) w = log hnn + ϕ+ λχ. CURVATURE ESTIMATES 7 From the equations (2.1), (2.5), (2.6) and (1.6), we infer, by observing the special form of Φ, i.e., Φ(F ) = F , Φ̇ = 1, f̃ = f and using the monotonicity and homgeneity of F (2.15) F = F (κi) = F ( , . . . , 1)κn ≤ F (1, . . . , 1)κn that in (t0, ξ0) (2.16) 0 ≤ − 1 Φ̇F ijhkih ṽ − θ − fhnn + c(θ)Φ̇F ijgij + λc − λc0Φ̇F gij − Φ̇F ϕiϕj + Φ̇F ij(log hnn)i(log h κn − κ1 (Fn − Fi)(h ni; ) 2(hnn) Similarly as in [6, p. 197], we distinguish two cases Case 1. Suppose that (2.17) |κ1| ≥ ǫ1κn, where ǫ1 > 0 is small, notice that the principal curvatures are labelled ac- cording to (1.5). Then, we infer from [6, Lemma 8.3] (2.18) F ijhkih F ijgijǫ (2.19) F ijgij ≥ F (1, . . . , 1), for a proof see e.e., [7, Lemma 2.2.19]. Since Dw = 0, (2.20) D log hnn = −Dϕ− λDχ, we obtain (2.21) Φ̇F ij(log hnn)i(log h n)j = Φ̇F ϕiϕj + 2λΦ̇F ϕiχj + λ χiχj , where (2.22) |ϕi| ≤ c|κi|+ c, as one easily checks. Hence, we conclude that κn is a priori bounded in this case. Case 2. Suppose that (2.23) κ1 ≥ −ǫ1κn, 8 CLAUS GERHARDT then, the last term in inequality (2.16) is estimated from above by (2.24) 1 + ǫ1 (Fn − Fi)(h ni; ) 2(hnn) 1 + 2ǫ1 (Fn − Fi)(h nn; ) 2(hnn) + c(ǫ1)Φ̇ (Fi − Fn)κ where we used the Codazzi equation. The last sum can be easily balanced. The terms in (2.16) containing the derivative of hnn can therefore be esti- mated from above by (2.25) 1− 2ǫ1 1 + 2ǫ1 nn; ) 2(hnn) 1 + 2ǫ1 (h inn; ) 2(hnn) ≤ Φ̇Fn (h inn; ) 2(hnn) = Φ̇Fn‖Dϕ+ λDχ‖ = Φ̇Fn{‖Dϕ‖ 2 + λ2‖Dχ‖2 + 2λ〈Dϕ,Dχ〉}. Hence we finally deduce (2.26) 0 ≤ −Φ̇1 ṽ − θ + cλ2Φ̇Fn(1 + κn)− fκn + λc + (c(θ) − λc0)Φ̇F ijgij Thus, we obtain an a priori estimate (2.27) κn ≤ const, if λ is chosen large enough. Notice that ǫ1 is only subject to the requirement 0 < ǫ1 < 2.3. Remark. Since the initial condition F ≥ f is preserved under the flow, a simple application of the maximum principle, cf. [4, Lemma 5.2], we conclude that the principal curvatures of the flow hypersurfaces stay in a compact subset of Γ . 2.4. Remark. These a priori estimates are of course also valid, if M is a stationary solution. CURVATURE ESTIMATES 9 3. Proof of Theorem 1.5 We consider the curvature flow (1.13) with initial hypersurface M0 = M2. The flow will exist in a maximal time interval [0, T ∗) and will stay in Ω̄. We shall also assume that M2 is not already a solution of the problem for otherwise the flow will be stationary from the beginning. Furthermore, the flow hypersurfaces can be written as graphs (3.1) M(t) = graphu(t, ·) over S0, since the initial hypersurface has this property and all flow hypersur- faces are supposed to be convex, i.e., uniform C1-estimates are guaranteed, cf. [4]. The curvature estimates from Theorem 1.4 ensure that the curvature op- erator is uniformly elliptic, and in view of well-known regularity results we then conclude that the flow exists for all time and converges in Cm+2,β(S0) for some 0 < β ≤ α to a limit hypersurface M , that will be a stationary solution, cf. [8, Section 6]. References [1] J. M. Ball, Differentiability properties of symmetric and isotropic functions, Duke Math. J. 51 (1984), no. 3, 699–728. [2] Klaus Ecker and Gerhard Huisken, Immersed hypersurfaces with constant Weingarten curvature., Math. Ann. 283 (1989), no. 2, 329–332. [3] Claus Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Diff. Geom. 32 (1990), 299–314. [4] , Closed Weingarten hypersurfaces in Riemannian manifolds, J. Diff. Geom. 43 (1996), 612–641, pdf file. [5] , Hypersurfaces of prescribed Weingarten curvature, Math. Z. 224 (1997), 167–194, pdf file. [6] , Hypersurfaces of prescribed scalar curvature in Lorentzian manifolds, J. reine angew. Math. 554 (2003), 157–199, math.DG/0207054. [7] , Curvature Problems, Series in Geometry and Topology, vol. 39, International Press, Somerville, MA, 2006, 323 pp. [8] , Curvature flows in semi-Riemannian manifolds, arXiv:0704.0236, 48 pages. [9] Weimin Sheng, John Urbas, and Xu-Jia Wang, Interior curvature bounds for a class of curvature equations, Duke Math. J. 123 (2004), no. 2, 235–264. Ruprecht-Karls-Universität, Institut für Angewandte Mathematik, Im Neuen- heimer Feld 294, 69120 Heidelberg, Germany E-mail address: [email protected] URL: http://www.math.uni-heidelberg.de/studinfo/gerhardt/ http://www.math.uni-heidelberg.de/studinfo/gerhardt/Gerhardt-JDG-96.pdf http://www.math.uni-heidelberg.de/studinfo/gerhardt/MZ224,97.pdf http://arXiv.org/pdf/math.DG/0207054 http://arXiv.org/abs/0704.0236 1. Introduction 2. Curvature estimates 3. Proof of Theorem 1.5 References

0704.1022

Almost sure functional central limit theorem for non-nestling random walk in random environment

ALMOST SURE FUNCTIONAL CENTRAL LIMIT THEOREM FOR NON-NESTLING RANDOM WALK IN RANDOM ENVIRONMENT FIRAS RASSOUL-AGHA1 AND TIMO SEPPÄLÄINEN2 Abstract. We consider a non-nestling random walk in a product random environ- ment. We assume an exponential moment for the step of the walk, uniformly in the environment. We prove an invariance principle (functional central limit theorem) under almost every environment for the centered and diffusively scaled walk. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively. 1. Introduction and main result We prove a quenched functional central limit theorem for non-nestling random walk in random environment (RWRE) on the d-dimensional integer lattice Zd in dimensions d ≥ 2. Here is a general description of the model, fairly standard since quite a while. An environment ω is a configuration of transition probability vectors ω = (ωx)x∈Zd ∈ Ω = PZd , where P = {(pz)z∈Zd : pz ≥ 0, z pz = 1} is the simplex of all probability vectors on Zd. Vector ωx = (ωx,z)z∈Zd gives the transition probabilities out of state x, denoted by πx,y(ω) = ωx,y−x. To run the random walk, fix an environment ω and an initial state z ∈ Zd. The random walk X0,∞ = (Xn)n≥0 in environment ω started at z is then the canonical Markov chain with state space Zd whose path measure P ωz satisfies P ωz (X0 = z) = 1 and P z (Xn+1 = y|Xn = x) = πx,y(ω). On the space Ω we put its product σ-fieldS, natural shifts πx,y(Tzω) = πx+z,y+z(ω), and a {Tz}-invariant probability measure P that makes the system (Ω,S, (Tz)z∈Zd,P) ergodic. In this paper P is an i.i.d. product measure on PZd . In other words, the vectors (ωx)x∈Zd are i.i.d. across the sites x under P. Date: November 21, 2018. 2000 Mathematics Subject Classification. 60K37, 60F05, 60F17, 82D30. Key words and phrases. Random walk, non-nestling, random environment, central limit theorem, invariance principle, point of view of the particle, environment process, Green function. 1Department of Mathematics, University of Utah. 1Supported in part by NSF Grant DMS-0505030. 2Mathematics Department, University of Wisconsin-Madison. 2Supported in part by NSF Grant DMS-0402231. http://arxiv.org/abs/0704.1022v2 2 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN Statements, probabilities and expectations under a fixed environment, such as the distribution P ωz above, are called quenched. When also the environment is av- eraged out, the notions are called averaged, or also annealed. In particular, the averaged distribution Pz(dx0,∞) of the walk is the marginal of the joint distribution Pz(dx0,∞, dω) = P z (dx0,∞)P(dω) on paths and environments. Several excellent expositions on RWRE exist, and we refer the reader to the lectures [3], [15] and [18]. We turn to the specialized assumptions imposed on the model in this paper. The main assumption is non-nestling (N) which guarantees a drift uniformly over the environments. The terminology was introduced by Zerner [19]. Hypothesis (N). There exists a vector û ∈ Zd \ {0} and a constant δ > 0 such that z · û π0,z(ω) ≥ δ There is no harm in assuming û ∈ Zd, and this is convenient. We utilize two auxiliary assumptions: an exponential moment bound (M) on the steps of the walk, and some regularity (R) on the environments. Hypothesis (M). There exist positive constants M and s0 such that es0|z|π0,z(ω) ≤ es0M Hypothesis (R). There exists a constant κ > 0 such that z: z·û=1 π0,z(ω) ≥ κ = 1. (1.1) Let J = {z : Eπ0,z > 0} be the set of admissible steps under P. Then P{∀z : π0,0 + π0,z < 1} > 0 and J 6⊂ Ru for all u ∈ Rd. (1.2) Assumption (1.1) above is stronger than needed. In the proofs it is actually used in the form (7.5) [Section 7] that permits backtracking before hitting the level x · û = 1. At the expense of additional technicalities in Section 7 quenched assumption (1.1) can be replaced by an averaged requirement. Assumption (1.2) is used in Lemma 7.10. It is necessary for the quenched CLT as was discovered already in the simpler forbidden direction case we studied in [10] and [11]. Note that assumption (1.2) rules out the case d = 1. However, the issue is not whether the walk is genuinely d-dimensional, but whether the walk can explore its environment thoroughly enough to suppress the fluctuations of the quenched mean. Most work on RWRE takes uniform ellipticity and nearest-neighbor jumps as standing assumptions, which of course imply Hypotheses (M) and (R). QUENCHED FUNCTIONAL CLT FOR RWRE 3 These assumptions are more than strong enough to imply a law of large numbers: there exists a velocity v 6= 0 such that n−1Xn = v = 1. (1.3) Representations for v are given in (2.5) and Lemma 5.1. Define the (approximately) centered and diffusively scaled process Bn(t) = X[nt] − [nt]v√ . (1.4) As usual [x] = max{n ∈ Z : n ≤ x} is the integer part of a real x. Let DRd[0,∞) be the standard Skorohod space of Rd-valued cadlag paths (see [6] for the basics). Let Qωn = P 0 (Bn ∈ · ) denote the quenched distribution of the process Bn on DRd[0,∞). The results of this paper concern the limit of the process Bn as n → ∞. As expected, the limit process is a Brownian motion with correlated coordinates. For a symmetric, non-negative definite d × d matrix D, a Brownian motion with diffusion matrix D is the Rd-valued process {B(t) : t ≥ 0} with continuous paths, independent increments, and such that for s < t the d-vector B(t)−B(s) has Gaussian distribution with mean zero and covariance matrix (t − s)D. The matrix D is degenerate in direction u ∈ Rd if utDu = 0. Equivalently, u · B(t) = 0 almost surely. Here is the main result. Theorem 1.1. Let d ≥ 2 and consider a random walk in an i.i.d. product random environment that satisfies non-nestling (N), the exponential moment hypothesis (M), and the regularity in (R). Then for P-almost every ω distributions Qωn converge weakly on DRd [0,∞) to the distribution of a Brownian motion with a diffusion matrix D that is independent of ω. utDu = 0 iff u is orthogonal to the span of {x − y : E(π0x)E(π0y) > 0}. Eqn (2.6) gives the expression for the diffusion matrix D, familiar for example from [14]. Before turning to the proofs we discuss briefly the current situation in this area of probability and the place of this work in this context. Several different approaches can be identified in recent work on quenched central limit theorems for multidimensional RWRE. (i) Small perturbations of classical ran- dom walk have been studied by many authors. The most significant results include the early work of Bricmont and Kupiainen [4] and more recently Sznitman and Zeitouni [16] for small perturbations of Brownian motion in dimension d ≥ 3. (ii) An aver- aged CLT can be turned into a quenched CLT by bounding certain variances through the control of intersections of two independent paths. This idea was introduced by Bolthausen and Sznitman in [2] and more recently applied by Berger and Zeitouni in [1]. Both utilize high dimension to handle the intersections. (iii) Our approach is based on the subdiffusivity of the quenched mean of the walk. That is, we show that the variance of Eω0 (Xn) is of order n 2α for some α < 1/2. We also achieve this through intersection bounds. Instead of high dimension we assume strong enough 4 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN drift. We introduced this line of reasoning in [9] and later applied it to the case of walks with a forbidden direction in [11]. The significant advance taken in the present paper over [9] and [11] is the elimination of restrictions on the admissible steps of the walk. Theorem 2.1 below summarizes the general principle for application in this paper. As the reader will see, the arguments in this paper are based on quenched expo- nential bounds that flow from Hypotheses (N), (M) and (R). It is common in this field to look for an invariant measure P∞ for the environment process that is mutu- ally absolutely continuous with the original P, at least on the part of the space Ω to which the drift points. In this paper we do things a little differently: instead of the absolute continuity, we use bounds on the variation distance between P∞ and P. This distance will decay exponentially in the direction û. In the case of nearest-neighbor, uniformly elliptic non-nestling walks in dimension d ≥ 4 the quenched CLT has been proved earlier: first by Bolthausen and Sznitman [2] under a small noise assumption, and recently by Berger and Zeitouni [1] without the small noise assumption. Berger and Zeitouni [1] go beyond non-nestling to more general ballistic walks. The method in these two papers utilizes high dimension crucially. Whether their argument can work in d = 3 is not presently clear. The approach of the present paper should work for more general ballistic walks in all dimensions d ≥ 2, as the main technical step that reduces the variance estimate to an intersection estimate is generalized (Section 6 in the present paper). We turn to the proofs. The next section collects some preliminary material and finishes with an outline of the rest of the paper. 2. Preliminaries for the proof. As mentioned, we can assume that û ∈ Zd. This is convenient because then the lattice Zd decomposes into levels identified by the integer value x · û. Let us summarize notation for the reader’s convenience. Constants whose exact values are not important and can change from line to line are often denoted by C and s. The set of nonnegative integers is N = {0, 1, 2, . . . }. Vectors and sequences are abbreviated xm,n = (xm, xm+1, . . . , xn) and xm,∞ = (xm, xm+1, xm+2, . . . ). Similar notation is used for finite and infinite random paths: Xm,n = (Xm, Xm+1, . . . , Xn) and Xm,∞ = (Xm, Xm+1, Xm+2, . . . ). X[0,n] = {Xk : 0 ≤ k ≤ n} denotes the set of sites visited by the walk. Dt is the transpose of a vector or matrix D. An element of Rd is regarded as a d× 1 column vector. The left shift on the path space (Zd)N is (θkx0,∞)n = xn+k. E, E0, and E 0 denote expectations under, respectively, P, P0, and P 0 . P∞ will denote an invariant measure on Ω, with expectation E∞. We abbreviate P 0 (·) = 0 (·) and E∞0 (·) = E∞Eω0 (·) to indicate that the environment of a quenched expectation is averaged under P∞. A family of σ-algebras on Ω that in a sense look towards the future is defined by Sℓ = σ{ωx : x · û ≥ ℓ}. QUENCHED FUNCTIONAL CLT FOR RWRE 5 Define the drift D(ω) = Eω0 (X1) = zπ0z(ω). The environment process is the Markov chain on Ω with transition kernel Π(ω,A) = P ω0 (TX1ω ∈ A). The proof of the quenched CLT Theorem 1.1 utilizes crucially the environment process and its invariant distribution. A preliminary part of the proof is summarized in the next theorem quoted from [9]. This Theorem 2.1 was proved by applying the arguments of Maxwell and Woodroofe [8] and Derriennic and Lin [5] to the environment process. Theorem 2.1. [9] Let d ≥ 1. Suppose the probability measure P∞ on (Ω,S) is invariant and ergodic for the Markov transition Π. Assume that z |z|2E∞(π0z) <∞ and that there exists an α < 1/2 such that as n→ ∞ |Eω0 (Xn)− nE∞(D)| = Ŏ (n2α). (2.1) Then as n → ∞ the following weak limit happens for P∞-a.e. ω: distributions Qωn converge weakly on the space DRd[0,∞) to the distribution of a Brownian motion with a symmetric, non-negative definite diffusion matrix D that is independent of ω. Another central tool for the development that follows is provided by the Sznitman- Zerner regeneration times [17] that we now define. For ℓ ≥ 0 let λℓ be the first time the walk reaches level ℓ relative to the initial level: λℓ = min{n ≥ 0 : Xn · û−X0 · û ≥ ℓ}. Define β to be the first backtracking time: β = inf{n ≥ 0 : Xn · û < X0 · û}. Let Mn be the maximum level, relative to the starting level, reached by time n: Mn = max{Xk · û−X0 · û : 0 ≤ k ≤ n}. For a > 0, and when β <∞, consider the first time by which the walker reaches level Mβ + a: λMβ+a = inf{n ≥ β : Xn · û−X0 · û ≥Mβ + a}. Let S0 = λa and, as long as β ◦ θSk−1 < ∞, define Sk = Sk−1 + λMβ+a ◦ θSk−1 for k ≥ 1. Finally, let the first regeneration time be Sℓ1I{β ◦ θSk <∞ for 0 ≤ k < ℓ and β ◦ θSℓ = ∞}. (2.2) Non-nestling guarantees that τ 1 is finite, and in fact gives moment bounds uniformly in ω as we see in Lemma 3.1 below. Consequently we can iterate to define τ 0 = 0, and for k ≥ 1 k = τ k−1 + τ 1 ◦ θτ k−1 . (2.3) 6 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN When the value of a is not important we simplify the notation to τk = τ k . Sznit- man and Zerner [17] proved that the regeneration slabs τk+1 − τk, (Xτk+n −Xτk)0≤n≤τk+1−τk , {ωXτk+z : 0 ≤ z · û < (Xτk+1 −Xτk) · û} ) (2.4) are i.i.d. for k ≥ 1, each distributed as τ1, (Xn)0≤n≤τ1 , {ωz : 0 ≤ z·û < Xτ1 ·û} under P0( · | β = ∞). Strictly speaking, uniform ellipticity and nearest-neighbor jumps were standing assumptions in [17], but these assumptions are not needed for the proof of the i.i.d. structure. From the renewal structure and moment estimates a law of large numbers (1.3) and an averaged functional central limit theorem follow, along the lines of Theorem 2.3 in [17] and Theorem 4.1 in [14]. These references treat walks that satisfy Kalikow’s condition, considerably more general than the non-nestling walks we study. The limiting velocity for the law of large numbers is E0(Xτ1 |β = ∞) E0(τ1|β = ∞) . (2.5) The averaged CLT states that the distributions P0{Bn ∈ · } converge to the distri- bution of a Brownian motion with diffusion matrix (Xτ1 − τ1v)(Xτ1 − τ1v)t|β = ∞ E0[τ1|β = ∞] . (2.6) Once we know that the P-a.s. quenched CLT holds with a constant diffusion matrix, this diffusion matrix must be the same D as for the averaged CLT. We give here the argument for the degeneracy statement of Theorem 1.1. Lemma 2.1. Define D by (2.6) and let u ∈ Rd. Then utDu = 0 iff u is orthogonal to the span of {x− y : E(π0x)E(π0y) > 0}. Proof. The argument is a minor embellishment of that given for a similar degeneracy statement on p. 123–124 of [10] for the forbidden-direction case where π0,z is supported by z · û ≥ 0. We spell out enough of the argument to show how to adapt that proof to the present case. Again, the intermediate step is to show that utDu = 0 iff u is orthogonal to the span of {x− v : E(π0x) > 0}. The argument from orthogonality to utDu = 0 goes as in [10, p. 124]. Suppose utDu = 0 which is the same as P0(Xτ1 · u = τ1v · u | β = ∞) = 1. Suppose z is such that Eπ0,z > 0 and z · û < 0. By non-nestling there must exist w such that Eπ0,zπ0,w > 0 and w · û > 0. Pick m > 0 so that (z + mw) · û > 0 but QUENCHED FUNCTIONAL CLT FOR RWRE 7 (z + (m− 1)w) · û ≤ 0. Take a = 1 in the definition (2.2) of regeneration. Then P0[Xτ1 = z + 2mw, τ1 = 2m+ 1 | β = ∞] [ ( m−1∏ πiw,(i+1)w πmw,z+mw ( m−1∏ πz+(m+j)w,z+(m+j+1)w P ωz+2mw(β = ∞) Consequently (z + 2mw) · u = (1 + 2m)v · u. (2.7) In this manner, by replacing σ1 with τ1 and by adding in the no-backtracking probabilities, the arguments in [10, p. 123] can be repeated to show that if Eπ0x > 0 then x · u = v · u for x such that x · û ≥ 0. In particular the very first step on p. 123 of [10] gives w · u = v · u. This combines with (2.7) above to give z · u = v · u. Now simply follow the proof in [10, p. 123–124] to its conclusion. � Here is an outline of the proof of Theorem 1.1. It all goes via Theorem 2.1. (i) After some basic estimates in Section 3, we prove in Section 4 the existence of the ergodic equilibrium P∞ required for Theorem 2.1. P∞ is not convenient to work with so we still need to do computations with P. For this purpose Section 4 proves that in the direction û the measures P∞ and P come exponentially close in variation distance and that the environment process satisfies a P0-a.s. ergodic theorem. In Section 5 we show that P∞ and P are interchangeable both in the hypotheses that need to be checked and in the conclusions obtained. In particular, the P∞-a.s. quenched CLT coming from Theorem 2.1 holds also P-a.s. Then we know that the diffusion matrix D is the one in (2.6). The bulk of the work goes towards verifying condition (2.1), but under P instead of P∞. There are two main stages to this argument. (ii) By a decomposition into martingale increments the proof of (2.1) reduces to bounding the number of common points of two independent walks in a common environment (Section 6). (iii) The intersections are controlled by introducing levels at which both walks regenerate. These common regeneration levels are reached fast enough and the pro- gression from one common regeneration level to the next is a Markov chain. When this Markov chain drifts away from the origin it can be approximated well enough by a symmetric random walk. This approximation enables us to control the growth of the Green function of the Markov chain, and thereby the number of common points. This is in Section 7 and in an Appendix devoted to the Green function bound. 3. Basic estimates for non-nestling RWRE This section contains estimates that follow from Hypotheses (N) and (M), all col- lected in the following lemma. These will be used repeatedly. In addition to the 8 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN stopping times already defined, let Hz = min{n ≥ 1 : Xn = z} be the first hitting time of site z. Lemma 3.1. If P satisfies Hypotheses (N) and (M), then there exist positive constants η, γ, κ, (Cp)p≥1, and s1 ≤ s0, possibly depending on M , s0, and δ, such that for all x ∈ Zd, n ≥ 0, s ∈ [0, s1], p ≥ 1, ℓ ≥ 1, for z such that z · û ≥ 0, a ≥ 1, and for P-a.e. ω, Eωx (e −sXn·û) ≤ e−sx·û(1− sδ/2)n, (3.1) Eωx (e s|Xn−x|) ≤ esMn, (3.2) P ωx (X1 · û ≥ x · û+ γ) ≥ κ, (3.3) Eωx (λ ℓ) ≤ Cpℓp, (3.4) Eωx (|Xλℓ − x|p) ≤ Cpℓp, (3.5) Eω0 [(MHz − z · û)p1I{Hz < n}] ≤ CpℓpP ω0 (Hz < n) + Cps−pe−sℓ/2, (3.6) P ωx (β = ∞) ≥ η, (3.7) Eωx (|τ 1 |p) ≤ Cp ap, (3.8) Eωx (|Xτ (a)1 +n −Xn| p) ≤ Cq aq, for all q > p. (3.9) The particular point in (3.8)–(3.9) is to make the dependence on a explicit. Note that (3.7)–(3.8) give E0(τj − τj−1)p <∞ (3.10) for all j ≥ 1. In Section 4 we construct an ergodic invariant measure P∞ for the environment chain in a way that preserves the conclusions of this lemma under P∞. Proof. Replacing x by 0 and ω by Txω allows us to assume that x = 0. Then for all s ∈ [0, s0/2] ∣∣Eω0 (e−sX1·û)− 1 + sEω0 (X1 · û) ∣∣ ≤ |û|2Eω0 (|X1|2es0|X1|/2) ≤ (2|û|/s0)2es0Ms2 = cs2, where we used moment assumption (M). Then by the non-nestling assumption (N) Eω0 (e −sXn·û|Xn−1) = e−sXn−1·ûEωXn−1(e −s(X1−X0)·û) ≤ e−sXn−1·û(1− sδ + cs2). Taking now the quenched expectation of both sides and iterating the procedure proves (3.1), provided s1 is small enough. To prove (3.2) one can instead show that Eω0 (e k=1 |Xk−Xk−1|) ≤ esnM . This can be proved by induction as for (3.1), using only Hypothesis (M) and Hölder’s inequality (to switch to s0). QUENCHED FUNCTIONAL CLT FOR RWRE 9 Concerning (3.3), we have P ω0 (X1 · û ≥ γ) ≥ (1− eγs(1− sδ/2))−→ sδ/2. So taking γ small enough and κ slightly smaller than sδ/2 does the job. Notice next that P ω0 (λ1 <∞) = 1 due to (3.1). P-a.s. Then Eω0 (λ (n+ 1)pP ω0 (λ1 > n) ≤ (n+ 1)pP ω0 (Xn · û ≤ 1) (n + 1)pEω0 (e −sXn·û). The last expression is bounded if s is small enough. Therefore, Eω0 (λ ℓ) ≤ Eω0 [ ∣∣∣ [ℓ]+1∑ (λi − λi−1) ≤ ([ℓ] + 1)p−1 [ℓ]+1∑ EωXλi−1 ≤ Cpℓp. Bound (3.5) is proved similarly: by the Cauchy-Schwarz inequality, Hypothesis (M) and (3.1), Eω0 (|Xλ1 |p) ≤ Eω0 (|Xn|2p)1/2P ω0 (Xn−1 · û < 1)1/2 [2p]! s −[2p] s0Mes )1/2 ∑ (1− sδ/2)(n−1)/2np ≤ Cp. To prove (3.6), write Eω0 [(MHz − z · û)p1I{Hz < n}] ℓp−1P ω0 (MHz − z · û ≥ ℓ,Hz < n) + Cpℓ 0 (Hz < n) ℓp−1Eω0 [P Xλz·û+ℓ (Xk · û−X0 · û ≤ −ℓ)] + Cpℓp0P ω0 (Hz < n) ℓp−1e−sℓ + Cpℓ 0 (Hz < n) ≤ Cps−pe−sℓ0/2 + Cpℓ 0 (Hz < n). To prove (3.7), note that Chebyshev inequality and (3.1) give, for s > 0 small enough, ℓ ≥ 1, and P-a.e. ω P ω0 (λ−ℓ+1 <∞) ≤ P ω0 (Xn · û ≤ −(ℓ− 1)) ≤ 2(sδ)−1e−s(ℓ−1). On the other hand, for an integer ℓ ≥ 2 we have P ω0 (λℓ < β) ≥ P ω0 (λℓ−1 < β,Xλℓ−1 = x)P x (λ−ℓ+1 = ∞). 10 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN Therefore, taking ℓ to infinity one has, for ℓ0 large enough, P ω0 (β = ∞) ≥ P ω0 (λℓ0 < β) (1− 2(sδ)−1e−sℓ). Markov property and (3.3) give P ω0 (λℓ0 < β) ≥ κℓ0/γ+1 > 0 and (3.7) is proved. Now we will bound the quenched expectation of λ 1I{β < ∞} uniformly in ω. To this end, for p1 > p and q1 = p1(p1 − p)−1, we have by (3.7) Eω0 (λ 1I{β <∞}) ≤ Eω0 (λ 1I{β = n}) Eω0 (λ )p/p1( P ω0 (β = n) )1/q1 By (3.4) one has, for p2 > p1 > p and q2 = p2(p2 − p1)−1, Eω0 (λ Eω0 (λ m+1+a) )p1/p2( P ω0 ([Mn] = m) )1/q2 (m+ 1 + a)p1 P ω0 (Xi · û ≥ m) )1/q2 where Cp really depends on p1 and p2, but these are chosen arbitrarily, as long as they satisfy p2 > p1 > p. Using (3.2) one has P ω0 (Xi · û ≥ m) ≤ 1 if m < 2M |û|i, e−smeM |û|si if m ≥ 2M |û|i. Hence, Eω0 (λ ) ≤ Cp (m+ 1 + a)p1(n1I{m < 2Mn|û|}+ e−sm/2)1/q2 ≤ Cpn(n+ a)p1n1/q2 + Cp (m+ 1)p1e−sm/2q2 + Cpa e−sm/2q2 ≤ Cpn1+1/q2(n+ a)p1 . Since {β = n} ⊂ {Xn · û ≤ 0}, one can use (3.1) to conclude that Eω0 (λ 1I{β <∞}) ≤ Cp np/p1+p/(p1q2)(n+ a)p(1− sδ/2)n/q1 ≤ Cpap. QUENCHED FUNCTIONAL CLT FOR RWRE 11 In the last inequality we have used the fact that a ≥ 1. Using, (3.7), the definition of the times Sk, and the Markov property, one has Eω0 [S ℓ 1I{β ◦ θSk <∞ for 0 ≤ k < ℓ and β ◦ θSℓ = ∞}] ≤ (ℓ+ 1)p−1 Eω0 [λ a1I{β ◦ θSk <∞ for 0 ≤ k < ℓ}] Eω0 [λ ◦ θSj1I{β ◦ θSk <∞ for 0 ≤ k < ℓ}] ≤ (ℓ+ 1)p−1 p(1− η)ℓ + (1− η)jCpap(1− η)ℓ−j−1 ≤ Cp(ℓ + 1)p(1− η)ℓ−1ap. Bound (3.8) follows then from (2.2). To prove (3.9) let q > p and write Eω0 (|Xτ (a)1 +n −Xn| Eω0 (|Xk+1+n −Xk+n|p|τ 1 |p−11I{k < τ k−1−q+pEω0 (|Xk+1+n −Xk+n|p|τ 1 |q) k−1−q+pEω0 (|τ 1 |2q)1/2 ≤ Cq aq, where we have used Hypothesis (M) along with the Cauchy-Schwarz inequality in the second to last inequality and (3.8) in the last. This completes the proof of the lemma. � 4. Invariant measure and ergodicity For ℓ ∈ Z define the σ-algebras Sℓ = σ{ωx : x · û ≥ ℓ} on Ω. Denote the restriction of the measure P to the σ-algebra Sℓ by P|Sℓ . In this section we prove the next two theorems. The variation distance of two probability measures is dVar(µ, ν) = sup{µ(A)− ν(A)} with the supremum taken over measurable sets A. Theorem 4.1. Assume P is product non-nestling (N) and satisfies the moment hy- pothesis (M). Then there exists a probability measure P∞ on Ω with these properties. (a) P∞ is invariant and ergodic for the Markov transition kernel Π. (b) There exist constants 0 < c, C <∞ such that for all ℓ ≥ 0 dVar(P∞|Sℓ ,P|Sℓ) ≤ Ce −cℓ. (4.1) (c) Hypotheses (N) and (M) and the conclusions of Lemma 3.1 hold P∞-almost surely. Along the way we also establish this ergodic theorem under the original environ- ment measure. E∞ denotes expectation under P∞. 12 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN Theorem 4.2. Assumptions as in Theorem 4.1 above. Let Ψ be a bounded S−a- measurable function on Ω, for some 0 < a <∞. Then Ψ(TXjω) = E∞Ψ P0-almost surely. (4.2) The ergodic theorem tells us that there is a unique invariant P∞ in a natural relationship to P, and that P∞ ≪ P on each σ-algebra S−a. Limit (4.2) cannot hold for all bounded measurable Ψ on Ω because this would imply the absolute continuity P∞ ≪ P on the entire space Ω. A counterexample that satisfies (N) and (M) but where the quenched walk is degenerate was given by Bolthausen and Sznitman [2, Proposition 1.5]. Whether regularity assumption (R) or ellipticity will make a difference here is not presently clear. For the simpler case of space-time walks (see description of model in [9]) with nondegenerate P ω0 absolute continuity P∞ ≪ P does hold on the entire space. Theorem 3.1 in [2] proves this for nearest-neighbor jumps with some weak ellipticity. The general case is no harder. Proof of Theorems 4.1 and 4.2. Let Pn(A) = P0(TXnω ∈ A). A computation shows fn(ω) = (ω) = P ωx (Xn = 0). By hypotheses (M) and (N) we can replace the state space Ω = PZd with the smaller space Ω0 = PZ 0 where P0 = {(pz) ∈ P : es0|z|pz ≤ es0M and z · û pz ≥ δ }. (4.3) Fatou’s lemma shows that the exponential bound is preserved by pointwise conver- gence in P0. Then the exponential bound shows that the non-nestling property is also preserved. Thus P0 is compact, and then Ω0 is compact under the product topology. Compactness gives a subsequence {nj} along which the averages nj−1 m=1 Pm converge weakly to a probability measure P∞ on Ω0. Hypotheses (N) and (M) transfer to P∞ by virtue of having been included in the state space Ω0. Thus the proof of Lemma 3.1 can be repeated for P∞-a.e. ω. We have verified part (c) of Theorem 4.1. Next we check that P∞ is invariant under Π. Take a bounded, continuous local function F on Ω0 that depends only on environments (ωx : |x| ≤ K). For ω, ω̄ ∈ Ω0 ∣∣ΠF (ω)− ΠF (ω̄) ∣∣Eω0 [F (TX1ω)]−Eω̄0 [F (TX1ω̄)] |z|≤C ∣∣∣π0,z(ω)F (Tzω)− π0,z(ω̄)F (Tzω̄) ∣∣∣+ ‖F‖∞ |z|>C π0,z(ω) + π0,z(ω̄) From this we see that ΠF is continuous. For let ω̄ → ω in Ω0 so that ω̄x,z → ωx,z at each coordinate. Since the last term above is controlled by the uniform exponential QUENCHED FUNCTIONAL CLT FOR RWRE 13 tail bound imposed on P0, continuity of ΠF follows. Consequently the weak limit m=1 Pm → P∞ together with Pn+1 = PnΠ implies the Π-invariance of P∞. We show the exponential bound (4.1) on the variation distance next because the ergodicity proof depends on it. On metric spaces total variation distance can be characterized in terms of continuous functions: dVar(µ, ν) = fdν : f continuous, sup |f | ≤ 1 This makes dVar(µ, ν) lower semicontinuous which we shall find convenient below. Fix ℓ > 0. Then dPn|Sℓ dP|Sℓ P ωx (Xn = 0,max Xj · û ≤ ℓ/2) E[P ωx (Xn = 0,max Xj · û > ℓ/2)|Sℓ]. (4.4) The L1(P)-norm of the second term is bounded by In,ℓ = P0(max Xj · û > Xn · û+ ℓ/2) and (3.1) tells us that In,ℓ ≤ e−sℓ/2(1− sδ/2)n−j ≤ Ce−sℓ/2. (4.5) The integrand in the first term of (4.4) is measurable with respect to σ(ωx : x·û ≤ ℓ/2) and therefore independent of Sℓ. The distance between the whole first term and 1 is then Ŏ (In,ℓ). Thus for large enough ℓ, dVar(Pn|Sℓ ,P|Sℓ) ≤ ∫ ∣∣∣ dPn|Sℓ dP|Sℓ ∣∣∣dP ≤ 2In,ℓ ≤ Ce−cℓ. By the construction of P∞ as the Cesàro limit and by the lower semicontinuity and convexity of the variation distance dVar(P∞|Sℓ ,P|Sℓ) ≤ lim dVar(Pm|Sℓ ,P|Sℓ) ≤ Ce Part (b) has been verified. As the last point we prove the ergodicity. Recall the notation E∞0 = E∞E 0 . Let Ψ be a bounded local function on Ω. It suffices to prove that for some constant b ∣∣∣n−1 Ψ(TXjω)− b ∣∣∣ = 0. (4.6) 14 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN By an approximation it follows from this that for all F ∈ L1(P∞) ΠjF (ω) → E∞F in L1(P∞). (4.7) By standard theory (Section IV.2 in [12]) this is equivalent to ergodicity of P∞ for the transition Π. We combine the proof of Theorem 4.2 with the proof of (4.6). For this purpose let Ψ be S−a+1-measurable with a <∞. Take a to be the parameter in the regeneration times (2.2). Let τi+1−1∑ Ψ(TXjω). From the i.i.d. regeneration slabs and the moment bound (3.10) follows the limit τm−1∑ Ψ(TXjω) = lim ϕi = b0 P0-almost surely, (4.8) where the constant b0 is defined by the limit. To justify this more precisely, recall the definition of regeneration slabs given in (2.4). Define a function Φ of the regeneration slabs by Φ(S0,S1,S2, . . . ) = τ2−1∑ Ψ(TXjω). Since each regeneration slab has thickness in û-direction at least a, the Ψ-terms in the sum do not read the environments below level zero and consequently the sum is a function of (S0,S1,S2, . . . ). Next one can check for k ≥ 1 that Φ(Sk−1,Sk,Sk+1, . . . ) = τ2(Xτk−1+ · −Xτk−1 )−1∑ j=τ1(Xτk−1+ · −Xτk−1) TXτk−1+j−Xτk−1 (TXτk−1ω) = ϕk. Now the sum of ϕ-terms in (4.8) can be decomposed into ϕ0 + ϕ1 + Φ(Sk,Sk+1,Sk+2, . . . ). The limit (4.8) follows because the slabs (Sk)k≥1 are i.i.d. and the finite initial terms ϕ0 + ϕ1 are eliminated by the m −1 factor. Let αn = inf{k : τk ≥ n}. Bounds (3.7)–(3.8) give finite moments of all orders to the increments τk−τk−1 and this implies that n−1(ταn−1−ταn) → 0 P0-almost surely. QUENCHED FUNCTIONAL CLT FOR RWRE 15 Consequently (4.8) yields the next limit, for another constant b: Ψ(TXjω) = b P0-almost surely. (4.9) By boundedness this limit is valid also in L1(P0) and the initial point of the walk is immaterial by shift-invariance of P. Let ℓ > 0 and choose a small ε0 > 0. Abbreviate Gn,x(ω) = E [ ∣∣∣n−1 Ψ(TXjω)− b ∣∣∣1I Xj · û ≥ X0 · û− ε0ℓ/2 I = {x ∈ Zd : x · û ≥ ε0ℓ, |x| ≤ Aℓ} for some constant A. Use the bound (4.1) on the variation distance and the fact that the functions Gn,x(ω) are uniformly bounded over all x, n, ω, and, if ℓ is large enough relative to a and ε0, for x ∈ I the function Gn,x is Sε0ℓ/3-measurable. P ω0 [Xℓ = x]Gn,x(ω) ≥ ε1 P∞{Gn,x(ω) ≥ ε1/(Cℓd)} ≤ Cℓdε−11 E∞Gn,x ≤ Cℓdε−11 EGn,x + Cℓ 2dε−11 e −cε0ℓ/3. By (4.9) EGn,x → 0 for any fixed x. Thus from above we get for any fixed ℓ, 1I{Xℓ ∈ I}Gn,Xℓ ≤ ε1 + Cℓ2dε−11 e−cε0ℓ/3. (4.10) The reader should bear in mind that the constant C is changing from line to line. Finally, we write ∣∣∣n−1 Ψ(TXjω)− b ≤ lim 1I{Xℓ ∈ I} ∣∣∣n−1 n+ℓ−1∑ Ψ(TXjω)− b ∣∣∣1I Xj · û ≥ Xℓ · û− ε0ℓ/2 + CP∞0 {Xℓ /∈ I} + CP∞0 Xj · û < Xℓ · û− ε0ℓ/2 ≤ lim 1I{Xℓ ∈ I}Gn,Xℓ + CP∞0 {Xℓ · û < ε0ℓ} + CP∞0 { |Xℓ| > Aℓ} + CE∞0 P ωXℓ Xj · û < X0 · û− ε0ℓ/2 As pointed out, P∞ satisfies Lemma 3.1 because hypotheses (N) and (M) were built into the space Ω0 that supports P∞. This enables us to make the error probabilities above small. Consequently, if we first pick ε0 and ε1 small enough, A large enough, then ℓ large, and apply (4.10), we will have shown (4.6). Ergodicity of P∞ has been shown. This concludes the proof of Theorem 4.1. 16 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN Thereom 4.2 has also been established. It follows from the combination of (4.6) and (4.9). � 5. Change of measure There are several stages in the proof where we need to check that a desired con- clusion is not affected by choice between P and P∞. We collect all instances of such transfers in this section. The standing assumptions of this section are that P is an i.i.d. product measure that satisfies Hypotheses (N) and (M), and that P∞ is the measure given by Theorem 4.1. We show first that P∞ can be replaced with P in the key condition (2.1) of Theorem 2.1. Lemma 5.1. The velocity v defined by (2.5) satisfies v = E∞(D). There exists a constant C such that |E0(Xn)− nE∞(D)| ≤ C for all n ≥ 1. (5.1) Proof. We start by showing v = E∞(D). The uniform exponential tail in the defi- nition (4.3) of P0 makes the function D(ω) bounded and continuous on Ω0. By the Cesàro definition of P∞, E∞(D) = lim nj−1∑ Ek(D) = lim nj−1∑ E0[D(TXkω)]. The moment bounds (3.7)–(3.9) imply that the law of large numbers n−1Xn → v holds also in L1(P0). From this and the Markov property v = lim E0(Xk+1 −Xk) = lim E0[D(TXkω)]. We have proved v = E∞(D). The variables (Xτj+1 −Xτj , τj+1−τj)j≥1 are i.i.d. with sufficient moments by (3.7)– (3.9). With αn = inf{j ≥ 1 : τj − τ1 ≥ n} Wald’s identity gives E0(Xταn−Xτ1) = E0(αn)E0(Xτ1 |β = ∞) and E0(ταn−τ1) = E0(αn)E0(τ1|β = ∞). Consequently, by the definition (2.5) of v, E0(Xn)− nv = vE0(ταn − τ1 − n)− E0(Xταn −Xτ1 −Xn). It remains to show that E0(ταn − τ1−n) and E0(Xταn −Xτ1 −Xn) are bounded by constants. We do this with a simple renewal argument. Let Yj = τj+1 − τj for j ≥ 1 and V0 = 0, Vm = Y1 + · · · + Ym. The quantity to bound is the forward recurrence time Bn = min{k ≥ 0 : n+ k ∈ {Vm}} because ταn − τ1 − n = Bn. We can write Bn = (Y1 − n)+ + 1I{Y1 = k}Bn−k ◦ θ QUENCHED FUNCTIONAL CLT FOR RWRE 17 where θ shifts the sequence {Yk} and makes Bn−k ◦ θ independent of Y1. The two main terms on the right multiply to zero, so for any integer p ≥ 1 Bpn = ((Y1 − n)+)p + 1I{Y1 = k}(Bn−k ◦ θ)p. Set z(n) = E0((Y1 − n)+)p. Moment bounds (3.7)–(3.8) give E0(Y p+11 ) < ∞ which implies z(n) <∞. Taking expectations and using independence gives the discrete renewal equation n = z(n) + P0(Y1 = k)E0B Induction on n shows that E0B k=1 z(k) ≤ C(p) for all n. In particular, E0(ταn − τ1 − n)p is bounded by a constant uniformly over n. To extend this to E0|Xταn − Xτ1 − Xn|p apply an argument like the one given for (3.9) at the end of Section 3. � Proposition 5.2. Assume that there exists an ᾱ < 1/2 such that |Eω0 (Xn)− E0(Xn)| = Ŏ (n2ᾱ). (5.2) Then condition (2.1) is satisfied for some α < 1/2. Proof. By (5.1) assumption (5.2) turns into |Eω0 (Xn)− nv| = Ŏ (n2ᾱ). (5.3) In the rest of this proof we use the conclusions of Lemma 3.1 under P∞ instead of P. This is justified by part (c) of Theorem 4.1. For k ≥ 1, recall that λk = inf{n ≥ 0 : (Xn − X0) · û ≥ k}. Take k = [nρ] for a small enough ρ > 0. The point of the proof is to let the walk run up to a high level k so that expectations under P∞ can be profitably related to expectations under P through the variation distance bound (4.1). Estimation is needed to remove the dependence on the environment on low levels. First compute as follows. |Eω0 (Xn − nv)| |Eω0 (Xn − nv, λk ≤ n) + Eω0 (Xn − nv, λk > n)| ≤ 2E∞ [ ∣∣Eω0 (Xn −Xλk − (n− λk)v, λk ≤ n)− Eω0 (λkv, λk ≤ n) + Eω0 (Xλk , λk ≤ n)2 n2E∞[P 0 (λk > n)] ≤ 8E∞ [ ∣∣∣ 0≤m≤n x·û≥k P ω0 (Xm = x, λk = m)E x {Xn−m − x− (n−m)v} + Ŏ (k2 + n2esk(1− sδ/2)n). (5.4) The last error term above is Ŏ (n2ρ). We used the Cauchy-Schwarz inequality and Hypothesis (M) to get the second term in the first inequality, and then (3.1), (3.4), and (3.5) in the last inequality. 18 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN To handle the expectation on line (5.4) we introduce a spanning set of vectors that satisfy the main assumptions that û does. Namely, let {ûi}di=1 span Rd and satisfy these conditions: |û − ûi| ≤ δ/(2M), where δ and M are the constants from Hypotheses (N) and (M), and αiûi with αi > 0. (5.5) Then non-nestling (N) holds for each ûi with constant δ/2, and all the conclusions of Lemma 3.1 hold when û is replaced by ûi and δ by δ/2. Define the event Ak = {infiXi · û ≥ k} and the set Λ = {x ∈ Zd : min x · ûi ≥ 1}. The point of introducing Λ is that the number of points x in Λ on level x · û = ℓ > 0 is of order Ŏ (ℓd−1). By Jensen’s inequality the expectation on line (5.4) is bounded by x∈Λ, x·û≥k 0≤m≤n P ω0 (Xm = x, λk = m) ∣∣Eωx {Xn−m − x− (n−m)v, Ak/2} (5.6) 0≤m≤n x 6∈Λ P ω0 (Xm = x, λk = m) ∣∣Eωx {Xn−m − x− (n−m)v} + 2E∞ 0≤m≤n x·û≥k P ω0 (Xm = x, λk = m) ∣∣Eωx {Xn−m − x− (n−m)v, Ack/2} By Cauchy-Schwarz, Hypothesis (M) and (3.1), the third term is Ŏ (n2e−sk/2) = Ŏ (1). The second term is of order n2max 0 (Xλk · ûi < 1)] ≤ n2max P ω0 (Xm · ûi < 1)P ω0 (Xm · û ≥ k) )1/2 ] ≤ es/2n2 (1− sδ/4)m/2e−µk/2eµM |û|m/2 = Ŏ (n2e−µk/2) = Ŏ (1), for µ small enough. It remains to bound the term on line (5.6). To this end, by Cauchy-Schwarz, (3.2) and (3.1), P ω0 (Xm = x, λk = m) ≤ {e−sx·û/2esM |û|m/2∧1}×{eµk/2(1−µδ/2)(m−1)/2∧1} ≡ px,m,k. Notice that ∑ {e−sx·û/2esM |û|m/2 ∧ 1} = Ŏ (md) QUENCHED FUNCTIONAL CLT FOR RWRE 19 and ∑ md{eµk/2(1− µδ/2)(m−1)/2 ∧ 1} = Ŏ (kd+1). Substitute these back into line (5.6) to eliminate the quenched probability coefficients. The quenched expectation in (5.6) is Sk/2-measurable. Consequently variation dis- tance bound (4.1) allows us to switch back to P and get this upper bound for line (5.6): x∈Λ, x·û≥k 0≤m≤n px,m,kE[|Eωx {Xn−m − x− (n−m)v, Ak/2}|2] + Ŏ (kd+1n2e−ck/2). The error term is again Ŏ (1). Now insert Ac back inside the quenched expectation, incurring another error term of order Ŏ (kd+1n2e−sk/2) = Ŏ (1). Using the shift-invariance of P, along with (5.3), and collecting all of the above error terms, we get |Eω0 (Xn − nv)| x∈Λ, x·û≥k 0≤m≤n px,m,kE[ |Eωx {Xn−m − x− (n−m)v}|2] + Ŏ (n2ρ) = Ŏ (kd+1n2ᾱ + n2ρ) = Ŏ (nρ(d+1)+2ᾱ). Pick ρ > 0 small enough so that 2α = ρ(d + 1) + 2ᾱ < 1. The conclusion (2.1) follows. � Once we have verified the assumptions of Theorem 2.1 we have the CLT under P∞-almost every ω. But we want the CLT under P-almost every ω. Thus as the final point of this section we prove the transfer of the central limit theorem from P∞ to P. This is where we use the ergodic theorem, Theorem 4.2. Let W be the probability distribution of the Brownian motion with diffusion matrix D. Lemma 5.3. Suppose the weak convergence Qωn ⇒ W holds for P∞-almost every ω. Then the same is true for P-almost every ω. Proof. It suffices to show that for any bounded uniformly continuous F on DRd[0,∞) and any δ > 0 Eω0 [F (Bn)] ≤ F dW + δ P-a.s. By considering also −F this gives Eω0 [F (Bn)] → F dW P0-a.s. for each such func- tion. A countable collection of them determines weak convergence. Fix such an F . Let c = F dW and h(ω) = lim Eω0 [F (Bn)]. 20 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN For ℓ > 0 define the events A−ℓ = {inf Xn · û ≥ −ℓ} and then hℓ(ω) = lim Eω0 [F (Bn), A−ℓ] and Ψ(ω) = 1I{ω : h̄ℓ(ω) ≤ c+ 12δ}. The assumed quenched CLT under P∞ gives P∞{h = c} = 1. By (3.1), and by its extension to P∞ in Theorem 4.1(c), there are constants 0 < C, s <∞ such that |h(ω)− hℓ(ω)| ≤ Ce−sℓ uniformly over all ω that support both P and P∞. Consequently if δ > 0 is given, E∞Ψ = 1 for large enough ℓ. Since Ψ is S−ℓ-measurable Theorem 4.2 implies that Ψ(TXjω) → 1 P0-a.s. By increasing ℓ if necessary we can ensure that {h̄ℓ ≤ c + 12δ} ⊂ {h̄ ≤ c + δ} and conclude that the stopping time ζ = inf{n ≥ 0 : h̄(TXnω) ≤ c+ δ} is P0-a.s. finite. From the definitions we now have 0 [F (Bn)] ≤ F dW + δ P0-a.s. Then by bounded convergence Eω0 E 0 [F (Bn)] ≤ F dW + δ P-a.s. Since ζ is a finite stopping time, the strong Markov property, the uniform continuity of F and the exponential moment bound (3.2) on X-increments imply Eω0 [F (Bn)] ≤ F dW + δ P-a.s. This concludes the proof. � 6. Reduction to path intersections The preceding sections have reduced the proof of the main result Theorem 1.1 to proving the estimate |Eω0 (Xn)−E0(Xn)| = Ŏ (n2α) for some α < 1/2. (6.1) The next reduction takes us to the expected number of intersections of the paths of two independent walks X and X̃ in the same environment. The argument uses a de- composition into martingale differences through an ordering of lattice sites. This idea QUENCHED FUNCTIONAL CLT FOR RWRE 21 for bounding a variance is natural and has been used in RWRE earlier by Bolthausen and Sznitman [2]. Let P ω0,0 be the quenched law of the walks (X, X̃) started at (X0, X̃0) = (0, 0) and P0,0 = P ω0,0 P(dω) the averaged law with expectation operator E0,0. The set of sites visited by a walk is denoted by X[0,n) = {Xk : 0 ≤ k < n} and |A| is the number of elements in a discrete set A. Proposition 6.1. Let P be an i.i.d. product measure and satisfy Hypotheses (N) and (M). Assume that there exists an ᾱ < 1/2 such that E0,0(|X[0,n) ∩ X̃[0,n)|) = Ŏ (n2ᾱ). (6.2) Then condition (6.1) is satisfied. Proof. For L ≥ 0, define B(L) = {x ∈ Zd : |x| ≤ L}. Fix n ≥ 1, c > |û|, and let (xj)j≥1 be some fixed ordering of B(cMn) satisfying ∀i ≥ j : xi · û ≥ xj · û. For B ⊂ Zd let SB = σ{ωx : x ∈ B}. Let Aj = {x1, . . . , xj}, ζ0 = E0(Xn), and for j ≥ 1 ζj = E(E 0 (Xn)|SAj). (ζj − ζj−1)j≥1 is a sequence of L2(P)-martingale differences and we have E[ |Eω0 (Xn)− E0(Xn)| 2] (6.3) |E0(Xn)− E{Eω0 (Xn)|SB(cMn)}|2 [ ∣∣Eω0 (Xn,max |Xi| > cMn) − E{Eω0 (Xn,max |Xi| > cMn) |SB(cMn)} ∣∣2 ] |B(cMn)|∑ E( |ζj − ζj−1|2 ) + Ŏ (n3e−sM(c−|û|)n). (6.4) In the last inequality we have used (3.2). The error is Ŏ (1). For z ∈ Zd define half-spaces H(z) = {x ∈ Zd : x · û > z · û}. Since Aj−1 ⊂ Aj ⊂ H(xj)c, E(|ζj − ζj−1|2) P(dωAj) P(dωAc )P(dω̃xj) Eω0 (Xn)− E 〈ω,ω̃xj 〉 0 (Xn) P(dωH(xj)c)P(dω̃xj) P(dωH(xj)) Eω0 (Xn)− E 〈ω,ω̃xj 〉 0 (Xn) . (6.5) Above 〈ω, ω̃xj〉 denotes an environment obtained from ω by replacing ωxj with ω̃xj . 22 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN We fix a point z = xj to develop a bound for the expression above, and then return to collect the estimates. Abbreviate ω̃ = 〈ω, ω̃xj〉. Consider two walks Xn and X̃n starting at 0. Xn obeys environment ω, while X̃n obeys ω̃. We can couple the two walks so that they stay together until the first time they visit z. Until a visit to z happens, the walks are identical. So we write P(dωH(z)) Eω0 (Xn)− Eω̃0 (Xn) (6.6) P(dωH(z)) P ω0 (Hz = m) Eωz (Xn−m − z)− Eω̃z (Xn−m − z) P(dωH(z)) P ω0 (Hz = m, ℓ− 1 ≤ max 0≤j≤m Xj · û− z · û < ℓ) Eωz (Xn−m − z)− Eω̃z (Xn−m − z) (6.7) Decompose H(z) = Hℓ(z) ∪H′ℓ(z) where Hℓ(z) = {x ∈ Zd : z · û < x · û < z · û+ ℓ} and H′ℓ(z) = {x ∈ Zd : x · û ≥ z · û+ ℓ}. Take a single (ℓ,m) term from the sum in (6.7) and only the expectation Eωz (Xn−m− P(dωH(z))P 0 (Hz = m, ℓ− 1 ≤ max 0≤j≤m Xj · û− z · û < ℓ) ×Eωz (Xn−m − z) P(dωH(z))P 0 (Hz = m, ℓ− 1 ≤ max 0≤j≤m Xj · û− z · û < ℓ) ×Eωz (Xτ (ℓ)1 +n−m −Xτ (ℓ)1 ) (6.8) P(dωH(z))P 0 (Hz = m, ℓ− 1 ≤ max 0≤j≤m Xj · û− z · û < ℓ) ×Eωz (Xn−m −Xτ (ℓ)1 +n−m +Xτ (ℓ)1 − z) (6.9) The parameter ℓ in the regeneration time τ 1 of the walk started at z ensures that the subsequent walkX 1 + · stays inH′ℓ(z). Below we make use of this to get independence from the environments in H′ℓ(z)c. By (3.9) the quenched expectation in (6.9) can be bounded by Cpℓ p, for any p > 1. QUENCHED FUNCTIONAL CLT FOR RWRE 23 Integral (6.8) is developed further as follows. P(dωH(z))P 0 (Hz = m, ℓ− 1 ≤ max 0≤j≤m Xj · û− z · û < ℓ) × Eωz (Xτ (ℓ)1 +n−m −Xτ (ℓ)1 ) P(dωHℓ(z))P 0 (Hz = m, ℓ− 1 ≤ max 0≤j≤m Xj · û− z · û < ℓ) P(dωH′ (z))E z (Xτ (ℓ)1 +n−m P(dωHℓ(z))P 0 (Hz = m, ℓ− 1 ≤ max 0≤j≤m Xj · û− z · û < ℓ) × Ez(Xτ (ℓ)1 +n−m −Xτ (ℓ)1 |SH′ℓ(z)c) P(dωHℓ(z))P 0 (Hz = m, ℓ− 1 ≤ max 0≤j≤m Xj · û− z · û < ℓ) × E0(Xn−m|β = ∞). (6.10) The last equality above comes from the regeneration structure, see Proposition 1.3 in Sznitman-Zerner [17]. The σ-algebra SH′ (z)c is contained in the σ-algebra G1 defined by (1.22) of [17] for the walk starting at z. The last quantity (6.10) above reads the environment only until the first visit to z, hence does not see the distinction between ω and ω̃. Hence when the integral (6.7) is developed separately for ω and ω̃ into the sum of integrals (6.8) and (6.9), integrals (6.8) for ω and ω̃ cancel each other. We are left only with two instances of integral (6.9), one for both ω and ω̃. The last quenched expectation in (6.9) we bound by p as was mentioned above. Going back to (6.6), we get this bound: P(dωH(z)) Eω0 (Xn)− Eω̃0 (Xn) P(dωH(z)) ℓpP ω0 (Hz < n, ℓ− 1 ≤ max 0≤j≤Hz Xj · û− z · û < ℓ) P(dωH(z))E 0 [(MHz − z · û)p1I{Hz < n}] ≤ Cpnpε P(dωH(z))P 0 (Hz < n) + Cps −pe−sn For the last inequality we used (3.6) with ℓ = nε and some small ε, s > 0. Square, take z = xj , integrate as in (6.5), and use Jensen’s inequality to bring the square inside the integral to get E( |ζj − ζj−1|2 ) ≤ 2Cpn2pεE[ |P ω0 (Hxj < n)|2 ] + 2Cps−2pe−sn 24 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN Substitute these bounds into line (6.4) and note that the error there is Ŏ (1). E[ |Eω0 (Xn)−E0(Xn)|2 ] ≤ Cpn2pε E[ |P ω0 (Hz < n)|2 ] + Ŏ (nds−2pe−sn ) + Ŏ (1) = Cpn P0,0(z ∈ X[0,n) ∩ X̃[0,n)) + Ŏ (1) = Cpn 2pεE0,0[ |X[0,n) ∩ X̃[0,n)| ] + Ŏ (1). Utilize assumption (6.2) and take ε > 0 small enough so that 2α = 2pε + 2ᾱ < 1. (6.1) has been verified. � 7. Bound on intersections The remaining piece of the proof of Theorem 1.1 is this estimate: E0,0( |X[0,n) ∩ X̃[0,n)| ) = Ŏ (n2α) for some α < 1/2. (7.1) X and X̃ are two independent walks in a common environment with quenched distribution P ωx,y[X0,∞ ∈ A, X̃0,∞ ∈ B] = P ωx (A)P ωy (B) and averaged distribution Ex,y(·) = EP ωx,y(·). To deduce the sublinear bound we introduce regeneration times at which both walks regenerate on the same level in space (but not necessarily at the same time). Intersections happen only within the regeneration slabs, and the expected number of intersections decays exponentially in the distance between the points of entry of the walks in the slab. From regeneration to regeneration the difference of the two walks operates like a Markov chain. This Markov chain can be approximated by a symmetric random walk. Via this preliminary work the required estimate boils down to deriving a Green function bound for a Markov chain that can be suitably approximated by a symmetric random walk. This part is relegated to an appendix. Except for the appendix, we complete the proof of the functional central limit theorem in this section. To aid our discussion of a pair of walks (X, X̃) we introduce some new notation. We write θm,n for the shift on pairs of paths: θm,n(x0,∞, y0,∞) = (θ mx0,∞, θ ny0,∞). If we write separate expectations for X and X̃ under P ωx,y, these are denoted by E x and Ẽωy . By a joint stopping time we mean pair (α, α̃) that satisfies {α = m, α̃ = n} ∈ σ{X0,m, X̃0,n}. Under the distribution P ωx,y the walks X and X̃ are independent. Consequently if α ∨ α̃ <∞ P ωx,y-almost surely then for any events A and B, P ωx,y[(X0,α, X̃0,α̃) ∈ A, (Xα,∞, X̃α̃,∞) ∈ B] = Eωx,y 1I{(X0,α, X̃0,α̃) ∈ A}P ωXα, eXα̃{(X0,∞, X̃0,∞) ∈ B} QUENCHED FUNCTIONAL CLT FOR RWRE 25 This type of joint restarting will be used without comment in the sequel. For this section it will be convenient to have level stopping times and running maxima that are not defined relative to the initial level. γℓ = inf{n ≥ 0 : Xn · û ≥ ℓ} and γ+ℓ = inf{n ≥ 0 : Xn · û > ℓ}. Since û ∈ Zd, γ+ℓ is simply an abbreviation for γℓ+1. Let Mn = sup{Xi · û : i ≤ n} be the running maximum. M̃n, γ̃ℓ and γ̃ ℓ are the corresponding quantities for the X̃ walk. The first backtracking time for the X̃ walk is β̃ = inf{n ≥ 1 : X̃n · û < X̃0 · û}. Define L = inf{ℓ > (X0 · û) ∧ (X̃0 · û) : Xγℓ · û = X̃γ̃ℓ · û = ℓ} as the first fresh common level after at least one walk has exceeded its starting level. Set L = ∞ if there is no such common level. When the walks are on a common level, their difference will lie in the hyperplane Vd = {z ∈ Zd : z · û = 0}. We start with exponential tail bounds on the time to reach the common level. Lemma 7.1. There exist constants 0 < a1, a2, C < ∞ such that, for all x, y ∈ Zd, m ≥ 0 and P-a.e. ω, P ωx,y(γL ∨ γ̃L ≥ m) ≤ Cea1|y·û−x·û|−a2m. (7.2) For the proof we need a bound on the overshoot. Lemma 7.2. There exist constants 0 < C, s < ∞ such that, for any level k, any b ≥ 1, any x ∈ Zd such that x · û ≤ k, and P-a.e. ω, P ωx [Xγk · û ≥ k + b] ≤ Ce−sb. (7.3) Proof. From (3.1) it follows that for a constant C, for any level ℓ, any x ∈ Zd, and P-a.e. ω, Eωx [number of visits to level ℓ] = P ωx [Xn · û = ℓ] ≤ C. (7.4) (This is certainly clear if x · û = ℓ. Otherwise wait until the process first lands on level ℓ, if ever.) 26 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN From this and the exponential moment hypothesis we deduce the required bound on the overshoots: for any k, any x ∈ Zd such that x · û ≤ k, and P-a.e. ω, P ωx [Xγk · û ≥ k + b] = z·û<k P ωx [γk > n, Xn = z, Xn+1 · û ≥ k + b] z·û=k−ℓ P ωx [γk > n, Xn = z]P z [X1 · û ≥ k + b] z·û=k−ℓ P ωx [Xn = z]Ce −s(ℓ+b) ≤ Ce−sb e−sℓ ≤ Ce−sb. � Proof of Lemma 7.1. Consider first γL, and let us restrict ourselves to the case where the initial points x, y satisfy x · û < y · û. Perform an iterative construction of stopping times ηi, η̃i and levels ℓ(i), ℓ̃(i). Let η0 = η̃0 = 0, x0 = x and y0 = y. ℓ(0) and ℓ̃(0) need not be defined. Suppose that the construction has been done to stage i − 1 with xi−1 = Xηi−1 , yi−1 = X̃η̃i−1 , and xi−1 · û < yi−1 · û. Then set ℓ(i) = Xγ(yi−1·û) · û, ℓ̃(i) = X̃γ̃(ℓ(i)) · û, ηi = γ(ℓ̃(i)) and η̃i = γ̃(Xηi · û+ 1). In words, starting at (xi−1, yi−1) with yi−1 above xi−1, let X reach the level of yi−1 and let ℓ(i) be the level X lands on; let X̃ reach the level ℓ(i) and let ℓ̃(i) be the level X̃ lands on. Now let X try to establish a new common level at ℓ̃(i) with X̃ : in other words, follow X until the time ηi it reaches level ℓ̃(i) or above, and stop it there. Finally, reset the situation by letting X̃ reach a level strictly above the level of Xηi , and stop it there at time η̃i. The starting locations for the next step are xi = Xηi , yi = X̃η̃i that satisfy xi · û < yi · û. We show that within each step of the iteration there is a uniform lower bound on the probability that a fresh common level was found. For this purpose we utilize assumption (1.1) in the weaker form P{ω : P ω0 (Xγ1 · û = 1) ≥ κ } = 1. (7.5) Pick b large enough so that the bound in (7.3) is < 1. For z, w ∈ Zd such that z · û ≥ w · û define a function ψ(z, w) = P ωz,w[ Xγk · û = k for each k ∈ {z · û, . . . , z · û+ b}, X̃γ̃(z·û) · û− z · û ≤ b ] ≥ κb(1− Ce−sb) ≡ κ2 > 0. QUENCHED FUNCTIONAL CLT FOR RWRE 27 The uniform lower bound comes from the independence of the walks, from (7.3) and from iterating assumption (7.5). By the Markov property P ωxi−1,yi−1[Xγ(ℓ̃(i)) · û = ℓ̃(i)] ≥ P ωxi−1,yi−1[ X̃γ̃(ℓ(i)) · û− ℓ(i) ≤ b, Xγk · û = k for each k ∈ {ℓ(i), . . . , ℓ(i) + b} ] ≥ Eωxi−1,yi−1 ψ(Xγ(yi−1·û), yi−1) ≥ κ2. The first iteration on which the attempt to create a common level at ℓ̃(i) succeeds I = inf{i ≥ 1 : Xγ ℓ̃(i) · û = ℓ̃(i)}. Then ℓ̃(I) is a new fresh common level and consequently L ≤ ℓ̃(I). This gives the upper bound γL ≤ γℓ̃(I). We develop an exponential tail bound for γℓ̃(I), still under the assumption x · û < y · û. From the uniform bound above and the Markov property we get P ωx,y[I > i] ≤ (1− κ2)i. Lemma 7.2 gives an exponential bound P ωx,y (X̃η̃i − X̃η̃i−1) · û ≥ b ≤ Ce−sb (7.6) because the distance (X̃η̃i − X̃η̃i−1) · û is a sum of four overshoots: X̃η̃i − X̃η̃i−1 · û = X̃γ̃(Xηi ·û+1) · û−Xηi · û− 1 + 1 + Xγ(ℓ̃(i)) · û− ℓ̃(i) X̃γ̃(ℓ(i)) · û− ℓ(i) γ( eXη̃i−1 ·û) · û− X̃η̃i−1 · û Next, from the exponential tail bound on X̃η̃i − X̃η̃i−1 · û and from ℓ̃(i) ≤ X̃η̃i · û = X̃η̃j − X̃η̃j−1 · û+ y · û we get the large deviation estimate P ωx,y[ℓ̃(i) ≥ bi+ y · û] ≤ e−sbi for i ≥ 1 and b ≥ b0, for some constants 0 < s < ∞ (small enough) and 0 < b0 < ∞ (large enough). Combine this with the bound above on I to write P ωx,y[ℓ̃(I) ≥ a] ≤ P ωx,y[I > i] + P ωx,y[ℓ̃(i) ≥ a] ≤ e−si + esy·û−sa ≤ 2esy·û−sa where we assume a ≥ 2b0 + y · û and set the integer i = ⌊b−10 (a− y · û)⌋. Recall that 0 < s <∞ is a constant whose value can change from line to line. 28 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN From (3.1) and an exponential Chebyshev P ωx,y[γk > m] ≤ P ωx [Xm · û ≤ k] ≤ esk−sx·û−h1m for all x, y ∈ Zd, k ∈ Z and m ≥ 0. Above and in the remainder of this proof h1, h2 and h3 are small positive constants. Finally we derive P ωx,y[γℓ̃(I) > m] ≤ P ωx,y[ℓ̃(I) ≥ k + x · û] + P ωx,y[γk+x·û > m] ≤ 2es(y−x)·û−sk + esk−h1m ≤ Ces(y−x)·û−h2m. To justify the inequalities above assume m ≥ 4sb0/h1 > 4s/h1 and pick k in the range + (y − x) · û ≤ k ≤ 3h1m + (y − x) · û. To summarize, at this point we have P ωx,y[γL > m] ≤ Ces(y−x)·û−h2m for x · û < y · û. (7.7) To extend this estimate to the case x · û ≥ y · û, simply allow X̃ to go above x and then apply (7.7). By an application of the overshoot bound (7.3) and (7.7) at the point (x, X̃γ̃(x·û+1)) P ωx,y[γL > m] ≤ Eωx,yP ωx, eXγ̃(x·û+1)[γL > m] ≤ P ωx,y[X̃γ̃(x·û+1) > x · û+ εm] + Cesεm−h2m ≤ Ce−h3m if we take ε > 0 small enough. We have proved the lemma for γL, and the same argument works for γ̃L. � Assuming that X0 · û = X̃0 · û define the joint stopping times (ρ, ρ̃) = (γ+ , γ̃+ (ν1, ν̃1) = (ρ, ρ̃) + (γL, γ̃L) ◦ θρ,ρ̃ if ρ ∨ ρ̃ <∞ ∞ if ρ = ρ̃ = ∞. (7.8) Notice that ρ and ρ̃ are finite or infinite together, and they are infinite iff neither walk backtracks below its initial level (β = β̃ = ∞). Let ν0 = ν̃0 = 0 and for k ≥ 0 define (νk+1, ν̃k+1) = (νk, ν̃k) + (ν1, ν̃1) ◦ θνk,ν̃k . Finally let (ν, ν̃) = (γL, γ̃L), K = sup{k ≥ 0 : νk ∨ ν̃k <∞}, and (µ1, µ̃1) = (ν, ν̃) + (νK , ν̃K) ◦ θν,ν̃ . (7.9) These represent the first common regeneration times of the two paths. Namely, Xµ1 · û = X̃µ̃1 · û and for all n ≥ 1, Xµ1−n · û < Xµ1 · û ≤ Xµ1+n · û and X̃µ̃1−n · û < X̃µ̃1 · û ≤ X̃µ̃1+n · û. QUENCHED FUNCTIONAL CLT FOR RWRE 29 Next we extend the exponential tail bound to the regeneration times. Lemma 7.3. There exist constants 0 < C < ∞ and η̄ ∈ (0, 1) such that, for all x, y ∈ Vd = {z ∈ Zd : z · û = 0}, k ≥ 0, and P-a.e. ω, we have P ωx,y(µ1 ∨ µ̃1 ≥ k) ≤ C(1− η̄)k. (7.10) Proof. We prove geometric tail bounds successively for γ+1 , γ ℓ , γ , ρ, ν1, νk, and finally for µ1. To begin, (3.1) implies that P ω0 (γ 1 ≥ n) ≤ P ω0 (Xn−1 · û ≤ 1) ≤ es2(1− η1)n−1 with η1 = s2δ/2, for some small s2 > 0. By summation by parts Eω0 (e 1 ) ≤ es2Js3 , for a small enough s3 > 0 and Js = 1 + (e s − 1)/(1 − (1 − η1)es). By the Markov property for ℓ ≥ 1, Eω0 (e ℓ ) ≤ x·û>ℓ−1 Eω0 (e ℓ−1 , Xγ+ = x)Eωx (e But if x · û > ℓ− 1, then Eωx (es3γ ℓ ) ≤ ETxω0 (es3γ 1 ). Therefore by induction Eω0 (e ℓ ) ≤ (es2Js3)ℓ for any integer ℓ ≥ 0. (7.11) Next for an integer r ≥ 1, Eω0 (e Mr ) = Eω0 (e ℓ ,Mr = ℓ) ≤ Eω0 (e ℓ )1/2P ω0 (Mr = ℓ) (es2J2s4) ℓ/2(1I{ℓ < 3Mr|û|}+ e−s5ℓ)1/2 ≤ C(es2J2s4)Cr, for some C and for positive but small enough s2, s4, and s5. In the last inequality above we used the fact that es2J2s4 converges to 1 as first s4 ց 0 and then s2 ց 0. In the second-to-last inequality we used (3.2) to get the bound P ω0 (Xi · û ≥ ℓ) ≤ e−sℓeM |û|si ≤ Ce−s5ℓ if ℓ ≥ 3M |û|r. Above we assumed that the walk X starts at 0. Same bounds work for any x ∈ Vd because a shift orthogonal to û does not alter levels, in particular P ωx (Mr = ℓ) = P Txω0 (Mr = ℓ). By this same observation we show that for all x, y ∈ Vd Eωx,y(e fMr ) ≤ C(es2J2s4)Cr by repeating the earlier series of inequalities. 30 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN Using (3.1) and these estimates gives for x, y ∈ Vd P ωx,y(ρ ≥ n, β ∧ β̃ <∞) = P ωx,y(γ Mr∨fMr ≥ n, β ∧ β̃ = r) ≤ e−s4n/2 1≤r≤εn Eωx (e Mr )1/2Eωx,y(e fMr )1/2 P ωx {Xr · û < x · û}+ P ωy {X̃r · û < y · û} ≤ Cεne−s4n/2(es2J2s4)Cεn + C(1− s6δ/2)εn. Taking ε > 0 small enough shows the existence of a constant η2 > 0 such that for all x, y ∈ Vd, n ≥ 1, and P-a.e. ω, P ωx,y(ρ ≥ n, β ∧ β̃ <∞) ≤ C(1− η2)n. Same bound works for ρ̃ also. We combine this with (7.2) to get a geometric tail bound for ν11I{β ∧ β̃ <∞}. Recall definition (7.8) and take ε > 0 small. P ωx,y[ν1 ≥ k, β ∧ β̃ <∞] ≤ P ωx,y[ρ ≥ k/2, β ∧ β̃ <∞] + P ωx,y[β ∧ β̃ <∞, |Xρ · û− X̃ρ̃ · û| > εk] + P ωx,y[γL ◦ θρ,ρ̃ ≥ k/2, β ∧ β̃ <∞, |Xρ · û− X̃ρ̃ · û| ≤ εk ]. On the right-hand side above we have an exponential bound for each of the three probabilities: the first probability gets it from the estimate immediately above, the second from a combination of that and (3.2), and the third from (7.2): P ωx,y[γL ◦ θρ,ρ̃ ≥ k/2, β ∧ β̃ <∞, |Xρ · û− X̃ρ̃ · û| ≤ εk ] = Eωx,y 1I{β ∧ β̃ <∞, |Xρ · û− X̃ρ̃ · û| ≤ εk}P ωXρ , eXρ̃{γL ≥ k/2} ≤ Cea1εk−a2k/2. The constants in the last bound above are those from (7.2), and we choose ε < a2/(2a1). We have thus established that Eωx,y(e s7ν11I{β ∧ β̃ <∞}) ≤ J̄s7 for a small enough s7 > 0, with J̄s = C(1− (1− η3)es)−1 and η3 > 0. To move from ν1 to νk use the Markov property and induction: Eωx,y(e s7νk1I{νk ∨ ν̃k <∞}) Eωx,y(e s7νk−11I{νk−1 ∨ ν̃k−1 <∞, Xνk−1 = z, X̃ν̃k−1 = z̃}) × Eωz,z̃(es7ν11I{β ∧ β̃ <∞}) ≤ J̄s7Eωx,y(es7νk−11I{νk−1 ∨ ν̃k−1 <∞}) ≤ · · · ≤ J̄ks7. QUENCHED FUNCTIONAL CLT FOR RWRE 31 Next, use the Markov property at the joint stopping times (νk, ν̃k), (7.2), (3.7), and induction to derive P ωx,y(K ≥ k) ≤ P ωx,y(νk ∨ ν̃k <∞) P ωx,y(νk−1 ∨ ν̃k−1 <∞, Xνk−1 = z, X̃ν̃k−1 = z̃)P ωz,z̃(β ∧ β̃ <∞) ≤ (1− η2)P ωx,y(νk−1 ∨ ν̃k−1 <∞) ≤ (1− η2)k. Finally use the Cauchy-Schwarz and Chebyshev inequalities to write P ωx,y(νK ≥ n) = P ωx,y(νk ≥ n,K = k) (1− η2)k + e−s7n 1≤k≤εn Eωx,y(e s7νk1I{νk ∨ ν̃k <∞}) ≤ C(1− η2)εn + Cεne−s7nJ̄εns7 . Looking at the definition (7.9) of µ1 we see that an exponential tail bound follows by applying (7.2) to the ν-part and by taking ε > 0 small enough in the last calculation above. Repeat the same argument for µ̃1 to conclude the proof of (7.10). � After these preliminaries define the sequence of common regeneration times by µ0 = µ̃0 = 0 and (µi+1, µ̃i+1) = (µi, µ̃i) + (µ1, µ̃1) ◦ θµi,µ̃i. (7.12) The next tasks are to identify suitable Markovian structures and to develop a cou- pling. Proposition 7.4. The process (X̃µ̃i−Xµi)i≥1 is a Markov chain on Vd with transition probability q(x, y) = P0,x[X̃µ̃1 −Xµ1 = y | β = β̃ = ∞]. (7.13) Note that the time-homogeneous Markov chain does not start from X̃0−X0 because the transition to X̃µ̃1 −Xµ1 does not include the condition β = β̃ = ∞. Proof. Express the iteration of the common regeneration times as (µi, µ̃i) = (µi−1, µ̃i−1) + (ν, ν̃) + (νK , ν̃K) ◦ θν,ν̃ ◦ θµi−1,µ̃i−1 , i ≥ 1. Let Ki be the value of K at the ith iteration: Ki = K ◦ θν,ν̃ ◦ θµi−1,µ̃i−1 . 32 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN Let n ≥ 2 and z1, . . . , zn ∈ Vd. Write P0,z[X̃µ̃i −Xµi = zi for 1 ≤ i ≤ n] (7.14) (ki,mi,m̃i,vi,ṽi)1≤i≤n−1∈Ψ Ki = ki, µi = mi, µ̃i = m̃i, Xmi = vi and X̃m̃i = ṽi for 1 ≤ i ≤ n− 1, (X̃µ̃1 −Xµ1) ◦ θmn−1,m̃n−1 = zn Above Ψ is the set of vectors (ki, mi, m̃i, vi, ṽi)1≤i≤n−1 such that ki is nonnegative and mi, m̃i, vi · û, and ṽi · û are all positive and strictly increasing in i, and ṽi − vi = zi. Define the events Ak,b,b̃ = {ν + νk ◦ θν,ν̃ = b , ν̃ + ν̃k ◦ θν,ν̃ = b̃} Bb,b̃ = {Xj · û ≥ X0 · û for 1 ≤ j ≤ b , X̃j · û ≥ X̃0 · û for 1 ≤ j ≤ b̃}. Let m0 = m̃0 = 0, bi = mi −mi−1 and b̃i = m̃i − m̃i−1. Rewrite the sum from above (ki,mi,m̃i,vi,ṽi)1≤i≤n−1∈Ψ [ n−1∏ 1I{Aki,bi,b̃i} ◦ θ mi−1,m̃i−1 1I{Bbi, b̃i} ◦ θ mi−1,m̃i−1 , Xmi = vi and X̃m̃i = ṽi for 1 ≤ i ≤ n− 1, β ◦ θmn−1 = β̃ ◦ θm̃n−1 = ∞ , (X̃µ̃1 −Xµ1) ◦ θmn−1,m̃n−1 = zn Next restart the walks at times (mn−1, m̃n−1) to turn the sum into the following. (ki,mi,m̃i,vi,ṽi)1≤i≤n−1∈Ψ Eω0,z [ n−1∏ 1I{Aki,bi,b̃i} ◦ θ mi−1,m̃i−1 1I{Bbi, b̃i} ◦ θ mi−1,m̃i−1 , Xmi = vi and X̃m̃i = ṽi for 1 ≤ i ≤ n− 1 × P ωvn−1 , ṽn−1 β = β̃ = ∞ , X̃µ̃1 −Xµ1 = zn Inside the outermost braces the events in the first quenched expectation force the level ℓ = Xmn−1 · û = vn−1 · û = X̃m̃n−1 · û = ṽn−1 · û QUENCHED FUNCTIONAL CLT FOR RWRE 33 to be a new maximal level for both walks. Consequently the first quenched expecta- tion is a function of {ωx : x · û < ℓ} while the last quenched probability is a function of {ωx : x · û ≥ ℓ}. By independence of the environments, the sum becomes (ki,mi,m̃i,vi,ṽi)1≤i≤n−1∈Ψ [ n−1∏ 1I{Aki,bi,b̃i} ◦ θ mi−1,m̃i−1 (7.15) 1I{Bbi, b̃i} ◦ θ mi−1,m̃i−1 , Xmi = vi and X̃m̃i = ṽi for 1 ≤ i ≤ n− 1 × Pvn−1 , ṽn−1 β = β̃ = ∞ , X̃µ̃1 −Xµ1 = zn By a shift and a conditioning the last probability transforms as follows. Pvn−1 , ṽn−1 β = β̃ = ∞ , X̃µ̃1 −Xµ1 = zn = P0,zn−1 X̃µ̃1 −Xµ1 = zn ∣∣ β = β̃ = ∞ Pvn−1 , ṽn−1 β = β̃ = ∞ = q(zn−1, zn)Pvn−1 , ṽn−1 β = β̃ = ∞ Now reverse the above use of independence to put the probability Pvn−1 , ṽn−1 [β = β̃ = ∞] back together with the expectation (7.15). Inside this expectation this furnishes the event β ◦ θmn−1 = β̃ ◦ θm̃n−1 = ∞ and with this the union of the entire collection of events turns back into X̃µ̃i −Xµi = zi for 1 ≤ i ≤ n−1. Going back to the beginning on line (7.14) we see that we have now shown P0,z[X̃µ̃i −Xµi = zi for 1 ≤ i ≤ n] = P0,z[X̃µ̃i −Xµi = zi for 1 ≤ i ≤ n− 1]q(zn−1, zn). Continue by induction. � The Markov chain Yk = X̃µ̃k − Xµk will be compared to a random walk obtained by performing the same construction of joint regeneration times to two independent walks in independent environments. To indicate the difference in construction we change notation. Let the pair of walks (X, X̄) obey P0 ⊗Pz with z ∈ Vd, and denote the first backtracking time of the X̄ walk by β̄ = inf{n ≥ 1 : X̄n · û < X̄0 · û}. Construct the common regeneration times (ρk, ρ̄k)k≥1 for (X, X̄) by the same recipe [(7.8), (7.9) and (7.12)] as was used to construct (µk, µ̃k)k≥1 for (X, X̃). Define Ȳk = X̄ρ̄k − Xρk . An analogue of the previous proposition, which we will not spell out, shows that (Ȳk)k≥1 is a Markov chain with transition q̄(x, y) = P0 ⊗ Px[X̄ρ̄1 −Xρ1 = y | β = β̄ = ∞]. (7.16) 34 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN In the next two proofs we make use of the following decomposition. Suppose x · û = y · û = 0, and let (x1, y1) be another pair of points on a common, higher level: x1 · û = y1 · û = ℓ > 0. Then we can write {(X0, X̃0) = (x, y), β = β̃ = ∞, (Xµ1 , X̃µ̃1) = (x1, y1)} (γ,γ̃) {X0,n(γ) = γ, X̃0,n(γ̃) = γ̃, β ◦ θn(γ) = β̃ ◦ θn(γ̃) = ∞}. (7.17) Here (γ, γ̃) range over all pairs of paths that connect (x, y) to (x1, y1), that stay between levels 0 and ℓ−1 before the final points, and for which a common regeneration fails at all levels before ℓ. n(γ) is the index of the final point along the path, so for example γ = (x = z0, z1, . . . , zn(γ)−1, zn(γ) = x1). Proposition 7.5. The process (Ȳk)k≥1 is a symmetric random walk on Vd and its transition probability satisfies q̄(x, y) = q̄(0, y − x) = q̄(0, x− y) = P0 ⊗ P0[X̄ρ̄1 −Xρ1 = y − x | β = β̄ = ∞]. Proof. It remains to show that for independent (X, X̄) the transition (7.16) reduces to a symmetric random walk. This becomes obvious once probabilities are decomposed into sums over paths because the events of interest are insensitive to shifts by z ∈ Vd. P0 ⊗ Px[β = β̄ = ∞ , X̄ρ̄1 −Xρ1 = y] P0 ⊗ Px[β = β̄ = ∞ , Xρ1 = w , X̄ρ̄1 = y + w] (γ,γ̄) P0[X0,n(γ) = γ, β ◦ θn(γ) = ∞]Px[X0,n(γ̄) = γ̄, β ◦ θn(γ̄) = ∞] (γ,γ̄) P0[X0,n(γ) = γ]Px[X0,n(γ̄) = γ̄] P0[β = ∞] (7.18) Above we used the decomposition idea from (7.17). Here (γ, γ̄) range over the appropriate class of pairs of paths in Zd such that γ goes from 0 to w and γ̄ goes from x to y + w. The independence for the last equality above comes from noticing that the quenched probabilities P ω0 [X0,n(γ) = γ] and P w [β = ∞] depend on independent collections of environments. The probabilities on the last line of (7.18) are not changed if each pair (γ, γ̄) is replaced by (γ, γ′) = (γ, γ̄−x). These pairs connect (0, 0) to (w, y−x+w). Because x ∈ Vd satisfies x · û = 0, the shift has not changed regeneration levels. This shift turns Px[X0,n(γ̄) = γ̄] on the last line of (7.18) into P0[X0,n(γ′) = γ ′]. We can reverse the steps in (7.18) to arrive at the probability P0 ⊗ P0[β = β̄ = ∞ , X̄ρ̄1 −Xρ1 = y − x]. This proves q̄(x, y) = q̄(0, y − x). QUENCHED FUNCTIONAL CLT FOR RWRE 35 Once both walks start at 0 it is immaterial which is labeled X and which X̄ , hence symmetry holds. � It will be useful to know that q̄ inherits all possible transitions from q. Lemma 7.6. If q(z, w) > 0 then also q̄(z, w) > 0. Proof. By the decomposition from (7.17) we can express Px,y[(Xµ1 , X̃µ̃1) = (x1, y1)|β = β̃ = ∞] = (γ,γ̃) EP ω[γ]P ω[γ̃]P ωx1 [β = ∞]P [β = ∞] Px,y[β = β̃ = ∞] If this probability is positive, then at least one pair (γ, γ̃) satisfies EP ω[γ]P ω[γ̃] > 0. This implies that P [γ]P [γ̃] > 0 so that also Px ⊗ Py[(Xµ1 , X̃µ̃1) = (x1, y1)|β = β̃ = ∞] > 0. � In the sequel we detach the notations Y = (Yk) and Ȳ = (Ȳk) from their original definitions in terms of the walks X , X̃ and X̄, and use (Yk) and (Ȳk) to denote canonical Markov chains with transitions q and q̄. Now we construct a coupling. Proposition 7.7. The single-step transitions q(x, y) for Y and q̄(x, y) for Ȳ can be coupled in such a way that, when the processes start from a common state x, Px,x[Y1 6= Ȳ1] ≤ Ce−α1|x| for all x ∈ Vd. Here C and α1 are finite positive constants independent of x. Proof. We start by constructing a coupling of three walks (X, X̃, X̄) such that the pair (X, X̃) has distribution Px,y and the pair (X, X̄) has distribution Px ⊗ Py. First let (X, X̃) be two independent walks in a common environment ω as before. Let ω̄ be an environment independent of ω. Define the walk X̄ as follows. Initially X̄0 = X̃0. On the sites {Xk : 0 ≤ k < ∞} X̄ obeys environment ω̄, and on all other sites X̄ obeys ω. X̄ is coupled to agree with X̃ until the time T = inf{n ≥ 0 : X̄n ∈ {Xk : 0 ≤ k <∞}} it hits the path of X . The coupling between X̄ and X̃ can be achieved simply as follows. Given ω and ω̄, for each x create two independent i.i.d. sequences (zxk )k≥1 and (z̄ k)k≥1 with distri- butions Qω,ω̄[zxk = y] = πx,x+y(ω) and Q ω,ω̄[z̄xk = y] = πx,x+y(ω̄). Do this independently at each x. Each time the X̃-walk visits state x, it uses a new zxk variable as its next step, and never reuses the same z k again. The X̄ walk operates the same way except that it uses the variables z̄xk when x ∈ {Xk} and the zxk variables when x /∈ {Xk}. Now X̄ and X̃ follow the same steps zxk until X̄ hits the set {Xk}. It is intuitively obvious that the walksX and X̄ are independent because they never use the same environment. The following calculation verifies this. Let X0 = x0 = x 36 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN and X̃ = X̄ = y0 = y be the initial states, and Px,y the joint measure created by the coupling. Fix finite vectors x0,n = (x0, . . . , xn) and y0,n = (y0, . . . , yn) and recall also the notation X0,n = (X0, . . . , Xn). The description of the coupling tells us to start as follows. Px,y[X0,n = x0,n, X̄0,n = y0,n] = P(dω) P(dω̄) P ωx (dz0,∞)1I{z0,n = x0,n} i:yi /∈{zk : 0≤k<∞} πyi,yi+1(ω) · i:yi∈{zk: 0≤k<∞} πyi,yi+1(ω̄) [by dominated convergence] = lim P(dω) P(dω̄) P ωx (dz0,N) 1I{z0,n = x0,n} i:yi /∈{zk : 0≤k≤N} πyi,yi+1(ω) · i:yi∈{zk: 0≤k≤N} πyi,yi+1(ω̄) = lim z0,N :z0,n=x0,n P(dω)P ωx [X0,N = z0,N ] i:yi /∈{zk: 0≤k≤N} πyi,yi+1(ω) P(dω̄) i:yi∈{zk: 0≤k≤N} πyi,yi+1(ω̄) [by independence of the two functions of ω] = lim z0,N :z0,n=x0,n P(dω)P ωx [X0,N = z0,N ] P(dω) i:yi/∈{zk : 0≤k≤N} πyi,yi+1(ω) P(dω̄) i:yi∈{zk: 0≤k≤N} πyi,yi+1(ω̄) = Px[X0,n = x0,n] · Py[X0,n = y0,n]. Thus at this point the coupled pairs (X, X̃) and (X, X̄) have the desired marginals Px,y and Px ⊗ Py. Next construct the common regeneration times (µ1, µ̃1) for (X, X̃) and (ρ1, ρ̄1) for (X, X̄) by the earlier recipes. Define two pairs of walks stopped at their common regeneration times: (Γ, Γ̄) ≡ (X0, µ1 , X̃0, µ̃1), (X0, ρ1 , X̄0, ρ̄1) . (7.19) Suppose the sets X[0, µ1∨ρ1) and X̃[0, µ̃1∨ρ̄1) do not intersect. Then the construction implies that the path X̄0, µ̃1∨ρ̄1 agrees with X̃0, µ̃1∨ρ̄1, and this forces the equalities (µ1, µ̃1) = (ρ1, ρ̄1) and (Xµ1 , X̃µ̃1) = (Xρ1 , X̄ρ̄1). We insert an estimate on this event. QUENCHED FUNCTIONAL CLT FOR RWRE 37 Lemma 7.8. There exist constants 0 < C, s < ∞ such that, for all x, y ∈ Vd and P-a.e. ω, P ωx,y(X[0, µ1∨ρ1) ∩ X̃[0, µ̃1∨ρ̄1) 6= ∅) ≤ Ce−s|x−y|. (7.20) Proof. Write P ωx,y(X[0, µ1∨ρ1) ∩ X̃[0, µ̃1∨ρ̄1) 6= ∅) ≤ P ωx,y(µ1 ∨ µ̃1 ∨ ρ1 ∨ ρ̄1 > ε|x− y|) + P ωx ( max 1≤i≤ε|x−y| |Xi − x| ≥ |x− y|/2) + P ωy ( max 1≤i≤ε|x−y| |Xi − y| ≥ |x− y|/2). By (7.10) and its analogue for (ρ1, ρ̄1) the first term on the right-hand-side decays exponentially in |x − y|. Using (3.2) the second and third terms are bounded by ε|x − y|e−s|x−y|/2eεs|x−y|M , for s > 0 small enough. Choosing ε > 0 small enough finishes the proof. � From (7.20) we obtain (Xµ1 , X̃µ̃1) 6= (Xρ1, X̄ρ̄1) ≤ Px,y Γ 6= Γ̄ ≤ Ce−s|x−y|. (7.21) But we are not finished yet: it remains to include the conditioning on no back- tracking. For this purpose generate an i.i.d. sequence (X(m), X̃(m), X̄(m))m≥1, each triple constructed as above. Continue to write Px,y for the probability measure of the entire sequence. Let M be the first m such that the paths (X(m), X̃(m)) do not backtrack, which means that k · û ≥ X 0 · û and X̃ k · û ≥ X̃ 0 · û for all k ≥ 1. Similarly define M̄ for (X(m), X̄(m))m≥1. M and M̄ are stochastically bounded by geometric random variables by (3.7). The pair of walks (X(M), X̃(M)) is now distributed as a pair of walks under the measure Px,y[ · |β = β̃ = ∞], while (X(M̄), X̄(M̄)) is distributed as a pair of walks under Px ⊗ Py[ · |β = β̄ = ∞]. Let also again Γ(m) = (X 0 , µ 0 , µ̃ ) and Γ̄(m) = (X 0 , ρ 0 , ρ̄ be the pairs of paths run up to their common regeneration times. Consider the two pairs of paths (Γ(M), Γ̄(M̄)) chosen by the random indices (M, M̄). We insert one more lemma. Lemma 7.9. For s > 0 as above, and a new constant 0 < C <∞, Γ(M) 6= Γ̄(M̄) ≤ Ce−s|x−y|/2. (7.22) 38 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN Proof. Let Am be the event that the walks X̃(m) and X̄(m) agree up to the maximum 1 ∨ ρ̄ 1 of their regeneration times. The equalities M = M̄ and Γ (M) = Γ̄(M̄) are a consequence of the event A1∩· · ·∩AM , for the following reason. As pointed out earlier, on the event Am we have the equality of the regeneration times µ̃(m)1 = ρ̄ 1 and of the stopped paths X̃ 0 , µ̃ 0 , ρ̄ . By definition, these walks do not backtrack after the regeneration time. Since the walks X̃(m) and X̄(m) agree up to this time, they must backtrack or fail to backtrack together. If this is true for each m = 1, . . . ,M , it forces M̄ = M , since the other factor in deciding M and M̄ are the paths X(m) that are common to both. And since the paths agree up to the regeneration times, we have Γ(M) = Γ̄(M̄). Estimate (7.22) follows: Γ(M) 6= Γ̄(M̄) ≤ Px,y Ac1 ∪ · · · ∪ AcM Px,y[M ≥ m, Acm ] ≤ Px,y[M ≥ m] )1/2( Px,y[Acm] ≤ Ce−s|x−y|/2. The last step comes from the estimate in (7.20) for each Acm and the geometric bound on M . � We are ready to finish the proof of Proposition 7.7. To create initial conditions Y0 = Ȳ0 = x take initial states (X 0 , X̃ 0 ) = (X 0 , X̄ 0 ) = (0, x). Let the final outcome of the coupling be the pair (Y1, Ȳ1) = − X(M) − X(M̄) under the measure P0,x. The marginal distributions of Y1 and Ȳ1 are correct [namely, given by the transitions (7.13) and (7.16)] because, as argued above, the pairs of walks themselves have the right marginal distributions. The event Γ(M) = Γ̄(M̄) implies Y1 = Ȳ1, so estimate (7.22) gives the bound claimed in Proposition 7.7. � The construction of the Markov chain is complete, and we return to the main development of the proof. It remains to prove a sublinear bound on the expected number E0,0|X[0,n)∩ X̃[0,n)| of common points of two independent walks in a common environment. Utilizing the common regeneration times, write E0,0|X[0,n) ∩ X̃[0,n)| ≤ E0,0|X[µi,µi+1) ∩ X̃[µ̃i,µ̃i+1)|. (7.23) The term i = 0 is a finite constant by bound (7.10) because the number of common points is bounded by the number µ1 of steps. For each 0 < i < n apply a decom- position into pairs of paths from (0, 0) to given points (x1, y1) in the style of (7.17): QUENCHED FUNCTIONAL CLT FOR RWRE 39 (γ, γ̃) are the pairs of paths with the property that (γ,γ̃) {X0,n(γ) = γ, X̃0,n(γ̃) = γ̃, β ◦ θn(γ) = β̃ ◦ θn(γ̃) = ∞} = {X0 = X̃0 = 0, Xµi = x1, X̃µ̃i = y1}. Each term i > 0 in (7.23) we rearrange as follows. E0,0|X[µi,µi+1) ∩ X̃[µ̃i,µ̃i+1)| x1,y1 (γ,γ̃) EP ω0,0[X0,n(γ) = γ, X̃0,n(γ̃) = γ̃] ×Eωx1,y1(1I{β = β̃ = ∞}|X[0 , µ1) ∩ X̃[0 , µ̃1)| ) x1,y1 (γ,γ̃) EP ω0,0[X0,n(γ) = γ, X̃0,n(γ̃) = γ̃]P x1,y1 [β = β̃ = ∞] ×Eωx1,y1( |X[0 , µ1) ∩ X̃[0 , µ̃1)| | β = β̃ = ∞ ) x1,y1 EP ω0,0[Xµi = x1, X̃µ̃i = y1]E x1,y1 ( |X[0 , µ1) ∩ X̃[0 , µ̃1)| | β = β̃ = ∞ ). The last conditional quenched expectation above is handled by estimates (3.7), (7.10), (7.20) and Schwarz inequality: Eωx1,y1( |X[0 , µ1) ∩ X̃[0 , µ̃1)| | β = β̃ = ∞ ) ≤ η −2Eωx1,y1( |X[0 , µ1) ∩ X̃[0 , µ̃1)| ) ≤ η−2Eωx1,y1(µ1 · 1I{X[0 , µ1) ∩ X̃[0 , µ̃1) 6= ∅} ) ≤ η−2 Eωx1,y1[µ )1/2( P ωx1,y1{X[0 , µ1) ∩ X̃[0 , µ̃1) 6= ∅} ≤ Ce−s|x1−y1|/2. Define h(x) = Ce−s|x|/2, insert the last bound back up, and appeal to the Markov property established in Proposition 7.4: E0,0|X[µi,µi+1) ∩ X̃[µ̃i,µ̃i+1)| ≤ E0,0 h(X̃µ̃i −Xµi) P0,0[X̃µ̃1 −Xµ1 = x] qi−1(x, y)h(y). In order to apply Theorem A.1 from the Appendix, we check its hypotheses in the next lemma. Assumption (1.2) enters here for the first and only time. Lemma 7.10. The Markov chain (Yk)k≥0 with transition q(x, y) and the symmet- ric random walk (Ȳk)k≥0 with transition q̄(x, y) satisfy assumptions (A.i), (A.ii) and (A.iii) stated in the beginning of the Appendix. Proof. From Lemma 7.3 and (3.2) we get moment bounds E0,x|X̄ρ̄k |m + E0,x|Xρk |m <∞ 40 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN for any power m < ∞. This gives assumption (A.i), namely that E0|Ȳ1|3 < ∞. The second part of assumption (A.ii) comes from Lemma 7.6. Assumption (A.iii) comes from Proposition 7.7. The only part that needs work is the first part of assumption (A.ii). We show that it follows from part (1.2) of Hypothesis (R). By (1.2) and non-nestling (N) there exist two non-zero vectors y 6= z such that z · û > 0 and Eπ0,yπ0,z > 0. Now we have a number of cases to consider. In each case we should describe an event that gives Y1−Y0 a particular nonzero value and whose probability is bounded away from zero, uniformly over x = Y0. Case 1: y is noncollinear with z. The sign of y · û gives three subcases. We do the trickiest one explicitly. Assume y · û < 0. Find the smallest positive integer b such that (y + bz) · û > 0. Then find the minimal positive integers k,m such that k(y + bz) · û = mz · û. Below Px is the path measure of the Markov chain (Yk) and then P0,x the measure of the walks (X, X̃) as before. Px{Y1 − Y0 = ky + (kb−m)z} ≥ P0,x X̃µ̃1 = x+ ky + (k + 1)bz , Xµ1 = (m+ b)z , β = β̃ = ∞ P Txω0 {Xi(b+1)+1 = i(y + bz) + z, . . . , Xi(b+1)+b = i(y + bz) + bz, X(i+1)(b+1) = (i+ 1)(y + bz) for 0 ≤ i ≤ k − 1, and then Xk(b+1)+1 = k(y + bz) + z , . . . , Xk(b+1)+b = k(y + bz) + bz } × P ω0 {X1 = z , X2 = 2z , . . . , Xm+b = (m+ b)z } × P ωx+ky+(k+1)bz{β = ∞}P ω(m+b)z{β = ∞} Regardless of possible intersections of the paths, assumption (1.2) and inequality (3.7) imply that the quantity above has a positive lower bound that is independent of x. The assumption that y, z are nonzero and noncollinear ensures that ky+(kb−m)z 6= 0. Case 2: y is collinear with z. Then there is a vector w 6∈ Rz such that Eπ0,w > 0. If w · û ≤ 0, then by Hypothesis (N) there exists u such that u · û > 0 and Eπ0,wπ0,u > 0. If u is collinear with z, then replacing z by u and y by w puts us back in Case 1. So, replacing w by u if necessary, we can assume that w · û > 0. We have four subcases, depending on whether x = 0 or not and y · û < 0 or not. (2.a) The case x 6= 0 is resolved simply by taking paths consisting of only w-steps for one walk and only z-steps for the other, until they meet on a common level and then never backtrack. (2.b) The case y · û > 0 corresponds to Case 3 in the proof of [11, Lemma 5.5]. (2.c) The only case left is x = 0 and y · û < 0. Let b and c be the smallest positive integers such that (y + bw) · û ≥ 0 and (y + cz) · û > 0. Choose minimal positive QUENCHED FUNCTIONAL CLT FOR RWRE 41 integeres m ≥ b and n > c such that m(w · û) = n(z · û). Then, P0{Y1 − Y0 = nz −mw} ≥ P0,0{X̃µ̃1 = y + bw + nz,Xµ1 = y + (b+m)w} P ω0 {Xi = iw for 1 ≤ i ≤ b and Xb+1+j = y + (b+ j)w for 0 ≤ j ≤ m} × P ω0 {Xi = iw for 0 ≤ i ≤ b,Xb+1 = bw + z and then Xb+1+j = y + bw + jz for 1 ≤ j ≤ n} × P ωy+(b+m)w(β = ∞)P ωy+bw+nz(β = ∞) Since w and z are noncollinear, mw 6= nz. For the same reason, w-steps are always taken at points not visited before. This makes the above lower bound positive. By the choice of b and z · û > 0, neither walk dips below level 0. We can see that the first common regeneration level for the two paths is (y+ bw+ nz)·û. The first walk backtracks from level bw ·û so this is not a common regeneration level. The second walk splits from the first walk at bw, takes a z-step up, and then backtracks using a y-step. So the common regeneration level can only be at or above level (y+ bw+(c+1)z) · û. The fact that n > c ensures that (y+ bw+nz) · û is high enough. The minimality of n ensures that this is the first such level. � Now that the assumptions have been checked, Theorem A.1 gives constants 0 < C <∞ and 0 < η < 1 such that qi−1(x, y)h(y) ≤ Cn1−η for all x ∈ Vd and n ≥ 1. Going back to (7.23) and collecting the bounds along the way gives the final estimate E0,0|X[0,n) ∩ X̃[0,n)| ≤ Cn1−η for all n ≥ 1. This is (6.2) which was earlier shown to imply condition (2.1) required by Theorem 2.1. Previous work in Sections 2 and 5 convert the CLT from Theorem 2.1 into the main result Theorem 1.1. The entire proof is complete, except for the Green function estimate furnished by the Appendix. Appendix A. A Green function type bound Let us write a d-vector in terms of coordinates as x = (x1, . . . , xd), and similarly for random vectors X = (X1, . . . , Xd). Let Y = (Yk)k≥0 be a Markov chain on Z d with transition probability q(x, y), and let Ȳ = (Ȳk)k≥0 be a symmetric random walk on Z d with transition probability q̄(x, y) = q̄(y, x) = q̄(0, y − x). Make the following assumptions. (A.i) A third moment bound E0|Ȳ1|3 <∞. 42 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN (A.ii) Some uniform nondegeneracy: there is at least one index j ∈ {1, . . . , d} and a constant κ0 such that the coordinate Y j satisfies Px{Y j1 − Y 0 ≥ 1} ≥ κ0 > 0 for all x. (A.1) (The inequality ≥ 1 can be replaced by ≤ −1, the point is to assure that a cube is exited fast enough.) Furthermore, for every i ∈ {1, . . . , d}, if the one-dimensional random walk Ȳ i is degenerate in the sense that q̄(0, y) = 0 for yi 6= 0, then so is the process Y i in the sense that q(x, y) = 0 whenever xi 6= yi. In other words, any coordinate that can move in the Y chain somewhere in space can also move in the Ȳ walk. (A.iii) Most importantly, assume that for any initial state x the transitions q and q̄ can be coupled so that Px,x[Y1 6= Ȳ1] ≤ Ce−α1|x| where 0 < C, α1 <∞ are constants independent of x. Throughout the section C will change value but α1 remains the constant in the assumption above. Let h be a function on Zd such that 0 ≤ h(x) ≤ Ce−α2|x| for constants 0 < α2, C < ∞. This section is devoted to proving the following Green function type bound on the Markov chain. Theorem A.1. There are constants 0 < C, η <∞ such that Ezh(Yk) = P0(Yk = y) ≤ Cn1−η for all n ≥ 1 and z ∈ Zd. To prove the estimate, we begin by discarding terms outside a cube of side r = c1 logn. Bounding probabilities crudely by 1 gives |y|>c1 logn Pz(Yk = y) ≤ n |y|>c1 logn h(y) ≤ Cn k>c1 logn kd−1e−α2k k>c1 logn e−(α2/2)k ≤ Cne−(α2/2)c1 logn ≤ Cn1−η as long as n is large enough so that kd−1 ≤ eα2k/2, and this works for any c1. B = [−c1 log n, c1 log n]d. Since h is bounded, it now remains to show that Pz(Yk ∈ B) ≤ Cn1−η. (A.2) For this we can assume z ∈ B since accounting for the time to enter B for the first time can only improve the estimate. QUENCHED FUNCTIONAL CLT FOR RWRE 43 Bound (A.2) will be achieved in two stages. First we show that the Markov chain Y does not stay in B longer than a time whose mean is a power of the size of B. Second, we show that often enough Y follows the random walk Ȳ during its excursions outside B. The random walk excursions are long and thereby we obtain (A.2). Thus our first task is to construct a suitable coupling of Y and Ȳ . Lemma A.1. Let ζ = inf{n ≥ 1 : Ȳ ∈ A} be the first entrance time of Ȳ into some set A ⊆ Zd. Then we can couple Y and Ȳ so that Px,x[ Yk 6= Ȳk for some 1 ≤ k ≤ ζ ] ≤ CEx e−α1|Ȳk|. The proof shows that the statement works also if ζ = ∞ is possible, but we will not need this case. Proof. For each state x create an i.i.d. sequence (Zxk , Z̄ k )k≥1 such that Z k has distri- bution q(x, x+ · ), Z̄xk has distribution q̄(x, x+ · ) = q̄(0, · ), and each pair (Zxk , Z̄xk ) is coupled so that P (Zxk 6= Z̄xk ) ≤ Ce−α1|x|. For distinct x these sequences are inde- pendent. Construct the process (Yn, Ȳn) as follows: with counting measures Ln(x) = 1I{Yk = x} and L̄n(x) = 1I{Ȳk = x} (n ≥ 0) and with initial point (Y0, Ȳ0) given, define for n ≥ 1 Yn = Yn−1 + Z Ln−1(Yn−1) and Ȳn = Ȳn−1 + Z̄ Ȳn−1 L̄n−1(Ȳn−1) In words, every time the chain Y visits a state x, it reads its next jump from a new variable Zxk which is then discarded and never used again. And similarly for Ȳ . This construction has the property that, if Yk = Ȳk for 0 ≤ k ≤ n with Yn = Ȳn = x, then the next joint step is (Zxk , Z̄ k ) for k = Ln(x) = L̄n(x). In other words, given that the processes agree up to the present and reside together at x, the probability that they separate in the next step is bounded by Ce−α1|x|. 44 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN Now follow self-evident steps. Px,x[ Yk 6= Ȳk for some 1 ≤ k ≤ ζ ] Px,x[ Yj = Ȳj ∈ Ac for 1 ≤ j < k, Yk 6= Ȳk ] 1I{ Yj = Ȳj ∈ Ac for 1 ≤ j < k }PYk−1,Ȳk−1(Y1 6= Ȳ1) 1I{ Yj = Ȳj ∈ Ac for 1 ≤ j < k }e−α1|Ȳk−1| ≤ CEx e−α1|Ȳm|. � For the remainder of this section Y and Ȳ are always coupled in the manner that satisfies Lemma A.1. Lemma A.2. Let j ∈ {1, . . . , d} be such that the one-dimensional random walk Ȳ j is not degenerate. Let r0 be a positive integer and w̄ = inf{n ≥ 1 : Ȳ jn ≤ r0} the first time the random walk Ȳ enters the half-space H = {x : xj ≤ r0}. Couple Y and Ȳ starting from a common initial state x /∈ H. Then there is a constant C independent of r0 such that Px,x[ Yk 6= Ȳk for some k ∈ {1, . . . , w̄} ] ≤ Ce−α1r0 for all r0 ≥ 1. The same result holds for H = {x : xj ≥ −r0}. Proof. By Lemma A.1 Px,x[ Yk 6= Ȳk for some k ∈ {1, . . . , w̄} ] ≤ CEx [ w̄−1∑ e−α1|Ȳk| ≤ CExj [ w̄−1∑ e−α1Ȳ t=r0+1 e−α1tg(xj, t) where for s, t ∈ [r0 + 1,∞) g(s, t) = Ps[Ȳ n = t , w̄ > n] is the Green function of the half-line (−∞, r0] for the one-dimensional random walk Ȳ j. This is the expected number of visits to t before entering (−∞, r0], defined on p. 209 in Spitzer [13]. The development in Sections 18 and 19 in [13] gives the bound g(s, t) ≤ C(1 + (s− r0 − 1) ∧ (t− r0 − 1)) ≤ C(t− r0), s, t ∈ [r0 + 1,∞). (A.3) QUENCHED FUNCTIONAL CLT FOR RWRE 45 Here is some more detail. Shift r0 + 1 to the origin to match the setting in [13]. Then P19.3 on p. 209 gives g(x, y) = u(x− n)v(y − n) for x, y > 0 where the functions u and v are defined on p. 201. For a symmetric random walk u = v. P18.7 on p. 202 implies that v(m) = P[Z1 + · · ·+ Zk = m] where c is a certain constant and {Zi} are i.i.d. strictly positive, integer-valued ladder variables for the underlying random walk. Now one can show inductively that v(m) ≤ v(0) for each m so the quantities u(m) = v(m) are bounded. This justifies (A.3). Continuing from further above we get the estimate claimed in the statement: [ w̄−1∑ e−α1|Ȳk| (t− r0)e−α1t ≤ Ce−α1r0. � For the next lemmas abbreviate Br = [−r, r]d for d-dimensional centered cubes. Lemma A.3. With α1 given in the coupling hypothesis (A.iii), fix any positive constant κ1 > 2α 1 . Consider large positive integers r0 and r that satisfy 2α−11 log r ≤ r0 ≤ κ1 log r < r. Then there exist a positive integer m0 and a constant 0 < α3 <∞ such that, for large enough r, x∈BrrBr0 Px[without entering Br0 chain Y exits Br by time r m0 ] ≥ α3 . (A.4) Proof. We consider first the case where x ∈ BrrBr0 has a coordinate xj that satisfies xj ∈ [−r,−r0 − 1] ∪ [r0 + 1, r] and Ȳ j is nondegenerate. For this case we can take m0 = 4. A higher m0 may be needed to move a suitable coordinate out of the interval [−r0, r0]. This is done in the second step of the proof. The same argument works for both xj ∈ [−r,−r0−1] and xj ∈ [r0+1, r]. We treat the case xj ∈ [r0 + 1, r]. One way to realize the event in (A.4) is this: starting at xj , the Ȳ j walk exits [r0 + 1, r] by time r 4 through the right boundary into [r + 1,∞), and Y and Ȳ stay coupled together throughout this time. Let ζ̄ be the time Ȳ j exits [r0+1, r] and w̄ the time Ȳ j enters (−∞, r0]. Then w̄ ≥ ζ̄. Thus the complementary probability of (A.4) is bounded by Pxj{ Ȳ j exits [r0 + 1, r] into (−∞, r0] } + Pxj{ζ̄ > r4} + Px,x{ Yk 6= Ȳk for some k ∈ {1, . . . , w̄} }. (A.5) 46 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN We treat the terms one at a time. From the development on p. 253-255 in [13] we get the bound Pxj{ Ȳ j exits [r0 + 1, r] into (−∞, r0] } ≤ 1− (A.6) for some constant α4 > 0. In some more detail: P22.7 on p. 253, the inequality in the third display of p. 255, and the third moment assumption on the steps of Ȳ give a lower bound Pxj{ Ȳ j exits [r0 + 1, r] into [r + 1,∞) } ≥ xj − r0 − 1− c1 r − r0 − 1 (A.7) for the probability of exiting to the right. Here c1 is a constant that comes from the term denoted in [13] by M s=0(1 + s)a(s) whose finiteness follows from the third moment assumption. The text on p. 254-255 suggests that these steps need the aperiodicity assumption. This need for aperiodicity can be traced back via P22.5 to P22.4 which is used to assert the boundedness of u(x) and v(x). But as we observed above in the derivation of (A.3) boundedness of u(x) and v(x) is true without any additional assumptions. To go forward from (A.7) fix any m > c1 so that the numerator above is positive for xj = r0 + 1 +m. The probability in (A.7) is minimized at x j = r0 + 1, and from xj = r0 + 1 there is a fixed positive probability θ to take m steps to the right to get past the point xj = r0 + 1 +m. Thus for all x j ∈ [r0 + 1, r] we get the lower bound Pxj{ Ȳ j exits [r0 + 1, r] into [r + 1,∞) } ≥ θ(m− c1) r − r0 − 1 and (A.6) is verified. As in (A.3) let g(s, t) be the Green function of the random walk Ȳ j for the half- line (−∞, r0], and let g̃(s, t) be the Green function for the complement of the interval [r0 + 1, r]. Then g̃(s, t) ≤ g(s, t), and by (A.3) we get this moment bound: Exj [ ζ̄ ] = t=r0+1 g̃(xj , t) ≤ t=r0+1 g(xj, t) ≤ Cr2. Consequently, uniformly over xj ∈ [r0 + 1, r], Pxj [ζ̄ > r 4] ≤ C . (A.8) From Lemma A.2 Px[ Yk 6= Ȳk for some k ∈ {1, . . . , w̄} ] ≤ Ce−α1r0 . (A.9) Putting bounds (A.6), (A.8) and (A.9) together gives an upper bound of 1 − α4 + Ce−α1r0 QUENCHED FUNCTIONAL CLT FOR RWRE 47 for the sum in (A.5) which bounds the complement of the probability in (A.4). By assumption r0 > 2α 1 log r, so for large enough r the sum above is not more than 1− α3/r for some constant α3 > 0. The lemma is now proved for those x ∈ Br r Br0 for which some j ∈ J ≡ {1 ≤ j ≤ d : the one-dimensional walk Ȳ j is nondegenerate} satisfies xj ∈ [−r,−r0 − 1] ∪ [r0 + 1, r]. Now suppose x ∈ Br r Br0 but all j ∈ J satisfy xj ∈ [−r0, r0]. Let T = inf{n ≥ 1 : Y jn /∈ [−r0, r0] for some j ∈ J}. The first part of the proof gives Px-almost surely PYT [without entering Br0 chain Y exits Br by time r 4/2] ≥ α3 Replacing r4 by r4/2 only affects the constant in (A.8). It can of course happen that YT /∈ Br but then we interpret the above probability as one. By the Markov property it remains to show that for a suitable m0 Px[T ≤ rm0/2] : x ∈ Br r Br0 but xj ∈ [−r0, r0] for all j ∈ J (A.10) is bounded below by a positive constant. Hypothesis (A.1) implies that for some constant b1, ExT ≤ br01 uniformly over the relevant x. This is because one way to realize T is to wait until some coordinate Y j takes 2r0 successive identical steps. By hypothesis (A.1) this random time is stochastically bounded by a geometrically distributed random variable. It is also necessary for this argument that during time [0, T ] the chain Y does not enter Br0. Indeed, under the present assumptions the chain never enters Br0 . This is because for x ∈ BrrBr0 some coordinate i must satisfy xi ∈ [−r,−r0−1]∪ [r0+1, r]. But now this coordinate i /∈ J , and so by hypothesis (A.ii) the one-dimensional process Y i is constant, Y in = x i /∈ [−r0, r0] for all n. Finally, the required positive lower bound for (A.10) comes by Chebychev. Take m0 ≥ κ1 log b1 +1 where κ1 comes from the assumptions of the lemma. Then, by the hypothesis r0 ≤ κ1 log r, Px[T > r m0/2] ≤ 2r−m0br01 ≤ 2rκ1 log b1−m0 ≤ 12 for r ≥ 4. � We come to one of the main auxiliary lemmas of this development. Lemma A.4. Let U = inf{n ≥ 0 : Yn /∈ Br} be the first exit time from Br for the Markov chain Y . Then there exist finite positive constants C1, m1 such that Ex(U) ≤ C1rm1 for all 1 ≤ r <∞. 48 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN Proof. First observe that supx∈Br Ex(U) < ∞ by assumption (A.1) because by a geometric time some coordinate Y j has experienced 2r identical steps in succession. Throughout, let r0 < r satisfy the assumptions of Lemma A.3. Once the statement is proved for large enough r, we obtain it for all r ≥ 1 by increasing C1. Let 0 = T0 = S0 ≤ T1 ≤ S1 ≤ T2 ≤ · · · be the successive exit and entrance times into Br0 . Precisely, for i ≥ 1 as long as Si−1 <∞ Ti = inf{n ≥ Si−1 : Yn /∈ Br0} and Si = inf{n ≥ Ti : Yn ∈ Br0} Once Si = ∞ then we set Tj = Sj = ∞ for all j > i. If Y0 ∈ Br r Br0 then also T1 = 0. Again by assumption (A.1) (and as observed in the proof of Lemma A.3) there is a constant 0 < b1 <∞ such that x∈Br0 Ex[T1] ≤ br01 . (A.11) So a priori T1 is finite but S1 = ∞ is possible. Since T1 ≤ U <∞ we can decompose as follows: Ex[U ] = Ex[U, Tj ≤ U < Sj ] Ex[Tj , Tj ≤ U < Sj] + Ex[U − Tj , Tj ≤ U < Sj]. (A.12) We first treat the last sum in (A.12). By an inductive application of Lemma A.3, for any z ∈ Br rBr0 , Pz[U > jr m0 , U < S1] ≤ Pz[ Yk ∈ Br r Br0 for k ≤ jrm0 ] 1I{ Yk ∈ Br r Br0 for k ≤ (j − 1)rm0 }PY(j−1)rm0 { Yk ∈ Br r Br0 for k ≤ r ≤ · · · ≤ (1− α3r−1)j . Utilizing this, still for z ∈ Br r Br0 , Ez[U, U < S1] = Pz[U > m , U < S1] ≤ rm0 Pz[U > jr m0 , U < S1] ≤ rm0+1α−13 . (A.13) Next we take into consideration the failure to exit Br during the earlier excursions in Br rBr0 . Let Hi = {Yn ∈ Br for Ti ≤ n < Si} be the event that in between the ith exit from Br0 and entrance back into Br0 the chain Y does not exit Br. We shall repeatedly use this consequence of Lemma A.3: for i ≥ 1, on the event {Ti <∞}, Px[Hi | FTi] ≤ 1− α3r−1. (A.14) QUENCHED FUNCTIONAL CLT FOR RWRE 49 Here is the first instance. Ex[U − Tj , Tj ≤ U < Sj ] = Ex [ j−1∏ 1IHk · 1I{Tj <∞} · EYTj (U, U < S1) ≤ rm0+1α−13 Ex [ j−1∏ 1IHk · 1I{Tj−1 <∞} ≤ rm0+1α−13 (1− α3r−1)j−1. Note that if YTj above lies outside Br then EYTj (U) = 0. In the other case YTj ∈ Br rBr0 and (A.13) applies. So for the last sum in (A.12): Ex[U − Tj , Tj ≤ U < Sj] ≤ rm0+1α−13 (1− α3r−1)j−1 ≤ rm0+2α−23 . (A.15) We turn to the second-last sum in (A.12). Utilizing (A.11) and (A.14), Ex[Tj , Tj ≤ U < Sj] ≤ [ j−1∏ 1IHk · 1I{Tj <∞} · (Ti+1 − Ti) ≤ br01 (1− α3r−1)j−1 [ i−1∏ 1IHk · (Ti+1 − Ti)1IHi · 1I{Ti+1 <∞} (1− α3r−1)j−1−i. (A.16) Split the last expectation as [ i−1∏ 1IHk · (Ti+1 − Ti)1IHi · 1I{Ti+1 <∞} [ i−1∏ 1IHk · (Ti+1 − Si)1IHi · 1I{Si <∞} [ i−1∏ 1IHk · (Si − Ti)1IHi · 1I{Ti <∞} [ i−1∏ 1IHk · 1I{Si <∞} · EYSi (T1) [ i−1∏ 1IHk · 1I{Ti <∞} ·EYTi (S1 · 1IH1) [ i−1∏ 1IHk · 1I{Ti−1 <∞} (br01 + r m0+1α−13 ) ≤ (1− α3r−1)i−1(br01 + rm0+1α−13 ). (A.17) In the second-last inequality above, before applying (A.14) to theHk’s, EYSi (T1) ≤ b comes from (A.11). The other expectation is estimated again by iterating Lemma 50 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN A.3 and again with z ∈ Br rBr0 : Ez(S1 · 1IH1) = Pz[S1 > m , H1] ≤ Pz[ Yk ∈ Br rBr0 for k ≤ m ] ≤ rm0 Pz[ Yk ∈ Br r Br0 for k ≤ jrm0 ] ≤ rm0+1α−13 . Insert the bound from line (A.17) back up into (A.16) to get the bound Ex[Tj , Tj ≤ U < Sj] ≤ (2br01 + rm0+1α−13 )j(1− α3r−1)j−2. Finally, bound the second-last sum in (A.12): Ex[Tj , Tj ≤ U < Sj] ≤ 2br01 r 2α−23 + r m0+3α−33 (1− α3r−1)−1. Taking r large enough so that α3r −1 < 1/2 and combining this with (A.12) and (A.15) gives Ex[U ] ≤ rm0+2α−21 + 4br01 r2α−23 + 2rm0+3α−33 . Since r0 ≤ κ1 log r for some constant C, the above bound simplifies to C1rm1 . � For the remainder of the proof we work with B = Br for r = c1 log n. The above estimate gives us one part of the argument for (A.2), namely that the Markov chain Y exits B = [−c1 log n, c1 logn]d fast enough. Let 0 = V0 < U1 < V1 < U2 < V2 < · · · be the successive entrance times Vi into B and exit times Ui from B for the Markov chain Y , assuming that Y0 = z ∈ B. It is possible that some Vi = ∞. But if Vi < ∞ then also Ui+1 < ∞ due to assumption (A.1), as already observed. The time intervals spent in B are [Vi, Ui+1) each of length at least 1. Thus, by applying Lemma A.4, Pz(Yk ∈ B) ≤ (Ui+1 − Vi)1I{Vi ≤ n} EYVi (U1)1I{Vi ≤ n} ≤ C(logn)m1Ez 1I{Vi ≤ n} (A.18) QUENCHED FUNCTIONAL CLT FOR RWRE 51 Next we bound the expected number of returns to B by the number of excursions outside B that fit in a time of length n: 1I{Vi ≤ n} (Vj − Vj−1) ≤ n (Vj − Uj) ≤ n (A.19) According to the usual notion of stochastic dominance, the random vector (ξ1, . . . , ξn) dominates (η1, . . . , ηn) if Ef(ξ1, . . . , ξn) ≥ Ef(η1, . . . , ηn) for any function f that is coordinatewise nondecreasing. If the {ξi : 1 ≤ i ≤ n} are adapted to the filtration {Gi : 1 ≤ i ≤ n}, and P [ξi > a|Gi−1] ≥ 1 − F (a) for some distribution function F , then the {ηi} can be taken i.i.d. F -distributed. Lemma A.5. There exist positive constants c1, c2 and γ such that the following holds: the excursion lengths {Vj − Uj : 1 ≤ j ≤ n} stochastically dominate i.i.d. variables {ηj} whose common distribution satisfies P[η ≥ a] ≥ c1a−1/2 for 1 ≤ a ≤ c2nγ. Proof. Since Pz[Vj − Uj ≥ a|FUj ] = PYUj [V ≥ a] where V means first entrance time into B, we shall bound Px[V ≥ a] below uniformly over x /∈ B : Pz[YU1 = x] > 0 Fix such an x and an index 1 ≤ j ≤ d such that xj /∈ [−r, r]. Since the coordinate Y j can move out of [−r, r], this coordinate is not degenerate, and hence by assumption (A.ii) the random walk Ȳ j is nondegenerate. As before we work through the case xj > r because the argument for the other case xj < −r is the same. Let w̄ = inf{n ≥ 1 : Ȳ jn ≤ r} be the first time the one-dimensional random walk Ȳ j enters the half-line (−∞, r]. If both Y and Ȳ start at x and stay coupled together until time w̄, then V ≥ w̄. This way we bound V from below. Since the random walk is symmetric and can be translated, we can move the origin to xj and use classic results about the first entrance time into the left half-line, T̄ = inf{n ≥ 1 : Ȳ jn < 0}. Pxj [w̄ ≥ a] ≥ Pr+1[w̄ ≥ a] = P0[T̄ ≥ a] ≥ for a constant α5. The last inequality follows for one-dimensional symmetric walks from basic random walk theory. For example, combine equation (7) on p. 185 of [13] with a Tauberian theorem such as Theorem 5 on p. 447 of Feller [7]. Or see directly Theorem 1a on p. 415 of [7]. 52 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN Now start both Y and Ȳ from x. Apply Lemma A.2 and recall that r = c1 log n. Px[V ≥ a] ≥ Px,x[V ≥ a, Yk = Ȳk for k = 1, . . . , w̄ ] ≥ Px,x[w̄ ≥ a, Yk = Ȳk for k = 1, . . . , w̄ ] ≥ Pxj [w̄ ≥ a]− Px,x[ Yk 6= Ȳk for some k ∈ {1, . . . , w̄} ] ≥ α5√ − Cn−c1α1 . This gives a lower bound Px[V ≥ a] ≥ if a ≤ α25(2C)−2n2c1α1 . This lower bound is independent of x. We have proved the lemma. � We can assume that the random variables ηj given by the lemma satisfy 1 ≤ ηj ≤ γ and we can assume both c2, γ ≤ 1 because this merely weakens the result. For the renewal process determined by {ηj} write S0 = 0 , Sk = ηj , and K(n) = inf{k : Sk > n} for the renewal times and the number of renewals up to time n (counting the renewal S0 = 0). Since the random variables are bounded, Wald’s identity gives EK(n) · Eη = ESK(n) ≤ n+ c2nγ ≤ 2n, while ∫ c2nγ ds ≥ c3nγ/2. Together these give EK(n) ≤ 2n ≤ C2n1−γ/2. Now we pick up the development from line (A.19). Since the negative of the function of (Vj − Uj)1≤i≤n in the expectation on line (A.19) is nondecreasing, the stochastic domination of Lemma A.5 gives an upper bound of (A.19) in terms of the i.i.d. {ηj}. Then we use the renewal bound from above. 1I{Vi ≤ n} (Vj − Uj) ≤ n ηj ≤ n = EK(n) ≤ C2n1−γ/2. QUENCHED FUNCTIONAL CLT FOR RWRE 53 Returning back to (A.18) to collect the bounds, we have shown that Pz(Yk ∈ B) ≤ C(logn)m1Ez 1I{Vi ≤ n} ≤ C(logn)m1C2n1−γ/2 and thereby verified (A.2). References [1] N. Berger and O. Zeitouni. A quenched invariance principle for certain ballistic random walks in i.i.d. environments. http://front.math.ucdavis.edu/math.PR/0702306. [2] E. Bolthausen and A.-S. Sznitman. On the static and dynamic points of view for certain random walks in random environment. Methods Appl. Anal., 9(3):345– 375, 2002. Special issue dedicated to Daniel W. Stroock and Srinivasa S. R. Varadhan on the occasion of their 60th birthday. [3] E. Bolthausen and A.-S. Sznitman. Ten lectures on random media, volume 32 of DMV Seminar. Birkhäuser Verlag, Basel, 2002. [4] J. Bricmont and A. Kupiainen. Random walks in asymmetric random environ- ments. Comm. Math. Phys., 142(2):345–420, 1991. [5] Y. Derriennic and M. Lin. The central limit theorem for Markov chains started at a point. Probab. Theory Related Fields, 125(1):73–76, 2003. [6] S. N. Ethier and T. G. Kurtz. Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1986. Characterization and convergence. [7] W. Feller. An introduction to probability theory and its applications. Vol. II. Second edition. John Wiley & Sons Inc., New York, 1971. [8] M. Maxwell and M. Woodroofe. Central limit theorems for additive functionals of Markov chains. Ann. Probab., 28(2):713–724, 2000. [9] F. Rassoul-Agha and T. Seppäläinen. An almost sure invariance principle for random walks in a space-time random environment. Probab. Theory Related Fields, 133(3):299–314, 2005. [10] F. Rassoul-Agha and T. Seppäläinen. Ballistic random walk in a random en- vironment with a forbidden direction. ALEA Lat. Am. J. Probab. Math. Stat., 1:111–147 (electronic), 2006. [11] F. Rassoul-Agha and T. Seppäläinen. Quenched invariance principle for multi- dimensional ballistic random walk in a random environment with a forbidden direction. Ann. Probab., 35(1):1–31, 2007. [12] M. Rosenblatt. Markov processes. Structure and asymptotic behavior. Springer- Verlag, New York, 1971. Die Grundlehren der mathematischen Wissenschaften, Band 184. [13] F. Spitzer. Principles of random walks. Springer-Verlag, New York, second edition, 1976. Graduate Texts in Mathematics, Vol. 34. 54 F. RASSOUL-AGHA AND T. SEPPÄLÄINEN [14] A.-S. Sznitman. Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. (JEMS), 2(2):93–143, 2000. [15] A.-S. Sznitman. Topics in random walks in random environment. In School and Conference on Probability Theory, ICTP Lect. Notes, XVII, pages 203–266 (electronic). Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004. [16] A.-S. Sznitman and O. Zeitouni. An invariance principle for isotropic diffusions in random environment. Invent. Math., 164(3):455–567, 2006. [17] A.-S. Sznitman and M. Zerner. A law of large numbers for random walks in random environment. Ann. Probab., 27(4):1851–1869, 1999. [18] O. Zeitouni. Random walks in random environments, volume 1837 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2004. Lectures from the 31st Summer School on Probability Theory held in Saint-Flour, July 8–25, 2001, Edited by Jean Picard. [19] M. P. W. Zerner. Lyapounov exponents and quenched large deviations for mul- tidimensional random walk in random environment. Ann. Probab., 26(4):1446– 1476, 1998. F. Rassoul-Agha, 155 S 1400 E, Salt Lake City, UT 84112 E-mail address : [email protected] URL: www.math.utah.edu/∼firas T. Seppäläinen, 419 Van Vleck Hall, Madison, WI 53706 E-mail address : [email protected] URL: www.math.wisc.edu/∼seppalai 1. Introduction and main result 2. Preliminaries for the proof. 3. Basic estimates for non-nestling RWRE 4. Invariant measure and ergodicity 5. Change of measure 6. Reduction to path intersections 7. Bound on intersections Appendix A. A Green function type bound References

0704.1023

The Effect of Annealing Temperature on Statistical Properties of $WO_3$ Surface

The Effect of Annealing Temperature on Statistical Properties of WO3 Surface G. R. Jafari a,b, A. A. Saberi c, R. Azimirad c, A. Z. Moshfegh c, and S. Rouhani c Department of Physics, Shahid Beheshti University, Evin, Tehran 19839, Iran Department of Nano-Science, IPM, P. O. Box 19395-5531, Tehran, Iran Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran We have studied the effect of annealing temperature on the statistical properties of WO3 surface using atomic force microscopy techniques (AFM). We have applied both level crossing and structure function methods. Level crossing analysis indicates an optimum annealing temperature of around 400oC at which the effective area of the WO3 thin film is maximum, whereas composition of the surface remains stoichiometric. The complexity of the height fluctuation of surfaces was charac- terized by roughness, roughness exponent and lateral size of surface features. We have found that there is a phase transition at around 400oC from one set to two sets of roughness parameters. This happens due to microstructural changes from amorphous to crystalline structure in the samples that has been already found experimentally. I. INTRODUCTION Transition metal oxides represent a large family of materials possessing various interesting properties, such as superconductivity, colossal magneto-resistance and piezoelectricity. Among them, tungsten oxide is of in- tense interest and has been investigated extensively for its distinctive properties. With outstanding electrochromic [1, 2, 3, 4, 5, 6], photochromic [7], gaschromic [8], gas sensor [9, 10, 11], photo-catalyst [12], and photolumines- cence properties [13], tungsten oxide has been used to construct ”smart-window”, anti-glare rear view mirrors of automobile, non-emissive displays, optical recording devices, solid-state gas sensors, humidity and tempera- ture sensors, biosensors, photonic crystals, and so forth. WO3 thin films can be prepared by various deposition techniques such as thermal evaporation [3, 8], spray py- rolysis [14], sputtering [6], pulsed laser ablation [10, 11], sol-gel coating [2, 5, 15], and chemical vapor deposition [16]. The gas sensitivity of WO3 heavily depends upon film parameters such as composition, morphology (e.g. grain size), nanostructure and microstructure (e.g. porosity, surface-to-volume ratio). Film parameters are related to the deposition technique used, the deposition conditions and the subsequent annealing process. Annealing, which is an essential process to obtain stable films with well-defined microstructure, causes stoichiometry and microstructural changes that have a high influence on the sensing characteristics of the films [17]. Moreover, the surface structure and surface morphology of the metal oxides are also important for different applications. In fact, the electrochromic devices are made of amorphous oxides [1], while crystalline phase plays a major role in catalysts and sensors [17]. This is because, the minor change in their chemical composition and crystalline structure could modify different properties of the metal oxides. In practice, one of the effective ways to modify the surface morphology is annealing process at various temperatures. So far, most of morphological analysis related to the WO3 surface were accessible through the experimental methods. Usually these analysis are rigorous and time consuming. Moreover, lack of the suitable analysis for AFM data to find the nano and microstructural properties of surfaces was feeling perfectly. In this article, we introduce the methods: roughness analysis and level crossing as suitable candidates and show that we can get easily the structure and morpho- logical properties of a surface in a fast manner, only using the AFM observation as an initial data. The roughness of a surface has been studied as a simple growth model using analytical and numerical methods [18, 19, 20, 21, 22, 23, 24, 25, 26]. These studies quite generally proposed that the height fluc- tuations have a self-similar character and their average correlations exhibit a dynamic scaling form. Also some authors recently use the average frequency of positive slope level crossing to provide further complete analysis on roughness of a surface [27]. This stochastic approach has turned out to be a promising tool also for other systems with scale dependent complexity, such as in surface growth where one would like to measure the roughness [28]. Some authors have applied this method to study the fluctuations of velocity fields in Burgers turbulence [29] and the Kardar-Parisi-Zhang equation in (d+1)-dimensions [30] and analyzing the stock market [31]. In this work, we have used the scaling analysis to determine the roughness, roughness exponent and the lateral size of surface features. Moreover, level crossing analysis has been utilized to estimate the effective area of a surface. This paper is organized as follows: In section II, we have discussed about the film preparation and experi- http://arxiv.org/abs/0704.1023v1 mental results obtained from AFM, XPS and UV-visible spectrophotometer for the annealed samples at the vari- ous temperatures. In section III, we have introduced the analytical methods briefly. Data description and data analysis based on the statistical parameters of WO3 sur- face as a function of annealing temperatures are given in section IV.Finally, section V concludes presented results. II. EXPERIMENTALS Thin films of WO3 were deposited on microscope slide glass using thermal evaporation method. The deposition system was evacuated to a base pressure of ∼ 4×10−3Pa. Thickness of the deposited films was considered about 200 nm measured by the stylus and optical techniques. More details about the other deposition parameters of the films are recently reported elsewhere [32]. To study the effect of annealing temperature on sur- face structure and optical properties of the samples, they were annealed at 200, 300, 350, 400, 450, and 500oC in air for a period of 60 min. Optical transmission and reflection measurements of the deposited films were per- formed in a range of 300-1100 nm wavelength using a Jascow V530 ultraviolet (UV)-visible spectrophotometer with resolution of 1 nm. X-ray photoelectron spectroscopy (XPS) using a Specs EA 10 Plus concentric hemispherical analyzer (CHA) with Al Kα anode at energy of 1486.6 eV was employed to study the atomic composition and chemical state of the tungsten oxide thin films. The pressure in the ultra high vacuum surface analysis chamber was less than 1.0×10−7Pa. All binding energy values were determined by calibration and fixing the C(1s) line to 285.0 eV . The XPS data analysis and deconvolution were performed by SDP (version 4.0) software. The nanoscale Surface topography of the deposited films was investigated by Thermo Microscope Autoprobe CP-Research atomic force microscopy (AFM) in air with a silicon tip of 10 nm radius in contact method. The AFM images were recorded with resolution of about 20 nm in a scale of 5× 5 µm. A. XPS Characterization The elemental and chemical characterizations of the films were performed by XPS. Figure 1a shows theW (4f) core level spectra recorded on the ”as deposited” WO3 sample, and the results of its fitting analysis. To repro- duce the experimental data, one doublet function was used for the W (4f) component. This contains W (4f7/2) at 35.6 eV and W (4f5/2) at 37.8 eV with a full-width at half-maximum (FWHM) of 1.75± 0.04 eV . The area ratio of these two peaks is 0.75 which is supported by the spin-orbit splitting theory of 4f levels. Moreover, the structure was shifted by 5 eV toward higher energy FIG. 1: W (4f) core level spectra of WO3 thin films: a) ”as deposited” and b) annealed at 500oC. relative to the metal state. It is thus clear that the main peaks in our XPS spectrum attributed to the W 6+ state on the surface [1, 2, 33]. In stoichiometricWO3, the six valence electrons of the tung- sten atom are transferred into the oxygen p-like bands, which are thus completely filled. In this case, the tung- sten 5d valence electrons have no part of their wave func- tion near the tungsten atom and the remaining electrons in the tungsten atom experience a stronger Coulomb in- teraction with the nucleus than in the case of tungsten atom in a metal, in which the screening of the nucleus has a component due to the 5d valence electrons. There- fore, the binding energy of the W (4f) level is larger in WO3 than in metallic tungsten. If an oxygen vacancy exists, the electronic density near its adjacent W atom increases, the screening of its nucleus is higher and, thus, the 4f level energy is expected to be at lower binding energy [1]. By increasing the annealing temperature it was ob- served that the position of W (4f) peak did not obviously change. But for WO3 thin film annealed at 500 oC (Fig. 1b), the W (4f) peak moved to a lower binding energy so that W (4f7/2) position was observed at 35.0 eV . This can be related to oxygen vacancy at this high annealing temperature and formation of W 5+. B. Optical Characterization The transmittance and reflectance spectra in the visible and infrared range recorded for the WO3 thin films before and after annealing at different temperatures (Fig. 2a). It is seen that, the transmittance of the ”as deposited” films in the visible range varies from about 80 up to nearly 100% (without considering the substrate contribution). Correspondingly, maximum value of the reflectance for both the film and the substrate is about 20% (the reflectance from the bare glass substrate was measured about 10%). The sharp reduction in the transmittance spectrum at the wavelength of ∼ 350nm is due to the fundamental absorption edge that was also reported previously [1, 2, 3]. The oscillations in the transmission and reflection spectra are caused by optical interference. The optical transmittance of WO3 films strongly depends on the oxygen content of the films. In fact, non-stoichiometric films with composition of WO3−x show a blue tinge for x > 0.03 [34]. The ”as deposited” pure tungsten oxide films were highly transparent with no observable blue coloration, under our experimental conditions. As can be seen from Fig. 2a after annealing process at 200 to 400oC, the transmittance and reflectance of the WO3 films have not changed significantly. Only, the position of the oscillations altered due to thickness reduction and film condensation after the heat-treatment process [1]. At 500oC transmittance and reflectance of the annealed WO3 film is reduced about 10%, therefore at this temperature, the film turn into non-stoichiometric composition, so that it could be seen from changing color of the film. The optical gap (Eg) was evaluated from the ab- sorption coefficient (α) using the standard relation: (αhν)1/η = A(hν − Eg), in which η depends on the kind of optical transition in semiconductors, and α was determined near the absorption edge using the simple relation: α = ln[(1 − R)2/T ]/d ,where d is thickness of the film. More useful explanation about the optical band gap calculation reported in [32]. The relationship between the optical band gap energy and annealing temperature for WO3 thin films has been shown in Fig. 2b. As can be seen from it, the optical band gap for the ”as deposited” WO3 evaluated 3.4 eV . Amorphous structure of the ”as deposited” WO3 causes to Eg is bigger than 2.7 eV . After annealing samples at 200 and 300oC, the optical band gap decreased slightly about 0.1 eV which can be related to condensation of the films. But the optical band gap of the WO3 annealed at 400 reduced to 3.1 eV due to crystallization of the film. This reduction continues to 2.5 eV for the sample annealed at 500oC. Reason of the Eg becomes smaller than 2.7 eV is oxygen vacancy at this temperature as was seen in Fig 1b. It is to note that for evaporated WO3 films one has found 2.7 < Eg < 3.5 eV [1]. λ ( ) 400 600 800 1000 27 oC 200 oC 300 oC 400 oC 500 oC 100 200 300 400 500 FIG. 2: a) Optical transmittance (T) and reflectance (R) and b) Optical band gap energy of the WO3 thin films annealed at different temperatures. C. AFM Analysis To study the effect of the annealing process on the sur- face morphology of the films, we have shown AFM images of the WO3 surfaces annealed at the different tempera- tures : 200, 300, 350, 400, 450, 500oC in Figure 1. As can be seen from Fig. 1, for the annealed film at 200oC, it seems that the surface morphology of the film is rela- tively the same with a smooth surface, amorphous struc- ture and nanometric grain size, as also reported by other investigators for WO3 films [35, 36]. We have also ob- served similar image for the ”as deposited” WO3 which is not shown here. For WO3 thin films, increasing an- nealing temperature to 350oC did not significantly ef- fect on surface parameters because it is low temperature for crystallization of WO3 [1]. But at higher annealing temperatures 400, 450 and 500oC, surface grain size and FIG. 3: AFM images of WO3 thin films annealed at various temperatures a) 200, b) 300, c) 350, d) 400, e) 450 and f) 500oC, respectively. roughness begin to increase. The more precise analysis of these surfaces are given in the next section. III. STATISTICAL QUANTITIES A. Roughness Analysis It is also known that to derive the quantitative infor- mation of the surface morphology one may consider a sample of size L and define the mean height of growing film h and its roughness σ by: σ(L, t) = (〈(h − h)2〉)1/2 (1) where t is growing time and 〈· · ·〉 denotes an averaging over different samples, respectively. Moreover, growing time is a factor which can be applied to control the sur- face roughness of thin films. Let us now calculate the roughness exponent of the growing surface. Starting from a flat interface (one of the possible initial conditions), it is conjectured that a scaling of lenght by factor b and of time by factor bz (z is the dynamical scaling exponent), rescales the roughness σ by factor bχ as follows [18]: σ(bL, bzt) = bασ(L, t) (2) which implies that σ(L, t) = Lαf(t/Lz). (3) For large t and fixed L (i.e.x = t/Lz → ∞) σ saturate. However, for fixed and large L and t ≪ Lz, one expects that correlations of the height fluctuations are set up only within a distance t1/z and thus must be independent of L. This implies that for x ≪ 1, f(x) ∼ xβg′(λ) with β = α/z. Thus, dynamic scaling postulates that σ(L, t) = tβ, t≪ Lz; Lα, t≫ Lz. The roughness exponent α and the dynamic exponent β characterize the self-affine geometry of the surface and its dynamics, respectively. In the present work, we see the surfaces at the limit t → ∞ and so we will only obtain the α exponent. The common procedure to measure the roughness ex- ponent of a rough surface is use of the surface structure function depending on the length scale l which is defined S2(l) = 〈|h(x+ l)− h(x)|2〉. (5) It is equivalent to the statistics of height-height corre- lation function C(l) for stationary surfaces, i.e. S2(l) = 2σ2(1−C(l)). The second order structure function S2(l), scales with l as l2α. B. Level Crossing Analysis Let ν+α denotes the number of positive slope crossing of h(x) − h̄ = α for interval L. Since the process is homogeneous, if we take a second time interval of L immediately following the first we shall obtain the same result, and for two intervals together we shall therefore obtain [28]: N+α (2L) = 2N α (L), (6) from which it follows that, for a homogeneous process, the average number of crossing is proportional to the in- terval L. Hence N+α (L) ∝ L, (7) N+α (L) = ν αL, (8) where ν+α is the average frequency of positive slope cross- ing of the level h(x) − h̄ = α. We now consider how the frequency parameter ν+α can be deduced from the under- lying probability distributions for h(x)− h̄. Consider a small length scale δx of a typical sample func- tion. Since we are assuming that the process h(x)− h̄ is a smooth function of x, with no sudden ups and downs, if δx is small enough, the sample can only cross h(x)−h̄ = α with positive slope if h(x) − h̄ < α at the beginning of the interval L. Furthermore, there is a minimum slope at x if the level h(x) − h̄ = α is to be crossed in interval ∆x depending on the value of h(x)− h̄ at position x. So there will be a positive crossing of h(x) − h̄ = α in the next interval ∆x if, at x, h(x)− h̄ < α and h(x) − h̄ h(x) − h̄ As shown in [28], the frequency ν+α can be written in terms of joint PDF (probability distribution function ) of p(α, y′) as follows: ν+α = p(α, y′)y′dy′. (10) and then the quantity N+tot which is defined as: N+tot = ν+α |α− ᾱ|dα. (11) will measure the total number of crossing the surface with positive slope. So, the N+tot and square area of growing surface are in the same order. Concerning this, it can be utilized as another quantity to study further the rough- ness of a surface [27]. l ( m) 0.5 1 1.5 2 FIG. 4: Log-Log plot of the selection structure function of various annealed temperature: a) 27, b) 200, c) 300, d) 350, e) 400, f) 450, g) 500oC. IV. RESULTS AND DISCUSSION Thin films of WO3 were deposited by using thermal evaporation method and then surface micrographs of WO3 samples were obtained by AFM technique after annealed at different temperatures (Fig.3). These micrographs were then analyzed using methods from stochastic data analysis have introduced in the last section. Figure 3 shows AFM images of WO3 thin films annealed at 200 ,300 ,350 ,400 ,450, and 500oC. The ”as deposited” and annealed sample at 200oC (Fig. 3a) have columnar structure, indicating that up to 200oC no significant changes in the microstructure occurs. However, at higher temperatures (figs. 3b-3f) we have observed increased grain size and rougher surface. Specifically at 500oC (Fig. 3f) we observe stark changes in the micrograph which is accompanied by composition changes in the surface. This can be related to the phase transition to Magneli phase e.g. WO3−x in the annealing process [36]. This is also confirmed by our XPS and UV-visible spectrophotometry analysis (Sec. II). These are shown the significant formation of W 5+ state in the surface at 500oC. Also our analysis shows that below 400oC the surfaces are in amorphous phase with the same behavior for all scales, but as soon as the crystalline phase appears the system behaves differently which diagnostics at small and large scale for temperatures above 400oC. By using pa- rameters of the analytical method given here, these tran- sitions can be quantified. h ( m) -0.02 -0.01 0 0.01 0.02 10000 200 oC 300 oC 400 oC 500 oC FIG. 5: The average frequency ν+α as a function of height h Now, we will use the statistical parameters introduced in the last section and will obtain some quantitative information about the effect of annealing temperature on the surface topography of the WO3 samples. The structure function S2(l) as defined in Eq.(5) can be used to quantify the topology of a rough surface. The structure function S2(l) is plotted against the length scale of the sample in Fig.4 . The saturated S2(l) is an indication of the surface roughness, as 2σ2. The most obvious observation indicates that roughness is raised with increasing annealing temperature. Roughness has a minimum of 0.91nm at 27 and 200oC and a maximum of 48nm at 500oC. This is because higher temperatures create higher peaks (i.e. peaks with more deviations from the average) . All exponents which is derivable from S2(l) have been summarized and given in Table I. As depicted in Fig.4 , the structure function S2(l), has a different behavior in the various temperatures. So that, in the annealing temperature range 27-350oC it has a typical behavior in all scales, but in the higher temperature range 400-500oC its behavior is different in the small and large scales. In the other words, the phase transition is occurred at 400oC, because for higher temperatures, there are two sets of roughness parameters needed to simulate the surface morphology. It can be related to the phase transition in the structure of the surface from amorphous to crystalline phase has been yielded from the band gap energy (see the section II.B). The slope of each S2(l) curves at the small and large scales yields the roughness exponents α and α′ of 100 200 300 400 500 FIG. 6: The normalized N+ behavior as a function of an- nealing temperature. The solid line is plotted according to Eq.(12) around 400oC . the corresponding surface. Hence, it is seen that the mono roughness exponent increases with the addition of annealing temperature up to 400oC. In the higher tem- peratures, we have obtained two roughness exponents( α-α′) equal to the 0.40-0.14, 0.71-0.20, and 0.69-0.24 for temperatures 400, 450 and 500oC, respectively. Difference in the α values, in these temperatures, are in agreement with changes of correlation length. Where the correlation length, is the distance at which the structure function behaves differently. The range of the scaling upon correlation length listed in the forth column in table I. The value of C∗s denotes the correlation length at small scales and C∗l for large scales. The higher C∗ value represents a smoother surface (as we expected from Fig.3). The correlation length obtained from the structure function is also a measure of minimum lateral size of surface features at each annealing temperature. The another important WO3 film parameter is the ef- fective area of the sample which has an important role in the gas sensitivity of WO3 surfaces. To obtain a mea- sure for this, we utilize the level crossing analysis. As shown in Fig.5, the average frequency ν+α as a function of height h, is plotted for the various annealing tempera- tures. The broad curves indicate the higher magnitude of height fluctuations around the average, and sharp curves show that the most of fluctuations are around the height average. This conclusion is in the correspondence with the results obtained from Fig.3. According to the Eq.(11) N+α i.e. The total number of the crossing surface with positive slope is proportional to the square of area of the growing surface. To obtain the optimum value of the effective area, we have calculated the ratio of effective areas with respect to the area of the ”as deposited” surface (27oC). The values are given TABLE I: The Roughness exponent, roughness, correlation length and effective area relative to the ”as deposited” sample area (27oC). T [oC] α− α′ σ[nm] C∗s -C [nm] N+/N+(27oC) 27 0.15− none 0.91 60− none 1.00 200 0.15− none 0.98 60− none 1.08± 0.02 300 0.61− none 2.20 100− none 1.17± 0.02 350 0.62− none 11.50 100− none 1.67± 0.02 400 0.40 − 0.15 17.00 100 − 300 2.00± 0.02 450 0.71 − 0.20 30.00 400− 1000 1.72± 0.02 500 0.69 − 0.24 48.00 200− 1400 1.41± 0.02 in the last column in Table I. It means, although the roughness increases by the annealing temperature but the effective square area of the rough surface has a maximum value of N+tot [27]. For more clarity, we have calculated the temperature dependence of normalized N+α numerically (Fig.6) around 400oC, and we have obtained the three following functions for this quantity : N+tot(T ) = (5.0718− 0.0223× T + 2.72× 10 −5 × T 2)−1(12) ln(N+tot(T )) = −6.3632+ 0.0344× T − 4.20× 10 −5 × T 2(13) N+tot(T ) = −8.7057+ 0.0520× T − 6.37× 10 −5 × T 2(14) According to this figure, the maximum value of the effective area is at 400oC (with respect to its value at 27oC) with the relative value equal to 2.00. Thus, applying this analysis easily shows that if one follows the condition which the effective area as an important parameter in the gas sensitivity of WO3 surfaces is optimum and furthermore, the film composition has not been changed (e.g. The Magneli phase transition has not been occurred), should choose the annealed surface at 400oC for better performance. V. CONCLUSIONS We have investigated the role of annealing tempera- ture, as an external parameter, to control the statisti- cal properties of a rough WO3 surface. The AFM mi- crostructure of the surfaces is just needed to apply in our analysis. We have computed the statistical quantities such as roughness exponent, roughness and lateral size of surface features of the ”as deposited” and annealed surfaces at 200, 300, 350, 400, 450, and 500oC, using the structure function. We have seen a phase transition at 400oC, because for higher temperatures there are two sets of roughness parameters, due to structural changes from amorphous to the crystalline phase. Moreover, using the level crossing analysis we have obtained an optimum an- nealing temperature, 400oC in which the surface of the WO3 has maximum value about twice relative to the ”as deposited” film without any changes in the film composi- tion that may increase surface reaction of the WO3 film as the gas sensor or photo-catalyst. VI. ACKNOWLEDGMENT GRJ and AAS would like to thank S.M.Fazeli for his useful comments and especially M.R.Rahimitabar for his useful lectures on ”stochastic data analysis”. AZM would like to acknowledge research council of Sharif University of Technology for financial support of the work. [1] C. G. Granqvist, Handbook of Electrochromic Materials (Elsevier, Amsterdam, 1995). [2] P. R. Bueno, F. M. Pontes, E. R. Leite, L. O. S. Bulhes, P. S. Pizani, P. N. Lisboa-Filho, and W. H. Schreiner, J. Appl. Phys. 96, 2102 (2004). [3] R. Azimirad, O. Akhavan, and A. Z. Moshfegh, J. Elec- trochem. Soc.153, E11(2006). [4] A. Siokou, S. Ntais, S. Papaefthimiou, G. Leftheriotis, and P. Yianoulis, Surface Science, 566/568, 1168 (2004). [5] S.-L. Kuai, G. Bader, and P. V. Ashrit, App. Phys. Lett., 86, 221110 (2005). [6] Y. Takeda, N. Kato, T. Fukano, A. Takeichi, and T. Mo- tohiro, J. Appl. Phys., 96, 2417 (2004). [7] C. O. Avellaneda and L. O. S. Bulhes, Solid State Ionics, 165, 117 (2003). [8] S.-H. 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0704.1024

Determination of Low-Energy Parameters of Neutron--Proton Scattering on the Basis of Modern Experimental Data from Partial-Wave Analyses

Determination of Low-Energy Parameters of Neutron–Proton Scattering on the Basis of Modern Experimental Data from Partial-Wave Analyses V. A. Babenko∗ and N. M. Petrov Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Metrologicheskaya ul. 14b, 03143 Kiev, Ukraine The triplet and singlet low-energy parameters in the effective-range expansion for neutron– proton scattering are determined by using the latest experimental data on respective phase shifts from the SAID nucleon–nucleon database. The results differ markedly from the analogous parameters obtained on the basis of the phase shifts of the Nijmegen group and contradict the parameter values that are presently used as experimental ones. The values found with the aid of the phase shifts from the SAID nucleon–nucleon database for the total cross section for the scattering of zero-energy neutrons by protons, σ0 = 20.426 b, and the neutron–proton coherent scattering length, f = −3.755 fm, agree perfectly with the experimental cross-section values obtained by Houk, σ0 = 20.436 ± 0.023 b, and experimental scattering-length values obtained by Houk andWilson, f = −3.756±0.009 fm, but they contradict cross-section values of σ0 = 20.491±0.014 b according to Dilg and coherent-scattering-length values of f = −3.7409± 0.0011 fm according to Koester and Nistler. PACS: 13.75.Cs, 21.30.-x, 25.40.Dn DOI: 10.1134/S1063778807040072 1. Along with the deuteron parameters, the low-energy parameters in the effective-range expansion for neutron–proton scattering, k cot δ = − rk2 + v2k 4 + v3k 6 + v4k 8 + . . . , (1) are fundamental quantities that play a key role in studying strong nucleon–nucleon interaction. ∗E-mail: [email protected] http://arxiv.org/abs/0704.1024v1 These parameters are of great importance for constructing various realistic nuclear-force models, which, in turn, form a basis for studying the structure of nuclei and various nuclear processes. For this reason, it is highly desirable to determine reliably and accurately the parameters in the effective-range expansion, including the scattering length a, the effective range r, the shape parameter v2, and higher order parameters vn. Although low-energy parameters for neutron–proton scattering have been determined and studied since the early 1950s, even the experimental values of such parameters as the scattering length a and the effective range r are ambiguous to date. As for the shape parameter v2, even its sign is unknown at the present time. The theoretical value of this parameter depends greatly on the nuclear-force model used: as we go over from one model to another, the parameter v2 in the triplet state changes within a broad interval, from −0.95 [1, 2] to 1.371 fm3 [3], whence it follows that the shape parameter is a very subtle and sensitive feature of nucleon–nucleon interaction. We would like to note that not only does the shape parameter v2 depend on the form of interaction, but it is also strongly dependent on the scattering length a and the effective range r. In particular, a change of only a few tenths of a percent in the scattering length a may lead to a severalfold change in the shape parameter v2 [4]. The shape parameters vn of order higher than that of v2 have been still more poorly determined and are more sensitive to details of nucleon– nucleon interaction. The aforesaid highlights once again the importance of reliably determining the scattering length a and the effective range r, the more so as these are quantities that are most frequently used as inputs in constructing various models of nucleon–nucleon interaction. 2. It is well known [5] that the neutron–proton system may occur either in the triplet (the total spin is S = 1) or the singlet (the total spin is S = 0) spin state. In determining the scattering lengths a and the effective ranges r in the triplet (t) and singlet (s) spin states, one employs the experimental dependence of the total (spin-averaged) cross section for the scattering of slow neutrons by free protons and data characterizing the scattering of zero- energy neutrons by para-hydrogen. In order to determine the triplet and singlet scattering lengths (at and as, respectively), use is usually made of equations that relate these quantities to the total cross section for the scattering of zero-energy neutrons by protons, σ0 = π , (2) and to the coherent scattering length, (3at + as) . (3) In this case, the cross section σ0 is determined from the results of experiments that study slow-neutron scattering on protons bound in various molecules (H2, H2O, C6H6, CH3OH), corrections associated with neutron capture by a proton and with effects of proton binding in molecules being subsequently eliminated. The elimination of binding-effect corrections is a nontrivial many-body problem, since, in addition to proton and neutron motion, it is necessary to take into account the motion of the molecular residue. A number of significant simplifications and approximations are made in solving this problem [6]. A compendium of experimental results from [7–13] on the total cross section for the scattering of zero-energy neutrons by free protons, σ0, is given in Table 1. Two values of the total cross section σ0 are recommended at the present time. These are the value obtained by Houk (1971) [12], σ0 = 20.436(23) b, (4) and the value obtained by Dilg (1975) [13], σ0 = 20.491(14) b. (5) Since these two values of σ0 are inconsistent, their weighted-mean value σ0 = 20.476(12) b (6) can also be used in determining the scattering lengths. It should be noted that the total cross section σ0 has not been measured since 1975. The coherent scattering length f , which is determined by relation (3), is found either from experiments where slow neutrons are scattered by pure para-hydrogen [8, 14, 15] or by crystals [16] or — and this is a more precise method — from experiments where neutrons are reflected by a liquid mirror and where use is made of a number of pure hydrocarbons [9, 10, 17–22]. Also, a method for determining the coherent scattering length by means of neutron interferometry from experiments to study neutron scattering on molecular hydrogen was proposed in [23]. The values found by various authors for the neutron–proton coherent scattering length f are quoted in Table 2, whence it can be seen that the value of this quantity is even more ambiguous than the value of σ0. In determining the scattering lengths in the triplet and the singlet state (at and as, respec- tively), one employs most frequently, at the present time, the coherent-length value obtained by Koester and Nistler [22], f = −3.7409(11) fm, (7) and the coherent-length value presented in the compilation of Dumbrajs et al. [24], f = −3.738(1) fm. (8) Recent experiments aimed at determining the neutron–proton coherent scattering length by means of neutron interferometry [23], which were mentioned above, yielded the value f = −3.7384(20) fm. (9) Within the experimental errors, the value in (9) agrees with the result of Koester and Nistler in (7) and with the value in (8), which was used by Dumbrajs et al. [24]. Table 3 presents values obtained in a number of previous studies [9, 10, 13, 18, 21, 22, 24–28] for the scattering lengths and effective ranges in the triplet and singlet spin states. All of them have been used as experimental values. The values of the triplet (at) and singlet (as) scattering lengths from Table 3 were obtained on the basis of formulas (2) and (3) by using various values for the total cross section σ0 and the neutron–proton coherent scattering length The values of the triplet effective range rt in Table 3 were determined primarily in an approximation that does not depend on the form of interaction; that is, rt ≡ ρ (−εd, 0) = 2R , (10) where ρ (−εd, 0) is the mixed effective radius of the deuteron; R = 1/α (11) is a parameter that characterizes the spatial dimensions of the deuteron; and α is the deuteron wave number, which is related to the deuteron binding energy εd by the equation εd = h̄ 2α2/mN . (12) In a number of studies [24, 26], the triplet effective range was determined in accordance with the formula rt = ρ (−εd, 0) + δrt , (13) where the correction δrt is a model-dependent quantity. According to the estimates obtained by Noyes on the basis of the dispersion relations [26], the correction δrt arising owing to one-pion exchange is δrt = −0.013 fm . (14) According to other estimates [24], this correction is δrt ≃ −0.001 fm , (15) which is an order of magnitude smaller in absolute value than the estimate in (14). In the latter case, the effective range rt is therefore nearly coincident with the mixed effective radius ρ (−εd, 0). The singlet effective range rs is usually determined on the basis of an analysis of the total cross section for neutron–proton scattering, σ (E), in the low-energy region at fixed values of the parameters at, as, and rt. The values found in this way for the singlet effective range rs appear to be even more ambiguous than the values of the triplet effective range. As can be seen from Table 3, the scattering-length and effective-range values used as experimental ones change within rather broad ranges. The scatter of these values is due first of all to the fact that different experimental values of the cross section for the scattering of zero-energy neutrons by free protons, σ0, and of the neutron–proton coherent scattering length f are used to determine these quantities. The ambiguity in determining the singlet effective range rs is also associated with an insufficient accuracy of the experimental total cross sections for neutron– proton scattering at energies below 5MeV. The values found by different authors for the singlet effective range rs change within a broad range, from 2.42 [9] to 2.81 fm [13]. Thus, the accuracy of the experiments performed in the 1950s–1970s is insufficient for unambiguously determining the low-energy parameters of neutron–proton scattering. At the same time, these parameters play an important role in the theory of few-nucleon systems, which is based on nucleon–nucleon interaction. As was shown in [29, 30], the binding energies of the 3H and 4He nuclei depend greatly on the singlet effective range rs, increasing as rs becomes smaller. By way of example, we indicate that, as rs decreases by 0.1 fm, the binding energies of the 3H and 4He nuclei increase by 0.3 and 1.5MeV, respectively. We note that the decrease of 0.01 fm in the triplet scattering length at also leads to the increase of 0.025MeV in the triton binding energy [31, 32]. At the same time, it is well known that, in calculations with realistic nucleon–nucleon potentials, the binding energies of few-nucleon systems prove to be underestimated. In such calculations, the 3H binding energy is as a rule underestimated by 1MeV. A reliable and precise determination of the low-energy parameters of neutron–proton scattering and their use in calculating the binding energies of systems that contain three or more nucleons may contribute to solving the problem of underestimating the binding energies of few-nucleon systems without introducing three-particle forces, quark degrees of freedom, and other concepts that would require revising basic points in the traditional theory of nuclear forces, which relies on pair nucleon–nucleon interaction. To conclude this section, we present, for low-energy parameters, values that are currently used as experimental ones. Most frequently, the present-day literature quotes two sets of low- energy parameters. These are the set from [24], at = 5.424(4) fm, rt = 1.759(5) fm; as = −23.748(10) fm, rs = 2.75(5) fm, which is matched with the experimental value (5) of the total cross section at zero energy due to Dilg [13] and with the value in (8) for the neutron–proton coherent scattering length from [24], and the set from [28], at = 5.419(7) fm, rt = 1.753(8) fm; as = −23.740(20) fm, rs = 2.77(5) fm, which corresponds to the weighted-mean value (6) of the cross sections presented by Houk [12] and Dilg [13] and to the value in (7) for the coherent length due to Koester and Nistler [22]. It should be noted that the experiments performed in the 1950–1970s were the main source of information used to deduce the values in (16) and (17) for the low-energy parameters of neutron–proton scattering. 3. In recent years, the accuracy of experimental data on nucleon–nucleon scattering has been improved considerably; moreover, methods of their partial-wave analysis, which make it possible to describe the results of scattering experiments in terms of phase shifts, have also been refined [33, 34]. Owing to this, the triplet and singlet low-energy parameters of neutron–proton scattering can be determined independently of one another by using the 3S1- and 1S0-state phase shifts [4, 35]. The results of the partial-wave analysis performed by the GWU group [33] (data from the well-known SAID nucleon–nucleon database) and by the Nijmegen group [34] are presently the most precise and most widely used data on the phase shifts for nucleon–nucleon scattering. The most popular modern realistic nucleon–nucleon potentials constructed within the last decade, which include the Nijm-I, Nijm-II, Reid93 [36], Argonne V18 [37], CD-Bonn [28, 38], and Moscow [39] potentials, are based on fits to data of the Nijmegen group [34]. However, it should be noted that the partial-wave analysis of the Nijmegen group is a result of processing and averaging experimental data on nucleon–nucleon scattering over a period from 1955 to 1992, but this analysis provides an insufficiently accurate description of modern experimental data on nucleon–nucleon scattering. Despite the proximity of the phase shifts for neutron– proton scattering that were obtained by the GWU and Nijmegen groups, the corresponding values of the low-energy parameters in the effective-range expansion are markedly different [4], this difference being not only quantitative but also qualitative. Using the approximation of the effective-range function k cot δ at low energies by polyno- mials and Padé approximants within the least squares method, we calculated the triplet and singlet low-energy parameters of neutron–proton scattering for the experimental data on the GWU [33] and Nijmegen [34] phase shifts. The results obtained for the low-energy parameters in the present study by employing the data from the partial-wave analysis of the GWU group, at = 5.4030 fm, rt = 1.7494 fm, v2t = 0.163 fm as = −23.719 fm, rs = 2.626 fm, v2s = −0.005 fm differ significantly from the parameter values at = 5.420 fm, rt = 1.753 fm, v2t = 0.040 fm as = −23.739 fm, rs = 2.678 fm, v2s = −0.48 fm which were obtained on the basis of the data from the partial-wave analysis of the Nijmegen group. The triplet low-energy parameters calculated here for the phase shifts of the Nijmegen group are virtually coincident with the analogous parameters obtained previously in [35]. Un- fortunately, the data presented by the Nijmegen group do not contain the singlet low-energy parameters of neutron–proton scattering. The value of the singlet shape parameter v2s for the Nijmegen phase shifts was calculated in [1], and it is in agreement with our value. Using expressions (18) and (19) for the scattering lengths and relying on formulas (2) and (3), we find for the cross section σ0 and for the coherent scattering length f that σ0 = 20.426 b , f = −3.755 fm (20) in the case of the GWU phase shifts and that σ0 = 20.473 b , f = −3.7395 fm (21) in the case of the Nijmegen phase shifts. The values in (21) are in good agreement with the weighted mean of the cross sections obtained by Houk and Dilg, σ0 = 20.476(12) b, and with the coherent-scattering-length value of f = −3.7409(11) fm according to Koester and Nistler [22]. It should be emphasized, however, that this agreement is not accidental; it is directly related to the fact that, in the partial-wave analysis of the Nijmegen group, the cross-section values obtained by Houk [12] and Dilg [13] and the coherent-scattering-length value obtained by Koester and Nistler [22] were used as input experimental parameters. It is precisely the reason why all of the experimental low- energy parameters in (17), with the exception of the singlet effective range, agree within the experimental error with the corresponding parameters in (19), which were calculated on the basis of the Nijmegen phase shifts. The singlet-effective-range value of rs = 2.678 fm, which was calculated for the phase shifts obtained by the Nijmegen group, is much smaller than the experimental value of rs = 2.77(5) fm, which was quoted by Dilg in [13]. In this connection, it should be noted that, in [13], the singlet effective range rs was determined from experimental data on the total cross section for neutron–proton scattering at energies below 5MeV at the scattering-length values fixed at at = 5.423(4) fm and as = −23.749 fm and the triplet-effective-range value fixed at rt = 1.760(5) fm, but, as was indicated above, this method for determining the effective range is highly unreliable (see Table 3). A determination of the singlet effective range rs directly from the singlet phase shift irrespective of the triplet parameters is more correct and consistent, which reduces substantially the uncertainty in this quantity. For the sake of comparison, the low-energy parameters for neutron–proton scattering that correspond to the GWU (GWU PWA) and Nijmegen (Nijm PWA) phase shifts are given in Ta- ble 4, along with the values of these parameters for a number of the realistic potentials (Argonne V18 [37], CD-Bonn [28, 38], and Moscow [39] potentials) whose parameters were matched with the Nijmegen nucleon–nucleon database. Also quoted there are the experimental values of the low-energy parameters. Table 4 shows that the values of the low-energy parameters obtained for the Nijmegen phase shifts are in perfect agreement with the corresponding parameters for the potentials fitted to the Nijmegen nucleon–nucleon database. A significant distinction between the values of the triplet low-energy parameters for the GWU and Nijmegen data was discussed in detail in our previous article [4]. Here, we only indicate that the difference of the triplet scattering lengths by 0.3% is in fact a more important circumstance than the fourfold distinction between the values of the triplet shape parameters. This is because many important features of the neutron–proton system — such as the asymp- totic deuteron normalization factor AS and the root-mean-square radius rd of the deuteron — are highly sensitive to variations in the triplet scattering length [40]. We also note that, although the triplet effective ranges obtained from experimental data of the two main groups are close to each other, the values of the difference δrt of the effective range rt and the mixed effective radius ρ (−εd, 0) for the GWU [33] and Nijmegen [34] phase shifts differ significantly. For example, the correction δrt for the phase shifts of the GWU group is positive, taking the value δrt = 0.0163 fm . (22) For the phase shifts of the Nijmegen group, this correction is negative and, in absolute value, is an order of magnitude smaller than the correction in (22): δrt = −0.001 fm. The value of the singlet effective range for the phase shifts of the GWU group also differs from its counterpart for the Nijmegen phase shifts (by about 2%), and the corresponding difference of the singlet shape parameters is formidable, reaching two orders of magnitude. In contrast to the partial-wave analysis of the Nijmegen group, the partial-wave analysis of the GWU group does not employ the values of the cross section σ0 and the coherent scattering length f as input parameters. The theoretical values of σ0 = 20.426 b and f = −3.755 fm, which we obtained here for the cross section in question and for the neutron–proton coherent scattering length from data of the partial-wave analysis performed by the GWU group, are in perfect agreement with the experimental cross-section value of σ0 = 20.436(23) b according to Houk [12], and the experimental coherent-scattering-length value obtained by Houk and Wilson [9, 10], f = −3.756(9) fm , (23) but they contradict the cross-section value of σ0 = 20.491(14) b according to Dilg [13] and the coherent-scattering-length value of f = −3.7384(20) fm, which was obtained recently by the neutron-interferometry method in [23]. Thus, we see that a reliable experimental determination of the total cross section for neutron–proton scattering at zero energy, σ0, and of the coherent scattering length, f , is now quite a pressing problem. 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Phys. 39, 546 (1967). 10. T. L. Houk and R. Wilson, Rev. Mod. Phys. 40, 672 (1968). 11. J. M. Neill, J. L. Russell, and J. R. Brown, Nucl. Sci. Eng. 33, 265 (1968). 12. T. L. Houk, Phys. Rev. C 3, 1886 (1971). 13. W. Dilg, Phys. Rev. C 11, 103 (1975). 14. G. L. Squires and A. T. Stewart, Proc. Roy. Soc. A 230, 19 (1955). 15. J. Callerame, D. J. Larson, S. J. Lipson, and R. Wilson, Phys. Rev. C 12, 1423 (1975). 16. C. G. Shull, E. O. Wollan, G. A. Morton, and W. L. Davidson, Phys. Rev. 73, 842 (1948). 17. D. J. Hughes, M. T. Burgy, and G. R. Ringo, Phys. Rev. 77, 291 (1950). 18. M. T. Burgy, G. R. Ringo, and D. J. Hughes, Phys. Rev. 84, 1160 (1951). 19. W. C. Dickinson, L. Passell, and O. Halpern, Phys. Rev. 126, 632 (1962). 20. L. Koester, Z. Phys. 198, 187 (1967). 21. L. Koester and W. Nistler, Phys. Rev. Lett. 27, 956 (1971). 22. L. Koester and W. Nistler, Z. Phys. A 272, 189 (1975). 23. K. Schoen, D. L. Jacobson, M. Arif et al., Phys. Rev. C 67, 044005 (2003). 24. O. Dumbrajs, R. Koch, H. Pilkuhn et al., Nucl. Phys. B 216, 277 (1983). 25. H. P. Noyes, Phys. Rev. 130, 2025 (1963). 26. H. P. Noyes, Ann. Rev. Nucl. Sci. 22, 465 (1972). 27. E. L. Lomon and R. Wilson, Phys. Rev. C 9, 1329 (1974). 28. R. Machleidt, Phys. Rev. C 63, 024001 (2001). 29. V. F. Kharchenko, N. M. Petrov, and S. A. Storozhenko, Nucl. Phys. A 106, 464 (1968). 30. V. F. Kharchenko, Fiz. Élem. Chastits At. Yadra 10, 884 (1979) [Sov. J. Part. Nucl. 10, 349 (1979)]. 31. V. F. Kharchenko and S. A. Storozhenko, Nucl. Phys. A 137, 437 (1969). 32. N. M. Petrov and I. V. Simenog, Yad. Fiz. 28, 381 (1978) [Sov. J. Nucl. Phys. 28, 193 (1978)]. 33. R. A. Arndt, W. J. Briscoe, I. I. Strakovsky, and R. L. Workman, Partial-Wave Analysis Facility SAID, The George Washington University [http://gwdac.phys.gwu.edu] ; R. A. Arndt, I. I. Strakovsky, and R. L. Workman, Phys. Rev. C 62, 034005 (2000). http://gwdac.phys.gwu.edu] 34. Nijmegen NN-Online program [http://nn-online.org] ; V. G. J. Stoks, R. A. M. Klomp, M. C. M. Rentmeester, and J. J. de Swart, Phys. Rev. C 48, 792 (1993). 35. J. J. de Swart, C. P. F. Terheggen, and V. G. J. Stoks, Invited talk at the 3rd International Symposium ”Dubna Deuteron 95”, Dubna, Russia, 1995, nucl-th/9509032. 36. V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen, and J. J. de Swart, Phys. Rev. C 49, 2950 (1994). 37. R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C 51, 38 (1995). 38. R. Machleidt, F. Sammarruca and Y. Song, Phys. Rev. C 53, 1483 (1996). 39. V. I. Kukulin, V. N. Pomerantsev, and A. Faessler, nucl-th/9903056. 40. V. A. Babenko and N. M. Petrov, Yad. Fiz. 66, 1359 (2003) [Phys. At. Nucl. 66, 1319 (2003)]. http://nn-online.org] http://arxiv.org/abs/nucl-th/9509032 http://arxiv.org/abs/nucl-th/9903056 Table 1. Total cross section for neutron scattering on a proton at zero energy No. References σ0 , b 1 Melkonian [7] (1949) 20.36(10) 2 Stewart and Squires [8] (1953) 20.41(14) 3 Houk and Wilson [9] (1967) 20.37(2) 4 Houk and Wilson [10] (1968) 20.442(23) 5 Neill et al. [11] (1968) 20.366(76) 6 Houk [12] (1971) 20.436(23) 7 Dilg [13] (1975) 20.491(14) Table 2. Amplitude for coherent neutron–proton scattering No. References f , fm 1 Shull et al. [16] (1948) −3.900(100) 2 Hughes et al. [17] (1950) −3.75(3) 3 Burgy et al. [18] (1951) −3.78(2) 4 Stewart and Squires [8] (1955) −3.80(5) 5 Dickinson et al. [19] (1962) −3.740(20) 6 Koester [20] (1967) −3.719(2) 7 Houk and Wilson [9, 10] (1967, 1968) −3.756(9) 8 Koester and Nistler [21] (1971) −3.740(3) 9 Koester and Nistler [22] (1975) −3.7409(11) 10 Callerame et al. [15] (1975) −3.733(4) 11 Schoen et al. [23] (2003) −3.7384(20) Table 3. Low-energy parameters of neutron–proton scattering from various studies No. References at , fm as , fm rt , fm rs , fm 1 Burgy et al. [18] (1951) 5.377(21) −23.690(55) 1.704(28) − 2 Noyes [25] (1963) 5.396(11) −23.678(28) 1.727(14) 2.51(11) 5.392(6) −23.689(13) 1.724(7) 2.42(9) 3 Houk and Wilson [9] (1967) 5.399(11) −23.680(28) 1.732(12) 2.48(11) 5.411(4) −23.671(12) 1.747(4) 2.59(8) 4 Houk and Wilson [10] (1968) 5.405(6) −23.728(13) 1.738(7) 2.56(10) 5 Koester and Nistler [21] (1971) 5.414(5) −23.719(13) − − 6 Noyes [26] (1972) 5.413(5) −23.719(13) 1.735 2.66 5.423(5) −23.712(13) 1.748(6) 2.75(10) 7 Lomon and Wilson [27] (1974) 5.414(5) −23.719(13) 1.750(5) 2.76(5) 2.77(5) 8 Dilg [13] (1975) 5.423(4) −23.749(9) 1.760(5) 2.81(5) 2.78(5) 9 Koester and Nistler [22] (1975) 5.424(3) −23.749(8) 1.760(5) 2.81(5) 10 Dumbrajs et al. [24] (1983) 5.424(4) −23.748(10) 1.759(5) 2.75(5) 11 Machleidt [28] (2001) 5.419(7) −23.740(20) 1.753(8) 2.77(5) Table 4. Low-energy parameters of neutron–proton scattering that were obtained on the basis of the present-day data of the partial-wave analysis and modern realistic models of nucleon– nucleon interaction No. Model at , fm as , fm rt , fm rs , fm σ0 , b f , fm 1 GWU PWA 5.4030 −23.719 1.7494 2.626 20.426 −3.755 2 Nijm PWA 5.420 −23.739 1.753 2.678 20.473 −3.7395 3 Argonne V18 5.419 −23.732 1.753 2.697 20.461 −3.7375 4 CD Bonn 5.4199 −23.738 1.751 2.671 20.471 −3.7392 5 Moscow 5.422 −23.740 1.754 2.66 20.476 −3.7370 6 Expt. [10, 12] 5.405(6) −23.728(13) 1.738(7) 2.56(10) 20.436(23) −3.756(9) 7 Expt. [24] 5.424(4) −23.748(10) 1.759(5) 2.75(5) 20.491(14) −3.738(1) 8 Expt. [28] 5.419(7) −23.740(20) 1.753(8) 2.77(5) 20.476(12) −3.7409(11)

0704.1025

Simulation of Ultra High Energy Neutrino Interactions in Ice and Water

October 31, 2018 Simulation of Ultra High Energy Neutrino Interactions in Ice and Water (the ACoRNE Collaboration)a S. Bevan1, S. Danaher2, J. Perkin3, S. Ralph3†, C. Rhodes4, L. Thompson3, T. Sloan5b and D. Waters1. 1 Physics and Astronomy Dept, University College London, UK. 2 School of Computing Engineering and Information Sciences, University of Northumbria, Newcastle-upon-Tyne, UK. 3 Dept of Physics and Astronomy, University of Sheffield, UK. 4 Institute for Mathematical Sciences, Imperial College London, UK. 5 Department of Physics, University of Lancaster, Lancaster, UK † Deceased a Work supported by the UK Particle Physics and Astronomy Research Council and by the Ministry of Defence Joint Grants Scheme b Author for correspondence, email [email protected] Abstract The CORSIKA program, usually used to simulate extensive cosmic ray air showers, has been adapted to work in a water or ice medium. The adapted CORSIKA code was used to simulate hadronic showers produced by neutrino interactions. The simulated showers have been used to study the spatial distribution of the deposited energy in the showers. This allows a more precise determination of the acoustic signals produced by ultra high energy neutrinos than has been possible previously. The properties of the acoustic signals generated by such showers are described. (Submitted to Astroparticle Physics) http://arxiv.org/abs/0704.1025v1 1 Introduction In recent years interest has grown in the detection of very high energy cosmic ray neutrinos [1]. Such particles could be produced in the cosmic particle accelerators which make the charged primaries or they could be produced by the interactions of the primaries with the Cosmic Mi- crowave Background, the so called GZK effect [2]. The flux of neutrinos expected from these two sources has been calculated [3,4]. It is found to be very low so that large targets are needed for a measurable detection rate. It is interesting to measure this neutrino flux to see if it is compatible with the values expected from these sources, incompatibility implying new physics. Searches for cosmic ray neutrinos are ongoing in AMANDA [5], IceCube [6], ANTARES [7] and NESTOR [8], detecting upward going muons from the Cherenkov light in either ice or water. In general, these experiments are sensitive to lower energies than discussed here since the Earth becomes opaque to neutrinos at very high energies. The experiments could detect almost horizontal higher energy neutrinos but have limited target volume due to the attenuation of the light signal in the ice. The Pierre Auger Observatory, an extended air shower array detector, will also search for upward and almost horizontal showers from neutrino interactions [9]. In addition to these detectors there are ongoing experiments to detect the neutrino interactions by either radio or acoustic emissions from the resulting particle showers [1]. These latter techniques, with much longer attenuation lengths, allow very large target volumes utilising either large ice fields or dry salt domes for radio or ice fields, salt domes and the oceans for the acoustic technique. In order to assess the feasibility of each technique the production of the particle shower from neutrino interactions needs to be simulated. Since experimental data on the interactions of such high energy particles do not exist it is necessary to use theoretical models to simulate them. The most extensive ultra high energy simulation program which has so far been developed is CORSIKA [10]. However, this program has been used previously only for the simulation of cosmic ray air showers. The program is readily available [10]. Different simulations are necessary for the radio and acoustic techniques. Radio emission occurs due to coherent Cherenkov radiation from the particles in the shower, the Askaryan Effect [11]. The emitted energy is sensitive to the distribution of the electron-positron asymme- try which develops in the shower and which grows for lower energy electromagnetic particles. Hence, to simulate radio emission, the electromagnetic component of the shower must be fol- lowed down to very low kinetic energies (∼ 100 keV) [12]. In contrast, an acoustic signal is generated by the sudden local heating of the surrounding medium induced by the particle shower [13]. Thus to simulate the acoustic signal the spatial distribution of the deposited en- ergy is needed. Once the electromagnetic energy in the shower reaches the MeV level (electron range ∼ 1 cm) the energy can be simply added to the total deposited energy and the simula- tion of such particles discontinued. Extensive simulations have been carried out for the radio technique [14]. However, the simulations for the acoustic technique are less advanced. Some work has been done [15,16] using the Geant4 package [17]. However, this work is restricted to energies less than 105 GeV for hadron showers since the range of validity of the physics models in this package does not extend to higher energy hadrons. In this paper the energy distributions of showers produced by neutrino interactions in sea water at energies up to 1012 GeV are discussed. The distributions are generated using the air shower program CORSIKA [10] modified to work in a sea water medium. The salt compo- nent of the sea water has a negligible effect1 and the results are presented in distance units of g cm−2, hence they should be applicable to ice also. The computed distributions have been pa- rameterised and this parameterisation is used to develop a simple program to simulate neutrino interactions and the resulting particle showers. The properties of the acoustic signals from the generated showers are also presented. 2 Adaptation of the CORSIKA program to a water medium The air shower program, CORSIKA (version 6204) [10], has been adapted to run in sea water i.e. a medium of constant density of 1.025 g cm−3 rather than the variable density needed for an air atmosphere. Sea water was assumed to consist of a medium in which 66.2% of the atoms are hydrogen, 33.1% of the atoms are oxygen and 0.7% of the atoms are made of common salt, NaCl. The salt was assumed to be a material with atomic weight and atomic number A=29.2 and Z=14, the mean of sodium and chlorine. The purpose of this is to maintain the structure of the program as closely as possible to the air shower version which had two principal atmospheric components (oxygen and nitrogen) with a trace of argon. The presence of the salt component had an almost undetectable effect on the behaviour of the showers. Other changes made to the program to accommodate the water medium include a modifica- tion of the stopping power formula to allow for the density effect in water 2. This only affects the energy loss for hadrons since the stopping powers for electrons are part of the EGS [18] package which is used by CORSIKA to simulate the propagation of the electromagnetic com- ponent of the shower. Smaller radial binning of the shower was also required since shower radii in water are much smaller than those in air. In addition the initial state energy for electrons and photons above which the LPM effect [19] was simulated in the program was reduced to the much lower value necessary for water 3. The LPM effect suppresses pair production from high energy photons and bremsstrahlung from high energy electrons. Similarly, the interactions of neutral pions had to be simulated at lower energy than in air because of the higher density water medium. In all about 100 detailed changes needed to be made to the CORSIKA program to accommodate the water medium. To test the implementation of the LPM effect [19] in the program 100 showers from incident gamma ray photons at several different energies were generated and the mean depth of the first interaction (the mean free path) calculated. The observed mean free path was found to be in agreement with the expected behaviour when both the suppression of pair production and photonuclear interactions were taken into account (see Figure 1). This showed that the LPM effect had been properly implemented in CORSIKA. Considerable fluctuations between showers occurred. These are expressed in terms of the ratio of the root mean square (RMS) deviation of a given parameter to its mean value: the 1The shower maximum was observed to peak at a depth 2.4 ± 1.1% less in sea water than in fresh water with the same peak energy deposited, for protons of energy 105 GeV. 2The stopping power was computed using the Bethe-Bloch formula [20] and the density effect from the formu- lae of Sternheimer et al [21]. 3The level was set at 1 TeV compared to the characteristic energy for water ELPM = 270 TeV [20]. RMS peak energy deposit to the mean peak energy deposit was observed to be 14% at 105 GeV reducing to 4% at 1011 GeV, that for the depth of the peak position varied from 19% to 7.4% and for the full width at half maximum of the shower from 63% to 18%. To smooth out such fluctuations averages of 100 generated showers will be taken in the following. The statistical error on the averages is then given by these RMS values divided by 10. The hadronic energy contributes only about 10% to the shower energy at the shower peak, the remainder being carried by the electromagnetic part of the shower. The simulations were all carried out in a vertical column of sea water 20 m long. The deposited energy generated by CORSIKA was binned into 20 g cm−2 slices longitudinally and 1.025 g cm−2 annular cylinders radially for 0 < r < 10.25 g cm−2 and 10.25 g cm−2 for 10.25 < r < 112.75 g cm−2 where r is the distance from the vertical axis. To reduce computing times, the thinning option was used i.e. below a certain fraction of the primary energy (in this case 10−4) only one of the particles emerging from the interaction is followed and an appropriated weight is given to it [22]. The simulation of particles continued down to cut- off energies of 3 MeV for electromagnetic particles and 0.3 GeV for hadrons. When a particle reached this cut-off, the energy was added to the slice where this occurred. The QGSJET [23] model was used to simulate the hadronic interactions. 3 Comparison with other simulations 3.1 Comparison with Geant4 Proton showers were generated in sea water using the program Geant4 (version 8.0) [17] and compared with those generated in CORSIKA. Unfortunately, the range of validity of Geant4 physics models for hadronic interactions does not extend beyond an energy of 105 GeV. Hence the comparison is restricted to energies below this. Figure 2 shows the longitudinal distributions of proton showers at energies of 104 and 105 GeV (averaged over 100 showers) as determined from Geant4 and CORSIKA. The showers from CORSIKA tend to be slightly broader and with a smaller peak energy than those generated by Geant4. The difference in the peak height is ∼ 5% at 104 GeV rising to ∼ 10% at energy 105 GeV. Figure 3 shows the radial distributions. The differences in the longitudinal distributions are reflected in the radial distributions. However, the shapes of the radial distributions are very similar between Geant4 and CORSIKA, with CORSIKA producing ∼ 10% more energy near the shower axis at depths between 450 and 850 g cm−2 where most of the energy is deposited. The acoustic signal from a shower is most sensitive to the radial distribution, particularly near the axis (r ∼ 0). It is relatively insensitive to the shape of the longitudinal distribution. 3.2 Comparison with the simulation of Alvarez-Muñiz and Zas The CORSIKA simulation was also compared with the longitudinal shower profile for protons computed in the simulation by Alvarez-Muñiz and Zas (AZ) [24]. There was a reasonable agreement between the longitudinal shower shapes from CORSIKA and those shown in Figure 2 of ref. [24]. However, the numbers of electrons and positrons at the peak of the CORSIKA showers was ∼ 20% lower than those from ref. [24]. This number is sensitive to the energy below which these particles are counted and this is not specified in [24]. Hence the agreement between CORSIKA and their simulation is probably satisfactory within this uncertainty. In conclusion, the modifications made to CORSIKA to simulate high energy showers in a water medium give results which are compatible with the predictions from the Geant4 simula- tions for energy less than 105 GeV and the simulation of AZ within 20%. This is taken to be the accuracy of the simulation program assuming that there are no unexpected and unknown interactions between the centre of mass energy explored at current accelerators and those stud- ied in these simulations. Studies of the sensitivity of the CORSIKA simulation to the different models of the hadronic interactions have been reported in reference [25]. They find that the peak number of electrons plus positrons varies by ∼ 20% for proton showers in air depending on the choice of the hadron interaction model used. These differences are similar in magnitude to the differences between the AZ, Geant4 and CORSIKA simulations reported here. Hence the observed differences between the Geant4, AZ and CORSIKA simulations in water could be within the uncertainties of the hadronic interaction models. 4 Simulation of neutrino induced showers Neutrinos interact with the nuclei of the detection medium by either the exchange of a charged vector boson (W+), i.e. charged current (CC) interactions or the exchange of the neutral vector boson (Z0), i.e. neutral current (NC) deep inelastic scattering interactions (see for example [26]). The ratio of the CC to NC interaction cross sections is approximately 2:1. The CC interactions produce charged secondary scattered leptons while the NC interactions produce neutrinos. The hadron shower carries a fraction y of the energy of the incident neutrino and the scattered lepton the remaining fraction 1 − y. We assume that the neutrino flavours are homogeneously mixed when they arrive at the Earth by neutrino oscillations. Hence in the CC interactions electrons, µ and τ leptons will be produced as the scattered leptons in equal proportions. At the energies we shall consider, these particles behave in a manner similar to minimum ionising particles for µ and τ leptons. This is almost true also for electrons for which the bremsstrahlung process will be suppressed by the LPM effect. Hence the charged scattered leptons contribute little to the energy producing an acoustic signal. In the case of NC interactions there is no contribution to this energy from the scattered lepton. For these reasons the contribution of the scattered lepton to the shower profile is ignored beyond z = 20 m in what follows. It is interesting to note that a τ lepton can decay to hadrons or a very high energy electron or muon can produce bremsstrahlung photons at large distances from the interaction point. These can initiate further distant showers, the so called “double bang” effect. The stochastic nature of such electron showers is studied in [15, 16]. These effects are not considered in this study. 4.1 Neutrino-nucleon interaction cross sections. A number of groups have computed the high energy neutrino-nucleon interaction cross sections, σ, [27–29]. In the quark parton model of the nucleon for the single vector boson exchange pro- cess, the differential cross section for CC interactions can be expressed in terms of the measured structure functions of the target nucleon F2 and xF3 as dQ2dy Q2 +M2W (F2(x,Q 2)(1− y + y2/2)± y(1− y/2)xF3(x,Q 2)) (1) where GF is the Fermi weak coupling, MW is the mass of the weak vector boson, Q 2 is the square of the four momentum transferred to the target nucleon, y = ν/E where ν is the energy transferred to the nucleon (ν = E−E ′ with E and E ′ the energies of the incident and scattered leptons) and x = Q2/2Mν is the fraction of the momentum of the target nucleon carried by the struck quark (here x and y are defined for a stationary target nucleon). The plus (minus) sign is for neutrino (anti-neutrino) interactions. It can be seen that y is the fraction of the neutrino’s energy which is converted into the energy of the hadron shower. A similar expression can be written down for the NC interaction (see for example [26]) which has a ratio to the CC cross section varying from 0.33 to 0.41 as the neutrino energy increases from 104 to 1013 GeV. The structure functions F2 and xF3 are the sum of the quark distribution functions which have been parameterised by fitting data [30, 31]. It can be shown that Q2 = sxy where s = 2ME is the square of the centre of mass energy (M is the target nucleon mass). To compute the cross sections the structure functions must be calculated at values of x . M2W/s i.e. at values well outside the region of the fits to the parton distribution functions (PDFs) which have been performed for x & 10−5, the range of current measurements. The extrapolation outside the measurement range is discussed in [27], [29] and [32, 33]. Here we adopt the procedure of extrapolating linearly on a log-log scale from the parameterised parton distribution functions of [30] computed at x = 10−4 and x = 10−5. By considering various theoretical evolution procedures it is estimated in [29] that the procedure has an accuracy of ∼ 32% per decade and we use this as an estimate of the accuracy of the calculation. However, this could be an underestimate [34]. The expression in equation 1 for charged current interactions and the one for neutral current interactions were integrated to obtain the total neutrino-nucleon interaction cross section, the value of the fraction of events per interval of y, 1/σdσ/dy, and the mean value of y. The total cross section was found to be in good agreement with the values in [27, 29] and in reasonable agreement with [28] which is based on a model different from the quark parton model. Fig- ure 4 shows the mean value of y obtained from this procedure (solid curve) and the effect of multiplying or dividing the PDFs by a factor 1.32 per decade (dashed curves) as an indication of the possible range of uncertainties in the extrapolation of the PDFs. Figure 5 shows the y dependence of the cross section for different neutrino energies. 4.2 A simple generator for neutrino interactions. A simple generator for neutrino interactions in a column of water of thickness 20 m was con- structed as follows. The neutrino interacts at the top of the water column (z=0, with the z axis along the axis of the column). The energy fraction transferred, y, for the interaction was gener- ated, distributed according to the curve for the energy of the neutrino shown in Figure 5. This allows the energy of the hadron shower to be calculated for the event. The assumption was made that these hadron showers will have approximately the same distributions as those of a proton interaction at z=0 (see Section 4.3 for a test of this assumption). A series of files of 100 such proton interactions were generated at energies in steps of half an order of magnitude between 105 and 1012 GeV. The hadron shower for each neutrino interaction was selected at random from the 100 showers in the file at the proton energy closest to the energy of the hadron shower. The deposited energy in each bin was then multiplied by the ratio of the energy of the hadron shower to that of the proton shower. This is made possible because the shower shapes vary slowly with shower energy. For example, the ratio of the peak energy deposit per 20 g cm−2 slice to the shower energy varies from 0.037 to 0.030 as the proton shower energy varies from 105 to 1012 GeV. 4.3 The HERWIG neutrino generator. The CORSIKA program has an option to simulate the interactions of neutrinos at a fixed point [35]. The first interaction is generated by the HERWIG package [36]. This option was adapted to our version of CORSIKA in sea water. Some problems were encountered with the y dependence of the resulting interactions due to the extrapolation of the PDFs to very small x at high energies. This only affects the rate of the production of the showers at different y and the distribution of the hadrons produced in the interaction at a given y should be unaffected. A total of 700 neutrino interactions were generated at an incident neutrino energy of 2 · 1011 GeV. These were divided into the shower energy intervals 0.5−2 ·1010, 2−4 ·1010, 4−7.5 ·1010, 0.75− 1.3 · 1011 and 1.3− 2 · 1011. The showers in which the scattered lepton energy disagreed with the shower energy by more than 20% were eliminated leading to a loss of 17% of the events with shower energy greater than 0.5 · 1010 GeV. This is due to radiative effects and misidentification of the scattered lepton. Approximately 70 events remained in each energy interval. The energy depositions from these were averaged and compared to the averages from proton showers scaled by the ratio of the shower energy to the proton energy. Figure 6 shows the longitudinal distributions of the hadronic shower energy deposited for the different energy intervals (labelled EW ) compared to the scaled proton distributions. Figure 7 shows a sample of the transverse distributions. There is a good consistency between the proton and neutrino induced showers. The proton showers peak, on average, 20 g cm−2 shallower in depth with a peak energy 2% larger than the neutrino induced showers. This is small compared to the overall uncertainty. The slight shift in the longitudinal distribution is reflected as a normalisation shift in the radial distributions. We conclude therefore that to equate a proton induced shower starting at the neutrino interaction point to that from a neutrino is a satisfactory approximation. 5 Parameterisation of showers In this section a parameterisation of the energy deposited by the showers generated by COR- SIKA (averaged over 100 showers depositing the same total energy) is described. Other avail- able parameterisations will then be compared with the showers generated by CORSIKA. The acoustic signal generated by a hadron shower depends mainly on the energy deposited in the inner core of the shower. This is illustrated in figure 8 which shows the contribution to the acoustic signal from cores of different radii. This figure shows that it is crucial to represent the deposited energy well at radius less than 2.05 g cm−2. The calculation of the acoustic signal from the deposited energy is described in section 6. 5.1 Parameterisation of the CORSIKA Showers The differential energy deposited was parameterised as follows = L(z, EL) · R(r, z, EL) (2) where the function L(z, EL) represents the longitudinal distribution of deposited energy and R(r, z, EL) the radial distribution. Here EL is log10E with E the total shower energy. The function L(z, EL) = dE/dz is a modified 4 version of the Gaisser-Hillas function [37]. This function represents the longitudinal distribution of the energy deposited. L(z, EL) = P1L z − P2L P3L − P2L (P3L−P2L) P4L+P5Lz+P6Lz P3L − z P4L + P5Lz + P6Lz2 Here the parameters PnL were fitted to quadratic functions of EL = log10E with values = 2.760 · 10−3 − 1.974 · 10−4EL + 7.450 · 10 −6E2L (4) P2L = −210.9− 6.968 · 10 −3EL + 0.1551E L (5) P3L = −41.50 + 113.9EL − 4.103E L (6) P4L = 8.012 + 11.44EL − 0.5434E L (7) P5L = 0.7999 · 10 −5 − 0.004843EL + 0.0002552E L (8) P6L = 4.563 · 10 −5 − 3.504 · 10−6EL + 1, 315 · 10 −7E2L. (9) The parameter P1L represents the peak energy deposited and P3L the depth in the z coordinate at this peak while P2L, P4L, P5L and P6L are related to the shower width and shape in z. The radial distribution was represented by the NKG function [37] R(r, z, EL) = )(P2R−1) )(P2R−4.5) where the integral )(P2R−1) )(P2R−4.5) dr = P1R Γ(4.5− 2P2R)Γ(P2R) Γ(4.5− P2R) 4 The modification is to replace the shape parameter λ in equation 3.5 of reference [37] by the quadratic expression in z in equation 3. The parameter P1R was found to vary strongly with depth while P2R was only a weak function of depth. The parameters PnR (with n = 1,2) were each represented by the quadratic form PnR = A +Bz + Cz 2 (11) and the quantities A,B,C parameterised as quadratic functions of EL. This gave for P1R A = 0.01287E2L − 0.2573EL + 0.9636 (12) B = −0.4697 · 10−4E2L + 0.0008072EL + 0.0005404 (13) C = 0.7344 · 10−7E2L − 1.375 · 10 −6EL + 4.488 · 10 −6 (14) and for the parameter P2R A = −0.8905 · 10−3E2L + 0.007727EL + 1.969 (15) B = 0.1173 · 10−4E2L − 0.0001782EL − 5.093 · 10 −6 (16) C = −0.1058 · 10−7E2L + 0.1524 · 10 −6EL − 0.1069 · 10 −8. (17) The fit was made in a depth range where dE/dz was greater than 10% of the peak value defined by equation 4. The program MINUIT [38] was used to minimise the squared fractional deviations Fi −Di Fi +Di where Fi and Di refer to the fitted value and the value observed in the ith bin from the COR- SIKA showers, respectively. In order to improve the fit at small radii the contributions to χ2 were arbitrarily weighted by 10 for r < 2.05 g cm−2, 4 for 2.05 < r < 3.075 g cm−2, unity for 3.075 < r < 51.25 g cm−2 and 0.25 for r > 51.25 g cm−2. The RMS value of the frac- tional deviations was 3.4% for radii less than 51.25 g cm−2 and for energies greater than 106.5 GeV. The fit becomes poorer at lower energies and greater radii than these. Integrating the pa- rameterisation shows that the fraction of the total energy computed from the fit within the fit range was 91% averaged over the deposited energy range 107 to 1012 GeV. The corresponding fraction directly from the CORSIKA distributions was 92.5%, averaged over the same energy range. When applying this parameterisation at depths with smaller energy deposit than 10% of the peak value, the energy was assumed to be confined to an annular radius of 1.025 g cm−2. There was a good agreement (within 5% at the peak) between the acoustic signal computed using this parameterisation and that taken directly from the CORSIKA showers. 5.2 The parameterisation used by the SAUND Collaboration The SAUND Collaboration [39] uses the following parameterisation [40], based on the NKG formulae (e.g. see reference [37]), for the energy deposited per unit depth, z, and per unit annular thickness at radius r from a shower of energy E = Ek( )t exp (t− z/λ) 2πrρ(r) (19) where zmax = 0.9X0 ln(E/Ec) is the maximum shower depth, X0 = 36.1 g cm −2 is the radia- tion length and Ec = 0.0838GeV. The constants t = zmax/λwhere λ = 130−5 log10(E/10 4GeV) g cm−2 and k = tt−1/ exp (t)λΓ(t). The radial density is given by ρ(r) = as−2(1 + a)s−4.5 Γ(4.5− s) 2πΓ(s)Γ(4.5− 2s) where a = r/rM with rM = 9.04 g cm −2, the Molière radius in water, and s = 1.25. Figure 9 shows the radial distributions from CORSIKA compared with the absolute predictions of this parameterisation. There is qualitative agreement between the parameterisation and the CORSIKA results. The difference in normalisation is explained by the somewhat different longitudinal profiles of the CORSIKA showers from the SAUND parameterisation. The latter are broader with a lower peak energy deposit and a depth of the maximum which is larger than the CORSIKA showers. CORSIKA predicts more energy at small r than the SAUND parameterisation. Quantitatively, 51% of the shower energy is contained within a cylinder of radius 4 cm for the CORSIKA showers compared to 35% from the SAUND parameterisation. These fractions are approxi- mately independent of energy. Hence, in acoustic detectors a harder frequency spectrum for the acoustic signals is predicted by CORSIKA than by the SAUND parameterisation. Note that in the fit described in Section 5.1 the values of the parameter P1R (equivalent to RM in equation 20) were strongly depth dependent and much lower than the Molière radius in water, assumed by the SAUND collaboration. In addition, the value of P2R (equivalent to s in equation 20) while relatively constant tended to be at a higher value (∼ 1.9) than that assumed by SAUND. 5.3 The parameterisation used by Niess and Bertin Hadron showers, generated by Geant4 (version 4.06 p03), were studied up to energies of 105 GeV and electromagnetic showers to higher energies by Niess and Bertin [15,16]. The hadronic showers were parameterised as follows. = rf(z)g(r, z) (21) f(z) = (bz′)a−1 exp−bz′ where E is the energy of the hadron shower, X0 is the radiation length in water, z ′ = z/X0, b = 0.56 as determined from the fit and a is chosen to satisfy z′max = (a − 1)/b. Here z max is the depth in radiation lengths at which the shower maximum occurs. This is parameterised as z′max = 0.65 log( ) + 3.93 (23) with Ec = 0.05427 GeV. The radial distribution function is parameterised as g(r, z) = g0 where ri = 3.5 cm, n = n1 = 1.66 − 0.29(z/zmax) for r < ri and n = n2 = 2.7 for r > ri. The constant g0 is chosen to be (2− n1)(n2 − 2)/((n2 − n1)r i ) so that the integral of the radial distribution is unity. Figure 10 shows the radial distributions from this parameterisation compared with the pre- dictions of CORSIKA. There is quite good agreement between the two. There is a difference in the normalisation with depth since Geant4, on which this parameterisation is based, produces showers which tend to develop more slowly with depth than those from CORSIKA (see Fig- ure 2). Furthermore, both this and the SAUND parameterisation (Section 5.2) assume a linear variation of the shower peak depth with logE whereas CORSIKA gives a clear parabolic shape (see equation 6). This is illustrated in Figure 11. The Niess-Bertin parameterisation predicts that 56% of the shower energy is contained within a cylinder of radius 4 cm in reasonable agreement with the value of 51% from CORSIKA (these values are almost independent of energy). 6 The acoustic signals from the showers. The pressure, P , from a hadron shower depositing total energy E at time t resulting from the deposition of relative energy density ǫ = (1/E)(1/2πr)d2E/drdz at a point distant d from the volume, dV , follows the form [13] P (d, t) = δ(t− d/c) dV (25) where the integral is over the total volume of the shower. Here β = 2.0 · 10−4 is the thermal expansion coefficient of the medium at 14◦C, Cp = 3.8 · 10 3 J kg−1 K−1 is the specific heat capacity and c = 1500 ms−1 is the velocity of sound in the sea water. Acoustic signals seen by an observer at distance r from the shower centre are computed from equation (25) as follows. Points are produced randomly throughout the volume of the shower with density proportional to the deposited energy density and the time of flight from every produced point to the observer calculated. The flight times to the observer are histogrammed over 2n bins (in this case n = 10 is chosen) centred on the mean flight time and with a suitable bin width, τ (chosen here to be 1µs). The counts in each bin of the histogram are divided by τ yielding the function Exyz(t). The Fourier transform of the pressure wave is then P (ω) = Exyz(t)e −iωtdt = Exyz(t)e −iωtdt = iωExyz(ω) using the standard Fourier transform theorem, that taking the derivative in the time domain is the same as multiplying by iω in the frequency domain. The Fourier transform Exyz(ω) at angular frequency ω is evaluated numerically by a fast Fourier Transform (FFT) from the histogram Exyz(t). A correction is applied for attenuation in the water by a factor A(ω) = e −α(ω)r where α(ω) is the frequency dependent attenuation coefficient. The pressure as a function of time is then evaluated numerically by an inverse FFT using frequency steps from zero to the sampling frequency (the inverse of the bin width τ i.e. 1 MHz in this case). This gives P (t) = n=511 n=−512 P (ωn)A(ωn)e inΩ (27) where Ω = 2π/1024 radians and ωn/2π = nΩ/2π MHz is the nth frequency. The attenuation coefficient α(ω) is computed either according to the formulae in [42] or using the complex attenuation given in [15, 16]. This method of calculation was computationally much faster than the evaluation of the space integral given in equation 18 of reference [13] and gave identical results. Acoustic pulses, computed with the complex attenuation described in [15, 16], using the parameterisations of the shower profile given above are shown in Figure 12. It can be seen that the parameterisation developed here gives similar results to that described in [15, 16] despite the fact that the latter was an extrapolation from low energy simulations. The parameterisation used by SAUND [39, 40] gives smaller signals concentrated at somewhat lower frequencies. Further properties of the acoustic signals are shown in Figures 13 to 16. The pulses tend to be somewhat asymmetric with the asymmetry defined by |Pmax| − |Pmin|/|Pmax| + |Pmin|. The complex nature of the attenuation enhances this asymmetry. This is most evident in the far field conditions e.g. at 5km where non complex attenuation would yield a totally symmetric pulse. Figure 13 shows the angular dependence of the peak pressure. Here the angle is that subtended by the acoustic detector relative to the plane, termed the median plane, through the shower maximum at right angles to the axis of the shower. The parameterisation derived here gives a somewhat narrower angular spread than the others. This could be due to the slightly longer showers predicted by CORSIKA than the others. Figure 13 also shows the asymmetry of the pulse as a function of this angle. The pulse initially becomes more symmetric moving out of the median plane and then the asymmetry becomes negative at larger angles. Figure 14 shows the decrease of the pulsed peak pressure with distance from the shower in the median plane and the asymmetry with distance in this plane. Figures 15 and 16 show the frequency composition of the pulses at different angles to the median plane at 1 km from the shower and at different distance in the median plane, respectively. 7 Conclusions The simulation program for high energy cosmic ray air showers, CORSIKA, has been modified to work in a water or ice medium. This allows both hadron and neutrino showers to be generated in the medium over a wide range of energy (105 to 1012 GeV). The properties of hadronic showers in water simulated by CORSIKA agree with those from other simulations to within 10 − 20%. A similar uncertainty has been noted previously from the variations in CORSIKA showers in air generated by different models of the hadron interactions. However, none of the other available simulations for water cover the range of energies accessible to CORSIKA. The hadronic showers produced by neutrino interactions are shown to have similar profiles to proton showers which deposit the same amount of energy to that from the neutrino and which start at the interaction point of the neutrino. The properties of the neutrino interactions are described. A parameterisation of the shower profiles generated by CORSIKA is given. There is reasonable agreement with the parameterisation based on the Geant4 simulations at low energy (< 105 GeV) developed by Niess and Bertin. However, the agreement with the parameterisation used by the SAUND Collaboration, which is based on the NKG formalism, is less good. The position of the shower maximum, determined from the CORSIKA program, is found to vary quadratically with logE rather than linearly as assumed in the latter two parameterisations. The acoustic signals generated by neutrino interactions using CORSIKA and by the two other parameterisations are described and their properties are studied. The acoustic signal is found to be very sensitive to the energy deposited close to the shower axis. 7.1 Acknowledgments We wish to thank Ralph Engel, Dieter Heck, Johannes Knapp and Tanguy Pierog for their assistance in modifying the CORSIKA program. We also thank Valentin Niess and Justin Van- denbroucke for valuable discussions. References [1] Proceedings of the Workshop on Acoustic and Radio EeV Neutrino Detection Activities (ARENA), DESY, Zeuthen (May 2005), Editors R. Nahnhauer and S. Böser [2] K. Griesen, Phys. Rev. Lett.16 (1966) 748, G.T. Zaptsepin, V.A. Kuzmin, JETP Lett. 4 (1966) 78. [3] E. Waxman and J. Bahcall, Phys. Rev. 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Seltzer and M.J.Berger, Atomic Data and Nuclear Data Tables 30 (1984) 261. [22] D. Heck et al., Forschungszentrum Karlsruhe GmbH, Karlsruhe, Report number FZKA 6019 (1998). [23] N.N. Kalmykov and S. Ostapchenko, Phys. Atom. Nucl. 56 (1993) 346, N.N. Kalmykov et al., Nucl. Phys. Proc. Suppl. 52B (1997) 17. [24] J. Alvarez-Muniz and E. Zas, Phys. Lett. B434 (1998) 396 (astro-ph/9806098). [25] Influence of Hadronic Interaction Model on the Development of EAS in Monte Carlo Simulations, D. Heck, J.Knapp and G. Schatz, Nucl. Phys. B (Proc. Suppl.) 52B (1997) 139-141. [26] “An Introduction to the Physics of Quarks and Leptons” by P. Renton (published by Cam- bridge University Press, 1990) [27] J. Kwiecinski, A.D. Martin and A.M. Stasto Acta Phys. Polon. B31 (2000) 1273 (hep-ph/0004109). [28] A.Z. Gazizov and S.I. Yanush Phys. Rev. D65 (2002) 093003 (hep-ph/0105368) [29] R. Ghandi, C. Quigg, M.H. Reno, I. Sarcevic, Astroparticle Physics 5 (1996) 81. [30] A.D. Martin, R.G. Roberts, W.J.Stirling and R.S. Thorne, Eur. Phys. J. C14 (2000) 133 (hep-ph/9907231). [31] http://www.phys.psu.edu/∼cteq/ [32] J. Kwiecinski, A.D. Martin and A.M. Stasto, Phys. Rev. D59 (1999) 093002. [33] A.D. Martin, M.G.Ryskin and A.M. Stasto, Acta Phys. Polon. B34 (2003)3273. [34] R.S. Thorne, private communication. [35] O. Pisanti, private communication, see also M. Ambrosio et al., astro-ph/0302062. [36] HERWIG, G. Corcella et al., hep-ph/0011363. http://arxiv.org/abs/astro-ph/9806098 http://arxiv.org/abs/hep-ph/0004109 http://arxiv.org/abs/hep-ph/0105368 http://arxiv.org/abs/hep-ph/9907231 http://www.phys.psu.edu/~cteq/ http://arxiv.org/abs/astro-ph/0302062 http://arxiv.org/abs/hep-ph/0011363 [37] “Introduction to Ultra High Energy Cosmic Rays” by P. Sokolsky (published by Addison- Wesley, 1989). [38] F. James and M. Roos, “Minuit, A System for Function Minimization and Analysis of the Parameter Errors and Correlations”, Comput. Phys. Commun. 10 (1975) 343. [39] J. Vandenbroucke, G. Gratta, N. Lehtenin, Astrophys. J. 621 (2005) 301. (astro-ph/0406105). [40] J. Vandenbroucke, private communication. [41] N.G. Lehtinen, S. Adam, G.Gratta, T.K. Berger and M.J. Buckingham (astro-ph/0104033). [42] M.A. Ainslie and J.G. McColm, J.Acoust. Soc. Am 103 (1998) 1671. http://arxiv.org/abs/astro-ph/0406105 http://arxiv.org/abs/astro-ph/0104033 Interaction length of photons in water (CORSIKA) Pair production length with LPM Effect (dash dotted curve) Total Interaction length including photonuclear interaction (solid curve) 9/7 X0 Water (No LPM effect) Log10 Photon Energy/GeV Corsika - LPM Effect On Corsika - LPM Effect Off 4 5 6 7 8 9 10 11 12 Figure 1: The interaction length for high energy gamma rays versus the photon energy measured in CORSIKA (data points with statistical errors). The dash dotted curve shows the pair produc- tion length computed from the LPM effect using the formulae of Migdal [19]. The solid curve shows the computed total interaction length, including both pair production and photonuclear interactions with the cross section from CORSIKA. The dashed line labelled 9/7X0 shows the expected pair production length without the LPM effect. Here X0 is the radiation length of the material. z (cm) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Average de/dz Geant4 seawater Corsika seawater Average dE/dz of 100 proton showers z (cm) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Average de/dz Geant4 seawater Corsika seawater Average dE/dz of 100 proton showers Figure 2: Averaged longitudinal energy deposited per unit path length of 100 proton showers at energy 104 GeV (upper plot) and 105 GeV (lower plot) generated in Geant4 and CORSIKA versus depth in the water. Geant4 seawater Corsika seawater Depth = 250cm 210 Depth = 450cm Depth = 650cm 210 Depth = 850cm 0 5 10 15 20 25 30 35 40 210 Depth = 1050cm Geant4 seawater Corsika seawater Depth = 250cm 10 Depth = 450cm ) Depth = 650cm 10 Depth = 850cm 0 5 10 15 20 25 30 35 40 10 Depth = 1050cm Figure 3: Averaged radial energy deposited per 20 g cm−2 vertical slice per unit radial distance for 100 proton showers at energy 104 GeV (left hand plots) and 105 GeV (right hand plots) generated in Geant4 and CORSIKA versus distance from the axis in the water for different depths of the shower. Average y for different structure function extrapolations Standard extrapolation (solid) Scaled extrapolations (dashed) log Eν/GeV 4 6 8 10 12 Figure 4: The mean value of y as a function of energy for νµ interactions computed according to the standard model with the PDFs of MRS99 [30], extrapolating x and Q2 out of the fit range from x = 10−4 linearly on a log-log scale. The upper dashed curve shows the result of multiplying the PDFs by 1.32log(10 −4/x) for PDFs with x < 10−4 and the lower dashed curve by dividing by this factor. The deviations of the dashed curve from the solid one is an indication of the precision of the standard model. Differential cross section for νµ Scattering E=104 GeV E=1013 GeV E=106 GeV 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 5: The fraction of events per unit y interval for different νµ energies computed by inte- grating the expressions for the CC and NC cross sections. x 10 5 10000 x 10 5 x 10 6 x 10 6 Mean EW = 10 10 GeV Proton (dashed) νµ (solid) showers of same energy EW = 3 10 10 GeV EW = 5.75 10 10 GeV EW = 10 11 GeV EW = 1.65 10 11 GeV Depth gm cm-2 x 10 6 200 400 600 800 1000 1200 1400 1600 1800 Figure 6: The longitudinal distribution of the deposited energy for neutrino showers (solid) generated by the Herwig-CORSIKA package and proton showers (dashed) scaled to the same values of shower energy EW . The scaling factors applied to the average of the protons showers with energy 1010 GeV were 1.0 and 3.0 for EW = 10 10 GeV and EW = 3 · 10 10 GeV, respec- tively. Those applied to proton showers with energy 1011 GeV were 0.575, 1.0 and 1.65 for EW = 5.75 · 10 10 EW = 10 11 and EW = 1.65 · 10 11 GeV, respectively. Depth 250 g/cm2 Proton (dashed) νµ (solid) EW = 3 10 10 GeV Depth 450 g/cm2 Depth 650 g/cm2 Depth 850 g/cm2 Depth 1050 g/cm2 Radius(cm) 5 10 15 20 25 30 35 40 Depth 250 g/cm2 Proton (dashed) νµ (solid) EW = 1.65 10 11 GeV Depth 450 g/cm2 Depth 650 g/cm2 Depth 850 g/cm2 Depth 1050 g/cm2 Radius(cm) 5 10 15 20 25 30 35 40 Figure 7: The solid curves show the averaged radial energy deposited per 20 g cm−2 vertical slice per unit radial distance for 70 neutrino showers with energy transfer EW = 3 · 10 10 GeV (left hand plots) and EW = 1.65 · 10 11 GeV (right hand plots). The incident neutrino energy was 2 ·1011 GeV. For comparison the dashed curves show the distributions from proton showers scaled to these energies. In the left (right) hand plots protons of energy 1010 GeV (1011 GeV) were scaled by a factor of 3 (1.65). −100 −80 −60 −40 −20 0 20 40 60 80 100 time (µs) Pulse @1km, z=8m (109GeV Primary) 0−1 cm 0−2 cm Total Figure 8: The acoustic signal at a distance of 1 km from the shower axis in the median plane computed from the average of 100 CORSIKA showers each depositing a total energy of 109 GeV in the water. The dotted, dashed and solid curves shows the signals computed from the deposited energies within cores of radius 1.025 g cm−2, 2.05 g cm−2 and the whole shower (solid curve), respectively. It can be seen that most of the amplitude of the signal comes from the energy within a core of radius 2.05 g cm−2. Solid CORSIKA, dash SAUND parameterisation Depth 250 g/cm2 106 GeV protons 100 shower average. Depth 450 g/cm2 Depth 650 g/cm2 Depth 850 g/cm2 Depth 1050 g/cm2 Radius(cm) 5 10 15 20 25 30 35 40 Solid CORSIKA, dash SAUND parameterisation Depth 250 g/cm2 1011 GeV protons 100 shower average. Depth 450 g/cm2 Depth 650 g/cm2 Depth 850 g/cm2 Depth 1050 g/cm2 Radius(cm) 5 10 15 20 25 30 35 40 Figure 9: The radial distributions of the deposited energy at different depths from CORSIKA compared to the parameterisation used by the SAUND collaboration [39] for 106 GeV and 1011 GeV proton induced showers. Solid CORSIKA, dash NB parameterisation Depth 250 g/cm2 106 GeV protons 100 shower average. Depth 450 g/cm2 Depth 650 g/cm2 Depth 850 g/cm2 Depth 1050 g/cm2 Radius(cm) 5 10 15 20 25 30 35 40 Solid CORSIKA, dash NB parameterisation Depth 250 g/cm2 1011 GeV protons 100 shower average. Depth 450 g/cm2 Depth 650 g/cm2 Depth 850 g/cm2 Depth 1050 g/cm2 Radius(cm) 5 10 15 20 25 30 35 40 Figure 10: The radial distributions of the deposited energy at different depths from CORSIKA compared to the parameterisation used by the Niess and Bertin [15, 16] (labelled NB parame- terisation) for 106 GeV and 1011 GeV proton induced showers. CORSIKA SAUND Log10E/GeV 6 7 8 9 10 11 12 Figure 11: The depth of the shower peak as a function of log10 E from CORSIKA (black points) for the showers starting at z = 0. The solid curve shows the parameterisation according to equation 6. The dashed (dash dotted) lines show the values assumed by the SAUND (Niess- Bertin) Collaborations. −100 −50 0 50 100 −0.05 time (µs) The Acoustic Pulse from a 1011GeV Shower at 1km ACoRNE SAUND 0 5 10 15 20 25 30 35 40 Frequency (kHz) Relative Power Spectrum of a Pulse from a 1011GeV Shower at 1km ACoRNE SAUND Figure 12: The left hand plot shows acoustic pulses generated from the parameterisation de- scribed in section 5.1 labelled ACoRNE, the parameterisation from reference [15, 16] labelled NB and that from reference [39,40] labelled SAUND. These pulses were evaluated for a hadron shower from a neutrino interaction depositing hadronic energy of 1011 GeV 1 km distant from an acoustic detector in a plane perpendicular to the shower axis at the shower maximum (the median plane). The right hand plot shows the frequency decomposition of the pulses in the left hand plot. −6 −4 −2 0 2 4 6 Angle (degrees) Peak Pressure vs Angle from a 1011GeV Shower at 1km ACoRNE SAUND 0 0.5 1 1.5 2 2.5 3 −0.15 −0.05 Angle (degrees) The Asymmetry of a Pulse from a 1011GeV Shower ACoRNE SAUND Figure 13: The left hand plot shows the variation of the peak pressure in the pulse with angle from the median plane at 1 km from the shower. The right hand plot shows the variation of the pulse asymmetry with this angle at the same distance. The curves were computed from the parameterisations labelled. Distance (meters) Peak Pressure vs Distance from a 1011GeV Shower ACoRNE SAUND Distance (Meters) The Asymmetry of a Pulse from a 1011GeV Shower ACoRNE SAUND Figure 14: The left hand plot shows the decrease of the pulse peak pressure and the right hand plot the pulse asymmetry, both in the median plane, as a function of distance from the shower computed from the parameterisations. 0 5 10 15 20 25 30 35 40 Frequency (kHz) Relative Power Spectrum of a Pulse from a 1011GeV Shower at 1km 0 5 10 15 20 25 30 35 40 Frequency (kHz) Relative Power Spectrum of a Pulse from a 1011GeV Shower at 1km Figure 15: The left hand plot shows the frequency decomposition of the acoustic signal, com- puted from the parameterisation of the CORSIKA showers, at different angles to the median plane at a distance of 1km from the shower and the right hand plot shows the cumulative fre- quency spectrum i.e. the integral of the left hand plot. 0 5 10 15 20 25 30 35 40 Frequency (kHz) Relative Power Spectrum of a Pulse from a 1011GeV Shower 1000m 2000m 0 5 10 15 20 25 30 35 40 Frequency (kHz) Relative Power Spectrum of a Pulse from a 1011GeV Shower 1000m 2000m Figure 16: The left hand plot shows the frequency decomposition of the acoustic signal, com- puted from the parameterisation of the CORSIKA showers, at different distances from the hy- drophone in the median plane and the right hand plot shows the cumulative frequency spectrum i.e. the integral of the left hand plot. April 8, 2007 Simulation of Ultra High Energy Neutrino Interactions in Ice and Water (the ACoRNE Collaboration)a S. Bevan1, S. Danaher2, J. Perkin3, S. Ralph3y, C. Rhodes4, L. Thompson3, T. Sloan5b and D. Waters1. 1 Physics and Astronomy Dept, University College London, UK. 2 School of Computing Engineering and Information Sciences, University of Northumbria, Newcastle-upon-Tyne, UK. 3 Dept of Physics and Astronomy, University of Sheffield, UK. 4 Institute for Mathematical Sciences, Imperial College London, UK. 5 Department of Physics, University of Lancaster, Lancaster, UK y Deceased a Work supported by the UK Particle Physics and Astronomy Research Council and by the Ministry of Defence Joint Grants Scheme b Author for correspondence, email [email protected] Abstract The CORSIKA program, usually used to simulate extensive cosmic ray air showers, has been adapted to work in a water or ice medium. The adapted CORSIKA code was used to simulate hadronic showers produced by neutrino interactions. The simulated showers have been used to study the spatial distribution of the deposited energy in the showers. This allows a more precise determination of the acoustic signals produced by ultra high energy neutrinos than has been possible previously. The properties of the acoustic signals generated by such showers are described. (Submitted to Astroparticle Physics) 1 Introduction In recent years interest has grown in the detection of very high energy cosmic ray neutrinos [1]. Such particles could be produced in the cosmic particle accelerators which make the charged primaries or they could be produced by the interactions of the primaries with the Cosmic Mi- crowave Background, the so called GZK effect [2]. The flux of neutrinos expected from these two sources has been calculated [3,4]. It is found to be very low so that large targets are needed for a measurable detection rate. It is interesting to measure this neutrino flux to see if it is compatible with the values expected from these sources, incompatibility implying new physics. Searches for cosmic ray neutrinos are ongoing in AMANDA [5], IceCube [6], ANTARES [7] and NESTOR [8], detecting upward going muons from the Cherenkov light in either ice or water. In general, these experiments are sensitive to lower energies than discussed here since the Earth becomes opaque to neutrinos at very high energies. The experiments could detect almost horizontal higher energy neutrinos but have limited target volume due to the attenuation of the light signal in the ice. The Pierre Auger Observatory, an extended air shower array detector, will also search for upward and almost horizontal showers from neutrino interactions [9]. In addition to these detectors there are ongoing experiments to detect the neutrino interactions by either radio or acoustic emissions from the resulting particle showers [1]. These latter techniques, with much longer attenuation lengths, allow very large target volumes utilising either large ice fields or dry salt domes for radio or ice fields, salt domes and the oceans for the acoustic technique. In order to assess the feasibility of each technique the production of the particle shower from neutrino interactions needs to be simulated. Since experimental data on the interactions of such high energy particles do not exist it is necessary to use theoretical models to simulate them. The most extensive ultra high energy simulation program which has so far been developed is CORSIKA [10]. However, this program has been used previously only for the simulation of cosmic ray air showers. The program is readily available [10]. Different simulations are necessary for the radio and acoustic techniques. Radio emission occurs due to coherent Cherenkov radiation from the particles in the shower, the Askaryan Effect [11]. The emitted energy is sensitive to the distribution of the electron-positron asymme- try which develops in the shower and which grows for lower energy electromagnetic particles. Hence, to simulate radio emission, the electromagnetic component of the shower must be fol- lowed down to very low kinetic energies (� 100 keV) [12]. In contrast, an acoustic signal is generated by the sudden local heating of the surrounding medium induced by the particle shower [13]. Thus to simulate the acoustic signal the spatial distribution of the deposited en- ergy is needed. Once the electromagnetic energy in the shower reaches the MeV level (electron range � 1 cm) the energy can be simply added to the total deposited energy and the simula- tion of such particles discontinued. Extensive simulations have been carried out for the radio technique [14]. However, the simulations for the acoustic technique are less advanced. Some work has been done [15,16] using the Geant4 package [17]. However, this work is restricted to energies less than 105 GeV for hadron showers since the range of validity of the physics models in this package does not extend to higher energy hadrons. In this paper the energy distributions of showers produced by neutrino interactions in sea water at energies up to 1012 GeV are discussed. The distributions are generated using the air shower program CORSIKA [10] modified to work in a sea water medium. The salt compo- nent of the sea water has a negligible effect1 and the results are presented in distance units of g cm�2, hence they should be applicable to ice also. The computed distributions have been pa- rameterised and this parameterisation is used to develop a simple program to simulate neutrino interactions and the resulting particle showers. The properties of the acoustic signals from the generated showers are also presented. 2 Adaptation of the CORSIKA program to a water medium The air shower program, CORSIKA (version 6204) [10], has been adapted to run in sea water i.e. a medium of constant density of 1.025 g cm�3 rather than the variable density needed for an air atmosphere. Sea water was assumed to consist of a medium in which 66:2% of the atoms are hydrogen, 33:1% of the atoms are oxygen and 0:7% of the atoms are made of common salt, NaCl. The salt was assumed to be a material with atomic weight and atomic number A=29.2 and Z=14, the mean of sodium and chlorine. The purpose of this is to maintain the structure of the program as closely as possible to the air shower version which had two principal atmospheric components (oxygen and nitrogen) with a trace of argon. The presence of the salt component had an almost undetectable effect on the behaviour of the showers. Other changes made to the program to accommodate the water medium include a modifica- tion of the stopping power formula to allow for the density effect in water 2. This only affects the energy loss for hadrons since the stopping powers for electrons are part of the EGS [18] package which is used by CORSIKA to simulate the propagation of the electromagnetic com- ponent of the shower. Smaller radial binning of the shower was also required since shower radii in water are much smaller than those in air. In addition the initial state energy for electrons and photons above which the LPM effect [19] was simulated in the program was reduced to the much lower value necessary for water 3. The LPM effect suppresses pair production from high energy photons and bremsstrahlung from high energy electrons. Similarly, the interactions of neutral pions had to be simulated at lower energy than in air because of the higher density water medium. In all about 100 detailed changes needed to be made to the CORSIKA program to accommodate the water medium. To test the implementation of the LPM effect [19] in the program 100 showers from incident gamma ray photons at several different energies were generated and the mean depth of the first interaction (the mean free path) calculated. The observed mean free path was found to be in agreement with the expected behaviour when both the suppression of pair production and photonuclear interactions were taken into account (see Figure 1). This showed that the LPM effect had been properly implemented in CORSIKA. Considerable fluctuations between showers occurred. These are expressed in terms of the ratio of the root mean square (RMS) deviation of a given parameter to its mean value: the 1The shower maximum was observed to peak at a depth 2:4� 1:1% less in sea water than in fresh water with the same peak energy deposited, for protons of energy 105 GeV. 2The stopping power was computed using the Bethe-Bloch formula [20] and the density effect from the formu- lae of Sternheimer et al [21]. 3The level was set at 1 TeV compared to the characteristic energy for water E = 270 TeV [20]. RMS peak energy deposit to the mean peak energy deposit was observed to be 14% at 105 GeV reducing to 4% at 1011 GeV, that for the depth of the peak position varied from 19% to 7:4% and for the full width at half maximum of the shower from 63% to 18%. To smooth out such fluctuations averages of 100 generated showers will be taken in the following. The statistical error on the averages is then given by these RMS values divided by 10. The hadronic energy contributes only about 10% to the shower energy at the shower peak, the remainder being carried by the electromagnetic part of the shower. The simulations were all carried out in a vertical column of sea water 20 m long. The deposited energy generated by CORSIKA was binned into 20 g cm�2 slices longitudinally and 1.025 g cm�2 annular cylinders radially for 0 < r < 10:25 g cm�2 and 10.25 g cm�2 for 10:25 < r < 112:75 g cm�2 where r is the distance from the vertical axis. To reduce computing times, the thinning option was used i.e. below a certain fraction of the primary energy (in this case 10�4) only one of the particles emerging from the interaction is followed and an appropriated weight is given to it [22]. The simulation of particles continued down to cut- off energies of 3 MeV for electromagnetic particles and 0.3 GeV for hadrons. When a particle reached this cut-off, the energy was added to the slice where this occurred. The QGSJET [23] model was used to simulate the hadronic interactions. 3 Comparison with other simulations 3.1 Comparison with Geant4 Proton showers were generated in sea water using the program Geant4 (version 8.0) [17] and compared with those generated in CORSIKA. Unfortunately, the range of validity of Geant4 physics models for hadronic interactions does not extend beyond an energy of 105 GeV. Hence the comparison is restricted to energies below this. Figure 2 shows the longitudinal distributions of proton showers at energies of 104 and 105 GeV (averaged over 100 showers) as determined from Geant4 and CORSIKA. The showers from CORSIKA tend to be slightly broader and with a smaller peak energy than those generated by Geant4. The difference in the peak height is � 5% at 104 GeV rising to � 10% at energy 105 GeV. Figure 3 shows the radial distributions. The differences in the longitudinal distributions are reflected in the radial distributions. However, the shapes of the radial distributions are very similar between Geant4 and CORSIKA, with CORSIKA producing � 10% more energy near the shower axis at depths between 450 and 850 g cm�2 where most of the energy is deposited. The acoustic signal from a shower is most sensitive to the radial distribution, particularly near the axis (r � 0). It is relatively insensitive to the shape of the longitudinal distribution. 3.2 Comparison with the simulation of Alvarez-Muñiz and Zas The CORSIKA simulation was also compared with the longitudinal shower profile for protons computed in the simulation by Alvarez-Muñiz and Zas (AZ) [24]. There was a reasonable agreement between the longitudinal shower shapes from CORSIKA and those shown in Figure 2 of ref. [24]. However, the numbers of electrons and positrons at the peak of the CORSIKA showers was � 20% lower than those from ref. [24]. This number is sensitive to the energy below which these particles are counted and this is not specified in [24]. Hence the agreement between CORSIKA and their simulation is probably satisfactory within this uncertainty. In conclusion, the modifications made to CORSIKA to simulate high energy showers in a water medium give results which are compatible with the predictions from the Geant4 simula- tions for energy less than 105 GeV and the simulation of AZ within 20%. This is taken to be the accuracy of the simulation program assuming that there are no unexpected and unknown interactions between the centre of mass energy explored at current accelerators and those stud- ied in these simulations. Studies of the sensitivity of the CORSIKA simulation to the different models of the hadronic interactions have been reported in reference [25]. They find that the peak number of electrons plus positrons varies by � 20% for proton showers in air depending on the choice of the hadron interaction model used. These differences are similar in magnitude to the differences between the AZ, Geant4 and CORSIKA simulations reported here. Hence the observed differences between the Geant4, AZ and CORSIKA simulations in water could be within the uncertainties of the hadronic interaction models. 4 Simulation of neutrino induced showers Neutrinos interact with the nuclei of the detection medium by either the exchange of a charged vector boson (W+), i.e. charged current (CC) interactions or the exchange of the neutral vector boson (Z0), i.e. neutral current (NC) deep inelastic scattering interactions (see for example [26]). The ratio of the CC to NC interaction cross sections is approximately 2:1. The CC interactions produce charged secondary scattered leptons while the NC interactions produce neutrinos. The hadron shower carries a fraction y of the energy of the incident neutrino and the scattered lepton the remaining fraction 1 � y. We assume that the neutrino flavours are homogeneously mixed when they arrive at the Earth by neutrino oscillations. Hence in the CC interactions electrons, � and � leptons will be produced as the scattered leptons in equal proportions. At the energies we shall consider, these particles behave in a manner similar to minimum ionising particles for � and � leptons. This is almost true also for electrons for which the bremsstrahlung process will be suppressed by the LPM effect. Hence the charged scattered leptons contribute little to the energy producing an acoustic signal. In the case of NC interactions there is no contribution to this energy from the scattered lepton. For these reasons the contribution of the scattered lepton to the shower profile is ignored beyond z = 20 m in what follows. It is interesting to note that a � lepton can decay to hadrons or a very high energy electron or muon can produce bremsstrahlung photons at large distances from the interaction point. These can initiate further distant showers, the so called “double bang” effect. The stochastic nature of such electron showers is studied in [15, 16]. These effects are not considered in this study. 4.1 Neutrino-nucleon interaction cross sections. A number of groups have computed the high energy neutrino-nucleon interaction cross sections, �, [27–29]. In the quark parton model of the nucleon for the single vector boson exchange pro- cess, the differential cross section for CC interactions can be expressed in terms of the measured structure functions of the target nucleon F and xF )(1� y + y =2)� y(1� y=2)xF )) (1) where G is the Fermi weak coupling, M is the mass of the weak vector boson, Q2 is the square of the four momentum transferred to the target nucleon, y = �=E where � is the energy transferred to the nucleon (� = E�E 0 with E and E 0 the energies of the incident and scattered leptons) and x = Q2=2M� is the fraction of the momentum of the target nucleon carried by the struck quark (here x and y are defined for a stationary target nucleon). The plus (minus) sign is for neutrino (anti-neutrino) interactions. It can be seen that y is the fraction of the neutrino’s energy which is converted into the energy of the hadron shower. A similar expression can be written down for the NC interaction (see for example [26]) which has a ratio to the CC cross section varying from 0.33 to 0.41 as the neutrino energy increases from 104 to 1013 GeV. The structure functions F and xF are the sum of the quark distribution functions which have been parameterised by fitting data [30, 31]. It can be shown that Q2 = sxy where s = 2ME is the square of the centre of mass energy (M is the target nucleon mass). To compute the cross sections the structure functions must be calculated at values of x . M2 =s i.e. at values well outside the region of the fits to the parton distribution functions (PDFs) which have been performed for x & 10�5, the range of current measurements. The extrapolation outside the measurement range is discussed in [27], [29] and [32, 33]. Here we adopt the procedure of extrapolating linearly on a log-log scale from the parameterised parton distribution functions of [30] computed at x = 10�4 and x = 10�5. By considering various theoretical evolution procedures it is estimated in [29] that the procedure has an accuracy of � 32% per decade and we use this as an estimate of the accuracy of the calculation. However, this could be an underestimate [34]. The expression in equation 1 for charged current interactions and the one for neutral current interactions were integrated to obtain the total neutrino-nucleon interaction cross section, the value of the fraction of events per interval of y, 1=�d�=dy, and the mean value of y. The total cross section was found to be in good agreement with the values in [27, 29] and in reasonable agreement with [28] which is based on a model different from the quark parton model. Fig- ure 4 shows the mean value of y obtained from this procedure (solid curve) and the effect of multiplying or dividing the PDFs by a factor 1.32 per decade (dashed curves) as an indication of the possible range of uncertainties in the extrapolation of the PDFs. Figure 5 shows the y dependence of the cross section for different neutrino energies. 4.2 A simple generator for neutrino interactions. A simple generator for neutrino interactions in a column of water of thickness 20 m was con- structed as follows. The neutrino interacts at the top of the water column (z=0, with the z axis along the axis of the column). The energy fraction transferred, y, for the interaction was gener- ated, distributed according to the curve for the energy of the neutrino shown in Figure 5. This allows the energy of the hadron shower to be calculated for the event. The assumption was made that these hadron showers will have approximately the same distributions as those of a proton interaction at z=0 (see Section 4.3 for a test of this assumption). A series of files of 100 such proton interactions were generated at energies in steps of half an order of magnitude between 105 and 1012 GeV. The hadron shower for each neutrino interaction was selected at random from the 100 showers in the file at the proton energy closest to the energy of the hadron shower. The deposited energy in each bin was then multiplied by the ratio of the energy of the hadron shower to that of the proton shower. This is made possible because the shower shapes vary slowly with shower energy. For example, the ratio of the peak energy deposit per 20 g cm�2 slice to the shower energy varies from 0.037 to 0.030 as the proton shower energy varies from 105 to 1012 GeV. 4.3 The HERWIG neutrino generator. The CORSIKA program has an option to simulate the interactions of neutrinos at a fixed point [35]. The first interaction is generated by the HERWIG package [36]. This option was adapted to our version of CORSIKA in sea water. Some problems were encountered with the y dependence of the resulting interactions due to the extrapolation of the PDFs to very small x at high energies. This only affects the rate of the production of the showers at different y and the distribution of the hadrons produced in the interaction at a given y should be unaffected. A total of 700 neutrino interactions were generated at an incident neutrino energy of 2 � 1011 GeV. These were divided into the shower energy intervals 0:5�2�1010, 2�4�1010, 4�7:5�1010, 0:75� 1:3 � 10 11 and 1:3� 2 � 1011. The showers in which the scattered lepton energy disagreed with the shower energy by more than 20% were eliminated leading to a loss of 17% of the events with shower energy greater than 0:5 � 1010 GeV. This is due to radiative effects and misidentification of the scattered lepton. Approximately 70 events remained in each energy interval. The energy depositions from these were averaged and compared to the averages from proton showers scaled by the ratio of the shower energy to the proton energy. Figure 6 shows the longitudinal distributions of the hadronic shower energy deposited for the different energy intervals (labelled E ) compared to the scaled proton distributions. Figure 7 shows a sample of the transverse distributions. There is a good consistency between the proton and neutrino induced showers. The proton showers peak, on average, 20 g cm�2 shallower in depth with a peak energy 2% larger than the neutrino induced showers. This is small compared to the overall uncertainty. The slight shift in the longitudinal distribution is reflected as a normalisation shift in the radial distributions. We conclude therefore that to equate a proton induced shower starting at the neutrino interaction point to that from a neutrino is a satisfactory approximation. 5 Parameterisation of showers In this section a parameterisation of the energy deposited by the showers generated by COR- SIKA (averaged over 100 showers depositing the same total energy) is described. Other avail- able parameterisations will then be compared with the showers generated by CORSIKA. The acoustic signal generated by a hadron shower depends mainly on the energy deposited in the inner core of the shower. This is illustrated in figure 8 which shows the contribution to the acoustic signal from cores of different radii. This figure shows that it is crucial to represent the deposited energy well at radius less than 2.05 g cm�2. The calculation of the acoustic signal from the deposited energy is described in section 6. 5.1 Parameterisation of the CORSIKA Showers The differential energy deposited was parameterised as follows = L(z; E ) �R(r; z; E ) (2) where the function L(z; E ) represents the longitudinal distribution of deposited energy and R(r; z; E ) the radial distribution. Here E is log E with E the total shower energy. The function L(z; E ) = dE=dz is a modified4 version of the Gaisser-Hillas function [37]. This function represents the longitudinal distribution of the energy deposited. L(z; E ) = P z � P z + P Here the parameters P were fitted to quadratic functions of E = log E with values = 2:760 � 10 � 1:974 � 10 + 7:450 � 10 = �210:9� 6:968 � 10 + 0:1551E = �41:50 + 113:9E � 4:103E = 8:012 + 11:44E � 0:5434E = 0:7999 � 10 � 0:004843E + 0:0002552E = 4:563 � 10 � 3:504 � 10 + 1; 315 � 10 : (9) The parameter P represents the peak energy deposited and P the depth in the z coordinate at this peak while P and P are related to the shower width and shape in z. The radial distribution was represented by the NKG function [37] R(r; z; E �4:5) where the integral �4:5) dr = P �(4:5� 2P �(4:5� P 4The modification is to replace the shape parameter � in equation 3.5 of reference [37] by the quadratic expres- sion in z in equation 3. The parameter P was found to vary strongly with depth while P was only a weak function of depth. The parameters P (with n = 1,2) were each represented by the quadratic form = A +Bz + Cz 2 (11) and the quantities A;B;C parameterised as quadratic functions of E . This gave for P A = 0:01287E � 0:2573E + 0:9636 (12) B = �0:4697 � 10 + 0:0008072E + 0:0005404 (13) C = 0:7344 � 10 � 1:375 � 10 + 4:488 � 10 �6 (14) and for the parameter P A = �0:8905 � 10 + 0:007727E + 1:969 (15) B = 0:1173 � 10 � 0:0001782E � 5:093 � 10 �6 (16) C = �0:1058 � 10 + 0:1524 � 10 � 0:1069 � 10 : (17) The fit was made in a depth range where dE=dz was greater than 10% of the peak value defined by equation 4. The program MINUIT [38] was used to minimise the squared fractional deviations where F and D refer to the fitted value and the value observed in the ith bin from the COR- SIKA showers, respectively. In order to improve the fit at small radii the contributions to �2 were arbitrarily weighted by 10 for r < 2:05 g cm�2, 4 for 2:05 < r < 3:075 g cm�2, unity for 3:075 < r < 51:25 g cm�2 and 0.25 for r > 51:25 g cm�2. The RMS value of the frac- tional deviations was 3:4% for radii less than 51.25 g cm�2 and for energies greater than 106:5 GeV. The fit becomes poorer at lower energies and greater radii than these. Integrating the pa- rameterisation shows that the fraction of the total energy computed from the fit within the fit range was 91% averaged over the deposited energy range 107 to 1012 GeV. The corresponding fraction directly from the CORSIKA distributions was 92:5%, averaged over the same energy range. When applying this parameterisation at depths with smaller energy deposit than 10% of the peak value, the energy was assumed to be confined to an annular radius of 1.025 g cm�2. There was a good agreement (within 5% at the peak) between the acoustic signal computed using this parameterisation and that taken directly from the CORSIKA showers. 5.2 The parameterisation used by the SAUND Collaboration The SAUND Collaboration [39] uses the following parameterisation [40], based on the NKG formulae (e.g. see reference [37]), for the energy deposited per unit depth, z, and per unit annular thickness at radius r from a shower of energy E = Ek( exp (t� z=�) 2�r�(r) (19) where z = 0:9X ln(E=E ) is the maximum shower depth, X = 36:1 g cm�2 is the radia- tion length andE = 0:0838GeV. The constants t = z =�where � = 130�5 log (E=10 g cm�2 and k = tt�1= exp (t)��(t). The radial density is given by �(r) = (1 + a) s�4:5 �(4:5� s) 2��(s)�(4:5� 2s) where a = r=r with r = 9:04 g cm�2, the Molière radius in water, and s = 1:25. Figure 9 shows the radial distributions from CORSIKA compared with the absolute predictions of this parameterisation. There is qualitative agreement between the parameterisation and the CORSIKA results. The difference in normalisation is explained by the somewhat different longitudinal profiles of the CORSIKA showers from the SAUND parameterisation. The latter are broader with a lower peak energy deposit and a depth of the maximum which is larger than the CORSIKA showers. CORSIKA predicts more energy at small r than the SAUND parameterisation. Quantitatively, 51% of the shower energy is contained within a cylinder of radius 4 cm for the CORSIKA showers compared to 35% from the SAUND parameterisation. These fractions are approxi- mately independent of energy. Hence, in acoustic detectors a harder frequency spectrum for the acoustic signals is predicted by CORSIKA than by the SAUND parameterisation. Note that in the fit described in Section 5.1 the values of the parameter P (equivalent to R in equation 20) were strongly depth dependent and much lower than the Molière radius in water, assumed by the SAUND collaboration. In addition, the value of P (equivalent to s in equation 20) while relatively constant tended to be at a higher value (� 1:9) than that assumed by SAUND. 5.3 The parameterisation used by Niess and Bertin Hadron showers, generated by Geant4 (version 4.06 p03), were studied up to energies of 105 GeV and electromagnetic showers to higher energies by Niess and Bertin [15,16]. The hadronic showers were parameterised as follows. = rf(z)g(r; z) (21) f(z) = exp�bz where E is the energy of the hadron shower, X is the radiation length in water, z0 = z=X b = 0:56 as determined from the fit and a is chosen to satisfy z0 = (a � 1)=b. Here z0 the depth in radiation lengths at which the shower maximum occurs. This is parameterised as = 0:65 log( ) + 3:93 (23) with E = 0:05427 GeV. The radial distribution function is parameterised as g(r; z) = g where r = 3:5 cm, n = n = 1:66 � 0:29(z=z ) for r < r and n = n = 2:7 for r > r The constant g is chosen to be (2� n � 2)=((n ) so that the integral of the radial distribution is unity. Figure 10 shows the radial distributions from this parameterisation compared with the pre- dictions of CORSIKA. There is quite good agreement between the two. There is a difference in the normalisation with depth since Geant4, on which this parameterisation is based, produces showers which tend to develop more slowly with depth than those from CORSIKA (see Fig- ure 2). Furthermore, both this and the SAUND parameterisation (Section 5.2) assume a linear variation of the shower peak depth with logE whereas CORSIKA gives a clear parabolic shape (see equation 6). This is illustrated in Figure 11. The Niess-Bertin parameterisation predicts that 56% of the shower energy is contained within a cylinder of radius 4 cm in reasonable agreement with the value of 51% from CORSIKA (these values are almost independent of energy). 6 The acoustic signals from the showers. The pressure, P , from a hadron shower depositing total energy E at time t resulting from the deposition of relative energy density � = (1=E)(1=2�r)d2E=drdz at a point distant d from the volume, dV , follows the form [13] P (d; t) = Æ(t� d= ) dV (25) where the integral is over the total volume of the shower. Here � = 2:0 � 10�4 is the thermal expansion coefficient of the medium at 14ÆC, C = 3:8 � 10 3 J kg�1 K�1 is the specific heat capacity and = 1500 ms�1 is the velocity of sound in the sea water. Acoustic signals seen by an observer at distance r from the shower centre are computed from equation (25) as follows. Points are produced randomly throughout the volume of the shower with density proportional to the deposited energy density and the time of flight from every produced point to the observer calculated. The flight times to the observer are histogrammed over 2n bins (in this case n = 10 is chosen) centred on the mean flight time and with a suitable bin width, � (chosen here to be 1�s). The counts in each bin of the histogram are divided by � yielding the function E (t). The Fourier transform of the pressure wave is then P (!) = using the standard Fourier transform theorem, that taking the derivative in the time domain is the same as multiplying by i! in the frequency domain. The Fourier transform E (!) at angular frequency ! is evaluated numerically by a fast Fourier Transform (FFT) from the histogram (t). A correction is applied for attenuation in the water by a factor A(!) = e��(!)r where �(!) is the frequency dependent attenuation coefficient. The pressure as a function of time is then evaluated numerically by an inverse FFT using frequency steps from zero to the sampling frequency (the inverse of the bin width � i.e. 1 MHz in this case). This gives P (t) = n=511 n=�512 (27) where = 2�=1024 radians and ! =2� = n =2� MHz is the nth frequency. The attenuation coefficient �(!) is computed either according to the formulae in [42] or using the complex attenuation given in [15, 16]. This method of calculation was computationally much faster than the evaluation of the space integral given in equation 18 of reference [13] and gave identical results. Acoustic pulses, computed with the complex attenuation described in [15, 16], using the parameterisations of the shower profile given above are shown in Figure 12. It can be seen that the parameterisation developed here gives similar results to that described in [15, 16] despite the fact that the latter was an extrapolation from low energy simulations. The parameterisation used by SAUND [39, 40] gives smaller signals concentrated at somewhat lower frequencies. Further properties of the acoustic signals are shown in Figures 13 to 16. The pulses tend to be somewhat asymmetric with the asymmetry defined by jP j � jP j + jP The complex nature of the attenuation enhances this asymmetry. This is most evident in the far field conditions e.g. at 5km where non complex attenuation would yield a totally symmetric pulse. Figure 13 shows the angular dependence of the peak pressure. Here the angle is that subtended by the acoustic detector relative to the plane, termed the median plane, through the shower maximum at right angles to the axis of the shower. The parameterisation derived here gives a somewhat narrower angular spread than the others. This could be due to the slightly longer showers predicted by CORSIKA than the others. Figure 13 also shows the asymmetry of the pulse as a function of this angle. The pulse initially becomes more symmetric moving out of the median plane and then the asymmetry becomes negative at larger angles. Figure 14 shows the decrease of the pulsed peak pressure with distance from the shower in the median plane and the asymmetry with distance in this plane. Figures 15 and 16 show the frequency composition of the pulses at different angles to the median plane at 1 km from the shower and at different distance in the median plane, respectively. 7 Conclusions The simulation program for high energy cosmic ray air showers, CORSIKA, has been modified to work in a water or ice medium. This allows both hadron and neutrino showers to be generated in the medium over a wide range of energy (105 to 1012 GeV). The properties of hadronic showers in water simulated by CORSIKA agree with those from other simulations to within 10 � 20%. A similar uncertainty has been noted previously from the variations in CORSIKA showers in air generated by different models of the hadron interactions. However, none of the other available simulations for water cover the range of energies accessible to CORSIKA. The hadronic showers produced by neutrino interactions are shown to have similar profiles to proton showers which deposit the same amount of energy to that from the neutrino and which start at the interaction point of the neutrino. The properties of the neutrino interactions are described. A parameterisation of the shower profiles generated by CORSIKA is given. There is reasonable agreement with the parameterisation based on the Geant4 simulations at low energy (< 105 GeV) developed by Niess and Bertin. However, the agreement with the parameterisation used by the SAUND Collaboration, which is based on the NKG formalism, is less good. The position of the shower maximum, determined from the CORSIKA program, is found to vary quadratically with logE rather than linearly as assumed in the latter two parameterisations. The acoustic signals generated by neutrino interactions using CORSIKA and by the two other parameterisations are described and their properties are studied. The acoustic signal is found to be very sensitive to the energy deposited close to the shower axis. 7.1 Acknowledgments We wish to thank Ralph Engel, Dieter Heck, Johannes Knapp and Tanguy Pierog for their assistance in modifying the CORSIKA program. We also thank Valentin Niess and Justin Van- denbroucke for valuable discussions. References [1] Proceedings of the Workshop on Acoustic and Radio EeV Neutrino Detection Activities (ARENA), DESY, Zeuthen (May 2005), Editors R. Nahnhauer and S. Böser [2] K. Griesen, Phys. Rev. Lett.16 (1966) 748, G.T. Zaptsepin, V.A. Kuzmin, JETP Lett. 4 (1966) 78. [3] E. Waxman and J. Bahcall, Phys. Rev. D59 (1999) 023002, (hep-ph/9807282). [4] R.D. Engel, D. Seckel and T.Stanev, Phys. Rev. D64 (2001) 093010 (astro-ph/0101216). [5] See for example M. Ackermann et al., (astro-ph/0412347) Phys. Rev. D71 (2005) 077102. [6] See for example Nucl. Instrum. Meth. A567 (2006) 438 [7] J.A. Aguilar et al., astro-ph/0606229. [8] See for example G. Aggouras et al., Nucl. Instrum. and Meth. A552 (2005) 420. [9] See for example Nucl. Phys. Proc. Suppl. 143 (2005) 373. [10] “CORSIKA: A Monte Carlo Code to Simulate Extensive Air Showers”, D. Heck et al., Karlsruhe Report FZKA 6019. (http://www-ik.fzk.de/corsika). [11] G. Askar’yan, Soviet Physics JETP 14 (1962) 441 and 21 (1965) 658. [12] J. Alvarez-Muñiz, E. Marqués, R.A. Vázquez and E. Zas Phys. Rev. D68 (2003) 043001 (astro-ph/0206043). [13] J.G. Learned Phys. Rev. D19 (1979) 3293. [14] J. Alvarez-Muñiz and E.Zas, Phys. Lett. B434 (1998) 396 (astro-ph/9806098). [15] V. Niess and V. Bertin astro-ph/0511617 and V. Niess, PhD Thesis, CPPM, Marseille. [16] V. Niess, PhD Thesis, CPPM, Marseille, see equations 1-55 and 1-56. [17] Geant4, J. Allison et al., Nucl. Inst. and Meths. in Phys. Research A506 (2003) 250 and IEEE Transactions on Nucl. Science 53 (2006) 270. [18] “The EGS4 Code System” W.R. Nelson, H.Hirayama and D.W.O. Rogers, report number SLAC-265. [19] L.D. Landau and I.J. Pomeranchuk, Dokl. Akad. Nauk. SSSR 92 (1953) 535 and 92 (1953) 735. These papers are available in English in L. Landau, “The Collected Papers of L.D. Landau”, Pergamon Press 1965. A.B. Migdal, Phys. Rev. 103 (1956) 1811. [20] Particle data table, Phys. Lett. 592 (2004) 1. [21] R. M. Sternheimer, S.M. Seltzer and M.J.Berger, Atomic Data and Nuclear Data Tables 30 (1984) 261. [22] D. Heck et al., Forschungszentrum Karlsruhe GmbH, Karlsruhe, Report number FZKA 6019 (1998). [23] N.N. Kalmykov and S. Ostapchenko, Phys. Atom. Nucl. 56 (1993) 346, N.N. Kalmykov et al., Nucl. Phys. Proc. Suppl. 52B (1997) 17. [24] J. Alvarez-Muniz and E. Zas, Phys. Lett. B434 (1998) 396 (astro-ph/9806098). [25] Influence of Hadronic Interaction Model on the Development of EAS in Monte Carlo Simulations, D. Heck, J.Knapp and G. Schatz, Nucl. Phys. B (Proc. Suppl.) 52B (1997) 139-141. [26] “An Introduction to the Physics of Quarks and Leptons” by P. Renton (published by Cam- bridge University Press, 1990) [27] J. Kwiecinski, A.D. Martin and A.M. Stasto Acta Phys. Polon. B31 (2000) 1273 (hep- ph/0004109). [28] A.Z. Gazizov and S.I. Yanush Phys. Rev. D65 (2002) 093003 (hep-ph/0105368) [29] R. Ghandi, C. Quigg, M.H. Reno, I. Sarcevic, Astroparticle Physics 5 (1996) 81. [30] A.D. Martin, R.G. Roberts, W.J.Stirling and R.S. Thorne, Eur. Phys. J. C14 (2000) 133 (hep-ph/9907231). [31] http://www.phys.psu.edu/ cteq/ [32] J. Kwiecinski, A.D. Martin and A.M. Stasto, Phys. Rev. D59 (1999) 093002. [33] A.D. Martin, M.G.Ryskin and A.M. Stasto, Acta Phys. Polon. B34 (2003)3273. [34] R.S. Thorne, private communication. [35] O. Pisanti, private communication, see also M. Ambrosio et al., astro-ph/0302062. [36] HERWIG, G. Corcella et al., hep-ph/0011363. [37] “Introduction to Ultra High Energy Cosmic Rays” by P. Sokolsky (published by Addison- Wesley, 1989). [38] F. James and M. Roos, “Minuit, A System for Function Minimization and Analysis of the Parameter Errors and Correlations”, Comput. Phys. Commun. 10 (1975) 343. [39] J. Vandenbroucke, G. Gratta, N. Lehtenin, Astrophys. J. 621 (2005) 301. (astro- ph/0406105). [40] J. Vandenbroucke, private communication. [41] N.G. Lehtinen, S. Adam, G.Gratta, T.K. Berger and M.J. Buckingham (astro-ph/0104033). [42] M.A. Ainslie and J.G. McColm, J.Acoust. Soc. Am 103 (1998) 1671. Interaction length of photons in water (CORSIKA) Pair production length with LPM Effect (dash dotted curve) Total Interaction length including photonuclear interaction (solid curve) 9/7 X0 Water (No LPM effect) Log10 Photon Energy/GeV Corsika - LPM Effect On Corsika - LPM Effect Off 4 5 6 7 8 9 10 11 12 Figure 1: The interaction length for high energy gamma rays versus the photon energy measured in CORSIKA (data points with statistical errors). The dash dotted curve shows the pair produc- tion length computed from the LPM effect using the formulae of Migdal [19]. The solid curve shows the computed total interaction length, including both pair production and photonuclear interactions with the cross section from CORSIKA. The dashed line labelled 9=7X shows the expected pair production length without the LPM effect. Here X is the radiation length of the material. z (cm) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Average de/dz Geant4 seawater Corsika seawater Average dE/dz of 100 proton showers z (cm) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Average de/dz Geant4 seawater Corsika seawater Average dE/dz of 100 proton showers Figure 2: Averaged longitudinal energy deposited per unit path length of 100 proton showers at energy 104 GeV (upper plot) and 105 GeV (lower plot) generated in Geant4 and CORSIKA versus depth in the water. Geant4 seawater Corsika seawater Depth = 250cm 210 Depth = 450cm Depth = 650cm 210 Depth = 850cm 0 5 10 15 20 25 30 35 40 210 Depth = 1050cm Geant4 seawater Corsika seawater Depth = 250cm 10 Depth = 450cm ) Depth = 650cm 10 Depth = 850cm 0 5 10 15 20 25 30 35 40 10 Depth = 1050cm Figure 3: Averaged radial energy deposited per 20 g cm�2 vertical slice per unit radial distance for 100 proton showers at energy 104 GeV (left hand plots) and 105 GeV (right hand plots) generated in Geant4 and CORSIKA versus distance from the axis in the water for different depths of the shower. Average y for different structure function extrapolations Standard extrapolation (solid) Scaled extrapolations (dashed) log Eν/GeV 4 6 8 10 12 Figure 4: The mean value of y as a function of energy for � interactions computed according to the standard model with the PDFs of MRS99 [30], extrapolating x and Q2 out of the fit range from x = 10�4 linearly on a log-log scale. The upper dashed curve shows the result of multiplying the PDFs by 1:32log(10 =x) for PDFs with x < 10�4 and the lower dashed curve by dividing by this factor. The deviations of the dashed curve from the solid one is an indication of the precision of the standard model. Differential cross section for νµ Scattering E=104 GeV E=1013 GeV E=106 GeV 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 5: The fraction of events per unit y interval for different � energies computed by inte- grating the expressions for the CC and NC cross sections. x 10 5 10000 x 10 5 x 10 6 x 10 6 Mean EW = 10 10 GeV Proton (dashed) νµ (solid) showers of same energy EW = 3 10 10 GeV EW = 5.75 10 10 GeV EW = 10 11 GeV EW = 1.65 10 11 GeV Depth gm cm-2 x 10 6 200 400 600 800 1000 1200 1400 1600 1800 Figure 6: The longitudinal distribution of the deposited energy for neutrino showers (solid) generated by the Herwig-CORSIKA package and proton showers (dashed) scaled to the same values of shower energy E . The scaling factors applied to the average of the protons showers with energy 1010 GeV were 1.0 and 3.0 for E 10 GeV and E = 3 � 10 10 GeV, respec- tively. Those applied to proton showers with energy 1011 GeV were 0.575, 1.0 and 1.65 for = 5:75 � 10 11 and E = 1:65 � 10 11 GeV, respectively. Depth 250 g/cm2 Proton (dashed) νµ (solid) EW = 3 10 10 GeV Depth 450 g/cm2 Depth 650 g/cm2 Depth 850 g/cm2 Depth 1050 g/cm2 Radius(cm) 5 10 15 20 25 30 35 40 Depth 250 g/cm2 Proton (dashed) νµ (solid) EW = 1.65 10 11 GeV Depth 450 g/cm2 Depth 650 g/cm2 Depth 850 g/cm2 Depth 1050 g/cm2 Radius(cm) 5 10 15 20 25 30 35 40 Figure 7: The solid curves show the averaged radial energy deposited per 20 g cm�2 vertical slice per unit radial distance for 70 neutrino showers with energy transfer E = 3 � 10 10 GeV (left hand plots) and E = 1:65 � 10 11 GeV (right hand plots). The incident neutrino energy was 2 �1011 GeV. For comparison the dashed curves show the distributions from proton showers scaled to these energies. In the left (right) hand plots protons of energy 1010 GeV (1011 GeV) were scaled by a factor of 3 (1.65). −100 −80 −60 −40 −20 0 20 40 60 80 100 time (µs) Pulse @1km, z=8m (109GeV Primary) 0−1 cm 0−2 cm Total Figure 8: The acoustic signal at a distance of 1 km from the shower axis in the median plane computed from the average of 100 CORSIKA showers each depositing a total energy of 109 GeV in the water. The dotted, dashed and solid curves shows the signals computed from the deposited energies within cores of radius 1.025 g cm�2, 2.05 g cm�2 and the whole shower (solid curve), respectively. It can be seen that most of the amplitude of the signal comes from the energy within a core of radius 2.05 g cm�2. Solid CORSIKA, dash SAUND parameterisation Depth 250 g/cm2 106 GeV protons 100 shower average. Depth 450 g/cm2 Depth 650 g/cm2 Depth 850 g/cm2 Depth 1050 g/cm2 Radius(cm) 5 10 15 20 25 30 35 40 Solid CORSIKA, dash SAUND parameterisation Depth 250 g/cm2 1011 GeV protons 100 shower average. Depth 450 g/cm2 Depth 650 g/cm2 Depth 850 g/cm2 Depth 1050 g/cm2 Radius(cm) 5 10 15 20 25 30 35 40 Figure 9: The radial distributions of the deposited energy at different depths from CORSIKA compared to the parameterisation used by the SAUND collaboration [39] for 106 GeV and 1011 GeV proton induced showers. Solid CORSIKA, dash NB parameterisation Depth 250 g/cm2 106 GeV protons 100 shower average. Depth 450 g/cm2 Depth 650 g/cm2 Depth 850 g/cm2 Depth 1050 g/cm2 Radius(cm) 5 10 15 20 25 30 35 40 Solid CORSIKA, dash NB parameterisation Depth 250 g/cm2 1011 GeV protons 100 shower average. Depth 450 g/cm2 Depth 650 g/cm2 Depth 850 g/cm2 Depth 1050 g/cm2 Radius(cm) 5 10 15 20 25 30 35 40 Figure 10: The radial distributions of the deposited energy at different depths from CORSIKA compared to the parameterisation used by the Niess and Bertin [15, 16] (labelled NB parame- terisation) for 106 GeV and 1011 GeV proton induced showers. CORSIKA SAUND Log10E/GeV 6 7 8 9 10 11 12 Figure 11: The depth of the shower peak as a function of log E from CORSIKA (black points) for the showers starting at z = 0. The solid curve shows the parameterisation according to equation 6. The dashed (dash dotted) lines show the values assumed by the SAUND (Niess- Bertin) Collaborations. −100 −50 0 50 100 −0.05 time (µs) The Acoustic Pulse from a 1011GeV Shower at 1km ACoRNE SAUND 0 5 10 15 20 25 30 35 40 Frequency (kHz) Relative Power Spectrum of a Pulse from a 1011GeV Shower at 1km ACoRNE SAUND Figure 12: The left hand plot shows acoustic pulses generated from the parameterisation de- scribed in section 5.1 labelled ACoRNE, the parameterisation from reference [15, 16] labelled NB and that from reference [39,40] labelled SAUND. These pulses were evaluated for a hadron shower from a neutrino interaction depositing hadronic energy of 1011 GeV 1 km distant from an acoustic detector in a plane perpendicular to the shower axis at the shower maximum (the median plane). The right hand plot shows the frequency decomposition of the pulses in the left hand plot. −6 −4 −2 0 2 4 6 Angle (degrees) Peak Pressure vs Angle from a 1011GeV Shower at 1km ACoRNE SAUND 0 0.5 1 1.5 2 2.5 3 −0.15 −0.05 Angle (degrees) The Asymmetry of a Pulse from a 1011GeV Shower ACoRNE SAUND Figure 13: The left hand plot shows the variation of the peak pressure in the pulse with angle from the median plane at 1 km from the shower. The right hand plot shows the variation of the pulse asymmetry with this angle at the same distance. The curves were computed from the parameterisations labelled. Distance (meters) Peak Pressure vs Distance from a 1011GeV Shower ACoRNE SAUND Distance (Meters) The Asymmetry of a Pulse from a 1011GeV Shower ACoRNE SAUND Figure 14: The left hand plot shows the decrease of the pulse peak pressure and the right hand plot the pulse asymmetry, both in the median plane, as a function of distance from the shower computed from the parameterisations. 0 5 10 15 20 25 30 35 40 Frequency (kHz) Relative Power Spectrum of a Pulse from a 1011GeV Shower at 1km 0 5 10 15 20 25 30 35 40 Frequency (kHz) Relative Power Spectrum of a Pulse from a 1011GeV Shower at 1km Figure 15: The left hand plot shows the frequency decomposition of the acoustic signal, com- puted from the parameterisation of the CORSIKA showers, at different angles to the median plane at a distance of 1km from the shower and the right hand plot shows the cumulative fre- quency spectrum i.e. the integral of the left hand plot. 0 5 10 15 20 25 30 35 40 Frequency (kHz) Relative Power Spectrum of a Pulse from a 1011GeV Shower 1000m 2000m 0 5 10 15 20 25 30 35 40 Frequency (kHz) Relative Power Spectrum of a Pulse from a 1011GeV Shower 1000m 2000m Figure 16: The left hand plot shows the frequency decomposition of the acoustic signal, com- puted from the parameterisation of the CORSIKA showers, at different distances from the hy- drophone in the median plane and the right hand plot shows the cumulative frequency spectrum i.e. the integral of the left hand plot. Introduction Adaptation of the CORSIKA program to a water medium Comparison with other simulations Comparison with Geant4 Comparison with the simulation of Alvarez-Muñiz and Zas Simulation of neutrino induced showers Neutrino-nucleon interaction cross sections. A simple generator for neutrino interactions. The HERWIG neutrino generator. Parameterisation of showers Parameterisation of the CORSIKA Showers The parameterisation used by the SAUND Collaboration The parameterisation used by Niess and Bertin The acoustic signals from the showers. Conclusions Acknowledgments

0704.1026

The Measure for the Multiverse and the Probability for Inflation

USTC-ICTS-07-07 The Measure for the Multiverse and the Probability for Inflation Miao Li1,2, Yi Wang2,1 1 The Interdisciplinary Center for Theoretical Study of China (USTC), Hefei, Anhui 230027, P.R.China 2 Institute of Theoretical Physics, Academia Sinica, Beijing 100080, P.R.China Abstract We investigate the measure problem in the framework of inflationary cos- mology. The measure of the history space is constructed and applied to inflation models. Using this measure, it is shown that the probability for the generalized single field slow roll inflation to last for N e-folds is suppressed by a factor exp(−3N), and the probability for the generalized n-field slow roll inflation is suppressed by a much larger factor exp(−3nN). Some non-inflationary models such as the cyclic model do not suffer from this difficulty. http://arxiv.org/abs/0704.1026v1 1 Introduction It is claimed that our vacuum is one of the 10500 possible meta-stable vacua in the string theory landscape [1]. If this is true, then the physical parameters labeling which vacuum we are living in can not be calculated from the first principle. Theoretically, these parameters may only be explained by some anthropic reasoning [2], or by pure chance. From the cosmological point of view, in the framework of eternal inflation [3], the vast landscape of vacua is not only a logic possibility but also the reality. If we demand that our observable universe is not too special in the multiverse, in principle, we can make predictions in the multiverse by calculating the probability of the corresponding universe history. A serious problem arises at this point. A measure of the history space is essentially needed in order to compare different histories of the universe. But in general relativity, it is not straightforward to construct such a measure. It is because there is no preferred space slicing and time notation in general relativity, and singularities commonly arise in the cosmic solutions. Even in the much simplified Friedmann-Robertson-Walker universe, the measure problem is not easy to solve. The construction of a measure of the history space is considered as one of the central problems in cosmology. Attempts for this problem can be found in [4, 5]. To analyze this problem in detail, let us construct the history space of the universe and discuss the measure. In general relativity, all the trajectories in the phase space should lie on the hypersurface H−1(0) due to the Hamiltonian constraint H = 0. Now we want to consider the history space, where a trajectory is represented by a single point. So we have to identify the points in H−1(0) which can be linked by the time evolution. Then, the history space, or the multiverse, takes the form M = H−1(0)/R. (1) The next step is to construct a measure on the history space. To make sense and to be natural in physics, a measure of the history space should satisfy three conditions [4, 6]: (i) It should be positive. (ii) It should depend only on the intrinsic dynamics and neither on any choice of time slicing nor on the choice of dependent variables. (iii) It should respect all the symmetries of the space of solutions. A measure of the history space satisfying these three requirements can be con- structed from the phase space symplectic form [4, 6, 7]. The symplectic form of the phase space ω can be written in terms of the canonical coordinates and momenta as dpi ∧ dq i. (2) where m is the number of canonical coordinates. If we choose pm = H, then from the Hamilton’s equations, q m = t is the time coordinate. And the symplectic form (2) can be written as dpi ∧ dq i + dH ∧ dt. (3) The Hamiltonian constraint H = 0 naturally yields a two-form transverse to the time evolution. This is the two-form in the history space, ωC ≡ ω|H=0 = dpi ∧ dq i. (4) The measure of the history space can be constructed by raising ωC to the (m−1)th power, (−1)(m−1)(m−2)/2 (m− 1)! ωm−1C . (5) Note that ΩM is an exact form. It can be globally written as ΩM ∼ dA with A ≡ p1dq dpi ∧ dq i. (6) This measure of the history space can be applied to the inflationary cosmology in determining the probability of inflation. At first, it was believed that the canonical measure favors inflation [6]. But soon it is realized that both inflationary and non- inflationary history have infinite measure [8]. So the measure problem in cosmology remained unsolved. Recently, Gibbons and Turok [4] suggested a solution to this measure problem. They noticed that a universe with a very small spacial curvature at the present time can not be distinguished from a flat one. So physically, it makes sense to cut off the history space by identifying a universe with a very small spacial curvature with a flat universe. As was shown in [4], the measure for some quantities, like the spacial curvature, is cutoff dependent, and dominated by the cutoff. While the measure for some other quantities, for example, the e-folding number of inflation, is cutoff independent. So by applying this cutoff, the question whether a N e-folds’ inflation is natural can be well defined, and investigated explicitly. It is shown that the history space volume for slow roll inflation is suppressed by a factor of exp(−3N), where N is the e-folding number. The work [4] concentrates on a single field minimal coupled inflation model. There is a vast variety of inflation models in addition to a single inflaton model, thus it is interesting to ask how other models weigh in this measure. Some of the inflation mod- els involve a modified Lagrangian density other than the minimal one, some involve multi-fields and some modify the Einstein gravity. We want to know whether these inflation models are also suppressed for a large e-folding number. An investigation of these models is the main task of this paper. This paper is organized as follows. In Section 2, we review the approach by Gibbons and Turok [4] for gravity minimally coupled to a scalar field. It is shown that the inflation probability can be calculated directly as a function of N . In Section 3, we discuss the measure for the scalar field with a more general Lagrangian. We find that in this generalized case, the measure for the slow roll inflationary history is suppressed by exactly the same factor exp(−3N). In Section 4, we consider the multi- field inflation. It can be shown that with the assumption of slow roll for the Hubble constant, the measure is a lot more suppressed by the exponential factor exp(−3nN), where n is the number of inflaton fields. So it seems much more unnatural for multi- field inflation to happen. In Section 5, we investigate the generalized Lagrangian for multi-field inflation. We find that the generalization of the Lagrangian can not solve the measure problem raised in Section 4. Finally, we summarize the paper in the last section. 2 Single Field Inflation Models In this section, we consider a single scalar field minimally coupled with gravity with the action −3a(N−2ȧ2 − k) + a3N−2ϕ̇2 − a3V (ϕ) , (7) where N is the lapse function, and k = 0,±1 represents the spacial curvature, and dot denotes the derivative with respect to time. For simplicity, we have set M2p ≡ 1/(8πG) = 1. By varying the action with respect to the lapse function N , we obtain the Fried- mann equation 3H2 = ϕ̇2 + V (ϕ)− , (8) where after the variation, we have set N = 1. Varying the action with respect to ϕ leads to the scalar field equation of motion ϕ̈+ 3Hϕ̇+ Vϕ = 0, (9) where the subscript ϕ in V denotes derivative with respect to ϕ. From the time derivative of (8) and using (9), we get Ḣ = − ϕ̇2 + . (10) To construct the history space, we need to slice the H = 0 hypersurface of the phase space. A good way to do this is to choose a constant H surface H = HS as a slicing [4], where HS is chosen low enough that it just above the end of inflation and the universe evolves adiabatically from then on. To choose a constant H slice is because for the flat or open universe, and non-negative potential V (ϕ), each history trajectory crosses a constant H surface exactly once. And the reason for choosing H low enough is that only this choice can result in a cutoff independent measure of e-folds, and this choice is in agreement with the anthropic “top down” approach to cosmology [9]. On a constant H surface, the measure for the history space takes the form dpϕ ∧ dϕ, (11) where pϕ ≡ a 3ϕ̇ denotes the canonical momentum for ϕ. It can be calculated that dϕda 3a2 6H2S − 2V + 4ka 6H2S − 2V + 6ka . (12) A divergence occurs in the large a limit of (12). This is the infinity discovered in [8]. Following [4], we set a cutoff for the spacial curvature to critical density ratio Ωk ≡ −k/(a 2H2S) as |Ωk| ≥ ∆Ωk, (13) The cutoff makes sense physically because a small enough Ωk is neither geometrically meaningful nor physically observable. As we are working on a constant HS surface, the cutoff can be translated into the cutoff of the scale factor a2 ≤ a2max ≡ . (14) Recall that ΩM = dA, The measure can be reduced to a surface integral around a constant a surface of the constant HS history space, pϕdϕ = a ϕ̇dϕ. (15) To investigate the probability distribution for inflation, now concentrate on an history space volume element A ∼ ϕ̇∆ϕ. Where we have dropped the a3max term as it is a constant. Since the variation operation ∆ is taken on a constant H surface, it is convenient to convert the time derivative ∂t to the derivative with respect to the Hubble constant ∂H , using ∂t = Ḣ∂H = − ϕ̇2∂H . (16) Then we can take the advantage that ∂H ·∆ = ∆ · ∂H . Note thatH do not change when we move on the history space. Then the equation (8) leads to a constraint for the history space variation ϕ̇∆ϕ̇ + Vϕ∆ϕ = 0, (17) Given this constraint, the Hubble evolution for A can be calculated as ∂HA = − A. (18) where we have neglected the spatial curvature energy density, because it have to be small during the last e-folds of inflation. Note that for the e-folding number N , −H = ∂tN = Ḣ∂HN , the equation (18) takes the form ∂HA = 3A∂HN, (19) which can be integrated out to give A = e3NA(HS). (20) The above equation tells us that as we stand at the end of inflation and track back- wards with time, a volume in the history space expands exponentially. In order not to break the slow roll condition along the whole 60 e-folds’ inflation, The volume el- ement A must lie in a exponentially narrow corner in the constant HS history space. So the probability for inflation is suppressed by the exp(−3N) factor. This suppres- sion shows that inflation is not as natural as we intuitively think. It may have not solved the naturalness problems of the hot big bang cosmology because of its unnat- ural nature, or there remains some unknown mechanism to produce a exponentially sharp peak for the possibility distribution of the history space. 3 Generalized Single Field Models In this section, we consider the action −3aN−2ȧ2 + a3f ϕ,N−1ϕ̇ . (21) A good many inflation models can be described using this action. For example, K-Inflation [10], Phantom Inflation [11], Inflation driven by the brane DBI action [12, 13], etc. Choosing ϕ as a canonical coordinate and using the proper time, the canonical momentum for ϕ takes the form p = πa3, where π ≡ fϕ̇. (22) Take variation with respect to N and ϕ, one obtains 3H2 = ϕ̇π − f, π̇ + 3Hπ − fϕ = 0, Ḣ = − ϕ̇π. (23) Using (23), the constraint for the variation in the history space can be written as ϕ̇∆π = fϕ∆ϕ. (24) And using the definition of π, we have the variation relation ∆ϕ̇ = f−1ϕ̇ϕ̇ (∆π − fϕ̇ϕ∆ϕ). (25) where we have assumed that fϕ̇ϕ̇ 6= 0, in order that ϕ can be treated as a dynamical degree of freedom. Now we take the cutoff as discussed in the last section, and reduce the integration of the history space to the boundary integration pdϕ = a3max πdϕ. (26) Then it can be calculated that the variation of the volume element in the history space A ∼ π∆ϕ evolves along the constant H surfaces as ∂HA = − A = 3A∂HN (27) So the conclusion is exactly the same as that of the last section. In order to get N e-folds’ slow roll inflation, the volume element in the history space should be exponentially fine turned. It should be noticed that in this general case, there is the possibility that even the history evolution is not slow rolling, accelerated expansion with a large e-folding number can be achieved in models such as the Kflation or the phantom inflation. But it is difficult to get a scale invariant perturbation spectrum if the slow roll condition is not satisfied [11]. 4 Multi-Field Inflation Models Multi-field inflation models take an important part in the inflationary model build- ing. In string theory, there can be a number of scalar fields at the inflation scale. Phenomenally, in multi-field models, slow roll condition is less stringent and can be satisfied in more models [14]. Moreover, there are interesting inflation models, like the hybrid inflation model [15], which requires essentially more than one field. So it is useful to study the measure for multi-field inflation and investigate the corresponding probability. The action for the multi-field inflation takes the form −3aN−2ȧ2 + a3N−2ϕ̇i 2 − a3V (ϕi) , (28) where the duplicate index i is summed over the n scalar fields. Choosing to use the proper time, the canonical momentum for ϕi is pi ≡ a 3ϕ̇i. And the equations of motion takes the form 3H2 = ϕ̇2i + V, Ḣ = ϕ̇2i , ϕ̈i + 3Hϕ̇i + Vϕi = 0. (29) The constraint for constant HS variation is ϕ̇i∆ϕ̇i + Vϕi∆ϕi = 0. (30) It can be checked by direct calculation that ∂H ·∆ = ∆ · ∂H is also true operating on ϕ̇i. In this multi-field case A ∼ ϕ̇1 ∆ϕ1 ∧∆ϕ̇2 ∧∆ϕ2 ∧ . . . ∧∆ϕ̇n ∧∆ϕn. (31) Using the constraint (30), each term in ∂HA is proportional to A, and ∂HA turns out to be ∂HA = 3n A∂HN. (32) In a multi-field inflation model, Ḧ/(HḢ) should also be small and rolling slowly as in the single field case. If one assumes that ϕ̈1/(Hϕ̇1) is also small and slow rolling, then the integration can be carried out as A = e3nNA(HS), (33) which shows that the departure from slow-roll evolves much faster than that in the single field case. As a result, multi-field inflation is much more unnatural then the single-field inflation with a much smaller measure in the history space. This result is not surprising. It is because from the first equation in (29), the Hubble constant has contribution from the energy density of all inflation fields. While from the third equation in (29), the Hubble constant appears as a friction in the evolution of each single inflaton field. So in the multi-field inflation case, the friction of each single field is contributed by all the fields, and the history space for slow roll inflation is much more concentrated then the single field models. As analyzed in [16], |ϕ̈1/(Hϕ̇1)| ≪ 1 may break down in some multi-field inflation models. Now let’s see whether a fast rolling ϕ̇1 can result in something more natural. If we want ϕ̈1/(Hϕ̇1) to cancel the exponential expansion of the history space volume, we need H, (34) which amounts to demanding that ∣ϕ̇1/a ∣ increases with time. As n ≥ 2, |ϕ̇1| must be increasing faster than a3 to make this cancellation possible. And this cancellation need to be valid along the whole 60 e-folds of inflation. It seems impossible for ϕ̇1 to behave like this. So even a fast rolling ϕ̇1 can not make the situation more natural. A few words are in order here. We have picked a specific field ϕ1 out of many other fields in studying the measure, this is just the result of integrating out pϕ1, namely, we have allowed ϕ̇1 to vary as much as possible. We could have picked out another field, then we would be discussing the differential measure in a different region on the history space. Now we see that the multi-field inflation is even more impossible than the single field inflation. Then, if for anthropic principle or some other reasons that a 60 e-folds’ inflation has to have happened in our history, it should be single field inflation rather than multi-field inflation, because the latter has much smaller measure. 5 Generalized Multi-Field Models In this section, we do the generalizations one step further to consider the action −3aN−2ȧ2 + a3f −1ϕ̇i , (35) which has the features of the actions in both Section 3 and Section 4. Using the proper time, the canonical momentum for ϕ takes the form pi = πia 3, where πi ≡ fϕ̇i, (36) with the equations of motion 3H2 = ϕ̇iπi − f, π̇i + 3Hπi − fϕi = 0, Ḣ = − ϕ̇iπi. (37) and the constraint for the history space variation ϕ̇i∆πi = fϕi∆ϕi. (38) We assume that the matrix fϕ̇iϕ̇j has inverse matrix. This should be true when all the constraints in (35) are solved and ϕi only denotes the dynamical degree of freedom. We use f ϕ̇iϕ̇j as the inverse matrix of fϕ̇iϕ̇j Then it can be shown that ∆ϕ̇i = f ϕ̇iϕ̇j (∆πj − fϕ̇jϕk∆ϕk). (39) In this generalized case, A ∼ π1 ∆ϕ1 ∧∆π2 ∧∆ϕ2 ∧ . . . ∧∆πn ∧∆ϕn. (40) using the same technique developed in Section 3 and Section 4, one finds ∂HA = f ϕ̇1ϕ̇iπ̇i ϕ̇1Ḣ ϕ̇iπ̇i ϕ̇iϕ̇k π̇k f ϕ̇1ϕ̇jfϕ̇jϕk ϕ̇k Ḣϕ̇1 ϕ̇iϕ̇jfϕ̇jϕkϕ̇k A (41) To see the implications of this equation, let us concentrate on the double field in- flation models. It is because it seems more difficult to cancel the −3nH term for lager n. Lagrangian densities like f = g(ϕ1)ϕ̇ 1 + h(ϕ2)ϕ̇ 2 are not of special interest here, because they can be transformed into the case discussed in Section 4 by a field redefinition. As another example, let us consider the Lagrangian density f = f ϕ1, ϕ2, (ϕ21 + ϕ , (42) in this case, the equation (41) takes the form ∂HA = −6A ∂HN, (43) where f ′ ≡ (ϕ̇21 + ϕ̇ ] . (44) So a fast rolling f ′ϕ̇21 is required to cancel the exponential expansion of the volume of the history space. To see the physical implications for this condition, consider the DBI inflation model by [13]. The action of the DBI inflation model is given by SDBI = − 1− f̃ ϕ̇2 − 1 + V (ϕ) where the angular motion of ϕi has been ignored, so ϕ̇ 2 ≡ ϕ̇2i . Then f ′ = 1/ 1− f̃ ϕ̇2 ≡ γ is just the relativistic factor defined in [13]. From the spectral index ns − 1 = 4 , (46) We conclude that γ should not be a fast rolling quantity along the whole history of observable inflation. Moreover, from the equation Ḣ = −γϕ̇2/2, we see again that γ can not be large for a long time during inflation. So the cancellation of the e−3nN factor can not be obtained. 6 Conclusion and Discussions In this paper, we have reviewed the measure problem in cosmology. We calculated the measure and the probability for inflation in single and multi-field models with gener- alized Lagrangian density. It is shown that the measure for the single field inflation and the corresponding generalizations are suppressed by a factor of exp(−3N). While the n-field and generalized multi-field inflation models has a measure proportional to exp(−3nN). This work can be understood in another way. Taking apart the discussion for the measure and the slow roll condition, other parts of this paper can be thought of as a proof of the attractor behavior of various kinds of generalized inflation models. On the one hand, it is a proof that the attractor behavior is very common in inflationary models. While on the other hand, to take the measure into consideration, we see that it is far from obvious for an attractor to be a natural solution in cosmology. And it is just this early time attractor combined with the requirement of slow roll that puts inflation into a highly unnatural situation. We did not study explicitly the inflation models with non-minimal coupling to gravity [17]. But these models do not seem to bring large correction for the sup- pression factor. It is because through conformal transformation, these non-minimal coupled models are generally equivalent with the corresponding minimal coupled in- flation models with the same number or one more inflation fields. Another reason not to consider these models in this work is that, as the energy scale commonly drops during inflation, near the end of inflation, the non-minimal coupling effect may not be so important. There are also inflation models with extra components or special spacetime prop- erties. Examples of this kind are inflation with holographic dark energy [18, 19] or in the non-commutative spacetime [20, 21, 22]. These models do not seem to change the results much either. Because in the former example, the holographic dark energy is diluted during inflation so do not seem to cause large corrections near the end of inflation. In the latter case, although the spectrum for perturbations is greatly modified in the non-commutative spacetime, the isotropic and homogeneous inflating background do not change much because it belongs to a lower energy scale. So the corrections to the probability can not be large. As a closing remark, we noticed that some non-inflationary models do not share the small measure problem. One example is the cyclic universe model [23]. Although the cyclic model is controlled by gravity coupled with a scalar field, it do not have slow roll behavior backwards in time in the cycle we live. So the key observation that the exponentially expansion of the phase space volume breaks the slow roll condition do not apply in the cyclic model. Nevertheless, the number of cycles in the cyclic universe must be finite [23], so it remains to explain how all the cycles begin in the first place. Acknowledgments This work was supported by grants of NSFC. 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Rev. 49 (2005) 43-57. http://arxiv.org/abs/hep-th/0304203 http://arxiv.org/abs/astro-ph/0308169 http://arxiv.org/abs/astro-ph/0308458 http://arxiv.org/abs/astro-ph/0311378 http://arxiv.org/abs/astro-ph/0603782 http://arxiv.org/abs/astro-ph/0604195 http://arxiv.org/abs/astro-ph/0605442 http://arxiv.org/abs/hep-th/0608207 http://arxiv.org/abs/astro-ph/0610389 http://arxiv.org/abs/hep-th/0702217 http://arxiv.org/abs/hep-th/0703173 http://arxiv.org/abs/astro-ph/0404480 Introduction Single Field Inflation Models Generalized Single Field Models Multi-Field Inflation Models Generalized Multi-Field Models Conclusion and Discussions

0704.1027

$B_s^0 \to \eta^{(\prime)} \eta^{(\prime)}$ decays in the pQCD approach

ZJOU-PHY-TH-07-01 NJNU-TH-07-05 Studies of B0s → η(′)η(′) decays in the pQCD approach Xin Liua∗, Zhen-Jun Xiaob†, Hui-Sheng Wangc a. Department of Physics, Zhejiang Ocean University, Zhoushan, Zhejiang 316000, P.R. China b. Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, Jiangsu 210097, P.R. China and c. Department of Applied Mathematics and Physics, Anhui University of Technology and Science, Wuhu, Anhui 241000, P.R. China (Dated: November 18, 2018) Abstract We calculate the CP averaged branching ratios and CP-violating asymmetries for B0s → ηη, ηη′ and η′η′ decays in the perturbative QCD (pQCD) approach here. The pQCD predictions for the CP-averaged branching ratios are Br(B0s → ηη) = 14.2+18.0−7.5 × 10−6, Br(B0s → ηη′) = 12.4+18.2−7.0 ×10−6, and Br(B0s → η′η′) = 9.2+15.3−4.9 ×10−6, which agree well with those obtained by employing the QCD factorization approach and also be consistent with available experimental upper limits. The gluonic contributions are small in size: less than 7% for Bs → ηη and ηη′ decays, and around 18% for Bs → η′η′ decay. The CP-violating asymmetries for three decays are very small: less than 3% in magnitude. PACS numbers: 13.25.Hw, 12.38.Bx, 14.40.Nd ∗ [email protected] † [email protected] http://arxiv.org/abs/0704.1027v1 Among various B → M1M2 decay channels ( here Mi refers to the light pseudo-scalar or vector mesons ), the decays involving the isosinglet η or η′ mesons in the final state are phenomenologically very interesting and have been studied extensively during the past decade because of the so-called Kη′ puzzle or other special features [1, 2, 3, 4]. Motivated by the large number of Bs production and decay events expected at the forthcoming LHC experiments, the studies about the Bs meson decays become more attractive than ever before. Very recently, some two-body Bs → Miη(′) decays, such as Bs → (π, ρ, ω, φ)η(′) decays have been studied in Refs. [5, 6] in the perturbative QCD (pQCD ) factorization approach [7, 8, 9]. In this paper, we would like to calculate the branching ratios and CP asymmetries for the three B0s → ηη, ηη′ and η′η′ decays by employing the low energy effective Hamiltonian [10] and the pQCD approach. Besides the usual factorizable contributions, we here are able to evaluate the non-factorizable and the annihilation contributions to these decays. On the experimental side, only the poor upper limit on Br(B0s → ηη) is available now [11] (upper limits at 90% C.L.): Br(B0s → ηη) < 1.5× 10−3 , (1) Of course, this situation will be improved rapidly when LHC experiment starts to run at the end of 2007. This paper is organized as follows. In Sec. I, we calculate analytically the related Feynman diagrams and present the various decay amplitudes for the studied decay modes. In Sec. II, we show the numerical results for the branching ratios and CP asymmetries of B0s → η(′)η(′) decays. A short summary and some discussions are also included in this section. I. PERTURBATIVE CALCULATIONS Since the b quark is rather heavy we consider the Bs meson at rest for simplicity. It is convenient to use light-cone coordinate (p+, p−,pT ) to describe the meson’s momenta: p± = (p0± p3)/ 2 and pT = (p 1, p2). Using the light-cone coordinates the Bs meson and the two final state meson momenta can be written as (1, 1, 0T ), P2 = (1, 0, 0T ), P3 = (0, 1, 0T ), (2) respectively, here the light meson masses have been neglected. Putting the light (anti-) quark momenta in Bs, η ′ and η mesons as k1, k2, and k3, respectively, we can choose k1 = (x1P 1 , 0,k1T ), k2 = (x2P 2 , 0,k2T ), k3 = (0, x3P 3 ,k3T ). (3) Then, after the integration over k−1 , k 2 , and k 3 , the decay amplitude for Bs → ηη′ decay, for example, can be conceptually written as A(Bs → ηη′) ∼ dx1dx2dx3b1db1b2db2b3db3 C(t)ΦBs(x1, b1)Φη′(x2, b2)Φη(x3, b3)H(xi, bi, t)St(xi) e −S(t) , (4) where ki are momenta of light quarks included in each meson, term Tr denotes the trace over Dirac and color indices, C(t) is the Wilson coefficient evaluated at scale t, the function H(k1, k2, k3, t) is the hard part and can be calculated perturbatively, the function ΦM is the wave function, the function St(xi) describes the threshold resummation [12] which smears the end-point singularities on xi, and the last term, e −S(t), is the Sudakov form factor which suppresses the soft dynamics effectively. We will calculate analytically the function H(xi, bi, t) for the considered decays in the first order in αs expansion and give the convoluted amplitudes in next section. For the two-body charmless Bs meson decays, the related weak effective Hamiltonian Heff can be written as [10] Heff = uq (C1(µ)O 1 (µ) + C2(µ)O 2 (µ))− VtbV ∗tq Ci(µ)Oi(µ) , (5) where Ci(µ) are Wilson coefficients at the renormalization scale µ and Oi are the four- fermion operators for the case of b → q (q = d, s) transition [5, 10]. For the Wilson coefficients Ci(µ) (i = 1, . . . , 10), we will use the leading order (LO) expressions, although the next-to-leading order (NLO) results already exist in the literature [10]. This is the consistent way to cancel the explicit µ dependence in the theoretical formulae. For the renormalization group evolution of the Wilson coefficients from higher scale to lower scale, we use the formulae as given in Ref.[13] directly. A. Decay amplitudes We firstly take Bs → ηη′ decay mode as an example, and then extend our study to Bs → ηη and η′η′ decays. Similar to the B0s → π0η(′) decays in [5], there are 8 type diagrams contributing to the Bs → ηη decays, as illustrated in Fig.1. We first calculate the usual factorizable diagrams (a) and (b). Operators O1,2,3,4,9,10 are (V − A)(V − A) currents, the sum of their amplitudes is given as Feη = 8πCFm dx1dx3 b1db1b3db3 φBs(x1, b1) (1 + x3)φ η (x3, b3) + (1− 2x3)rη(φPη (x3, b3) + φTη (x3, b3)) ·αs(t1e) he(x1, x3, b1, b3) exp[−Sab(t1e)] +2rηφ η (x3, b3)αs(t e)he(x3, x1, b3, b1) exp[−Sab(t2e)] . (6) where rη = m 0/mB; CF = 4/3 is a color factor. The explicit expressions of the function he, the scales t e and the Sudakov factors Sab can be found Ref. [5]. The form factors of Bs to η decay, F Bs→ηss̄ 0,1 (0), can thus be extracted from the expression in Eq. (6). The operators O5,6,7,8 have a structure of (V − A)(V + A). Some of these operators can contribute to the decay amplitude in a factorizable way, but others may contribute after making a Fierz transformation in order to get right flavor and color structure for FIG. 1: Typical Feynman diagrams contributing to the Bs → ηη′ decays, where diagram (a) and (b) contribute to the Bs → η form factor FBs→η0,1 . factorization to work. Such kinds of contributions can be written as F P1eη = −Feη . (7) F P2eη = 16πCFm (f sη′ − fuη′)m2η′ 2msmBs dx1dx3 b1db1b3db3 φBs(x1, b1) φAη (x3, b3) + rη((2 + x3)φ η (x3, b3)− x3φTη (x3, b3)) ·αs(t1e)he(x1, x3, b1, b3) exp[−Sab(t1e)] η (x3, b3)− 2(x1 − 1)rηφPη (x3, b3) ·αs(t2e)he(x3, x1, b3, b1) exp[−Sab(t2e)] . (8) For the non-factorizable diagrams 1(c) and 1(d), the corresponding decay amplitudes can be written as Meη = dx1dx2 dx3 b1db1b2db2 φBs(x1, b1)φ η′(x2, b2) 2x3rηφ η (x3, b1)− x3φη(x3, b1) ·αs(tf)hf (x1, x2, x3, b1, b2) exp[−Scd(tf )]} , (9) MP1eη = 0, (10) MP2eη = −Meη . (11) For the non-factorizable annihilation diagrams 1(e) and 1(f), we find Maη = dx1dx2 dx3 b1db1b2db2 φBs(x1, b1) η (x3, b2)φ η′(x2, b2) +rηrη′ (x2 + x3 + 2)φ η′(x2, b2) + (x2 − x3)φTη′(x2, b2) φPη (x3, b2) +rηrη′ (x2 − x3)φPη′(x3, b2) + (x2 + x3 − 2)φTη′(x2, b2) φTη (x3, b2) ·αs(t3f)h3f (x1, x2, x3, b1, b2) exp[−Sef (t3f )] η (x3, b2)φ η′(x2, b2) +rηrη′ (x2 + x3)φ η′(x2, b2) + (x3 − x2)φTη′(x2, b2) φPη (x3, b2) +rηrη′ (x3 − x2)φPη′(x2, b2) + (x2 + x3)φTη′(x2, b2) φTη (x3, b2) ·αs(t4f)h4f (x1, x2, x3, b1, b2) exp[−Sef (t4f )] , (12) MP1aη = dx1dx2 dx3 b1db1b2db2 φBs(x1, b1) (x3 − 2)rηφAη′(x2, b2)(φPη (x3, b2) + φTη (x3, b2))− (x2 − 2)rη′φAη (x3, b2) (φPη′(x2, b2) + φ η′(x2, b2)) · αs(t3f)h3f (x1, x2, x3, b1, b2) exp[−Sef(t3f )] x3rηφ η′(x2, b2)(φ η (x3, b2) + φ η (x3, b2)) −x2rη′φAη (x3, b2)(φPη′(x2, b2) + φTη′(x2, b2)) ·αs(t4f )h4f(x1, x2, x3, b1, b2) exp[−Sef (t4f)] , (13) MP2aη = dx1dx2 dx3 b1db1b2db2 φBs(x1, b1) η (x3, b2)φ η′(x2, b2) +rηrη′ (x2 + x3 + 2)φ η′(x2, b2) + (x3 − x2)φTη′(x2, b2) φPη (x3, b2) +rηrη′ (x3 − x2)φPη′(x3, b2) + (x2 + x3 − 2)φTη′(x2, b2) φTη (x3, b2) ·αs(t3f )h3f(x1, x2, x3, b1, b2) exp[−Sef (t3f)] η (x3, b2)φ η′(x2, b2) +rηrη′ (x2 + x3)φ η′(x2, b2) + (x2 − x3)φTη′(x2, b2) φPη (x3, b2) +rηrη′ (x2 − x3)φPη′(x2, b2) + (x2 + x3)φTη′(x2, b2) φTη (x3, b2) ·αs(t4f )h4f(x1, x2, x3, b1, b2) exp[−Sef (t4f)] . (14) For the factorizable annihilation diagrams 1(g) and 1(h), we have Faη = F aη = −8πCFm4Bs dx2 dx3 b2db2b3db3 η (x3, b3)φ η′(x2, b2) +2rηrη′((x3 + 1)φ η (x3, b3) + (x3 − 1)φTη (x3, b3))φPη′(x2, b2) ·αs(t3e)ha(x2, x3, b2, b3) exp[−Sgh(t3e)] η (x3, b3)φ η′(x2, b2) +2rηrη′((x2 + 1)φ η′(x2, b2) + (x2 − 1)φTη′(x2, b2))φPη (x3, b3) ·αs(t4e)ha(x3, x2, b3, b2) exp[−Sgh(t4e)] F P2aη = −16πCFm4Bs dx2 dx3 b2db2b3db3 x3rη(φ η (x3, b3)− φTη (x3, b3))φAη′(x2, b2) + 2rη′φAη (x3, b3)φPη′(x2, b2) ·αs(t3e)ha(x2, x3, b2, b3) exp[−Sgh(t3e)] x2rη′(φ η′(x2, b2)− φTη′(x2, b2))φAη (x3, b3) + 2rηφAη′(x2, b2)φPη (x3, b3) ·αs(t4e)ha(x3, x2, b3, b2) exp[−Sgh(t4e)] . (16) For the Bs → ηη′ decay, besides the Feynman diagrams as shown in Fig. 1 where the upper emitted meson is the η′, the Feynman diagrams obtained by exchanging the position of η and η′ also contribute to this decay mode. The corresponding expressions of amplitudes for new diagrams will be similar with those as given in Eqs.(6-14), since the η and η′ are all light pseudoscalar mesons and have the similar wave functions. The expressions of amplitudes for new diagrams can be obtained by the replacements φAη ←→ φη′ , φPη ←→ φPη′ , φTη ←→ φtη′ , rη ←→ rη′ . (17) For example, we find that: Feη′ = Feη, Faη′ = −Faη, F P1aη′ = −F P1aη , F P2aη′ = F P2aη . (18) Before we write down the complete decay amplitude for the studied decay modes, we firstly give a brief discussion about the η − η′ mixing and the gluonic component of the η′ meson. There exist two popular mixing basis for η − η′ system, the octet-singlet and the quark flavor basis, in literature. Here we use the SU(3)F octet-singlet basis with the two mixing angle (θ1, θ8) scheme [14] to describe the mixing of η and η ′ mesons. In the numerical calculations, we will use the following mixing parameters [14] θ8 = −21.2◦, θ1 = −9.2◦, f1 = 1.17fπ, f8 = 1.26fπ. (19) In this paper, we firstly take η and η′ as a linear combination of light quark pairs uū, dd̄ and ss̄, and then estimate the possible gluonic contributions to Bs → η(′)η(′) decays by using the formulae as presented in Ref. [4]. We found that the possible gluonic contributions are indeed small. B. Complete decay amplitudes For B0s → ηη′ decay, by combining the contributions from different diagrams, the total decay amplitude can be written as M(ηη′) = FeηF2f η′ + Feη′F C4 − C5 − FeηF2f η′ + Feη′F C4 − 2C5 − F P2eη F2 + F C5 + C6 − + (Meη +Meη′)F2F ξuC2 − ξt C3 + 2C4 − − [MeηF1F ′2 +Meη′F ′1F2] ξt MP2eη +M 2C6 + MP2eη F1F F ′1F2 + (Maη +Maη′)F1F ξuC2 − ξt C3 + 2C4 − − (Maη +Maη′)F2F ′2ξt MP1aη +M MP2aη +M 2C6 + MP2aη +M −fBs · F P2aη + F C5 + C6 − , (20) where ξu = V ubVus and ξt = V tbVts, and the relevant mixing parameters and decay con- stants are cos θ8 − sin θ1, F2 = − sin θ8 + cos θ1, (21) F ′1 = sin θ8 + cos θ1, F 2 = − sin θ8 + cos θ1, (22) f dη = cos θ8 − sin θ1, f η = − cos θ8 − sin θ1, (23) f dη′ = sin θ8 + cos θ1, f η′ = − sin θ8 + cos θ1. (24) Similarly, the decay amplitudes for B0s → ηη and B0s → η′η′ decay can be obtained easily from Eq.(20) by the following replacements f dη , f η ←→ f dη′ , f sη′ ; F1(θ1, θ8)←→ F ′1(θ1, θ8); F2(θ1, θ8)←→ F ′2(θ1, θ8). (25) Note that the contributions from the possible gluonic component of η′ meson have not been included here. II. NUMERICAL RESULTS AND DISCUSSIONS In this section, we will calculate the CP-averaged branching ratios and CP violating asymmetries for those considered decay modes. The input parameters and the wave functions to be used are given in Appendix A. In numerical calculations, central values of input parameters will be used implicitly unless otherwise stated. Using the decay amplitudes obtained in last section, it is straightforward to calculate the branching ratios. By employing the two mixing angle scheme of η − η′ system and using the mixing parameters as given in Eq. (19), one finds the CP-averaged branching ratios for the considered three decays as follows Br( B0s → ηη) = 14.2+6.6−4.2(ωb) +16.7 −6.2 (m × 10−6, (26) Br( B0s → ηη′) = 12.4+5.7−3.6(ωb) +17.3 −6.0 (m × 10−6, (27) Br( B0s → η′η′) = 9.2+3.0−2.0(ωb) +15.0 −4.5 (m × 10−6, (28) where the main errors are induced by the uncertainties of ωb = 0.50 ± 0.05 GeV, and 0 = [1.49−2.38] GeV (corresponding to ms = 130±30 MeV), respectively. The above pQCD predictions agree well with those obtained in the QCD facterization approach [2]. As for the gluonic contributions, we follow the same procedure as being used in Ref. [4] to include the possible gluonic contributions to the Bs → η(′) transition form factors 0,1 and found that the gluonic contributions to the branching ratios are less than 3% for B → ηη decay, ∼ 7% for B → ηη′ decay, and around 18% for B → η′η′ decay. The central values of the pQCD predictions for Bs → η(′)η(′) decays after the inclusion of possible gluonic contributions are the following Br(B0s → ηη) = 13.7+6.4−4.0(ωb) +16.5 −6.1 (m × 10−6, Br( B0s → ηη′) = 11.6+5.3−3.4(ωb) +16.8 −5.7 (m × 10−6, Br( B0s → η′η′) = 10.8+3.7−2.4(ωb) +16.2 −5.2 (m × 10−6. (29) Now we turn to the evaluations of the CP-violating asymmetries of Bs → η(′)η(′) decays in pQCD approach. For B0s meson decays, a non-zero ratio (∆Γ/Γ)Bs is expected in the SM [15, 16]. For Bs → η(′)η(′) decays, three quantities to describe the CP violation can be defined as follows [16]: AdirCP = |λCP |2 − 1 1 + |λCP |2 , AmixCP = 2Im(λCP ) 1 + |λCP |2 , A∆Γs = 2Re(λCP ) 1 + |λCP |2 , (30) λCP = ηf V ∗tbVts〈f |Heff |B̄0s〉 ts〈f |Heff |B0s 〉 〈f |Heff |B̄0s〉 〈f |Heff |B0s〉 , (31) in a very good approximation. Here AdirCP and AmixCP means the direct and mixing-induced CP violation respectively, while the third term A∆Γs is related to the presence of a non- negligible ∆Γs. By using the mixing parameters in Eq. (19) and the input parameters as given in Appendix A, one found the pQCD predictions for AdirCP , AmixCP and Hf AdirCP (B0s → ηη) = −0.2± 0.1(γ)± 0.1(ωb)+0.4−0.2(m × 10−2, AdirCP (B0s → ηη′) = +0.6+0.1−0.2(γ)± 0.1(ωb)± 0.3(m × 10−2, AdirCP (B0s → η′η′) = −0.8+0.2−0.1(γ)± 0.1(ωb)± 0.7(m × 10−2, (32) AmixCP (B0s → ηη) = [−0.3± 0.1(γ)± 0.2(ωb)± 0.5(m 0 )]× 10−2, AmixCP (B0s → ηη′) = [−0.8± 0.2(γ)± 0.1(ωb)± 0.2(m 0 )]× 10−2, AmixCP (B0s → η′η′) = +1.8+0.3−0.5(γ)± 0.0(ωb)+0.5−0.3(m × 10−2, (33) A∆Γs(ηη) ≈ A∆Γs(ηη′) ≈ A∆Γs(η′η′) ≈ 1, (34) where the dominant errors come from the variations of CKM angle γ = 60◦ ± 20◦, ωb = 0.50± 0.05 GeV and mηss̄0 = [1.49− 2.38] GeV ( corresponding to ms = 130 ± 30 MeV), respectively. It is easy to see that both the direct and mixing-induced CP violations of the considered Bs decays are very small in magnitude, and thus almost impossible to measure them even in the LHC experiments. The above pQCD predictions are also consistent with the QCDF predictions [1, 2]. In short, we calculated the branching ratios and CP-violating asymmetries of B0s → ηη, ηη′ and η′η′ decays at the leading order by using the pQCD factorization approach. Besides the usual factorizable diagrams, the non-factorizable and annihilation diagrams are also calculated analytically in the pQCD approach. From our calculations and phenomeno- logical analysis, we found the following results: • Using the two mixing angle scheme, the pQCD predictions for the CP-averaged branching ratios are Br(B0s → ηη) = 14.2+18.0−7.5 × 10−6, Br(B0s → ηη′) = 12.4+18.2−7.0 × 10−6, Br(B0s → η′η′) = 9.2+15.3−4.9 × 10−6, (35) where the various errors as specified previously have been added in quadrature. The pQCD predictions for the three decay channels agree well with those obtained by employing the QCDF approach. • The gluonic contributions are small in size: less than 7% for Bs → ηη and ηη′ decays, and around 18% for Bs → η′η′ decay. • The direct and mixing-induced CP violations of the considered three decay modes are very small: less than 3% in magnitude. Note added: After completion of this paper, the paper in Ref.[18] appeared, and where a systematic study for the Bs → M1M2 decays in the pQCD factorization approach has been done. Since different mixing-scheme of η − η′ system have been used, the explicit expressions of the decay amplitudes of the relevant decays are different in these two papers, but the numerical predictions for branching ratios and CP violations agree well with each other. The possible gluonic contributions are estimated here. Acknowledgments X. Liu would like to acknowledge the financial support of The Scientific Research Start-up Fund of Zhejiang Ocean University under Grant No.21065010706. This work was partially supported by the National Natural Science Foundation of China under Grant No.10575052, and by the Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) under Grant No. 20050319008. APPENDIX A: INPUT PARAMETERS AND WAVE FUNCTIONS In this Appendix we show the input parameters and the light meson wave functions to be used in the numerical calculations. The masses, decay constants, QCD scale and B0s meson lifetime are (f=4) = 250MeV, fπ = 130MeV, fBs = 230MeV, 0 = 1.4GeV, ms = 130MeV, fK = 160MeV, MBs = 5.37GeV, MW = 80.41GeV, τB0s = 1.46× 10 −12s (A1) For the CKM matrix elements, here we adopt the Wolfenstein parametrization for the CKM matrix, and take λ = 0.2272, A = 0.818, ρ = 0.221 and η = 0.340 [11]. For the B meson wave function, we adopt the model φBs(x, b) = NBsx 2(1− x)2exp M2Bs x (ωbb) , (A2) where ωb is a free parameter and we take ωb = 0.50± 0.05 GeV in numerical calculations, and NBs = 63.67 is the normalization factor for ωb = 0.50. For the distribution amplitudes φAη , φPη and φTη , we utilize the result from the light-cone sum rule [17] including twist-3 contribution. For the corresponding Gegenbauer moments and relevant input parameters, we here use a 2 = 0.115, a 4 = −0.015, ρηdd̄ = 0 , η3 = 0.015 and ω3 = −3.0. We also assume that the wave function of uū is the same as the wave function of dd̄ [3]. For the wave function of the ss̄ components, we also use the same form as dd̄ but with mss̄0 and fy instead of m 0 and fx, respectively: fx = fπ, fy = 2f 2K − f 2π . (A3) These values are translated to the values in the two mixing angle method: f1 = 152.1MeV, f8 = 163.8MeV, θ1 = −9.2◦, θ8 = −21.2◦. (A4) The parameters mi0 (i = ηdd̄(uū), ηss̄) are defined as: dd̄(uū) 0 ≡ mπ0 ≡ (mu +md) 2M2K −m2π (2ms) . (A5) [1] M. Beneke and M. Neubert, Nucl. Phys. B 675, 333 (2003). [2] J.F. Sun, G.H. Zhu, D.S. Du, Phys. Rev. D 68 (2003) 054003. [3] E. Kou, Phys. Rev. D 63, 054027 (2001); E. Kou and A.I. Sanda, Phys. Lett. B 525, 240 (2002). [4] Y.Y. Charng, T. Kurimoto, and H.N. Li, Phys. Rev. D 74, 074024 (2006). [5] Z.J. Xiao, X. Liu, and H.S. Wang, Phys. Rev. D 75,034017 (2007). [6] X.F. Chen, D.Q. Guo and Z.J. Xiao, hep-ph/0701146. [7] C.-H. V. Chang and H.N. Li, Phys. Rev. D 55, 5577 (1997); T.-W. Yeh and H.N. Li, Phys. Rev. D 56, 1615 (1997). [8] H.N. Li, Prog.Part.& Nucl.Phys. 51, 85 (2003), and reference therein. [9] G.P. Lepage and S.J. Brodsky, Phys. Rev. D 22, 2157 (1980). [10] G. Buchalla, A.J. Buras, and M.E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996). [11] Particle Data Group, W.-M. Yao et al., J. Phys. G 33, 1 (2006). [12] H.N. Li, Phys. Rev. D 66, 094010 (2002). [13] C.-D. Lü, K. Ukai and M.Z. Yang, Phys. Rev. D 63, 074009 (2001). [14] Th. Feldmann, P. Kroll and B. Stech, Phys. Rev. D 58, 114006 (1998); R. Escribano and J.M. Frere, J. High Energy Phys. 0506 (2005) 029. [15] M. Beneke, G. Buchalla, C. Greub, A. Lenz and U. Nierste, Phys. Lett. B 459, 631 (1999). [16] L. Fernandez, Ph.D Thesis, CERN-Thesis-2006-042. [17] P. Ball, J. High Energy Phys. 9809, 005 (1998); P. Ball, J. High Energy Phys. 9901, 010 (1999); P. Ball and R. Zwicky, Phys. Rev. D 71, 014015 (2005). [18] A. Ali, G. Kramer, Y. Li, C.D. Lü, Y.L. Shen, W. Wang, and Y.M. Wang, hep-ph/0703162. http://arxiv.org/abs/hep-ph/0701146 http://arxiv.org/abs/hep-ph/0703162 Perturbative calculations Decay amplitudes Complete decay amplitudes Numerical results and Discussions Acknowledgments Input parameters and wave functions References

0704.1028

A neural network approach to ordinal regression

A Neural Network Approach to Ordinal Regression Jianlin Cheng [email protected] School of Electrical Engineering and Computer Science, University of Central Florida, Orlando, FL 32816, USA Abstract Ordinal regression is an important type of learning, which has properties of both clas- sification and regression. Here we describe a simple and effective approach to adapt a traditional neural network to learn ordinal categories. Our approach is a generaliza- tion of the perceptron method for ordinal regression. On several benchmark datasets, our method (NNRank) outperforms a neural network classification method. Compared with the ordinal regression methods using Gaussian processes and support vector machines, NNRank achieves comparable performance. Moreover, NNRank has the advantages of traditional neural networks: learning in both online and batch modes, handling very large training datasets, and making rapid predictions. These features make NNRank a useful and complementary tool for large-scale data processing tasks such as information retrieval, web page ranking, collaborative filtering, and protein ranking in Bioinformatics. 1. Introduction Ordinal regression (or ranking learning) is an impor- tant supervised problem of learning a ranking or or- dering on instances, which has the property of both classification and metric regression. The learning task of ordinal regression is to assign data points into a set of finite ordered categories. For example, a teacher rates students’ performance using A, B, C, D, and E (A > B > C > D > E) (Chu & Ghahramani, 2005a). Ordinal regression is different from classification due to the order of categories. In contrast to metric re- gression, the response variables (categories) in ordinal regression is discrete and finite. The research of ordinal regression dated back to the ordinal statistics methods in 1980s (McCullagh, 1980; McCullagh & Nelder, 1983) and machine learning re- search in 1990s (Caruana et al., 1996; Herbrich et al., 1998; Cohen et al., 1999). It has attracted the con- siderable attention in recent years due to its poten- tial applications in many data-intensive domains such as information retrieval (Herbrich et al., 1998), web page ranking (Joachims, 2002), collaborative filtering (Goldberg et al., 1992; Basilico & Hofmann, 2004; Yu et al., 2006), image retrieval (Wu et al., 2003), and pro- tein ranking (Cheng & Baldi, 2006) in Bioinformatics. A number of machine learning methods have been de- veloped or redesigned to address ordinal regression problem (Rajaram et al., 2003), including perceptron (Crammer & Singer, 2002) and its kernelized gener- alization (Basilico & Hofmann, 2004), neural network with gradient descent (Caruana et al., 1996; Burges et al., 2005), Gaussian process (Chu & Ghahramani, 2005b; Chu & Ghahramani, 2005a; Schwaighofer et al., 2005), large margin classifier (or support vec- tor machine) (Herbrich et al., 1999; Herbrich et al., 2000; Joachims, 2002; Shashua & Levin, 2003; Chu & Keerthi, 2005; Aiolli & Sperduti, 2004; Chu & Keerthi, 2007), k-partite classifier (Agarwal & Roth, 2005), boosting algorithm (Freund et al., 2003; Dekel et al., 2002), constraint classification (Har-Peled et al., 2002), regression trees (Kramer et al., 2001), Naive Bayes (Zhang et al., 2005), Bayesian hierarchical ex- perts (Paquet et al., 2005), binary classification ap- proach (Frank & Hall, 2001; Li & Lin, 2006) that de- composes the original ordinal regression problem into a set of binary classifications, and the optimization of nonsmooth cost functions (Burges et al., 2006). Most of these methods can be roughly classified into two categories: pairwise constraint approach (Herbrich et al., 2000; Joachims, 2002; Dekel et al., 2004; Burges et al., 2005) and multi-threshold approach (Cram- mer & Singer, 2002; Shashua & Levin, 2003; Chu & Ghahramani, 2005a). The former is to convert the full ranking relation into pairwise order constraints. The latter tries to learn multiple thresholds to divide data A Neural Network Approach to Ordinal Regression into ordinal categories. Multi-threshold approaches also can be unified under the general, extended binary classification framework (Li & Lin, 2006). The ordinal regression methods have different advan- tages and disadvantages. Prank (Crammer & Singer, 2002), a perceptron approach that generalizes the bi- nary perceptron algorithm to the ordinal multi-class situation, is a fast online algorithm. However, like a standard perceptron method, its accuracy suffers when dealing with non-linear data, while a quadratic kernel version of Prank greatly relieves this problem. One class of accurate large-margin classifier approaches (Herbrich et al., 2000; Joachims, 2002) convert the ordinal relations into O(n2) (n: the number of data points) pairwise ranking constraints for the structural risk minimization (Vapnik, 1995; Schoelkopf & Smola, 2002). Thus, it can not be applied to medium size datasets (> 10,000 data points), without discarding some pairwise preference relations. It may also overfit noise due to incomparable pairs. The other class of powerful large-margin classifier methods (Shashua & Levin, 2003; Chu & Keerthi, 2005) generalize the support vector formulation for or- dinal regression by finding K − 1 thresholds on the real line that divide data into K ordered categories. The size of this optimization problem is linear in the number of training examples. However, like support vector machine used for classification, the prediction speed is slow when the solution is not sparse, which makes it not appropriate for time-critical tasks. Simi- larly, another state-of-the-art approach, Gaussian pro- cess method (Chu & Ghahramani, 2005a), also has the difficulty of handling large training datasets and the problem of slow prediction speed in some situations. Here we describe a new neural network approach for ordinal regression that has the advantages of neural network learning: learning in both online and batch mode, training on very large dataset (Burges et al., 2005), handling non-linear data, good performance, and rapid prediction. Our method can be considered a generalization of the perceptron learning (Crammer & Singer, 2002) into multi-layer perceptrons (neural network) for ordinal regression. Our method is also related to the classic generalized linear models (e.g., cumulative logit model) for ordinal regression (Mc- Cullagh, 1980). Unlike the neural network method (Burges et al., 2005) trained on pairs of examples to learn pairwise order relations, our method works on individual data points and uses multiple output nodes to estimate the probabilities of ordinal cate- gories. Thus, our method falls into the category of multi-threshold approach. The learning of our method proceeds similarly as traditional neural networks using back-propagation (Rumelhart et al., 1986). On the same benchmark datasets, our method yields the performance better than the standard classifica- tion neural networks and comparable to the state-of- the-art methods using support vector machines and Gaussian processes. In addition, our method can learn on very large datasets and make rapid predictions. 2. Method 2.1. Formulation Let D represent an ordinal regression dataset consist- ing of n data points (x, y) , where x ∈ Rd is an input feature vector and y is its ordinal category from a fi- nite set Y . Without loss of generality, we assume that Y = 1, 2, ...,K with ”<” as order relation. For a standard classification neural network without considering the order of categories, the goal is to pre- dict the probability of a data point x belonging to one category k (y = k). The input is x and the target of encoding the category k is a vector t = (0, ..., 0, 1, 0, ..., 0), where only the element tk is set to 1 and all others to 0. The goal is to learn a function to map input vector x to a probability distribution vector o = (o1, o2, ...ok, ...oK), where ok is closer to 1 and other elements are close to zero, subject to the constraint i=1 oi = 1. In contrast, like the perceptron approach (Crammer & Singer, 2002), our neural network approach considers the order of the categories. If a data point x belongs to category k, it is classified automatically into lower- order categories (1, 2, ..., k − 1) as well. So the target vector of x is t = (1, 1, .., 1, 0, 0, 0), where ti (1 ≤ i ≤ k) is set to 1 and other elements zeros. Thus, the goal is to learn a function to map the input vector x to a probability vector o = (o1, o2, ..., ok, ...oK), where oi (i ≤ k) is close to 1 and oi (i ≥ k) is close to 0.∑K i=1 oi is the estimate of number of categories (i.e. k) that x belongs to, instead of 1. The formulation of the target vector is similar to the perceptron ap- proach (Crammer & Singer, 2002). It is also related to the classical cumulative probit model for ordinal re- gression (McCullagh, 1980), in the sense that we can consider the output probability vector (o1, ...ok, ...oK) as a cumulative probability distribution on categories (1, ..., k, ..., K), i.e., is the proportion of cate- gories that x belongs to, starting from category 1. The target encoding scheme of our method is related to but, different from multi-label learning (Bishop, 1996) and multiple label learning (Jin & Ghahramani, 2003) A Neural Network Approach to Ordinal Regression because our method imposes an order on the labels (or categories). 2.2. Learning Under the formulation, we can use the almost exactly same neural network machinery for ordinal regression. We construct a multi-layer neural network to learn ordinal relations from D. The neural network has d inputs corresponding to the number of dimensions of input feature vector x and K output nodes correspond- ing to K ordinal categories. There can be one or more hidden layers. Without loss of generality, we use one hidden layer to construct a standard two-layer feedfor- ward neural network. Like a standard neural network for classification, input nodes are fully connected with hidden nodes, which in turn are fully connected with output nodes. Likewise, the transfer function of hid- den nodes can be linear function, sigmoid function, and tanh function that is used in our experiment. The only difference from traditional neural network lies in the output layer. Traditional neural networks use soft- max e (or normalized exponential function) for output nodes, satisfying the constraint that the sum of outputs i=1 oi is 1. zi is the net input to the output node Oi. In contrast, each output node Oi of our neural net- work uses a standard sigmoid function 1 1+e−zi , with- out including the outputs from other nodes. Output node Oi is used to estimate the probability oi that a data point belongs to category i independently, with- out subjecting to normalization as traditional neural networks do. Thus, for a data point x of category k, the target vector is (1, , 1, .., 1, 0, 0, 0), in which the first k elements is 1 and others 0. This sets the target value of output nodes Oi (i ≤ k) to 1 and Oi (i > k) to 0. The targets instruct the neural network to ad- just weights to produce probability outputs as close as possible to the target vector. It is worth pointing out that using independent sigmoid functions for out- put nodes does not guaranteed the monotonic relation (o1 >= o2 >= ... >= oK), which is not necessary but, desirable for making predictions (Li & Lin, 2006). A more sophisticated approach is to impose the inequal- ity constraints on the outputs to improve the perfor- mance. Training of the neural network for ordinal regres- sion proceeds very similarly as standard neural net- works. The cost function for a data point x can be relative entropy or square error between the tar- get vector and the output vector. For relative en- tropy, the cost function for output nodes is fc =∑K i=1 (ti log oi + (1− ti) log(1− oi)). For square er- ror, the error function is fc = i=1 (ti − oi) 2. Pre- vious studies (Richard & Lippman, 1991) on neural network cost functions show that relative entropy and square error functions usually yield very similar re- sults. In our experiments, we use square error function and standard back-propagation to train the neural net- work. The errors are propagated back to output nodes, and from output nodes to hidden nodes, and finally to input nodes. Since the transfer function ft of output node Oi is the independent sigmoid function 1 1+e−zi , the deriva- tive of ft of output node Oi is (1+e−zi )2 1+e−zi (1 − 1 1+e−zi ) = oi(1 − oi). Thus, the net error propagated to output node Oi is = ti−oi oi(1−oi) oi(1 − oi) = ti − oi for relative entropy cost function, = −2(ti−oi)×oi(1−oi) = −2oi(ti−oi)(1−oi) for square error cost function. The net errors are prop- agated through neural networks to adjust weights us- ing gradient descent as traditional neural networks do. Despite the small difference in the transfer function and the computation of its derivative, the training of our method is the same as traditional neural networks. The network can be trained on data in the online mode where weights are updated per example, or in the batch mode where weights are updated per bunch of examples. 2.3. Prediction In the test phase, to make a prediction, our method scans output nodes in the order O1, O2, ..., OK . It stops when the output of a node is smaller than the predefined threshold T (e.g., 0.5) or no nodes left. The index k of the last node Ok whose output is bigger than T is the predicted category of the data point. 3. Experiments and Results 3.1. Benchmark Data and Evaluation Metric We use eight standard datasets for ordinal regres- sion (Chu & Ghahramani, 2005a) to benchmark our method. The eight datasets (Diabetes, Pyrimidines, Triazines, Machine CUP, Auto MPG, Boston, Stocks Domain, and Abalone) are originally used for metric regression. Chu and Ghahramani (Chu & Ghahra- mani, 2005a) discretized the real-value targets into five equal intervals, corresponding to five ordinal cat- egories. The authors randomly split each dataset into training/test datasets and repeated the partition 20 times independently. We use the exactly same parti- tions as in (Chu & Ghahramnai, 2005a) to train and test our method. A Neural Network Approach to Ordinal Regression We use the online mode to train neural networks. The parameters to tune are the number of hidden units, the number of epochs, and the learning rate. We create a grid for these three parameters, where the hidden unit number is in the range [1..15], the epoch number in the set (50, 200, 500, 1000), and the initial learning rate in the range [0.01..0.5]. During the training, the learning rate is halved if training errors continuously go up for a pre-defined number (40, 60, 80, or 100) of epochs. For experiments on each data split, the neural network parameters are fully optimized on the training data without using any test data. For each experiment, after the parameters are opti- mized on the training data, we train five models on the training data with the optimal parameters, start- ing from different initial weights. The ensemble of five trained models are then used to estimate the general- ized performance on the test data. That is, the average output of five neural network models is used to make predictions. We evaluate our method using zero-one error and mean absolute error as in (Chu & Ghahramani, 2005a). Zero-one error is the percentage of wrong assignments of ordinal categories. Mean absolute error is the root mean square difference between assigned categories (k′) and true categories (k) of all data points. For each dataset, the training and evaluation process is repeated 20 times on 20 data splits. Thus, we com- pute the average error and the standard deviation of the two metrics as in (Chu & Ghahramani, 2005a). 3.2. Comparison with Neural Network Classification We first compare our method (NNRank) with a stan- dard neural network classification method (NNClass). We implement both NNRank and NNClass using C++. NNRank and NNClass share most code with minor difference in the transfer function of output nodes and its derivative computation as described in Section 2.2. As Table 1 shows, NNRank outperforms NNClass in all but one case in terms of both the mean-zero error and the mean absolute error. And on some datasets the improvement of NNRank over NNClass is sizable. For instance, on the Stock and Pyrimidines datasets, the mean zero-one error of NNRank is about 4% less than NNClass; on four datasets (Stock, Pyrimidines, Triazines, and Diabetes) the mean absolute error is reduced by about .05. The results show that the or- dinal regression neural network consistently achieves the better performance than the standard classifica- tion neural network. To futher verify the effectiveness of the neural network ordinal regression approach, we are currently evaluating NNRank and NNclass on very large ordinal regression datasets in the bioinformatics domain (work in progress). 3.3. Comparison with Gaussian Processes and Support Vector Machines To further evaluate the performance of our method, we compare NNRank with two Gaussian process meth- ods (GP-MAP and GP-EP) (Chu & Ghahramani, 2005a) and a support vector machine method (SVM) (Shashua & Levin, 2003) implemented in (Chu & Ghahramani, 2005a). The results of the three meth- ods are quoted from (Chu & Ghahramani, 2005a). Ta- ble 2 reports the zero-one error on the eight datasets. NNRank achieves the best results on Diabetes, Tri- azines, and Abalone, GP-EP on Pyrimidines, Auto MPG, and Boston, GP-MAP on Machine, and SVM on Stocks. Table 3 reports the mean absolute error on the eight datasets. NNRank yields the best results on Diabetes and Abalone, GP-EP on Pyrimidines, Auto MPG, and Boston, GP-MAP on Triazines and Machine, SVM on Stocks. In summary, on the eight datasets, the performance of NNRank is comparable to the three state-of-the-art methods for ordinal regression. 4. Discussion and Future Work We have described a simple yet novel approach to adapt traditional neural networks for ordinal regres- sion. Our neural network approach can be consid- ered a generalization of one-layer perceptron approach (Crammer & Singer, 2002) into multi-layer. On the standard benchmark of ordinal regression, our method outperforms standard neural networks used for classi- fication. Furthermore, on the same benchmark, our method achieves the similar performance as the two state-of-the-art methods (support vector machines and Gaussian processes) for ordinal regression. Compared with existing methods for ordinal regres- sion, our method has several advantages of neural net- works. First, like the perceptron approach (Crammer & Singer, 2002), our method can learn in both batch and online mode. The online learning ability makes our method a good tool for adaptive learning in the real-time. The multi-layer structure of neural network and the non-linear transfer function give our method the stronger fitting ability than perceptron methods. Second, the neural network can be trained on very A Neural Network Approach to Ordinal Regression large datasets iteratively, while training is more com- plex than support vector machines and Gaussian pro- cesses. Since the training process of our method is the same as traditional neural networks, average neural network users can use this method for their tasks. Third, neural network method can make rapid prediction once models are trained. The ability of learning on very large dataset and predicting in time makes our method a useful and competitive tool for ordinal regression tasks, particularly for time-critical and large-scale ranking problems in information retrieval, web page ranking, collaborative filtering, and the emerging fields of Bioinformat- ics. We are currently applying the method to rank proteins according to their structural rele- vance with respect to a query protein (Cheng & Baldi, 2006). To facilitate the application of this new approach, we make both NNRank and NNClass to accept a general input format and freely available at http://www.eecs.ucf.edu/∼jcheng/cheng software.html. There are some directions to further improve the neu- ral network (or multi-layer perceptron) approach for ordinal regression. One direction is to design a trans- fer function to ensure the monotonic decrease of the outputs of the neural network; the other direction is to derive the general error bounds of the method under the binary classification framework (Li & Lin, 2006). Furthermore, the other flavors of implemen- tations of the multi-threshold multi-layer perceptron approach for ordinal regression are possible. Since ma- chine learning ranking is a fundamental problem that has wide applications in many diverse domains such as web page ranking, information retrieval, image re- trieval, collaborative filtering, bioinformatics and so on, we believe the further exploration of the neural net- work (or multi-layer perceptron) approach for ranking and ordinal regression is worthwhile. References Agarwal, S., & Roth, D. (2005). Learnability of bipar- tite ranking functions. In Proc. of the 18th annual conference on learning theory (colt-05). Aiolli, F., & Sperduti, A. (2004). 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Schwaighofer, A., Tresp, V., & Yu, K. (2005). Hiear- achical bayesian modelling with gaussian processes. In Advances in neural information processing sys- tems 17 (nips). MIT press. Shashua, A., & Levin, A. (2003). Ranking with large margin principle: two approaches. In Advances in neural information processing systems 15 (nips). Vapnik, V. (1995). The nature of statistical learning theory. Berlin, Germany: Springer-Verlag. Wu, H., Lu, H., & Ma, S. (2003). A practical svm- based algorithm for ordinal regression in image re- trieval. 612–621. Yu, S., Yu, K., Tresp, V., & Kriegel, H. P. (2006). Collaborative ordinal regression. In Proc. of 23rd international conference on machine learning, 1089– 1096. A Neural Network Approach to Ordinal Regression Zhang, H., Jiang, L., & Su, J. (2005). Augmenting naive bayes for ranking. In International conference on machine learning (icml-05). A Neural Network Approach to Ordinal Regression Table 1. The results of NNRank and NNClass on the eight datasets. The results are the average error over 20 trials along with the standard deviation. Mean zero-one error Mean absolute error Dataset NNRank NNClass NNRank NNClass Stocks 12.68±1.8% 16.97± 2.3% 0.127±0.01 0.173±0.02 Pyrimidines 37.71±8.1% 41.87±7.9% 0.450±0.09 0.508±0.11 Auto MPG 27.13±2.0% 28.82±2.7% 0.281±0.02 0.307±0.03 Machine 17.03±4.2% 17.80±4.4% 0.186±0.04 0.192±0.06 Abalone 21.39±0.3% 21.74± 0.4% 0.226±0.01 0.232±0.01 Triazines 52.55±5.0% 52.84±5.9% 0.730±0.06 0.790±0.09 Boston 26.38±3.0% 26.62±2.7% 0.295±0.03 0.297±0.03 Diabetes 44.90±12.5% 43.84±10.0% 0.546±0.15 0.592±0.09 Table 2. Zero-one error of NNRank, SVM, GP-MAP, and GP-EP on the eight datasets. SVM denotes the support vector machine method (Shashua & Levin, 2003; Chu & Ghahramani, 2005a). GP-MAP and GP-EP are two Gaussian process methods using Laplace approximation (MacKay, 1992) and expectation propagation (Minka, 2001) respectively (Chu & Ghahramani, 2005a). The results are the average error over 20 trials along with the standard deviation. We use boldface to denote the best results. Data NNRank SVM GP-MAP GP-EP Triazines 52.55±5.0% 54.19±1.5% 52.91±2.2% 52.62±2.7% Pyrimidines 37.71±8.1% 41.46±8.5% 39.79±7.2% 36.46±6.5% Diabetes 44.90±12.5% 57.31±12.1% 54.23±13.8% 54.23±13.8% Machine 17.03±4.2% 17.37±3.6% 16.53±3.6% 16.78±3.9% Auto MPG 27.13±2.0% 25.73±2.2% 23.78±1.9% 23.75±1.7% Boston 26.38±3.0% 25.56±2.0% 24.88±2.0% 24.49±1.9% Stocks 12.68±1.8% 10.81±1.7% 11.99±2.3% 12.00±2.1% Abalone 21.39±0.3% 21.58±0.3% 21.50±0.2% 21.56±0.4% Table 3. Mean absolute error of NNRank, SVM, GP-MAP, and GP-EP on the eight datasets. SVM denotes the support vector machine method (Shashua & Levin, 2003; Chu & Ghahramani, 2005a). GP-MAP and GP-EP are two Gaussian process methods using Laplace approximation and expectation propagation respectively (Chu & Ghahramani, 2005a). The results are the average error over 20 trials along with the standard deviation. We use boldface to denote the best results. Data NNRank SVM GP-MAP GP-EP Triazines 0.730±0.07 0.698±0.03 0.687±0.02 0.688±0.03 Pyrimidines 0.450±0.10 0.450±0.11 0.427±0.09 0.392±0.07 Diabetes 0.546±0.15 0.746±0.14 0.662±0.14 0.665±0.14 Machine 0.186±0.04 0.192±0.04 0.185±0.04 0.186±0.04 Auto MPG 0.281±0.02 0.260±0.02 0.241±0.02 0.241±0.02 Boston 0.295±0.04 0.267±0.02 0.260±0.02 0.259±0.02 Stocks 0.127±0.02 0.108±0.02 0.120±0.02 0.120±0.02 Abalone 0.226±0.01 0.229±0.01 0.232±0.01 0.234±0.01

0704.1029

Controlling surface statistical properties using bias voltage: Atomic force microscopy and stochastic analysis

Controlling surface statistical properties using bias voltage: Atomic force microscopy and stochastic analysis P. Sangpour,1 G. R. Jafari,1 O. Akhavan,1 A.Z. Moshfegh,1 and M. Reza Rahimi Tabar1, 2 1Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran 2CNRS UMR 6529, Observatoire de la Côte d’Azur, BP 4229, 06304 Nice Cedex 4, France The effect of bias voltages on the statistical properties of rough surfaces has been studied using atomic force microscopy technique and its stochastic analysis. We have characterized the complexity of the height fluc- tuation of a rough surface by the stochastic parameters such as roughness exponent, level crossing, and drift and diffusion coefficients as a function of the applied bias voltage. It is shown that these statistical as well as microstructural parameters can also explain the macroscopic property of a surface. Furthermore, the tip convolution effect on the stochastic parameters has been examined. PACS numbers: 05.10.Gg, 02.50.Fz, 68.37.-d I. INTRODUCTION As device dimensions continue to shrink into the deep sub- micron size regime, there will be increasing attention for un- derstanding the thin-film growth mechanism and the kinet- ics of growing rough surfaces in various deposition methods. To perform a quantitative study on surfaces roughness, an- alytical and numerical treatments of simple growth models propose, quite generally, the height fluctuations have a self- similar character and their average correlations exhibit a dy- namic scaling form1,2,3,4,5,6. In these models, roughness of a surface is a smooth function of the sample size and growth time (or thickness) of films. In addition, other statistical quan- tities such as the average frequency of positive slope level crossing, the probability density function (PDF), as well as drift and diffusion coefficients provide further complete analy- sis on roughness of a surface. Very recently, it has been shown that, by using these statistical variables in the Langevin equa- tion, regeneration of rough surfaces with the same statistical properties of a nanoscopic imaging is possible7,8. In practice, one of the effective ways to modify roughness of surfaces is applying a negative bias voltage during deposi- tion of thin films10, while their sample size and thickness are constant. In bias sputtering, electric fields near the substrate are modified to vary the flux and energy of incident charged species. This is achieved by applying either a negative DC or RF bias to the substrate. Due to charge exchange processes in the anode dark space, very few discharge ions strike the substrate with full bias voltage. Rather a broad low energy distribution of ions bombard the growing films. Generally, bias sputtering modifies film properties such as surface morphology, resistivity, stress, density, adhesion, and so on through roughness improvement of the surface, elimi- nation of interfacial voids and subsurface porosity, creation of a finer and more isotropic grain morphology, and the elimina- tion of columnar grains10. In this work, the effect of bias voltage on the statistical properties of a surface, i.e., the roughness exponent, the level crossing, the probability density function, as well as the drift and diffusion coefficients has been studied. In this regard, we have analyzed the surface of Co(3 nm)/NiO(30 nm)/Si(100) structure (as a base structure in the magnetic multilayers, e.g., spin valves operated using giant magnetoresistance (GMR) effect11,12) fabricated by bias sputtering method at different bias voltages. The behavior of statistical characterizations ob- tained by nanostructural analysis have been also compared with behavior of sheet resistance measurement of the films deposited at the different bias voltages, as a macroscopic anal- ysis. II. EXPERIMENTAL The substrates used for this experiment were n-type Si(100) wafers with resistivity of about 5-8 Ω-cm and the dimension of 5×11 mm2 . After a standard RCA cleaning procedure and a short time dip in a diluted HF solution, the wafers were loaded into a vacuum chamber. The chamber was evacuated to a base pressure of about 4×10−7 Torr. To deposit nickel oxide thin film, first high purity NiO powder was pressed and baked over night at 1400 ◦C in an atmospheric oven yielded a green solid disk suitable for thermal evaporation. Before each NiO deposition, a pre-evaporation was done for about 5 min- utes. Then a 30 nm thick NiO layer was deposited on the Si substrate with applied power of about 350 watts resulted in a deposition rate of 0.03 nm/s at a pressure of 2×10−6 Torr. After that, without breaking the vacuum, a thin Co layer of 3 nm was deposited on the NiO surface by using DC sputtering technique. During the deposition, a dynamic flow of ultra- high purity Ar gas with pressure of 70 mTorr was used for sputtering discharge. The discharge power to grow Co layers was considered around 40 watts that resulted in a deposition rate of about 0.01 nm/s. The thickness of the deposited films was measured by styles technique, and controlled in-situ by a quartz crystal oscillator located near the substrate. The dis- tance between the target (50 mm in diameter) and substrate was 70 mm. Before each deposition, a pre-sputtering was also performed for about 10 minutes. The deposition of Co layers was done at various negative bias voltages ranging from zero to -80 V at the same sputtering conditions. The schematic http://arxiv.org/abs/0704.1029v1 details about the way of exerting the bias voltage to the Si substrate can be found in13. In order to analyze the deposited samples, we have used atomic force microscopy (AFM) on contact mode to study the surface topography of the Co layer. The surface topogra- phy of the films was investigated using Park Scientific Instru- ments (model Autoprobe CP). The images were collected in a constant force mode and digitized into 256× 256 pixels with scanning frequency of 0.6 Hz. The cantilever of 0.05 N m−1 spring constant with a commercial standard pyramidal Si3N4 tip with an aspect ratio of about 0.9 was used. A variety of scans, each with size L were recorded at random locations on the Co film surface. The electrical property of the deposited films was examined by four-point probe sheet resistance (Rs) measurement at room temperature. III. STATISTICAL QUANTITIES A. roughness exponents It is known that to derive a quantitative information of a surface morphology one may consider a sample of size L and define the mean height of growing film h and its roughness w by the following expressions14: h(L, t, λ) = ∫ L/2 h(x, t, λ)dx (1) w(L, t, λ) = (〈(h− h)2〉)1/2 (2) where t is proportional to deposition time and 〈· · · 〉 denotes an averaging over different samples, respectively. Moreover, we have introduced λ as an external factor which can apply to control the surface roughness of thin films. In this work, λ ≡ V/Vopt is defined where V and Vopt are the applied and the optimum bias voltages, so that at λ = 1 the surface shows its optimal properties. For simplicity, we assume that h = 0, without losing the generality of the subject. Starting from a flat interface (one of the possible initial conditions), we con- jecture that a scaling of space by factor b and of time by factor bz (z is the dynamical scaling exponent), rescales the rough- ness w by factor bχ as follows: w(bL, bzt, λ) = bχ(λ)w(L, t, λ) (3) which implies that w(L, t, λ) = Lχ(λ)f(t/Lz, λ). (4) If for large t and fixed L (t/Lz → ∞) w saturate, then f(x, λ) −→ g(λ), as x −→ ∞. However, for fixed large L and t << Lz , one expects that correlations of the height fluctuations are set up only within a distance t1/z and thus must be independent of L. This implies that for x << 1, f(x) ∼ xβg′(λ) with β = χ/z. Thus dynamic scaling postu- lates that w(L, t, λ) = tβ(λ)g(λ) ∼ tβ(λ), t≪ Lz; Lχ(λ)g′(λ) ∼ Lχ(λ), t≫ Lz. The roughness exponent χ and the dynamic exponent z char- acterize the self-affine geometry of the surface and its dynam- ics, respectively. The dependence of the roughness w to the h or t shows that w has a fixed value for a given time. The common procedure to measure the roughness exponent of a rough surface is use of a surface structure function de- pending on the length scale △x = r which is defined as: S(r) = 〈|h(x+ r) − h(x)|2〉. (6) It is equivalent to the statistics of height-height correlation function C(r) for stationary surfaces, i.e. S(r) = 2w2(1 − C(r)). The second order structure function S(r), scales with r as rξ2 where χ = ξ2/2 [1]. B. The Markov nature of height fluctuations We have examined whether the data of height fluctuations follow a Markov chain and, if so, determine the Markov length scale lM . As is well-known, a given process with a degree of randomness or stochasticity may have a finite or an infi- nite Markov length scale9. The Markov length scale is the minimum length interval over which the data can be consid- ered as a Markov process. To determine the Markov length scale lM , we note that a complete characterization of the sta- tistical properties of random fluctuations of a quantity h in terms of a parameter x requires evaluation of the joint PDF, i.e. PN (h1, x1; ....;hN , xN ), for any arbitrary N . If the pro- cess is a Markov process (a process without memory), an im- portant simplification arises. For this type of process, PN can be generated by a product of the conditional probabili- ties P (hi+1, xi+1|hi, xi), for i = 1, ..., N − 1. As a nec- essary condition for being a Markov process, the Chapman- Kolmogorov equation, P (h2, x2|h1, x1) = d(hi)P (h2, x2|hi, xi)P (hi, xi|h1, x1) (7) should hold for any value of xi, in the interval x2 < xi < The simplest way to determine lM for stationary or homo- geneous data is the numerical calculation of the quantity, S = |P (h2, x2|h1, x1)− dh3P (h2, x2|h3, x3)P (h3, x3|h1, x1)|, for given h1 and h2, in terms of, for example, x3 − x1 and considering the possible errors in estimating S. Then, lM = x3 − x1 for that value of x3 − x1 such that, S = 0. It is well-known that the Chapman-Kolmogorov equation yields an evolution equation for the change of the distribu- tion function P (h, x) across the scales x. The Chapman- Kolmogorov equation formulated in differential form yields a master equation, which can take the form of a Fokker-Planck equation15: P (h, x) = [− D(1)(h, x) + D(2)(h, x)]P (h, x).(8) The drift and diffusion coefficients D(1)(h, r), D(2)(h, r) can be estimated directly from the data and the moments M (k) of the conditional probability distributions: D(k)(h, x) = limr→0M M (k) = dh′(h′ − h)kP (h′, x+ r|h, x). (9) The coefficients D(k)(h, x)‘s are known as Kramers-Moyal coefficients. According to Pawula‘s theorem15, the Kramers- Moyal expansion stops after the second term, provided that the fourth order coefficient D(4)(h, x) vanishes15. The forth order coefficients D(4) in our analysis was found to be about D(4) ≃ 10−4D(2). In this approximation, we can ignore the coefficients D(n) for n ≥ 3. Now, analogous to equation (8), we can write a Fokker- Planck equation for the PDF of h which is equivalent to the following Langevin equation (using the Ito interpretation)15: h(x, λ) = D(1)(h, x, λ) + D(2)(h, x, λ)f(x) (10) where f(x) is a random force, zero mean with gaussian statis- tics, δ-correlated in x, i.e. 〈f(x)f(x′)〉 = δ(x− x′). Further- more, with this last expression, it becomes clear that we are able to separate the deterministic and the noisy components of the surface height fluctuations in terms of the coefficients D(1) and D(2). C. The level crossing analysis We have utilized the level crossing analysis in the context of surface growth processes, according to19,20. In the level crossing analysis, we are interested in determining the aver- age frequency (in spatial dimension) of observing of the defi- nite value for height function h = α in the thin films grown at different bias voltages, ν+α (λ). Then, the average number of visiting the height h = α with positive slope in a sample with size L will be N+α (λ) = ν α (λ)L. It can be shown that the ν+α can be written in terms of the joint PDF of h and its gra- dient. Therefore, the quantity ν+α carry the whole information of surface which lies in P (h, h′), where h′ = dh/dx, from which we get the following result for the frequency parameter ν+α in terms of the joint probability density function ν+α = p(α, h′)h′dh′. (11) The quantity N+tot which is defined as N tot = ν+α dα will measure the total number of crossing the surface with positive slope. So, the N+tot and square area of growing surface are in the same order. Concerning this, it can be utilized as an- other quantity to study further the roughness of a surface. It is expected that in the stationary state the N+tot depends on bias voltages. IV. RESULTS AND DISCUSSION To study the effect of the bias voltage on the surface sta- tistical characteristics, we have utilized AFM method for ob- taining microstructural data from the Co layer deposited at the different bias voltages in the Co/NiO/Si(100) system. Figure 1 shows AFM micrographs of the Co layer deposited at various negative bias voltages of -20, -40, -60, and -80 V, as compared with the unbiased samples. For the unbiased very thin Co layer, Fig. 1a shows a colum- nar structure of the Co grains grown over the evaporated NiO underlaying surface. However, Fig. 1b shows that by applying the negative bias voltage during the Co deposition, the colum- nar growth is eliminated. Moreover, Figs. 1c and 1d show that by increasing the bias voltage up to -60 V the grain size of the Co layer is increased which means a more uniform and smoother surface is formed. But, for the bias voltage of -80 V, due to initiation of resputtering of the Co surface by the high energy ion bombardment, we have observed a non-uniform surface, even at the macroscale of the samples. Therefore, based on the AFM micrographs, the optimum surface mor- phology of the Co/NiO/Si(100) system was achieved at the bias voltage of -60 V for our experimental conditions21. Now, by using the introduced statistical parameters in the last section, it is possible to obtain some quantitative infor- mation about the effect of bias voltage on surface topography of the Co/NiO/Si(100) system. Figure 2 presents the struc- ture function S(r) of the surface grown at the different bias voltages, using Eq. (6). The slope of each curve at the small scales yields the roughness exponent (χ) of the correspond- ing surface. Hence, it is seen that the surface grown at the optimum bias voltage (-60 V) shows a minimum roughness with χ = 0.60, as compared with the other biased samples with χ =0.75, 0.70, and 0.64 for V=-20, -40, and -80 V, re- spectively. For the unbiased sample, we have obtained two roughness exponent values of 0.73 and 0.36, because of the non-isotropic structure of the surface (see Fig. 1a). In any case, at large scales where the structure function is saturated, Fig. 2 shows the maximum and the minimum roughness val- ues for the bias voltages of 0 and -60 V, respectively. It is also possible to evaluate the grain size dependence to the applied bias voltage, using the correlation length achieved by the structure function represented in Fig. 2. For the unbi- ased sample, we have two correlation lengths of 30 and 120 nm due to the columnar structure of the grains. However, by applying the bias voltage, we can attribute just one correla- tion length to each curve showing elimination of the columnar structure in the biased samples. For the bias voltage of -20 V, the correlation length of r∗ is found to be 56 nm. By increas- ing the value of the bias voltage to -40 and -60 V, we have measured r∗=76 and 95 nm, respectively. However, at -80 V, due to initiation of the destructive effects of the high energy ions on the surface, the correlation length is reduced to 76 nm. Now, based on the above analysis, if we assume that Vopt=-60 V, then the roughness exponent and the correlation length can be expressed in terms of the λ as follows, respectively, χ(λ) = 0.61 + 0.16 sin2(2πλ/3.31 + 1.07) (12) FIG. 1: (Color online) AFM surface images (all 1× 1µm2) of Co(3 nm)/NiO(30 nm)/Si(100) thin films deposited at the bias voltages of a) 0, b) -20, c) -40, and d) -60 V. (from top to bottom corresponding a to d, respectively) r (nm) 50 100 150 200 FIG. 2: (Color online) Log-Log plot of structure function of the sur- face at different bias voltages. r∗(λ) = 53.50 + 40.23 sin2(2πλ/3.00 + 2.62)(nm) (13) where for λ = 0 with the columnar structure, we have consid- ered the average values. To obtain the stochastic behavior of the surface, we need to measure the drift coefficient D(1)(h) and diffusion coefficient D(2)(h) using Eq. (9). Figure 3 shows D(1)(h) for the sur- faces at the different bias voltages. It can be seen that the drift coefficient shows a linear behavior for h as: D(1)(h, λ) = −f (1)(λ)h (14) where f (1)(λ) = [0.55 + 1.30 sin2(2πλ/3.50 + 1.40)]× 10−4.(15) The minimum value of f (1)(λ) for the biased samples at λ = 1 shows that the deterministic component of the height fluctuations for these samples is lower than the other biased and unbiased ones. Figure 4 presents D(2)(h) for the differ- ent bias voltages. At λ = 0, the maximum value of diffusion has been obtained for any h, as compared with the other cases. By increasing the bias voltage, the value of D(2) is decreased, as can be seen for λ = 1/3 and 2/3. The minimum value of D(2), which is nearly independent of h, is achieved when λ = 1. This shows that the noisy component of the surface height fluctuation at λ = 1 is negligible as compared with the unbiased and the other biased samples. The behavior of D(2) at λ = 4/3 becomes similar to its behavior at λ = 2/3. It is seen that the diffusion coefficient D(2) is approximately a quadratic function of h. Using the data analysis, we have found that D(2)(h, λ) = f (2)(λ)h2 (16) -30 -20 -10 0 10 20 30 -0.04 -0.02 FIG. 3: (Color online) Drift coefficient of the surface at different bias voltages. FIG. 4: (Color online) Diffusion coefficient of the surface at different bias voltages. where f (2)(λ) = [3.20 + 3.53 sin2(2πλ/3.33 + 1.34)]× 10−6 Now, using the Langevin equation (Eq. (10)) and the mea- sured drift and diffusion coefficients, we can conclude that the height fluctuation has the minimum value at λ = 1 which means a smoother surface at the optimum condition. More- over, the obtained equations for the coefficients (Eqs. (14) and h (A) -20 0 20 FIG. 5: (Color online) Level crossing of the surface at different bias voltages. (16)) can be used to regenerate the rough surfaces the same as AFM images shown in Fig. 17,8. To complete the study, roughness of a surface can be also evaluated by the level crossing analysis, as another procedure. Figure 5 shows the observed average frequency ν+α as a func- tion of h for the different bias voltages. As λ is increased from 0 to 1, the value of ν+α is decreased at any height. Once again, the optimum situation is observed for the bias voltage of -60 V showing the surface formed at λ = 1 condition is a smoother surface with lower height fluctuations than the surface formed at the other conditions. It is seen that, at λ = 4/3, the height fluctuation of the surface finds a maximum value, as compared with the other surfaces. The same as the roughness exponent and the correlation length behavior in terms of λ, the N+tot can be also expressed as: N+tot(λ) = [1.20 + 0.17 sin 2(2πλ/3.52 + 1.40)]. (18) Since the system under investigation has a thin Co layer which is the only conductive layer, thus, it is obvious that lower height fluctuation corresponds to smaller electrical re- sistivity of the surface. Concerning this, we have measured sheet resistance of the Co surface grown at the different bias voltages, as shown in Fig. 6. For the bias voltage ranging from 0 to -60 V, the Rs value is reduced from 432 to 131 Ω/sq.. The minimum value of Rs is measured at the optimum condition of -60 V (λ = 1) which can be related to modified and smooth surface roughness. Elimination of interfacial voids as well as porosities, and reduction of impurities in the Co layer. A sim- ilar behavior was also observed at Vopt=-50 V for Ta/Si(111) system18. By increasing the applied bias voltage to values greater than its optimum value, surface roughness is increased because of surface bombardment by high energy ions. This can be seen by the observed increase in the Rs value at the 0 0.5 1 1.5 FIG. 6: (Color online) Sheet resistance measurement of the Co thin layer as a function of the applied bias voltage. bias voltage of -80 V (λ = 4/3). It is easy to examine that the variation of Rs as a function of λ can be expressed as below: Rs(λ) = [135.48 + 307.74 sin 2(2πλ/3.93 + 1.77)] (19) It behaves similar to the behavior of roughness characteristics of the surfaces. Therefore, we have shown that the roughness behavior explained by the statistical characterizations of the surface, which have been obtained by using microstructural analysis of AFM, can be related to the sheet resistance mea- surement of rough surfaces, as a macrostructural analysis. V. THE TIP CONVOLUTION EFFECT It is well-known that images acquired with AFM are a con- volution of tip and sample interaction. In fact, using scanning probe techniques for determining scaling parameters of a sur- face leads to an underestimate of the actual scaling dimension, due to the dilation of tip and surface. Concerning this, Aue and Hosson22 showed that the underestimation of the scaling exponent depends on the shape and aspect ratio of the tip, the actual fractal dimension of the surface, and its lateralvertical ratio. In general, they proved that the aspect ratio of the tip is the limiting factor in the imaging process. Here, we want to study the aspect ratio effect of the tip on the investigated stochastic parameters. To do this, using a computer simulation program, we have generated a rough surface by using a Brownian motion type algorithm23,24 with roughness and its exponent of 10.00 nm and 0.67, respectively. We have assumed these roughness parameters in order to have some similarity between the generated surface and our ana- lyzed surface by AFM. In the simulation program, the gener- ated surface has been scanned using a sharp cone tip with an 250 500 750 1000 -1500 -1000 FIG. 7: (Color online) Height profile of a rough generated surface before dilation (real) and after dilation (tip) caused by a tip with the aspect ratio of 0.73. assumed aspect ratio of 0.73 which is also nearly similar to the applied tip in our AFM analysis with the aspect ratio of 0.9. Moreover, this assumption does not limit the generality of our discussion, because it is shown that the fractal behav- ior of a rough surface presents an independent tip aspect ratio behavior (saturated behavior) for the aspect ratios greater than about 0.422. Figure 7 shows a line profile of a generated surface which is dilated by a tip with the known aspect ratio. It is clearly seen that the scanned image (the image affected by the tip convolution) does not completely show the generated surface topography (real surface). Now, it is possible to study the dependance of the examined surface stochastic parameters on the geometrical characteristic of the tip, i.e. aspect ratio. In this regard, Fig. 8 shows variation of the one- dimensional structure function of the generated rough surface due to the tip convolution effect. It can be seen that by increas- ing the aspect ratio the tip convolution results in obtaining a surface image whit a decreased roughness. Since the aspect ratio of the applied AFM tip was around 0.9, so the measured roughness exponents at the different bias voltages might be corrected by a 1.07 factor. In other words, the relative change (the difference between the real and measured values com- paring the real one) of the roughness exponent is about 7.2%. Moreover, Fig. 8 shows that the correlation length is increased by the tip convolution effect. It should be noted that, in our simulation, we have assumed that the apex of the tip is com- pletely sharp (the tip radius is assumed zero). However, it is well-known that the radius of the pyramidal tips is ∼ 20 nm. Therefore, the real correlation lengths are even roughly 20 nm larger than the measured ones by the sharp tip. The same tip convolution effect can be also presented for the drift and diffusion coefficients. Figure 9 presents the 100 101 102 tip 40 FIG. 8: (Color online) The one-dimensional structure function anal- ysis, plotting log[S(r)] vs. log(r) in which r is pixel position along the x-axis. This results in the roughness values of 10 and 7.36 nm for the generated surface before dilation and after dilation using a tip with the aspect ratios of 0.73, respectively. -1 0 1 -0.05 tip 40 FIG. 9: (Color online) The calculated drift coefficient for the gener- ated surface before dilation (real) and after dilation with a tip having aspect ratio of 0.73 (tip). calculated drift coefficient for the generated surface and the scanned surface. One can see that the tip convolution results in decreasing of the drift coefficient, corresponding to decreas- ing of the surface roughness. This means that after dilation the correlation length will increase and hence the measured value for f (1)(λ) would be smaller than its value for the original sur- -1 0 1 0.0005 0.001 0.0015 0.002 tip 40 FIG. 10: (Color online) The calculated diffusion coefficient for the generated surface before dilation (real) and after dilation with a tip having aspect ratio of 0.73 (tip). -3000 -2000 -1000 0 1000 2000 3000 10000 12000 14000 tip 40 FIG. 11: (Color online) Level crossing analysis of the generated sur- face before dilation (real) and after dilation (tip). face. Therefore, the magnitude of slope of the drift coefficient must decrease after using the tip. For our generated surface, the measured value of the drift coefficient should be modified by a factor of around 2. The variation of the diffusion coefficient of the generated rough surface due to the tip convolution effect has been also shown in Fig. 10. The reduction of the diffusion coefficient of the scanned surface as compared to its values for the generated surface, due to the tip convolution, can be easily seen. In fact to compensate the tip effect on the diffusion coefficient, we should modify its measured values by a factor of about 4, for the assumed generated surface. Finally, we remind that the total number of crossing the sur- face with positive slope (N+tot) has been defined as a parame- ter describing the rough surfaces. Hence, we have also studied the effect of the tip convolution on this parameter, as shown in Fig. 11. It is seen that N+tot decreased due to the tip con- volution effect. For the assumed generated surface, we have obtained that the N+tot of the surface before dilation is about 1.7 times larger than its value after the dilation. In this figure, we have also shown the variation of the average height due to the tip effect. One has to note that our generated surface is a pure two- dimensional one which presents no line-to-line interaction. So, for this simple model, differently shaped tips with the same aspect ratio yield the same results. Therefore, for the three-dimensional case one can expect to obtain a larger dis- tortion of the surface due to stronger line-to-line interac- tion leading to an even larger underestimation of the studied stochastic parameters. These analysis showed that, although the measured values of the surface parameters by AFM method are different from the real ones, the general behavior of these parameters as a function of the bias voltage are not affected by the tip convo- lution. Therefore, our general conclusions about the variation of the studied stochastic parameters by applying the bias volt- age is intact. VI. CONCLUSIONS We have investigated the role of bias voltage, as an exter- nal parameter, to control the statistical properties of a rough surface. It is shown that at an optimum bias voltage (λ = 1), the stochastic parameters describing a rough surface such as roughness exponent, level crossing, drift and diffusion coeffi- cient must be found in their minimum values as compared to an unbiased and the other biased samples. In fact, dependence of the height fluctuation of a rough surface to different kinds of the external control parameters, such as bias voltage, tem- perature, pressure, and so on, can be expressed by AFM data which are analyzed using the surface stochastic parameters. In addition, this characterization enable us to regenerate the rough surfaces grown at the different controlled conditions, with the same statistical properties in the considered scales, which can be useful in computer simulation of physical phe- nomena at surfaces and interfaces of, especially, very thin lay- ers. It is also shown that these statistical and microstructural parameters can explain well the macroscopic properties of a surface, such as sheet resistance. Moreover, we have shown that the tip-sample interaction does not change the physical behavior of the stochastic parameters affected by the bias volt- Acknowledgments AZM would like to thank Research Council of Sharif Uni- versity of Technology for financial support of this work. We also thank F. Ghasemi for useful discussions. 1 A.L. Barabasi and H.E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, New York, 1995). 2 T. Halpin-Healy and Y.C. Zhang, Phys. Rep. 254, 218 (1995); J. Krug, Adv. Phys. 46, 139 (1997). 3 J. Krug and H. Spohn “In Solids Far From Equilibrium Growth, Morphology and Defects”, edited by C. Godreche (Cambridge University Press, New York, 1990). 4 P. Meakin Fractals, Scaling and Growth Far From Equilibrium (Cambridge University Press, Cambridge, 1998). 5 M. Kardar, Physica A 281, 295 (2000). 6 A.A. Masoudi, F. Shahbazi, J. Davoudi, and M. Reza Rahimi Tabar, Phys. Rev. E 65, 026132 (2002). 7 G.R. Jafari, S.M. Fazeli, F. Ghasemi, S.M. Vaez Allaei, M. Reza Rahimi Tabar, A. Irajizad, and G. Kavei, Phys. Rev. Lett. 91, 226101 (2003). 8 M. Waechter, F. Riess, Th. Schimmel, U. Wendt and J. Peinke, Eur. Phys. J. B 41, 259-277 (2004). 9 J. Peinke, M. Reza Rahimi Tabar, and M. Sahimi, Phys. Rev. Lett., submitted, (2004). 10 M. Ohring, Materials Science of Thin Films: Deposition and Structure (Academic Press, New York, 2002). 11 W.F. Egelhoff, Jr., P.J. Chen, C.J. Powell, M.D. Stiles, R.D. McMichael, C.L. Lin, J.M. Sivertsen, J.H. Judy, K. Takano, A.E. Berkowitz, T.C. Anthony, and J.A. Brug, J. Appl. Phys. 79, 5277 (1996). 12 B. Dai, J.N. Coi, and W. Lai, J. Magn. Magn. Mat. 27, 19 (2003). 13 A.Z. Moshfegh and O. Akhavan, J. Phys. D 34, 2103 (2001). 14 M. Marsilli, A. Maritan, F. Toigoend, and J.R. Banavar, Rev. Mod. Phys. 68, 963 (1996). 15 H. Risken, The Fokker-Planck equation (Springer, Berlin, 1984). 16 J. Davoudi and M. Reza Rahimi Tabar, Phys. Rev. Lett. 82, 1680 (1999). 17 A. Irajizad, G. Kavei, M. Reza Rahimi Tabar, and S.M. Vaez Al- laei, J. Phys.: Condens. Matter 15, 1889 (2003). 18 A.Z. Moshfegh and O. Akhavan, Thin Solid Films 370, 10 (2000). 19 S.N. Majumdar and A.J. Bray, Phys. Rev. Lett. 86, 3700 (2001). 20 F. Shahbazi, S. Sobhanian, M. Reza Rahimi Tabar, S. Khorram, G.R. Frootan, and H. Zahed, J. Phys. A 36, 2517 (2003). 21 A.Z. Moshfegh and P. Sangpour, Phys. Stat. Sol. C 1, 1744 (2004). 22 J. Aue and J.Th.M. De Hosson, Appl. Phys. Lett. 71, 1347 (1997). 23 H. Peitgen and D. Saupe, The Science of Fractal Images (Springer, Berlin, 1988). 24 J. Peinke, M. Reza Rahimi Tabar, M. sahimi and F. Ghasemi, cond-mat/0411529, (2004). http://arxiv.org/abs/cond-mat/0411529 Introduction Experimental Statistical quantities roughness exponents The Markov nature of height fluctuations The level crossing analysis Results and discussion The tip convolution effect Conclusions Acknowledgments References

0704.1030

Etched Glass Surfaces, Atomic Force Microscopy and Stochastic Analysis

Etched Glass Surfaces, Atomic Force Microscopy and Stochastic Analysis G. R. Jafari a,b, M. Reza Rahimi Tabar c,d, A. Iraji zad c, G. Kavei f 1 1a Department of Physics, Shahid Beheshti University, Evin, Tehran 19839, Iran b Department of Nano-Science, IPM, P. O. Box 19395-5531, Tehran, Iran c Department of Physics, Sharif University of Technology, P. O. Box 11365-9161, Tehran, Iran d CNRS UMR 6529, Observatoire de la Côte d’Azur, BP 4229, 06304 Nice Cedex 4, France e Material and Energy, Research Center, P.O. Box 14155-4777, Tehran, Iran The effect of etching time scale of glass surface on its statistical properties has been studied using atomic force microscopy technique. We have characterized the complexity of the height fluctuation of a etched surface by the stochastic parameters such as intermittency exponents, roughness, roughness exponents, drift and diffusion coefficients and find their variations in terms of the etching time. PACS numbers: I. INTRODUCTION The complexity of rough surfaces is subject of a large va- riety of investigations in different fields of science1,2. Sur- face roughness has an enormous influence on many important physical phenomena such as contact mechanics, sealing, ad- hesion, friction and self-cleaning paints and glass windows,3,4. A surface roughness of just a few nanometers is enough to re- move the adhesion between clean and (elastically) hard solid surfaces3. The physical and chemical properties of surfaces and interfaces are to a significant degree determined by their topographic structure. The technology of micro fabrication of glass is getting more and more important because glass sub- strates are currently being used to fabricate micro electro me- chanical system (MEMS) devices5. Glass has many advan- tages as a material for MEMS applications, such as good me- chanical and optical properties. It is a high electrical insulator, and it can be easily bonded to silicon substrates at tempera- tures lower than the temperature needed for fusion bonding6. Also micro and nano-structuring of glass surfaces is impor- tant for the production of many components and systems such as gratings, diffractive optical elements, planar wave guide de- vices, micro-fluidic channels and substrates for (bio) chemical applications7. Wet etching is also well developed for some of these applications8,9,10,11,12,13,14. One of the main problems in the rough surface is the scal- ing behavior of the moments of height h and evolution of the probability density function (PDF) of h, i.e. P (h, x) in terms of the length scale x. Recently some authors have been able to obtain a Fokker-Planck equation describing the evolution of the probability distribution function in terms of the length scale, by analyzing some stochastic phenomena, such as rough surfaces15,16,17, turbulent system18, financial data19, cosmic background radiation20 and heart interbeats21 etc. They no- ticed that the conditional probability density of field incre- ment satisfies the Chapman-Kolmogorov equation. Mathe- matically, this is a necessary condition for the fluctuating data to be a Markovian process in the length (time) scales22. In this work, we investigate the etching process as a stochastic process. We measure the intermittency exponents of height structure function, roughness, roughness exponents and Kramers-Moyal‘s (KM) coefficients. Indeed we consider the etching time t, as an external parameter, to control the sta- tistical properties of a rough surface and find their variations with t. It is shown that the first and second KM‘s coefficients have well-defined values, while the third and fourth order co- efficients tend to zero. The first and second KM‘s coefficients for the fluctuations of h(x), enables us to explain the height fluctuation of the etched glass surface. II. EXPERIMENTAL We started with glass microscope slides as a sample. Only one side of samples was etched by HF solution for different etching time (less than 20 minutes). HF concentration was %40 for all the experiments. The surface topography of the etched glass samples in the scale (< 5µm) was obtained us- ing an AFM (Park Scientific Instruments). The images in this scale were collected in a constant force mode and digi- tized into 256×256 pixels. A commercial standard pyramidal Si3N4 tip was used. A variety of scans, each with size L, were recorded at random locations on the surface. Figure 1 shows typical AFM image with resolutions of about 20nm. III. STATISTICAL QUANTITIES A. Multifractal Analysis and the Intermittency Exponent Assuming statistical translational invariance, the structure functions Sq(l) =< |h(x + l) − h(x)|q >, (moments of the increment of the rough surface height fluctuation h(x)) will depend only on the space deference of heights l, and has a power law behavior if the process has the scaling property: Sq(l) =< |h(x + l)− h(x)|q >∝ Sq(L0)( )ξ(q) (1) where L0 is the fixed largest length scale of the system, < ··· > denotes statistical average (for non-overlapping incre- ments of length l), q is the order of the moment (we take here q > 0), and ξ(q) is the exponents of structure function. The http://arxiv.org/abs/0704.1030v1 FIG. 1: AFM surface image of etched glass film with size 5× 5µm2 after 12 minutes. second moment is linked to the slope β of the Fourier power spectrum: β = 1 + ξ2. The main property of a multifractal processes is that it is characterized by a non-linear ξq function verses q. Monofractals are the generic result of this linear be- havior. For instance, for Brownian motion (Bm) ξq = q/2, and for fractional Brownian motion (fBm) ξq ∝ q. B. Roughness and Roughness Exponents It is also known that to derive the quantitative information of the surface morphology one may consider a sample of size L and define the mean height of growing film h and its vari- ance, σ by: σ(L, t) = (〈(h − h)2〉)1/2 (2) where t is etching time and 〈· · · 〉 denotes an averaging over different samples, respectively. Moreover, etching time is a factor which can apply to control the surface roughness of thin films. Let us now calculate also the roughness exponent of the etched glass. Starting from a flat interface (one of the possible initial conditions), it is conjectured that a scaling of space by factor b and of time by factor bz (z is the dynamical scaling exponent), rescales the variance, σ by factor bχ as follows1: σ(bL, bzt) = bασ(L, t) (3) which implies that σ(L, t) = Lαf(t/Lz). (4) If for large t and fixed L (x = t/Lz → ∞) σ saturate. How- ever, for fixed large L and t ≪ Lz , one expects that corre- lations of the height fluctuations are set up only within a dis- tance t1/z and thus must be independent of L. This implies that for x ≪ 1, f(x) ∼ xβ with β = α/z. Thus dynamic scaling postulates that σ(L, t) ∝ tβ , t≪ Lz; Lα, t≫ Lz. The roughness exponent α and the dynamic exponent β char- acterize the self-affine geometry of the surface and its dynam- ics, respectively. The common procedure to measure the roughness exponent of a rough surface is use of the surface structure function de- pending on the length scale l which is defined as: S2(l) = 〈|h(x + l)− h(x)|2〉. (6) It is equivalent to the statistics of height-height correlation function C(l) for stationary surfaces, i.e. S2(l) = 2σ2(1 − C(l)). The second order structure function S(l), scales with l as l2α1. C. The Markov Nature of Height Fluctuations: Drift and Diffusion Coefficients We check whether the data of height fluctuations follow a Markov chain and, if so, measure the Markov length scale lM . As is well-known, a given process with a degree of randomness or stochasticity may have a finite or an infinite Markov length scale23. The Markov length scale is the min- imum length interval over which the data can be considered as a Markov process. To determine the Markov length scale lM , we note that a complete characterization of the statisti- cal properties of random fluctuations of a quantity h in terms of a parameter x requires evaluation of the joint PDF, i.e. PN (h1, x1; ....;hN , xN ), for any arbitrary N . If the process is a Markov process (a process without memory), an im- portant simplification arises. For this type of process, PN can be generated by a product of the conditional probabili- ties P (hi+1, xi+1|hi, xi), for i = 1, ..., N − 1. As a nec- essary condition for being a Markov process, the Chapman- Kolmogorov equation, P (h2, x2|h1, x1) = d(hi)P (h2, x2|hi, xi)P (hi, xi|h1, x1) (7) should hold for any value of xi, in the interval x2 < xi < The simplest way to determine lM for homogeneous sur- face is the numerical calculation of the quantity, S = |P (h2, x2|h1, x1)− dh3P (h2, x2|h3, x3)P (h3, x3|h1, x1)|, for given h1 and h2, in terms of, for example, x3 − x1 and considering the possible errors in estimating S. Then, lM = x3 − x1 for that value of x3 − x1 such that, S = 0 It is well-known, the Chapman-Kolmogorov equation yields an evolution equation for the change of the distribu- tion function P (h, x) across the scales x. The Chapman- Kolmogorov equation formulated in differential form yields a l (nm) 50 100 150 200 FIG. 2: Scaling of the structure functions in log-log plot for mo- ments less than 8. (from bottom to top). master equation, which can take the form of a Fokker-Planck equation22,23: P (h, x) = [− D(1)(h, x) + D(2)(h, x)]P (h, x).(8) The drift and diffusion coefficients D(1)(h, r), D(2)(h, r) can be estimated directly from the data and the moments M (k) of the conditional probability distributions: D(k)(h, x) = limr→0M M (k) = dh′(h′ − h)kP (h′, x+ r|h, x). (9) The coefficients D(k)(h, x)‘s are known as Kramers-Moyal coefficients. According to Pawula‘s theorem22, the Kramers- Moyal expansion stops after the second term, provided that the fourth order coefficient D(4)(h, x) vanishes22. The forth order coefficients D(4) in our analysis was found to be about D(4) ≃ 10−4D(2). In this approximation, we can ignore the coefficients D(n) for n ≥ 3. We note that this Fokker-Planck equation is equivalent to the following Langevin equation (us- ing the Ito interpretation)22: h(x) = D(1)(h, x) + D(2)(h, x)f(x) (10) where f(x) is a random force, zero mean with gaussian statis- tics, δ-correlated in x, i.e. 〈f(x)f(x′)〉 = 2δ(x−x′). Further- more, with this last expression, it becomes clear that we are able to separate the deterministic and the noisy components of the surface height fluctuations in terms of the coefficients D(1) and D(2). 2 4 6 8 10 FIG. 3: The results of scaling exponent ξq which is clearly linear vs. l (nm) 500 1000 6 min 8 min 10 min 12 min 15 min FIG. 4: Log-Log plot of selection structure function of the etched glass surfaces. IV. RESULTS AND DISCUSSION Now, using the introduced statistical parameters in the pre- vious sections, it is possible to obtain some quantitative infor- mation about the effect of etching time on surface topography of the glass surface. To study the effect of the etching time on the surface statistical characteristics, we have utilized AFM imaging technique in order to obtain microstructural data of the etched glass surfaces at the different etching time in the HF. Figure 1 shows the AFM image of etched glass after 12 minuets etched. To investigate the scaling behavior of the mo- ments of δhl = h(x + l) − h(x), we consider the samples that they reached to the stationary state. This means that their statistical properties do not change with time. In our case the -0.2 0 0.2 -0.08 -0.06 -0.04 -0.02 2 min 6 min 8 min 10 min 12 min 15 min FIG. 5: Drift coefficients of the surfaces at different etching time less than 20 minutes. samples with etching time more than 20 minutes are almost stationary. Figure 2 shows the log-log plot of the structure functions verses length scale l for different orders of moments. The straight lines show that the moments of order q have the scaling behavior. We have checked the scaling relation up to moment q = 10. The resulting intermittency exponent ξq is shown in figure 3. It is evident that ξq has a linear behavior. This means that the height fluctuations are mono-fractal be- havior. We also directly estimated the scaling exponent of the linear term lqH/ < (h(x + l)− h(x))q > and obtain the fol- lowing values for the samples with 20 minuets etching time, ξ1 = 0.70± 0.04 and ξ2 = 1.40 ± 0.04. This means etching memorize fractal feature during etching. Therefore using the scaling exponent ξ2 we obtain the roughness exponent α as ξ2/2 = 0.70± 0.04. Figure 4 presents the structure function S(l) of the surface at the different etching time, using equation (6). It is also possible to evaluate the grain size dependence to the etching time, using the correlation length achieved by the structure function represented in figure 4. The correlation lengths increase with etching time. Its value has a exponen- tial behavior 448(1 − exp(−0.15t))nm. Also we find that the dynamical exponent is given by β = 0.6 ± 0.1. Also we measured the variation of the Markov length with etching time t (min), and obtain lM = 40 + 3t (nm) for time scales t < 20min. Finally to obtain the stochastic equation of the height fluc- tuations behavior of the surface, we need to measure the Keramer- Moyal Coefficients. In our analysis the forth order coefficients D(4) is less than Second order coefficients, D(2), about D(4) ≃ 10−4D(2). In this approximation, we ignore the coefficients D(n) for n ≥ 3. So, to discuss the surfaces it just needs to measure the drift coefficient D(1)(h ) and diffusion coefficient D(2)(h ) using Eq. (9). Figures 5 and 6 show the drift coefficient D(1)(h ) and diffusion coefficients D(2)(h for the surfaces at the different etching time, respectively. It -0.2 0 0.2 0.001 0.002 0.003 0.004 2 min 6 min 8 min 10 min 12 min 15 min FIG. 6: Diffused coefficients of the surface at different etching time less than 20 minutes. can be shown that the drift and diffusion coefficients have the following behavior, D(1)( , t) = −f (1)(t) D(2)( , t) = f (2)(t)( )2 (12) The two coefficients f (1)(t) and f (2)(t) increase with the then is saturated. Using the data analysis we obtain that they are linear verses time (min): f (1)(t) = 0.005t and f (2)(t) = 0.0003t for time scales t < 20 min. To better comparing the parameter of samples we divided the heights to their variances. In this case, maximum and minimum of heights are about plus 1 and mines 1, respectively. Compar- ing samples with etching times 2 and 6 minutes, shows f (1) increases 300 percent after 4 minutes (from 2 min to 6 min) from f (1)(t = 2×60) = 0.6 to f (1)(t = 6×60) = 1.8. Also, f (2) is 0.006 and 0.018 after 2 and 6 minutes, respectively. V. CONCLUSIONS We have investigated the role of etching time, as an exter- nal parameter, to control the statistical properties of a rough surface. We have shown that in the saturate state the struc- ture of topography has fractal feature with fractal dimension Df = 1.30. In addition, Langevin characterization of the etched surfaces enable us to regenerate the rough surfaces grown at the different etching time, with the same statistical properties in the considered scales15. VI. ACKNOWLEDGMENT We would like to thank S. M. Mahdavi for his useful com- ments and discussions and Also P. Kaghazchi and M. Shirazi for samples preparation. 1 A.L. Barabasi and H.E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, New York, 1995). 2 S. Davies, P. Hall , J. Roy. Stat. Soc. B 61 (1999) 3. 3 A. G. Peressadko, N. Hosoda, and B. N. J. Persson, PRL 95, 124301 (2005), B N J Persson, O Albohr, U Tartaglino, A I Volok- itin and E Tosatti, J. Phys.: Condens. Matter 17 (2005) R1R62. 4 Zhao Y-P,Wang L S and Yu T X, J. Adhes. Sci. Technol. 17 519 (2003) 5 Won Ick Jang, Chang Auck Choi, Myung Lae Lee, Chi Hoon Jun and Youn Tae Kim, J. Micromech. Microeng. 12 (2002) 297306. 6 M. Bu , T. Melvin a, G. J. Ensell, J. S. Wilkinson, A. G.R. Evans, Sensors and Actuators A 115:pp. 476-482 (2004). 7 Yu-Cheng Lin, Hsiao-Ching Ho, Chien-Kai Tseng and Shao-Qin Hou 2001 J. Micromech. Microeng. 11 189-194 8 D.M. Knotter, J. Am. Chem. Soc. 122 (2000) 4345. 9 G.A.C.M. Spierings, J. Mater. Sci. 28 (1993) 6261. 10 R. Schuitema, et al, Light scattering at rough interfaces of thin film solar cells to improve the efficiency and stability, IEEE/ProRISC99, pp 399-404 (1999). 11 L. B. Glebov, et al, Photo induced chemical etching of silicate and borosilicate glasses, Glasstech. Ber. Glass Sci. Technol. 75 C2 pp 298 - 301 (2002). 12 G. R. Jafari, S. M. Mahdavi, A. Iraji zad, and P. Kaghazchi, Sur- face And Interface Analysis; 37: 641 645 (2005). 13 N. Silikas, k.E.R. England, D.C Wattes, K.D Jandt, J. Dentistry 27 (1999) 137. 14 A. Irajizad, G. Kavei, M. Reza Rahimi Tabar, and S.M. Vaez Al- laei, J. Phys.: Condens. Matter 15, 1889 (2003). 15 G.R. Jafari, S.M. Fazeli, F. Ghasemi, S.M. Vaez Allaei, M. Reza Rahimi Tabar, A. Irajizad, and G. Kavei, Phys. Rev. Lett. 91, 226101 (2003). 16 M. Waechter, F. Riess, Th. Schimmel, U. Wendt and J. Peinke, Eur. Phys. J. B 41, 259-277 (2004). 17 P. Sangpour, G. R. Jafari, O. Akhavan, A.Z. Moshfegh, and M. Reza Rahimi Tabar, Phys. Rev. B 71, 155423 (2005). 18 Christoph Renner, Joachim Peinke, and Rudolf Friedrich, Journal of Fluid Mechanics, 433:383409, 2001. 19 Ch. Renner, J. Peinke, R. Friedrich, Physica A 298, 499 (2001). 20 F. Ghasemi, A. Bahraminasab, S. Rahvar, and M. Reza Rahimi Tabar, Preprint arxiv:astro-phy/0312227, 2003. 21 F. Ghasemi, J. Peinke, M. Sahimi and M. Reza Rahimi Tabar, Eur. Phys. J. B 47, 411(2005) 22 H. Risken, The Fokker-Planck equation (Springer, Berlin, 1984). 23 R. Friedrich, J. Zeller, and J. Peinke, Europhysics Letters 41, 153 (1998). http://arxiv.org/abs/astro-phy/0312227

0704.1031

Measurements of Single and Double Spin Asymmetry in \textit{pp} Elastic Scattering in the CNI Region with Polarized Hydrogen Gas Jet Target

Measurements of Single and Double Spin Asymmetry in pp Elastic Scattering in the CNI Region with Polarized Hydrogen Gas Jet Target H. Okada∗,†, I. Alekseev∗∗, A. Bravar‡, G. Bunce§,¶, S. Dhawan‖, K.O. Eyser††, R. Gill§, W. Haeberli‡‡, H. Huang§, O. Jinnouchi§§, Y. Makdishi§, I. Nakagawa¶¶, A. Nass∗∗∗, N. Saito∗,§§, E. Stephenson†††, D. Sviridia∗∗, T. Wise‡‡, J. Wood§ and A. Zelenski§ ∗Kyoto University, Kyoto Japan †RIKEN, Wako, JAPAN ∗∗Institute for Theoretical and Experimental Physics (ITEP), 117259 Moscow, Russia ‡University of Geneva , 1205 Geneva, Switzerland §Brookhaven National Laboratory, Upton, NY 11973, USA ¶RIKEN BNL Research Center, Upton, NY 11973, USA ‖Yale University, New Haven, CT 06520, USA ††University of California, Riverside, CA 92521, USA ‡‡University of Wisconsin, Madison, WI 53706, USA §§KEK, Tukuba, Japan ¶¶RIKEN, Wako, JAPAN ∗∗∗University of Erlangen, 91058 Erlangen, Germany †††Indiana University Cyclotron Facility, Bloomington, IN 47408, USA Abstract. Precise measurements of the single spin asymmetry, AN and the double spin asymmetry, ANN , in proton-proton (pp) elastic scattering in the region of four-momentum transfer squared 0.001 < −t < 0.032 (GeV/c)2 have been performed using a polarized atomic hydrogen gas jet target and the RHIC polarized proton beam at 24 GeV/c and 100 GeV/c. The polarized gaseous proton target allowed us to achieve the measurement of ANN in the CNI region for the first time. Our results of AN and ANN provide significant constraints to determine the magnitude of poorly known hadronic single and double spin-flip amplitudes at this energy. Keywords: Elastic scattering, spin, coulomb nuclear interference PACS: 13.88.+e, 13.85.Dz, 29.27.Pj, 29.27.Hj Introduction. pp elastic scattering is one of the most fundamental reactions in particle-nuclear physics and is described in transition amplitudes by use of helicity of initial and final states. Requiring that the interaction is invariant under space inversion, time reversal and rotation in spin space, pp scattering in a given spin state is described in five independent transition amplitudes (φi, i = 1− 5) as functions of the center-of- mass energy squared, s, and t [1]. The understanding of these amplitudes would provide crucial guidelines to investigate the reaction mechanism. Each transition amplitude is described as a sum of the hadronic amplitude (φ hadi ) and the electro-magnetic amplitude (φ emi ). In the small −t region, φ emi and φ hadi become similar in strength and interfere with each other. We call this interference the Coulomb Nuclear Interference (CNI). Thanks to the great successes of QED and the past pre- http://arxiv.org/abs/0704.1031v1 RHIC proton beam Forward scattered proton proton target recoil proton recoil proton measure!measure! ( ) 0Tm2ppt Rp2inout <-=-= -t (GeV/c) -310 -210 -0.01 FIGURE 1. Left: Example of "parallel" case, p↑p↑ → pp. Right: εNN , εN and εb at s = 13.7 GeV as a function of −t. cisely measured quantities (ex. magnetic moment), φ emi is precisely described. On the other hand, φ hadi is not fully described by theory, because the perturbative QCD is not applicable in the CNI region. By use of the experimental data of total and differen- tial cross-section of unpolarized pp elastic scattering, we can determine the sum of two non-spin-flip hadronic amplitudes (φ had+ = φ 3 ) [3]. In order to approach to the hadronic single and double spin-flip amplitudes (φ had5 and φ 2 ), we measure AN and ANN in the CNI region. AN is defined by the asymmetry of cross-section with up-down transverse polarization for one of the protons. Similarly ANN is defined by the asymme- try of cross-section for parallel and anti-parallel transverse polarization for both of the protons. The left side of Fig. 1 depicts "parallel" case. We define the scattering plane from 3-momenta of incident and recoil particles, which is normal to the spin directions. By use of transition amplitudes, AN is expressed as, −Im[φ em5 (s, t)φ + (s, t)+φ 5 (s, t)φ + (s, t)] |φ+(s, t)|2 . (1) The first term of Eq. 1 is calculable and has a peak around −t ≃ 0.003 (GeV/c)2 [2] which is generated by proton’s anomalous magnetic moment. Because the presence of φ had5 introduces a deviation in shape and magnitude from the first term, a measurement of AN in the CNI region, therefore, can be a sensitive probe for φ had5 . ANN is expressed as, ANN ≈ 2|φ had5 (s, t)| 2+Re[(φ+(s, t))∗φ had2 (s, t)] |φ+(s, t)|2 . (2) Because the first term is 2nd order of φ had5 and the second term is 1st order of φ 2 , ANN is sensitive to φ had2 [4]. From a consequence of angular momentum conservation at small −t and large s, we use φ had4 ∝ t → 0 for these expressions. Experiment. Experiment has been performed using a polarized hydrogen gas jet target and polarized RHIC proton beam at 24 GeV/c ( s = 6.7 GeV) and 100 GeV/c s = 13.7 GeV). We detect the recoil protons by silicon detectors which are located on both sides of the target. The details of experimental setup are described in [5, 6]. In the pp elastic scattering process, both forward-scattered particle and recoil particle are protons and there are no other particles involved nor new particles produced in the process. Since initial states are well defined, the elastic process can be, in principle, identified by detecting the recoil particle only. By measuring kinetic energy TR, time of flight, and recoil angle of recoil particle, we measure the mass of recoil particle and all the forward scattered rest particles. We collected 4.3 M events at s = 13.7 GeV and 0.8 M events at s = 6.7 GeV, respectively. The details of event selection is described in [5]. The selected event yield is sorted by −t bins, which is obtained measuring the kinetic energy of the recoil particle: −t = 2mpTR, and spin states. mp is the proton mass. Then we calculate two types of single spin raw asymmetries, εN for the target spin state and εb for the beam spin state. We also calculate double spin raw asymmetry, εNN for the target and beam spin states. The right side of Fig. 1 displays these raw asymmetries of s = 13.7 GeV data as a function of −t in the region 0.001 ≤ −t ≤ 0.035 (GeV/c)2 (0.5 ≤ TR ≤ 17 MeV). The polarized gaseous proton target allowed us to achieve the measurement in the CNI region for the first time. In order to cancel out the asymmetries of up-down luminosity, and detector acceptance, we employed so-called "square root formula" for εN and εb calculation. On the other hand, εNN needs to be corrected by the luminosity asymmetry. The target spin flips every 5 minutes and the density of both spin states are the same and stable during the experimental period (∼ 90 hours). The beam intensity , which is measured by the wall current monitor [7], varies by bunch (every 106 nsec in 2004) and fill (every several hours). By accumulating intensity for the experimental period, the variation is compensated. Therefore the luminosity asymmetry is quite small compared to statistical error of εNN . Results and discussions. AN is measured normalizing εN by well measured the target polarization Pt [6], . (3) Utilizing the measured AN , we also measure the beam polarization, Pb = εb/AN 1. Normalizing εNN by Pt and Pb, ANN is obtained via ANN = . (4) The left and right plots of Fig. 2 display the results of AN and ANN at s = 6.7 GeV with filled circles and 13.7 GeV with open circles, respectively. The errors on the data 1 This experimental setup also plays an important role in the RHIC spin program to measure the absolute beam polarization [8]. -t (GeV/c) -310 -210 =6.7 GeVs =13.7 GeVs = 6.7 GeVsNo had. func. = 13.7 GeVsNo had. func. -t (GeV/c) -310 -210 -0.06 -0.04 -0.02 =6.7 GeVs =13.7 GeVs FIGURE 2. The left and right plots display the results of AN and ANN at s = 6.7 GeV with filled circles and 13.7 GeV with open circles, respectively. The errors on the data points are statistical. The lower bands represents the total systematic errors. The solid and dashed lines in the left plots correspond to the first term in Eq. 1 for these s, respectively. points are statistical. The lower bands represent the total systematic errors. The solid and dashed lines correspond to the first term in Eq. 1 for these s, respectively. AN at s = 13.7 GeV are consistent with the dashed line (χ2/ndf=13.4/14). On the other hand, although the accuracy is statistically limited, AN at s = 6.7 GeV are not consistent with the solid line (χ2/ndf=35.5/9) and this discrepancy implies the presence of φ had5 . ANN for these s have no clear −t dependence and the average values are consistent with zero within 1.5 σ . In summary, measurements of AN and ANN provide experimental knowledge to poorly known φ had5 and φ 2 . The s dependence of φ had5 is provided by AN results and the theoretical interpretation is under way [9]. However, there is no comprehensive understanding of φ had2 and φ 5 yet. Further measurements at different s are required to fully describe the behavior of φ had2 and φ REFERENCES 1. M. Jacob and G.C. Wick , Annals Phys. 7, 404 (1959). 2. N.H. Buttimore et al., Phys. Rev. D 59, 114010 (1999). 3. M.M. Block and R.N. Cahn, Czech. J. Phys. 40, 164-175 (1990). 4. T.L. Trueman, RHIC Spin Note, September 27, (2005), hep-ph/0604153. 5. H. Okada et al., Phys. Lett. B 638, 450 (2006); H. Okada, Doctororal thesis, July (2006); http://www.star.bnl.gov/∼hiromi/HiromiOkadaThesis.pdf 6. A. Zelenski et al., Nucl. Inst. and Meth. A 536, 248 (2005); T. Wise et al., Nucl. Inst. and Meth. A 559, 1 (2006). 7. P.R. Cameron et al., Nucl.Instrum.Meth. A345, 226-229 (1994). 8. K.O. Eyser, these proceedings (2006). 9. Private discussion with L. Trueman. http://arxiv.org/abs/hep-ph/0604153 http://www.star.bnl.gov/~hiromi/HiromiOkadaThesis.pdf

0704.1032

Theta constants identities for Jacobians of cyclic 3-sheeted covers of the sphere and representations of the symmetric group

THETA CONSTANTS IDENTITIES FOR JACOBIANS OF CYCLIC 3-SHEETED COVERS OF THE SPHERE AND REPRESENTATIONS OF THE SYMMETRIC GROUP BY YAACOV KOPELIOVICH To my friend Elizabeth Drake Abstract. We find identities between theta constants with rational charac- teristics evaluated at period matrix of R, a cyclic 3 sheeted cover of the sphere with 3k branch points λ1...λ3k. These identities follow from Thomae formula [BR]. This formula expresses powers of theta constants as polynomials in λ1...λ3k. We apply the representation of the symmetric group to find relations between the polynomials and hence between the associated theta constants. 1. Introduction Let R be a Riemann surface with the equation: (z − λi)(∗). We find relations that are satisfied by theta constants with rational characteristics evaluated at τR, the period matrix of R. Special type identities for period matrices are known in the case of a general Riemann surface ( Schottky-Jung identities). For hyperelliptic curves there are vanishing theta constants of even characteristics that characterize the associated period matrix. According to Mumford, [Mu] spe- cial relations of non vanishing of theta constants evaluated at period matrices of hyperelliptic curves were obtained by Frobenius. The original Schottky problem seeks special relations among theta constants that characterize the entire moduli space of algebraic curves of genus g. In this note we seek special relations that are satisfied by n-sheeted cyclic covers of the sphere. When n = 2 cyclic covers are just hyperelliptic curves. The next case is n = 3 and we find relations between theta constants with rational characteristics evaluated at τR the period matrices of such curves . These identities are a result of Thomae formula for cyclic n sheeted covers of the sphere. This formula expresses powers of such theta constants evaluated at the period matrix τR through polynomial expression of λi. A relation between these polynomials produces a relation between associated theta constants. Applying the representation theory of the symmetric group, S3m we produce a basis for the vector space spanned by the polynomials and as a result relations between the associated theta constants. For the simplest case of 6 branch points our results overlap with results of Mat- sumoto [Ma]. In his paper Matsumoto finds the explicit action of S6 on theta Date: 04 April 2007. http://arxiv.org/abs/0704.1032v1 2 KOPELIOVICH constants evaluated at τR and expresses branch points λi as rational functions of theta constants. As a result he writes identities between cubic powers of these constants which essentially coincide with the identities obtained by us in the last section of our note. Using the representation theory of S6 we see that the space generated by theta constants is 5 dimensional. This seems to be a new result even in this case. We note that the Algebraic dimension of this particular family of curves is 3. This work was partially done during a visit to the TAMU math department and the author thanks the department for the invitation and kind hospitality. I thank Samuel Grushevsky and Mike Fried for constructive remarks on this note. 2. Thomae formula for cyclic covers and relations between theta constants We explain the general Thomae formula following [Na] for an algebraic curve R given by the equation: (z − λi)(∗) We denote f : R 7→ CP1 the projection (z, y) 7→ z. Define Qi = f −1(λi), to be the unique branch point on R that is the pre image of λi. Fix a homology basis a1, a2...a3m−2, b1, b2, ..., b3m−2 on R such that the intersections are aiaj = 0 = bibj and aibj = 1. Let v1...v3m−2 be a basis of standard holomorphic differentials dual to the basis a1, a2...a3m−2, b1, b2, ..., b3m−2 i.e. vj = 0, vj = δij . Now fix an ordering of λi. Let φ be the automorphism of order 3 defined by (z, y) 7→ (z, ωy) for ω3 = 1.We write α ≡ β for linear equivalent of divisors, i.e. if there exists a function g : R 7→ CP 1 and div(g) = α − β. The group Div0/ ≡ is Jac(R), the Jacobian of R. (Div0 - divisors of degree 0.) Let ψ be the mapping ψ : Div 7→ Div/ ≡ . Then the following lemma is true: Lemma 2.1. Let P1, P2 ∈ R,P1 6= P2 and Di = Pi + φ(Pi) + φ 2(Pi), i = 1, 2 then ψ(D1) ≡ ψ(D2). Proof. Let f1(P ) = f(P )−f(P1) f(P )−f(P2) , then div(f1) = D1 −D2. � Define D = ψ P + φ(Pi) + φ 2(Pi) as the equivalence class in the Jacobian. Lemma 2.2. Let K be the canonical divisor of R Then the following holds: D ≡ 3Qi ≡ ∞1 +∞2 +∞3 K ≡ (2m− 2)D Qi ≡ mD Proof. The first item follows exactly as in the previous lemma. To show the rest, note that z dz is a holomorphic differential with the divisor Q6m−61 . � THETA CONSTANTS FOR 3-SHEETED COVERS OF THE SPHERE 3 Now let Λ = {Λ1,Λ2,Λ3} be a partition of {1, 2, 3, 4, 5, ...3m} with |Λi| = m for i = 1, 2, 3. We are interested in the following divisor eΛ associated with the partition: eΛ = XΛ1 + 2XΛ2 −D −∆ where for each subset S of {1, 2...3m} we set Fix a point P0 ∈ R and let ΦP0 : R→ Jac(R) be given by ΦP0 (P ) = v1... v3m−2 Definition 2.3. Let Hg denote the set of g × g symmetric matrices, τ such that the imaginary part of τ is positive definite. For ε, ε′,∈ Rg and τ ∈ Hg we denote (τ) = lεZ2g exp 2πi )t ε′ This series is uniformly and absolutely convergent on compact subsets of Cg×Hg. To each w ∈ C3m−2 associate a unique w1, w2 ∈ R g such that w = w1 + τw2. [Na] proves the following formula for theta constants with characteristics associ- ated to divisors eΛ. see [BR] as well: Theorem 2.4. The divisor eΛ is a point of order 6 on the Jacobian and (1) θ[eΛ] 6 (τR) = CΛ(detA) 3((Λ0Λ0)(Λ1Λ1)(Λ2Λ2)) (Λ0Λ1)(Λ1Λ2)(Λ0Λ2) Here A is the matrix of certain differentials integrated with respect to ai. and if Λi = {i1 < ... < im} ,Λj = {j1 < ... < jm} (ΛiΛi) = (λik − λil) , (ΛiΛj) = k=1,l=1 (λik − λjl) We apply the theorem to generate special relations between theta functions with characteristics eΛ, evaluated at τR. For each partition Λ denote the poly- nomial on the right hand side of the last equation by pΛ. To obtain identities for θ[eΛ] we search for identities between pΛ. The key observation that allows us to simplify the problem is the following form of the polynomials: choose Λ = {{1, 2...,m}, {m+ 1, ..., 2m}, {2m, ..., 3m}}. Then by definition of pΛ the factor ΛiΛj is the discriminant and a common factor for each pΛ which does not depend on the partition Λ. Thus identities between θ6[eΛ] are equivalent to identi- ties between the polynomials ((Λ0Λ0)(Λ1Λ1)(Λ2Λ2)) Consequently, identities between θ6[eΛ] are equivalent to identities between the polynomials: ((Λ0Λ0)(Λ1Λ1)(Λ2Λ2)) . To get a hint for the result observe that the group S3m acts naturally on the polynomials ((Λ0Λ0)(Λ1Λ1)(Λ2Λ2)) via its action on the partitions of {1...3m} . Thus Span(((Λ0Λ0)(Λ1Λ1)(Λ2Λ2)) , is a vector space and has a representation of S3m on it. 4 KOPELIOVICH 3. Explicit Basis In this section we provide an explicit basis for the space of polynomials from the previous section. We imitate the process described in [J] to construct a basis for the irreducible representation of the symmetric group of Sn. For complex numbers these representations are completely classified. We describe the construction for any representation of the symmetric group and obtain the relevant case of cyclic covers as an immediate corollary of the general case. We remind the reader some facts from the representation theory of Sn. Let n be a natural number and let k1...km be a partition of n. i.e. i=1 ki = n and k1 ≥ k2 ≥ k3... ≥ km. Definition 3.1. A Young diagram associated to a partition consists of m rows such that i’th row has ki elements. Definition 3.2. Let Y be a Young diagram; a tableau is obtained by distributing the numbers {1...n} within the m rows with the following properties • Each row contains exactly ki elements • The numbers in each row form an increasing sequence Assume that Λ = {Λ0, ...,Λk} is a tableau of n. Define the polynomial: (ΛiΛi) = ik<il,{ik,il}∈Λi (λik − λil ), where pΛ = (ΛiΛi). The symmetric group, Sn acts on Λ and therefore acts on the polynomials pΛ. To find the basis for pΛ we use a modification of Garnier relation [J] (7.1) to construct a basis for the polynomials.1 Arrange the tableau in columns ( i.e. the first column will be elements of Λ1 the second column elements of Λ2 etc). Overall we have k columns for Λ. Let X be a subset of the i − th column of Λ and Y is a subset of the i+1− th column of Λ. Let σ1...σk be coset representatives for SX×Y in SX Then we have the Garnier relations: Theorem 3.3. Let µi denote the number of elements in the i− th column of Λ. if Y | > µi then sign(σm) (pσmΛ) = 0. Proof. If |X Y | > µi, by the pigeon hole principle there exists an involution δ such that σmΛ is invariant under it. Thus sign(σm) (pσmΛ) = sign(σm) (pδσmΛ) = signσm (pσmΛ) = 0 In order to exhibit an explicit basis we define a standard Young tableau 1We were not able to find a reference to our approach of constructing Specht modules though we are confident its a folklore. THETA CONSTANTS FOR 3-SHEETED COVERS OF THE SPHERE 5 Definition 3.4. A standard tableau is a tableau where the rows and the columns are arranged in an increasing order. Definition 3.5. We define an ordering on the set of tableaux by setting Λ1 < Λ2 if there is an i such that • if j > i than j is in the same column of Λ1,Λ2 • i is in more left column in Λ1 than Λ2. Theorem 3.6. Let Λ1...Λk be the collection of standard tableaux for a given parti- tion. Then pΛ1 ...pΛk is a basis for the vector space spanned by Λ. Proof. We follow [J] in the proof. We show that pΛk spans any other polynomial corresponding to our partition. Let t be a tableau and suppose by induction that the theorem is proved for each t1 tableau such that t1 < t. If t is non standard there exists adjacent columns a1 < ... < aq < ... < ar and b1 < b2... < bq < ...bs such that aq > bq. Apply Garnier relation for X = a1...ar, Y = b1...bq. For each σ a representative in SX Y in SX×Y we have that [tσ] < t by the definition of the order < . The result follows immediately from the induction hypothesis. � Definition 3.7. For an element k of the tableau t Let Ck, Rk be the unique column and row k belongs to. The hook of k, hk is the number of elements beneath k in Ck plus the number of elements to the right of k in Rk (include the element itself in the row but not in the column.) It is well known that the number of standard tableaux equals to See [J]. 4. The ideal of theta identities We apply the theory of the previous paragraph to cyclic covers of order 3. Ac- cording to the theory, the hooks of the partitions correspond to tableau with 3 rows and m elements in each row. Our first corollary is Corollary 4.1. The dimension of the polynomials pΛ (and hence the vector space spanned by θ6[eΛ] (τR) corresponding to them) is: (3m)!×2 (m+2)!(m+1)!m! Hence we can also give a basis for θ6[eΛ] (τR) that correspond to the different partitions eΛ. Corollary 4.2. The set of θ6[eΛS ] (τR), ΛS is a standard partition is a basis for a vector space spanned by θ6[eΛ] (τR). In particular each θ6[eΛ] (τR), Λ can be written as a linear combination of elements from the set θ6[eΛS ] (τR). 5. Example Let us revisit the case when there are 6 branch points and the genus of the surface is 4. In this case, by the formula for the dimension, the number of basis functions, θ3[eΛ] is : 2 × 4!3!2! = 5. We enumerate the 15 partitions as well as the the polynomials that correspond to them: (1) Λ = {(1, 2), (3, 4), (5, 6)}pΛ = (λ1 − λ2)(λ3 − λ4)(λ5 − λ6) (2) Λ = {(1, 2), (3, 5), (4, 6)}pΛ = (λ1 − λ2)(λ3 − λ5)(λ4 − λ6) 6 KOPELIOVICH (3) Λ = {(1, 2), (3, 6), (4, 5)}pΛ = (λ1 − λ2)(λ3 − λ6)(λ4 − λ5) (4) Λ = {(1, 3), (2, 4), (5, 6)}pΛ = (λ1 − λ3)(λ2 − λ4)(λ5 − λ6) (5) Λ = {(1, 3), (2, 5), (4, 6)}pΛ = (λ1 − λ3)(λ2 − λ5)(λ4 − λ6) (6) Λ = {(1, 3), (2, 6), (4, 5)}pΛ = (λ1 − λ3)(λ2 − λ6)(λ4 − λ5) (7) Λ = {(1, 4), (2, 5), (3, 6)}pΛ = (λ1 − λ4)(λ2 − λ5)(λ3 − λ6) (8) Λ = {(1, 4), (2, 6), (3, 5)}pΛ = (λ1 − λ4)(λ2 − λ6)(λ3 − λ5) (9) Λ = {(1, 4), (2, 3), (5, 6)}pΛ = (λ1 − λ4)(λ2 − λ3)(λ5 − λ6) (10) Λ = {(1, 5), (2, 3), (4, 6)}pΛ = (λ1 − λ5)(λ2 − λ3)(λ4 − λ6) (11) Λ = {(1, 5), (2, 4), (3, 6)}pΛ = (λ1 − λ5)(λ2 − λ4)(λ3 − λ6) (12) Λ = {(1, 5), (2, 6), (3, 4)}pΛ = (λ1 − λ5)(λ2 − λ6)(λ3 − λ4) (13) Λ = {(1, 6), (2, 3), (4, 5)}pΛ = (λ1 − λ6)(λ2 − λ3)(λ4 − λ5) (14) Λ = {(1, 6), (2, 4), (3, 5)}pΛ = (λ1 − λ6)(λ2 − λ4)(λ3 − λ5) (15) Λ = {(1, 6), (2, 5), (3, 4)}pΛ = (λ1 − λ6)(λ2 − λ5)(λ3 − λ4) The basis for the vector space of the polynomials corresponds to the following standard tableaux: (1) Λ = {(1, 2), (3, 4), (5, 6)}pΛ = (λ1 − λ2)(λ3 − λ4)(λ5 − λ6) (2) Λ = {(1, 2), (3, 5), (4, 6)}pΛ = (λ1 − λ2)(λ3 − λ5)(λ4 − λ6) (3) Λ = {(1, 3), (2, 4), (5, 6)}pΛ = (λ1 − λ3)(λ2 − λ4)(λ5 − λ6) (4) Λ = {(1, 3), (2, 5), (4, 6)}pΛ = (λ1 − λ3)(λ2 − λ5)(λ4 − λ6) (5) Λ = {(1, 4), (2, 5), (3, 6)}pΛ = (λ1 − λ4)(λ2 − λ5)(λ3 − λ6) The rest of the 10 polynomials can be rewritten as a linear combination of the set above applying Garnier’s algorithm as in Theorem 3.7. For example we have: (λ1−λ2)(λ3−λ6)(λ4−λ5) = −(λ1−λ2)(λ3−λ4)(λ5−λ6)+(λ1−λ2)(λ3−λ5)(λ4−λ6) (λ1−λ3)(λ2−λ6)(λ4−λ5) = −(λ1−λ3)(λ2−λ4)(λ5−λ6)+(λ1−λ3)(λ2−λ5)(λ4−λ6) (λ1−λ6)(λ2−λ5)(λ3−λ4) = (λ1−λ4)(λ2−λ5)(λ3−λ6)−(λ1−λ3)(λ2−λ5)(λ4−λ6) The others polynomials can be expressed in a similar way leading to identities between θ6[eΛ] (τR) in this case. Let us conclude with the following remarks on the identities above: In the hyperelliptic curve case the identities between integral characteristics of theta functions evaluated at period matrix of hyperelliptic curves arise from vanishing properties of theta functions. In our case it is interesting to investigate whether an analogous situation can arise. The only source of cubic theta identities known to the author, is the following theorem in [Ko]: Theorem 5.1. Let be an odd integral theta characteristics in genus 3m− 2 Then for any τ ∈ H3m−2: 0≤νi≤3 µiνiθ3 µ′ + 2ν (0, τ) = 0 It is plausible that the vanishing of theta constants with rational characteristics of order 3 on τR will produce a new proof for the special identities obtained in this note using Thomae formula. Finally note that for all the identities (4) the coefficients are ±1 It is plausible that this a general phenomenon. THETA CONSTANTS FOR 3-SHEETED COVERS OF THE SPHERE 7 6. conclusion There exists an extensive literature on Schottky-Jung identities and on theta constants for hyperelliptic curves. In this note we obtained special identities for other classes of algebraic curves. In subsequent notes we plan to pursue and develop further the themes touched in this note, especially applications of similar methods to general Hurwitz spaces and their mapping class groups. References [AK] R.Adin, Y.Kopeliovich, Short Eigenvectors and Multidimensional Theta Functions, Linear Algebra and Appl. 257(1)(1997) 49-63 [BR] M. Bershadsky, A. Radul, Fermionic fields on Zn curves Comm. in Mathematical Phys. 116(4)(1988) 689-700 [FK1] H. Farkas, Y. Kopeliovich, New Theta Constant Identities Israel Journal of Mathematics 82(1)(1993) 133-140 [FK2] H. Farkas, Y.Kopeliovich, New Theta Constant Identities II Proceeding of AMS.123(4)(1995) 1009-1020 [J] G.D.James The representation theory of the Symmetric Groups Lecture Notes in Math. vol. 682 (Springer Verlag 1978) [Ko] Y. Kopeliovich, Multi Dimensional Theta Constant Identities Journal of Geometric Analysis 8 (4)(1998) 571-581 [Ma] K.Matsumoto Theta constants associated with the cyclic triple coverings of the complex projective line branching at six points Publ. Res. Inst. Math. Sci. 37 (3) (2001) 419-440 [Mu] D. Mumford, Tata Lectures on Theta II (Progress in Mathematics, Birkhauser 1984) [Na] A. Nakayashiki, On the Thomae formula for ZN curves Publ. Res. Inst. Math Sci. 33 (6)(1997) 987-1015 [Th] J. Thomae, Beitrag zur Bestimmung θ(0, 0...,0) durch di klassenmoduln algebraicer Funk- tionen Crelle’s Journal 71(1870) 201-222 5736 Las Virgenes Rd. Calabasas CA 91302 1. Introduction 2. Thomae formula for cyclic covers and relations between theta constants 3. Explicit Basis 4. The ideal of theta identities 5. Example 6. conclusion References

0704.1033

Topology of spaces of equivariant symplectic embeddings

Topology of spaces of equivariant symplectic embeddings Alvaro Pelayo Abstract We compute the homotopy type of the space of Tn-equivariant symplectic embeddings from the standard 2n-dimensional ball of some fixed radius into a 2n-dimensional symplectic– toric manifold (M, σ), and use this computation to define a Z≥0-valued step function on R≥0 which is an invariant of the symplectic–toric type of (M, σ). We conclude with a discussion of the partially equivariant case of this result. 1 The main theorem Let (M, σ) be a 2n-dimensional symplectic manifold and write Br for the compact 2n-ball of radius r > 0 in the complex space Cn equipped with the restriction of the standard symplectic form σ0 of C n. (The proofs of the results in this paper hold verbatim for the open ball.) Recently a lot of effort has been put into understanding the topological and geometric properties of the space of symplectic embeddings from Br into M . This question is not only intriguing, but it is also very fundamental because it acknowledges one of the main differences that exist between Riemannian and symplectic geometry, e.g. Gromov’s non–squeezing theorem [12]. Figure 1: An equivariant and symplectic embedding B2r → S2s with r/s = This question, posed with such generality, has proven to be extremely difficult to answer. Significant progress has been made by McDuff [17], [18], Biran [3], [5] and most recently by Lalonde–Pinsonnault [14], among other authors. One of the most general results is due to McDuff; she showed the connectedness of the space of 4-balls into 4-manifolds with non-simple Seiberg– Witten type, in particular rational or ruled surfaces. Recall that we say that a symplectic 4-manifold Q has simple Seiberg–Witten type or just simple type if the only non–zero Gromov invariants of Q occur in classes A ∈ H2(Q) for which k(A) = K · A + A2 = 0. It follows from work of Taubes http://arxiv.org/abs/0704.1033v1 and Li–Liu that the symplectic 4-manifolds with non–simple type are blow–ups of (i) rational and ruled manifolds; (ii) manifolds with b1 = 0, b 2 = 1, like the Enriques or Barlow surface; and (iii) manifolds with b1 = 2 and (H1(X)) 2 6= 0; examples (with K = 0) are hyperelliptic surfaces, some non–Kähler T2-bundles over T2 and quotients T2 × Σg/G where Σg is a surface of genus g greater than 1, and G is certain finite group. See [17] for further references and examples. McDuff’s techniques are unique to dimension 4 and do not extend at all to higher dimensions— this is also the case in the other authors’ work—existence of J-holomorphic curves with special homological properties is essential in their proofs. Although J-holomorphic curves exist in all even dimensions, it is only in dimension 4 where these homological properties hold. In the present paper we study a special case of this question: M is a symplectic–toric manifold of arbitrary dimension, and the symplectic embeddings that we consider preserve the toric struc- ture, see Figure 1. Precisely this means that there exists an automorphism Λ of the n-torus Tn such that the following diagram commutes: n × Br Λ×f // f // M , (1.1) where ψ is a fixed effective and Hamiltonian Tn-action on M and · denotes the standard action by rotations on Br (component by component). In this case we say that f is a Λ-equivariant mapping. (1,0)(0,0) (1,1)(0,1)(1,1)(0,1) (2,0)(0,0)(1,0)(0,0) (0,1) Figure 2: The momentum polytope of CP2 and B21 (left), of a Hirzebruch surface (center) and of (CP1)2 (right). The feature that makes the study of symplectic manifolds equipped with Hamiltonian torus actions richer than the study of generic symplectic manifolds is the presence of the smooth mo- mentum map µM : M → Lie(Tn)∗, whose image ∆M is a convex polytope (called the momentum polytope of M , cf. Figure 2) as shown independently by Atiyah and Guillemin–Sternberg [1], [8]. Here we are identifying the Lie algebra Lie(Tn) and its dual Lie(Tn)∗ with Rn. Since this identi- fication is not canonical, we need to specify the convention we adopt in this paper. This amounts to choosing an epimorphism R → T1 which we take to be x 7→ e2 −1x. This epimorphism in- duces an isomorphism between Lie(T1) and R via ∂ 7→ 1/2, giving rise to a new isomorphism Lie(Tn) → Rn, ∂ 7→ 1/2 ek, by canonically identifying Lie(Tn) with the product of n copies of Lie(T1) (see [9] for more details). For example, under the convention of the previous paragraph, the momentum map µBr of Br is a mapping from Br into R n with components µBrk (z) = |zk|2, for all integer k with 1 ≤ k ≤ n. (There are a number of different conventions used in the literature, and our choice is intended to give the simplest formula for the momentum map of Br.) The simplest symplectic manifolds which admit Hamiltonian effective torus actions are called symplectic–toric. Definition 1.1 A symplectic–toric manifold M , also called a Delzant manifold, is a compact connected symplectic manifold equipped with an effective Hamiltonian action of a torus of dimen- sion half of the dimension of the manifold. In this case the momentum polytope ∆M is called the Delzant polytope of M . ⊘ Symplectic–toric manifolds were classified by Delzant in [7]. In particular, he showed that the momentum image of such a manifold under the momentum map completely determines M up to equivariant symplectomorphisms. The main result of this paper, Theorem 1.2 below, describes the topology of the space of equiv- ariant embeddings of symplectic balls into a symplectic–toric manifold. We denote by χ(M) the Euler characteristic of M . Theorem 1.2. For every symplectic–toric 2n-manifold M there is an associated Z-valued non– increasing step function Emb(M,σ) : R≥0 → [0, n!χ(M)] such that for each r ≥ 0 the space of equivariant symplectic embeddings from the 2n-ball Br into M is homotopically equivalent to a disjoint union of Emb(M,σ)(r) subspaces, each of which is homeomorphic to the n-torus T As a matter of fact we can explicitly and easily read Emb(M,σ) from the polytope ∆ Example 1.3 Let (M, σ) equal the blow–up of S2r0×S with r0 = 1/ 2 whose Delzant polytope has vertices at (0, 0), (2, 0), (2, 1), (1, 2) and (0, 2) (see Figure 4). Then Emb(M,σ) = 10χ[0,1) + 2χ[1, 2), where χA denotes the characteristic function of A ⊂ R. We identify the 2-sphere of radius r equipped with the standard area form with (CP1, 4r2 ·σFS), where σFS is the Fubini–Study form. ⊘ Proposition 1.4. The function Emb(M,σ)(r) given in Theorem 1.2 is an invariant of the symplectic– toric type of M and is given by the formula Emb(M,σ)(r) = n! p∈MTn cp(r), (1.2) where for each fixed point p ∈M , cp(r) = 1 if the infimum of the SL(n, Z)-lengths of the edges of ∆M meeting at µM(p) is strictly greater than r2, and cp(r) = 0 otherwise. Example 1.5 Let σFS be the Fubini–Study form on CP n and observe that Tn acts naturally on (CPn, λ · σFS), λ > 0, with n + 1 fixed points. The momentum polytope is a tetrahedrum with vertices at 0 and λ ei, where the ei are the canonical basis vectors in R n. So if M = CPn × CPm, the space of equivariant symplectic embeddings from Br into M is homotopically equivalent to (n+m)!(n+1)(m+1)⊔ if r < λ, and it is empty otherwise. ⊘ The study of the space of symplectic embeddings is directly related to the study of the sym- plectic ball packing problem cf. [4], the equivariant version of which was treated in [19]. 2 Proof of Theorem 1.2 In this section we prove Theorem 1.2. For clarity, the proof is divided in three steps, which we describe next. We start by introducing the notation and making the following observations: i) Throughout the proof Rwill denote the space of rotations by matrices of the form (δi τ(i)θi j)ni, j=1 with τ ∈ Sn (the symmetric group) and θi j ∈ T1, and ET x will denote the space of equivari- ant symplectic embeddings f from Br into M such that f(0) = p and µ M(p) = x ∈ ∆M , equipped with the Cm-Whitney topology (m ≥ 0). Throughout the present section we fix f . Since each component of R is canonically identified with the n-torus Tn (cf. Corollary 2.5), Theorem 1.2 amounts to prove that if p ∈MTn (MTn denotes the Tn-fixed point set) is such that µM(p) = x, then the space ETnx gets identified with R via a homotopy equivalence. ii) Secondly let BTnr denote the space of equivariant symplectomorphisms of Br (again with respect to the Cm-Whitney topology). Recall that, for example, the Cm-Whitney topology on BTnr is given by the well–known norm ‖ φ ‖Cm= max 0≤k≤m ‖ Tkφ ‖M(C), where we are taking the norm ‖ · ‖M(C) on the right–hand side of this expression to be the canonical Euclidean norm on the space of n× n matrices with complex entries. iii) We identify the automorphism group Aut(Tn) with the matrix group GL(n, Z). iv) The elements αpi ’s, 1 ≤ i ≤ n, denote the weights of the isotropy representation of Tn on TpM ; the canonical basis vectors ei ∈ Rn represent the weights of the isotropy representa- tion of Tn on T0Br. Step 1: Invariance of the image f(Br). In this step we first show how to go from smooth maps on manifolds to affine maps on polytopes (see diagram (2.3)), and secondly we use this to show the invariance of the image f(Br) ⊂ M . Precisely, one can think of an embedding being equivariant in the sense of commuting with the n-action, and it is when we reparametrize the torus that Λ appears. Lemma 2.1. Let g be any Λ-equivariant and symplectic embedding such that the normalization condition f(0) = g(0) = p holds. Then for all z ∈ Br, if Tn · z denotes the Tn-orbit that passes through z, the identity f(Tn · z) = g(Tn · z) holds, and therefore f(Br) = g(Br). Proof. Let f, g be Λ-equivariant and symplectic embeddings from Br into M with f(0) = g(0) = p. Under the identifications described in Section 1, the following diagram commutes, where the top arrow stands for the affine map with linear part (Λt)−1, which takes 0 to x: (Λt)−1+x // ∆M . (2.3) In order to prove the commutativity of diagram (2.3), we denote by ξM the vector field induced by the element ξ ∈ Lie(Tn) via the exponential map and note that from the definition of the momentum maps µM and µBr , the Λ-equivariance of f , and the fact that f ∗σ = σ0, we have the following sequence of equalities, where Tf(z)µ M and Tzµ Br denote, respectively, the tangent mapping of µM at f(z) and of µBr at z, 〈TzµBr(v), ξ〉z = (σ0)z(v, ξBr(z)) = σf(z)(Tzf(v), Λ(ξ)M(f(z))) = 〈Tf(z)µM(Tzf(v)), Λ(ξ)〉f(z) = 〈Λt ◦ Tf(z)µM(Tzf(v)), ξ〉z, (2.4) where z ∈ Br, v ∈ TzBr and ξ ∈ Lie(Tn). Therefore by equation (2.4) and by using the chain rule we obtain that for all z ∈ Br and v ∈ TzBr Br(v) = Tz(Λ t ◦ µM ◦ f)(v). (2.5) Considering equation (2.5), f(0) = p and µM(p) = x and composing with (Λt)−1, after integration we obtain the commutativity condition on diagram (2.3). Notice that diagram (2.3) also holds for the embedding g. Then it follows from the conjunction of diagram (2.3) and diagram (1.1) that for all t ∈ Tn the following identities hold: µM(ψ(Λ(t), f(z))) = µM(f(t · z)) = (Λt)−1 ◦ µBr(z) + x = µM(g(t · z)) = µM(ψ(Λ(t), g(z))). (2.6) Expression (2.6) is clearly equivalent to µM(Tn · f(z)) = µM(Tn · g(z)), since Λ is an auto- morphism. Now since M is symplectic–toric, by the proof of the Atiyah–Guillemin–Sternberg convexity theorem we know that each fiber of the momentum map µM consists of a single con- nected orbit which together with the last equality implies that Tn · f(z) = Tn · g(z). Since f(Br) is the union of the orbit images f(Tn · z), we immediately obtain that f(Br) = g(Br), which concludes the proof. It is possible to explicitly describe the momentum image µM(f(Br)), and for this purpose we recall the notion of SL(n, Z)–length: if x, y ∈ Rn, we say that a segment line [x, y] joining x to y (0,0) (0,1) (1,1) (1,0) (0,1) (1,1) (0,0) (1,0) Figure 3: Equivariant symplectic ball embeddings in (CP1)2 (left); the triangle on the right does not come from such embedding. in Rn has SL(n, Z)-length d if there exists a matrix A ∈ SL(n, Z) such that A(d e1) = y − x (d is not defined in general, only for segments of rational slope). The Euclidean length of a segment line agrees with its SL(n, Z)-length if and only if the segment is parallel to one of the coordinate axes in Rn. In their article [16], Karshon and Tolman made the following two definitions. Let (Q, σQ) be a connected symplectic 2m-dimensional manifold with momentum map µQ for an action of an m-torus Tm on Q, and let Γ ⊂ (Lie(Tm))∗ be an open convex subset which contains the image of Q under the momentum map µQ. The quadruple (Q, σQ, µQ, Γ) is a proper Hamiltonian Tm- manifold if the momentum map µQ is proper as a map to Γ. The proper Hamiltonian Tm-manifold (Q, σ, µQ, Γ) is said to be centered about a point α ∈ Γ if α is contained in the momentum map image of every component of QK , for each K ⊂ Tm. Here QK := {q ∈ Q |ψQ(a, q) = q, ∀a ∈ K}, where ψQ : T m ×Q→ Q denotes the action of Tm on Q. The following lemma is Proposition 2.8 in [16]. Lemma 2.2 (Karshon–Tolman, [16]). Let the quadruple (Q, σQ, µQ, Γ) be a proper Hamiltonian 2m-dimensional Tm-manifold. Suppose that (Q, σQ, µQ, Γ) is centered about α ∈ Γ and that the preimage (µQ)−1({α}) consists of a single fixed point q. Then Q is equivariantly symplectomor- phic to {z ∈ Cm |α+ |zj |2 ηqj ∈ Γ}, where ηq1, . . . , η m are the weights of the isotropy representation of T m on TqQ. We use Lemma 2.2 in order to prove the following lemma. Lemma 2.3. The momentum image µM(f(Br)) equals the subset of R n given by the convex hull of x and x+ r2 αpi , where 1 ≤ i ≤ n. Furthermore, the infimum of the SL(n, Z)-lengths of the edges of ∆M meeting at x is greater than or equal to r2, if and only if for all 0 < s < r there exists an embedding h : Bs → M which is Λ-equivariant and symplectic, satisfying h(0) = p. Proof. The first observation is that ∆Br equals the convex hull in Rn of 0 and r2 e1, . . . , r 2 en. Secondly, since µBr : Br → ∆Br is onto, it follows from diagram (2.3) that µM(f(Br)) = (Λ t)−1(∆Br) + x. Since Λ is an automorphism, (Λt)−1 is an automorphism of the corresponding dual spaces and therefore there exists a permutation τ ∈ Sn such that (Λt)−1(ei) = αpτ(i). Then the linearity of Λ implies that µM(f(Br)) equals the the convex hull in R n of the points x and x+ r2 αpi , 1 ≤ i ≤ n, which proves the first claim. Suppose that the infimum of the SL(n, Z)-lengths of the edges meeting at x is greater than or equal to r2. Let Σ be the convex hull of x and x + r2 αpi , with 1 ≤ i ≤ n, and let Z be the convex hull of x+r2 αpi , with 1 ≤ i ≤ n. Notice that Σ ⊂ ∆M , Σ\Z is open in ∆M , and let Γ ⊂ Rn be the open half–space of Rn, whose closure’s boundary ∂(cl(Γ)) is the hyperplane of Rn that contains Z, and such that Σ \ Z ⊂ Γ. Let N := (µM)−1(Σ \ Z) and let σN be the symplectic form obtained by restricting σ to N . The set N is open in M because it is the preimage of the open set Σ \Z under the momentum map µM : M → ∆M . By the proof of Atiyah–Guillemin–Sternberg convexity theorem, cf. [1], [8], N is a connected manifold. Since M is compact, the momentum map µM : M → ∆M is a proper map and therefore its restriction µM : N → Σ \Z is a proper map, which means that µM : N → Γ is proper, since (µM)−1(Γ \ (Σ \ Z)) = ∅. Therefore (N, σN , ψN) is a connected symplectic manifold with momentum map µM , and the quadruple (N, σN , µM , Γ) is a proper Hamiltonian n-space. On the other hand, notice that the quadruple (N, σN , µM , Γ) is centered about the point x, and (µM)−1({x}) = p, so we can apply Lemma 2.2, and conclude that N is equivariantly symplecto- morphic to the submanifold X ⊂ Cn given by X := {z ∈ Cn | x+ |zi|2 αpi ∈ Σ \ Z} = {z ∈ Cn | x+ |zi|2 αpi ∈ Σ} \ {z ∈ Cn | x+ |zi|2 αpi ∈ Z} = Br \ ∂Br = Int(Br). Hence there exists an equivariant symplectomorphism φ : Int(Br) → N , and by letting j : Bs → Int(Br) be the standard inclusion, if s < r, the map h := jN ◦φ◦ j : Bs → M , where jN : N →M is the inclusion map, is an equivariant symplectic embedding for all s < r with h(0) = p. The converse follows from the first statement of the lemma. Note that µM(f(Br)) only depends on the fixed point p and the radius r (which was fixed a priori) and not on f . In Figure 3 several momentum ball images are drawn using Lemma 2.3. Note that the shaded triangle on the right picture is not a Delzant polytope since it fails to be smooth at (0, 0). Delzant polytopes are simple, edge–rational and smooth polytopes, cf. Figure 2 (see [10] or [6] for a definition of these notions). Step 2: A deformation retraction on BTnr . In this step we use Alexander’s trick to construct a deformation retraction from the space of equivariant symplectomorphisms of the 2n-dimensional ball Br in C n onto a disjoint union of copies of Tn. The continuity of this deformation is standard and may be found in [13]. Lemma 2.4. The space BTnr of equivariant symplectomorphisms of the 2n-dimen- sional ball Br in C n, with respect to the standard symplectic form σ0 and the canonical action of Tn by rotations, deformation retracts onto its subspace of linear, equivariant and symplectic rotations given by matrices in R. Proof. We define the transformationHT from BTnr × [0, 1] into BT r , by the formulaH (φ, t) := φt, where φt is the composite map φt := (mt) −1 ◦ φ ◦mt, t 6= 0. (2.7) The map mt in expression (2.7) denotes the linear contraction of factor 0 ≤ t ≤ 1 on Br, mt(z) = t z; and when t = 0, φt = φ0 is defined to be the tangent mapping Tφ of the map φ, evaluated at 0. (This expression for φt is known as Alexander’s trick.) It is easy to check that H is continuous and that the evaluation map [0, 1] × Br → Br given by (t, z) 7→ φt(z) is smooth. Since HT the identity on linear maps, we conclude that it is a deformation retraction, not only a homotopy, onto the space of ball rotations by matrices (δi τ(i)θi j) i, j=1 with τ ∈ Sn (the symmetric group) and θi j ∈ T1. We have left to check that HT is well defined, i.e. that φt ∈ BT r . Indeed, the equivariance of the mapping φt follows directly from formula (2.7); explicitely we have that if φ is equivariant with respect to Λ ∈ Aut(Tn), then φt(s · z) = 1/t φ(s · tz) = Λ(s) · φt(z) for all s ∈ Tn. By differentiating formula (2.7) we obtain that T(φt) = (mt) −1 ◦ Tφ ◦mt, (2.8) and since mt is a linear isomorphism, the mapping φt is a diffeomorphism. Furthermore, since the mapping φ is symplectic, it follows from expression (2.8) that for all z ∈ Br we have that (φ∗tσ0)z(u, v) = (σ0)φ(z)(Tzφ(u), Tzφ(v)) = (σ0)z(u, v), for every pair of vectors u, v ∈ TzBr, and hence φt is a symplectic mapping. Therefore φt is a diffeomorphism, which is equivariant and symplectic, or equivalently φt ∈ BT r . We have been assuming that t 6= 0, but if t = 0, it is trivial that φ0 ∈ BT Corollary 2.5. The space BTnr of equivariant symplectomorphisms of the 2n-dim- ensional ball Br in C n, with respect to the standard symplectic form σ0 and the canonical action of Tn by rotations, is homotopically equivalent to a disjoint union of n! copies of Tn. Proof. Apply Lemma 2.4 and observe that the space of ball rotations by matrices (δi τ(i)θi j) i, j=1 with τ ∈ Sn and θi j ∈ T1 is homotopically equivalent to a disjoint union of n! copies of Tn. We conclude the proof with Step 3, in which Lemma 2.1, Lemma 2.3 and Lemma 2.4 are combined in order to prove Theorem 1.2. The proof of Proposition 1.4 will follow from the proof of Theorem 1.2, since the function Emb(M,σ) will be explicitly computed. Step 3: Lifting the deformation φt to ET x and conclusion. In this final step we show that ETnx is homotopically equivalent to a disjoint union of copies of Lemma 2.6. Suppose that the infimum of the SL(n, Z)-lengths of the edges of ∆M meeting at x is strictly greater than r2. Then there exists an equivariant and symplectic embedding u : Br → M with u(0) = p such that if ρ is the identification map on ETnx which takes values on BT r and is given by formula ρ(h) := u−1 ◦ h, where h ∈ ETnx , the space ET x is homotopically equivalent to the space ρ−1(R). Proof. The first observation is that by Lemma 2.3 there exists a Λ-equivariant and symplectic embedding u from Br intoM with u(0) = p. In order to construct homotopy equivalences between ETnx and R, we define ρ to be the identification map on ET x which takes values in BT r and is given by formula ρ(h) := u−1◦h for every h ∈ ETnx . Now we claim that the mapHT x from ET x ×[0, 1] to ETnx , given by the commutative diagram (2.9) below, is a well–defined and continuous homotopy satisfying HT x (ET x × {0}) = ρ−1(R), while ET x is preserved at time t = 1, i.e. we have that x (ET x × {1}) = ET x . The diagram is the following: ETnx × [0, 1] BTnr × [0, 1] Br // BTnr . (2.9) The mapping HT x is well defined by Lemma 2.1. Note that H x is continuous, since the identifi- cations ρ and ρ−1 are obviously continuous and we showed in Lemma 2.4 that HT is continuous. We can therefore conclude, from the previous considerations and the fact that that HT is a de- formation retraction in the Cm-Whitney topology, that HT x induces homotopy equivalences ρ and δ(f) := HT x (f, 0) between ET x and ρ −1(R), with ρ◦δ homotopic to idETn and δ◦ρ = idρ−1(R). In order to conclude the proof of Theorem 1.2 we simply make the following observations: • First, the space described in it is precisely the disjoint union of the ETnx , x being a vertex of ∆M , because 0 is to be mapped to a fixed point of ψ. • The number of Tn-fixed points, which is the same as the number of vertices of ∆M , is pre- cisely χ(M). This follows from the analysis of the momentum map as in Atiyah–Delzant– Guillemin–Sternberg theory (see for example [10], [11]). • If we denote by Emb(M,σ)(r) the number of copies of Tn onto which the space considered in Theorem 1.2 retracts (see formula (1.2)), Emb(M,σ)(r) is obtained by multiplying the number of fixed points that admit such an embedding (see Lemma 2.3) by the number of copies of n onto which ETnx (for the particular point) retracts; this latter number is n! (see Corollary 2.5), i.e. as many copies of Tn as possible ways that the canonical basis vectors ei may be mapped onto the basis of weights αpi (for the particular point). Also, the former number is by Lemma 2.3 controlled by the Boolean variable cp(r) defined in Proposition 1.4. Therefore Emb(M,σ)(r) is given by Emb(M,σ)(r) = n! p∈MTn cp(r), as we wanted to show. • It is obviously true that if (M, σ) is equivariantly symplectomorphic to (M̃, σ̃), then Emb(M,σ)(r) = (M̃, σ̃) (r), so the integer Emb(M,σ)(r) is a symplectic–toric invariant. (2,1) (1,2)(0,2) (2,0)(0,0) (2,1) (1,2)(0,2) (2,0)(0,0) Figure 4: Polytope corresponding to the Delzant manifold (M, σ) obtained by blowing up S2r0×S2r0 with r0 = 1/ 2. Observe that Emb(M,σ)( 2) = 0 (proof in left figure) and Emb(M,σ)(1/ 10 (proof in right figure). See Lemma 2.7. As a final remark we observe that the invariant function Emb(M,σ) associated to the Delzant manifold (M, σ) always reaches its minimum and maximum values on an interval of strictly posi- tive length. Lemma 2.7. There exist numbers r0, s0 > 0 such that if r ≤ r0, then the space of equivariant symplectic embeddings from Br into M is homotopically equivalent to a disjoint union of n!χ(M) copies of Tn, and if s ≥ s0, then it is empty. Proof. It follows easily from Lemma 2.3, Corollary 2.5 and the previous observations. This concludes the proof of Theorem 1.2 (and hence by construction the proof of Proposition 1.4). 3 Remarks on the partially equivariant case of Theorem 1.2 In this section we initiate a discussion on the topology of the space of partially equivariant sym- plectic embeddings and sketch some suggestions to answer a question in this direction. First the notion of Λ-equivariance (Λ ∈ Aut(Tn)) in Section 1 has a natural extension: we say that an embedding from the 2n-ball Br into the 2n-dimensional Delzant manifold M is γ- -equivariant with respect to a monomorphism γ : Tn−k → Tn, 1 ≤ k ≤ n − 1, if the following diagram commutes: n−k × Br γ×f // f // M For example, Mγ is the set of p ∈ M such that ψ(γ(t), p) = p for all t ∈ Tn−k, and the rest of terminology is also analogous. This definition extends naturally to the case when k = n, in which the embeddings considered are purely symplectic, as well as to the case when k = 0, in which the embeddings are fully equivariant, case which we treated previously in the paper. Unless otherwise specified we do not consider these two cases in the discussion that follows. The question we would like to address is the following: Question 3.1 Let r be such that any connected component C ofMγ admits a Darboux–Weinstein neighborhood of radius r, and by this we mean a neighborhood that is equivariantly symplecto- morphic to a bundle over C with fiber the standard ball of radius r. Is the space of γ-equivariant symplectic embeddings from Br intoM homotopically equivalent to the space of purely symplectic embeddings from B2kr into M γ up to reparametrization groups (as explained below)? ⊘ To analyze Question 3.1 first define B̂2kr to be the embedded 2k-ball in Br, i.e. the set of points (z1, . . . , zk, 0) in Br so that i=1 |zi|2 ≤ r2. The preimage under the momentum map of the k-face corresponding to γ is the fixed point locus Mγ . Now consider any symplectic embed- ding f : B̂2kr → Mγ . We want to find a canonical way to extend f to an equivariant symplectic embedding can(f) : Br →M up to homotopy. Here is an attempt to construct can(f): near the image of f , we can apply the equivariant ver- sion of the Darboux–Weinstein’s theorem in order to find a neighborhood of Im(f) in M which is symplectomorphic to Im(f)× B2(n−k), with the action of Tn−k given by the standard action on 2(n−k), and the symplectic form coinciding with the product symplectic form. Note that the sym- plectic normal bundle toMγ is trivial over Im(f) because Im(f) is contractible, so a neighborhood of Im(f) looks like Im(f)× B2(n−k) with a product symplectic form, and the action of Tn−k on it is conjugate to the standard one. Using this identification M is described as a product, and we can define can(f)(z) := (f(z1, . . . , zk), zk+1, . . . , zn). This expression for can(f) is clearly symplec- tic and equivariant with respect to Tn−k-actions on the last n− k coordinates but is not canonical because the local symplectomorphism given by Darboux–Weinstein’s theorem is not unique. We cannot expect it to always be the same independently of f , because it is not true that globally the normal bundle to Mγ is symplectically trivial, it only becomes true over a neighborhood of Im(f). So this construction depends on choices of parameters. Calling CAN the space of canonical embeddings can(f) : Br → ∆M , where f : B̂2kr → Mγ is a symplectic embedding, observe that CAN is naturally identified with the space of purely sym- plectic embeddings from the standard B2kr into M γ , up to homotopy. The question then becomes whether any γ-equivariant symplectic embedding f : B2kr → M may be deformed through a con- tinuous family of equivariant symplectic embeddings to an embedding in CAN. Equivalently, we ask the question: is the natural map between the space of partially equivariant embeddings from Br into M and the space of symplectic embeddings from B r into the fixed point set Mγ (given by the restriction to the fixed ball B2kr ) a fibration? Note that the construction of can(f) would give a section of this fibration. Conjecture 3.1. Question 3.1 has an affirmative answer. Example 3.2 If M = S2 × S2 with a product symplectic form and product T2-action (this space has been carefully studied by Lalonde–Pinsonnault [14] and Anjos [2] among other authors), the fixed point locus of the second S1 factor is S2×{a, b}, where a, b are the fixed points of the action of S1 on S2. Now, given a symplectic embedding f of the ball B2 into S2, it is easy to build an 1-equivariant embedding of B4 into S2 × S2 canonically by (z1, z2) 7→ (f(z1), z2), where z2 is taken to be a coordinate centered at the fixed point a. In this case the normal bundle to the fixed point component S2 × {a} is globally trivial. ⊘ The combination of purely symplectic results of Biran, Lalonde–Pinsonnault and others and an affirmative answer to Question 3.1 would give insight into the partially equivariant case in higher dimensions; for example McDuff showed that if M is a symplectic 4-manifold with non- simple Seiberg–Witten type, then the space of symplectic embeddings from Br into M is path connected (which extends results of Biran). This is a consequence of the non–trivial result: any two cohomologous and deformation equivalent symplectic forms on M are isotopic (proved in [17]). Examples are known in dimensions 6 and above of cohomologous symplectic forms that are deformation equivalent but not isotopic, so these techniques do not help to understand the topology of the space of symplectic embeddings from Br into M . A positive answer to Question 3.1 would give the first non–trivial result in dimension 6. Another way of trying to generalize Theorem 1.2 is to consider embeddings equivariant with respect to a complexity one action, that is, an action of Tn−1 on M2n. This is a hopeful approach since a complete classification of complexity one actions has been recently achieved by Karshon and Tolman [15]. Acknowledgments The author is grateful to D. Auroux and V. Guillemin for discussions, and for hosting him at the M.I.T. regularly during the Fall and Spring semesters of 2003 and 2004. He thanks D. Auroux, J.J. Duistermaat, Y. Karshon and M. Pinsonnault for making comments on a preliminary version of this paper. Finally, the author is grateful to an anonymous referee for helpful suggestions that have shortened the proof in Section 2, as well as for his/her interesting comments on Section 3. References [1] M. Atiyah, Convexity and commuting Hamiltonians. Bull. London Math. Soc., 1 (1982) 1–15. [2] S. Anjos, Homotopy type of symplectomorphism groups of S2 × S2. Geom. Topol. 6 (2002) 195–218. [3] P. Biran, Connectedness of spaces of symplectic embeddings, Int. Math. Res. Lett., 10 (1996) 487–491. [4] P. Biran, From symplectic packing to algebraic geometry and back, Proc. 3rd Europ. Congr. of Math., Barcelona 2, Prog. Math., Birkhauser (2001) 507–524. [5] P. Biran, Geometry of symplectic intersections, Proc. ICM Beijing III (2004) 1-3. [6] A. Cannas da Silva, Lectures in symplectic geometry, Springer-Verlag, Berlin (2000). [7] T. Delzant, Periodic Hamiltonians and convex images of the momentum mapping, Bull. Soc. Math. France, 116 (1988) 315–339. [8] V. Guillemin and S. Sternberg, Convexity properties of the momentum mapping, Invent. Math., 67 (1982) 491–513. [9] V. Guillemin and S. Sternberg, Symplectic techniques in physics, Cambr. Univ. Press (1984). [10] V. Guillemin, Moment maps and combinatorial invariants of Hamiltonian Tn spaces, Prog. Math., Birkhauser (1994). [11] V. Guillemin, V. Ginzburg and Y. Karshon, Moment maps, cobordisms and Hamiltonian group actions, Amer. Math. Soc. Monograph 98, (2002). [12] M. Gromov, Pseudoholomorphic curves in symplectic geometry, Invent. Math., 82 (1985) 307–347. [13] M. Hirsch, Differential topology. Springer-Verlag 33 (1980). [14] F. Lalonde and M. Pinsonnault, The topology of the space of symplectic balls in rational 4-manifolds, Duke Math. J. 122 (2004), no. 2, 347–397. [15] Y. Karshon and S. Tolman, Centered complexity one Hamiltonian torus actions, Trans. Amer. Math. Soc. 353 (2001), no. 12, 4831–4861 (electronic). [16] Y. Karshon and S. Tolman, The Gromov width of complex Grasmannians, Alg. and Geom. Topol. 5 paper 38, 911–922. [17] D. McDuff, From deformation to isotopy, Topics in Symplectic 4-manifolds, Internat. Press, Cambridge, MA (1998). [18] D. McDuff, The structure of rational and ruled symplectic 4-manifolds, J. Amer. Math. Soc. 3(3) (1990) 679–712. [19] A. Pelayo, Toric symplectic ball packing, Top. and its Appl. 153 (2006) 3633–3644. A. Pelayo Department of Mathematics, University of Michigan 2074 East Hall, 530 Church Street, Ann Arbor, MI 48109–1043, USA e-mail: [email protected] The main theorem Proof of Theorem ?? Remarks on the partially equivariant case of Theorem ??

0704.1034

Toric symplectic ball packing

Toric symplectic ball packing Alvaro Pelayo Abstract We define and solve the toric version of the symplectic ball packing problem, in the sense of listing all 2n-dimensional symplectic–toric manifolds which admit a perfect packing by balls embedded in a symplectic and torus equivariant fashion. In order to do this we first describe a problem in geometric–combinatorics which is equiva- lent to the toric symplectic ball packing problem. Then we solve this problem using arguments from Convex Geometry and Delzant theory. Applications to symplectic blowing–up are also presented, and some further questions are raised in the last section. 1 The Main Theorem Loosely speaking, the “symplectic packing problem” asks how much of the volume of a symplectic manifold (M, σ) may be filled up with disjoint embedded open symplectic balls. A lot of progress on this and intimately related questions has been made by a number of authors, among them Biran [2], [3], [6], McDuff–Polterovich [19], Traynor [23] and Xu [24]. Several authors have made progress on directly related questions, like the topology of the space of symplectic ball embeddings, among them McDuff [18], Biran [4] and most recently Lalonde– Pinsonnault [17] (the equivariant version of this question was studied in [22]). Despite the fact that these significant contributions have appeared in recent years, the symplec- tic packing problem remains largely not understood; for more details and a survey of known results see the paper by Biran [5] and for some nice examples see [21]. Outstanding progress has been made in dimension four (see for example [2], [19], [23]), but nothing is understood in dimension six or above. The underlying reason for this dimensional barrier is that the techniques used by the previous authors are unique to dimension four and do not extend to higher dimensions. For more details see Section 1 of [22] or Section 3 of Biran’s paper [6]. The present paper is devoted to the study of a particular case of the symplectic packing problem, the torus equivariant case. In this case both the symplectic manifold M and the standard open symplectic ball (Br, σ0) in C n are equipped with a Hamiltonian action of an n-dimensional torus n, and the symplectic embeddings of this ball intoM that we consider are equivariant with respect to these actions. Our main result is Theorem 1.7, which provides the list of symplectic–toric manifolds which admit a full packing by balls embedded in such a way, i.e. we prove existence and a uniqueness of http://arxiv.org/abs/0704.1034v1 (0,0) (2,0,0) (0,0,2) (0,2,0) (3,0) (0,3) Figure 1: Momentum polytope of (CP2, 3 · σFS) (left) and (CP3, 2 · σFS) (right). such manifolds. Our proofs rely on the discovery by Delzant [8] that symplectic–toric manifolds are classified by their convex images under the momentum map. This allows us to solve an a priori symplectic–geometric problem using techniques from Convex Geometry and Delzant theory (for a treatment of this theory see for example the book by Guillemin [14]). Definition 1.1 ([7]). A compact connected symplectic manifold M = (M, σ) is a symplectic– toric manifold, or a Delzant manifold, if it is equipped with an effective and Hamiltonian action of a torus of dimension half of the dimension of the manifold. ⊘ Delzant manifolds come equipped with a momentum map µM : M → Rn which satisfies iξMσ = d〈µM , ξ〉 for all ξ in the Lie algebra of the torus (see [14]), and where ξM is the vec- tor field on M induced by ξ (the existence of µ may be taken as definition of Hamiltonian action). µM carries M to a convex polytope in Rn (here we identify Rn with the dual of the Lie algebra of the torus; see Section 2 for details on how we make this non–canonical identification), and this polytope, which is called the momentum polytope of M , determines M up to equivariant symplec- tomorphism (see Theorem 2.4). Example 1.2 (Projective Spaces). The projective space (CPn, λ ·σFS) equipped with a λ multiple of the Fubini–Study form σFS = z̄izi) j 6=k(z̄jzj dzk ∧ dz̄k − z̄jzk dzj ∧ dz̄k) and the rotational action of Tn, (eiθ1 , . . . , eiθn) · [z0 : . . . : zn] = [z0 : e−2πiθ1 z1 : . . . : e−2πiθn zn], is a 2n–dimensional Delzant manifold with momentum map components µCP k (z) = λ|zk|∑ |zi|2 whose momentum polytope equals the convex hull in Rn of 0 and the scaled canonical vectors λe1, . . . , λen, see Figure 2. ⊘ Example 1.3 The product ( λi · σFS) is a Delzant manifold of dimension 2 ni. ⊘ Example 1.4 (Open Balls). The open symplectic ball (Br, σ0) in C n with the Tn action by rota- tions (component by component) has momentum map components µBrk (z) = |zk|2 and momentum polytope equal to the convex hull in Rn of 0 and the scaled canonical basis vectors r2e1, . . . , r The momentum polytope of (CPn, λ · σFS) equals the momentum polytope of B√λ minus the face of the momentum polytope of the former which faces the origin, which does not belong to the momentum polytope ∆Br of Br. ∆ Br is an integral simplex in the sense of Definition 2.5. ⊘ (0,0) (0,0)(1,0) (1,0) (1,1) (1,1) (2,1)(2,1) Figure 2: A manifold equivariantly symplectomorphic to (CP1 × CP1, σFS ⊕ σFS) packed by two equivariant symplectic balls of radius 1 in two different ways (see Lemma 2.12 for an explanation). Definition 1.5 An embedding f of the 2n-ball Br into a 2n–dimensional Delzant manifold M is equivariant if there exists an automorphism Λ of Tn such that the diagram n × Br Λ×f // f // M commutes, where ψ is a fixed effective and Hamiltonian Tn-action onM and · denotes the standard action by rotations on Br (component by component). In this case we say that f is a Λ–equivariant embedding. ⊘ Next we give a precise notion of “perfect equivariant symplectic ball packing”. In the following definition we use the term “family of maps” to mean a “collection of maps” in a set-theoretical fashion, i.e. we do not assume that this collection of maps has any additional structure. Definition 1.6 We define the real–valued mapping Ω on the space of 2n–dimensional Delzant manifolds by Ω(M) := (volσ(M)) −1 supE∈F f∈E volσ(f(Brf )), where F is the set of families E of equivariant symplectic ball embeddings such that if f, g ∈ E then f(Brf ) ∩ g(Brg) = ∅, and rf ≥ 0 for all f ∈ E . We say that M admits a perfect equivariant and symplectic ball packing if there exists a family E0 ∈ F such that Ω(M) = 1 at E0. ⊘ A version of Definition 1.6 in the “general” symplectic case, as well as the general symplectic packing problem, were introduced by McDuff and Polterovich in [19]. They denote Ω(M) by v(M, k) where k is the (a priori fixed) number of balls that we are embedding in M . McDuff and Polterovich consider that all balls have the same (a priori fixed) radius r > 0. This is in contrast to Definition 1.6 above, where neither the number of balls nor the radius are fixed. Theorem 1.7. LetM be a 2n–dimensional Delzant manifold. ThenM admits a perfect equivariant symplectic ball packing if and only if there exists λ > 0 such that 1. (n = 2) M is equivariantly symplectomorphic to either (CP2, λ ·σFS) or the product (CP1× 1, λ · (σFS ⊕ σFS)). 2. (n 6= 2) M is equivariantly symplectomorphic to (CPn, λ · σFS). (0,0,−1) (0,0,1) Figure 3: A one parameter family of perfect packings of (S2, 1 · dθ ∧ dh) ≃ (CP1, 2 · σFS) by a ball of radius 0 ≤ R ≤ 2 and a ball of radius 2−R2, see Remark 1.9. In the figure r = 0 and R = 1. In general, r = 1− R2. Equivalently, (CPn, λ · σFS) and (CP1 ×CP1, λ · (σFS ⊕ σFS)) with n ≥ 1 and λ > 0 are the only Delzant manifolds manifolds which admit a perfect equivariant and symplectic ball packing. The proof of Theorem 1.7 in any dimension follows from the abstract combinatorial structure of Delzant polytopes, the abstract notion of convexity, and its properties. In addition, a number of figures are presented along with the proof to suggest some intuition of the solution. In Section 3 we analyze in how many different ways the spaces which appear in Theorem 1.7 may be packed – this is summarized in Proposition 1.8 below; the existence statement in Proposition 1.8 is a simple construction, c.f. Remark 2.1, Figure 2, Figure 3, and it is independent from the proof of the uniqueness statement in Theorem 1.7, this last one being the part which occupies most of Section 3. The existence part of the statement of Theorem 1.7 follows from Proposition 1.8. If Proposition 1.8 is assumed, Theorem 1.7 becomes a uniqueness theorem. Proposition 1.8. For all λ > 0 and n ≥ 1, the complex projective space (CPn, λ · σFS) may be perfectly packed by one equivariant symplectic ball, and it may not be perfectly packed by two or more equivariant symplectic balls for n ≥ 2. If n = 1, λ > 0, (CPn, λ · σFS) may be perfectly packed only by one or two equivariant symplectic balls, and there is a one parameter family of packings by two equivariant symplectic balls, c.f. Figure 3. For all λ > 0, the 4–dimensional Delzant manifold (CP1×CP1, λ·(σFS⊕σFS)) may only be perfectly packed by two equivariant symplectic balls, and this in precisely two distinct ways, c.f. Figure 2. Remark 1.9 We identify the 2-sphere of radius R equipped with the standard area form dθ ∧ dh with (CP1, 4R2 · σFS), where σFS is the Fubini–Study form on CP1. Under the conventions that we use throughout the paper, the momentum map for (S2, 1 · dθ ∧ dh) is equal to (θ, h) 7→ h, and the momentum polytope is [−1, 1]. In the literature it seems (far) more common to have [−1, 1] as momentum polytope for S2 with the standard area form dθ ∧ dh; we do not follow this convention in order to have a simpler expression for µM(f(Br)) in Remark 2.1. Notice that the area of (S2, 1 · dθ ∧ dh) is 2 π, while the length of the associated momentum polytope is 2. ⊘ The paper is divided into five sections: in Section 2 we describe a problem in geometric– combinatorics equivalent to the toric symplectic ball packing problem; in Section 3 we solve it; in Section 4 we relate our results to the theory of blowing up; we end by raising some further questions in Section 5. 2 From Symplectic Geometry to Combinatorics Let M be a Delzant manifold of dimension 2n, and denote by Br the 2n–dimensional ball in C equipped with the restriction of the standard symplectic form σ0, and with the standard action by rotations of the n–torus Tn (see Definition 1.1; for the main properties of Delzant manifolds see Section 1 of [22], or for more details [7], [8], [14]). Recall that the main feature that makes the study of symplectic manifolds equipped with torus actions richer than the study of generic symplectic manifolds is the existence of the momentum map µM : M → Lie(Tn)∗ whose image ∆M is a convex polytope, called the momentum polytope of M , as shown independently by Atiyah and Guillemin–Sternberg [1], [12]. The momentum map is unique up to addition of a constant in (Lie(Tn))∗, and it is in this sense that we say “the” momentum map instead of “a” momentum map. Here we are identifying the Lie algebra Lie(Tn) and its dual Lie(Tn)∗ with Rn. We denote by χ(M) the Euler characteristic of M . Since this identification is not canonical we need to specify the convention we adopt in this paper. This amounts to choosing an epimorphism R → T1 which we take to be x 7→ e2 −1x. This epimorphism induces an isomorphism between Lie(T1) and R via ∂ 7→ 1/2 giving rise to a new isomorphism Lie(Tn) → Rn, ∂ 7→ 1/2 ek, by canonically identifying Lie(Tn) with the product of n copies of Lie(T1) (see Section 32 in [13] for more details). Remark 2.1 In Section 2 of [22] we proved that if f : Br →M is a Λ–equivariant and symplectic embedding with f(0) = p and µM(p) = x, then the following diagram is commutative: (Λt)−1+x // ∆M (2.1) It follows from diagram (2.1) that ifM is a 2n–dimensional Delzant manifold and f is a symplectic Λ–equivariant embedding from Br into M with f(0) = p and µ M(p) = x, then the momentum image µM(f(Br)) equals the subset of R n given by the convex hull of the points x and x + r2 αpi , i = 1, . . . , n, minus the convex hull of x+ r2 αpi , i = 1, . . . , n, where the α i are the characters of the isotropy representation of Tn on the tangent space at p to M . ⊘ In the case when M is a Delzant manifold, the momentum polytope ∆M of M is the so called Delzant polyope of M , and it satisfies specific properties as we see from Definition 2.2 below, which is a purely combinatorial definition – in this respect Delzant polytopes may be defined without reference to Delzant manifolds. (2,0)(0,0) (1,1)(0,1) (1,0)(0,0) (1,1)(0,1) Figure 4: Two Delzant polytopes (left) and a non-Delzant polytope (right). Definition 2.2 [[14], [7]] A Delzant polytope ∆ of dimension n in Rn is a simple, rational and smooth polytope. Here simple means that there are exactly n edges meeting at each vertex of ∆; rational means that the edges of ∆ meeting at the vertex x are rational in the sense that each edge is of the form x+t ui, t ≥ 0, ui ∈ Zn; smooth means that u1, . . . , un may be chosen to be a Z-basis of the integer lattice Zn. ⊘ Remark 2.3 A similar class of polytopes, Newton polytopes, have long been considered in alge- braic geometry [11]; the (only) difference between Newton and Delzant polyopes is that the former are required to have all of their vertices lying in the integer lattice Zn. We learned from Yael Karshon that “Delzant polytopes” have also been refered to as “non– singular”, “torsion–free” or “unimodular” by other authors. ⊘ Theorem 2.4 (Delzant, [8]). Two Delzant manifolds are equivariantly symplectomorphic if and only if they have the same Delzant polytope up to a transformation in SL(n, Z), and translation by an element in Rn. For every Delzant polytope ∆ there exists a Delzant manifold M∆ whose momentum polytope is precisely ∆. Definition 2.5 [Integral simplex] If Υ is an n–dimensional simplex (i.e. closed convex hull in n of n + 1 linearly independent points), an open simplex with respect to a vertex x of Υ is the convex region obtained from Υ by removing the only face of Υ which does not contain x. A region Σ ⊂ Rn is an open simplex if there is a simplex Υ such that Σ is an open simplex with respect to x for some vertex x of Υ. If Σ is an open simplex, then its closure in Rn is denoted by Σc. We say that an n–dimensional open simplex Σ ⊂ Rn is integral if: 1. The SL(n, Z)-length of each edge of Σ is the same for all edges, 2. Σ has the same Euclidean volume as a simplex d∆0 for some d ≥ 0, where ∆0 is the convex hull of 0 and the canonical basis vectors e1, . . . , en. Remark 2.6 If Σ is a (closed) simplex of Euclidean volume equal to the Euclidean volume of d∆0, where ∆0 was defined in Definition 2.5, then all of the edges of Σ meeting at a common vertex x of Σ have equal SL(nZ)-length if and only if there exists a transformation A ∈ SL(nZ) (0,0) (2,0) (0,1) (1,1) (0,0) (1,0) (0,1) (1,1) (0,0) (1,0) (0,1) (1,1) Figure 5: The closures of a coherent (left) and two non-coherent families of simplices (right). such that A(d∆0) = Σ. Similarly for open simplices – notice that an open simplex has a unique vertex. ⊘ Definition 2.7 [Coherent family of simplices] Let M a 2n–dimensional Delzant manifold with Delzant polytope ∆M . We say that a family E∆ of n–dimensional open simplices contained in ∆M is coherent if for every Σ ∈ E∆ the following three properties are satisfied: 1. The vertex xΣ of Σ is a vertex of ∆ 2. Every (n− 1)-dimensional face of Σ is contained in ∂(∆M ), 3. Σ is an integral simplex. The family E∆c of closed simplices Σc, where Σ ∈ E∆, is called the closure of E∆. ⊘ Remark 2.8 It follows from Definition 2.7 that coherent families of disjoint simplices contain at most χ(M) simplices, where χ(M) stands for Euler characteristic of M . Therefore the closure of such a family contains at most χ(M) simplices. ⊘ Theorem 2.9 (Atiyah [1], Guillemin–Sternberg [12]). Let M be a 2n–dimensional symplectic manifold equipped with an effective and Hamiltonian action of an m–dimensional torus Tm. Then the fibers (µM)−1(x), where x ∈ ∆M , of the momentum mapping µM , are connected subsets of M . The following is Theorem 2.10 in Guillemin’s book [14], adapted to fit the conventions for µM : M → Rn, introduced on the third paragraph of the section. Theorem 2.10 ([14]). The symplectic volume of a Delzant manifold Q of dimension 2m is equal to m! πm times the Euclidean volume of its momentum polytope ∆Q. Additionally, unlike in [14] we use volσ(S) = n, i.e. we do not normalize the integral by dividing by n!. Although Theorem 2.10 is stated only for Delzant manifolds, the result extends to Br ⊂ Cn. Corollary 2.11. The symplectic volume of Br is equal to n! πn times the Euclidean volume of ∆Br . If f : Br → M is an equivariant symplectic embedding, the symplectic volume of f(Br) is equal to n! πn times the Euclidean volume of µM(f(Br)), and to n! π n times the Euclidean volume of ∆Br . Both Theorem 2.10 and Corollary 2.11 are particular versions of the Duistermaat–Heckman theorem in [10], or Section 2 in [14]. We can now describe Ω in combinatorial terms; we denote the Euclidean volume measure in Rn by voleuc. Lemma 2.12. Let M a 2n–dimensional Delzant manifold and let Ω be the mapping defined in Definition 1.6. Let F∆ be the set of coherent families E∆ of pairwise disjoint simplices contained in ∆M . Then: Ω(M) = voleuc(∆M) E∆∈F∆ voleuc(Σ). (2.2) Furthermore,M admits a perfect equivariant and symplectic ball packing if and only if there exist a coherent family E∆ of pairwise disjoint simplices contained in ∆M and such that ∆M = ⋃Σ∈E∆ Σc. Proof. Write ν for the right hand–side of (2.2). It follows from Theorem 2.9 that a pairwise disjoint family of equivariant symplectic ball embeddings E0 = {fi} gives rise to a pairwise disjoint family {Σi := µM(fi(Bri))} ⊂ ∆M . By Remark 2.1, each Σi is an integral simplex contained in ∆M and the family E0 is coherent. Without loss of generality we assume that the supremum in the formula given in Definition 1.6 is achieved at the family E0. Since µM(fi(Bri)) = Σi and fi is symplectic and equivariant, by Corollary 2.11 volσ(fi(Bri)) = n! π n voleuc(Σi) and volσ(M) = n! πn voleuc(∆ M), which by plugging these values into the formula of Ω in Definition 1.6 evaluated at the family E0, implies that Ω(M) ≤ ν. One shows that Ω(M) ≥ ν, by starting with a coherent family of pairwise disjoint simplices and repeating this same argument. Now suppose that M admits a perfect equivariant and symplectic ball packing. Then by Def- inition 1.6, volσ(M) = i volσ(fi(Bri)) at certain family of embedded balls, and in this case the equality ∆M = M(fi(Bri)) holds, and by Remark 2.1 each momentum image µ M(fi(Bri)) is an integral simplex; finally since the images f(Bri) are pairwise disjoint, by Theorem 2.9 these simplices are pairwise disjoint. The converse is proved similarly. We will use Lemma 2.12 in Step 1 of the proof of Theorem 1.7. 3 Proof of Theorem 1.7 and Proposition 1.8 For clarity the proof is divided into six steps: Step 1. In this step we analyze the type of simplices which both form a coherent family as well as give rise to a perfect equivariant and symplectic packing. Lemma 3.1. If the coherent family E∆ contains only one open simplex, then M is equivariantly symplectomorphic to (CPn, λ · σFS) for some λ > 0. Otherwise there exist at least two disjoint simplices in E∆, and every simplex in the family has exactly one face which is not contained in ∂(∆M ). Proof. By Lemma 2.12, there exist a coherent family E∆ consisting of pairwise disjoint open simplices Σ contained in ∆M such that ∆M = Σ∈E∆ Σc. The coherence of E∆ implies that for each Σ ∈ E∆ there exists a vertex x ∈ ∆M such that there are exactly n faces of Σc of dimension n−1 which contain x, and each of them is contained in the boundary ∂(∆M ) of ∆M , which leaves only the (n − 1)-dimensional face of Σc which does not contain x, say the face F̂Σ, as possibly not contained in the boundary ∂(∆M ) of ∆M . Call U0 to the subfamily of E∆ consisting of those simplices such that one of their (n − 1)-dimensional faces is not contained in ∂(∆M ), and U1 to the subfamily of E∆ such that all simplices of this subfamily have all of their (n− 1)-dimensional faces contained in the boundary ∂(∆M ). Denoting by Θi = Σ∈Ui Σc , i = 0, 1, the observation made in the previous paragraph implies that Θ0 ∪Θ1 = ∆M and Θ0 ∩Θ1 = ∅. Now we distinguish two cases, according to whether Θ1 = ∅ or Θ1 6= ∅. Let us first assume that Θ1 6= ∅; then there exists an open simplex Σ contained in ∆M such that all of the (n − 1)- -dimensional faces of Σc are contained in ∂(∆ M ), and therefore ∂(Σc) ⊂ ∂(∆M ). This being the case, it follows from the convexity of ∆M that Σc = ∆ M . Since Σc = ∆ M , and by construction Σ ∈ E∆, where E∆ is a coherent family by Theorem 2.4, we conclude that ∆M is the Delzant polytope of a Delzant manifold M equivariantly symplectomorphic to the n-dimensional complex projective space (CPn, λ · σFS) for some λ > 0 (depending on the volume of M). If otherwise Θ1 = ∅, then Θ0 = ∆M and there are two cases: when |U| = 1 and when |U| > 1. Suppose first that |U| = 1. Then if Σ is the only open simplex in the family U0 = U the simplex Σc has a face which is not contained in ∂(∆M ), and therefore using the same argument as earlier in the proof we obtain that ∆M 6= Θ0∪Θ1, which contradicts the fact that E∆ = U0∪U1 is a coherent family. So |U| = 1 may not happen. Therefore we can pick two different simplices Σi ∈ U0, i = 0, 1. The statement of the lemma follows. From this point on, throughout this proof, we assume that the family E∆ contains at least two simplices, since the case where E∆ consists of precisely one simplex is solved in Lemma 3.1. In what follows let F̂Σi , i = 0, 1, be the only (n − 1)-dimensional face of (Σi)c which is not contained in ∂(∆M ). Step 2. We give a formula for F̂Σ0 as a disjoint union of two subpolytopes of ∆ M , one of which intersects F̂Σ0 at dimension n− 1 while the other intersects it at dimension < n− 1. Lemma 3.2. For each x ∈ F̂Σ0 there exists Σ′ ∈ E∆ such that x ∈ (Σ′)c and Σ′ 6= Σ0. Proof. First notice that there exists a unique hyperplane HΣi in R n which contains F̂Σi , i = 0, 1. For each positive integer n let Un := B1/n(x) ∩ (∆M \ (Σ0)c), where 〈·, ·〉 denotes the standard scalar product in Rn, and for each x ∈ Rn, δ > 0 we write Bδ(x) for the standard open ball in Rn centered at x and of radius δ with respect to 〈·, ·〉. We claim that Un 6= ∅ for all n. Indeed, writeHΣ0 = {x : 〈x, v〉 = λ} for some vector v in Rn and some constant λ ∈ R, and suppose that ∆M ⊂ H+Σ0 or ∆M ⊂ H , where H±Σ0 denote the closed subspaces of n at both sides of HΣ0 . Recall the following fact: Generic fact. Two subsets C1 and C2 of R n are separated by a hyperplane H if each lies in a different closed half–space H±. If y belongs to the closure of C1, a hyperplane that separates C1 and {y} is called a supporting hyperplane of C1 at y. In this case it is a generic (and easy to see) fact that H ∩ Int(C1) = ∅. Then HΣ0 is a supporting hyperplane of ∆ M with respect to any point which is in F̂Σ0 for ∆M and therefore HΣ0 ∩ Int(∆M) = ∅, which contradicts IntF̂Σ0 (F̂Σ0) ⊂ Int(∆ M) (the fact that (F̂Σ0) ⊂ Int(∆M ) follows from convexity). Therefore ∆M 6⊂ H+Σ0 and ∆ M 6⊂ H−Σ0 , which then implies the existence of zi ∈ Int(H±Σ0). Since ∆ M and B1/n(x) are convex, their intersection ∆M ∩ B1/n(x) is convex and so we may pick ǫ > 0 small enough such that yi := (1− ǫ) x+ ǫ zi ∈ ∆M ∩ B1/n(x), i = 0, 1. Since 〈v, z0〉 > λ and 〈v, z1〉 < λ, a computation then gives 〈v, y0〉 > λ and 〈v, y1〉 < λ, so precisely one of y0, y1 lies in (Σ0)c while the other lies in Un, so Un 6= ∅ as we wanted to show. For each integer n, pick yn ∈ Un, and observe that by construction the sequence {yn}∞n=1 converges to x. Since E∆ is finite, there exists a convergent subsequence {ynk}∞k=1 of {yn}∞n=1, and a simplex Σ′ ∈ E∆ such that ynk ∈ (Σ′)c for all k ≥ 1. Now Σ′ 6= Σ0 because ynk /∈ Σ0 but ynk ∈ Σ′, k ≥ 1, by construction. Finally since (Σ′)c is compact, x ∈ (Σ′)c as we wanted to show. Corollary 3.3. Let F∆ be the sufamily of E∆ consisting of those simplices Σ such that both F̂Σ0 ∩ F̂Σ 6= ∅ and dim(F̂Σ0 ∩ F̂Σ) < n− 1, and (F∆)′ the subfamily of E∆ consisting of those simplices Σ̂ 6= Σ0 such that dim(F̂Σ0 ∩ F̂Σ̂) = n − 1. Then F̂Σ0 may be expressed as the following union of two subsets of ∆M : F̂Σ0 = (F̂Σ0 ∩ ( ,Σc)) ∪ (F̂Σ0 ∩ ( Σ̂∈(F∆)′ Σ̂c)) (3.1) and the union is a disjoint one. Proof. The union given in expression (3.1) is clearly disjoint and we only need to show that F̂Σ0 ⊂ (F̂Σ0 ∩ ( Σc)) ∪ (F̂Σ0 ∩ ( Σ̂∈(F∆)′ Σ̂c)), (3.2) since the reverse inclusion is trivially true. Notice that showing that expression (3.2) holds is equivalent to showing that F̂Σ0 ⊂ ( Σc) ∪ ( Σ̂∈(F∆)′ Σ̂c), expression which is precisely equivalent to the statement of Lemma 3.2, which concludes the proof. Step 3. We prove that formula (3.1) implies (by coherence of E∆) that for all Σ ∈ E∆, the only faces of Σ0 and of Σ which are not contained in ∆ M are identical, i.e. we have the following. Lemma 3.4. F̂Σ0 = F̂Σ for all Σ ∈ E∆. Proof. Since E∆ is a coherent family of pairwise disjoint open simplices, every (n−1)-dimensional face of every closed simplex Σc ∈ E∆c , but the face F̂Σ, is contained in ∂(∆M ), and IntF̂Σ(F̂Σ) ⊂ Int(∆M). Therefore F̂Σ0 ∩ Σc = F̂Σ0 ∩ F̂Σ for all Σ ∈ E∆, which by expression (3.1) then implies that: F̂Σ0 = (F̂Σ0 ∩ F̂Σ) ∪ Σ̂∈(F∆)′ (F̂Σ0 ∩ F̂Σ̂). (3.3) Let us assume by contradiction that (F∆)′ = ∅ and notice that the left–most member of the right hand side of expression (3.3) is a union of convex polytopes of dimension strictly less that n − 1; furthermore since F∆ is a subfamily of the coherent family E∆, by Remark 2.8 F∆ is finite, and therefore we conclude that F̂Σ0 is a finite union of convex polytopes, the dimension of each of which is, by construction of F∆, strictly less than n − 1, which is a contradiction since by definition of Σ0 we have that dim(F̂Σ0) = n− 1; here we are using the following generic property of polytopes in Rn: Generic fact. Let ∆0, ∆1, . . . ,∆k be a finite family of polytopes in R n such that ∆0 = i=1∆i Then there exists j with 1 ≤ j ≤ k such that dim(∆0) = dim(∆j). Therefore (F∆)′ 6= ∅ and hence there exists Σ1 ∈ E∆ such that both Σ1 6= Σ0 and dim(F̂Σ0 ∩ F̂Σ1) = n − 1. Without loss of generality we may assume that Σ1 = Σ1. By definition of HΣi , F̂Σ0 ∩ F̂Σ1 ⊂ HΣ0 ∩HΣ1 which in particular implies that n− 1 = dim(F̂Σ0 ∩ F̂Σ1) ≤ dim(HΣ0 ∩HΣ1) ≤ dim(HΣ0) = n− 1 and therefore must have HΣ0 = HΣ1 – here we are using: Generic fact. If L, L′ are two hyperplanes in Rn whose intersection is (n − 1)–dimensional, then L = L′. Since ∆M ∩HΣi = F̂Σi and HΣ0 = HΣ1 we must have F̂Σ0 = F̂Σ1 . Step 4. Recall that from Step 3 onwards we have been assuming that the the coherent family E∆ contains at least two simplices Σ0, Σ1. Next we show that the fact that ∆ M is a Delzant polytope implies that ∆M equals the union of Σ0, Σ1, and hence there are no other simplices in the coherent family E∆. Lemma 3.5. E∆ contains precisely two simplices Σ0, Σ1 joined at their unique face (Σ0)c ∩ (Σ1)c, and ∆M = (Σ0)c ∪ (Σ1)c, where (Σ0)c and (Σ1)c are joined at their unique common face F̂Σ0 . Proof. Let us assume that there exists Σ′ 6= Σ0, Σ1 with Σ′ ∈ E∆. By assumption E∆ is a family of pairwise disjoint open simplices, so by Lemma 3.4 we have that (Σ0)c∩(Σ1)c = (Σ0)c∩(Σ′)c = (Σ1)c ∩ (Σ′)c = F̂Σ0 . On the other hand, since F̂Σ0 ⊂ ∂(Σ0) ∩ ∂(Σ1) ∩ ∂(Σ′) by construction, we have that ∅ = Int((Σ′)c) ∩ Int((Σ0)c ∪ (Σ1)c), and therefore taking the closure of both sides of this expression we obtain ∅ = (Σ′)c ∩ ((Σ0)c ∪ (Σ1)c) = ((Σ′)c ∩ (Σ0)c) ∪ ((Σ′)c ∩ (Σ1)c). We glue along horizontal faces to get = Figure 6: A 3-dimensional polytope obtained by gluing two 3-dimensional simplices S and S’ along a face does not satisfy the Delzant condition at those vertices contained in the hyperplane along which they are glued. Therefore F̂Σ0 = ∅, which is a contradiction. Hence there does not exist such Σ′, which then implies that E∆ = {Σ0,Σ1}, which proves the first claim of the lemma. Therefore, since the family E∆ is coherent, and by assumption it gives a perfect and equivariant symplectic packing of M , it follows that ∆M = (Σ0)c ∪ (Σ1)c, where (Σ0)c and (Σ1)c are joined at their unique common face F̂Σ0 which is in the interior of ∆ Step 5. In this step we analyze which Delzant polytopes in Rn may be obtained as the union of the closures of two open simplices. Lemma 3.6. Let ∆ be a convex polytope in Rn obtained as the union of the closures of two n- dimensional open simplices Σ− and Σ+ joined uniquely by the (n − 1)-dimensional face F̂ = (Σ−)c ∩ (Σ−)c, whose relative interior is contained in the interior of ∆. Then ∆ is unique, and if n > 2, ∆ is not a Delzant polytope, i.e. there does not exist a 2n–dimensional Delzant manifold M such that ∆ = ∆M . If n = 1, 2, ∆ is a Delzant polytope if and only if Σ− and Σ+ are integral simplices. Proof. First observe that ∆ is unique because ∆ ⊂ Rn, and ∆ is n-dimensional, and therefore the face F̂ is (n − 1)-dimensional, and the plane in which F̂ is contained is uniquely determined by any of its orthogonal vectors, see Remark 3.7. By assumption ∆ is equal to the union (Σ−)c ∪ (Σ+)c of the closures of the simplices Σ− and Σ+, the intersection of which equals an (n − 1)-dimensional simplex F̂ := (Σ−)c ∩ (Σ+)c, and hence F̂ has precisely n vertices. Every vertex of (Σ−)c and (Σ+)c outside of F̂ is also a vertex of ∆. Each other vertex of F̂ is also a vertex of ∆, but the converse need not be the case – if the converse holds, then ∆ has precisely n + 2 vertices, see Figure 6, of which precisely n vertices belong to F̂ , and of the two remaining vertices, one belongs to the simplex Σ− but does not belong to F̂ , while the other one belongs to the simplex Σ+ but does not belong to F̂ . Each individual vertex of F̂ is connected with each of the other n + 1 vertices of ∆ by an edge, and this is in contradiction with the Delzant property of ∆ unless n ≤ 2, specifically it contradicts the simplicity property that Delzant polytopes exhibit, c.f. Definition 2.2. Notice that it follows from Definition 2.5 that any interval of finite length which contains precisely one of its two endpoints is an integral simplex. If n = 1, all three of ∆, (Σ−)c, (Σ+)c are closed intervals of finite length and ∆ is the union of (Σ−)c and (Σ+)c, the intersection of which consists of exactly one point in the interior of ∆, c.f. Figure 3. If n = 2, see Figure 2, where the packings are explicitly presented. The integrality of Σ− and Σ+ is an essential requirement in order for ∆ to satisfy Definition 2.2. Remark 3.7 If in the statement of Lemma 3.6, the simplices Σ− and Σ+ where m-dimensional, with m < n, there are infinitely many different ways of gluing them in this fashion; this gluing leads to a convex polytope if and only if Σ− and Σ+ are contained in the same m-dimensional subspace of Rn. ⊘ Step 6. This is the conclusion step. A combination of the previous lemmas gives the proof of Theorem 1.7 and Proposition 1.8. Write Xn, λ = (CP n, λ · σFS) and Yλ = (CP1 × CP1, λ · σFS). Proof of Proposition 1.8. Clearly X1, λ may be packed either by one 2-ball or by two 2-balls by prescribing a point in its momentum polytope, which is an interval, c.f. Example 1.2, whose length depends on the real parameter λ, and it cannot be packed in any other way, c.f. Figure 3. Similarly, Yλ may be perfectly packed by two equivariant symplectic 2-balls, c.f. Figure 2. Now we show that Xn, λ, n > 2, may not be packed by two or more balls; if otherwise, there exists a coherent family of at least two balls, which realizes the perfect packing, and let us call Σ0 and Σ1 to the corresponding simplices to these two n-balls. Let ∆ be the momentum polytope of Xn, λ (a simplex in R n) . By Lemma 3.5, ∆ = (Σ0)c ∪ (Σ1)c, and hence by Lemma 3.6, ∆ is not a Delzant polytope because we are assuming that n > 2, which is a contradiction. It is left to show that Yλ may only be perfectly packed by two equivariant symplectic balls, and this is in precisely two ways. Recall that the existence part is clear. The fact that there are precisely two ways to use two balls to perfectly pack Yλ is also clear from Figure 2, since the ball images are integral simplices, see Definition 2.5. Notice that one equivariantly symplectically embedded ball fills up at most half of the volume of Yλ, and therefore it does not give a perfect packing, c.f. Figure 2. Now by Lemma 3.5, Yλ does not admit a perfect packing by three or more balls, which concludes the proof. Conclusion of the proof of Theorem 1.7. Lemma 3.5, Lemma 3.6 and Theorem 2.4, imply that the only symplectic–toric manifolds which admit a perfect packing are Xn, λ for arbitrary n ≥ 1 and Yλ for λ > 0. The sufficiency condition is implied by Proposition 1.8, which concludes the proof of Theorem 4 A Remark on blowing up The connection between symplectic ball embeddings and blowing up was first explored by D. McDuff in [18]. Let us first outline McDuff’s construction and afterwards we will state the blow up version of Theorem 1.1 in [22]. Let (M, σ) be a Delzant manifold and let J be a σ–tamed almost complex structure on M . Recall that we say that σ is J-standard near p ∈ M if the pair (σ, J) is diffeomorphic to the standard pair (σ0, −1) of R2n near 0. Choose a σ-standard almost complex structure J for which σ is J-standard near p and denote by Θ : (M̃, J̃) → (M, J) to the complex blow up of M at p. Let f be a symplectic embedding from Br into M which is holomorphic with respect to the standard multiplication by −1 on Br and J , near 0, and such that f(0) = p. Such an embedding gives rise to a symplectic form σ̃f on M̃ which lies in the cohomology class [Θ∗σ]−πr2e where e is the Poincaré dual to the homology class of the exceptional divisor Θ−1(p). The form σ̃f is called the symplectic blow up of σ with respect to f , and is uniquely determined up to isotopy of forms. For the specific construction see [20] pages 223-225. McDuff and Polterovich showed that the same construction extends for arbitrary symplectic embeddings from Br into M without having to assume holomorphicity near 0. Roughly speaking, one perturbs the embedding slightly to make it holomorphic near 0, and define its blow up as the blow up of the perturbed embedding. They also showed that the isotopy class of the form σ̃f depends only on the embedding f and the germ of J at p and that if two symplectic embeddings f1 and f2 are isotopic through a family of symplectic embeddings of Br which take 0 to p, then the corresponding blow up forms are isotopic. On the other hand a (more general) version of the following result was proved in [21]: Theorem 4.1. ([22]). LetM be a 2n–dimensional Delzant manifold. Then the space of equivariant symplectic embeddings from the 2n–ball Br into M which send the origin to the same fixed point p ∈M , is homotopically equivalent to the n–torus Tn. And from this result we are able to describe equivariant blow up at a fixed point. Corollary 4.2. Let f1 and f2 be equivariant symplectic embeddings from the 2n–dimensional ball Br into a 2n–dimensional Delzant manifold M . If the normalization condition f1(0) = f2(0) = p holds, then the corresponding blow up manifolds (M̃, σ̃f1) and (M̃, σ̃f2) at p are isotopic, in the sense that the symplectic forms σ̃f1 and σ̃f2 may be joined by a continuous path σt, 0 ≤ t ≤ 1, of symplectic forms with σ0 = σ̃f1 and σ1 = σ̃f2 . Proof. Follows by observing that if f1 and f2 are equivariant symplectic embeddings from the 2n–dimensional ball Br into a 2n–dimensional Delzant manifold M such that f1(0) = f2(0) = p, then by Theorem 4.1 they are isotopic. 5 Further questions The following questions regard generalizations of the work presented in this paper. Question 5.1 Given 0 ≤ r ≤ 1, find all Delzant manifolds M such that Ω(M) = r. For r = 0 it is a trivial question, and we answered the case r = 1 in Theorem 1.7. What can we say for r = 1/2? In other words, to what extent does Ω encodes the geometry of a Delzant manifold? Are there special values of r other than 0 and 1 for which the list of Delzant manifolds M such that Ω(M) = r is finite, up to equivariant symplectomorphism? ⊘ In [22] we discussed on the topology of the space of partially equivariant embeddings and suggested a result in this direction. Recall from [22] that the notion of Λ–equivariance (Λ ∈ Aut(Tn)) has a natural extension: Definition 5.2 An embedding from Br into M is σ-equivariant if there is a monomorphism γ : Tn−k → Tn, k ∈ {1, . . . , n− 1} such that the following diagram commutes n−k × Br γ×f // f // M Mγ is the set of p ∈ M such that ψ(γ(t), p) = p for all t ∈ Tn−k, and the rest of terminology is also analogous to that of [22]. Definition 5.3 We define the real–valued mapping Ωn−k on the space of 2n–dimensional Delzant manifolds by Ωn−k(M) := (volσ(M)) −1 sup En−k∈Fn−k volσ(f(Br)) where Fn−k is the family of sets En−k of partially equivariant embeddings of degree n− k and such that their images f(Br) are pairwise disjoint (degree n − k in the sense of the monomorphism γ defined at the begining of this section having domain a (n − k)-torus) We say that M admits a perfect equivariant and symplectic ball packing of degree n− k if Ωn−k(M) = 1 at certain family E0n−k. ⊘ Question 5.4 Does for every natural number n, every 2n–dimensional Delzant manifold admit perfect ball packing of degree k for all k such that 1 < k ≤ n? Or suppose that M admits a perfect symplectic ball packing, is there k > 1 such that M admits a perfect equivariant symplectic ball packing of degree n− k? ⊘ Question 5.4 is directly related to the partially equivariant version of Theorem 1.2, which was introduced in [22]. Finally, Delzant’s theorem has been recently generalized in [9] to “symplectic manifolds whose principal torus orbits are coisotropic”, and one could ask the same question treated in the present paper for the symplectic manifolds in [9]. Acknowledgments. The author is grateful to J.J. Duistermaat for discussions, and for his hos- pitality while the author visited Utrecht University (December 2004 and May 2005). He is also thankful to D. Auroux and V. Guillemin for discussions and for hosting him at the MIT in a number of occasions (during Fall 2003, Winter 2004). Finally, he is grateful E. Veomett for comments on a preliminary version, and the referee of the paper for his/her suggestions, which have improved the paper. References [1] M. Atiyah, Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc., 1, (1982) 1–15. [2] P. Biran, A stability property of symplectic packing, Invent. Math. 136 (1999), no. 1, 123– [3] P. Biran, Symplectic packing in dimension 4.Geom. Funct. Anal. 7 (1997), no. 3, 420–437. [4] P. Biran, Connectedness of spaces of symplectic embeddings, Int. Math. Res. Lett., 10 (1996) 487–491. [5] P. Biran, From symplectic packing to algebraic geometry and back, Proc. 3rd Europ. Congr. of Math., Barcelona 2, Prog. in Math., Birkhauser (2001) 507–524. [6] P. Biran, Geometry of symplectic intersections, Proc. ICM Beijing III (2004) 1-3. [7] A. Cannas da Silva, Lectures in symplectic geometry, Springer–Verlag, Berlin, (2000). [8] T. Delzant, Periodic Hamiltonians and convexity images of the momentum mapping, Bull. Soc. Math. France, 116 (1988) 315–339. [9] J.J. Duistermaat, A. Pelayo, Symplectic torus actions with coisotropic principal orbits, Preprint of January 2006. [10] J.J. Duistermaat, G.J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69 (1982), no. 2, 259–268. [11] W. Fulton, Introduction to toric varieties, Ann. Math. Stud. 131, Princ. Univ. Press 1993. [12] V. Guillemin and S. Sternberg, Convexity properties of the momentum mapping, Invent. Math., 67 (1982) 491–513. [13] V. Guillemin and S. Sternberg, Symplectic techniques in physics, Camb. Univ. Press, 1984 [14] V. Guillemin, Moment maps and combinatorial invariants of Hamiltonian Tn spaces, Prog. in Math., Birkauser, (1994). [15] V. Guillemin, V. Ginzburg and Y. Karshon, Moment maps, cobordisms and Hamiltonian group actions, Amer. Math. Soc. Monograph 98, (2002). [16] B.S. Kruglikov, A remark on symplectic packings. (Russian) Dokl. Akad. Nauk 350 (1996), no. 6, 730–734. [17] F. Lalonde, M. Pinsonnault. The topology of the space of symplectic balls in rational 4- manifolds. Duke Math. J. 122 (2004), no. 2, 347–397. [18] D. McDuff, From deformation to isotopy, Topics in Symplectic 4-manifolds, Internat. Press, Cambridge, MA (1998). [19] D. McDuff, L. Polterovich, Symplectic packings and algebraic geometry. With an appendix by Y. Karshon. Invent. Math. 115 (1994), no. 3, 405–434. [20] D. McDuff and D.A. Salamon, Introduction to Symplectic Topology, 2nd edition, OUP, (1998). [21] F. M. Maley, J. Mastrangeli, L. Traynor, Symplectic packings in cotangent bundles of tori. Experiment. Math. 9 (2000), no. 3, 435–455. [22] A. Pelayo, Topology of spaces of equivariant symplectic embeddings, Proc. Amer. Math. Soc. 135 (2007) 277–288. [23] L. Traynor, Symplectic packing constructions. J. Diff. Geom. 41 (1995), no. 3, 735–751. [24] G. Xu, Curves in P 2 and symplectic packings. Math. Ann. 299 (1994), no. 4, 609–613. A. Pelayo Department of Mathematics, University of Michigan 2074 East Hall, 530 Church Street, Ann Arbor, MI 48109–1043, USA e-mail: [email protected] The Main Theorem From Symplectic Geometry to Combinatorics Proof of Theorem ?? and Proposition ?? A Remark on blowing up Further questions

0704.1035

Anisotropic brane gravity with a confining potential

arXiv:0704.1035v1 [gr-qc] 8 Apr 2007 Anisotropic brane gravity with a confining potential M. Heydari-Fard 1∗and H. R. Sepangi† Department of Physics, Shahid Beheshti University, Evin, Tehran 19839, Iran December 4, 2018 Abstract We consider an anisotropic brane world with Bianchi type I and V geometries where the mecha- nism of confining the matter on the brane is through the use of a confining potential. The resulting equations on the anisotropic brane are modified by an extra term that may be interpreted as the x-matter, providing a possible phenomenological explanation for the accelerated expansion of the universe. We obtain the general solution of the field equations in an exact parametric form for both Bianchi type I and V space-times. In the special case of a Bianchi type I the solutions of the field equations are obtained in an exact analytic form. Finally, we study the behavior of the observationally important parameters. 1 Introduction The type Ia supernovae (SNe Ia) [1] observations provide the first evidence for the accelerating expan- sion of the present universe. These results, when combined with the observations on the anisotropy spectrum of cosmic microwave background (CMB) [2] and the results on the power spectrum of large scale structure (LSS) [3], strongly suggest that the universe is spatially flat and dominated by a component, though arguably exotic, with large negative pressure, referred to as dark energy [4]. The nature of such dark energy constitutes an open and tantalizing question connecting cosmology and particle physics. Different mechanisms have been suggested over the past few years to accommodate dark energy. The simplest form of dark energy is the vacuum energy (the cosmological constant). A tiny positive cosmological constant which can naturally explain the current acceleration would encounter many theoretical problems such as the fine-tuning problem and the coincidence problem. The former can be stated as the existence of an enormous gap between the vacuum expectation value, in other words the cosmological constant, in particle physics and that observed over cosmic scales. The absence of a fundamental mechanism which sets the cosmological constant to zero or to a very small value is also known as the cosmological constant problem. The latter however relates to the question of the near equality of energy densities of the dark energy and dark matter today. Another possible form of dark energy is a dynamical, time dependent and spatially inhomogeneous component, called the quintessence [5]. An example of quintessence is the energy associated with a scalar field φ slowly evolving down its potential V (φ) [6, 7]. Slow evolution is needed to obtain a negative pressure, pφ = φ̇2+V (φ), so that the kinetic energy density is less than the potential energy density. Yet another phenomenological explanation based on current observational data is given by the x-matter (xCDM) model which is associated with an exotic fluid characterized by an equation of state px = wxρx (wx < − is the necessary condition to make a universe accelerate), where the parameter wx can be a constant or more generally a function of time [8]. ∗email: [email protected] †email: [email protected] http://arxiv.org/abs/0704.1035v1 Over the past few years, we have been witnessing a phenomenal interest in the possibility that our observable four-dimensional (4D) universe may be viewed as a brane hypersurface embedded in a higher dimensional bulk space. Physical matter fields are confined to this hypersurface, while gravity can propagate in the higher dimensional space-time as well as on the brane. The most popular model in the context of brane world theory is that proposed by Randall and Sundrum (RS). In the so-called RSI model [9], the authors proposed a mechanism to solve the hierarchy problem with two branes, while in the RSII model [10], they considered a single brane with a positive tension, where 4D Newtonian gravity is recovered at low energies even if the extra dimension is not compact. This mechanism provides us with an alternative to compactification of extra dimensions. The cosmological evolution of such a brane universe has been extensively investigated and effects such as a quadratic density term in the Friedmann equations have been found [11]-[13]. This term arises from the imposition of the Israel junction conditions which is a relationship between the extrinsic curvature and energy-momentum tensor of the brane and results from the singular behavior in the energy-momentum tensor. There has been concerns expressed over applying such junction conditions in that they may not be unique. Indeed, other forms of junction conditions exist, so that different conditions may lead to different physical results [14]. Furthermore, these conditions cannot be used when more than one non-compact extra dimension is involved. To avoid such concerns, an interesting higher-dimensional model was introduced in [15] where particles are trapped on a 4-dimensional hypersurface by the action of a confining potential V. In [16], the dynamics of test particles confined to a brane by the action of such potential at the classical and quantum levels were studied and the effects of small perturbations along the extra dimensions investigated. Within the classical limits, test particles remain stable under small perturbations and the effects of the extra dimensions are not felt by them, making them undetectable in this way. The quantum fluctuations of the brane cause the mass of a test particle to become quantized and, interestingly, the Yang-Mills fields appear as quantum effects. Also, in [17], a braneworld model was studied in which the matter is confined to the brane through the action of such a potential, rendering the use of any junction condition unnecessary and predicting a geometrical explanation for the accelerated expansion of the universe. In brane theories the covariant Einstein equations are derived by projecting the bulk equations onto the brane. This was first done by Shiromizu, Maeda and Sasaki (SMS) [18] where the Gauss- Codazzi equations together with Israel junction conditions were used to obtain the Einstein field equations on the 3-brane. In a series of recent papers a number of authors [23, 24] have presented detailed descriptions of the dynamic of homogeneous and anisotropic brane worlds in the SMS model. The study of anisotropic homogeneous brane world cosmological models has shown an important difference between these models and standard 4D general relativity, namely, that brane universes are born in an isotropic state. For the anisotropic Bianchi type I and V geometries, with a conformally flat bulk (vanishing Weyl tensor), this type of behavior has been found by exactly solving the field equations [25]. In this paper, we follow [17] and consider an m-dimensional bulk space without imposing the Z2 symmetry. As mentioned above, to localize the matter on the brane, a confining potential is used rather than a delta-function in the energy-momentum tensor. The resulting equations on the anisotropic brane are modified by an extra term that may be interpreted as the x-matter, providing a possible phenomenological explanation for the accelerated expansion of the universe. The behavior of the observationally important physical quantities such as anisotropy and deceleration parameter is studied in this scenario. We should emphasize here that there is a difference between the model presented in this work and models introduced in [19, 20] in that in the latter no mechanism is introduced to account for the confinement of matter to the brane. 2 The model In this section we present a brief review of the model proposed in [16]. Consider the background manifold V 4 isometrically embedded in a pseudo-Riemannian manifold Vm by the map Y : V 4 → Vm such that ,ν = ḡµν , GABY a = 0, GABN b = gab = ±1, (1) where GAB (ḡµν) is the metric of the bulk (brane) space Vm(V 4) in arbitrary coordinates, {Y A} ({xµ}) is the basis of the bulk (brane) and NAa are (m − 4) normal unit vectors, orthogonal to the brane. Perturbation of V̄4 in a sufficiently small neighborhood of the brane along an arbitrary transverse direction ξ is given by ZA(xµ, ξa) = YA + (LξY) A, (2) where L represents the Lie derivative and ξa (a = 1, 2, ...,m− 4) is a small parameter along NAa that parameterizes the extra noncompact dimensions. By choosing ξ orthogonal to the brane, we ensure gauge independency [16] and have perturbations of the embedding along a single orthogonal extra direction N̄a giving local coordinates of the perturbed brane as ZA,µ(x ν , ξa) = YA,µ + ξ aN̄Aa,µ(x ν). (3) In a similar manner, one can find that since the vectors N̄A depend only on the local coordinates xµ, they do not propagate along the extra dimensions. The above assumptions lead to the embedding equations of the perturbed geometry Gµν = GABZ ,ν , Gµa = GABZ a, GABN b = gab. (4) If we set NAa = δ a , the metric of the bulk space can be written in the following matrix form GAB = gµν +AµcA ν Aµa Aνb gab , (5) where gµν = ḡµν − 2ξ aK̄µνa + ξ aξbḡαβK̄µαaK̄νβb, (6) is the metric of the perturbed brane, so that K̄µνa = −GABY a;ν , (7) represents the extrinsic curvature of the original brane (second fundamental form). We use the notation Aµc = ξ dAµcd, where Aµcd = GABN c = Āµcd, (8) represents the twisting vector fields (the normal fundamental form). Any fixed ξa signifies a new perturbed geometry, enabling us to define an extrinsic curvature similar to the original one by K̃µνa = −GABZ a;ν = K̄µνa − ξ K̄µγaK̄ νb +AµcaA . (9) Note that definitions (5) and (9) require K̃µνa = − . (10) In geometric language, the presence of gauge fields Aµa tilts the embedded family of sub-manifolds with respect to the normal vector NA. According to our construction, the original brane is orthogonal to the normal vector NA. However, equation (4) shows that this is not true for the deformed geometry. Let us change the embedding coordinates and set XA,µ = Z ,µ − g abNAa Abµ. (11) The coordinates XA describe a new family of embedded manifolds whose members are always or- thogonal to NA. In this coordinates the embedding equations of the perturbed brane is similar to the original one, described by equation (1), so that YA is replaced by XA. This new embedding of the local coordinates is suitable for obtaining induced Einstein field equations on the brane. The extrinsic curvature of a perturbed brane then becomes Kµνa = −GABX a;ν = K̄µνa − ξ bK̄µγaK̄ νb = − , (12) which is the generalized York’s relation and shows how the extrinsic curvature propagates as a result of the propagation of the metric in the direction of extra dimensions. The components of the Riemann tensor of the bulk written in the embedding vielbein {XA,α,N a }, lead to the Gauss-Codazzi equations Rαβγδ = 2g abKα[γaKδ]βb +RABCDX ,δ , (13) 2Kα[γc;δ] = 2g abA[γacKδ]αb +RABCDX ,δ , (14) where RABCD and Rαβγδ are the Riemann tensors for the bulk and the perturbed brane respectively. Contracting the Gauss equation (13) on α and γ, we find Rµν = (KµαcK ν −KcK µν ) +RABX ,ν − g abRABCDN b , (15) and the Einstein tensor of the brane is given by Gµν = GABX ,ν +Qµν + g abRABN b gµν − g abRABCDN b , (16) where Qµν = −g KγµaKγνb −KaKµνb KαβaK αβa −KaK gµν . (17) As can be seen from the definition of Qµν , it is independently a conserved quantity, that is Q ;ν = 0 [19]. Using the decomposition of the Riemann tensor into the Weyl curvature, the Ricci tensor and the scalar curvature RABCD = CABCD − (m− 2) GB[DRC]A − GA[DRC]B (m− 1)(m − 2) R(GA[DGC]B), (18) we obtain the 4D Einstein equations as Gµν = GABX ,ν +Qµν − Eµν + (m− 2) gabRABN b gµν (m− 2) (m− 1)(m − 2) Rgµν , (19) where Eµν = g abCABCDN ,ν , (20) is the electric part of the Weyl tensor CABCD. Now, let us write the Einstein equation in the bulk space as AB + Λ (b)GAB = α ∗SAB, (21) where SAB = TAB + VGAB , (22) here α∗ = 1 (M∗ is the fundamental scale of energy in the bulk space), Λ (b) is the cosmological constant of the bulk and TAB is the energy-momentum tensor of the matter confined to the brane through the action of the confining potential V. We require V to satisfy three general conditions: firstly, it has a deep minimum on the non-perturbed brane, secondly, depends only on extra coordi- nates and thirdly, the gauge group representing the subgroup of the isometry group of the bulk space is preserved by it [16]. The vielbein components of the energy-momentum tensor are given by Sµν = SABX ,ν , Sµa = SABX a , Sab = SABN b . (23) Use of equation (22) then gives RAB = − (m− 2) GABS + (m− 2) Λ(b)GAB + α ∗SAB, (24) R = − (α∗S −mΛ(b)). (25) Substituting RAB and R from the above into equation (19) and using equation (23), we obtain Gµν = Qµν − Eµν + (m− 3) (m− 2) α∗gabSabgµν + (m− 2) Sµν − (m− 4)(m− 3) (m− 1)(m− 2) α∗Sgµν (m− 7) (m− 1) Λ(b)gµν . (26) On the other hand, again from equation (21), the trace of the Codazzi equation (14) gives the “gravi- vector equation” Kδaγ;δ −Ka,γ −AbaγK b +AbaδK bδ +Baγ = 3(m− 4) α∗Saγ , (27) where Baγ = g mnCABCDN n . (28) Finally, the “gravi-scalar equation” is obtained from the contraction of (15), (19) and using equation S − gmnSmn gab = (Q+R+W ) gab − Λ(b)gab, (29) where W = gabgmnCABCDN a . (30) Equations (26)-(30) represent the projections of the Einstein field equations on the brane-brane, bulk-brane, and bulk-bulk directions. As was mentioned in the introduction, localization of matter on the brane is realized in this model by the action of a confining potential. Since the potential V is assumed to have a minimum on the brane, which can be taken as zero, localization of matter may simply be realized by taking equation (22) and consider its components on the brane, in which case we may write ατµν = (m− 2) Tµν , Tµa = 0, Tab = 0, (31) where α is the scale of energy on the brane. Now, the induced Einstein field equations on the original brane can be written as Gµν = ατµν − (m− 4)(m− 3) 2(m− 1) ατgµν − Λgµν +Qµν − Eµν , (32) where Λ = − (m−7) (m−1) Λ(b) and Qµν is a conserved quantity which according to [19] may be considered as an energy-momentum tensor of a dark energy fluid representing the x-matter, the more common phrase being ‘x-Cold-Dark Matter’ (xCDM). This matter has the most general form of the equation of state which is characterized by the following conditions [21]: violation of the strong energy condition at the present epoch for ωx < −1/3 where px = ωxρx, local stability i.e. c s = δpx/δρx ≥ 0 and preservation of causality i.e. cs ≤ 1. Ultimately, we have three different types of ‘matter’ on the right hand side of equation (32), namely, ordinary confined conserved matter represented by τµν , the matter represented by Qµν which will be discussed later and finally, the Weyl matter represented by Eµν . 3 Field equations on the anisotropic brane In the following we will investigate the influence of the extrinsic curvature terms on the anisotropic universe described by Bianchi type I and V geometries. We restrict our analysis to a constant curvature bulk, so that Eµν = 0. The constant curvature bulk is characterized by the Riemann tensor RABCD = k∗(GACGBD − GADGBC), (33) where GAB denotes the bulk metric components in arbitrary coordinates and k∗ is either zero for the flat bulk, or proportional to a positive or negative bulk cosmological constant respectively, corre- sponding to two possible signatures (4, 1) for the dS5 bulk and (3, 2) for the AdS5 bulk. We take, in the embedding equations, g55 = ε = ±1. With this assumption the Gauss-Codazzi equations reduce Rαβγδ = (KαγKβδ −KαδKβγ) + k∗(gαγgβδ − gαδgβγ), (34) Kα[β;γ] = 0. (35) The effective equations derived in the previous section with constant curvature bulk can be written Gµν = ατµν − λgµν +Qµν . (36) Here, λ is the effective cosmological constant in four dimensions with Qµν being a completely geo- metrical quantity given by Qµν = (KKµν −KραK ν ) + αβ −K2 , (37) where K = gµνKµν . To proceed further, the confined source on the brane should now be specified. The most common matter source used in cosmology is that of a perfect fluid which, in co-moving coordinates, is given by τµν = (ρ+ p)uµuν + pgµν , uµ = −δ µ, p = (γ − 1)ρ, 1 ≤ γ ≤ 2, (38) where γ = 2 represents the stiff cosmological fluid describing the high energy density regime of the early universe. From a formal point of view the Bianchi type I and V geometries are described by the line element ds2 = −dt2 + a21(t)dx 2 + a22(t)e −2βxdy2 + a23(t)e −2βxdz2. (39) The metric for the Bianchi type I geometry corresponds to the case β = 0, while for the Bianchi type V case we have β = 1. Here ai(t), i = 1, 2, 3 are the expansion factors in different spatial directions. For later convenience, we define the following variables [25] ai, Hi = , i = 1, 2, 3, 3H = Hi, ∆Hi = Hi −H, i = 1, 2, 3. (40) In equation (40), v is the volume scale factor, Hi, i = 1, 2, 3 are the directional Hubble parameters, and H is the mean Hubble parameter. From equation (40) we also obtain H = v̇ . The physical quantities of observational importance in cosmology are the expansion scalar Θ, the mean anisotropy parameter A, and the deceleration parameter q, which are defined according to Θ = 3H, 3A = , q = H−1 − 1 = −H−2 Ḣ +H2 . (41) We note that A = 0 for an isotropic expansion. Moreover, the sign of the deceleration parameter indicates how the universe expands. A positive sign for q corresponds to the standard decelerating models whereas a negative sign indicates an accelerating expansion in late times. Using the York’s relation Kµνa = − , (42) we realize that in a diagonal metric, Kµνa is diagonal. After separating the spatial components, the Codazzi equations reduce to (here α, β, γ, σ = 1, 2, 3) Kαγa,σ +K βσ = K σa,γ +K βγ , (43) Kαγa,0 + Kαγa = K00a, i = 1, 2, 3. (44) The first equation gives K11a,σ = 0 for σ 6= 1, since K 1a does not depend on the spatial coordinates. Repeating the same procedure for α, γ = i, i = 2, 3, we obtain K22a,σ = 0 for σ 6= 2 and K 3a,σ = 0 for σ 6= 3. Assuming K11a = K 2a = K 3a = ba(t), where ba(t) are arbitrary functions of t, the second equation gives ḃa + K00a, i = 1, 2, 3. (45) Summing equations (45) we find K00a = − 3ḃav . (46) For µ, ν = 1, 2, 3 we obtain Kµνa = bagµν . (47) Assuming further that the functions ba are equal and denoting ba = b, θ = and Θ = v̇ , we find from equation (37) that αβ = b2 , K = b , (48) Qµν = − gµν , µ, ν = 1, 2, 3, Q00 = . (49) Now, using these relations and equation (36), the modified Friedmann equations become 3Ḣ + H2i = λ− ρ(3γ − 2) + , (50) (vHi) = 2β 2v−2/3 + λ− ρ(γ − 2) + , i = 1, 2, 3. (51) For β = 0 we obtain the field equations for Bianchi type I geometry, while β = 1 gives the Bianchi type V equations on the brane world. These equations are modified with respect to the standard equations by the components of the extrinsic curvature. Such term may be used to offer an explanation for the x-matter. In the next section we discuss the ramifications of this term on the cosmology of our model. We also note the implicit effects of the bulk signature ε on the expansion of the universe. For the sake of completeness, let us compare the model presented in this work to the usual brane worlds models where the Israel junction condition is used to calculate the extrinsic curvature in terms of the energy-momentum tensor on the brane and its trace, that is Kµν = − α∗2(τµν − τgµν), (52) where α∗ is proportional to the gravitational constant in the bulk. If we did that, we would obtain b(t) = −1 α∗2ρ, which, upon its substitution in equation (50), gives 3Ḣ + H2i = λ+ ρ(2− 3γ) + ρ2(1− 3γ), (53) which is same as equation (16) in [25]. Therefore, the emergence of a ρ2 term in the Friedmann equations is a consequence of the discontinuity in the bulk and the brane system. The existence of this term either does not agree with observations or requires extra parameters and fine tuning. 4 Dark energy and role of extrinsic curvature As we noted before, Qµν is an independently conserved quantity, suggesting that an analogy with the energy momentum of an uncoupled non-conventional energy source would be in order. To evaluate the compatibility of such geometrical model with the present experimental data, we identify Qµν with x-matter [21] by defining Qµν as a perfect fluid and write Qµν = [(ρx + px)uµuν + pxgµν ] , px = (γx − 1)ρx. (54) Comparing Qµν , µ, ν = 1, 2, 3 and Q00 from equation (54) with the components of Qµν and Q00 given by equation (49), we obtain px = − , ρx = . (55) Equation (54) was chosen in accordance with the weak-energy condition corresponding to the positive energy density and negative pressure with ε = +1. Use of the above equations leads to an equation for b(t) . (56) If γx is taken as a constant, the solution for b(t) is b(t) = b0v −γx/2. (57) Using equation (55) and this solution, the energy density of xCDM becomes v−γx . (58) A brief discussion on the energy-momentum conservation on the brane would be appropriate at this point. The contracted Bianchi identities in the bulk space, GAB;A = 0, using equation (21), imply TAB + = 0. (59) Since the potential V has a minimum on the brane, the above conservation equation reduces to τµν;µ = 0, (60) and gives γρ = 0. (61) Thus, the time evolution of the energy density of the matter is given by ρ = ρ0v −γ . (62) Using the geometrical energy density for Qµν and the evolution law of the matter energy density, the field equations (50)-(51) become 3Ḣ + H2i = λ+ ρ0(2− 3γ)v b20(2− 3γx)v −γx , (63) (vHi) = 2β 2v−2/3 + λ+ ρ0(2− γ)v b20(2− γx)v −γx , i = 1, 2, 3. (64) Summing equations (64) we find (vH) = 2β2v−2/3 + λ+ ρ0(2− γ)v b20(2− γx)v −γx . (65) Now, substituting back equation (65) into equations (64) we obtain Hi = H + , i = 1, 2, 3, (66) with hi, i = 1, 2, 3 being constants of integration satisfying the consistency condition i=1 hi = 0. The basic equation describing the dynamics of the anisotropic brane world with a constant curvature bulk can be written as v̈ = 6β2v1/3 + 3λv + ρ0(2− γ)v 1−γ + b20(2− γx)v 1−γx . (67) Here, we note that for a stiff fluid (γ = 2), the dynamics of the matter on the brane is solely determined by the geometrical matter (xCDM). The general solution of equation (67) becomes t− t0 = 9β2v4/3 + 3λv2 + 3αρ0v 2−γ + 9b20v 2−γx + C )−1/2 dv, (68) where C is a constant of integration. The time variation of the physically important parameters described above in the exact parametric form, with v taken as a parameter, is given by Θ = 3H = 9β2v4/3 + 3λv2 + 3αρ0v 2−γ + 9b20v 2−γx + C , (69) ai = a0iv 1/3 exp 9β2v4/3 + 3λv2 + 3αρ0v 2−γ + 9b20v 2−γx +C )−1/2 , i = 1, 2, 3, (70) A = 3h2 9β2v4/3 + 3λv2 + 3αρ0v 2−γ + 9b20v 2−γx + C , (71) 9β2v4/3 + 2−γ + 2−γx + 3C 9β2v4/3 + 3λv2 + 3αρ0v2−γ + 9b 2−γx + C ) − 1, (72) where h2 = i=1 h i . In addition, the integration constants hi and C must satisfy the consistency condition h2 = 2 C. For β = 0 we obtain the general solutions for Bianchi type I geometry, while β = 1 gives the Bianchi type V solutions on the anisotropic brane world. In a matter dominated Bianchi type I universe, γ = 1 with γx = 0, equation (68) becomes t− t0 = 3αρ0v + 9b 2 + C , (73) where for later convenience, we take C = 3α2̺2 . The time dependence of the volume scale factor of the Bianchi type I universe is given by v(t) = e3b0(t−t0) − , (74) which for t = t0+ becomes zero. By reparameterizing the initial value of the cosmological time according to e−3b0t0 = αρ0 , the evolution of the anisotropic brane universe starts at t = 0 from a singular state v(t = 0) = 0. Therefore the expansion scalar, scale factors, mean anisotropy, and decelerating parameter are given by Θ(t) = 3b0(t−t0) e3b0(t−t0) − αρ0 , (75) ai = a0i e3b0(t−t0) − ]1/3 [ e−3b0(t−t0) ] 2b0hi , i = 1, 2, 3, (76) A(t) = 3b20e 6b0(t−t0) , (77) q(t) = e−3b0(t−t0) − 1. (78) We consider λ = 0 and show that, within the context of the present model, the extrinsic curvature can be used to account for the accelerated expansion of the universe. In figure 1 we have plotted the behavior of the deceleration parameter for different values of γ. The behavior of this parameter shows that when the geometrical energy density is positive and the pressure is negative the AdS5 bulk is not compatible with the expansion of the universe. Also, this behavior is much dependent on the range of the values that γx can take. The use of the de Sitter bulk with ρx > 0 allows us to use the wealth of available data from the recent measurements to determine limits on the values of γx in our geometric model. For having an accelerating universe we distinguish γx < . As mentioned before, q(t) > 0 corresponds to the standard decelerating models whereas q(t) < 0 indicates an accelerating expansion at late times. Therefore, the universe undergoes an accelerated expansion at late times in 0.5 1 1.5 2 0.5 1 1.5 2 Figure 1: Left, the deceleration parameter of the Bianchi type I universe and right, the same parameter in the Bianchi type V universe for the de Sitter bulk with γ = 1 (solid line), γ = 4/3 (dashed line), γ = 2 (dot-dashed line), γx = 0.3 and λ = 0. the absence of a positive cosmological constant. As has been noted in [20], it should be emphasised here too that the geometrical approach considered here is based on three basic postulates, namely, the confinement of the standard gauge interactions, the existence of quantum gravity in the bulk and finally, the embedding of the brane world. All other model dependent properties such as warped metric, mirror symmetries, radion or extra scalar fields, fine tuning parameters like the tension of the brane and the choice of a junction condition are left out as much as possible in our calculations. To understanding the behavior of the mean anisotropy parameter in our model, let us consider it as a function of the volume scale factor A(v) = 9β2v4/3 + 3λv2 + 3αρ0v 2−γ + 9b20v 2−γx +C . (79) The behavior of the anisotropy parameter at the initial state depends on the values of γ and γx. For an accelerating universe we obtain γx < . From equation (79), in the limit v → 0 (singular state) and taking γx < , we find A(v) = , 1 ≤ γ ≤ 2. (80) Therefore for a brane world scenario with a confining potential, the initial state is always anisotropic. In our model the behavior of the anisotropy parameter coincides with the standard 4D cosmology and is different from the brane world models where a delta-function in the energy-momentum tensor is used [26] to confine the mater on the brane. The behavior of the mean anisotropy parameter of the Bianchi type I and V geometries are illustrated, for γx = 0.3 and different values of γ, in figure 2. The behavior of this parameter shows that the universe starts from a singular state with maximum anisotropy and ends up in an isotropic de Sitter inflationary phase at late times. In figure 3 we have plotted the deceleration parameter and the anisotropy parameter of the Bianchi type I geometry for γ = 1 and different values of γx = 0, 0.3, 0.5. At this point it would be appropriate to compare our model with other forms of dark energy such as the 4D quintessence. One may consider a 4D effective Lagrangian whose variation with respect to gµν would result in the dynamical equations (36) compatible with the embedding and with the confined matter hypotheses [20]. In contrast to the standard model, our model corresponds to an Einstein-Hilbert Lagrangian which is modified by extrinsic curvature terms. The resulting Einstein equations are thus modified by the term Qµν . Since, as was mentioned before, this quantity is independently conserved, there is no exchange of energy between this geometrical correction and the confined matter source. Such an aspect has one important consequence however; if Qµν is to be related to dark energy, as we did in this paper, it does not exchange energy with ordinary matter, like the coupled quintessence models [27]. The coupled scalar field models may avoid the cosmic coincidence problem with the available data being used to fix the corresponding dynamics and, consequently, the scalar field potential responsible for the present accelerating phase of the universe. 0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 2: Left, the anisotropy parameter of the Bianchi type I universe and right, the same parameter in the Bianchi type V universe for the de Sitter bulk with γ = 1 (solid line), γ = 4/3 (dashed line), γ = 2 (dot-dashed line), γx = 0.3 and λ = 0. 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 Figure 3: Left, the deceleration parameter of the Bianchi type I geometry and right, the anisotropy parameter in the Bianchi type I geometry for the de Sitter bulk with γx = 0 (solid line), γx = 0.3 (dashed line), γx = 0.5 (dot-dashed line), γ = 1 and λ = 0. 5 Conclusions In this paper, we have studied an anisotropic brane world model in which the matter is confined to the brane through the action of a confining potential, rendering the use of any junction condition redundant. 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0704.1036

Maximal ball packings of symplectic-toric manifolds

Maximal ball packings of symplectic-toric manifolds Alvaro Pelayo and Benjamin Schmidt August 26, 2021 Abstract Let (M, σ, ψ) be a symplectic-toric manifold of dimension at least four. This paper investigates the so called symplectic ball packing problem in the toral equivariant setting. We show that the set of toric symplectic ball packings of M admits the structure of a convex polytope. Previous work of the first author shows that up to equivalence, only (CP1)2 and CP2 admit density one packings when n = 2 and only CPn admits density one packings when n > 2. In contrast, we show that for a fixed n ≥ 2 and each δ ∈ (0, 1), there are uncountably many inequivalent 2n-dimensional symplectic-toric manifolds with a maximal toric packing of density δ. This result follows from a general analysis of how the den- sities of maximal packings change while varying a given symplectic-toric manifold through a family of symplectic-toric manifolds that are equivariantly diffeomorphic but not equivariantly symplectomorphic. 1 Introduction and Main Result 1.1 Motivation What portion of a manifold can be filled by disjointly embedded balls? Answers to this ball packing question depend considerably on the manifold being packed, the types of embeddings allowed, the number of balls allowed, and the radii of balls allowed. In this paper, we consider a packing problem for a specific class of compact symplectic manifolds. Throughout, we let (M2n, σ) denote a 2n-dimensional compact, connected, smooth manifold equipped with a symplectic form σ. Before stating our main result, we discuss some of the foundational results concerning ball packings of symplectic manifolds. In [9], M. Gromov considered the problem of finding the supremum of the densities νN (M, σ) = sup N vol(Br) volσ(M) of packings of a symplectic manifold (M2n, σ) by a fixed number N of disjoint symplectic embeddings of the standard open radius r symplectic ball Br ⊂ C n. Therein, Gromov found obstructions to a full (also http://arxiv.org/abs/0704.1036v2 called perfect) packing by too few balls of the same radius; specifically, he established that νN (B1) ≤ N/2 for each 1 < N < 2n. Motivated by techniques from algebraic geometry, D. McDuff and L. Polterovich established a correspondence between symplectically embedded balls and symplectic blowing-up in [11]. They showed, for instance, that when N is a square, CP2 is fully packable by N disjoint symplectically embedded balls. In contrast, they also established that νN (CP 2) < 1 for nonsquare 1 < N ≤ 8. In [1], P. Biran demonstrated that for any symplectic four manifold (M4, σ) with rational cohomology class [σ] ∈ H2(M, Q), νN(M, σ) = 1 for all sufficiently large N . We refer the reader to [2] for more in this direction. In this article, we consider the symplectic packing problem in a toral equivariant setting. Specifically, we consider toric ball packings of symplectic-toric manifolds (see Definition 2.4). In this particular ball packing problem, the Euler characteristic χ(M) gives an upper bound on the number of disjoint balls in any possible packing. For this reason, we allow the balls in an equivariant packing to have varying radii as opposed to the packings described in the previous paragraph where all balls have the same radius. The study of toric ball packings of symplectic-toric manifolds was initiated by the first author in [12] where the homotopy type of the space of equivariant and symplectic embeddings of a fixed ball is described, and in [13] where the symplectic-toric manifolds admitting a full toric ball packing are classified. The present work was motivated by the latter: Theorem 1.1. [13, Thm. 1.7] A symplectic-toric manifold (M2n, σ, ψ) admits a full toric ball packing if and only if there exists λ > 0 such that • if n = 2, (M4, σ, ψ) is equivariantly symplectomorphic to either (CP2, λ · σFS) or a product (CP 1, λ · (σFS ⊕ σFS)) (where σFS denotes the Fubini-Study form and these manifolds are equipped with the standard actions of T2), or • if n = 1 or n > 2, (M2n, σ, ψ) is equivariantly symplectomorphic to (CPn, λ · σFS) (where σFS denotes the Fubini-Study form and this manifold is equipped with the standard action of Tn). 1.2 Main Results Our motivation for this paper was to understand when it is possible to list (up to equivalence) all symplectic- -toric manifolds admitting a maximal toric ball packing of a specified density, such as in Theorem 1.1. As it turns out, this is possible only when that density is one. To be more precise, let S2n denote the set of equivalence classes of 2n-dimensional symplectic-toric manifolds (see Section 2 for definitions). The maximal density function Ω2n : S 2n → (0, 1] associates to each equivalence class [(M2n, σ, ψ)] the largest density from all possible symplectic-toric packings of M . Theorem 1.1 classifies the set Ω−12n ({1}). In contrast with Theorem 1.1, we prove the following: Theorem 1.2. Let S2n denote the set of equivalence classes of 2n-dimensional symplectic-toric manifolds and let Ω2n : S 2n → (0, 1] be the maximal density function. Then Ω−1({x}) is uncountable for all x ∈ (0, 1). Theorem 1.2 answers [13, Quest. 5.1]. The proof of Theorem 1.2 follows from the proof of the next theorem, which asserts that there are uncountable families of equivariantly diffeomorphic symplectic toric manifolds with the same maximal density that are not equivariantly symplectomorphic. Theorem 1.3. Let S2n denote the set of equivalence classes of 2n-dimensional symplectic-toric manifolds and let Ω2n : S 2n → (0, 1] be the maximal density function, n ≥ 2. Suppose that (M, σ, ψ) is a symplectic-toric manifold with Euler characteristic χ(M) ≥ ⌊(n + 2)/2⌋ · ⌈(n + 2)/2⌉ + 1. Then for any ǫ > 0, there exists a constant c > 0 and a family F of equivariantly diffeomorphic symplectic-toric manifolds satisfying • | volσ′(M ′)− volσ(M)| < ǫ for all (M ′, σ′, ψ′) ∈ F , • Ω−12n ({x}) ∩ F is uncountable for all x ∈ (δ − c, δ) or for all x ∈ (δ, δ + c). To prove Theorem 1.3 we exploit the following: Proposition 1.4. Let (M2n, σ, ψ) be a symplectic-toric manifold of dimension 2n ≥ 4. The set of symplectic- -toric packings of (M2n, σ, ψ) has the structure of a convex polytope in RV , where V denotes the Euler characteristic χ(M) of M . Moreover, (M2n, σ, ψ) admits finitely many maximal toric ball packings. The proof of Theorem 1.3 essentially follows from the fact that there are certain linear perturbations of a given symplectic-toric manifold along which the maximal density function is locally convex. The proof of its local convexity follows from the Brunn-Minkowski inequality once an explicit description of the maximal density function in terms of the polytope from Proposition 1.4 is given. In the next section, we collect together preliminary material and reduce Theorem 1.3 to a proposition concerning polytopes. In the last section we prove these propositions using basic convexity techniques. Acknowledgements: We are indebted to Professor Novik for bringing to our attention an explicit example showing that the density function is surjective during the first author’s visit to the University of Washington, as well as for providing us with the statement and proof of Lemma 2.16. The first author is grateful to her for stimulating conversations and for hospitality at the University of Washington. Part of the work of the first author was funded by Rackham Fellowships from the University of Michigan. His research was partially conducted while visiting Oberlin College. The second author was partially funded by the Clay Mathematics Institute as a Liftoff Fellow as well as by an NSF Postdoctoral Fellowship during the period this research was conducted. 2 Reduction to Convex Geometry Throughout this section (M2n, σ) denotes a 2n-dimensional compact connected smooth manifold equipped with a symplectic form. We let Tk ∼= (S1)k denote the k-dimensional torus and identify its Lie algebra t with Rk. This identification is not unique and throughout we make the convention that the identification comes from the isomorphism Lie(S1) ∼= R given by ∂ 7→ 1/2. Using the standard inner product on Rk, we identify the dual space t∗ with t. Definition 2.1. A σ-preserving action ψ : Tk ×M → M of a k-dimensional torus is Hamiltonian if for each ξ ∈ t there exists a smooth function µξ : M → R such that iξMσ = dµξ, and the map t → C∞(M, R), ξ 7→ µξ, is a Lie algebra homomorphism. Here, ξM is the vector field on M infinitesimally generating the one parameter action coming from ξ and the Lie algebra structure on C∞(M, R) is given by the Poisson bracket. It follows from Definition 2.1 that a σ-preserving action ψ : Tk ×M → M of a k-dimensional torus is Hamiltonian if and only if there exists a momentum map µM :M → t∗ satisfying Hamilton’s equation iξMσ = d〈µ M , ξ〉, for all ξ ∈ t. Note that a momentum map is well defined up to translation by an element of t∗. Nevertheless, we will ignore this ambiguity and refer to the momentum map. It is well known that if (M2n, σ) admits an effective and Hamiltonian action of Tk, then k ≤ n (see for instance [3, Theorem 27.3]). The maximal case, usually referred to as a symplectic-toric manifold or Delzant manifold, is a triple (M2n, σ, ψ) consisting of a compact connected symplectic manifold (M, σ) equipped with an effective and Hamiltonian action ψ : Tn ×M →M . Example 2.2. Equip the open radius r ball Br ⊂ Cn with the standard symplectic form σ0 = j dzj∧dzj . The action Rot : Tn×Br → Br of T n given by (θ1, . . . , θn)·(z1, . . . , zn) = (θ1 z1, . . . , θn zn) is Hamiltonian. Its momentum map µBr has components µBrk = |zk| 2. Its image, which we shall denote by ∆n(r), is given ∆n(r) = ConvHull(0, r2 e1, . . . , r 2 en) \ ConvHull(r 2 e1, . . . , r 2 en), where {ei} i=1 is the standard basis of R n. When the dimension n is clear from the context, we shall write ∆(r) = ∆n(r). ⊘ The next two definitions define the packings considered in this paper. These definitions first appeared in [13] in a slightly different but equivalent form. Definition 2.3. Let (M2n, σ, ψ) be a symplectic-toric manifold, let Λ ∈ Aut(Tn) and let r > 0. A subset B ⊂ M is said to be a Λ-equivariantly embedded symplectic ball of radius r if there exists a symplectic embedding f : Br →M with image B and such that the following diagram commutes: Tn × Br Λ×f // Tn ×M f // M We say that the Λ-equivariantly embedded symplectic ball B has center f(0) ∈M . We shall say that a subset B′ ⊂ M is an equivariantly embedded symplectic ball of radius r′ if there exists Λ′ ∈ Aut(Tn) such that B′ is a Λ′-equivariantly embedded symplectic ball of radius r′ (although this is a slight abuse of the standard use of the word “equivariantly”). ⊘ We define the symplectic volume of a subset A ⊂M by volσ(A) := Definition 2.4. Let (M2n, σ, ψ) be a symplectic-toric manifold. A toric ball packing of M is a disjoint union P := α∈ABα of equivariantly embedded symplectic balls Bα (of possibly varying radii) in M . The density Ω(P) of a packing P is defined by Ω(P) := volσ(P)/ volσ(M). The density Ω(M 2n, σ, ψ) of a symplectic-toric manifold (M2n, σ, ψ) is defined by Ω(M, σ, ψ) := sup{Ω(P) | P is a toric packing ofM}. A packing achieving this density is said to be a maximal density packing. If in addition, this density is one, then (M2n, σ, ψ) is said to admit a full or perfect toric ball packing. ⊘ For a symplectic-toric manifold (M2n, σ, ψ), the number of fixed points of ψ is known to coincide with the Euler characteristic χ(M) (see e.g. [5]). It follows that a toric ball packing P consists of at most χ(M) disjoint equivariant balls. By a well known theorem of Atiyah and Guillemin-Sternberg, the image of the momentum map of a toral Hamiltonian action is the convex hull of the images of the fixed points (see for instance [3, Theorem 27.1]). The images of momentum maps for symplectic-toric manifolds are a particular class of polytopes. Recall that an n-dimensional polytope is simple if there are precisely n edges meeting at each one of its vertices. Definition 2.5. A simple n-dimensional convex polytope ∆ ⊂ Rn is said to be Delzant if for each vertex v, the edges meeting at v are all of the form v + ti ui where ti > 0 and {u1, . . . , un} define a basis of the Z-module Zn. ⊘ A polytope is describable as the intersection of closed half-spaces {x ∈ Rn | 〈x, ui〉 ≥ λi}, where the vector ui is an inward pointing normal vector to the i th facet of ∆ and each λi is a real scalar. In this notation, the polytope ∆ is Delzant if and only if there are precisely n facets incident to each vertex of ∆ and the inward pointing normals to these facets u1, . . . , un can be chosen to be a Z-basis of Z Given a symplectic-toric manifold M , the symplectic-toric manifolds obtained from M by scaling the symplectic form, changing the time parameter in the acting torus by an automorphism, and any others that are equivariantly symplectomorphic to one of these will clearly have the same maximal density. Therefore, for the purpose of this paper we will say that two symplectic-toric manifolds (M1, σ1, ψ1) and (M2, σ2, ψ2) are equivalent if there exists an automorphism Λ ∈ Aut(Tn) ∼= GL(n, Z), a positive number λ > 0, and a symplectomorphism h : (M1, σ1) → (M2, λ · σ2) such that the following diagram commutes: Tn ×M1 Λ×h // Tn ×M2 h // M2 We recall the result of Delzant [4]: Theorem 2.6. Suppose that (M1, σ1, ψ1) and (M2, σ2, ψ2) are two 2n-dimensional symplectic-toric man- ifolds with momentum maps µM1 and µM2 respectively. Then there exists a (ψ1, ψ2)-equivariant symplecto- morphism h : (M1, σ1) → (M2, σ2) such that µ M1 = µM2 ◦ h if and only if µM1(M1) = µ M2(M2). In view of this theorem, there is a natural equivalence relation one can put on the set of Delzant polytopes so that momentum maps will induce a bijective correspondence between equivalence classes of symplectic- -toric manifolds as above and equivalence classes of Delzant polytopes. To be more specific, first note that scaling Rn leaves invariant the Delzant polytopes. Define two Delzant polytopes ∆1,∆2 ⊂ R n to be in the same projective class if there exists λ > 0 such that λ∆1 = ∆2. The group AGL(n, Z) consisting of affine transformations of Rn with linear part in GL(n, Z) acts on the set of projective classes of Delzant polytopes in Rn. For the purpose of this paper we say that two Delzant polytopes ∆1 and ∆2 are equivalent if the projective classes of ∆1 and ∆2 are in the same AGL(n, Z) orbit. By applying Theorem 2.6, it is standard to show that there is a bijective correspondence between equivalence classes of symplectic-toric manifolds and equivalence classes of Delzant polytopes as defined here. We exploit this correspondence in order to reduce Theorem 1.3 and Proposition 1.4 to propositions concerning packing Delzant polytopes. The next few definitions are translations of the above definitions into the appropriate definitions concerning Delzant polytopes. Again, they first appeared in [13] in a slightly different but equivalent form. Definition 2.7. Let ∆ be a Delzant polytope. A subset Σ ⊂ ∆ is said to be an admissible simplex of radius r with center at the vertex v ∈ ∆ if Σ is the image of ∆(r1/2) by an element of AGL(n, Z) which takes the origin to v and the edges of ∆(r1/2) to the edges of ∆ meeting v. For a vertex v ∈ ∆, we put rv := max{r > 0 | ∃ an admissible simplex of radius r with center v}. ⊘ (1) Remark 2.8. In view of Example 2.2, the simplex ∆(r1/2) may be identified with the set obtained by removing from ConvHull(0, re1, . . . , ren) the facet not containing the origin. For this reason we say that AGL(n, Z) images of ∆(r1/2) as in the above definition have radius r instead of radius r1/2. ⊘. We denote the Euclidean volume of a subset A ⊂ ∆ by voleuc(A). Definition 2.9. Let ∆ be a Delzant polytope. An admissible packing of ∆ is a disjoint union P := α∈A Σα of admissible simplices (of possibly varying radii) in ∆. The density Ω(P) of a packing P is defined by Ω(P) := voleuc(P)/ voleuc(∆). The density Ω(∆) of a Delzant polytope ∆ is defined by Ω(∆) := sup{Ω(P) | P is an admissible packing of∆}. A packing achieving this density is said to be a maximal density packing. If in addition, this density is one, then ∆ is said to admit a full or perfect packing. ⊘ The next lemma shows that admissible simplices in ∆ are parametrized by their centers and radii. The rational or SL(n, Z)-length of an interval I ⊂ Rn with rational slope is the unique number l := lengthQ(I) such that I is AGL(n, Z)-congruent to an interval of length l on a coordinate axis. For a vertex v in a Delzant polytope, we denote the n edges leaving v by e1v, . . . , e v . By Definition 2.5, each e v is of the form v+ tiv u v with t v > 0 and {u i=1 defining a Z-basis of Z n. In this notation, we have that lengthQ(e v) = t Lemma 2.10. Let ∆ be a Delzant polytope. Then for each vertex v ∈ ∆, rv = min{lengthQ(e v), . . . , lengthQ(e v )}. There is an admissible simplex Σ(v, r) of radius r with center v if and only if 0 ≤ r ≤ rv. Moreover this admissible simplex is unique and voleuc(Σ(v, r)) = r n/n!. Proof. We first argue the uniqueness of an admissible simplex Σ of radius r and center v, assuming its existence. Suppose Σ1 and Σ2 were two such. By definition, there exist two affine transformationsA1, A2 ∈ AGL(n, Z) satisfying A1(∆(r 1/2)) = Σ1 and A2(∆(r 1/2)) = Σ2. As both Σi are centered at v, both Ai have translational part given by v. Write Ai(·) = Λi(·)+ v with Λi ∈ GL(n, Z). By the Delzant property of ∆, c.f. Definition 2.5, the automorphisms Λ1 and Λ2 both take the standard basis {ei} i=1 of Z n bijectively onto the basis {uiv} i=1 of Z n as unordered sets. Therefore Λ−11 Λ2 leaves invariant the standard basis of Z as an unordered set and hence leaves ∆(r1/2) invariant as a set. It follows that Σ1 = Σ2. Next we argue existence. As above, write eiv = v + t v with {u i=1 forming a Z-basis of Z n. Let Λ ∈ GL(n, Z) be the automorphism of Zn defined by Λ(ei) = u v, with {ei} i=1 the standard basis of Z n. The affine transfor- mation A : Rn → Rn defined by A(x) = Λ(x) + v satisfies A(t ei) = t u v + v for each t > 0. There- fore, A(∆(r1/2)) defines an admissible simplex for ∆ with center v if and only if r ≤ min{t1v, . . . , t min{lengthQ(e v), . . . , lengthQ(e v )}, concluding the proof of existence. Finally, for each 0 ≤ r ≤ rv, let Σ(v, r) be the unique admissible simplex with radius r and center v guaranteed by the previous two paragraphs. Since elements in AGL(n, Z) preserve Euclidean volume, we have voleuc(Σ(v, r)) = voleuc(∆(r 1/2)) = rn/n!, completing the proof. Lemmas 2.13 and 2.14 below reduce the problem of analyzing the densities of toric ball packings of symplectic-toric manifolds to analyzing the densities of admissible packings of Delzant polytopes. These lemmas appear in [12] and [13] respectively, in slightly different form. We include the arguments here for the reader’s convenience. We first recall some definitions and a result due to Y. Karshon and S. Tolman. Let (N2n, σN) denote a connected (and not necessarily compact) symplectic manifold equipped with an effective Hamiltonian action of Tn with momentum map µN : N → t∗. Suppose that T ⊂ t∗ is an open and convex subset containing µN(N) with the property that µN : N → T is a proper map. The quadruple (N, σN , µN , T ) is said to be a proper Hamiltonian Tn-manifold. For a subgroup K ⊂ Tn, denote the fixed point set for the Tn action on N by NK . A proper Hamiltonian Tn-manifold (N, σN , µN , T ) is said to be centered at α ∈ T provided that α is in the momentum map image of every component of NK for all subgroups K ⊂ Tn. For a fixed point p ∈ NT , there exist isotropy weights η1, . . . , ηn ∈ t ∗ such that the induced linear symplectic action on Tp(N) is isomorphic to the action on (C n, σ0) generated by the momentum map (x) = µN(p) + |zj | 2ηj . For a symplectic-toric manifold M with fixed point p ∈ MT , and corresponding vertex v := µ(p) ∈ ∆, and edges eiv = v + t v (i = 1, . . . , n) emanating from v , the Z-basis {u i=1 of Z n coincides with the set of isotropy weights at p. Proposition 2.11. [10, Prop. 2.8] Let (N2n, σN , µN , T ) be a proper Hamiltonian Tn-manifold. Assume that N is centered about α ∈ T and that (µN)−1({α}) consists of a single fixed point p. Then N is equivariantly symplectormorphic to {z ∈ Cn |α+ |zj | 2ηj ∈ T }, where η1, . . . , ηn ∈ t ∗ are the isotropy weights at p. Remark 2.12. The reader who consults [10] will find an additional factor of π in the definition of isotropy weights and in the statement of their Proposition 2.8. This factor does not appear in the present work because of the particular identification chosen between t and Rn. ⊘ Lemma 2.13. Let (M2n, σ, ψ) be a symplectic-toric manifold with momentum map µM : M → t∗ and associated Delzant polytope ∆ := µM(M). Let B ⊂ M be an equivariantly embedded symplectic ball of radius r and center p ∈ M . Then µM(B) is an admissible simplex of radius r2 in ∆ with center µM(p). Conversely, if Σ ⊂ ∆ is an admissible simplex of radius r, then there exists an equivariantly embedded symplectic ball B ⊂M of radius r1/2 satisfying µM(B) = Σ. Proof. First suppose that B ⊂ M is an equivariantly embedded ball of radius r. By definition, there is an automorphism Λ ∈ Aut(Tn) ∼= GL(n, Z) and a symplectic embedding f : (Br, σ0) → (M, σ) with image B such that the following diagram commutes: Tn × Br Λ×f // Tn ×M f // M We denote by ξBr and ξM the vector fields on Br and M infinitesimally generating the actions of the one paramater group coming from ξ ∈ t. Fix z ∈ Br and a tangent vector v ∈ TzBr. Note that from the definition of the momentum maps µM and µBr , commutativity of the above diagram, and the fact that f ∗σ = σ0, we have the following sequence of equalities: 〈dµBrz (v), ξ〉µBr (z) = σ0(v, ξBr)z = σ(dfz(v), dfz(ξBr))f(z) = σ(dfz(v), (Λ ξ)M)f(z) = 〈dµMf(z)(dfz(v)), Λ ξ〉µM (f(z)) = 〈Λt dµMf(z)(dfz(v)), ξ〉µBr (z). (2) By equation (2) and the chain rule we obtain that for all z ∈ Br and v ∈ TzBr, dµBrz (v) = d(Λ t ◦ µM ◦ f)z(v). (3) As Br is connected, (3) implies that there exists x ′ ∈ Rn such that µBr + x′ = Λt ◦ µM ◦ f as maps Br → R n. Letting x = (Λt)−1(x′) yields commutativity of the following diagram: (Λt)−1+x // ∆M . (4) It follows from commutativity of this diagram that x = µM(f(0)) = µM(p) so that µ(B) = µ(f(Br)) is an admissible simplex of radius r2 and center µM(p), completing the proof of the first statement. Next suppose that Σ ⊂ ∆ is an admissible simplex of radius r. By applying a translation, we may assume that Σ is centered at the origin. Identify Σ with the set ConvHull(0, r η1, . . . , r ηn) \ ConvHull(r η1, . . . , r ηn). Let T ⊂ t∗ be the the unique open half space of t∗ containing Σ with bounding hyperplane containing ConvHull(r η1, . . . r ηn). Denote by σ N , µN the restrictions of the symplectic form σ, and of the momen- tum map µM , to the open submanifold N := (µM)−1(Σ) ⊂ M . The quadruple (N, σN , µN , T ) is a proper Hamiltonian Tn manifold centered at 0 ∈ T . It now follows from Proposition 2.11, that (N, σN ) is equivariantly symplectomorphic to {z ∈ Cn | |zj | 2ηj ∈ T } = Br1/2 . In other words, the set N ⊂ M is an equivariantly embedded symplectic ball of radius r1/2 satisfying µM(N) = Σ (c.f. [13, Lem 2.3] for an explicit verification). Lemma 2.14. Let (M2n, σ, ψ) be a symplectic-toric manifold with momentum map µM :M → Rn and as- sociated Delzant polytope ∆ := µM(M). Then for each toric ball packing P of M , µM(P) is an admissible packing of ∆ satisfying Ω(P) = Ω(µM(P)). Moreover, given an admissible packing Q of ∆, there exists a toric ball packing P of M satisfying µM(P) = Q. Proof. Let P be a toric ball packing ofM . For each equivariant symplectic ball B in the packing P , µM(B) is an admissible simplex in ∆ by the previous lemma. Since the fibers of the momentum map µM :M → ∆ are connected (see for instance [3, Theorem 27.1]), disjoint equivariant symplectic balls in the packing P are sent to disjoint admissible simplices in ∆. Hence, µM(P) is a toric packing of ∆. By the Duistermaat- -Heckman Theorem (see for instance [3, Theorem 30.3]) the push forward of symplectic volume under the momentum map satisfies µM∗ (volM) = K(n) voleuc, where K(n) > 0 is a dimensional constant. It follows that Ω(P) = Ω(µM(P)). It remains to argue that given an admissible packing Q of ∆, there exists a toric ball packing P of M satisfying µM(P) = Q. By the Lemma 2.13, for each admissible simplex Σ there exists an equivariant symplectic ball B ⊂ M with µM(B) = Σ. Choosing one such equivariant symplectic ball for each admissible simplex in Q defines a disjoint collection P of equivariant symplectic balls mapping onto Q under the momentum map. Hence P is a toric packing of M satisfying µM(P) = Q. Before concluding this section by reformulating our main results in terms of polytopes, we must first introduce the complete regular n–dimensional fan associated to an n-dimensional Delzant polytope ∆, as in [4, Sec. 5]. As above, write {x ∈ Rn | 〈x, ui〉 ≥ λi}, where F is the number of facets of ∆ and ui is the unique primitive integral normal vector to the i th facet. For each face ∆′ of ∆ of codimension k, there is a unique multi-index I∆′ of length k, I∆′ = {i1, . . . , ik}, 1 ≤ i1 < . . . < ik ≤ F , such that ∆′ = {x ∈ Rn | 〈x, uj〉 = λj , ∀j ∈ I∆′}. Letting σ∆′ denote the cone in R n generated by the vectors {uj | j ∈ I∆′}, the complete regular n- -dimensional fan associated to ∆ is given by {σ∆′ |∆ ′ is a face of∆}. For our purposes here, we state the well known fact that if two Delzant polytopes have the same associated fan, their associated symplectic- -toric manifolds are equivariantly diffeomorphic. This is a standard fact that follows from the construction of a symplectic-toric manifold starting from a given Delzant polytope. It now follows from Lemma 2.13 and Lemma 2.14 that proving Theorem 1.2 is reduced to establishing the following proposition: Proposition 2.15. Let Dn denote the set of equivalence classes of n-dimensional Delzant polytopes and Ω : Dn → (0, 1] be the maximal density function, n ≥ 2. Then Ω−1({x}) is uncountable for all x ∈ (0, 1). Recall that the number of fixed points of the Tn-action on M equals the Euler Characteristic of M and the number of vertices of the momentum polytope µ(M), c.f. [5]. We are grateful to Professor Novik for the following observation: Lemma 2.16. An n-dimensional Delzant polytope ∆ with at least ⌊(n+2)/2⌋ · ⌈(n+2)/2⌉+1 vertices has at least n + 3 facets. Moreover, this bound is sharp in the sense that there exists an n-dimensional Delzant polytope with ⌊(n+ 2)/2⌋ · ⌈(n+ 2)/2⌉ vertices and n+ 2 facets. Proof. Indeed, to see that a fewer number of vertices is not enough, let ∆ be the direct product of two regular simplexes, one of dimension ⌊n/2⌋ and another one of dimension ⌈n/2⌉. Their product is an n- dimensional Delzant polytope that has ⌊(n+ 2)/2⌋ · ⌈(n+ 2)/2⌉ vertices and only (n+ 2) facets. It is well known, see e.g. [7, pp. 98-100] for the proof of (the dual) statement, that ∆ has the maximal number of vertices amongst all simple n-polytopes with n+ 2 facets. It then follows from Lemma 2.13, Lemma 2.14 and Lemma 2.16 that proving Theorem 1.3 is reduced to establishing the following proposition: Proposition 2.17. Let Dn denote the set of equivalence classes of n-dimensional Delzant polytopes and Ω : Dn → (0, 1] be the maximal density function, n ≥ 2. Suppose that ∆ is a Delzant polytope having at least n + 3 facets and let Ω(∆) := δ ∈ (0, 1). Then for any ǫ > 0, there exists a constant c > 0 and a family F of Delzant polytopes satisfying • the polytopes in F determine a common fan, • | voleuc(∆ ′)− voleuc(∆)| < ǫ for all ∆ ′ ∈ F , • Ω−1({x}) ∩ F is uncountable for all x ∈ (δ − c, δ) or for all x ∈ (δ, δ + c). Similarly we have reduced proving Proposition 1.4 to showing: Proposition 2.18. Let ∆ ⊂ Rn be a Delzant polytope. The set of admissible packings of ∆ has the structure of a convex polytope in RV , where V is the number of vertices of ∆. Moreover, ∆ admits finitely many maximal density packings. We prove these propositions in the next section. 3 Proofs of propositions 2.15, 2.17, and 2.18. In this section, we prove Propositions 2.15, 2.17, and 2.18 using convexity arguments. First we recall some preliminary notions. For a set A ⊂ Rn, denote the closure of A by A. Now suppose that A is a convex set. A function f : A→ R is said to be convex if for all x1, x2 ∈ A and t ∈ (0, 1), f((1− t) x1 + t x2) ≤ (1− t) f(x1) + t f(x2) and strictly convex if the inequality is always strict. Similarly, the function f is said to be concave if for all x1, x2 ∈ A and t ∈ (0, 1), f((1− t) x1 + t x2) ≥ (1− t) f(x1) + t f(x2) and strictly concave if the inequality is always strict. It follows from the definitions that if f is a convex function on A and g is a positive concave function on A, then f/g is a convex function which is strict if one of f or g is strict. Moreover, if f1, . . . , fk are convex functions onA then so is the function max{f1, . . . , fk}. We let C(Rn) denote the space of compact convex subsets of Rn and endow it with the Hausdorff metric dH given by dH(A, B) := inf{ǫ > 0 |A ⊂ Nǫ(B) andB ⊂ Nǫ(A)}, where Nǫ(X) denotes the open ǫ-neighborhood of a subset X ⊂ R n. A compact convex set A ∈ C(Rn) with non-empty interior is said to be a convex body. If λ > 0 and A and B are convex bodies then so are the λA := {λ a | a ∈ A} A+B = {a+ b | a ∈ A and b ∈ B}. Subsets A, B ⊂ Rn are said to be homothetic if there exists v ∈ Rn and λ > 0 such that λA + {v} = B. We recall the celebrated Brunn-Minkowski inequality (see [8] for a detailed survey): Theorem 3.1 (Brunn-Minkowski). Let A, B be convex bodies in Rn and 0 < λ < 1. Then vol1/neuc ((1− λ)A+ λB) ≥ (1− λ) vol euc (A) + λ vol euc (B), with equality if and only if A and B are homothetic. In the remainder of this section, we let ∆n = i=1{x ∈ R n | 〈x, ui〉 ≥ λi} denote an n-dimensional Delzant polytope with F facets and V vertices. We enumerate the vertices v1, v2, . . . , vV and facets F1, . . . ,FF . When two vertices vj and vk are adjacent in ∆, we denote their common edge by ej,k. By the Delzant property, each vertex vi is the unique intersection point of n facets, F jvi = {x ∈ R n | 〈x, uvi〉 = λvi} ∩∆, j = 1, . . . , n, where the inward normal vectors ujvi to the j th facet F jvi collectively define a Z-basis of Z n. In particular, ∆ is simple and there are precisely n edges ejvi , j = 1, . . . n leaving each vertex vi of ∆. We shall denote the set of admissible packings of ∆ by AP(∆). Proposition 2.18. Let ∆ ⊂ Rn be a Delzant polytope. The set AP(∆) of admissible packings of ∆ has the structure of a polytope in RV , where V is the number of vertices of ∆. Moreover, ∆ admits finitely many maximal toric ball packings. Proof. Each packing Q ∈ AP(∆) consists of a disjoint union i=1Σ(vi, Ri(Q)) of admissible simplices Σ(vi, Ri(Q)) centered at the vertex vi with (possibly zero) radius Ri(Q). Define the map R : AP(∆) → ΠVi=1[0, rvi], Q 7→ (R1(Q), . . . , RV (Q)). By Lemma 2.10, the map R is injective so that we can identify AP(∆) with its image in ΠVi=1[0, rvi ]. Note that R is not surjective as the admissible simplices with given radii (x1, . . . , xV ) ∈ Π i=1[0, r vi] will not in general be disjoint. We must argue that the image set of R is precisely the solution set to finitely many linear inequalities. As a first step, we give a criterion for admissible simplices to be disjoint in terms of their intersections along the edges of ∆. To this end, let F denote a finite family of admissible simplices with pairwise distinct centers and fix an admissible simplex Σ from the family. Let v denote the center of Σ. Disjointness Condition: For Σ to be disjoint from the rest of the familyF , it is a necessary and sufficient condition that its closure Σ intersects the closure of the other admissible simplices in the family in at most one point in each of the edges ejv, j = 1, . . . , n. To see that the above disjointness condition is necessary, suppose that the closure of another admissible simplex in F , say Σ′, intersects Σ in more than a single point along some edge ejv. Then Σ ∩ Σ ′ ∩ ejv is a convex subset of ejv with at least two points and is therefore a closed subsegment e ⊂ e i with nonempty interior. The interior of e is contained in both Σ and Σ′, whence Σ ∩ Σ′ 6= ∅. Next we argue that the above condition is sufficient. To see this, let Σ′ be another admissible simplex in the family F . Denote by v′ ∈ ∆ the center of Σ′ and let xjv′ denote the unique point in the set Σ ′ ∩ e v′ \ Σ ′ ∩ e v′ for each j = 1, . . . , n. If the above condition holds, then {v′, x1v′ , . . . , x v′} ⊂ ∆ \Σ. But since ∆ \Σ is a convex set and Σ′ = ConvHull(v′, x1v′ , . . . , x v′), it follows that Σ ′ ⊂ ∆ \ Σ, concluding the proof of sufficiency. For distinct vertices vi, vj ∈ {v1, . . . , vV }, define Li,j by Li,j = lengthQ(ei,j) if vi and vj are adjacent rvi + rvj otherwise It follows from (1) and Lemma 2.10 and the preceding disjointness condition that a point (x1, . . . , xV ) ∈ ΠVi=1[0, rvi] lies in the image of R if and only if the equations xi + xj ≤ Li,j , hold for all i 6= j ∈ {1, . . . , V }. Therefore, R(AP(∆)) = i 6=j∈{1,...,V } {(x1, . . . , xV ) ∈ R ≥0 | xi + xj ≤ Li,j}. (5) By Lemma 2.10, the density function in the coordinates of RV≥0 is given by Ω((x1, . . . , xV )) = (Σ i )/(n! voleuc(∆)). As n > 1, Ω is a strictly convex function on AP(∆) so that its maximum value can only be obtained at its vertices, establishing the last part of the proposition. In view of the injectivity of R, we will henceforth identify AP(∆) with the polytope R(AP(∆)). Before proving Proposition 2.17, we need to set up some additional notation and prove a lemma. A natural way to perturb the Delzant polytope ∆ is by perturbing its defining linear equations. Define the map F → C(Rn), s = (s1, . . . , sF ) 7→ ∆s (6) ∆s := {x ∈ Rn | 〈x, ui〉 ≥ λi + s This map is continuous and has image in the set of polytopes with not more than F facets (we regard the empty set as a polytope). For small enough r > 0, the polytopes ∆s with s ∈ B(0, r) ⊂ R F still have F facets, are Delzant, and all determine the same fan. Let B∆ denote the largest open ball centered at the origin with these three properties and let Dn denote the set of equivalence classes of Delzant polytopes. The map B∆ → D n induced by (6) induces an equivalence relation on B∆ by declaring points in the fibers to be equivalent. We denote the set of equivalence classes by B̂∆ and endow it with the quotient topology. As GL(n, Z) is discrete, the dimension of a fixed equivalence class of Delzant polytopes is n+1, the dimension of the group of homotheties of Rn. It follows that the dimension of B̂∆ satisfies dim(B̂∆) ≥ F − (n+ 1). Define the function Ω : B∆ → (0, 1] by Ω(s) := Ω(∆s), and for s1, s2 ∈ B∆, define the function Ωs1, s2 : [0, 1] → (0, 1] by Ωs1, s2(t) := Ω((1 − t) s1 + t s2) = Ω(∆(1−t) s1+t s2). Lemma 3.2. The function Ω : B∆ → (0, 1] is continuous. For distinct s1, s2 ∈ B∆, there exists a suitably small ǫ(s1, s2) > 0 such that the restriction Ω s1, s2 |[0, ǫ] : [0, ǫ] → (0, 1] is convex. Furthermore, if ∆s1 and ∆s2 are not homothetic, then Ω s1, s2 |[0, ǫ] is strictly convex. Proof. Define || · ||n : R ≥0 → R by ||(x1, . . . , xV )||n = ( i=1 x 1/n. It is well known that || · ||n is a strictly convex function. For s ∈ B∆, we have that Ω1/n(s) = max || · ||n|AP(∆s) (n!)1/n vol1/neuc (∆s) Each of the vertices v1, . . . , vV of ∆ are defined as the unique solution to the linear system: 〈ujvi, vi〉 = λ j = 1, . . . n. Similarly, ∆s has V vertices v1(s), . . . , vV (s), each defined by the linear system: 〈ujvi, vi(s)〉 = λ + sjvi j = 1, . . . n. Hence, the vertices define affine maps vi : B∆ → R n. It follows that the edges of ∆s and their rational lengths also vary linearly with s ∈ B∆ and that, as we have already remarked, s 7→ ∆s defines a continuous map B∆ → C(R By the proof of Proposition 2.18, c.f. expression (5), AP(∆s) = i 6=j∈{1,...,V } {(x1, . . . , xV ) ∈ R ≥0 | xi + xj ≤ Li,j(s)}, where Li,j(s) = lengthQ(ei,j(s)) if vi(s) and vj(s) are adjacent; rvi(s) + rvj (s) otherwise, rvi(s) = min{lengthQ(e (s)), . . . , lengthQ(e (s))}. Hence, the defining equations of AP(∆s) vary continuously with s ∈ B∆ so that s 7→ AP(∆s) defines a continuous map B∆ → C(R ≥0). Since vol 1/n and max || · ||n define continuous maps on C(R n) and C(RV≥0), it follows that Ω is continuous. Now fix s1, s2 ∈ B∆ and t ∈ (0, 1). We first claim that ∆(1−t) s1+t s2 = (1− t)∆s1 + t∆s2. Note that ∆(1−t) s1+t s2 = ConvHull(v1((1− t) s1 + t s2), . . . , vV ((1− t) s1 + t s2)). Since vi((1− t) s1 + t s2) = (1− t) vi(s1) + t vi(s2), {v1((1− t) s1 + t s2), . . . , vV ((1− t) s1 + t s2)} ⊂ (1− t)∆s1 + t∆s2 , whence ∆(1−t) s1+t s2 ⊂ (1− t)∆s1 + t∆s2 . Now consider the (n+ 1)-dimensional polytope ∆(s1, s2) := {(x, t) ∈ Rn × [0, 1] | 〈x, ui〉 ≥ λi + (1− t) s1 + t s2}. Let Ht = ∆(s1, s2) ∩ {xn+1 = t} and note that Ht is naturally identified with ∆(1−t) s1+t s2 . Now, if (x, 0) ∈ H0, (y, 1) ∈ H1, and t ∈ (0, 1), then (1 − t) (x, 0) + t (y, 1) ∈ Ht, concluding the proof of the claim. By Theorem 3.1, the map [0, 1] → R given by t 7→ vol1/neuc (∆(1−t) s1+t s2) is concave and strictly concave if and only if ∆s1 and ∆s2 are not homothetic. To conclude the proof of the Lemma, we must argue that there is a suitably small ǫ = ǫ(s1, s2) > 0 such that the map [0, 1] → R given by t 7→ max || · ||n|AP(∆(1−t) s1+t s2) is convex when restricted to the interval [0, ǫ). Let p1, p2, . . . , pk ∈ R V denote the vertices of AP(∆s1). It follows from the description above that for t sufficiently small AP(∆(1−t) s1+t s2) also has k vertices v1(t), . . . , vk(t) ∈ R V and that the maps t 7→ vi(t) define (possibly constant) line segments in R V . By convexity of || · ||n, max || · ||n|AP(∆(1−t) s1+t s2 ) = max{||v1(t)||n, . . . , ||vk(t)||n}, and the result follows. We are now ready to prove: Proposition 2.17. Let Dn denote the set of equivalence classes of n-dimensional Delzant polytopes and Ω : Dn → (0, 1] be the maximal density function, n ≥ 2. Suppose that ∆ is a Delzant polytope having at least n + 3 facets and let Ω(∆) := δ ∈ (0, 1). Then for any ǫ > 0, there exists a constant c > 0 and a family F of Delzant polytopes satisfying • the polytopes from F determine a common fan, • | voleuc(∆ ′)− voleuc(∆)| < ǫ for all ∆ ′ ∈ F , • Ω−1({x}) ∩ F is uncountable for all x ∈ (δ − c, δ) or for all x ∈ (δ, δ + c). Proof. Let ∆ be a Delzant polytope with with at least n + 3 facets and with Ω(∆) = δ and let ǫ > 0. By continuity of the volume function with respect to the Hausdorff metric, there is a suitably small connected neighborhood N ⊂ B∆ of the origin for which | voleuc(∆s)− voleuc(∆)| < ǫ for each s ∈ N . We define the desired family F by F = {∆s | s ∈ N} and remark that by construction all ∆′ ∈ F determine the same fan. Therefore, it remains to establish the third item of the proposition. As dim(N) ≥ n + 3 and since the space of homotheties of Rn has dimension n + 1, there exists s ∈ N\{0} such that ∆s is not homothetic to∆. By Lemma 3.2, there exists ǫ ′ > 0 such that Ω 0, s : [0, ǫ [0, 1) is strictly convex and therefore Ω: N → (0, 1] is not the constant map. Since Ω is continuous and N is connected, there exists c > 0, such that (δ, δ + c) ⊂ Ω(N) or (δ − c, δ) ⊂ Ω(N). Suppose that (δ − c, δ) ⊂ Ω(N). Note that Ω : B∆ → (0, 1] descends to a continuous map Ω̂ : B̂∆ → (0, 1]. Suppose that for some x ∈ (δ − c, δ), Ω̂−1({x}) is countable. As dim(B̂∆) ≥ 2, B̂∆ \ Ω̂ −1({x}) is connected. Hence, Ω̂(B̂∆ \ Ω̂ −1({x})) is connected, a contradiction. In case that (δ, δ + r) ⊂ Ω(N) an analogous argument concludes the proof. Remark 3.3. The proof of Proposition 2.17 above also establishes the following: suppose that ∆ is an n-dimensional Delzant polytope with at least n+ 3 facets and with Ω(∆) := δ ∈ (0, 1). Also suppose that Ω(B∆) contains an open neighborhood (δ−c, δ+c) of δ. Then for each x ∈ (δ−c, δ+c), Ω −1({x}) ⊂ Dn is uncountable. Proposition 2.15. Let Dn denote the set of equivalence classes of n-dimensional Delzant polytopes and Ω : Dn → (0, 1] be the maximal density function, n ≥ 2. Then Ω−1({x}) is uncountable for all x ∈ (0, 1). Proof. By Remark 3.3, it suffices to show that there exists an n-dimensional Delzant polytope ∆ with at least n + 3 facets for which Ω(B∆) = (0, 1). Consider the polytope ∆(ǫ1, ǫ2) obtained by removing from the standard n-dimensional simplex an admissible simplex of radius ǫi at the vertex ei for i = 1, 2. For compatible choices of ǫ1 and ǫ2, we obtain a Delzant polytope with n + 3 facets. Fix ǫ 1 and ǫ 2 both very close to zero and let E ⊂ [0, 1]2 be the set of pairs (x, y) for which ∆(x, y) = ∆s for some s ∈ B∆(ǫ0 By definition, Ω({∆(x, y) | (x, y) ∈ E}) ⊂ Ω(B∆(ǫ01, ǫ02)), a connected subset of the interval (0, 1). We conclude by showing Ω({∆(x, y)|(x, y) ∈ E}) contains deleted open neighborhoods of 0 and 1 in [0, 1]. To obtain a deleted neighborhood of 1 we remark that as (x, y) → (0, 0), (x, y) ∈ E and Ω(∆(x, y)) → 1. Similarly, to obtain a deleted neighborhood of 0 we remark that as y → 0, the pairs of the form (1− 2y, y) are in E and Ω(1− 2y, y) → 0. We thank Professor Novik for bringing the example above to our attention. We conclude with the following: Question. Let [(M2n, σ, ψ)] be an equivalence class of symplectic-toric manifolds. Is there a formula for the number of different maximal toric packings of M in terms of its equivariant symplectic invariants? References [1] P. Biran: A stability property of symplectic packing, Invent. Math. 136 (1999), no. 1, 123–155. [2] P. Biran: From symplectic geometry to algebraic geometry and back, Proceedings of the 3’rd Eu- ropean Congress of Mathematics (Barcelona 2000), Vol. II, 507-524, Progr. Math, 202, Birkhauser 2001. [3] A. Cannas da Silva: Lectures in symplectic geometry, Springer–Verlag, Berlin, (2000). [4] T. Delzant: Hamiltoniens périodiques et image convexe de l’application moment, Bull. Soc. Math. France, 116 (1988) 315–339. [5] W. Fulton: Introduction to toric varieties, Princeton Univ. Press, Ann. Math. Stud. 131, (1993). [6] V. Guillemin: Moment Maps and Combinatorial Invariants of Hamiltonian T n-spaces. Birkhäuser, Boston, etc., 1994. [7] B. Grünbaum, Convex polytopes, Second edition, (prepared and with a preface by V. Kaibel, V. Klee and G. M. Ziegler), Graduate Texts in Mathematics, 221, Springer-Verlag, New York, 2003. [8] R. Gardner: The Brunn-Minkowski inequalty, Bull. Amer. Math. Soc. 39(2002), 355-405. [9] M. Gromov: Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307- [10] Y. Karshon and S. Tolman: The Gromov width of complex Grassmannians, Alg. Geom. Top. 5 (2005), 911-922. [11] D. McDuff and L. Polterovich: Symplectic packings and algebraic geometry. With an appendix by Y. Karshon, Invent. Math. 115 (1994), no. 3, 405–434. [12] A. Pelayo: Topology of spaces of equivariant symplectic embeddings, Proc. Amer. Math. Soc. 135 (2007) 277–288. [13] A. Pelayo: Toric symplectic ball packing, Top. and its Appl. 153 (2006) 3633–3644. Alvaro Pelayo Department of Mathematics, University of Michigan 2074 East Hall, 530 Church Street, Ann Arbor, MI 48109–1043, USA e-mail: [email protected] Benjamin Schmidt Department of Mathematics, University of Chicago 5734 South University Avenue, Chicago, Illinois 60637 e-mail: [email protected] Introduction and Main Result Motivation Main Results Reduction to Convex Geometry Proofs of propositions ??, ??, and ??.

0704.1037

Controlled collisions of a single atom and ion guided by movable trapping potentials

Controlled collisions of a single atom and ion guided by movable trapping potentials Zbigniew Idziaszek CNR-INFM BEC Center, I-38050 Povo (TN), Italy and Center for Theoretical Physics, Polish Academy of Sciences, 02-668 Warsaw, Poland Tommaso Calarco CNR-INFM BEC Center and ECT*, I-38050 Povo (TN), Italy and ITAMP, Harvard University, Cambridge, MA 02138, USA Peter Zoller Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences and Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria We consider a system composed of a trapped atom and a trapped ion. The ion charge induces in the atom an electric dipole moment, which attracts it with an r−4 dependence at large distances. In the regime considered here, the characteristic range of the atom-ion interaction is comparable or larger than the characteristic size of the trapping potential, which excludes the application of the contact pseudopotential. The short-range part of the interaction is described in the framework of quantum-defect theory, by introducing some short-range parameters, which can be related to the s-wave scattering length. When the separation between traps is changed we observe trap-induced shape resonances between molecular bound states and vibrational states of the external trapping potential. Our analysis is extended to quasi-one-dimensional geometries, when the scattering exhibit confinement-induced resonances, similar to the ones studied before for short-range interactions. For quasi-one-dimensional systems we investigate the effects of coupling between the center of mass and relative motion, which occurs for different trapping frequencies of atom and ion traps. Finally, we show how the two types of resonances can be employed for quantum state control and spectroscopy of atom-ion molecules. PACS numbers: 32.80.Pj, 34.90.+q I. INTRODUCTION Techniques developed in atomic physics during the last two decades allow the preparation of single atoms and ions in the laboratory. Single neutral atoms can be stored and laser cooled in far off-resonance laser traps (FORT) [1, 2], and arrays of atoms can be prepared in an op- tical lattice via a Mott insulator phase, where exactly one atom is stored per lattice site [3]. Similar techniques are being pursued in the context of atom chips [4]. Fur- thermore, single ions and arrays of ions can be stored in Paul and Penning traps, and sideband laser cooling allows to prepare ions in the vibrational ground state of these trapping potentials [5]. The controlled prepa- ration, manipulation and measurement of electronic in- ternal and motional states of laser driven single atoms and ions, combined with the possibility of controlled en- tanglement of atoms and ions provides the basic ingredi- ents to for investigations of fundamental aspects of quan- tum mechanics, and applications such as high-precision measurements and quantum information. In quantum information processing entanglement of qubits stored in internal states of single atoms or ions is achieved by con- trolled interactions between particles, either in the form of switchable qubit-dependent two-particle interactions or via an auxiliary collective mode of the systems, which serves as a quantum data bus. In particular, for neutral atoms cold controlled collisions of atoms stored in mov- able spin-(i.e. qubit-)dependent optical lattices has been proposed as a means to entangle atomic pairs, and has experimentally implemented to generate N -atom cluster states [6]. In the present work we study the controlled cold colli- sion of a single atom and a single ion, where the atom (ion) is prepared in a given motional state of an atom (ion) trap, and we move the traps to guide the atom and ion wave packet to “collide” for a given time (see Fig. 1). Our focus is the development of a quantum- defect formalism for trapped atoms and ions, in a form which is convenient for future applications and exten- sions, in particular in the context of quantum informa- tion processing to swap qubits stored in atoms and ions, and the entanglement of these qubits in a controlled colli- sion. The atom-ion collision is governed by the potential V (r) → −αe2/(2r4) (r → ∞), where α is the dipolar polarizability – to be contrasted to a van der Waals po- tential V (r) → −C6/(r 6) which represents the collisional interactions between e.g. alkali atoms in their electronic ground state. Below we will calculate the dynamics of the interacting atom-ion system for a given time-dependence of the motion of the trap. This includes the transition to excited trap states after the collision, the possible forma- tion of an excited atom-ion (molecule) complex, and the description of trap induced resonances. To study the controlled atom-ion collisions we first derive an effective Hamiltonian, describing effective net forces acting on the particles moving in rapidly changing http://arxiv.org/abs/0704.1037v2 (d) (c) FIG. 1: (Color online) Schematic drawing of a controlled collision between a single atom and a single ion, whose center- of-mass wavepackets are guided by time-dependent atom trap and ion trap, respectively. Labels distinguish different phases of the process: a) initially particles prepared in the motional ground state; b) collision (overlap of the wave packets); c) excitation of the motional states in the traps; d) particles in some excited states after the collision. laser and rf fields. In our approach, we include in the Hamiltonian only the asymptotic part of the atom-ion potential, whereas the short-range interactions are taken into account by imposing appropriate boundary condi- tions on the wave function at r → 0. In the regime of cold collisions considered here, the boundary conditions can be expressed in terms of a single quantum-defect param- eter, independent of the collisional energy and angular momentum. The quantum-defect approach is dictated by the relatively long-range character of the atom-ion in- teraction, which exceeds the typical size of the trapping potentials, and, in contrast to the neutral atoms, excludes the applicability of the pseudopotential. Our method to describe the dynamics of the controlled collision is based on the application of the correlation di- agrams, i.e. energy spectra as a function of the trap separation. Such correlation diagrams, widely used in quantum chemistry to characterize reactions of diatomic molecules, connect in our case the asymptotic vibrational states with the molecular and vibrational states at zero trap separations. At the intermediate distances the en- ergy curves exhibit avoided or diabatic crossings, depend- ing on the symmetry of eigenstates and of the coupling term in the Hamiltonian. In the atom-ion system the avoided crossing can be attributed to the resonances be- tween molecular and vibrational states, that appear when the energy of a vibrational level, coincide with the energy of a molecular state shifted by the external trapping. The dynamics in the vicinity of such avoided crossings can be accurately described in the framework of the Landau- Zener theory. In this paper we determine the level split- ting at the avoided crossing, at different trap separations, and for different symmetries of the molecular states. In this way we can characterize the time scale appropriate for adiabatic or diabatic traversing of a given avoided crossing. In our paper we consider two different geometries of the trapping potentials: spherically symmetric traps and very elongated cigar-shape traps (quasi-1D traps). The former case requires full three-dimensional treatment, while the latter one can be described using an effec- tive one-dimensional Hamiltonian. In the latter case, the tight transverse confinement effectively renormalizes the one-dimensional quantum-defect parameters. In ad- dition, very elongated cigar-shape traps exhibit another type of resonances, that can be observed at zero-trap sep- arations, but for changing ratio of the transverse confine- ment to the s-wave scattering length of the atom-ion in- teraction. Such confinement-induced resonances appear when the energy of the colliding particles coincide with energy of a bound state lifted up by the tight transverse potential. As already outlined above, the counterpart to the idea of ultracold controlled trap-guided collisions has been im- plemented in optical lattices, where application of spin- dependent potentials has allowed to entangle atoms be- tween neighboring sites [6]. Moreover, resonance phe- nomena in trapped systems, similar to the ones consid- ered here, have been already thoroughly investigated for interactions between neutral atoms. In quasi-1D systems, collisions between neutral atoms exhibit confinement- induced resonances [10], whereas displacement of the trapping potentials leads to trap-induced resonances [11]. The paper is organized as follows. In section II we present our model that we use to describe the controlled atom-ion collisions. In particular, in section II A we in- troduce an effective Hamiltonian of a single trapped atom and ion. Section II B discuss the quantum-defect treat- ment of the short-range part of the interaction potential. In sections II C and IID we discuss our model for two par- ticular geometries of the trapping potentials, spherically symmetric traps of the same frequency and elongated cigar-shape traps with the same transverse trapping fre- quency, respectively. We extend the quantum-defect treatment to quasi-1D systems in section II E. Our ap- proach to describe the dynamics of the atom-ion system is discussed in section II F. Section III presents the results of our calculations. We start from discussing the proper- ties of the scattering in quasi-1D systems in the presence of long-range r−4 potential, in particular, the problem of determining of 1D quantum-defect parameters. Sec- tions III B and III C present adiabatic energy curves and adiabatic eigenstates in, respectively, 1D and 3D traps with the same trapping frequencies for the atom and the ion. The dynamics in the vicinity of the avoided crossing is discussed in section III D, where we apply the Landau- Zener theory and semiclassical methods to address this problem. Section III E discuss most complicated case of different trapping frequencies, focusing on the 1D atom- ion motion. The outlook and final conclusions are pre- sented in section IV. Finally appendix A presents the microscopic derivation of the effective Hamiltonian and appendix B describes the details of the pseudopotential treatment of the scattering in quasi-1D traps. II. BASIC SETUP AND MODEL We consider a system consisting of a single atom and single ion, stored in their respective trapping potentials. Single atoms can be trapped in experiments with optical tweezers [1, 2] or optical lattices [3]. Such potentials are created in far-detuned laser fields due to the AC Stark effect. In another experimental technique, on atom chips [4], single atoms can be trapped in microtraps created by electric and magnetic fields around wires and electrodes. On the other hand, single ions can be confined in radio- frequency traps, which use a rapidly oscillating electric field to create an adiabatic trapping potential for the ion charge [5]. Typically, the trapping potentials close to the trap center are with a good approximation harmonic, and in our approach we consider the simple picture of the atom and ion confined in the harmonic potentials with frequencies ωa and ωi, respectively, with d denoting the displacement between the traps (c.f. Fig. 2.a). We assume that the atom and ion are prepared initially in a given trap state, e.g. the ground state of the potential, and that the atom and ion are brought to collision by overlapping their center-of-mass (COM) wave packets, i.e. we change d(t) as a function of time as outlined in Fig. 1. In studying this collision with COM wave packets guided by movable trapping potentials we will study two cases of (i) three-dimensional (3D) dynamics, where both traps are assumed to be spherically symmetric, and (ii) the case of quasi-1D traps, i.e. elongated, cigar-shaped traps which can be described as an effectively 1D colli- sion dynamics. This second case is conceptually simpler, but also provides collisional features specific to 1D sit- uation (e.g. the appearance of 1D confinement-induced resonances, reminiscent of resonances in [10]). The val- ues of trapping frequencies in atom and ion traps are typically quite different, thus it is natural to consider ωa 6= ωi. For such condition, however, the COM and relative motion are coupled, which significantly compli- cates the theoretical description. Apart from this general regime, we consider also the special case of ωa = ωi, when COM and relative motions can be separated. In this case we perform the full diagonalization for the 3D problem, which is numerically very demanding in the general case of ωa 6= ωi. At large distances, the interaction between atom and ion is given by a potential V (r) ∼ −αe2/(2r4), where α is the dipolar polarizability. This potential originates from the attraction between the ion charge and the electric dipole that it induces on the atom [24]. At short distances the interaction is more complicated; however, it turns out that a detailed knowledge of the form of core region of the potential is not necessary, and that it can be subsumed in our model by introducing a set of (energy independent) quantum-defect parameters. Let us focus now on characteristic scales that can be associated with the atom-ion interaction. For the sake of clarity table I summarizes all the length scales that a d Ri, Ra (c)(b) FIG. 2: (Color online) Panel (a): Schematic drawing of the considered setup consisting of a trapped atom and ion. Panels (b) and (c): Long and short-range regimes, respectively, in the controlled collisions between a trapped atom and ion. are introduced in the course of the paper. We define its characteristic length as R∗ ≡ αe2µ/~2, and char- acteristic energy as E∗ = ~2/ 2µ(R∗)2 , where µ is the reduced mass. Other characteristic lengths and energies are given by the ground states of the trapping poten- tials. We associate to the trapping frequencies the har- monic oscillator lengths for atom (ion): la = ~/(maωa) (li = ~/(miωi)). For optical and rf traps, the trapping frequencies are typically ωa = 2π × 10 − 100kHz and ωi = 2π × 0.1 − 10MHz. In table II we collected some example values of R∗, li, la, and E ∗ for a few systems of alkali atoms and alkaline earth ions. One can observe that the range of the polarization potential is compara- ble or larger than the size of the trapped ground state. Therefore, for the trapped particles the polarization in- teraction cannot be replaced with the standard contact pseudopotential familiar from s-wave atom-atom scatter- ing in the context, e.g., of Bose-Einstein condensation. In the interactions between trapped atom and ion, one can distinguish two different regimes depending on the relative distance d between traps: (i) d ≫ Ri, Ra, (ii) d . Ri ∼ Ra, where Rν = αe2/mνω for ν = i, a is some characteristic distance at which the atom-ion in- teraction becomes comparable to the trapping potential of the ion and atom, respectively. Because of the weak dependence of Rν on ων and mν , the characteristic dis- tances Ri and Ra are roughly the same. The two regimes (i) and (ii) correspond to the cases where the motion in the traps is weakly or strongly affected by the atom-ion interactions, respectively. The discussed regimes are il- lustrated schematically in panels (b) and (c) of Fig. 2, and they correspond to snap shots in the collision pro- cess when the wavepackets of the particles do not over- lap / do overlap during the controlled collision. In the regime (i) of large distances, the atom-ion interaction can be treated perturbatively as a distortion of the trapping potential. In this limit the system can be described in terms of two coupled harmonic oscillators, and all the dy- Definition Description αe2µ/~2 characteristic length of the polarization potential ~/(miωi) harmonic oscillator length of ion trap ~/(maωa) harmonic oscillator length of atom trap ~/(µω) harmonic oscillator length for relative degrees of freedom (axial direction in case of quasi 1D traps) ~/(µω⊥) harmonic oscillator length in the trans- verse direction for relative degrees of freedom distance at which the polarization po- tential becomes equal to the ion trapp- ping potential maω2a distance at which the polarization po- tential becomes equal to the atom trappping potential Rrel = distance at which the polarization po- tential becomes equal to the trappping potential for relative degrees of freedom distance at which the polarization po- tential becomes equal to the trapping potential in the transverse direction for relative degrees of freedom R1D = max(R⊥, l⊥) boundary of the quasi-1D regime R0 size of the core region of the atom-ion complex TABLE I: Definitions of the length scales used throughout the paper. R∗(a0) li(a0) la(a0) E ∗/h (kHz) 40Ca+ + 87Rb 3989 300 644 4.143 9Be+ + 87Rb 2179 632 644 46.59 40Ca+ + 23Na 2081 300 1252 28.56 TABLE II: Characteristic distance R∗, characteristic energy E∗, harmonic oscillator length li for ion trap of ωi = 1MHz, and harmonic oscillator length la for atom trap of ωa = 100kHz, for some combinations of alkali earth ions and al- kali atoms. namics can be solved analytically. We discuss this case in more details in section III F. On the other hand, in the regime (ii) the description is more difficult, since it requires inclusion of the short-range part of the interac- tion, and full treatment of the long-range r−4 part. In the present paper we focus mainly on the latter case. A. Effective Hamiltonian We adopt the following time-dependent Hamiltonian to describe the system of a single trapped atom and a single trapped ion (c.f. Fig. 2): H(t) = ν=i,a ν(zν − dν(t)) + V (|ri − ra|), (1) Here the label i (a) refers to the ion (atom) respectively, p and r are the momentum and position operators, dν(t) denotes the positions of the atom and ion traps, respec- tively, that can be controlled in the course of dynamics, and ρ2 = x2 + z2. We assume that the trapping po- tentials are axially symmetric and displaced along the axis of symmetry. The trapping frequencies are denoted by ων and ω⊥ν for the axial and transverse directions, respectively. Finally, V (r) denotes the interaction po- tential between the atom and the ion. At large distances the main contribution to this interaction comes from the polarization of the atomic cloud, V (r) ∼ −αe2/(2r4). A microscopic derivation of the Hamiltonian (1) is pre- sented in appendix A. The basic idea of this derivation is based on the application of the Born-Oppenheimer ap- proximation to the motion of the outer shell electrons, and followed by a time averaging over fast time scale of the rf and laser frequencies. In this picture, the Hamilto- nian (1) represents effective net interactions felt by parti- cles moving in the rapidly changing time-dependent po- tentials. The interaction potential V (r) can be identi- fied with the adiabatic Born-Oppenheimer curve of the electronic ground-state, that depends on the difference between the atom and ion COM coordinates. B. Quantum-defect theory We denote by R0 the characteristic distance at which V (r) starts to deviate from the asymptotic r−4 law. Typically, R0 is the size of the core region of the ion- atom complex, and is much smaller than all the other length scales in our problem. Therefore, we can describe the short-range part of interaction in the spirit of the quantum-defect theory. For R0 ≪ R ∗, the parameters describing the short-range potential become independent of energy and angular momentum, since for r ≪ R∗ the interaction potential is much larger than typical kinetic energies and heights of the angular-momentum barrier. This feature is a key ingredient of quantum-defect theory, where the complicated short-range dynamics can be sum- marized in terms of few, energy-independent constants (phase shifts or quantum defects). To analyze the short-distance behavior of the wave functions, we omit for the moment the trapping poten- tials in the Hamiltonian (1), and consider only the part describing the scattering of atom from the ion in free space. In this case, the relative and COM degrees of freedom are decoupled, and the relative motion is gov- erned by the Hamiltonian H0 = p 2/2µ+V (r). We apply the partial wave expansion to the relative wave function: Ψrel(r) = l Rl(r)Ylm(θ, φ) with l denoting the angular momentum and Ylm(θ, φ) the spherical harmonics. For r ≫ R0, we set V (r) = −αe 2/(2r4), which allows to solve radial Schrödinger equation in terms of the Math- ieu functions of the imaginary argument [7, 8, 9]. In this way we obtain the following short-distance behavior of the radial wave functions Rl(r, k) ∼ sin (R ∗/r + ϕl(k)) , r ≪ R∗/k, (2) where ~k is the relative momentum, and ϕl(k) are some short-range phases. As one can easily verify, the asymp- totic solution (2) fulfills the radial Schrödinger equa- tion with the energy ~2k2/(2µ) and centrifugal barrier 2l(l+1)/(2µr2) terms neglected. The short-range phases constitute our quantum-defect parameters. For R0 ≪ R we can assume that ϕl(k) are independent of the en- ergy and angular momentum: ϕl(k) ≡ ϕ, which reduces description of the short-range interaction to the single quantum-defect parameter φ. In the calculations we re- place V (r) by its asymptotic r−4 behavior, effectively letting R0 → 0 and imposing boundary condition stated by (2) with the short-range phase φ. For k = 0 the solution (2) becomes valid at all dis- tances. Utilizing the fact that zero-energy solution be- haves asymptotically as Ψ(r) ∼ 1 − as/r (r → ∞), we can relate the short-range phase to the s-wave scattering length as as = −R ∗ cotϕ (3) In this way the knowledge of the scattering length as allows us to calculate the quantum defect parameter ϕ. C. Symmetric 3D trapping potential and identical trapping frequencies In this section we consider 3D dynamics assuming, spherically symmetric trapping potentials ων = ω⊥ν for ν = i, a and identical trapping frequencies for atom and ion: ωi = ωa = ω. For such conditions the relative and CM motions are decoupled. The COM motion is governed by the Hamiltonian of a harmonic oscillator with mass M = mi +ma and fre- quency ω, while the Hamiltonian of the relative motion is given by Hrel = µω2(r− d)2 − , (4) where p and r are the relative-motion momentum and position operators, respectively, and d = da − di. In the case of relative motion we define Rrel = αe2/µω2 = (R∗)1/3l2/3 (5) 0 1 2 3 -5.000 5.000 10.00 15.00 20.00 25.00 30.00 U(r)/ FIG. 3: (Color online) Contour plot of the total potential U(r) = V (r) + 1 µω2(r − d)2 for the relative motion in the atom-ion system. The figure shows the case of spherically symmetric trapping potentials with the same trapping fre- quencies, separated by d = (0, 0, 2R∗), and for R∗ = 2l ~/(µω)). as a characteristic length at which the atom-ion inter- action is comparable to the trapping potential for the relative degrees of freedom, where l = ~/(µω) denotes harmonic oscillator length. Fig. 3 illustrates a typical potential for the atom-ion relative motion. In addition this figure depicts differ- ent lengths scales characteristic of our problem. In the presented case R∗ = 2l, and the distance between trap centers is d = 2R∗. D. Quasi-1D trapping potentials As a second geometry we consider cigar-shape traps with a transverse trapping frequency much larger than the axial one: ω⊥ν ≫ ων (quasi-1D traps) for ν = i, a. For energies smaller than the excitation energy in the transverse direction, the motion in the transverse direc- tion is frozen to zero-point oscillations, and the dynamics takes place along the weakly confined direction. Never- theless, the transverse motion plays an important role at short distances, effectively renormalizing the short-range phase, as we will show in the next section. In this way quasi-1D traps offer the additional possibility of tuning the interactions, similarly to the case of neutral atoms exhibiting confinement-induced resonances [10]. Here, we consider only some particular situation when ω⊥i = ω⊥a is the same for atom and ion. This simpli- fies our description of the renormalization effects, since in this case the transverse COM and relative motions can be separated. We expect that our results are also qual- itatively valid in the general case of different transverse trapping frequencies. To obtain an effective 1D Hamiltonian, describing the evolution of the wave packets along the z axis, we de- compose the total wave function of the atom and ion into a series over eigenmodes of the transverse part of the Hamiltonian. For total energies E < 3~ω⊥ only the transverse ground-state mode contributes to the total wave function at large distances. In addition, condition |zi − za| ≫ R⊥ assures that the axial and the transverse motions are decoupled and the wave function can be writ- ten as a product of the axial and the transverse compo- nents. Here, R⊥ = αe2/µω2⊥ is some characteris- tic distance at which the atom-ion interaction becomes comparable to the transverse trapping potential, and the condition |zi − za| ≫ R⊥ describes the regime where the transverse oscillation frequency is weakly modified by the atom-ion interaction. Hence, we have Ψ(ri, ra) → ψ0(ρi, ρa)Ψ1D(zi, za) (|zi − za| → ∞) for E < 3~ω⊥. Here, ψ0 is the ground state of the transverse part of the Hamiltonian: ψ0(ρi, ρa) = e −ω⊥(miρ )/2~/π1/2, and Ψ1D(zi, za) denotes the axial part of the wave func- tion. Substituting the decomposition over transverse modes into the Schrödinger equation with the Hamilto- nian (1), and retaining only the lowest transverse mode we obtain an effective 1D Hamiltonian, that governs the dynamics of Ψ1D(zi, za) H1D = ν=i,a ν(zν − dν) + V1D(|zi − za|). Here V1D(|z|) is the 1D interaction potential obtained by integrating out the transverse degrees of freedom: V1D(|zi − za|) = dρidρa|ψ0(ρi, ρa)| 2V (|ri − ra|) At sufficiently large distances the effective 1D interaction has similar power dependence as in 3D V1D(|z|) = − , |z| ≫ l⊥, (8) where l⊥ = ~/(µω⊥). Summarizing, in our 1D cal- culations we apply the Hamiltonian (6) with the ap- proximation (8), that are valid for |zi − za| ≫ R1D ≡ max(R⊥, l⊥). Finally in the case of equal longitudinal trapping fre- quencies: ωi = ωa = ω, the relative and the COM de- grees of freedom can be separated, and the dynamics is described by the Hamiltonian of the relative motion Hrel1D = µω2(z − d)2 − , (9) where z = zi − za and p = pi − pa. In the next Section we shall solve the problem in the different regimes de- scribed before, and investigate the peculiar phenomena that arise from the interplay between the trapping and the interaction potentials involving the two particles. E. Quantum-defect theory in 1D The quantum-defect treatment of the short-range in- teractions can be extended to the 1D dynamics, described by the Hamiltonian (6). In 1D the asymptotic behavior of the relative wave function at short distances is governed Ψerel(z, k) ∼ |z| sin (R ∗/|z|+ ϕe(k)) , z ≪ R∗/k, Ψorel(z, k) ∼ z sin (R ∗/|z|+ ϕo(k)) , z ≪ R∗/k, where labels e and o refer to the even and odd solu- tions respectively. In our model of 1D dynamics we treat Eqs. (10)-(11) as boundary conditions for |z| → 0. In section IIIA we show that the phases ϕe and ϕo are uniquely determined by l⊥ and ϕ, and that their values can be calculated by solving the scattering problem in the quasi-1D geometry. This requires that l⊥ ≫ R0, which allows to use the quantum-defect description of the short- range potential. As we discuss later, in contrast to 3D traps, in quasi-1D traps the values of ϕ are in general different for scattering waves of different symmetry. F. Time-dependent problem The atom-ion collision by moving the trapping po- tential d(t) is an intrinsically time-dependent problem which requires the integration of the time dependent Schrödinger equation for the given initial condition to predict the transition probabilities for the possible final states. In our approach to the dynamics we first calcu- late the correlation diagrams, showing the energy levels as a function of the trap separations, and on the ba- sis of these diagrams we predict the possible scenario of the atom-ion collision. When the motion of the trap is sufficiently adiabatic, the evolution of the system pro- ceeds along one of the energy curves, therefore the basis of adiabatic eigenstates, dependent parametrically on d, is particularly useful in the analysis of the collision pro- cess. Of course, the adiabaticity is usually broken in the vicinity of avoided crossings. In such cases, however, one can apply e.g. the Landau-Zener theory to calculate the probability of the adiabatic and diabatic passage through an avoided crossing. In our approach to the atom-ion collisions we consider that initially the traps are well separated. In this limit, the asymptotic states of (1) are given by products of the harmonic oscillator states in the two traps. In the course of dynamics, the distance d(t) decreases, particle interact for some definite time, and finally they are again separated, and the final state evolves into some superpo- sition of the harmonic oscillator states (see Fig. 1 for a schematic picture). For such a scheme, we are interested in predicting the final state for some particular realiza- tion of d(t). We stress that in the intermediate phase, the system may evolve into atom-ion molecular complex, and such possibility is fully accounted for in our model. We start from the time-dependent Schrödinger equa- tion with the Hamiltonian (1). We expand the time- dependent wave function in the basis of the energy- ordered adiabatic eigenstates Ψn(x1,x2|d) of the Hamil- tonian (1) H(d)Ψn(xi,xa|d) = En(d)Ψn(xi,xa|d), (12) where we explicitly point out its dependence on the dis- tance d. Substituting the expansion Ψ(xi,xa, t) = cn(t) exp dτ En(d(τ)) ×Ψn (xi,xa|d(t)) (13) into the Schrödinger equation, we obtain a set of coupled differential equations that govern the dynamics of the expansion coefficients cn: ċn = −ḋ m 6=n cm(t) exp dτ (En(d(τ)) − Em(d(τ))) × 〈Ψn(d)| |Ψm(d)〉, (14) In the case of fully adiabatic evolution the coefficients cn(t) remain constant, and the evolution of the sys- tem proceeds along the adiabatic energy curves. From Eqs. (14) one can derive the condition for the adiabaticity of the transfer process. Adiabaticity requires that ḋ mul- tiplied by the nonadiabatic coupling 〈Ψn(d)| |Ψm(d)〉 be much smaller than the frequency of the oscillating factor in the exponential of (14). In this case the oscillat- ing factor effectively cancels out the contribution due to changes of d. The adiabaticity condition can be written as ~ḋ〈Ψn(d)| |Ψm(d)〉 ≪ (En−Em) 2∀m,n [13]. At trap separations where atom and ion can be approximately described by harmonic oscillator states, one can easily estimate the nonadiabatic couplings 〈Ψn(d)| |Ψm(d)〉, which gives the following constraint on the adiabatic changes of the trap separation: (En − Em) , k = i, a, (15) where k stands for i (a) when the ion trap (atom trap) is moved. We stress that the latter condition is valid for transitions between different vibrational states, that may occur during the transfer of the atom or of the ion. In the case of avoided crossings between vibrational and molecular states, the adiabaticity of the transfer is char- acterized by the condition that can be determined from the Landau-Zener theory. III. RESULTS We turn now to the analysis of the adiabatic eigenen- ergies and eigenstates as a function of the trap separa- tion d, and of the trapping potential geometry. We start our analysis from the simplest case of quasi-1D traps, where the dynamics takes place effectively in 1D, while the assumptions of equal trapping frequencies allows us to consider COM and relative motions separately. Before discussing this problem, we first study the dependence of the 1D short-range phases ϕe and ϕo on l⊥, R ∗ and ϕ. We argue that in the considered range of parameters they are practically independent of the kinetic energy of the scattering particles, which is assumed in our quantum- defect-theory approach. The reader not interested in de- tails of the derivation may start at Section III B, where we analyze 1D relative motion eigenenergies and eigen- states assuming some particular values of ϕe and ϕo. In Section III C we switch to 3D geometries, analyzing the system of two spherically symmetric traps with the same trapping frequency for atom and ion. Section IIID is de- voted to the properties of adiabatic energy spectra in the vicinity of an avoided crossing. Using the semiclassical theory we calculate the level separation at the avoided crossing, and then applying Landau-Zener theory we in- vestigate the conditions for the adiabatic and diabatic transfer, depending on the distance between the traps. Finally, in Section III E we address the most complicated case of different trapping frequencies for atom and ion, coupling COM and relative degrees of freedom. Because of the complexity of this problem, we limit our analysis only to 1D dynamics; we argue, however, that the ob- served behavior should be qualitatively valid also for the 3D system. A. Short-range phases in quasi-1D traps As a preliminary technical step in preparation for the calculations of adiabatic energy curves that will be re- ported in the following, the present subsection deals with calculating the short-range phases ϕe and ϕo in quasi- one-dimensional geometries. To find the 1D short-range phases we solve the Schrödinger equation for the relative part of the Hamil- tonian (1), assuming the same transverse trapping fre- quency for atom and ion: ω⊥i = ω⊥a = ω⊥, and ne- glecting axial trapping frequencies, which is valid for z ≪ Ri, Ra. The latter condition requires Ri, Ra ≫ R1D, where R1D = max(R⊥, l⊥) determines the boundaries of the 1D regime. In this way we obtain the following equa- tion for the relative wave function Ψ(r) µω2⊥ρ 2 − E Ψ(r) = 0, (16) In the case of interaction potentials at distances R1D much larger than typical kinetic energies in the trapping potential: |V (R1D)| ≫ ~ωi, ~ωa, we can neglect the en- ergy dependence of the short-range phases, and solve (16) for E = ~ω⊥. At small distances, r ≪ R1D, the atom-ion interac- tion dominates over the transverse trapping potential, and the solution of (16) behaves according to (2). At large distances, z ≫ R1D, the motion in the transverse direction is frozen to its zero-point oscillations and in the -1.0 -0.5 0.0 0.5 1.0 1D regime 1D regime 2l R1D V(r)/ l = 0.09 R* (z) |z| sin(|z|/R*+ ) (r) sin(r/R 3D regime FIG. 4: (Color online) Potential for the atom-ion relative motion (upper panel) and the relative wave function (lower panel) for the quasi-one dimensional system with l⊥ = 0.09R and without trapping in the axial direction. asymptotic regime the scattering wave function assumes the form Ψ(r) → ψ0(ρ)Ψ1D(z) (|z| → ∞), where ψ0(ρ) is the ground-state wave function of the 2D harmonic os- cillator: ψ0(ρ) = e −ρ2/(2l2 )/π1/2, and Ψ1D(z) is a linear combination of odd and even waves: Ψ1D(z) = ceΨe(z) + coΨo(z), (17) with Ψe(z), Ψg(z) given by (10) and (11) respectively [25]. Fig. 4 shows the potential for the relative motion, and the axial profile (ρ = 0) of the relative wave function for l⊥ = 0.09R ∗, which corresponds to a system of 40Ca+ and 87Rb in a trap with ω⊥ = 2π × 1MHz. In addition, the figure illustrates different length scales present in the quasi-1D problem, and indicates the 1D (|z| ≫ R1D) and 3D regimes (r ≪ R1D) in the behavior of the wave function. Fig. 5 shows an example of the dependence of ϕe and ϕo on ϕ, for l⊥ = R ∗. It compares the results of nu- merical calculations with predictions based on the pseu- dopotential method, that is dicsussed in appendix B. We observe that for even waves the agreement is fairly good, while for odd waves the agreement is poorer, which is probably due to the fact that the p-wave energy- dependent pseudopotential does not work already in the regime of R∗ ∼ l⊥, or contributions of odd partial waves with l > 1 can be important. The pseudopotentials are expected to give accurate predictions for R∗ ≪ l⊥. When R∗ ≫ l⊥, the pseudopotential approach is not applicable at all, and one has to resort to numerical cal- culations. As an example, we present in Fig. 6 values for the short-range phases for l⊥ = 0.1R ∗. We note the presence of several resonances in the dependence of ϕe and ϕo. This behavior is related to the contribution of several partial waves for R∗ ≫ l⊥, leading to resonances when the energy of a bound state in the combined har- monic and r−4 potential becomes equal to the energy of the scattered wave. To analyze this issue more carefully -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 l = R* numerical energy-dependent pseudopotential l = R* numerical energy-dependent pseudopotential FIG. 5: (Color online) Even and odd short-range phases ϕe and ϕo calculated for l⊥ = R ∗. Numerical results (red solid lines) are compared with predictions of the model replacing r−4 with the energy dependent pseudopotential. we have calculated the energies of some particular bound states in in the combined harmonic and r−4 potential. Fig. 7 shows the relation between the bound-state energy and the short-range phase for few bound states that are responsible for the scattering resonances close to φ = 0 for φe (cf. Fig. 6). For comparison we include the bound states of pure r−4 potential and we label them by the angular momentum l. We note that, due to the limited resolution of our numerical calculations, Fig. 6 does not allow to resolve the single resonances close to φ = 0, however, one can observe the rapid changes of φe in this region. For the numerical calculations we have transformed (16) into cylindrical coordinates and we were solving 2D elliptic partial differential equation using a finite- element method. To fix the short-range phase we im- pose the boundary conditions on the logarithmic deriva- tive of Ψ(r), ∂rΨ(r)/Ψ(r), at rmin, where rmin = 0.09R (rmin = 0.022R ∗) for l⊥ = R ∗ (l⊥ = 0.1R ∗) . At large distances we impose the Dirichlet boundary condition on the rectangle with boundaries |z| = zmax, ρ = ρmax, as- suming that the wave function has a Gaussian transverse profile at |z| = zmax and vanishes at ρ = ρmax. For l⊥ = R ∗ we took zmax = 6.3R ∗, ρmax = 3R ∗, while for l⊥ = 0.1R ∗ we have used zmax = R ∗, ρmax = 0.4R To determine the values of ϕe and ϕo, we fit at large distances (|z| ≫ R1D) the symmetric and antisymmetric solutions of (16) for E = ~ω⊥, to the asymptotic formula (17). B. Relative motion in a 1D system: adiabatic eigenenergies and eigenstates In this section we consider the 1D motion of an atom and an ion, assuming ωi = ωa. In this case we can focus only on the relative motion described by the Hamilto- nian (9), with the boundary conditions at z → 0, stated by (10) and (11). To find eigenenergies and eigenfunc- -0.4 -0.2 0.0 0.2 0.4 0.4 (b) FIG. 6: (Color online) Even short-range phase ϕe (upper panel) and odd short-range phase ϕo (lower panel) calculated numerically for l⊥ = 0.1R ∗, versus 3D short-range phase ϕ. -50 -5 -0.5 0.0 0.5 1.0 FIG. 7: (Color online) 3D short-range phase ϕ versus energy of bound states in the presence of the transverse confinement with l⊥ = 0.1R ∗. The bound states in the combined harmonic and r−4 potentials (solid lines) are compared with the bound states of pure r−4 interaction (dashed lines). tions for arbitrary value of d we diagonalize the Hamil- tonian (9) numerically in the basis of its eigenstates for d = 0, that are found by numerical integration of the 1D Schrödinger equation. In the basis we include the low- est 50 odd and even eigenfunctions, which is sufficient to perform the diagonalization for d/R∗ . 2.5. Fig. 8 present the adiabatic energy spectrum for some example parameters: ϕe = −π/4, ϕo = π/4 and R ∗ = 3.48l. The latter value corresponds to the system of 40Ca+ and 87Rb in the trap with ω = 2π × 100kHz, while the particular choice of the short-range phases ϕe and ϕo is explained later in this section. Points with labels correspond to the wave functions shown in Fig. 9 presented together with their eigenenergies and potential energy curves. The three panels shown in Fig. 9 illustrate three dif- ferent regimes, where the system exhibits qualitatively different behavior. In the first regime, represented by the eigenstate Ψa and realized at large distances between traps, d≫ Rrel with Rrel defined in (5), the main effect of the atom-ion interaction is the distortion of the trapping potential, and in this limit the Hamiltonian can be diago- nalized analytically in the model of two coupled harmonic oscillators (see Section III E for more details). The value of the short-range phase is not important for this model. At large separations the eigenstate Ψa is typically only weakly perturbed with respect to the eigenstate of the harmonic oscillator. In the second regime, represented by the eigenstates Ψb and Ψc, and realized at distances d ∼ Rrel, the system exhibits resonances between vibrational (Ψb) and molec- ular (Ψc) states, manifesting themselves as avoided cross- ings in the adiabatic energy spectrum. The resonances appear when the energies of the two eigenstates become equal, and the energy splitting at the avoided crossing is proportional to the tunneling rate through the poten- tial barrier separating the two regions of the potential. In section IIID we calculate this splitting in the frame- work of a semiclassical approximation, and describe the dynamics in the vicinity of the avoided crossing. The value of the short-range phase determines the energy of the molecular states, but it is practically not important for the atom-ion vibrational states. Since the point of the avoided crossing depends on the short-range phases, the controlled collisions can provide some information on the short-range interaction potential. Finally, at distances d ≪ Rrel the barrier separating the external trap from the well given by the atom-ion at- traction disappears and all eigenstates have some short- range component behaving at short distances according to Eqs. (10) and (11). Hence, in this regime all eigen- states depend on the value of the short-range phases. The molecular states with energies below the dissociation threshold (E = 0) are mainly localized in the well of the atom-ion attractive potential, however for the tight traps considered here, they can be strongly affected by the ex- ternal trapping (e.g. state Ψe). The other type of states, with energies E > 0, are analogs of the vibrational states and they are localized mainly at distances where the ex- ternal trapping potential dominates (e.g. state Ψd). So far we have discussed the properties of the energy spectrum for some particular choice of ϕe and ϕo. It turns out that a qualitatively similar behavior can be observed in all the systems with |ϕe − ϕo| = π/2. In the general case, however, the adiabatic energy spec- trum has a slightly more complicated structure, as it is illustrated in Fig. 10, showing the eigenenergies for ϕe = ϕo = −π/4. We note that by going from d = 0 0.0 0.5 1.0 1.5 2.0 2.5 = + / 4 = - / 4 FIG. 8: (Color online) Energy spectrum of the relative motion for an atom and an ion in 1D traps versus the distance d between traps, calculated for ϕe = −π/4, ϕo = π/4 and R 3.48l (see text for details). The dashed curve shows the line µω2d2, giving the approximate shift of the bound state in the trapping potential (see section IIIC for derivation). to positive d 6= 0 all the energies split into two branches. To understand the nature of this splitting in Fig. 11 we present the wave functions for d = 0.1R∗, with the cor- responding eigenvalues marked with stars in Fig. 10. We observe that the branches represent the eigenstates local- ized on the left-hand side and right-hand side of the point z = 0. The left-localized eigenstates correspond to the rising branches, because they are mainly localized in the region of strong atom-ion attraction. We note that this behavior is similar to the properties of eigenstates in a double-well potential, where appropriate wave functions are constructed by taking symmetric and antisymmetric combination of the states in two wells. In the particular case of ϕe = ϕo, the symmetric and antisymmetric com- bination leads to the states localized on the positive and negative z semi axes, respectively. C. Relative motion in 3D for spherical ωi = ωa traps: adiabatic eigenenergies and eigenstates In this section we extend our analysis to 3D, and con- sider spherically symmetric traps for the atom and the ion with the same trapping frequencies: ωi = ωa. In this case the COM and relative motion can be separated, and in the following we focus only on the relative motion governed by the Hamiltonian (4). We have diagonalized the Hamiltonian (4) taking dif- ferent values of the short-range phase ϕ, and we have observed that the adiabatic energy spectra exhibit quali- tatively the same features as in 1D. In the numerical cal- culations we first calculated the eigenstates for d = 0, by solving the radial Schrödinger equation, for angular mo- menta l ≤ 75 and for energies E ≤ 75~ω. These states -3 -2 -1 0 1 2 3 4 d = 2.5 R* = - /4 = + /4 d = 1.2 R* = - /4 = + /4 d = 0 = - /4 = + /4 FIG. 9: (Color online) Eigenstates of the relative motion for an atom and an ion in 1D traps at different separations d between the traps, calculated for ϕe = −π/4, ϕo = π/4 and R∗ = 3.48l. Presented eigenstates correspond to labeled points in Fig. 8. The horizontal lines present the correspond- ing eigenenergies, and the thick black line shows the potential energy. 0.0 0.5 1.0 1.5 2.0 2.5 = - / 4 = - / 4 FIG. 10: (Color online) Energy spectrum of the relative mo- tion for trapped atom and ion in 1D traps versus distance d between traps, calculated for ϕe = ϕo = −π/4 and R ∗ = 3.48l (see text for details). were used as a basis in the numerical diagonalization of the Hamiltonian (4). A sample adiabatic energy spectrum is shown in Fig. 12, presenting the results for ϕ = −π/4 and R∗ = 3.48l. At d = 0 the angular momentum l is a good quantum number and the states have definite angular -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 d = 0.1 R* = - /4 = - /4 FIG. 11: (Color online) Eigenstates of the relative motion for trapped atom and ion in 1D at trap separation d = 0.1R∗, calculated for ϕe = ϕo = −π/4 and R ∗ = 3.48l. The hori- zontal lines present the corresponding eigenenergies, and the thick black line shows the potential energy. 0.0 0.5 1.0 1.5 2.0 l = 0 l = 1 l = 2 l = 3 l = 4 = - /4 FIG. 12: (Color online) Energy spectrum of the relative mo- tion for an atom and an ion confined in spherically symmet- ric traps versus the distance d between traps, calculated for ϕ = −π/4, and R∗ = 3.48lr (see text for details). The blue dashed line shows the shift of the bound states with d, given approximately by 1 µω2d2. symmetry, which is depicted by the appropriate symbols in Fig. 12. For nonzero d, angular momentum is not conserved, hence in the dynamics the state can change its angular symmetry. We note that the energies of the bound states shift to a good approximation according to µω2d2/2. This behavior can be explained by noting that bound states |Ψmol〉 are concentrated around r = 0, and 〈Ψmol(d)|Hrel(d)|Ψmol(d)〉 ≈ Eb + µω2d2, where Eb is the binding energy at d = 0. D. Avoided crossings: semiclassical analysis In the considered setup, level anticrossings reflect res- onances between molecular and trap states. A simple picture of such an avoided crossing is shown in Fig. 13. If one passes an avoided crossing from the direction of the asymptotic trap state, then for adiabatic evolution the system at small distances evolves into a molecular state. On the other hand, for diabatic passage, the particles remain in their traps, and the state basically does not change, apart from the modification due to the smaller trap separation. In addition, for fast changes of the trap positions, the particles can be excited to higher motional states. To describe quantitatively the dynamics in the vicinity of the avoided crossing one can apply the Landau-Zener theory. Assuming that close to the avoided crossing the eigenenergies are linear in d, and that d(t) varies linearly in time, the probability that the crossing is traversed diabatically is given by [14, 15] p|1〉→|1〉′ = exp |〈Ψ1|H |Ψ2〉| ~|ḋ||∂E12/∂d| , (18) where the labels |1〉, |2〉 (|1〉′, |2〉′) respectively refer to the vibrational and molecular states before (after) pass- ing the avoided crossing (cf. Fig. 13), and E12(d) = E1(d) − E2(d). For |〈Ψ1|H |Ψ2〉| 2 ≪ ~|ḋ||∂E12/∂d|, the probability (18) is close to unity, and the avoided crossing is passed diabatically. In the opposite case: |〈Ψ1|H |Ψ2〉| 2 ≫ ~|ḋ||∂E12/∂d|, p is small, and the avoided crossing is traversed adiabatically. The matrix element 〈Ψ1|H |Ψ2〉 can be related to the energy gap at the avoided crossing, and in this way we obtain the fol- lowing constraint on the adiabaticity of the transfer close to the avoided crossing: ≪ (E1(d) − E2(d)) 2 (19) The position of the avoided crossings is directly related to the energies of the bound states; thus, a measurement of the final state after controlled collisions, provides tech- nique for spectroscopy of the trapped atom-ion complex. Since the avoided crossings become weaker as the separa- tion between traps increases, this scheme allows to probe only excited molecular states, having sufficiently small binding energies, comparable to the energy scales of the trapping potentials. In the case when the tunneling barrier is sufficiently large, the energy splitting at the avoided crossings can be estimated using a semiclassical approximation. We first focus on the relative motion in the 1D system, assuming ωi = ωa = ω. Using the standard WKB method, one can derive the following result for the energy splitting ∆E ∆E = EΨmol(x1)Ψvib(x2) v(x1)v(x2)Te −W/~. (20) Here x1, x2 are some arbitrary points located in the classically forbidden region close to the classical turning FIG. 13: (Color online) Schematic drawing of the avoided crossing between vibrational and molecular states in the sys- tem of a trapped atom and an ion. An adiabatic change of the distance d between traps induces a transition from the vibrational to the molecular states, while a diabatic process leaves the particles in the vibrational states. points, W (x1, x2, E) = 2m(V (x) − E) is the ac- tion along the classical trajectory from x1 to x2, T = is the tunneling time, Ψmol(x1), Ψvib(x2) denote, respec- tively, the molecular and vibrational states (cf. Fig. 13) with the same eigenvalues E (the resonance case), and v(x) = 2(V (x)− E)/m is the velocity of a particle with energy −E in the inverted potential −V (x). In princi- ple the choice of x1 and x2 is arbitrary, but in our cal- culations we take x1 and x2 at fixed distance from the location of the molecular (x = 0) and vibrational states (x = d), close to the classical turning points. In this way at sufficiently large separations Ψmol(x1), Ψvib(x2) becomes independent of d. We turn now to the case of relative motion in 3D traps. Similar to the analysis for 1D systems, the energy split- ting at the avoided crossings can be calculated semiclassi- cally by applying the instanton technique. In our calcula- tion we adopt the formulation based on the path decom- position expansion developed by Auerbach and Kivelson [16]. We obtain the following formula describing the level splitting: ∆E = EAΨmol(x1)Ψvib(x2) v(x1)v(x2)Te −W/~ (21) The meaning of all the quantities is the same as in the 1D case, the only difference is the prefactorA which accounts for the fluctuations around the instanton path (see [16] for details). In the case ωi = ωa the instanton path that minimizes the classical action is simply a straight line connecting the trap centers. Fig. 14 compares the semiclassical formulas Eqs. (20) and (21) with the exact energy splitting determined from the numerical energy spectra, as those presented in Fig. 8. The presented results correspond to the avoided crossings between the vibrational ground state molecular states. In 1D the energy splitting at the avoided crossing does not 0.8 1.0 1.2 1.4 1.6 1.8 WKB 1D instanton 3D (s-wave) 1D numerical 3D numerical (s-wave) R* = 3.48 l FIG. 14: (Color online) Energy splitting at the avoided cross- ing between vibrational ground-state and the molecular state in 1D and 3D systems. Semiclassical results (solid lines) are compared to the numerical values extracted from the adia- batic energy spectra for different combinations of ϕe and ϕo. 3D results present avoided crossings widths for s-wave bound states. depend on the symmetry of the molecular state, while in 3D it does depend on its angular momentum l, and we present here only the case l = 0. In the first ap- proximation, the splitting depends on the short-range phases only through the critical trap separation at which the resonance occurs. The numerical data are obtained for different combinations of the short-range phases, for which the adiabatic spectra exhibit avoided crossings at different values of d. In our approach, instead of calculat- ing Ψmol(x1) and Ψvib(x2), we fix the overall amplitude in (20) by fitting to the single point at the largest trap separation, where we expect the WKB approximation to be most accurate. The semiclassical curves stop at the distances where the potential barrier disappears. We ob- serve that for the same ratio ofR∗ to l the splittings in 1D are larger than in 3D. Fig. 14 shows the prediction of the semiclassical formula Eq. (20), and the energy splittings calculated numerically for molecular states with different l. We note that separations at the avoided crossing are largest for the spherically symmetric molecular states, with l = 0. The knowledge of the energy splitting ∆E can be used to calculate the probabilities of an adiabatic and dia- batic passage of avoided crossings. Assuming that the avoided crossing is traversed at constant rate, we apply the formula (18) with ∂E1(d)/∂d ≈ 0 and ∂E2(d)/∂d = 〈Ψmol(d)|∂Hrel(d)/∂d|Ψmol(d)〉 ≈ µω 2d (c.f. Figs. 8 and 12), which leads to p|1〉→|1〉′ = exp (∆E)2 ~|ḋ|µω2d , (22) where we use the same notation as in Fig. 13. Analyzing Figs. 14 and 15 we can now estimate the rates ḋ required 1.0 1.2 1.4 1.6 s-wave semiclassical s-wave numerical p-wave semiclassical p-wave numerical d-wave semiclassical d-wave numerical R* = 3.48 l FIG. 15: (Color online) Energy splitting at the avoided cross- ing between vibrational ground state and molecular states with different angular momenta, for a 3D spherically sym- metric trap. Semiclassical results obtained by means of the instanton technique (solid lines) are compared to the numeri- cal values extracted from the adiabatic energy spectra. Semi- classical calculations stop at the distance d = 1.1R∗ where the potential barrier disappears. for adiabatic and diabatic transitions. For instance in 3D, for the parameters of Fig. 14, the diabatic transfer of particles across the avoided crossings up to distances d ≈ R∗ (∆E . 0.1~ω) can be realized by keeping ḋ/R∗ ≫ 0.001ω . In summary, the analysis carried out here provides the basis for a description of the dynamics of various pro- cesses of interaction between an atom and an ion manip- ulated through external trapping potentials, and it gives a way to estimate quantitatively with simple analytical means the outcome of controlled interaction experiments in the different regimes described in Sect. II. E. Center of mass coupled to the relative motion: 1D analysis for ωi 6= ωa In this section we consider the effects of coupling be- tween COM and relative motions in 1D, that appear when the trapping frequencies for atom and ion are not equal. Fig. 16 shows the adiabatic levels as a function of trap separation for some example parameters: ωi = 5.5ωa and la = 0.9R ∗. This choice corresponds to the interaction of 40Ca+ and 87Rb in the traps with ωa = 2π × 10kHz and ωi = 2π × 55kHz. To obtain the adiabatic spectrum presented in Fig. 16, we performed the diagonalization of the Hamiltonian (6) in the product basis of the COM and relative motion eigenstates evaluated at d = 0. In the calculations we consider all the states with a total energy E ≤ 460~ωa, which leads to about 7100 states in the basis. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 = 5.5 = - / 4 = + / 4 FIG. 16: Energy spectrum for atom and ion confined in har- monic traps with ωi = 5.5ωa, as a function of the distance d. Calculations are performed for ϕe = −π/4, ϕo = π/4 and la = 0.9R ∗ (see text for details). The arrows on the right-hand side of Fig. 16 indi- cate the asymptotic states for large separations, which can be labeled by the number of phonons for the atom and ion trap, respectively. In comparison to the case where relative and COM motions are decoupled, we ob- serve the following new features: (i) the molecular spec- trum contains states with different numbers of excita- tions in the COM degree of freedom. This can be ob- served at d = 0, when the molecular levels are with a good approximation equally separated by ~ωCM where ω2CM = (maω a +miω i )/M . [26]; (ii) the avoided cross- ings between molecular and vibrational states are weaker for the states involving vibrational excitations of the ion. This can be observed by comparing the avoided cross- ings for the state |0〉a|1〉i with the avoided crossings for the neighboring states. This behavior can be understood when we notice that for weaker atom trap (for ωa < ωi), the particles have larger probability to tunnel when the atom is excited. F. Large distances between traps: two coupled oscillators approximation Finally we turn to the limit of large distances: d ≫ Ri, Ra. Expansion of the interaction up to second-order terms in the distance r leads to H1D ≃ ν=i,a ν(zν − z̄ν) (zi − z̄i)(za − z̄a), (23) where z̄ν denotes the equilibrium position of the parti- cles and we neglect the second-order terms modifying the trapping frequencies. The Hamiltonian (23) describes the system of two coupled harmonic oscillators, which can be written in the form H1D = ~ωaa †a+ ~ωib †b+ ~ωc(a+ a †)(b+ b†), (24) where we have introduced the usual annihilation and cre- ation operators zi + i za + i . (26) Here ωc denotes the coupling frequency: ~ωc = 10E∗(R∗)4lali/d 6. The validity of the considered model is limited by the assumption that a stable equilibrium position exists, which is fulfilled for Ri, Ra . 0.57d, as it can be easily verified. Typical, maximal values of ωc for atoms in optical potentials and ions in rf traps are of the order of 10kHz. In practice the model ceases to be valid already at weaker conditions, when the terms higher than the second order cannot be neglected. In any case, this model is relevant for an important class of processes involving coherent transfer of quanta between the atom and the ion, of direct application in a quantum information processing context. IV. CONCLUSIONS AND OUTLOOK In this paper we analyzed in detail the interaction be- tween a single atom and a single ion guided by external trapping potentials. This kind of work is motivated by re- cently opened experimental possibilities within combined systems – currently being built in several groups world- wide – where magneto-optical traps or optical lattices for neutral atoms coexist with electromagnetic traps for ions. Tight confinement of single particles, associated with independent control of the respective confining poten- tials, allows for exploring different regimes of the two- body dynamics involving one atom and one ion at a time. At large distances, the interaction is dominated by the inverse quartic term arising from the attraction between the ion’s charge and the electric dipole induced by it on the atomic electron wave function. A pseudopotential approximation, similar to that familiar from ultracold- atom collision theory, is not applicable in our case, as the characteristic range of the atom-ion molecular potential often exceeds the size of the tight traps experimentally available. To describe the interaction at short distances, smaller than that range, we employ quantum-defect the- ory, which allows to deal with different geometries from isotropic three-dimensional traps to very elongated quasi- one-dimensional ones. A good description of adiabatic dynamics, involving processes where the traps are moved toward or across each other at a rate much slower than the trapping fre- quencies, can be obtained from quasistatic eigenenergy curves, which we calculate for various trapping configu- rations based on the methods outlined above. A remark- able feature displayed by the system is the presence of resonances between molecular-ion bound states and mo- tional excitations within the trap. These trap-induced resonances are similar in nature to Feshbach resonances driven by external fields, and they could as well be spec- troscopically probed in simple experiments where the in- teraction is controlled via the external guiding potentials. In addition to the above aspects, an important moti- vation for the interest in systems of trapped atoms and ions and their trap-induced resonances resides in possible applications to quantum information processing. In this context, one can utilize controlled atom-ion interactions to effect coherent transfer of qubits, thereby creating in- terfaces between atoms and ions. By storing quantum in- formation in internal atomic states, and performing gate operations with ions, one would combine the advantages of both: (i) long decoherence times for neutral atoms and (ii) short gate-operation times for charged particles due to the relatively strong interactions. In the present work we have focused on the motional degrees of free- dom, which can serve as auxiliary degrees of freedom for quantum gates involving internal-state qubits, provided an appropriate coupling mechanism between internal and external degrees of freedom is employed – e.g., sideband excitation via a laser. Another potential application of our results is cooling of the atomic motion. In typical neutral-atom quantum computation schemes with qubits stored in internal states of atoms trapped in optical-lattice sites, motional excita- tions constitute a serious source of errors, and the atomic motion needs to be cooled in a state-insensitive manner between computational steps in order to avoid qubit de- coherence. Since the long-range part of the atom-ion in- teraction is not sensitive to the internal state [27], our setup can be applied for sympathetic cooling of atoms, through the exchange of energy with laser cooled ions [12]. Beside these applications in the context of quantum information processing, our work opens broader perspec- tives for the study of new interesting collisional physics in a physical situation never explored before. In prin- ciple, our scheme allows for production of ultracold trapped atom-ion molecules (molecular complexes), when the trapping potentials are lowered adiabatically at the stage when the particles remain close to each other. In this way our method can be regarded as a way to per- form cold chemical reactions, where the final state of the molecule can be well controlled. Beyond sufficiently large separation between traps, however, the survival probability of such molecules in the final state is neg- ligible. Indeed, in the calculations presented here, we have only regarded processes leading to a final state in which the atom and the ion are still separated, and no molecule has been formed as an outcome of the inter- action. In other words, the Hamiltonian describing our single-channel model does not include the possibility of transitions to other Born-Oppenheimer curves. While this is a justified assumption for the dynamics considered here, it is in a sense a limitation of our current approach. Investigation of molecular formation in traps beyond this approximation is certainly among the interesting devel- opments that can arise from our work. Its long-term motivation is really to open, beyond the present exam- ples, a new paradigm for cold collision physics, which can be described as the mechanical control of single-particle chemical reactions. Acknowledgments The authors thank J. Denschlag, P. Schmidt, R. Stock, and A. Simoni for helpful discussions. This work was supported by the Austrian Science Foundation (FWF), the European Union projects OLAQUI (FP6-013501- OLAQUI), CONQUEST (MRTN-CT-2003-505089), the SCALA network (IST-15714), the Institute for Quantum Information GmbH, the EU Marie Curie Outgoing In- ternational Fellowship QOQIP, and the National Science Foundation through a grant for the Institute for Theoret- ical Atomic, Molecular and Optical Physics at Harvard University and Smithsonian Astrophysical Observatory. APPENDIX A: DERIVATION OF THE EFFECTIVE HAMILTONIAN (1) While the Hamiltonian (1) is intuitively obvious, we find it nonetheless worthwhile to summarize the micro- scopic derivation in an adiabatic approximation, and to discuss its validity. For simplicity we consider an atom with a single outer-shell electron (alkali atom) and an ion with single positive charge +e. In addition we do not consider the internal structure of the ion. The total Hamiltonian can be written as +He +Hlas +Hrf . (A1) Here the labels 1 and 2 correspond, respectively, to atomic nucleus and ion, and He is the Hamiltonian of the electron, which includes the Coulomb interactions, |xe − x1| |xe − x2| |x2 − x1| . (A2) Here, for simplicity, we omit the contributions of the core regions for both the atom and ion. In this way our model refers in fact to H+2 molecule. For alkali atoms and alkaline-earth ions one should treat the complete struc- ture of core regions, however, our approach can be readily generalized to this more complicated case. The term Hlas describes the interaction of the atom with a laser beam creating an optical potential, written in the electric dipole representation Hlas = −dE⊥(x1, t), (A3) where d = e(x1 − xe) is the dipole moment of the atom, and for simplicity we neglected the influence of the laser on the ion, which has typically a different electronic structure that the atom. In addition we have applied the long-wavelength approximation, neglecting changes of the electric field on the scale of the atom. In the case of the optical lattice the transverse part of the electric field E⊥ can be assumed to have the form of a standing E⊥(x, t) = E0 cos(ωLt) [cos(kLx+ φx) + (x→ y, z)] , where for simplicity we have assumed the same amplitude E0, the same wave vector kL, and the same frequency of the laser light ωL for all three laser beams creating the optical lattice potential. The abbreviation (x → y, z) denotes sum of the terms with x replaced by y and z, and ϕk for k = x, y, z is the phase factor characterizing the standing wave. Finally, Hrf is the electric potential creating the rf-trap Hrf = eΦ(x1, t) + eΦ(x2, t)− eΦ(xe, t), (A5) where Φ(x1, t) is the time-dependent electric field of the rf trap Φ(x, t) = 2 + uyy 2 + uzz 2 + vyy 2 + vzz 2) cosωrft. (A6) Here, ωrf is the frequency of the time-dependent part of the electric potential, and uk, vk (k = x, y, z) are am- plitudes depending on the geometry of the trap [5]. The electric field at every instant of time has to fulfill the Laplace equation: ∆Φ = 0, hence, the coefficients uk, vk are subject to the following conditions: ux+uy+uz = 0, vx + vy + vz = 0. Below we indicate the basic steps of the derivation. Expansion in the basis of Born-Oppenheimer wave functions for electron motion. We start from generating a complete set of electronic wave functions, parametrized by the positions of the atomic core and of the ion HeΦn(xe|x1,x2) = En(|x2 − x1|)Φn(xe|x1,x2) (A7) In this way the total wave function can be expanded in the basis of Born-Oppenheimer electronic wave functions Ψ(x1,x2,xe, t) = cn(x1,x2, t)Φn(xe|x1,x2). (A8) Since the basis is complete, the expansion of the wave function does not involve any approximations. Retaining in the expansion only the modes coupled by the laser. In the expansion (A8) we keep only two modes coupled by the laser light creating the optical lattice: the electronic ground state Φg(xe|x1,x2) and the electronic excited state Φe(xe|x1,x2). In this way we utilize the Born-Oppenheimer approximation, treating the electron motion in the adiabatic approximation. This assumes that the time scale of the electron dynamics is much faster than the dynamics of the atomic nucleus and of the ion, which is typically fulfilled since the electron is much lighter than the other two particles. The approximation of the two coupled channels can be easily generalized to more, or even infinite number of channels, since the other channels are weakly populated and we treat them within the perturbation theory. Adiabatic elimination of the excited electronic state coupled through the laser: derivation of the optical trap potential. The expansion coefficients cg(x1,x2, t) and ce(x1,x2, t) fulfill the following set of coupled equations: ∂cg(x1,x2, t) + Eg(|x2 − x1|) + eΦ(x2, t) cg(x1,x2, t)− degE⊥(x1, t)ce(x1,x2, t) (A9) ∂ce(x1,x2, t) + Ee(|x2 − x1|) + eΦ(x2, t) ce(x1,x2, t)− degE⊥(x1, t)cg(x1,x2, t), (A10) where deg(x1,x2) = 〈Φe|d|Φg〉 is the dipole matrix ele- ment between the ground and excited electronic states, which in general depends on the position of atom and ion. In the derivation of (A9)-(A10) we have neglected the action of the rf field on the atom core and the elec- tron, putting 〈Φk|Hrf |Φk〉 ≈ eΦ(x2, t), k = e, g (A11) 〈Φe|Hrf |Φg〉 ≈ 0. (A12) We focus on the regime of far-detuned laser: ∆ ≫ ΩL, where ∆ denotes the detuning ∆(|x2 − x1|) = Ee(|x2 − x1|)−Eg(|x2−x1|)−ωL, and ΩL = −degE0/~ is the the Rabi frequency. For such conditions, the excited state is only weakly populated and can be adiabatically elimi- nated. Additional simplification comes from the fact that the transitions between states g and e, due to the laser light, occur on a time scale much shorter than the mo- tion of atom and ion, and the dynamics of the atom and ion centers of masses can be decoupled from the internal dynamics, in full analogy to the Born-Oppenheimer ap- proximation. Eliminating the excited state, in basically the same manner as in the standard derivation of the AC Stark shift, we obtain the following equation that governs the dynamics of the atom and ion center of masses: ∂cg(x1,x2, t) + Eg(|x2 − x1|) + Vopt(x1) + eΦ(x2, t) cg(x1,x2, t), (A13) where Vopt(x) = − cos(kLx+ φx) 2 + (x→ y, z) (A14) is the effective potential due to the laser field. In the for- mula (A14) we have neglected the position dependence of the detuning ∆(|x2 −x1|) and of the dipole matrix el- ement deg(x1,x2), assuming that they are modified only at very short distances between the particles. This can be justified, since in the atomic collisions, the probabil- ity of finding the particles at short distances, comparable to the range of chemical binding forces, is typically very small. Time averaging over fast oscillations of the rf field: derivation of the Paul trapping potential. We replace the time-dependent rf field by an effective, adiabatic poten- tial. In this way we neglect the fast micromotion of the ion on the time scale of ωrf [5]. Finally we obtain the fol- lowing equation that describes the dynamics of the wave function cg(x1,x2, t) dependent on the atom and ion po- sitions + Eg(|x2 − x1|) + Vopt(x1) Vrf(x2) cg(x1,x2, t). (A15) Here Vrf denotes the effective potential Vrf(x) = 2 + ωyy 2 + ωzz , (A16) where ωk = (ak + q 1/2ωrf/2, ak = 4euk/(m2ω and qk = 2evk/(m2ω rf) for k = x, y, z. The interac- tion between atom and ion is given by Eg(|x2 − x1|) – the ground-state energy of the electron motion, which at large distances behaves as Eg(r) ∼ −αe 2/(2r4). APPENDIX B: DETERMINING OF ϕe AND ϕo FROM PSEUDOPOTENTIALS In principle, the replacement of the r−4 interaction by the pseudopotential is strictly valid when R∗ ≪ l⊥. Nevertheless, it is possible to try to determine ϕe and ϕo forR∗ ∼ l∗ using the energy-dependent pseudopotentials. To this end we replace the r−4 interaction with [17, 18] Vs(r) = 2π~2a(k) r (B1) for even scattering, and [19] Vp(r) = π~2ap(k) ∇ δ(r) r2, (B2) for odd scattering. Here, a(k) = − tan δ0(k)/k is the energy-dependent s-wave scattering length, ap is the p- wave scattering length: ap(k) 3 = − tan δ1(k)/k 3, the symbol ∇) denotes the gradient operator that acts to the left (right) of the pseudopotential, and δ0(k), δ1(k) are the s- and p-wave phase shifts, respectively. In 1D the even (ψe) and odd (ψo) scattering waves have the following asymptotic behavior: ψe(z, k) ∼ sin(k|z|+ δe(k)), and ψo(z, k) ∼ z/|z| sin(k|z| + δo(k)) (|z| → ∞), where δe(k) and δo(k) denote even and odd scattering phases, respectively. In analogy to the three-dimensional scattering theory, we define 1D even (ae1D) and 1D odd (ao1D) scattering lengths: a 1D = limk→0 −(tan δe,o(k))/k. From this definition it follows that for k = 0 the even and odd scattering waves behaves as: ψe(z, k = 0) ∼ |z|−a and ψo(z, k = 0) ∼ z − a 1Dz/|z| (|z| → ∞). The latter result applied to Eqs. (10) and (11) leads to ae1D/R ∗ = − cotϕe (B3) ao1D/R ∗ = − cotϕo (B4) In the pseudopotential approximation one can solve the quasi-1D scattering problem exactly, and calculate values of the 1D scattering lengths for even [20, 21] and odd [22] waves ae1D(k) = − 2as(E) , (B5) ao1D(k) = 12ap(E)3 where E = ~ω⊥+~ 2k2/(2µ) and ζ(s, a) denotes the Hur- witz Zeta function: ζ(s, a) = k=0(k + a) −s [23]. Fi- nally, to relate the energy-dependent scattering lenghts as(E) and ap(E) to ϕ, we apply the exact solutions of the Schrödinger equation for r−4 potential, given by the Mathieu functions [7, 8, 9]. [1] N. Schlosser, G. Reymond, I.E. Protsenko and P. Grang- ier, Nature (London) 411, 1024 (2001). [2] Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Förster, M. Khudaverdyan, D. Meschede, D. Schrader, and A. Rauschenbeutel, Nature (London), 442, 151 (2006). [3] M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch, Nature 415, 39-44 (2002). [4] R. Folman et al., Adv. At. Mol. Opt. Phys. 48 263 (2002); R. Dumke et al., Phys. Rev. Lett. 89, 97903 (2002). [5] D. Leibfried, R. Blatt, C. Monroe, D. Wineland, Rev. Mod. Phys. 75, 281 (2003). [6] O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hänsch, and I. Bloch, Nature 425, 937-940 (2003). [7] E. Vogt, and G.H. Wannier, Phys Rev. 95, 1190 (1954). [8] T.F. O’Malley, L. Spruch, L. Rosenberg, J. Math. Phys. 2, 491 (1961). [9] R.M. Spector, J. Math. Phys. 5, 1185 (1964). [10] M. Olshanii, Phys. Rev. Lett. 81, 938 (1998). [11] R. Stock, I. H. Deutsch, and E. L. Bolda Phys. Rev. Lett. 91, 183201 (2003). [12] Z. Idziaszek, T. Calarco, P. Zoller, in preparation. [13] D.J. Tannor Introduction to Quantum Mechanics: A Time-Dependent Perspective, (University Science Books, 2006). [14] L.D. Landau, and E.M. Lifshitz Quantum Mechan- ics: Non-relativistic Theory, 3rd edition (Butterworth- Heinemann, Oxford, 2003). [15] C. Zener, Proc. R. Soc. London A137, 696 (1932). [16] A. Auerbach, and S. Kivelson, Nucl. Phys. B 257, 799 (1985). [17] D. Blume, C.H. Greene, Phys. Rev. A 65, 043613 (2002). [18] E.L. Bolda, E. Tiesinga, and P.S. Julienne, Phys. Rev. A 66, 013403 (2002). [19] Z. Idziaszek, and T. Calarco, Phys. Rev. Lett. 96, 013201 (2006). [20] T. Bergeman, M.G. Moore, and M. Olshanii, Phys. Rev. Lett. 91, 163201 (2003). [21] Z. Idziaszek, and T. Calarco, Phys. Rev. A 74, 022712 (2006). [22] B.E. Granger, D. Blume, Phys. Rev. Lett. 92, 133202 (2004). [23] M. Abramowitz and I.E. Stegun Handbook of Mathemat- ical Functions, 10th edition, (Department of Commerce, Washington DC, 1972). [24] For atomic states with a permanent quadrupole moment, the long-range part of the potential comes from the inter- action of ion charge with the atom quadrupole moment. [25] 1D solutions (10) and (11) are valid at all distances for k → 0 (E → ~ω⊥). [26] COM and relative motions can be decoupled in the lowest order when the size of the molecule is much smaller than the harmonic oscillator lengths li and la. [27] At least for the internal states with the total angular momentum J = 1/2.

0704.1038

Counting BPS operators in N=4 SYM

arXiv:0704.1038v3 [hep-th] 2 Nov 2007 DIAS-STP-07-05 arXiv:0704.1038 [hep-th] Counting BPS Operators in N = 4 SYM F.A. Dolan Institiúid Ard-léinn Bhaile Átha Cliath, (Dublin Institute for Advanced Studies,) 10 Burlington Rd., Dublin 4, Ireland The free field partition function for a generic U(N) gauge theory, where the fun- damental fields transform in the adjoint representation, is analysed in terms of symmetric polynomial techniques. It is shown by these means how this is related to the cycle polynomial for the symmetric group and how the large N result may be easily recovered. Higher order corrections for finite N are also discussed in terms of symmetric group characters. For finite N , the partition function involving a single bosonic fundamental field is recovered and explicit counting of multi-trace quarter BPS operators in free N = 4 super Yang Mills discussed, including a general result for large N . The partition function for quarter BPS operators in the chiral ring of N = 4 super Yang Mills is analysed in terms of plane partitions. Asymptotic counting of BPS primary operators with differing R-symmetry charges is discussed in both free N = 4 super Yang Mills and in the chiral ring. Also, general and explicit expressions are derived for SU(2) gauge theory partition functions, when the fundamental fields transform in the adjoint, for free field theory. Keywords: Characters, Partition Functions, Gauge Theory, N = 4 Super Yang Mills E-mail: [email protected] http://arxiv.org/abs/0704.1038v3 1. Introduction For the past while there has been intense interest in finite N partition functions for Yang Mills theories, especially in super-symmetric ones, particularly with regard to their construction for BPS states and the counting thereof [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. Much attention has been devoted to this issue for N = 4 super Yang Mills theory, it being by now the archetypal example of a conformal field theory for which we have a dual description in terms of string theory, by means of the AdS/CFT conjecture [16]. An example is in the assiduous efforts that have been made to explain the entropy of certain BPS black holes in AdS5×S5 [17,18,19,20] in terms of microscopic counting of dual operators in N = 4 super Yang Mills, with gauge group SU(N), by means of partition functions [1]. While this problem remains unsolved to date, essentially due to the difficulty of defining what is meant by these dual operators for finite N , there are other equally interesting sectors of (super) Yang Mills theories where more progress with counting has been made, examples being in free N = 4 super Yang Mills and in chiral ring sectors involving BPS operators. For N = 4 super Yang Mills, the half BPS sector consists of multi-trace operators involving a single bosonic operator, Z. Similarly, the quarter BPS sector consists of multi- trace operators involving two bosonic operators Z, Y while the eighth consists of ones involving three bosonic operators Z, Y,X and two fermionic operators λ, λ̄. (All these operators are here assumed to belong to the Lie algebra of U(N).) For the chiral ring, the commuting/anti-commuting of these operators is at the heart of why we can write very concise and elegant generating functions for the finite N multi- trace partition functions [1]. Analysis of these partition functions, in terms of the counting of operators, has become a sophisticated industry where such approaches as the so-called ‘Plethystic Program’ have provided substantial results [3,9]. This paper is devoted largely to the issue of partition functions for free field theory and particularly to the counting of gauge invariant multi-trace operators for the case of two bosonic fundamental fields. This is of relevance to the quarter BPS sector of N = 4 super Yang Mills in the free field limit when the operators Z, Y do not commute, in contrast to the chiral ring when they do. The partition function for a free, massless quark-gluon gas was computed long ago [21]. This involved taking particle statistics into account using coherent state techniques and then imposing the gauge singlet condition by integrating over the relevant gauge group. With some modifications to the expression thus derived we may write the multi-trace partition function for some generic bosonic/fermionic fundamental fields in terms of an integral over the gauge group, involving the single particle partition function [22,23]. This is the starting point here. For U(N) gauge theories we may easily write down the integral, though, even for this case, its evaluation is far from simple. One approach (which we adapt here for the SU(2) case) is to rewrite the expression in terms of an N -fold contour integral, whereby it may in principle be evaluated by summing the contributions from poles inside origin centred unit discs, in each of the N complex planes - similar techniques have been used in [10]. Due to the number of poles this becomes unfeasible for higher values of N . Another approach is to use the fact that the complex integral that interests us provides for an inner product for symmetric polynomials - see Macdonald [24], pp. 363 - 372, for a related discussion. Taking this point more seriously reveals an alternative route to evaluating the free field theory partition function which exposes not only the large N case in an almost trivial way, but also how and where this differs from the finite N case. This treatment also reveals an alternative interpretation of the free field partition function at finite N - it is related to a gauge group average of the cycle polynomial for the symmetric permutation group (after a certain identification of ‘letters’ with gauge group valued variables). This point is not dwelt on further here though makes the connection between the partition function and Polya enumeration explicit. For single trace operators at large N this connection has already been made [22,25] for N = 4 super Yang Mills, whereby the partition function for single trace operators is related to the cycle polynomial for the cyclic permutation group.1 Another issue is how to use the expression for the free field partition function to give explicit counting of gauge invariant multi-trace operators in a Yang-Mills theory, with gauge group U(N). The case for one bosonic fundamental field has been widely discussed and, for finite N , the operators are counted by partition numbers of non-negative integers into at most N parts (which is here denoted by pN (n) - no closed formula for these numbers for arbitrary N, n exist, though they have a ‘nice’ generating function). Here, this result is re-derived, from a symmetric function perspective, by employing the well known Cauchy- Littlewood formula. To proceed further with counting, for the quarter BPS sector of N = 4 super Yang Mills, for instance, character methods prove to be both natural and indispensable. Char- acters in relation to conformal field theories prove to be very convenient for encoding the allowable representations [15,29,30,31,32,33] and for studying related partition func- tions, [34,35,36,33]. For N = 4 super Yang Mills, it was shown in [33] that, if we are 1 For more recent applications of symmetric polynomials and Polya enumeration to super- symmetric quantum mechanical models, analysed in terms of Fock space methods, see [26,27,28]. to distinguish among primary operators with differing conformal dimensions, spins and R-symmetry charges, such counting is most easily achieved using reductions of the full N = 4 superconformal characters, in certain limits that isolate corresponding sectors of short/semi-short operators. (One such limit corresponds to the index constructed in [1].) This point may be easily illustrated for quarter BPS primary operators in N = 4 su- per Yang Mills, the case of two bosonic fundamental fields here. (See [37] for an explicit construction and counting of quarter BPS operators.) For N = 4 super Yang Mills, the counting of quarter BPS primary operators is complicated by the fact that, if we are to keep track of differing R-symmetry representations, any partition function restricted to this sector must be expanded in terms of U(2) characters (or two-variable Schur polyno- mials). Denoting some partition function restricted to this sector by Z(t, u), where t, u are letters corresponding to the fields Z, Y , then by expanding Z(t, u) = N(n,m) s(n,m)(t, u) , s(n,m)(t, u) = tn+1um − tmun+1 t− u , (1.1) in terms of two-variable Schur polynomials s(n,m)(t, u), we obtain the numbers N(n,m) of gauge invariant quarter BPS operators belonging to the [m,n−m,m] SU(4)R R-symmetry representation and having conformal dimension n + m (so that they are superconformal primary highest weight states in the corresponding quarter BPS supermultiplets). (The case m = 0 counts gauge invariant half BPS primary operators.) Here, the free field partition function is thus expanded in terms of Schur polynomials, depending on the same variables as the one particle partition function, the two boson case being a specialisation. This is quite naturally achieved using the Cauchy-Littlewood formula (and, if we include fermions, another formula due to Littlewood). Generally, we may obtain a result that relates the counting numbers to a sum over Kronecker coefficients. These arise naturally in the theory of the symmetric permutation group, though remain somewhat mysterious from a combinatorial perspective. Specialising to the two boson case, a recursive procedure is employed here for the counting of multi-trace quarter BPS operators in free field theory at finite N . This issue was given considerable discussion for the large N case in [33] - here the results of [33] are generalised in terms of a generating function that may be employed to count quarter BPS operators for any R-symmetry charges, at large N . Asymptotic counting is also addressed in the latter case for the numbers N(n,m) in (1.1) for large n and fixed m. To complete the discussion, counting of quarter BPS operators in the chiral ring of N = 4 super Yang Mills is investigated in terms of expanding over U(2) characters as in (1.1). An explicit formula is given for the corresponding finite N partition function, with a short combinatorial interpretation given in terms of plane partitions, and specialised to large N . For the latter case, the exponential behaviour of the numbers N(n,m) in (1.1) is found for n and m both comparably large. This behaviour is consistent with a special case addressed in [3,9]. By way of completion, a similar discussion for an arbitrary number of bosonic fundamental fields in the chiral ring is included. Two appendices are included; the first establishes some notation used for partitions and gives some standard results for the symmetric group and symmetric polynomials, the second gives some tables of numbers of quarter BPS operators in free N = 4 super Yang Mills, with gauge group U(N), for which explicit formulae are given in the main text. Footnotes contain further details and points of clarification. 2. Free Field Partition Functions We start from the single particle partition function which is here denoted by f(t) for some variables2 t = (t1, t2 . . .). The general form of f(t) is f(t) = aiti , (2.1) where each ti is a letter corresponding to a fundamental field and ai are signs, being +1 for a bosonic field, or −1 for a fermionic field. For compact gauge Lie group G, the multi-trace partition function is then given by [21], (see [22,23] for refinements,) ZG(t) = dµG(g) exp f(tn)χR(g , (2.2) involving the Haar (or G-invariant, or Hurwitz) measure dµG(g) for g ∈ G (so that∫ dµG(g)F (g) = dµG(g)F (gh) = dµG(g)F (hg) for all h ∈ G and dµG(g) = 1) and where χR(g) is the character for the R representation of G, assuming that the funda- mental fields transform in identical gauge group representations R. For G = U(N), so that for any matrix U ∈ U(N) we may write U = VΘV †, where V is a unitary matrix and Θ = diag.(eiθ1 , . . . , eiθN ), 0 ≤ θi < 2π, and for some F (U) = F (Θ), independent of V , then we may write dµU(N)(U)F (U) = (2π)NN ! 1≤k<l≤N |eiθk − eiθl |2 F (Θ) , (2.3) 2 In what follows roman letters are used to denote a collection of variables and, for x = (x1, . . . , xi), y = (y1, . . . , yj) for example, the shorthand x α is used to mean (x1 α, . . . , xi α) and z = xy to mean z = (z11, . . . , zij) where zrs = xrys. The latter convenient notation has been used by Macdonald [24]. which is, of course, related to the Weyl parametrisation of U(N). Thus, for such F (U), the left-hand side of (2.3) simplifies to an integral over the N torus. Of course, as χR(U) gen- erally depends on linear combinations of tr(U j)ktr(U †l)m for various non-negative integers j, k, l,m then any function of χR(U) is an example of such an F (U). We are interested in the case where R = Adj. is the adjoint representation so that for U(N) we have that χAdj.(U) = trU trU † (while for SU(N) then χAdj.(U) = trU trU †−1). For U(N) we then find that, using (2.2) with (2.3), ZU(N)(t) = (2π)NN ! 1≤k<l≤N |eiθk − eiθl |2 exp f(tn) j,k=1 ein(θj−θk) (2.4) We may write (2.4) as an N -fold contour integral by first making the variable change zj = e iθj so that the integrals in (2.4) are around unit circles in each zj complex plane and then we obtain ZU(N)(t) = (2πi)NN ! ∆(z)∆(z−1) exp f(tn)pn(z)pn(z , (2.5) where ∆(z) = 1≤i<j≤N (zi − zj) is the Vandermonde determinant and pn(z) = i=1 zi is a power symmetric polynomial - see appendix A for a brief discussion of symmetric polynomials. This integral may then in principle be evaluated by deforming the contours so as to extract the residues at poles within the discs |zj | < 1, 1 ≤ j ≤ N . A crucial observation is that, for some N variable symmetric polynomials g(z), h(z), (2πi)NN ! ∆(z)∆(z−1)g(z)h(z−1) , (2.6) acts as an inner product - this is easy to see in terms of Schur polynomials which provide an orthonormal basis for symmetric polynomials. The reader may now wish to peruse appendix A where notation regarding partitions and a short discussion of symmetric poly- nomials is included. The General and Large N Cases for U(N) For application of inner products to (2.5) we have that, in terms of power symmetric polynomials pλ(z) for partitions λ, f(tn)pn(z)pn(z nanan! f(tn)anpn(z) anpn(z −1)an , fλ(t)pλ(z)pλ(z −1) , (2.7) with the definitions of nanan! , fλ(t) = f(tn)an , (2.8) being in terms of the frequency representation of λ, (1a1 , 2a2 , . . .) with n≥1 n an = |λ|, the weight of the partition λ (note that the frequency representation of λ is simply a convenient re-ordering of the parts of λ). In (2.7) the numbers zλ have a standard combinatorial interpretation - for a given permutation σ ∈ Sm with a1 1-cycles, a2 2-cycles etc., so that n≥1 n an = m = |λ|, then n≥1 n anan! is the size of the centraliser Zσ = {τ ; τ ∈ Sm, τστ−1 = σ} of σ ∈ Sm. (This may be easily seen as under conjugation of σ by τ then τ can permute the cycles of length n among themselves in an! ways and/or render a cyclic rotation on each of the individual cycles in nan ways.) More details of the symmetric group are to be found in appendix A. We may immediately observe that (2.7) represents a sum over cycle polynomials of the symmetric group Sm. This is given by, for letters u1, u2, . . . um,3 Cm(u) = a1,...,am≥0 δa1+2a2+...+mam,m nanan! uλ , (2.9) where ‘λ ⊢ m’ means that λ is any partition of m - see appendix A for notation - and an , (2.10) in terms of the frequency representation of λ above. Identifying un = f(t n)pn(z)pn(z then we may rewrite ZU(N)(t) = (2πi)NN ! ∆(z)∆(z−1)Cm(u) , (2.11) 3 This formula is easy to see from the definition of the cycle polynomial for a subgroup G of Sm. This is given by j1(g) · · ·um jm(g) = |Kg|u1 j1(g) · · ·um jm(g) , where ji(g) denotes the number of i cycles in the unique decomposition of g into disjoint cycles and Kg denotes the conjugacy classes of G with class representatives g. The size of the conjugacy class Kg is given by |Kg| = |G|/|Zg| where Zg is the centraliser of g ∈ G. For the present case then, G = Sm, |Sm| = m!, and |Zσ| = zλ, where λ gives the cycle structure of σ ∈ Sm, and thus, for the corresponding conjugacy class Kλ, |Kλ| = m!/zλ. the sum of the U(N) group averages of each of the cycle polynomials Cm(u). (Physically, the interpretation is that the cycle index for the symmetric permutation group accounts for particle statistics while integration over the gauge group imposes the gauge singlet condition. For purposes of clarity, the U(N) case has been focused upon here, though from the form of (2.2) it is easy to see how this generalises for other gauge groups whereby the letters un = f(t n)χR(g n) for the fundamental fields transforming in identical gauge group representations, R.) Directly from (2.7), in terms of the inner product (2.6), then ZU(N)(t) = fλ(t) pλ, pλ fλ(t) µ⊢|λ| ℓ(µ)≤N , (2.12) where on the right-hand side of (2.12) we have used an expression for the inner product of two power symmetric polynomials expressed in terms of the characters of the symmet- ric group, given in appendix A. (Here ‘ℓ(µ)’ means the number of non-zero parts of the partition µ.) Using a result of appendix A (essentially orthogonality relations for symmetric group characters), (2.12) may be rewritten as ZU(N)(t) = |λ|≤N fλ(t) + |λ|>N fλ(t) µ⊢|λ| ℓ(µ)≤N . (2.13) In the large N limit, ZU(N)(t) simplifies considerably as only the first term in (2.13) need be considered. Using the frequency representation of λ then ZU(∞)(t) = fλ(t) = f(tn)an = 1− f(tn) , (2.14) a result which has been obtained using Polya counting methods for single trace operators and saddle point approximations [22,23]. Higher order corrections in |λ|, the weight of the partition λ, to (2.13) may be obtained by successive evaluation of µ⊢|λ| ℓ(µ)≤N . One method is to employ the Murnaghan- Nakayama Rule, used to compute χ λ using skew hooks and Young diagrams. (A readable account of the Murnaghan-Nakayama Rule may be found in [38], though of course it is explained in many standard textbooks that discuss the symmetric group.) For the case of |λ| = N+1 then we may observe that, µ⊢N+1 ℓ(µ)≤N µ⊢N+1 ℓ(µ)≤N+1 )2 − (χνλ)2 = zλ − (χνλ)2 , ν = (1N+1) , (2.15) since the partition ν = (1N+1) is the only one excluded among those partitions µ of N+1 with ℓ(µ) ≤ N . By applying the Murnaghan-Nakayama Rule we may determine, for ν = (1L), (χνλ) 1 for |λ| = L 0 otherwise , (2.16) since χνλ in this case is just a sign. (This may be easily seen as there is only one possible way to remove successive skew hooks, which in this case are just column Young diagrams of length λi, from the (1 L) column Young diagram to leave one of normal shape, in this case, another column Young diagram.) Thus, using (2.15) with (2.16) in (2.13), we obtain ZU(N)(t) = |λ|≤N+1 fλ(t)− |λ|=N+1 fλ(t) + |λ|>N+1 fλ(t) µ⊢|λ| ℓ(µ)≤N . (2.17) By a similar line of argument we may do the same for the case of |λ| = N+2. We have that, for ν1 = (1 N+2) and ν2 = (2, 1 N+1), µ⊢N+2 ℓ(µ)≤N µ⊢N+2 ℓ(µ)≤N+2 )2 − (χν1λ ) 2 − (χν2λ ) 2 = zλ − (χν1λ ) 2 − (χν2λ ) 2 . (2.18) We may determine, for ν = (2, 1L), (χνλ) (a1 − 1)2 for |λ| = n≥1 n an = L+2 0 otherwise . (2.19) Using (2.16), (2.19) with (2.18) in (2.17) then we obtain ZU(N)(t) = |λ|≤N+2 fλ(t)− N+1≤|λ|≤N+2 fλ(t)− |λ|=N+2 fλ(t) |λ|>N+2 fλ(t) µ⊢|λ| ℓ(µ)≤N (2.20) where, in the frequency representation of λ, (a1−1)2 (a1 − 1)! (a1 − 2)! )/ ∞∏ nanan! . (2.21) We may proceed in this manner to compute explicit higher order corrections though this becomes cumbersome save for the first few cases as shown. (For |λ| > N + 2 the corrections will always involve contributions from (2.16) and (2.19) as well as extra ones coming from µ⊢|λ| ℓ(µ)≤|λ| µ⊢|λ| ℓ(µ)≤N The One Boson Case for U(N) For the case of one bosonic fundamental field (applicable to half BPS operators for N = 4 super Yang Mills), we have f(t) = t in (2.5), so that we may write ZU(N)(t) = (2πi)NN ! ∆(z)∆(z−1) j,k=1 1− tzjzk−1 . (2.22) To evaluate this integral we may use the Cauchy-Littlewood formula, 1− xiyj ℓ(λ)≤min.{L,M} sλ(x1, . . . , xL)sλ(y1, . . . , yM) , (2.23) where the sum on the right-hand side is over all partitions λ such that the corresponding Young diagrams have no more than min.{L,M} rows, ℓ(λ) ≤ min.{L,M}. With xi = tzi, yi = zi −1, i = 1, . . . , N , in (2.23), so that sλ(tz1, . . . , tzN ) = t |λ|sλ(z), and employing also (2.6) and the orthonormality of Schur polynomials, we may easily obtain, ZU(N)(t) = ℓ(λ)≤N sλ, sλ ℓ(λ)≤N t|λ| . (2.24) By changing summation variables so that λi − λi+1 = ai, i = 1, . . . , N−1, λN = aN then we may write ZU(N)(t) = a1,...,aN=0 ta1+2a2+...+NaN = PN (t) , (2.25) where4 PN (t) = 1− ti . (2.26) Of course this is nothing other than the generating function for the number pN (n) of partitions of n into no more than N parts since, by definition, ℓ(λ)≤N δ|λ|,n = pN (n) , (2.27) so that by the above pN (n)t n = PN (t) . (2.28) 4 1/P∞(t) = (1−tn) is commonly called the Euler function, denoted by Φ(t). P∞(t) =∑∞ p(n)tn acts as a generating function for the number of unordered partitions of n, p(n). Note that pN (n) = p(n) for n ≤ N , i.e. the number of partitions of n into no more than N parts is the same as the total number of partitions of n so long as n ≤ N . This makes explicit the connection between ZU(N)(t) and the partition numbers pN (n). The SU(2) Gauge Group Case Here we first consider f(t) = j=1 tj in (2.1) so that the variables 0 ≤ ti < 1 represent k bosons in the single particle partition function. For such fields transforming in the adjoint representation of SU(2) then (2.2) simplifies significantly. For any U ∈ SU(2) we may write U = VΘV †, where V is unitary and Θ = diag.(eiθ, e−iθ), for 0 ≤ θ < 2π, so that for F (U) = F (θ), then in usual Weyl parametrisation, SU(2) dµSU(2)(U)F (U) = dθ sin2 θ F (θ) 1− cos θ F ( θ 1− cos θ F ( θ (2.29) where F (θ) = F (θ + π) is assumed in writing the last line. In the present case, F (U) = F (θ) = n≥1 f(t n)χAdj.(U n)/n, where χAdj.(U) = tr(U)tr(U †) − 1 = e2iθ + e−2iθ + 1 = 2 cos 2θ + 1, so that ZSU(2)(t) = dθ (1− cos θ) (1− tj)(1− tjeiθ)(1− tje−iθ) . (2.30) Making the variable change z = eiθ, and using that F (θ) = F (−θ) is even, then ZSU(2)(t) = (1− z) (1− tj)(1− tjz)(1− tjz−1) , (2.31) where the integral is around the unit circle |z| = 1. The residues in (2.31) may be easily computed since all the relevant (simple) poles in the disc |z| < 1 occur at the points z = tj . ZSU(2)(t) = 1− ti2 j 6=i (ti − tj)(1− titj)(1− tj) . (2.32) This partition function has an interesting interpretation from a group theory perspec- tive. We may write, 1 + t (1− tz)(1− tz−1) = χn(z)t n , (2.33) where χj(z) = 2 − z−j− 12 2 − z− 12 , j ∈ 1 Z , (2.34) is an SU(2) character, corresponding to the spin j irreducible representation, Rj. Now the integral in (2.31) acts as an SU(2) inner product, χj , χk (1− z)χj(z)χk(z−1) = δjk , (2.35) for j, k ∈ N. Thus, from (2.31) with (2.33) and (2.35), (1− tj2)ZSU(2)(t) = n1,...,nk=0 χn1 · · ·χnk , 1 n1 · · · tknk , (2.36) acts as a generating function for the number of singlets in the decomposition of the SU(2) representation Rn1 ⊗· · ·⊗Rnk .5 By using that χn(z) = j=−n z j we may use the Cauchy residue theorem to compute explicitly that χn1 · · ·χnk , 1 · · · (δj1+···+jk,n1+···+nk − δj1+···+jk,n1+···+nk+1) . (2.37) If we modify the one particle partition function to include k bosons and k̄ fermions and hence consider (2.1) in the form f(t, t̄) = j=1 tj − ̄=1 t̄̄ then we may similarly as above evaluate ZSU(2)(t, t̄) = (1− z) 1≤j≤k 1≤̄≤k̄ (1− t̄̄)(1− t̄̄z)(1− t̄̄z−1) (1− tj)(1− tjz)(1− tjz−1) , (2.38) where the contour is around the unit disc |z| = 1. So long as k > k̄ then (2.38) receives contributions from only those simple poles at z = tj so that for this case we obtain ZSU(2)(t, t̄) k−k̄−1 1− ti2 1≤̄≤k̄ 1≤j≤k,j 6=i (ti − t̄̄)(1− tit̄̄)(1− t̄̄) (ti − tj)(1− titj)(1− tj) . (2.39) For k ≤ k̄ then (2.38) also receives contributions from poles at z = 0. For instance ZSU(2)(t, t̄) ti(1− ti2) 1≤j,̄≤k j 6=i (ti − t̄̄)(1− tit̄̄)(1− t̄̄) (ti − tj)(1− titj)(1− tj) 1≤j,̄≤k t̄̄(1− t̄̄) tj(1− tj) (2.40) 5 Generating functions for products of Lie algebra representations have been considered elsewhere, in [39] for instance. A generating function for the number of singlets in n products of the fundamental times n products of the anti-fundamental representations for SU(N) was found by Gessel [40] in terms of Toeplitz determinants involving Bessel functions. See also [41] for a nice physics oriented discussion of similar issues. The special case of R 1 ⊗ · · · ⊗R 1 (2n products of the fundamental) for SU(2), contains a Catalan number, 1 , of singlet representations. where the last term on the right-hand side of (2.40) comes from the simple pole at z = 0. These formulae should be useful for computing the multi-trace partition functions, for fundamental fields transforming in an SU(2) gauge group, in other sectors of N = 4 super Yang Mills. For instance, after a suitable identification of the variables tj , t̄̄ with variables in single particle partition functions for semi-short sectors of N = 4 super Yang Mills, described in detail in [33], then (2.40) should allow for an explicit expression for corresponding multi-trace partition functions. They may also be useful for computing the N = 4 superconformal index of [1] for SU(2) gauge group, or at least for restrictions of it such as described in [33] or [42]. 3. Counting Operators in Free N = 4 Super Yang Mills In this section, the counting of half and quarter BPS operators for free N = 4 super Yang Mills, when the fundamental fields transform in the adjoint representation of U(N), is discussed in some detail. Counting Operators Directly We may, of course, proceed to count multi-trace half and quarter BPS primary oper- ators directly, in terms of the fundamental fields, Z, for half BPS operators and Z, Y , for quarter BPS operators. (Z, Y ) forms a U(2) doublet, where U(2) has generators given by a subset of the SU(4)R generators, Hi, Ei±, 1 ≤ i ≤ 3, where Hi are the Cartan sub-algebra genera- tors and Ei± are ladder operators satisfying (in the Chevalley-Serre basis) [Hi, Ej±] = ±KijEj±, with [Kij] being the usual SU(4) Cartan matrix. The U(2) generators consist of the SU(2) generators H2, E2±, where explicitly [(H1, H2, H3), E2±] = ∓(1,−2, 1)E2±, along with the generator H1+H2+H3, whose eigenvalues give the conformal dimensions in this case, [H1+H2+H3, (Z, Y )] = (Z, Y ) [33]. Explicitly, we have that [E2+, Z] = 0, [E2−, Z] = Y , [(H1, H2, H3), Z] = (0, 1, 0)Z, [(H1, H2, H3), Y ] = (1,−1, 1)Y so that an operator involving n Z’s and m Y ’s transforms in the [m,n−m,m] SU(4)R R-symmetry representation. For k-trace half BPS primary operators transforming in the [0, n, 0] SU(4)R R- symmetry representation, with conformal dimension n, then in terms of the fundamental field Z a basis is provided by, tr(Zn1) · · · tr(Znk) , ni = n . (3.1) We have that, due to trace identities for finite N , tr(Zn) for n > N is expressible in terms of a sum over multi-trace operators of the form (3.1), for k > 1, and thus, a minimal basis for multi-trace half BPS primary operators consists of (3.1) for all 1 ≤ k ≤ n and with every ni ≤ N , ordered so that n1 ≥ n2 ≥ . . . ≥ nk ≥ 0, i.e. so that (n1, . . . , nk) is a partition of n where each part ni ≤ N . With this restriction, the number of multi-trace half BPS primary operators for a given n is N(n) = pN (n) , (3.2) since the number pN (n), in (2.27), of partitions of n into ≤ N parts is the same as the number of partitions of n in which each part is ≤ N - see [43] for a simple proof employing generating functions. For quarter BPS operators belonging to the [m,n−m,m] SU(4)R R-symmetry repre- sentation, a basis for k-trace operators is Zn1jY m1j · · · tr ZnkjY mkj nij = n , mij = m, (3.3) where there is a choice of ordering in each trace. (Note that the m = 0 case corresponds to the half BPS case already considered.) Using the basis provided by (3.3) for all allowable k, then to avoid over-counting of multi-trace quarter BPS operators, the cyclicity of each trace and also trace identities for finite N must be accounted for. Assuming that this is done, let M(n,m) denote the number of elements in this minimal basis for multi-trace operators of the form (3.3). Then, to obtain the number N(n,m) of multi-trace quarter BPS primary operators in the SU(4)R representation [m,n−m,m], the number of U(2) descendants, in the SU(4)R representation [m,n−m,m], of multi-trace quarter BPS primary operators, in SU(4)R representations [j, n+m−2j, j], 0 ≤ j ≤ m−1, must be subtracted from M(n,m). (These descendants arise due to the relation [E2−, Z] = Y . Acting with (E2−) m−j on the highest weight state in the SU(4)R representation [j, n+m−2j, j] we obtain a descendant in the SU(4)R representation [m,n−m,m].) The number of such U(2) descendants coincides with N(n+m−j,j), the number of corresponding primary operators. In this way, we obtain M(n,m) = N(n,m) +N(n+1,m−1) + . . .+N(n+m−1,1) +N(n+m) , (3.4) so that N(n,m) = M(n,m) −M(n+1,m−1) may be obtained recursively for each m. We may illustrate by counting all multi-trace quarter BPS primary operators in the [1, n−1, 1] R-symmetry representation. In this case a basis for k+1-trace operators is provided by tr(Zn1) · · · tr(Znk)tr(ZjY ) , ni = n− j . (3.5) Cyclicity of traces implies that we may arrange Y as shown, to avoid over-counting. U(N) trace identities imply, similarly as for the half BPS case, that a minimal basis for multi- trace operators requires j < N and each ni ≤ N in (3.5) for every 1 ≤ k ≤ n−j, so that (n1, . . . , nk) forms a partition of n−j, with every part ≤ N . Thus, by a similar argument as for the half BPS case, M(n,1) = j=0 pN (n−j). Finally, to ensure that only primary operators are counted then we must subtract off contributions from descendants of half BPS primary operators in the [0, n+1, 0] SU(4)R representation, of which there are pN (n+1). Using (3.4) with (3.2) we then conclude that N(n,1) = pN (n− j)− pN (n+1) , (3.6) gives the number of multi-trace quarter BPS primary operators in the [1, n−1, 1] R- symmetry representation. Counting in this fashion becomes more difficult for greater m and now a procedure is described employing symmetric polynomials to find a generating function for the numbers of multi-trace quarter BPS primary operators in the [m,n−m,m] SU(4)R representation, for m = 0, 1, 2 at finite N and for any n,m at large N . This generating function is subsequently used to provide asymptotic counting for fixed m, large n in the large N limit. Counting Operators via Expansion of Partition Functions in Schur Polynomials For k bosonic fundamental fields, we may take f(t) = j=1 tj in (2.1) so that (2.5) may be written as ZU(N)(t) = (2πi)NN ! ∆(z)∆(z−1) r,s=1 1− tjzrzs−1 . (3.7) Often it is the case that such partition functions should be expanded in terms of sλ(t), the k variable Schur polynomial labelled by partitions λ. An example is provided by (1.1) for counting multi-trace quarter BPS operators. We may use the Cauchy-Littlewood formula (2.23) to expand in this way, to obtain ZU(N)(t) = ℓ(λ)≤min.{k,N2} Nλ sλ(t) , (3.8) where (2πi)NN ! ∆(z)∆(z−1) sλ(zz −1) , (3.9) where zz−1 has components zizj −1, 1 ≤ i, j ≤ N . From Macdonald [24] we have that sλ(xy) = µ,ν⊢|λ| γλµνsµ(x)sν(y) , (3.10) in terms of Kronecker coefficients, γλµν = σ∈S|λ| χλ(σ)χµ(σ)χν(σ) ρ⊢|λ| χλρ χ (3.11) being a sum over irreducible S|λ| characters evaluated at σ ∈ S|λ|, related to a sum over irreducible S|λ| characters evaluated on the conjugacy classes labelled by the partitions ρ in the second line. Using (2.6) along with the orthonormality property of Schur polynomials we find that µ⊢|λ| ℓ(µ)≤N γλµµ . (3.12) The situation becomes much more involved if we include also k̄ fermionic fields, so that (2.1) may be written in the form f(t, t̄) = j=1 tj − ̄=1 t̄̄, and attempt to ex- pand ZU(N)(t, t̄) in terms of products of Schur polynomials sλ(t)sµ(t̄). Such expansion is required for counting, for instance, for the free field partition function in the eighth BPS sector of N = 4 super Yang Mills. In this case the partition function is expanded, analogous to (1.1), in terms of SU(2|3) characters, which may be expressed in terms of a linear combination of products of two-variable and three-variable Schur polynomials. (See [33] for a discussion of counting for the eighth BPS sector along these lines.) Including fermions, (3.7) becomes modified by ZU(N)(t, t̄) = (2πi)NN ! ∆(z)∆(z−1) 1≤j≤k 1≤̄≤k̄ r,s=1 1− t̄̄zrzs−1 1− tjzrzs−1 . (3.13) To achieve the expansion, we may use the Cauchy-Littlewood formula (2.23) along with another formula of Littlewood, (1 + xiyj) = ℓ(λ)≤L,ℓ(λ̃)≤M sλ(x1, . . . , xL) sλ̃(y1, . . . , yM ) , (3.14) where λ̃ is the partition conjugate to λ (where the rows and columns of the Young diagram corresponding to λ are interchanged) and where the sum is restricted to those λ whereby the corresponding Young diagrams have at most L rows, ℓ(λ) ≤ L, and M columns, ℓ(λ̃) ≤ M . We may thus write ZU(N)(t, t̄) = ℓ(λ)≤min.{k,N2} ℓ(µ)≤k̄,ℓ(µ̃)≤N2 Nλ,µ sλ(t) sµ(t̄) , (3.15) where Nλ,µ = (−1)|µ| (2πi)NN ! ∆(z)∆(z−1) sλ(zz −1)sµ̃(zz −1) . (3.16) Obviously these numbers are considerably more involved than those in (3.9). We may of course use (3.10) again to interpret (3.16) in terms of Kronecker coefficients. Counting Quarter BPS Operators by Symmetric Polynomial Methods The two bosonic fundamental field case is now focused upon.6 In particular, the numbers N(n,m) in (1.1) are evaluated using results of the last sub-section. We may proceed to evaluate Nλ recursively. The simplest case is for Nλ = N(n), whereby introducing a formal variable t then it is clear, by (2.23) with (2.22), (2.25) and (2.26), that N(n)tn = (2πi)NN ! ∆(z)∆(z−1) j,k=1 1− tzjzk−1 = PN (t) , (3.17) so that, by (2.28), N(n) is given by (3.2). More generally to evaluate N(n,m) from (3.9) we may use, for y = zz−1, s(m)(y)s(n)(y) = s(n,m)(y) + s(n+1,m−1)(y) + . . .+ s(n+m−1,1)(y) + s(n+m)(y) , s(m)(zz −1) = ℓ(µ)≤N sµ(z)sµ(z −1) , (3.18) 6 The two boson case leads to an interesting generalisation of an identity in [44] involv- ing Littlewood-Richardson coefficients cνλµ, the coefficients that appear in the decomposition sλ(x)sµ(x) = cνλµsν(x). With f(t) in (2.1) given by f(t) = t1 + t2, and expanding appro- priately the corresponding integrand in (3.7) using (2.23); then using (2.6), the orthonormality of Schur polynomials and the result (2.14), we obtain (note that cνλµ = 0 if |ν| 6= |λ|+ |µ|) ZU(∞)(t1, t2) = 〈sλsµ, sλsµ〉∞ = λ,µ,ν ν⊢|λ|+|µ| 1− t1n − t2n which reduces to Theorem 4.1 of [44] if we take t1 = t2 = t. where the expression in the first line of (3.18) may be easily seen using Young tableaux multiplication rules while (2.23) determines the expression in the second line. From (3.9) with (3.18), we may find a useful generating function, in terms of a formal variable t, for the numbers in (3.4) as follows, F (m)N (t) = M(n,m)tn (2πi)NN ! ∆(z)∆(z−1) s(m)(zz s(n)(zz −1)tn (2πi)NN ! ∆(z)∆(z−1) ℓ(µ)≤N sµ(z)sµ(z j,k=1 1− tzjzk−1 ℓ(µ)≤N sλsµ, sλsµ (3.19) so that we may write N(n,m) = F (m)N (t)− F (m−1)N (t) , (3.20) which allows for recursive determination of N(n,m). Applying this to the case of Nλ = N(n,1) we have, from (3.18) s(1)(zz −1) = s(1)(z)s(1)(z −1) , (3.21) so that, from (3.19), F (1)N (t) = N(n,1) +N(n+1) ℓ(λ)≤N s(1)sλ, s(1)sλ . (3.22) Using (again, this may be easily seen from Young tableaux multiplication rules) s(1)(z)sλ(z) = sλ+er (z) , (3.23) for {er; 1 ≤ r ≤ N, er · es = δrs} being usual orthonormal vectors, we find that F (1)N (t) = N(n,1) +N(n+1) ℓ(λ)≤N r,s=1 sλ+er , sλ+es . (3.24) Now for any partition λ, sλ+er , sλ+es vanishes unless er = es for any r, s and λr−1−λr > 0 for r = 2, . . . , N , due to s(λ1,...,λr−1,λr+1,...,λN )(z) = 0 for λr−1 = λr , r > 1 . (3.25) Changing summation variables to those in (2.25) then we have, with the definition (2.26), F (1)N (t) = N(n,1) +N(n+1) a1,...,aN≥0 ta1+...+NaN + a1,...,aN≥0 ta1+...+NaN a1,...,aN≥0 ta1+...+NaN = 1− tPN−1(t) . (3.26) Thus, using (2.28), (3.2) with (3.26),7 N(n,1) = pN−1(j)−N(n+1) = pN−1(j)− pN (n+ 1) . (3.27) For the case of Nλ = N(n,2) we have that, from (3.18), s(2)(zz −1) = s(2)(z)s(2)(z −1) + s(1,1)(z)s(1,1)(z −1) , (3.28) so that, from (3.19), we have F (2)N (t) = ℓ(λ)≤N s(2)sλ, s(2)sλ s(1,1)sλ, s(1,1)sλ . (3.29) Using s(2)(z)sλ(z) = sλ+2er (z) + 1≤r<s≤N sλ+er+es(z) , s(1,1)(z)sλ(z) = 1≤r<s≤N sλ+er+es(z) , (3.30) 7 This formula agrees with (3.6) due to pN (n − j) = pN−1(j) which follows because the corresponding generating functions match, pN (n− j)t = (1 + t+ . . .+ t )PN (t) = PN−1(t) = pN−1(j)t along with (3.25) and s(λ1,...,λr−1,λr+2,...,λN )(z) = −s(λ1,...,λr+1,λr−1+1,...,λN )(z) , (3.31) for the cases where λr−1 = λr, we may obtain, with the definition (2.26), F (2)N (t) = 1−tN+1 (1−t)(1−t2)PN−1(t) + (1−t)(1−t2)PN−2(t) , (3.32) where the first contribution comes from |λ|〈s(2)sλ, s(2)sλ while the second comes |λ|〈s(1,1)sλ, s(1,1)sλ . Since the partition number pk(−n) = 0 for n = 1, 2, . . . we may write, using (2.28),8 (1− t)(1− t2)Pk(t) = n,i,j=0 pk(n− i− 2j)tn . (3.33) Thus, from (3.32) with (3.27), N(n,2) = − pN−1(j) i,j=0 (pN−2(n−i−2j) + pN−1(n−i−2j)) if n ≤ N ,∑∞ i,j=0 (pN−2(n−i−2j) + pN−1(n−i−2j)− pN−1(N+1−n−i−2j)) if n ≥ N+1. (3.34) Tables of the numbers (3.2), (3.27) and (3.34) are given in appendix B for some few cases of n,N . Notice from these tables that the numbers N(n,m) below the diagonal line N ≥ n+m for a given n are the same for all N . This is a general feature that derives from values of N(n,m) for N ≥ n + m, which numbers may be obtained from a corresponding generating function that is now constructed. Using these techniques, we may provide a consistency check of (3.17), (3.26), (3.32) along with a general result for N(n,m) for high enough values of N , N ≥ m + n. This employs the orthogonality property of power symmetric polynomials pλ(z) (in the large N 8 This is a special case of the following: for any f(n), n ∈ Z, that satisfies f(−n) = 0, n = 1, 2, . . ., then we may (at least formally) write Pk(t) f(n)t n,i1,...,ik=0 f(n− i1 − 2i2 − . . .− kik)t limit) along with s(n)(z) = pλ(z) i1,...,in=0 i1!i2! · · · in! δi1+2i2+...+nin,n p1(z) p2(z) )i2 · · · pn(z) (3.35) Using the trivial identity pλ(xy) = pλ(x)pλ(y) then from (2.6), (3.19) with (3.35) we have F (m)∞ (t) = pλpµ, pλpµ pν , pν (3.36) where for (1a1 , 2a2 , . . .) being the frequency representation of λ and (1b1 , 2b2 , . . .) being that of µ then ν has frequency representation (1a1+a2 , 2a2+b2 , . . .) so that |ν| = n + m. This agrees with F (m)N (t) in a series expansion up to O(tN−m) (since the last equation in (3.36) is also valid for finite N so long as |ν| = n +m ≤ N , by a result of appendix A). Now, since (aj+bj)! aj !bj! , (3.37) we obtain from (3.36) that, F (m)∞ (t) = a1,...,an=0 b1,...,bm=0 δa1+···+nan,nδb1+···+mbm,m (aj + bj)! aj !bj! b1,...,bm=0 δb1+···+mbm,m (aj + bj)! aj!bj ! 1− tj b1,...,bm=0 δb1+···+mbm,m (1− tj)bj+1 1− tj . (3.38) For the first few cases we have that, with PN (t) as defined in (2.26), F (m)∞ (t) = P∞(t) for m = 0 1−tP∞(t) for m = 1 (1−t)(1−t2)P∞(t) for m = 2 , (3.39) whose series expansion agrees with (3.17), (3.26), (3.32) up to O(tN−m) for, respectively, m = 0, 1, 2. We may use (3.20) with (3.36) to determine N(n,m) exactly for N ≥ n+m. Asymptotic Counting of Quarter BPS Operators at Large N Asymptotic counting for the one boson case in the large N limit, for which, with PN (t) as defined in (2.26), with p(n) being the total number of (unordered) partitions of n, ZU(∞)(t) = P∞(t) = p(n)tn , (3.40) is the multi-trace partition function, entails finding an asymptotic value for the partition number p(n) for ‘large’ n. This may be achieved by performing a saddle point approxima- tion of p(n) = 1 dtP∞(t)t −n−1. The function P∞(t) has a ‘large’ singularity at t = 1, but in addition has singularities at all other roots of unity - see [45] on the validity of ignor- ing these contributions asymptotically. This method was used by Hardy and Ramanujan to find their celebrated formula, here given in a less detailed form as, p(n) ∼ 1 , (3.41) which was improved by Rademacher to give p(n) exactly. Their method relied crucially on the modular properties of P∞(t). Focusing now on the two bosonic fundamental field case for which, in the large N limit, (3.20) with (3.38) gives exact counting, at issue is first finding asymptotic values for the numbers Q(n,m, b) = Q(n,m, b1, . . . , bm), with constraint equation j=1 jbj = m, defined by (1− tj)bj+1 1− tj = 1 + Q(n,m, b)tn . (3.42) Having found these we may then attempt to find the dominant contribution to (3.20) with (3.38) for large N . In order to give asymptotic values for Q(n,m, b) we may follow [46] and apply a formula due to Meinardus which gives a general result for the generating function (1− tn)−an = 1 + r(n)tn . (3.43) A detailed version of Meinardus’ theorem may be found in [46] but for purposes of brevity we may note that it implies that, as n → ∞, r(n) ∼ C nκ exp AΓ(α+ 1)ζ(α+ 1)nα )1/(α+1) (α+ 1)/α , (3.44) where ζ(s) = j=1 j −s is the Riemann zeta function and the constants C, κ, α, A are determined by the auxiliary Dirichlet series, D(s) = , (3.45) which must converge for Re(s) > α, a positive real number, and possess an analytic continuation in the region Re(s) ≥ c, −1 < c < 0, such that, in this region, D(s) is analytic except at a simple pole at s = α where it has residue A. In terms of α,A then 2π(1+α) AΓ(α+ 1)ζ(α+ 1) )(1−2D(0))/2(α+1) expD′(0) , κ = (D(0)− 1− 1 α)/(α+ 1) . (3.46) Applying Meinardus’ theorem to the case of (3.42), clearly we have D(s) = + ζ(s) , (3.47) so that, assuming m = j=1 jbj is fixed, D(s) has a simple pole at s = α = 1 where it has residue A = 1. Using D(0) = −1 bj , expD ′(0) = , (3.48) then, from (3.44) with (3.46), we may easily determine that, as n → ∞, Q(n,m, b) ∼ 1 bj m∏ . (3.49) This reduces to (3.41) when bj = 0, 1 ≤ j ≤ m, whereby Q(n, 0, . . . , 0) = p(n). Using (3.20) for (3.38) with (3.42) and (3.49) then, as n → ∞, N(n,m) = b1,...,bm≥0∑ jbj=m Q(n,m, b)− b1,...,bm−1≥0∑ jbj=m−1 Q(n+1, m−1, b) (3.50) since Q(n,m,m, 0, . . . , 0), for b1 = m, bj = 0, j > 1, dominates over all other terms in (3.50). This gives asymptotic values for the numbers in (1.1), for counting quarter BPS operators, transforming in [m,n−m,m] SU(4)R representations, in the large N limit of free N = 4 super Yang Mills, as previously described. 4. Counting Operators in the Chiral Ring of N = 4 Super Yang Mills For the purposes of counting operators in the chiral ring of N = 4 super Yang Mills, we denote corresponding multi-trace partition functions by CU(N)(t). The generating function for CU(N)(t) for the case of one bosonic fundamental field has been written in the form [1,3,9] C(ν, t) = 1− νtn = νNCU(N)(t) , (4.1) so that ν acts as a chemical potential for the rank of the gauge group U(N). The equiv- alence CU(N)(t) = ZU(N)(t) = PN (t), with ZU(N)(t) as in (2.25), is actually a special case of the q-Binomial theorem. Writing - see [43] for notation - (a; q)k = (1− a)(1− aq) · · · (1− aqk−1) , (4.2) then the q-Binomial theorem is, for |x|, |q| < 1, (a; q)k (q; q)k (ax; q)∞ (x; q)∞ . (4.3) (Identifying ν = x and q = t and setting a = 0 in (4.3), so that 1/(ν; t)∞ = C(ν, t) above and 1/(t; t)N = PN (t) in (2.26), then CU(N)(t) = ZU(N)(t) = PN (t) straightforwardly. This special case of the q-Binomial theorem is due to Euler.) For the two boson case, so that the single particle partition function is given by f(t, u) = t + u for some t, u, then the generating function for the finite N chiral ring partition function CU(N)(t, u) is given by [1,3,9] C(ν, t, u) = n,m=0 1− νtnum = νNCU(N)(t, u) . (4.4) This function is more difficult to analyse in terms of counting though has been investigated by Stanley [47] in relation to partitions - there it has been dubbed the ‘double Eulerian’ generating function. Through use of the Cauchy-Littlewood formula, then we may expand CU(N)(t, u) in terms of partitions of N as, CU(N)(t, u) = hλ(t)hλ(u) , (4.5) where hλ(t) = sλ(1, t, t 2, . . .) , (4.6) so that using an identity for Schur polynomials to be found in [24,47] then CU(N)(t, u) = 1≤i<j≤N (1− tλi−λj+j−i)(1− uλi−λj+j−i)∏N i=1(t; t)λi+N−i(u; u)λi+N−i (i−1)λi . (4.7) (4.5) with (4.6) has a natural interpretation in terms of plane partitions in that, for π being all column-strict plane partitions of shape λ, |π| = i,j πij , hλ(t) = sλ(1, t, t 2, . . .) = t|π| . (4.8) Obviously, (4.5) with (4.8) generalise for other chiral ring sectors. (For a different connec- tion between the ‘double Eulerian’ generating function and major indices of permutations see [47], p. 385.) As an illustration of (4.5) with (4.8), we may consider the case N = 2 whereby λ = (2, 0), (1, 1) gives the two possible partitions of 2. For λ = (2, 0) (corre- sponding to a Young diagram with a single row of two boxes) π11 ≥ π12 ≥ 0 gives all column-strict plane partitions of shape (2, 0), while for λ = (1, 1) (corresponding to a Young diagram with a single column of two boxes) then π11 > π21 ≥ 0 gives all column- strict plane partitions of shape (1, 1). Thus, h(2,0)(t) = π11,π12≥0 π11≥π12 tπ11+π12 = (1− t)(1− t2) , h(1,1)(t) = π11,π21≥0 π11>π21 tπ11+π21 = (1− t)(1− t2) , (4.9) so that, from (4.5) for N = 2, CU(2)(t, u) = h(2,0)(t)h(2,0)(u)+h(1,1)(t)h(1,1)(u) = 1 + ut (1− t)(1− t2)(1− u)(1− u2) , (4.10) which is the correct result as may be verified by extracting the ν2 coefficient in an expansion of (4.4) up to O(ν2). 9 See [47] for a detailed description of plane partitions. Briefly, a column-strict plane parti- tion of shape λ is an array π = (πij) of non-negative integers with finitely many non-zero entries, that is arranged in a Young tableaux with shape λ - see appendix A - such that the numbers πij are weakly decreasing along each row, πij ≥ πi j+1 ≥ 0, and strictly decreasing down each column, πij > πi+1 j ≥ 0. The sum of the parts of π is given by |π| = πij. (Note that in contrast to the definition in [47], here we are allowing πij = 0, for some i, j, to be a part of the plane partition π with shape λ.) In the large N limit, CU(∞)(t, u) = n1,n2≥0 n1+n2>0 1− tn1un2 , (4.11) upon which attention is shortly focused. For the numbers N(n,m) → N̂(n,m) counting quarter BPS primary operators for the chiral ring of N = 4 super Yang Mills, belonging to [m,n − m,m] SU(4)R R-symmetry representations, as in (1.1), we have 10 N̂(n,m) = dt du CU(N)(t, u) s(n,m)(t−1, u−1) (t−1 − u−1)2 . (4.12) These may be more conveniently evaluated in terms of the numbers in (3.4) M(n,m) → M̂(n,m), counting all chiral ring quarter BPS operators in the [m,n−m,m] SU(4)R repre- sentation, given by M̂(n,m) = (2πi)2 dt du CU(N)(t, u) t−n−1u−m−1 , (4.13) so that N̂(n,m) = M̂(n,m)−M̂(n+1,m−1). Defining Pλ(n) to be the number of column-strict plane partitions π of shape λ so that |π| = i,j πij = n, then, from (4.5) with (4.8) and (4.13), M̂(n,m) = λ⊢N Pλ(n)Pλ(m). Thus, N̂(n,m) = (Pλ(n)Pλ(m)− Pλ(n+1)Pλ(m−1)) , (4.14) counts chiral ring quarter BPS primary operators in SU(4)R representations [m,n−m,m] for any n,m at finite N . Asymptotic Counting for Chiral Ring BPS Operators at Large N For asymptotic counting of operators in the chiral ring of N = 4 super Yang Mills at large N , a relatively crude method is employed here which nevertheless captures the exponential behaviour of counting numbers of interest. This method is based on saddle point approximations of functions near a dominant singularity - see [45] for a useful sum- mary. (Often for physical applications in thermodynamics, e.g. for entropy formulae, we are interested only in the exponential behaviour of such numbers anyhow.) 10 This formula employs the orthonormality relation of Schur polynomials described here and has appeared in a similar context in [33], appendix B. To illustrate, we consider the one boson case in the large N limit again. We first find a convenient ‘approximating function’ as follows, P∞(t) = 1− tn = exp ln(1− tn) ∼ exp ds ln(1− ts) = exp 6 ln t (4.15) which has an ‘easier’ singularity structure. (The approximation in the second step may be justified by the Euler-Maclaurin formula for approximating sums by integrals.) Using (4.15) then for large enough n, p(n) = dt P∞(t)t −n−1 ∼ 1 dt eg(t) , g(t) = − π 6 ln t − n ln t . (4.16) We may approximate the latter integral for large n by noting that the dominant contribu- tion is at the saddle point t′ = e−π/ 6n ∼ 1 for which g(t′) = π n , g′(t′) = 0 , g′′(t′) = 6n3 eπ 2/3n = α , (4.17) so that, for t′′ = t− t′, p(n) ∼ eπ 2n/3 1 dt′′ e αt′′2 ∼ eπ 2n/3 1 ds e− αs2 = 2n/3 . (4.18) Thus, ln p(n) ∼ π n , (4.19) which captures the correct behaviour of ln p(n) for large n, according to (3.41). We may proceed analogously for the quarter BPS chiral ring multi-trace partition function at large N , (4.11), which we approximate by CU(∞)(t, u) = n1,n2≥0 n1+n2>0 1− tn1un2 = exp n1,n2≥0 n1+n2>0 ln(1− tn1un2) ∼ exp dv dw ln(1− tvuw) = exp ( ζ(3) ln t lnu (4.20) In this case we have, from (4.13), M̂(n,m) ∼ (2πi)2 dt du eg(t,u) , (4.21) where g(t, u) = ln t lnu − n ln t−m lnu , (4.22) for n,m large. The dominant contribution to the integral, for n,m both large and of the same order, occurs about the point (t′, u′) ∼ (1, 1) where t′ = e−(ζ(3)mn −2)1/3 , u′ = e−(ζ(3)nm −2)1/3 , (4.23) for which g(t′, u′) = 3 3 ζ(3)nm , g(t, u) (t′,u′) g(t, u) (t′,u′) = 0 , g(t, u) (t′,u′) = 2(ζ(3)−1n5m−1) 3 e2(ζ(3)mn −2)1/3 = α , g(t, u) (t′,u′) = 2(ζ(3)−1m5n−1) 3 e2(ζ(3)nm −2)1/3 = β , g(t, u) (t′,u′) = (ζ(3)−1n2m2) 3 e(ζ(3)mn −2)1/3+(ζ(3)nm−2)1/3 = γ . (4.24) So long as m,n are both large and of the same order, the saddle point approximation is justified and we obtain M̂(n,m) ∼ e3 ζ(3)nm 1 dv dw e− (αv2+βw2+2γvw) = h(α, β, γ)e3 ζ(3)nm , (4.25) where h(α, β, γ) = αβ − γ2 (ζ(3)m−2n−2) 3 e−(ζ(3)mn −2)1/3−(ζ(3)nm−2)1/3 . (4.26) We thus have that for n,m both comparably large, lnM̂(n,m) ∼ 3 3 ζ(3)nm , (4.27) so that ln N̂(n,m) = ln M̂(n,m) − M̂(n+1,m−1) ∼ lnM̂(n,m) ∼ 3 3 ζ(3)nm . (4.28) It is difficult to check the consistency of this result given the dearth of literature on these types of multi-variable generating functions and their asymptotic behaviour, however, we may consider the simpler function, also considered in [9], CU(∞)(t, t) = (1− tn)n+1 = E(r) tr , (4.29) where, in terms of the counting numbers N̂(n,m), from (1.1), CU(∞)(t, t) = (n−m+1)N̂(n,m) tn+m ⇒ E(r) = (r−2m+1)N̂(r−m,m) . (4.30) Hence, E(r) counts quarter BPS primary operators in the chiral ring of N = 4 super Yang Mills, transforming in [m,n−m,m] SU(4)R representations, with the same conformal dimensions r = n+m. Extracting the dominant contribution to ln E(r) from (4.30), which occurs at the maximum value of m, mM = [ r], and using (4.28), we obtain ln E(r) ∼ ln N̂(r−mM,mM) ∼ 32 2ζ(3)r2 . (4.31) By considering (4.29) directly, we may employ Meinardus’ theorem (described in the third section) to find the behaviour of ln E(r) as r → ∞. Note, however that Meinardus’ theorem may not be applied directly to (4.29) since the corresponding auxiliary Dirichlet series (3.45), with aj = j+1, has two simple poles. To overcome this difficulty we split (4.29) into a product of two functions, both separately amenable to application of Meinar- dus’ theorem. One is the reciprocal of the Euler function, P∞(t) in (4.15). The other, the MacMahon function, is given by M(t) = (1− tn)n = q(r) tr , (4.32) and has been considered in a similar context as here in [3].11 Writing CU(∞)(t, t) = P∞(t)M(t) , (4.33) with P∞(t) as in (4.15), so that, using (4.29), E(r) = r1,r2≥0 r1+r2=r p(r1)q(r2) , (4.34) we may find the asymptotic behaviour of ln E(r), as r → ∞, by extracting the dominant contribution from (4.34) using the asymptotic behaviour of p(r), q(r). The auxiliary Dirichlet series for M(t) in (4.32) is, from (3.45) with aj = j, D(s) = ζ(s− 1) , (4.35) 11 The relation of M(t) in (4.32) to plane partitions is given a description in [46]. Briefly, q(r) gives the number of ordinary plane partitions π, so that πij ≥ πi+1 j > 0, πij ≥ πi j+1 > 0, with |π| = πij = r. p(r) < q(r) as ordinary partitions λ are a special case of plane partitions. In fact, the formula for ln q(r) found here is a special case of a more exact asymptotic formula first found by Wright [48] for the number of plane partitions q(r) of the number r. which has a simple pole at s = α = 2, at which the residue is A = 1. Thus, from (3.44), ln q(r) ∼ 3 2ζ(3)r2 . (4.36) This is consistent with (4.31) as the dominant contribution to lnE(r) comes from the r1 = 0 term in (4.34) (since p(r) ≪ q(r) as r → ∞) so that lnE(r) ∼ ln q(r). It has not escaped attention that the method used here, to capture the exponential behaviour of asymptotic values for the numbers M̂(n,m), may be easily extended to chiral ring sectors other than the quarter BPS one. Suppose, for simplicity, that Zj , 1 ≤ j ≤ k−1, are commuting bosonic fundamental fields, in the U(N) Lie algebra, so that the single particle partition function is given by f(t) = j=1 tj , in terms of the corresponding letters tj . Let M̂(m1,...,mk−1) denote the number of independent operators involving products of m1 Z1’s, m2 Z2’s etc. in corresponding multi-trace operators. The multi-trace partition function, in the large N limit, is given by, CU(∞)(t) = n1,...,nk−1≥0 n1+...+nk−1>0 1− t1n1 · · · tk−1nk−1 , (4.37) which may be crudely approximated by, similarly as before,12 CU(∞)(t) ∼ exp dvj ln(1−t1v1 · · · tk−1vk−1) = exp (−1)k+1ζ(k) ln t1 · · · ln tk−1 . (4.38) Thus, without going into as much detail, for the analogue of (4.27) we have, (assuming mj are all comparably large,) lnM̂(m1,...,mk−1) ∼ g(t 1, . . . , t k−1) = k ζ(k)m1 · · ·mk−1 , (4.39) where g(t′1, . . . , t k−1) is the value of g(t1, . . . , tk−1) = (−1)k+1ζ(k) ln t1 · · · ln tk−1 −m1 ln t1 − . . .−mk−1 ln tk−1 , (4.40) 12 For Lin(x) = xj/jn being the usual Polylogarithm, with Lin(1) = ζ(n), n > 1, Lin(0) = 0, then with the convention Li1(x) = − ln(1− x), the following integral Lin(zx) = Lin+1(z) , may be useful for showing this, after a suitable change of variables. at the saddle point (t′1, . . . , t k−1) ∼ (1, . . . , 1), where (ln t′1, . . . , ln t k−1) = − k ζ(k)m1 · · ·mk−1 (1/m1, . . . , 1/mk−1) , (4.41) so that g(t1, . . . , tk−1) ,...,t′ = 0 , j = 1, . . . , k−1 . (4.42) (4.39) is consistent with a result implied by Meinardus’ theorem. The function,13 CU(∞)(t, . . . , t) ∼ (1− tn)−n k−2/(k−2)! = c(k, r)tr , (4.43) has auxiliary Dirichlet series, from (3.45) with aj = j k−2/(k − 2)!, D(s) = (k − 2)! ζ(s+ 2− k) , (4.44) which has a simple pole at s = α = k−1 at which the residue is A = 1/(k − 2)!, so that, from (3.44), ln c(k, r) ∼ k (k−1) ζ(k) rk−1 . (4.45) (4.45) is precisely the result that may be obtained from (4.39) if we maximise the product m1 · · ·mk−1, subject to the constraint j=1 mj = r, for which the solution is mj = mM = r/(k − 1) (relaxing the constraint that mj be non-negative integers, which is irrelevant asymptotically), so that lnM̂(mM,...,mM) ∼ ln c(k, r). This is applicable to counting multi-trace operators in the eighth BPS chiral ring sec- tor for N = 4 super Yang Mills with fundamental fields Z, Y,X involving m1 Z’s, m2 Y ’s, m3 X ’s. Expanding the corresponding partition function (4.37), with k = 4, in terms of Schur polynomials s(m1,m2,m3)(t), m1 ≥ m2 ≥ m3 ≥ 0, similar to (1.1), the expansion coefficients N̂(m1,m2,m3) count spinless multi-trace primary operators transforming in the [m2 +m3, m1 −m2, m2 −m3] SU(4)R R-symmetry representation, with conformal dimen- sionsm1+m2+m3 [33]. Just as in (4.28), asymptotically lnN(m1,m2,m3) ∼ lnM(m1,m2,m3). This counting, however, ignores contributions of the fermionic fields λ, λ̄, which it may be important to include in order to give correct counting of eighth BPS chiral ring operators. 13 This may be easily seen from (4.37), as the number of solutions to mj = n, where mj are non-negative integers, is the binomial number n+k−2 which, to leading order in large n, behaves like nk−2/(k−2)!. More properly, we should split the product CU(∞)(t, . . . , t) into pieces separately amenable to Meinardus’ theorem, as for the prior case for k = 3, however, just as for that case, the numbers c(k, r) dominate, and so other contributions are ignored here. 5. Conclusions There are some obvious questions not answered by this work. The first is whether or not the approach in the second section using symmetric polynomials can give insight into thermodynamics at finite N , such as for the Hagedorn transition, for example. While it gives the large N expression (2.14) in an elementary way, its wider applicability or usefulness to such questions is unclear. The approach is undoubtedly useful for finding exact expressions for counting numbers (as in (3.2), (3.27) and (3.34) for quarter BPS operators) and (3.9), (3.16) may be useful for analysing counting for more complicated sectors of N = 4 super Yang Mills, with gauge group U(N). The second question is how the arguments employing symmetric polynomial tech- niques here may be extended to other gauge groups, the most pertinent being perhaps SU(N). Arguments here employing (2.6) and the orthonormality property of Schur poly- nomials should remain largely unaffected for SU(N). Exact values for counting numbers obtained here should require some modification for SU(N), though asymptotic values may be unchanged. The third question concerns asymptotic values for counting numbers and how these may be improved. The asymptotic counting formulae given in such papers as [3,9] for chiral ring sectors are special cases of formulae such as those of Hardy and Ramanujan, Meinardus, etc., all of which derive from single variable generating functions. It is hoped that the expressions (3.50), (4.28), (4.39), given here for asymptotic counting of BPS operators, that distinguishes between differing R-symmetry charges, represents a serious attempt at going beyond consideration of single variable generating functions.14 Improving upon these formulae will require more sophisticated techniques, perhaps along the lines used to find those of Hardy and Ramanujan or Meinardus and employing any modular properties of the multi-variable functions involved. This issue may also be important for microscopic counting for Black Holes, as the BPS solutions found thus far, for N = 4 superconformal symmetry, depend on special values of R-symmetry charges [17,18,19,20] - see [50] for a related detailed discussion. Thus far, the elegant results for finite N partition functions for chiral ring sectors have been interpreted from a largely geometric perspective - it may be interesting to investigate more how such results are related to the theory of random matrices and/or symmetric polynomials. 14 After submission of the first version of this paper to the electronic archive, I received an e-mail from Hai Lin pointing out an interesting comparison between (4.28) here and (2.14) of [49], obtained in quite a different context. The two formulae are essentially the same given the numerical value 3 3 ζ(3) = 3.189 . . ., correct to three decimal places. Acknowledgements I warmly thank Yang-Hui He, Paul Heslop, Hugh Osborn, Christian Romelsberger and Christian Saemann for useful comments and discussions. This work is supported by an IRCSET (Irish Research Council for Science, Engineering and Technology) Post-doctoral Fellowship. Appendix A. Partitions, symmetric group characters, symmetric polynomials and inner products A generic partition λ is any finite or infinite sequence λ = (λ1, λ2, . . .) of non-negative integers in decreasing order λ1 ≥ λ2 ≥ . . . ≥ 0 containing only finitely many non-zero terms. Often it is convenient to omit zero entries. The non-zero entries are called the parts of λ the number of which we denote by ℓ(λ). The sum of the parts of λ is called the weight of λ which we denote by |λ| = i λi. If |λ| = L then λ is a partition of L and we write λ ⊢ L. For convenience we sometimes write λ in its frequency representation which is a reordering of the entries in λ, indicating the number of times each successive non-negative integer occurs, (1a1 , 2a2 , . . .) so that exactly an of the parts of λ equal n and |λ| = n≥1 n an. In terms of standard Young diagrams, λ corresponds to a Young diagram of shape λ, with λ1 boxes in the first row, λ2 boxes in the second row etc.; the number of parts ℓ(λ) is simply the number of rows and the weight |λ| is the total number of boxes. For the symmetric group, SN , the irreducible representations are labelled by par- titions λ ⊢ N - see [38] for a useful summary - so that, for Xλ(σ), σ ∈ SN , being a corresponding matrix representation, then the character of σ ∈ SN in the representation Xλ is χλ(σ) = tr(Xλ(σ)). The characters are class functions so that they take a constant value on conjugacy classes and, recalling that for SN the conjugacy classes Kµ are labelled by partitions µ ⊢ N , corresponding to the cycle structure of a class representative, then χλ(σ) = χλµ for all σ ∈ Kµ. With zλ as defined in (2.8), a crucial property of SN charac- ters is the orthogonality of the matrix [zµ −1/2χλµ]λµ. This gives rise to the orthogonality relations, for λ, µ ⊢ N , (see also Ch. IV of [51] for a related discussion,) χλ(σ)χµ(σ) = χλν χ ν = δλµ , (A.1) and ∑ χνλ χ µ = zλδλµ . (A.2) A convenient basis for N variable symmetric polynomials are Schur polynomials sλ(z) = sλ(z1, . . . , zN ) labelled by λ = (λ1, . . . , λN ). They may be expressed in a number of ways [24,47]. For convenience we write them as sλ(z) = aλ+ρ(z)/aρ(z) , (A.3) where ρ, the Weyl vector, is given by ρ = (N − 1, N − 2, . . . , 1, 0) and aλ+ρ(z) = sgn(σ) zσ(1) λ1+N−1 · · · zσ(j)λj+N−j · · · zσ(N)λN = det[ziλj+N−j ] , (A.4) aρ(z) = det[zi N−j ] = 1≤i<j≤N (zi − zj) = ∆(x) , (A.5) being the Vandermonde determinant. Schur polynomials sλ(z) have a standard interpre- tation as corresponding to the characters of irreducible U(N) (or, for i zi = 1, SU(N)) Lie algebra representations. Here, λ gives the shape of the Young tableaux for the corre- sponding U(N) Lie algebra representation. For λ = (λ1, . . . , λN ) and µ = (µ1, . . . , µN ) where λi, µi ∈ Z then, from the definition of (2.6) along with (A.3), sλ, sµ sgn(σ)δλσµ = sgn(σ)δλµσ , (A.6) where, for any λ′ = (λ′1, . . . , λ N ), λ ′σ = σ(λ′+ρ)−ρ is the shifted Weyl reflection of λ′ by σ, with the action of SN on λ′ being given by σ(λ′1, . . . , λ′N ) = (λ′σ(1), . . . , λ′σ(N)). (Equation (A.6) is a reflection of sλ(x) = sgn(σ)sλσ(x) for any partition λ and σ ∈ SN - note that this property is useful for showing (3.25), (3.31). λσ has a standard interpretation in terms of U(N) Lie algebra representations - for the Verma module with dominant integral highest weight having orthonormal basis labels λ, λ1 ≥ λ2 ≥ . . . ≥ λN ≥ 0, then λσ, for σ 6= idSN , are the orthonormal basis labels for the highest weights of all invariant sub-modules. This fact may be exploited to derive the Weyl character formula (A.3) for the irreducible U(N) Lie algebra representation with dominant integral highest weight having orthonormal basis labels λ, or, alternatively, Young tableaux of shape λ.) When λ, µ are partitions so that λ1 ≥ . . . ≥ λN ≥ 0 and µ1 ≥ . . . ≥ µN ≥ 0 then (A.6) reduces to a well defined inner product, sλ, sµ = δλµ , (A.7) so that in this case the Schur polynomials are orthonormal. Note that in order that sλ(x) be non-zero for some arbitrary partition λ then ℓ(λ) ≤ N , so that (A.7) is zero for ℓ(λ) > N or ℓ(µ) > N . Another basis for symmetric polynomials are the power symmetric polynomials, pλ(z), for λ = (λ1, . . . , λL) ⊢ L, which are defined by pλ(z) = pλ1(z)pλ2(z) · · ·pλL(z) , pn(z) = n . (A.8) Note that there is no longer the restriction that ℓ(λ) ≤ N as for Schur polynomials. Symmetric group characters may be used to relate the two bases for symmetric poly- nomials [24,47] so that, with the definition of zλ in (2.8), sλ(z) = χλµ pµ(z) , (A.9) (a theorem of Frobenius) and, for λ ⊢ L, pλ(z) = ℓ(µ)≤N λ sµ(z) . (A.10) ((A.9) with χ λ = 1 for all λ ⊢ N is useful for obtaining (3.35).) Regarding the inner product (2.6), then using (A.10) along with (A.7), we then have that, for λ ⊢ L, µ ⊢ M , 〈 pλ, pµ = δLM ℓ(ν)≤N µ . (A.11) Orthogonality of symmetric group characters implies, from (A.2), that for |λ|, |µ| ≤ N then (A.11) simplifies to, with the definition of zλ in (2.8), pλ, pµ = zλδλµ . (A.12) Appendix B. Tables N n 2 3 4 5 6 7 8 9 10 11 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 4 4 5 5 6 6 3 2 3 4 5 7 8 10 12 14 16 4 2 3 5 6 9 11 15 18 23 27 5 2 3 5 7 10 13 18 23 30 37 6 2 3 5 7 11 14 20 26 35 44 Numbers of multi-trace half BPS primary operators, with conformal dimension n and belonging to [0, n, 0] R-symmetry representations, for free N = 4 SYM with U(N) gauge group. 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0704.1039

Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces

Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces Frédéric Bourgeois Alexandru Oancea Université Libre de Bruxelles Université Louis Pasteur B-1050 Bruxelles, Belgium F-67084 Strasbourg, France [email protected] [email protected] 11 March 2008 Abstract We define Floer homology for a time-independent, or autonomous Hamiltonian on a symplectic manifold with contact type boundary, under the assumption that its 1-periodic orbits are transversally nondegenerate. Our construction is based on Morse-Bott techniques for Floer trajectories. Our main motivation is to understand the relationship between linearized contact homology of a fillable contact manifold and symplectic homology of its filling. 2000 Mathematics Subject Classification: 53D40. Contents 1 Introduction 2 2 Symplectic homology 6 3 The Morse-Bott chain complex 11 4 Morse-Bott moduli spaces 21 4.1 Transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 Compactness for Morse-Bott trajectories . . . . . . . . . . . . . . 26 4.3 Gluing for Morse-Bott moduli spaces . . . . . . . . . . . . . . . . 30 4.4 Coherent orientations . . . . . . . . . . . . . . . . . . . . . . . . 69 A Appendix: Asymptotic estimates 84 http://arxiv.org/abs/0704.1039v2 Symplectic homology for autonomous Hamiltonians 2 1 Introduction One crucial hypothesis in the definition of Floer homology [12] of a Hamiltonian H on a symplectic manifold (W,ω) is that the 1-periodic orbits of the Hamilto- nian vector field XH are nondegenerate. Unless they are all constant – which happens if the Hamiltonian is C2-small – this forces H to be time-dependent. The purpose of this paper is to define Floer homology for a time-independent, or autonomous Hamiltonian H :W → R under the assumption that its 1-periodic orbits are transversally nondegenerate. This last condition is generic in the space of autonomous Hamiltonians. Although this generalization of Floer homology is interesting by itself, our main motivation is to understand the relationship between linearized contact homology of a fillable contact manifold (M, ξ) and symplectic homology of the filling (W,ω). In this case there is a natural class of time-independent Hamilto- nians on W whose nonconstant 1-periodic orbits correspond precisely to closed Reeb orbits on M = ∂W , and for which the Floer trajectories can be related to holomorphic cylinders in the symplectization M × R [3]. The goal of the present paper is to relate the Floer trajectories of a specific time-dependent perturbation to the Floer trajectories of the unperturbed Hamiltonian. Thus Floer homology for time-independent Hamiltonians serves as a bridge between symplectic homology and linearized contact homology. Moreover, the moduli spaces of Floer trajectories for autonomous Hamiltonians are related to the moduli spaces defining S1-equivariant symplectic homology [4, 3]. The Morse-Bott analysis in this paper is, to the best of our knowledge, new to the literature, being based on ideas contained in the first author’s Ph.D. dissertation [1] within the context of contact homology. Although our situation is that of critical manifolds of dimension one, the complexity of the analytical setup is the same as that of the higher dimensional case. We must mention at this point Frauenfelder’s inspired approach [16, Ap- pendix A] in which he defines a complex for a Morse-Bott function on a finite dimensional manifold via “flow lines with cascades” – these being our Floer trajectories with gradient fragments – and in which, without proving the cor- respondence with gradient trajectories for some perturbed Morse function, he directly shows deformation invariance of the resulting chain complex. We now describe the structure of the paper. We give in the introduction only a loose statement of our main Correspondence Theorem 3.7 and we recall in Section 2 the construction of symplectic homology. Although this is well-known to specialists we still need to establish notations, and we seize the occasion to set up a general framework using the Novikov ring and nontrivial homotopy classes of periodic orbits. Section 3 describes the Morse-Bott complex and formally states the Corre- spondence Theorem 3.7. The latter is complemented by Proposition 3.9 which describes how the coherent orientation signs for the Morse-Bott complex are related to the ones for the Floer complex. Section 4 contains the proofs of the previous transversality, compactness, Symplectic homology for autonomous Hamiltonians 3 gluing and orientation statements. Finally, the Appendix contains the state- ments concerned with the asymptotic behaviour of the various types of Floer trajectories that we use. These asymptotic estimates enter crucially in the proof of the compactness statements, as well as in the definition of the Fredholm setup for gluing. We end the introduction with an informal presentation of our results. Let H : W → R be an autonomous Hamiltonian defined on a symplectic manifold (W,ω). We assume that H is a Morse function and that the nonconstant 1- periodic orbits ofH are transversally nondegenerate. The set P(H) of 1-periodic orbits of H is the set of critical points of the Hamiltonian action functional and consists of isolated elements γep corresponding to critical points p̃ ∈ Crit(H), and of nonisolated elements coming in families Sγ which are Morse-Bott non- degenerate circles. These correspond to reparametrizations of some given orbit γ ∈ P(H), with γ : S1 = R/Z →W . For each circle Sγ we choose a perfect Morse function fγ : Sγ → R with exactly one maximum Max and one minimum min . We denote by γmin , γMax the orbits in Sγ corresponding to the minimum and the maximum of fγ respec- tively. We choose a chart S1 × R2n−1 ∋ (τ, p) and a smooth cut-off function ργ : S 1 × R2n−1 → R in the neighbourhood of each γ(S1) ⊂W , and we denote by ℓγ ∈ Z+ the maximal positive integer such that γ(θ + 1/ℓγ) = γ(θ), θ ∈ S1. Following [8], for δ > 0 small enough the time-dependent Hamiltonian Hδ : S 1 ×W → R, Hδ(θ, τ, p) := H − δ ργ(τ, p)fγ(τ − ℓγθ) has only nondegenerate 1-periodic orbits. Moreover, these are of the following two types: they are either constant orbits γep corresponding to critical points p̃ ∈ Crit(H), or they are nonconstant orbits of the form γp ∈ P(H) for p ∈ Crit(fγ). Thus, out of each circle Sγ of periodic orbits for H there are exactly two orbits surviving for Hδ, namely γmin and γMax . Let J be a generic time-dependent almost complex structure on W . Given p ∈ Crit(fγ), q ∈ Crit(fγ) we denote by M(γp, γq;Hδ, J) the moduli space of Floer trajectories for the pair (Hδ, J) modulo reparametriza- tion, with negative asymptote γp and positive asymptote γq. We also denote M(Sγ , Sγ ;H, J) the moduli space of Floer trajectories for the pair (H, J) modulo reparametriza- tion, with negative asymptote in Sγ and positive asymptote in Sγ . Our goal is to describe the moduli spaces of the first type in terms of moduli spaces of the second type. Symplectic homology for autonomous Hamiltonians 4 We denote by M(p, q;H, {fγ}, J) the moduli space of Floer trajectories for the pair (H, J) with intermediate gradient fragments, consisting of tuples [u] = (cm, [um], cm−1, [um−1], . . . , [u1], c0), m ≥ 0 such that: (i) [ui] ∈ M(Sγi , Sγi−1 ;H, J), i = 1, . . . ,m with γm := γ, γ0 := γ; (ii) cm is a semi-infinite gradient trajectory of fγ = fγm connecting γp to the endpoint of um; (iii) cj , j = 1, . . . ,m − 1 is a finite gradient trajectory of fγj connecting the endpoints of uj+1 and uj ; (iv) c0 is a semi-infinite gradient trajectory of fγ = fγ0 connecting the endpoint of u1 to γq. We give a pictogram of such an element [u] with m ≥ 1 in Figure 4 on page 45, where one should read ci instead of vi. If m = 0 such an element [u] is simply an infinite gradient trajectory of some fγ . Let us note that, just as the space of Floer trajectories for a nondegenerate Hamiltonian can be compactified by adding “broken” Floer trajectories, the space of Floer trajectories with interme- diate gradient fragments can be compactified by adding “broken” such objects, with an obvious meaning. We denote by M(p, q;H, {fγ}, J) these compactified moduli spaces. Our main result is the following comparison theorem. Theorem. The following assertions hold. (i) any sequence [vn] ∈ M(γp, γq;Hδn , J), δn → 0 converges to an element of M(p, q;H, {fγ}, J); (ii) any element of M(p, q;H, {fγ}, J) can be obtained as such a limit; (iii) there is a bijective correspondence between elements of M(γp, γq;Hδ, J) and elements of M(p, q;H, {fγ}, J) if the difference of index of the end- points is equal to one, or equivalently if the moduli spaces have dimension zero. The rigorous forms for the statements (i), (ii), (iii) are given in Proposi- tion 4.7, Proposition 4.22 and Theorem 3.7 respectively. Unsurprisingly, the Fredholm setup for the previous theorem uses Sobolev norms with exponential weights since we have degenerate asymptotics. Similarly, due to the convergence estimates in the Appendix, there are such weights centered on the portions of the Floer cylinders approaching gradient fragments. For each peak in the weight, there is a special section supported around this peak which has constant norm Symplectic homology for autonomous Hamiltonians 5 with respect to δ → 0. For each gradient fragment this section corresponds to the reparametrization shift of the underlying gradient trajectory. As δ → 0, the corresponding peak explodes and thus forbids all infinitesimal variations except for the single degree of freedom coming from Morse theory. To be useful for homological calculations the above theorem needs to be complemented by a statement concerning signs. We describe in Section 4.4 how to construct coherent orientations on the relevant spaces of Fredholm operators and how to obtain signs ǫ(u) and ǫ(uδ) for elements [u] ∈ M(p, q;H, {fγ}, J) and uδ ∈ M(γp, γq;Hδ, J) when the corresponding moduli spaces are zero- dimensional. We recall in Remark 3.2 the definition of good orbits borrowed from Symplectic Field Theory, where it plays a crucial role in all orientation and signs problems. In the following statement we denote again by m ≥ 0 the number of nonconstant Floer trajectories involved in u. Proposition 3.9. Assume the moduli spaces under consideration have dimen- sion zero. The bijective correspondence between elements uδ ∈ M(γp, γq;Hδ, J) and [u] ∈ M(p, q;H, {fγ}, J) changes signs as follows: (i) If m ≥ 1 we have ǫ(u) = (−1)m−1ǫ(uδ); (ii) If m = 0 we have u = uδ and ǫ(u) = ǫ(uδ), p is the minimum and q is the maximum of the same function fγ , the moduli space M(p, q;H, {fγ}, J) consists of the two gradient lines of fγ running from p to q, and their signs are different if and only if the underlying orbit γ is good. This result has two pleasant consequences. On the one hand we can construct a “Morse-Bott” chain complex which computes symplectic homology by count- ing with suitable signs rigid elements in the moduli spaces M(p, q;H, {fγ}, J). On the other hand, this chain complex singles out algebraically the good orbits and can be used to relate the symplectic homology of a manifold (W,ω) with contact type boundary to the linearized contact homology of its boundary – the latter being defined by a chain complex involving only good orbits. As already mentioned at the beginning of this section, this is achieved in [3]. Acknowledgements. A.O. has benefited from a Swiss National Fund grant under the supervision of Prof. Dietmar Salamon at ETH Zürich. Both authors acknowledge financial support from the Fonds National de la Recherche Scien- tifique, Belgium, the Forschungsinstitut für Mathematik, Zürich, the Institut de Recherche Mathématique Avancée at Université Louis Pasteur, Strasbourg, as well as from the Mathematisches Forschungsinstitut, Oberwolfach. The authors would like to thank an anonymous referee for having carefully read through the arguments in the paper and for having kindly suggested a solution to a gap in the original proof of Proposition 4.9. Symplectic homology for autonomous Hamiltonians 6 2 Symplectic homology We define in this section the symplectic homology groups of a symplectically aspherical manifold with contact type boundary. Our construction is modelled on those of Cieliebak, Floer, Hofer and Viterbo [6, 7, 13, 28]. We consider nontrivial homotopy classes of loops and we use the Novikov ring. Let (W,ω) be a compact symplectic manifold with contact type boundary M := ∂W . This means that there exists a vector field X defined in a neigh- bourhood of M , transverse and pointing outwards along M , and such that LXω = ω. Such an X is called a Liouville vector field. The 1-form λ := (ιXω)|M is a contact form on M . We denote by ξ the contact distribution defined by λ. The Reeb vector field Rλ is uniquely defined by the conditions ker ω|M = 〈Rλ〉 and λ(Rλ) = 1. We denote by φλ the flow of Rλ. The action spectrum of (M,λ) is defined by Spec(M,λ) := {T ∈ R+ | there is a closed Rλ-orbit of period T }. We assume throughout this paper the condition f∗ω = 0 for all smooth f : T 2 →W. (1) This guarantees that the energy of a Floer trajectory does not depend on its homology class, but only on its endpoints (see below). Condition (1) plays an important role in the Morse-Bott description of the symplectic homology groups. Our main class of examples is provided by exact symplectic forms. Let φ be the flow of X . We parametrize a neighbourhood U of M by G :M × [−δ, 0] → U, (p, t) 7→ φt(p). Then d(etλ) is a symplectic form on M ×R+ and G satisfies G∗ω = d(etλ). We denote Ŵ :=W M × R+ and endow it with the symplectic form ω̂ := ω, on W, d(etλ), on M × R+. Given a time-dependent HamiltonianH : S1×Ŵ → R, we define theHamil- tonian vector field XθH by ω̂(XθH , ·) = dHθ, θ ∈ S1 = R/Z, where Hθ := H(θ, ·). We denote by φH the flow of XθH , defined by φ0H = Id and φθH(x) = X H(x)), θ ∈ R. Symplectic homology for autonomous Hamiltonians 7 Let H be the set of admissible Hamiltonians, consisting of functions H : S1 × Ŵ → R which satisfy (i) H < 0 on W ; (ii) H(θ, p, t) = αet + β for t large enough, with α /∈ Spec(M,λ); (iii) every 1-periodic orbit γ : S1 → Ŵ of XθH is nondegenerate, i.e. 1l− dφ1H(γ(0)) 6= 0. We denote by P(H) the set of 1-periodic orbits of XθH and by Pa(H) the set of 1-periodic orbits in a given free homotopy class a in Ŵ . Let J denote the set of admissible almost complex structures J : S1 → End(TŴ ), J2 = −1l which are compatible with ω̂ and have the following standard form for t large enough: { J(p,t)|ξ = J0, J(p,t) = Rλ. Here J0 is any compatible complex structure on the symplectic bundle (ξ, dλ) which is independent of θ and t. Let us fix a reference loop la : S 1 → Ŵ for each free homotopy class a in Ŵ such that [la] = a. If a is the trivial homotopy class we choose la to be a constant loop. Recall that free homotopy classes of loops in Ŵ are in one-to-one correspondence with conjugacy classes in π1(Ŵ ). As a consequence, the inverse a−1 of a free homotopy class is well-defined. We require that la−1 coincides with the loop la with the opposite orientation. The Hamiltonian action functional acts on pairs (γ, [σ]) consisting of a loop γ ∈ C∞(S1, Ŵ ) and the homology class (rel boundary) of a map σ : Σ → Ŵ defined on a Riemann surface Σ with two boundary components ∂0Σ (with the opposite boundary orientation) and ∂1Σ (with the boundary orientation), which satisfies σ|∂0Σ = l[γ], σ|∂1Σ = γ. (3) Its values are defined by AH(γ, [σ]) := − σ∗ω̂ − H(θ, γ(θ)) dθ. (4) The differential dAH(γ, [σ]) : C∞(S1, γ∗TŴ ) → R is given by dAH(γ, [σ])ζ := ω̂(γ̇ −XθH(γ), ζ) dθ. Therefore the critical points of AH are pairs (γ, [σ]) such that γ ∈ P(H). We fix from now on, for each γ ∈ P(H), a map σγ satisfying (3); then the set of all pairs (γ, [σ]) can be identified with H2(W ;Z) for fixed γ. Symplectic homology for autonomous Hamiltonians 8 Let us choose a symplectic trivialization Φa : S 1 × R2n → l∗aTŴ for each free homotopy class a in Ŵ . If a is the trivial homotopy class we choose the trivialization to be constant. Moreover, we require that Φa−1(θ, ·) = Φa(−θ, ·), θ ∈ S1 = R/Z. For each γ ∈ P(H) there exists a unique (up to homotopy) trivialization Φγ : Σ× R2n → σ∗γTŴ such that Φγ = Φ[γ] on ∂0Σ× R2n. Let Ψ : [0, 1] → Sp(2n), Ψ(θ) := Φ−1γ ◦ dφθH(γ(0)) ◦ Φγ . (5) Because γ is nondegenerate we can define the Conley-Zehnder index µ(γ) by µ(γ) := µ(γ, σγ) := −µCZ(Ψ), (6) where µCZ(Ψ) is the Conley-Zehnder index of a path of symplectic matrices [23]. Remark 2.1. If, in the previous construction, we replace σγ with σγ#A for some A ∈ H2(W ;Z), then the resulting index will be µ(γ, σγ#A) = µ(γ, σγ)− 2〈c1(TW ), A〉. (7) We define the Novikov ring Λω as the set of formal linear combinations A∈H2(W ;Z) A, λA ∈ Z such that # {A |λA 6= 0, ω(A) ≤ c} <∞ for all c > 0. The multiplication in Λω is given by λ ∗ λ′ := A,B∈H2(W ;Z) We note that, if ω is exact, then Λω = Z[H2(W ;Z)]. We define a grading on Λω by |eA| := −2〈c1(TW ), A〉. For each free homotopy class a in Ŵ and each admissible Hamiltonian H we define the symplectic chain group SCa∗ (H) as the free Λω-module generated by elements γ ∈ Pa(H). The grading is given by |eAγ| := µ(γ)− 2〈c1(TW ), A〉. We define the space of Floer trajectories M̂A(γ, γ;H, J) as the set of solu- tions u : R× S1 → Ŵ of the equation ∂su+ Jθ(∂θu−XθH) = 0, (8) such that u(s, θ) = γ(θ), lim u(s, θ) = γ(θ), lim ∂su = 0 (9) uniformly in θ and [σγ#u] = [σγ#A]. (10) Symplectic homology for autonomous Hamiltonians 9 Remark 2.2. Under the nondegeneracy assumption on γ, γ condition (9) is equivalent to the finiteness of the energy E(u) := EJ,H(u) := |∂su|2θ + |∂tu−XθH |2θ dsdθ. (11) Because γ, γ are nondegenerate the linearized operator Du : W 1,p(R × S1, u∗TŴ ) → Lp(R× S1, u∗TŴ ), p > 2 given by Duζ := ∇sζ + Jθ∇θζ + (∇ζJθ)∂θu−∇ζ , u ∈ M̂A(γ, γ;H, J) (12) is Fredholm with index ind(Du) = µ(γ)− µ(γ) + 2〈c1(TW ), A〉. (13) An almost complex structure J ∈ J is called regular for u ∈ M̂A(γ, γ;H, J) if Du is surjective, and it is called regular if Du is surjective for all γ, γ ∈ P(H), A ∈ H2(W ;Z) and u ∈ M̂A(γ, γ;H, J). It is proved in [15] that the space Jreg(H) of regular almost complex structures is of the second category in J . For every J ∈ Jreg(H) the space M̂A(γ, γ;H, J) is a smooth manifold of dimension µ(γ)− µ(γ) + 2〈c1(TW ), A〉. From now on we fix some J ∈ Jreg(H). If γ 6= γ or A 6= 0, the additive group R acts freely on M̂A(γ, γ;H, J) by s0 · u(·, ·) := u(s0 + ·, ·). We define the moduli space of Floer trajectories MA(γ, γ;H, J) := M̂A(γ, γ;H, J)/R. Its dimension is dim MA(γ, γ;H, J) := µ(γ)− µ(γ) + 2〈c1(TW ), A〉 − 1. If γ = γ and A = 0, the space M̂0(γ, γ;H, J) consists of a single point, cor- reponding to a constant solution (i.e. independent of s). The R action is then trivial and we define the moduli space by M0(γ, γ;H, J) := M̂0(γ, γ;H, J). A straightforward application of the maximum principle [28] using the special form of admissible Hamiltonians for large t shows that all solutions of equa- tions (8) and (9) are contained in a compact set. Moreover, by condition (1), there are no J-holomorphic spheres that can bubble off. Therefore the mod- uli space MA(γ, γ;H, J) can be compactified [12] to a space MA(γ, γ;H, J) consisting of all tuples ([uk], [uk−1], . . . , [u1]), [ui] ∈ MAi(γi, γi;H, J) such that γ = γ, γi = γi+1, γk = γ and iAi = A. We call such a tuple ([uk], [uk−1], . . . , [u1]) a broken trajectory of level k. The topology of the compactified moduli space is described by the following notion of convergence: a sequence [uν ] ∈ MA(γ, γ;H, J) is said to converge to the broken trajectory Symplectic homology for autonomous Hamiltonians 10 γ = γ γ1 = γ2 γ2 = γ3 γ = γ3 Figure 1: Broken trajectory. ([uk], [uk−1], . . . , [u1]) if there exist sequences s i ∈ R, 1 ≤ i ≤ k such that sνi ·uν converges uniformly on compact sets to ui. If the space M̂A(γ, γ;H, J) is nonempty then its dimension is strictly positive due to the action of R. In this case, the broken trajectories involved in the compactification have level at most dim M̂A(γ, γ;H, J). In particular, when µ(γ) − µ(γ) + 2〈c1(TW ), A〉 = 1 the moduli space MA(γ, γ;H, J) is compact and consists of a finite number of points. In this situation one can associate a sign ǫ(u) to each element [u] of this moduli space [13] (see also Section 4.4). We define the Floer differential ∂ : SCa∗ (H) → SCa∗−1(H) ∂γ := µ(γ)−µ(γ)+2〈c1(TW ),A〉=1 [u]∈MA(γ,γ;H,J) ǫ(u)eAγ. (14) According to Floer [12] we have ∂2 = 0. We define the symplectic homol- ogy groups of the pair (H, J) by SHa∗ (H, J) := H∗(SC ∗ (H), ∂). Remark 2.3. In view of condition (1) the Novikov ring Λω can be replaced by Z[H2(W ;Z)], or even by Z at the price of losing the grading. Indeed, the energy of a Floer trajectory depends only on its endpoints, hence the moduli spaces M(γ, γ;H, J) := (γ, γ;H, J) are compact. Therefore the sum (14) in- volves only a finite number of classes A. By a standard argument [12] the groups SHa∗ (H, J) do not depend on J ∈ Jreg(H). Nevertheless, they do depend onH and, in order to obtain an invariant of (W,ω), we need an additional algebraic limit construction. We define an Symplectic homology for autonomous Hamiltonians 11 admissible homotopy of Hamiltonians as a map H : R×S1×Ŵ → R with the following properties: (i) H(s, ·, ·) = H− ∈ H for s ≤ −1, H(s, ·, ·) = H+ ∈ H for s ≥ 1; (ii) H < 0 on W and there exist t0 ≥ 0 and functions α, β : R → R such that, for all t ≥ t0, we have H(s, t, p) = α(s)et + β(s); (iii) ∂sH ≥ 0. An admissible homotopy of almost complex structures is a map J : R → J such that J(s) = J− for s ≤ −1 and J(s) = J+ for s ≥ 1. Given an admissible homotopy of Hamiltonians one defines regular admissible homotopies of almost complex structures in the usual way, by linearizing the equation ∂su(s, θ) + J(s, θ, u(s, θ))(∂θu(s, θ)−XθH(s, θ, u(s, θ))) = 0, (15) subject to the limit conditions u(s, ·) = γ ∈ P(H−), lim u(s, ·) = γ ∈ P(H+). (16) Regular admissible homotopies of almost complex structures form again a set of the second category in the space of admissible homotopies and the rigid behaviour of H for t ≥ t0, together with the condition ∂sH ≥ 0, ensures again that solutions of (15) and (16) stay in a compact set (see [22]). The usual count of solutions of (15) and (16) induces the monotonicity morphism σ : SHa∗ (H−) → SHa∗ (H+), (17) which does not depend on the choice of admissible homotopy connecting H− and H+. These morphisms form a direct system on the set {SHa∗ (H), H ∈ H} and we define the symplectic homology groups of (W,ω) by SHa∗ (W,ω) := lim→ SHa∗ (H). According to [6, Lemma 3.7] and [28, Theorem 1.7] these groups do not depend on the choice of the Liouville vector field X . 3 The Morse-Bott chain complex In this section we apply the Morse-Bott formalism of [1] to the case of Hamil- tonians H : Ŵ → R having circles of 1-periodic orbits. We denote by Pλ the set of closed unparametrized Rλ-orbits in M . For each free homotopy class of loops b in M we denote by Pbλ the set of all γ ∈ Pλ in the homotopy class b. The inclusion i : M →֒ W induces a map (still denoted by Symplectic homology for autonomous Hamiltonians 12 i) between the sets of free homotopy classes of loops in M and W respectively. For each free homotopy class a in W we denote −1(a) b∈i−1(a) We assume in this section that the closed Reeb orbits on M are transversally nondegenerate in M . This means that, for every orbit γ of period T > 0, we 1l− dφTλ (γ(0))|ξ 6= 0. This can always be achieved by an arbitrarily small perturbation of λ or, equiv- alently, of X , and such perturbations do not change the symplectic homology groups. If all orbits γ ∈ Pλ are transversally nondegenerate one can assign to each of them a Conley-Zehnder index µCZ(γ) according to the following recipe. We fix a reference loop lb : S 1 → M for each free homotopy class b in M such that [lb] = b. If b is the trivial homotopy class we choose lb to be a constant loop and we require that lb−1 coincides with lb with the opposite orientation. We define symplectic trivializations Φb : S 1 × R2n−2 → l∗bξ as follows. For each class b we choose a homotopy hab : S 1 × [0, 1] → W from la, a = i(b) to lb such that ha−1b−1(τ, ·) = hab(−τ, ·). (18) We extend the trivialization Φa : S 1 × R2n → l∗aTŴ over the homotopy hab to get a trivialization Φ′b : S 1 × R2n → l∗bTŴ . This trivialization is homotopic to another one, still denoted Φ′b, such that Φ′b(S 1 × R2n−2 × {0} × {0}) = l∗bξ, Φ′b(S 1 × {0} × R× {0}) = l∗b 〈 〉, (19) Φ′b(S 1 × {0} × {0} × R) = l∗b 〈Rλ〉. We define Φb := Φ b|S1×R2n−2×{0}×{0}. If b is the trivial homotopy class we choose hab to be a path of constant loops, so that Φb is constant. We fix for each γ ∈ Pλ a map σγ : Σ → M , where Σ is a Riemann surface with two boundary components ∂0Σ (with the opposite boundary orientation) and ∂1Σ (with the boundary orientation), satisfying σ|∂0Σ = l[γ], σ|∂1Σ = γ. (20) For each γ ∈ Pλ there exists a unique (up to homotopy) trivialization Φγ : Σ× R2n−2 → σ∗γξ Symplectic homology for autonomous Hamiltonians 13 such that Φγ = Φ[γ] on ∂0Σ× R2n−2. Let Ψ : [0, T ] → Sp(2n− 2), Ψ(τ) := Φ−1γ ◦ dφτλ(p) ◦ Φγ , p ∈ im γ. (21) Because γ is nondegenerate we can define the Conley-Zehnder index µ(γ) by µ(γ) := µ(γ, σγ) := µCZ(Ψ), (22) where µCZ(Ψ) is the Conley-Zehnder index of a path of symplectic matrices [23]. Remark 3.1. If, in the previous construction, we replace σγ with σγ#A for some A ∈ H2(M ;Z), then the resulting index will be µ(γ, σγ#A) = µ(γ, σγ) + 2〈c1(ξ), A〉. (23) Note that c1(ξ) = i ∗c1(TW ) because i ∗TW = ξ⊕〈 ∂ , Rλ〉. Moreover, the parity of µ(γ) is well-defined independently of the trivialization of ξ along γ. Remark 3.2. For each simple orbit γ ∈ Pλ we denote by γk, k ∈ Z+ its positive iterates. The parity of the Conley-Zehnder index of all the odd, respectively even iterates is the same. If these two parities differ we say that all even iterates γ2k, k ∈ Z+ are bad orbits. It is proved in [27, Lemma 3.2.4] that the even iterates of a simple orbit γ of period T are bad if and only if dφTλ (p)|ξ, p ∈ im γ has an odd number of real negative eigenvalues strictly smaller than −1 (see also Lemma 4.25). The orbits in Pλ which are not bad are called good orbits. We define a new class H′ of admissible Hamiltonians consisting of elements H : Ŵ → R such that (i) H |W is a C2-small Morse function and H < 0 on W ; (ii) H(p, t) = h(t) outside W , where h(t) is a strictly increasing function with h(t) = αet + β, α, β ∈ R, α /∈ Spec(M,λ) for t bigger than some t0, and such that h′′ − h′ > 0 on [0, t0[. Note that the 1-periodic orbits of XH in W are constant and nondegenerate by assumption (i). A direct computation shows that Xh(p, t) = −e−th′(t)Rλ. (24) The 1-periodic orbits of XH fall in two classes: (1) critical points of H in W ; (2) nonconstant 1-periodic orbits of Xh, located on levels M × {t}, t ∈]0, t0[, which are in one-to-one correspondence with closed −Rλ-orbits of period e−th′(t). Symplectic homology for autonomous Hamiltonians 14 Recall that, for every critical point p̃ ∈ Crit(H), the corresponding constant XH-orbit γep has Conley-Zehnder index µ(γep) = ind(p̃;−H)− n, n = dim W, where ind(p̃;−H) is the Morse index of p̃ with respect to −H [25, Lemma 7.2]. Let α := limt→∞ e −tH(p, t). We denote by P≤αλ the set of all γ ∈ Pλ such γ∗λ ≤ α. BecauseH is independent of θ, every orbit γ ∈ P≤αλ gives rise to a whole circle of nonconstant 1-periodic orbits γH of XH . We denote by Sγ the set of such orbits and identify Sγ with its image under the natural embedding Sγ → Ŵ given by γH 7→ γH(0). Note that all elements of Sγ differ by a shift in the parametrization, and that the γH are noninjective if their minimal period is smaller than 1. Lemma 3.3. Let H ∈ H′. Every nonconstant 1-periodic orbit γH of H is transversally nondegenerate in Ŵ . Proof. We have to show that the only eigenvector of dφ1H(γH(0)) corresponding to the eigenvalue 1 is γ̇H(0). To this effect we note that ξ is an invariant space and that dφ1H(γH(0))|ξ = e−th′(t) (γH(0))|ξ. Because we have assumed that all Rλ-orbits are transversally nondegenerate in M , it follows that dφ1H(γH(0))|ξ has no eigenvalue equal to one. On the other hand we have dφ1H(γH(0)) − e−t(h′′ − h′)Rλ. The conclusion follows because h′′(t)− h′(t) > 0. For each γ ∈ P≤αλ we choose a Morse function fγ : Sγ → R with exactly one maximum M and one minimum m. We fix from now on an element H ∈ H′ and, for each γ ∈ Pλ corresponding to a nonconstant γH ∈ P(H), we denote by ℓγ ∈ Z+ the maximal positive integer such that γH(θ + 1ℓγ ) = γH(θ) for all θ ∈ S1. We choose a symplectic trivialization ψ := (ψ1, ψ2) : Uγ V ⊂ S1 × R2n−1 between open neighbourhoods Uγ ⊂ Ŵ of γH(S1) and V of S1 × {0}, such that ψ1(γ(θ)) = ℓγθ. Here S1 × R2n−1 is endowed with the symplectic form ω0 := i=1 dqi ∧ dpi, q1 ∈ S1, (p1, q2, p2, . . . , qn, pn) ∈ R2n−1. Let ρ : S1 × R2n−1 → [0, 1] be a smooth cutoff function supported in a small neighbourhood of S1 × {0} such that ρ|S1×{0} ≡ 1. For δ > 0 and (θ, p, t) ∈ S1 × Uγ we define Hδ(θ, p, t) := h(t) + δρ(ψ(p, t))fγ(ψ1(p, t)− ℓγθ). (25) The Hamiltonian Hδ coincides with H outside the open sets S 1 × Uγ . This is precisely the perturbation described in [8, Proposition 2.2]. It is shown therein that, for δ sufficiently small, the set P(Hδ) consists of the following elements: Symplectic homology for autonomous Hamiltonians 15 (1) constant orbits, which are the same as those of H ; (2) nonconstant orbits, which are nondegenerate and form pairs (γmin , γMax ), where γ ∈ P≤αλ and γmin , γMax coincide with the orbits in Sγ starting at the minimum and the maximum of fγ respectively. Lemma 3.4. The periodic orbits γmin , γMax ∈ P(Hδ) satisfy µ(γmin) = µ(γ) + 1, µ(γMax ) = µ(γ). (26) Proof. We denote by γH the 1-periodic orbit of XH corresponding to γ ∈ P≤αλ . We define the Robbin-Salamon index of γH by µRS(γH) := µ(Ψ), where Ψ : [0, 1] → Sp(2n) is given by (5) and µ(Ψ) is the Robbin-Salamon index of an arbitrary path of symplectic matrices [23, §4]. It is shown in [8, Proposition 2.2] that − µ(γmin) = µRS(γH)− , −µ(γMax ) = µRS(γH) + . (27) Note that γH has the orientation of −Rλ. Define Ψ̃ : [0, 1] → Sp(2n) by Ψ̃(θ) := Φ−1γ (−θ) ◦ dφ−θH (γ(0)) ◦Φγ(0), where Φγ : R/Z × R2n → γ∗HTŴ is the trivialization involved in (5). Then µRS(−γH) = −µRS(γH) = µ(Ψ̃). Let Sp∗(2n) ⊂ Sp(2n) be the set of symplectic matrices with no eigenvalue equal to 1 and recall that we have denoted by a free homotopy classes of loops in W and by b free homotopy classes of loops in M . By our choice (18) and (19) of trivializations of TŴ over the reference loops lb, b ∈ i−1(a) we deduce that the path Ψ̃ is homotopic with endpoint in Sp∗(2n) to the path [0, 1] → Sp(2n) : θ 7→ Ψλ (Tθ)⊕ Here T := e−t(h′′(t) − h′(t)) and Ψλ : [0, T ] → Sp(2n − 2) is defined by (21). By the symplectic shear axiom for the Robbin-Salamon index [23, Theorem 4.1] the index of the above path is µ(γ) + 1 . As a consequence −µRS(γH) = µRS(−γH) = µ(γ) + Together with (27) this yields the conclusion of the Lemma. Let p ∈ Crit(fγ); then γp ∈ P(Hδ) for all δ ∈]0, δ0] if δ0 is small enough, and Lemma 3.4 says that µ(γp) = µ(γ) + ind(p; fγ). If p̃ is a critical point of H in Symplectic homology for autonomous Hamiltonians 16 W we denote by γ ∈ P(H) the corresponding constant orbit. Our goal is to describe the boundary points as δ → 0 of MA]0,δ0[(γp, γq;H, {fγ}, J) := 0<δ<δ0 {δ} ×MA(γp, γq;Hδ, J), (28) µ(γp)− µ(γq) + 2〈c1(TW ), A〉 = 1, where γ, γ ∈ P≤αλ , p ∈ Crit(fγ), q ∈ Crit(fγ), A ∈ H2(W ;Z), J ∈ J γ ∈ P≤αλ , p ∈ Crit(fγ), q ∈ Crit(H), A ∈ H2(W ;Z), J ∈ J . Our description is very similar to that of [1] within the setting of contact homology. We fix J ∈ J , γ, γ ∈ P≤αλ and q̃ ∈ Crit(H). We define two Morse- Bott spaces of Floer trajectories M̂A(Sγ , Sγ ;H, J) and M̂A(Sγ , q̃;H, J) as follows. For γ, γ ∈ P≤αλ we denote by M̂A(Sγ , Sγ ;H, J) the set of solutions u : R× S1 → Ŵ of the Floer equation (8) subject to the asymptotic conditions u(s, θ) = γH(θ), lim u(s, θ) = γ (θ), lim ∂su = 0 (29) uniformly in θ, with γH ∈ Sγ , γH ∈ Sγ (30) [σγ#u] = [σγ#A]. It is implicit in the above definition that the orbits γH and γH may vary for different elements of M̂A(Sγ , Sγ ;H, J). For γ ∈ P≤αλ and q̃ ∈ Crit(H) we denote by M̂A(Sγ , q̃;H, J) the set of solutions u : R× S1 → Ŵ of the Floer equation (8) subject to the asymptotic conditions u(s, θ) = γH(θ), lim u(s, θ) = q̃, lim ∂su = 0 (31) uniformly in θ, with γH ∈ Sγ (32) [σγ#u] = A. Again, the orbit γH may vary for different elements of M̂A(Sγ , q̃;H, J). Symplectic homology for autonomous Hamiltonians 17 If γ 6= γ or A 6= 0, the additive group R acts freely on M̂A(Sγ , Sγ ;H, J) and M̂A(Sγ , q̃;H, J) by s0 · u(·, ·) := u(s0 + ·, ·). We define the Morse-Bott moduli spaces of Floer trajectories by MA(Sγ , Sγ ;H, J) := M̂A(Sγ , Sγ ;H, J)/R MA(Sγ , q̃;H, J) := M̂A(Sγ , q̃;H, J)/R. If γ = γ and A = 0, the space M̂0(Sγ , Sγ ;H, J) is diffeomorphic to Sγ , con- sists of constant cylinders (i.e. independent of s) and the R action is trivial. In this case, we define the Morse-Bott moduli spaces by M0(Sγ , Sγ ;H, J) := M̂0(Sγ , Sγ ;H, J). We have natural evaluation maps ev : MA(Sγ , Sγ ;H, J) → Sγ , ev : MA(Sγ , Sγ ;H, J) → Sγ ev : MA(Sγ , q̃;H, J) → Sγ defined by ev([u]) := lim u(s, ·), ev([u]) := lim u(s, ·). In the statement of the next result we denote by J ′ the set of almost complex structures J ∈ J which are independent of θ ∈ S1. Proposition 3.5. (i) Given H ∈ H′, let J ′(H) ⊂ J ′ be the (nonempty and open) set of almost complex structures J such that, for any x ∈ Ŵ located on a simple 1-periodic orbit of XH , we have [XH , JXH ](x) 6= 0 and [XH , JXH ](x) /∈ 〈XH , JXH〉. (33) There exists a set of second category J ′reg(H) ⊂ J ′(H) consisting of almost complex structures J that are regular for all u ∈ M̂A(Sγ , Sγ ;H, J) or u ∈ M̂A(Sγ , q̃;H, J) with γ or γ being a simple orbit, and such that Jξ = ξ, = Rλ outside a fixed open neighbourhood of the nonconstant periodic orbits of XH . (ii) Given H ∈ H′, there exists a set of second category Jreg(H) ⊂ J con- sisting of regular almost complex structures J which, outside a fixed open neighbourhood of the nonconstant periodic orbits of XH , are independent of θ and satisfy Jξ = ξ, J ∂ = Rλ. In each of the previous cases the relevant moduli spaces MA(Sγ , Sγ ;H, J), MA(Sγ , q̃;H, J) are smooth manifolds of dimension dim MA(Sγ , Sγ ;H, J) = µ(γ)− µ(γ) + 2〈c1(TW ), A〉, dim MA(Sγ , q̃;H, J) = µ(γ)− µ(γeq) + 2〈c1(TW ), A〉, and the evaluation maps ev, ev are smooth. Symplectic homology for autonomous Hamiltonians 18 The proof of this statement is given in Section 4. Unless the contrary is explicitly mentioned, all statements in this section hold both for J ∈ Jreg(H) or J ∈ J ′reg(H), provided one considers moduli spaces with at least one simple asymptotic orbit in the latter case. Let now J ∈ Jreg(H) and fix for each γ ∈ P≤αλ a metric on Sγ such that Rλ has length one. Let Freg(H, J) be the set of regular Morse functions, consisting of families {fγ}, γ ∈ P≤αλ of perfect Morse functions fγ : Sγ → R such that all the maps ev are transverse to the unstable manifolds Wu(p), p ∈ Crit(fγ), all the maps ev are transverse to the stable manifolds W s(p), p ∈ Crit(fγ) and all pairs (ev, ev) : MA(Sγ , Sγ ;H, J) → Sγ × Sγ , (ev, ev) : MA1(Sγ , Sγ1 ;H, J) ev ×ev MA2(Sγ1 , Sγ ;H, J) → Sγ × Sγ (34) are transverse to productsWu(p)×W s(q), p ∈ Crit(fγ), q ∈ Crit(fγ). Here and in the sequel the unstable and stable manifolds are understood with respect to ∇fγ . Denote by C∞p (Sγ ,R) the set of perfect Morse functions on Sγ . Lemma 3.6. The set Freg(H, J) is of the second Baire category in the space∏ C∞p (Sγ ,R). Proof. The first two transversality conditions on ev, ev are satisfied if and only if the maximum of each function fγ is a regular value of all the evaluation maps ev having Sγ as target space, and if the minimum of each fγ is a regular value of all the evaluation maps ev mapping to Sγ . The third transversality condition requires in addition that each pair (M,m) ∈ Sγ × Sγ , with M the maximum of fγ and m the minimum of fγ , is a regular value of (ev, ev). By Sard’s theorem the minimum and maximum of each fγ can be chosen inside a set of second category in Sγ . The conclusion follows. Let now J ∈ Jreg(H) and {fγ} ∈ Freg(H, J). For p ∈ Crit(fγ) we denote the Morse index by ind(p) := dim Wu(p;∇fγ). Let γ, γ ∈ P≤αλ and p ∈ Crit(fγ), q ∈ Crit(fγ). For m ≥ 0 we denote by MAm(p, q;H, {fγ}, J) (35) the union for γ1, . . . , γm−1 ∈ P≤αλ and A1+ . . .+Am = A of the fibered products Wu(p)×ev (MA1(Sγ , Sγ1)×R+)ϕfγ1◦ev×ev (M A2(Sγ1 , Sγ2)×R+)ϕfγ2◦ev×ev . . . ϕfγm−1 ◦ev ×ev MAm(Sγm−1 , Sγ)ev×W s(q), Symplectic homology for autonomous Hamiltonians 19 with the convention γ0 = γ. This is well defined as a smooth manifold of dimension dim MAm(p, q;H, {fγ}, J) = ind(p)− 1 + (dim MA1(Sγ , Sγ1) + 1)− 1 +(dim MA2(Sγ1 , Sγ2) + 1)− 1 + ... +dim MAm(Sγm−1 , Sγ)− 1 + (1− ind(q)) = µ(γ) + ind(p)− µ(γ)− ind(q) + 2〈c1(TW ), A1 + ...+Am〉 − 1 = µ(γp)− µ(γq) + 2〈c1(TW ), A〉 − 1. The last equality follows from Lemma 3.4. Note that MA0 (p, q;H, {fγ}, J) is naturally a submanifold of MA(Sγ , Sγ ;H, J). We denote MA(p, q;H, {fγ}, J) := MAm(p, q;H, {fγ}, J) and we call this the moduli space of Morse-Bott broken trajectories, whereasMAm(p, q;H, {fγ}, J) is called the moduli space of Morse-Bott bro- ken trajectories with m sublevels (see also Definition 4.1 and Figure 4). Similarly, given γ ∈ P≤αλ , p ∈ Crit(fγ), q̃ ∈ Crit(H), we define mod- uli spaces of Morse-Bott broken trajectories MAm(p, q̃;H, {fγ}, J), m ≥ 0 and MA(p, q̃;H, {fγ}, J) by replacing the last term MAm(Sγm−1 , Sγ)ev×W s(q) in the definition (35) with MAm(Sγm−1 , q̃;H, J). This is again well defined as a smooth manifold of dimension dim MA(p, q̃;H, {fγ}, J) = µ(γp)− µ(γeq) + 2〈c1(TW ), A〉 − 1. Again, MA0 (p, q̃;H, {fγ}, J) is naturally a submanifold of MA(Sγ , q̃;H, J). The significance of the above moduli spaces of broken Morse-Bott trajec- tories is explained by the following theorem, which describes the boundary of ]0,δ0[ (γp, γq;H, {fγ}, J) in (28) as δ → 0. Theorem 3.7 (Correspondence Theorem). Let H ∈ H′ be fixed and let α := limt→∞ e −tH(p, t) be the maximal slope of H. Let J ∈ Jreg(H) and {fγ} ∈ Freg(H, J). There exists δ1 := δ1(H, J) ∈ ]0, δ0[ such that, for any γ, γ ∈ P≤αλ , p ∈ Crit(fγ), q ∈ Crit(fγ), γ ∈ P≤αλ , p ∈ Crit(fγ), q ∈ Crit(H), and any A ∈ H2(W ;Z) with µ(γp)− µ(γq) + 2〈c1(TW ), A〉 = 1, the following hold: Symplectic homology for autonomous Hamiltonians 20 (i) J is regular for MA(γp, γq;Hδ, J) for all δ ∈]0, δ1[; (ii) the space MA ]0,δ1[ (γp, γq;H, {fγ}, J) is a 1-dimensional manifold having a finite number of components which are graphs over ]0, δ1[, i.e. the natural projection MA ]0,δ1[ (γp, γq;H, {fγ}, J) →]0, δ1[ is a submersion; (iii) there is a bijective correspondence between points [u] ∈ MA(p, q;H, {fγ}, J) and connected components of MA]0,δ1[(γp, γq;H, {fγ}, J). The proof of this statement, including a discussion of gluing and compactness for Morse-Bott moduli spaces, is given in Section 4. We assume in the remainder of this section that the conclusions of Theo- rem 3.7 are satisfied. For each [u] ∈ MA(p, q;H, {fγ}, J) the sign ǫ(uδ) is con- stant on the corresponding connected component C[u] for continuity reasons. We define a sign ǭ(u) by ǭ(u) := ǫ(uδ), δ ∈]0, δ1[, (δ, [uδ]) ∈ C[u]. (36) We define the Morse-Bott chain groups by BCa∗ (H) := i−1(a),≤α Λω〈γmin , γMax 〉, a 6= 0, (37) BC0∗(H) := ep∈Crit(H) Λω〈p̃〉 ⊕ i−1(0),≤α Λω〈γmin , γMax 〉. (38) where α := limt→∞ e −tH(p, t) and P i −1(0),≤α λ = P λ ∩ P i−1(0) λ . The grading is defined by |eAp̃| := ind(p̃;−H)− n− 2〈c1(TW ), A〉, |eAγmin | := µ(γ) + 1− 2〈c1(TW ), A〉, |eAγMax | := µ(γ)− 2〈c1(TW ), A〉. We define the Morse-Bott differential ∂ : BCa∗ (H) → BCa∗−1(H) ∂p̃ := eq∈Crit(H) |ep|−|eq|=1 [u]∈M0(ep,eq;H,{fγ},J) ǭ(u)q̃, (39) ∂γp := eq∈Crit(H) |γp|−|e eq|=1 [u]∈MA(γp,eq;H,{fγ},J) ǭ(u)eAq̃ (40) ,q∈Crit(fγ) |γp|−|e [u]∈MA(γp,γ ;H,{fγ},J) ǭ(u)eAγ , p ∈ Crit(fγ). Symplectic homology for autonomous Hamiltonians 21 The sums (39) and (40) clearly involve only periodic orbits in the same free homotopy class as that of p̃ or γp respectively. Remark 3.8. Since H is C2-small, the moduli spaces MA(p̃, q̃;Hδ, J), p̃, q̃ ∈ Crit(H) of expected dimension ind(p̃;−H)−ind(q̃;−H)+2〈c1(TW ), A)〉−1 = 0 are independent of δ and consist exclusively of gradient trajectories of H in W [17, Theorem 6.1](see also [25, Theorem 7.3]). As a consequence, these moduli spaces are empty whenever A 6= 0. We have, following directly from the definitions, an obvious isomorphism of free Λω-modules SCa∗ (Hδ) ≃ BCa∗ (H), δ ∈]0, δ1[. It follows now from Theorem 3.7 and the definition (36) of signs in the Morse- Bott complex that the corresponding differentials, defined by (14) and (39-40), also coincide. Here we use the fact that the Hamiltonian action functional decreases along Floer trajectories, hence the differential (14) applied to elements p̃ ∈ Crit(H) does not involve nonconstant elements of P(Hδ) and reduces to (39) by Remark 3.8. As a consequence, we have H∗(BC ∗ (H), ∂) = SH ∗ (Hδ, J). We shall construct in Section 4.4 a system of coherent orientations on the Morse-Bott moduli spaces MA(Sγ , Sγ ;H, J), MA(Sγ , q̃;H, J) whenever γ, γ ∈ P≤αλ are good orbits. This in turn determines signs ǫ(u) via an orientation rule for fiber products (see (87)). Proposition 3.9. Assume dim MA(p, q;H, {fγ}, J) = 0. The bijective corre- spondence between elements [u] ∈ MAm(p, q;H, {fγ}, J), m ≥ 1 and elements of [uδ] ∈ MA(γp, γq;Hδ, J) given by Theorem 3.7 changes the signs by the rule ǫ(u) = (−1)m−1ǫ(uδ). Moreover, if m = 0 then u = uδ and ǫ(u) = ǫ(uδ), p is a minimum and q is a maximum, the moduli space MA0 (p, q;H, {fγ}, J) consists of the two gra- dient lines running from p to q and their signs are different if and only if the underlying orbit is good. In view of (36) and (39–40), this identification of signs between ǫ(u) and ǭ(u) allows to define the Morse-Bott differential exclusively in terms of Morse-Bott data. 4 Morse-Bott moduli spaces The structure of this section is as follows. We give in §4.1 the proof of Proposi- tion 3.5, whereas Theorem 3.7 is proved in §4.2–§4.3, which treat compactness and gluing and correspond to assertions (i-ii) and (iii) respectively. Finally §4.4 contains a full discussion of orientation issues and the proof of Proposition 3.9. Symplectic homology for autonomous Hamiltonians 22 4.1 Transversality Proof of Proposition 3.5. We first prove (ii). Let J ℓ ⊂ J be the space of admissible almost complex structures of class Cℓ, ℓ ≥ 1, and let J ℓ(H) ⊂ J ℓ be the set of almost complex structures J which, outside a fixed neighbourhood of the nonconstant periodic orbits of XH , are independent of θ and satisfy Jξ = ξ, = Rλ. By a standard trick of Taubes [15, Theorem 5.1] it is enough to show that there exists an open and dense set J ℓreg(H) ⊂ J ℓ(H) consisting of regular elements. We define the universal moduli spaces MA(Sγ , Sγ ;H,J ℓ(H)) = {(u, J) | J ∈ J ℓ(H), u ∈ MA(Sγ , Sγ ;H, J)} MA(Sγ , q̃;H,J ℓ(H)) = {(u, J) | J ∈ J ℓ(H), u ∈ MA(Sγ , q̃;H, J)}. The main point is to show that these universal moduli spaces are Banach man- ifolds. Then the sets J ℓreg(H) consist of the regular values of the natural pro- jections from the universal moduli spaces to J ℓ(H). We only treat the case of MA(Sγ , Sγ ;H,J ℓ(H)) since the second case is entirely similar, and we assume without loss of generality that γ 6= γ. This universal moduli space is the zero set of a distinguished section of a Banach vector bundle E → BA×J ℓ(H) which we now define. Let p > 2 and d > 0. Let BA = B1,p,d(Sγ , Sγ , A;H) be the space of proper maps u : R× S1 → Ŵ which are locally in W 1,p and satisfy (i) the map u converges uniformly in θ as s→ ±∞ to γ(·+ θ0), respectively γ(· + θ0), for some θ0, θ0 ∈ S1, and represents the homology class A ∈ H2(W ;Z); (ii) there exist tubular neighbourhoods U and U of γ and γ respectively, to- gether with parametrizations Ψ : U → S1×R2n−1 and Ψ : U → S1×R2n−1 such that Ψ ◦ γ(θ) = {θ} × {0}, Ψ ◦ γ(θ) = {θ} × {0}, Ψ ◦ γ(θ + θ0)−Ψ ◦ u(s, θ) ∈ W 1,p(]−∞,−s0], ed|s|ds dθ), Ψ ◦ γ(θ + θ0)−Ψ ◦ u(s, θ) ∈ W 1,p([s0,∞[, ed|s|ds dθ), for some s0 > 0 sufficiently large. Then BA is a Banach manifold and, for d/p strictly smaller than the constant r in Proposition A.1, it contains the moduli spaces MA(Sγ , Sγ ;H, J) for all J ∈ J ℓ. Let E → BA ×J ℓ(H) be the Banach vector bundle with fiber E(u,J) = Lp(R×S1, u∗TŴ ; ed|s|ds dθ). Let ∂̄H : BA×J ℓ(H) → E be the section defined ∂̄H(u, J) := ∂su+ Jθ(∂θu−XH). (41) Symplectic homology for autonomous Hamiltonians 23 Then MA(Sγ , Sγ ;H,J ℓ(H)) = ∂̄−1H (0) and it remains to show that ∂̄H is transverse to the zero section. This means that the vertical differential D∂̄H(u, J) : TuBA × TJJ ℓ(H) → E(u,J) is surjective for all (u, J) ∈ ∂̄−1H (0). We have TuBA =W 1,p(R× S1, u∗TŴ ; ed|s|ds dθ)⊕ V ⊕ V , where V , V are the one-dimensional real vector spaces generated by two sections of u∗TŴ of the form (1 − β(s, θ))XH(γ(θ)) and β(s, θ)XH(γ(θ)) respectively, with β(s, θ) = β(s) a smooth cutoff function which vanishes for s ≤ 0 and is equal to 1 for s ≥ 1. The space TJJ ℓ(H) consists of matrix valued functions Y : S1 → End(TŴ ) of class Cℓ satisfying the conditions JθYθ + YθJθ = 0, ω̂(Yθv, w) + ω̂(v, Yθw) = 0, ∀v, w ∈ TŴ , (42) and such that, outside fixed neighbourhoods of the nonconstant periodic orbits of XH , they are independent of θ and have the form with respect to the splitting ξ ⊕ Span(Rλ, ∂∂t ). The operator D∂̄H(u, J) can be written D∂̄H(u, J) · (ζ, Y ) = Duζ + Yθ(u)(∂θu−XH(u)). Du : W 1,p(R× S1, u∗TŴ ; ed|s|ds dθ)⊕ V ⊕ V → Lp(R× S1, u∗TŴ ; ed|s|ds dθ) is the linearization of the Cauchy-Riemann operator associated to the pair (H, J) and is explicitly given by formula (12). It is proved in [5, Proposition 4] that Du is a Fredholm operator. It is at this point that the exponential weight plays a crucial role, due to the degeneracy of the asymptotic orbits. As a consequence the range of D∂̄H(u, J) is closed and we are left to prove that it is also dense. Let q > 1 be such that 1/p + 1/q = 1. We show that every η ∈ Lq(R× S1, u∗TŴ ; ed|s|ds dθ) satisfying 〈η,Duζ〉ed|s|ds dθ = 0, 〈η, Yθ(u)(∂θu−XH(u))〉ed|s|ds dθ = 0 for all ζ and Y vanishes. The first equation implies, by elliptic regularity, that η is of class Cℓ and has the unique continuation property. Assume by contradiction that η does not vanish. Then the set {(s, θ) : η(s, θ) 6= 0} is open and dense. On the other hand, it is proved in [15, Theorem 4.3] that the set R(u) := {(s, θ) : ∂su(s, θ) 6= 0, u(s, θ) 6= γ(θ), γ(θ), u(s, θ) /∈ u(R \ {s}, θ)} of regular points of u is open and dense (although nondegeneracy of the asymp- totic orbits is a standing assumption in [15], it does not play any role in the Symplectic homology for autonomous Hamiltonians 24 proof of this result). Let z0 = (s0, θ0) be a point in R(u) with η(z0) 6= 0 and u(z0) belonging to the fixed open neighbourhood of γ (such a point exists since we have assumed γ 6= γ). One can choose a matrix Yθ0(u(z0)) satisfying (42) such that 〈η(z0), Yθ0(u(z0))J(u(z0))∂su(z0)〉 6= 0. Because z0 is a regular point we can choose a time-dependent cutoff function ρ : S1 × Ŵ → [0, 1] supported near (θ0, u(z0)) such that Y := ρYθ0(u(z0)) satisfies ∫ 〈η, Yθ(u)(∂θu−XH(u))〉ed|s|ds dθ 6= 0. This contradicts (43) and shows thatD∂̄H(u, J) is surjective, hence the universal moduli space MA(Sγ , Sγ ;H,J ℓ(H)) is a Banach manifold as claimed. We now prove (i). The set J ′(H) is obviously open. The fact that it is nonempty can be seen as follows. The space S1 ×R2 admits the “skating ring” contact form α = sin θdx− cos θdy, (θ, x, y) ∈ S1×R2 for which ∂ ∈ ξ = kerα. If J denotes the almost complex structure on ξ satisfying J ∂ = cos θ ∂ sin θ ∂ , then [ ∂ , J ∂ ] 6= 0 and [ ∂ , J ∂ ] /∈ ξ = 〈 ∂ , J ∂ 〉. This simple model can be adapted to our situation as follows. We can symplectically trivialize a neighbourhood of the simple orbit γ as S1 × R2n−1 ∋ (θ, t, q2, p2, . . . , qn, pn) with the standard symplectic form dθ ∧ dt + dq2 ∧ dp2 + . . . + dqn ∧ dpn, so that XH corresponds to . Let J be a compatible almost complex structure such that J ∂ + cos θ ∂ + sin θ ∂ . Since ∂ and ∂ commute we have [XH , JXH ] = [ , cos θ ∂ + sin θ ∂ ] 6= 0 and [XH , JXH ] /∈ 〈XH , JXH〉, so that J ∈ J ′(H). Let J ′ℓ ⊂ J ′ be the space of admissible almost complex structures of class Cℓ, ℓ ≥ 1 which are independent of θ ∈ S1, and let J ′ℓ(H) ⊂ J ′ℓ be the space of almost complex structures J which, outside a fixed neighbourhood of the nonconstant periodic orbits of XH , satisfy Jξ = ξ, J = Rλ. It is enough to show that there exists an open and dense set J ′ℓreg(H) ⊂ J ′ℓ(H) consisting of elements which are regular for Floer trajectories with one nontrivial simple asymptote. We have J ′ℓ ⊂ J ℓ and the main point is to show that the corresponding universal moduli spaces MA(Sγ , Sγ ;H,J ′ℓ(H)) ⊂ MA(Sγ , Sγ ;H,J ℓ(H)) and MA(Sγ , q̃;H,J ′ℓ(H)) ⊂ MA(Sγ , q̃;H,J ℓ(H)) are Banach manifolds. We again treat only MA(Sγ , Sγ ;H,J ′ℓ(H)) and assume without loss of generality that γ is a simple orbit and γ 6= γ. This universal moduli space is the zero set of the section of the restricted bundle E → BA × J ′ℓ(H) defined by (41), and we have to show that the vertical differential D∂̄H(u, J) : TuBA × TJJ ′ℓ(H) → E(u,J) is surjective. Arguing by contradiction, we get an element η ∈ Lq(R × S1, u∗TŴ ; ed|s|ds dθ) of class Cℓ which does not vanish on an open and dense subset of R×S1 and satisfies 〈η, Y (u)(∂θu−XH(u))〉ed|s|ds dθ = 0 for any Y ∈ TJJ ′ℓ(H). The main difference with respect to (ii) is that TJJ ′ℓ(H) ⊂ TJJ ℓ(H) consists of elements Y ∈ End(TŴ ) which are independent of θ. Symplectic homology for autonomous Hamiltonians 25 I(u) := (s, θ) : ∂su(s, θ) 6= 0, u−1(u(s, θ)) = {(s, θ)} be the set of injective points, and denote IR(u) := I(u)∩ ]R,∞[×S1, R > 0. The main observation is that our special choice of J ∈ J ′(H) implies that IR(u) is open and dense in ]R,∞[×S1 for R large enough. This is proved exactly as in [15, §7], and the main steps are the following. Since γ is simple, every u as above is simple, i.e. for every integer m > 1 there exists a point (s, θ) ∈ R × S1 = R × R/Z such that u(s, θ + 1 ) 6= u(s, θ). Let U be a neighbourhood of γ in which [XH , JXH ] 6= 0 and [XH , JXH ] /∈ 〈XH , JXH〉. We call a point (s, θ) regular if ∂su, ∂θu, XH(u), JXH(u) are linearly independent at (s, θ), and we denote by R(u) the set of regular points. Then [15, Lemma 7.6] holds and [15, Lemma 7.7] shows that the set {(s, θ) ∈ R(u) : u(s, θ) ∈ U} is open and dense in u−1(U). Note that we crucially use here our hypothesis J ∈ J ′(H), which plays the role of the hypothesis J ∈ Jad(M,ω,X) in [15]. Finally [15, Lemma 7.8] shows that the set of points which are regular and injective is open and dense in u−1(U), and in particular IR(u) is open and dense in ]R,∞[×S1 for R large enough. We can then choose a point z0 = (s0, θ0) ∈ IR(u) such that η(z0) 6= 0 and a matrix Y (u(z0)) satisfying (42) and 〈η(z0), Y (u(z0))J(u(z0))∂su(z0)〉 6= 0. Since z0 is an injective point we can choose a cutoff function ρ : Ŵ → R supported near u(z0) such that Y := ρY (u(z0)) satisfies 〈η, Y (u)(∂θu − XH(u))〉ed|s|ds dθ 6= 0. This contradiction shows that D∂̄H(u, J) is surjective and therefore MA(Sγ , Sγ ;H,J ′ℓ(H)) is a Banach manifold as claimed. The dimension of the moduli space MA(Sγ , Sγ ;H, J), J ∈ Jreg(H) is equal to ind(Du) − 1. The restriction of the operator Du to the subspace W 1,p(R × S1, u∗TŴ ; ed|s|ds dθ) is conjugated to a Cauchy-Riemann operator Du :W 1,p(R× S1, u∗TŴ ; ds dθ) → Lp(R× S1, u∗TŴ ; ds dθ) via multiplication by e |s|. If the asymptotics of Du were nondegenerate, the Fredholm index of Du would be given by [26] µRS(γ)− µRS(γ) + 2〈c1(TW ), A〉. Due to the one-dimensional degeneracy of γ and γ, the actual index of Du is obtained by a calculation analogous to [5, Proposition 4] (see also Lemma 3.4) : (µRS(γ)− )− (µRS(γ) + ) + 2〈c1(TW ), A〉. (44) We have proved in Lemma 3.4 that µRS(γ) = µ(γ) + , hence ind(Du) = ind(Du) + 2 = µ(γ)− µ(γ) + 2〈c1(TW ), A〉+ 1. Finally note that the evaluation maps ev, ev are well-defined and smooth on BA. Hence their restrictions to the moduli spaces are smooth as well. Symplectic homology for autonomous Hamiltonians 26 4.2 Compactness for Morse-Bott trajectories Definition 4.1. Let H, {fγ} and J be fixed as above, and let p ∈ Crit(fγ), q ∈ Crit(fγ). The space M̂A(p, q;H, {fγ}, J) of parametrized Morse-Bott broken trajectories consists of tuples u = (cm, um, cm−1, um−1, . . . , u1, c0) such that (i) ui ∈ M̂Ai(Sγi , Sγi−1 ;H, J), i = 1, . . . ,m with γm = γ, γ0 = γ and A1 + . . .+Am = A; (ii) c0 : [−1,+∞[→ Sγ0 , ci : [−Ti/2, Ti/2] → Sγi , i = 1, . . . ,m − 1 and cm :]−∞, 1] → Sγm satisfy ċi = ∇fγi ◦ ci, i = 0, . . . ,m; (iii) ev(ui) = ev(ci), ev(ui) = ev(ci−1), i = 1, . . . ,m and c0(+∞) = q, cm(−∞) = p. The space MA(p, q;H, {fγ}, J) of unparametrized Morse-Bott broken tra- jectories consists of equivalence classes [u] = (cm, [um], cm−1, [um−1], . . . , [u1], c0) such that u ∈ M̂A(p, q;H, {fγ}, J). Definition 4.2. Let uk = (cmk,k, umk,k, cmk−1,k, . . . , u1,k, c0,k) ∈ M̂A(pk, qk;H, {fγ}, J) with k = 1, . . . , ℓ, and satisfying qk = pk−1 for k = 2, . . . , ℓ. We denote p := pℓ, q := q1. A sequence vn ∈ M̂A(γp, γq;Hδn , J) with δn → 0, n → ∞ is said to converge to u := (uℓ, . . . ,u1) if there exist shifts (s i,k) ∈ R, i = 1, . . . ,mk such vn(·+ sni,k , ·) → ui,k, n→ ∞ uniformly on compact sets in R× S1. We write in this case vn → u. A sequence [ṽn] ∈ MA(γp, γq;Hδn , J) with δn → 0, n → ∞ is said to converge to [u] ∈ MA(p, q;H, {fγ}, J) if there exist representatives vn and v such that vn → v (this condition is obviously independent on the choice of representatives). We write in this case [ṽn] → [u]. We call u a broken Floer trajectory with gradient fragments. We call each of the uk’s a Floer trajectory with gradient fragments. Each uk is a level of u and each ui,k is a sublevel of uk. Definition 4.3. An element u = (cm, um, cm−1, . . . , u1, c0) ∈ M̂A(p, q;H, {fγ}, J) with m ≥ 1 is stable if each ui, i = 1, . . . ,m is a nonconstant Floer trajectory and if each ci, i = 1, . . . ,m − 1 defined on an interval of nonzero length is Symplectic homology for autonomous Hamiltonians 27 nonconstant. An element u = (c0) ∈ MA(p, q;H, {fγ}, J) is stable if p 6= q. A broken Floer trajectory with gradient fragments u = (uℓ, . . . ,u1) is stable if each uk, k = 1, . . . , ℓ is stable. Remark 4.4. A convergent sequence vn of nonconstant Floer trajectories has a stable limit u which is unique up to shifts on the ci,k and ui,k. The proofs of the next two lemmas use the asymptotic estimates proved in the Appendix. The relevant notation is introduced at the beginning of the Appendix, and we briefly recall it here for the reader’s convenience. For each γ ∈ P(H) we choose coordinates (ϑ, z) ∈ S1 × R2n−1 parametrizing a tubular neighbourhood of γ, such that ϑ ◦ γ(θ) = θ and z ◦ γ(θ) = 0. Given a smooth function fγ : Sγ → R, we denote by ϕ s the gradient flow of fγ with respect to the natural metric on S1. In a neighbourhood of γ ∈ P(H) the Floer equation ∂su+ J∂θu− JXH = 0 becomes ∂sZ+J∂θZ+J −JXH = 0, where Z(s, θ) := (ϑ◦u(s, θ)−θ, z◦u(s, θ)). Since XH = on {z = 0} this can be rewritten as ∂sZ + J∂θZ + Sz = 0 for some matrix-valued function S = S(ϑ, z). The matrix S∞(θ) := S(θ, 0) is symmetric. Let A∞ : H k(S1,R2n) → Hk−1(S1,R2n) be the operator defined by A∞Z := J Z + S∞(θ)z. The kernel of A∞ has dimension one and is spanned by the constant vector e1 := (1, 0, . . . , 0). We denote by Q∞ the orthogonal projection onto (ker A∞) ⊥ and we set P∞ := 1l−Q∞. Lemma 4.5. Let vn ∈ M̂A(γp, γq;Hδn , J) with δn → 0, n→ ∞ and s 1 < s shifts such that vn(·+ sn1 , ·) → u1, vn(·+ sn2 , ·) → u2 uniformly on compact sets, with u1 ∈ M̂A1(Sγ1 , Sγ ;H, J) and u2 ∈ M̂A2(Sγ , Sγ2 ;H, J). Any two sequences of shifts sn1 < s + < s − < s 2 satisfying s + − sn1 → ∞, sn2 − sn− → ∞ and + − sn1 ) → 0, δn(sn2 − sn−) → 0, (45) have the property that vn(·+ sn+, ·) → ev(u1), vn(·+ sn−, ·) → ev(u2) uniformly on compact sets. Proof. We claim that there exists K > 0 such that vn([s 1 +K, s 2 −K] × S1) is contained in a given small neighbourhood of Sγ . If that was not the case, we could find a sequenceKn → ∞ and a sequence (sn, θn) ∈ [sn1 +Kn, sn2 −Kn]×S1 such that dist(vn(sn, θn), Sγ) is bounded away from zero. Up to a subsequence, vn(· + sn, ·) converges to some Floer trajectory v which must be nonconstant. On the other hand, for any s ∈ R and for any K > 0 we have, for n large enough, vn(s+ s 2 −K, ·) ≤AHδn vn(s+ sn, ·) ≤AHδn vn(s+ s 1 +K, ·) and in the limit AH(u2(s − K, ·)) ≤ AH(v(s, ·)) ≤ AH(u1(s + K, ·)). We let K go to infinity and obtain AH(γ) ≤ AH(v(s, ·)) ≤ AH(γ). This holds for all Symplectic homology for autonomous Hamiltonians 28 s ∈ R and therefore the cylinder v is constant over some element of P(H), a contradiction which proves the claim. By (98) in the proof of Proposition A.3 applied to vn on [s 1 + K, s 2 −K] we get |Q∞vn(s, θ)| ≤ Cmax(‖Q∞vn(sn1 +K)‖, ‖Q∞vn(sn2 −K)‖). (46) Let γ+ be the limit in Sγ of vn(s +, ·), and let In(ǫ) := [sn+(ǫ), sn+] ⊂ [sn1 + K, sn+] be the maximal subinterval containing s + such that P∞vn(s), s ∈ In(ǫ) is at distance at least ǫ from the critical points of fγ , except maybe γ+ (if the latter is a critical point). By the second part of Proposition A.3 applied to vn on In(ǫ) we obtain |ϑ ◦ vn(s, θ)− θ−ϕ δs (θ0)| ≤ Cmax(‖Q∞vn(s +(ǫ))‖, ‖Q∞vn(sn+)‖)eMδn(s Since δn(s + − sn1 ) → 0 and taking into account (46) we get |ϑ ◦ vn(s, θ)− ϑ ◦ γ+(θ)| ≤ C1 max(‖Q∞vn(sn1 +K)‖, ‖Q∞vn(sn2 −K)‖). (47) For K large enough the right hand term becomes so small that the distance between P∞vn(s), s ∈ In(ǫ) and the critical points of fγ , except possibly γ+, is strictly bigger than ǫ, hence In(ǫ) = [s 1 +K, s +] by maximality (this holds for K large enough). Applying (47) to s = sn1 +K we obtain |ϑ ◦ vn(sn1 +K, θ)− ϑ ◦ γ+(θ)| ≤ C1 max(‖Q∞vn(sn1 +K)‖, ‖Q∞vn(sn2 −K)‖). Passing to the limit in the above inequality we obtain |ϑ ◦ u1(K, θ)− ϑ ◦ γ+(θ)| ≤ C1 max(‖Q∞u1(K)‖, ‖Q∞u2(−K)‖). Letting K → ∞ we obtain ev(u1) = γ+. That this implies uniform convergence on compact sets to the constant cylinder over ev(u1) can be seen in two ways: either one notices that the above estimates hold uniformly when sn+ is replaced with sn++K+, whereK+ is a bounded constant, or one uses the fact that a Floer trajectory passing through a periodic orbit is necessarily a constant cylinder, by unique continuation applied to the infinite jet at that orbit [21, Theorem 2.3.2]. A similar argument proves the assertion involving ev(u2). Lemma 4.6. Let vn ∈ M̂A(p, q;Hδn , J) with δn → 0, n→ ∞. Assume we are given two sequences of shifts sn1 < s 2 such that vn(· + sni , ·), i = 1, 2 converge uniformly on compact sets to constant cylinders uγi over orbits γi belonging to the same family Sγ . Then there exists a (possibly broken) gradient trajectory of fγ starting at γ1 and ending at γ2. Moreover, the length of the gradient trajectory is T = limn→∞ δn(s 2 − sn1 ). Proof. We claim that, for n large enough, vn([s 1 , s 2 ]×S1) is entirely contained in an arbitrarily small neighbourhood of γ. By contradiction, if this fails we can reparametrize the sequence vn so that it converges to a nonconstant Floer Symplectic homology for autonomous Hamiltonians 29 trajectory v ∈ MB(γ′, γ′;H, J) for some class B ∈ H2(M ;Z) and some γ′, γ′ ∈ P(H) such that their actions satisfy AH(γ) ≤ AH(γ′) < AH(γ′) ≤ AH(γ), which is impossible. We first assume that γ1 is not a critical point of fγ . Let ǫ > 0 be fixed and denote by In(ǫ) = [s 1 , s 2 (ǫ)] ⊂ [sn1 , sn2 ] the maximal subinterval containing sn1 such that the distance between P∞vn(s), s ∈ In(ǫ) and Crit(fγ) is at least ǫ. We can apply Proposition A.3 to vn and In(ǫ). In particular, for some sequence θn ∈ S1 we have (s,θ)∈In(ǫ)×S1 |ϑ ◦ vn(s, θ)− θ − ϕδnfγs (θn)| = 0. Since vn(s 1 , ·) converges to γ1, we also have (s,θ)∈In(ǫ)×S1 |ϑ ◦ vn(s, θ)− θ − ϕ δn(s−s (γ1)| = 0. (48) Modulo passing to a subsequence we know that vn(s 2 (ǫ), ·) converges, which together with (48) implies that δn(s 2 (ǫ)−sn1 ) converges to T (ǫ) ∈ R+. This holds for each ǫ > 0 and, since sn2 (ǫ) < s ′) if ǫ > ǫ′, the limit limǫ→0 T (ǫ) = T ∈ R+ exists. Then ϕ s (γ1), s ∈ [0, T ] is a gradient trajectory starting at γ1. If T is finite then this trajectory, and therefore vn(In(ǫ)×S1) stay at a fixed distance from Crit(fγ) for n large enough. Hence In(ǫ) = In for ǫ sufficiently small and we are done. If T is infinite and the limit lims→∞ ϕ s (γ1) is equal to γ2, we are also done. Otherwise we are in the next case, with shifts s̃ limǫ→0 s 2 (ǫ) and s̃ 2 := s We now assume that γ1 is a critical point of fγ and γ1 6= γ2. Given ǫ > 0 we denote by In(ǫ) = [s 1 , s 2 (ǫ)] ⊂ [sn1 , sn2 ] the maximal subinterval containing sn1 such that the distance between P∞vn(s), s ∈ In(ǫ) and Crit(fγ)\{γ1} is at least ǫ. For ǫ > 0 small enough the loops P∞vn(s 2 (ǫ)) are at a distance bigger than ǫ from γ1 and, up to a subsequence, vn(s 2 (ǫ), ·) converges to some γ̃2 ∈ Sγ which is not a critical point of fγ . The same argument as in the previous case applied “backwards” to the shifts sn1 < s 2 (ǫ) produces a negative gradient trajectory running from γ̃2 to some critical point γ̃1. By definition of In(ǫ), we must have γ̃1 = γ1 and we thus obtain a gradient trajectory from γ1 to γ̃2. We are now in the first case with shifts s̃n1 := s 2 (ǫ) and s̃ 2 := s We successively apply the above two cases in order to produce a broken gra- dient trajectory from γ1 to γ2. This is a finite process since a broken trajectory has a finite number of nonconstant fragments. Proposition 4.7. Let vn ∈ M̂A(p, q;Hδn , J) with δn → 0, n → ∞. There exists a broken Floer trajectory with gradient fragments u and a subsequence (still denoted by vn) such that vn → u. Proof. The energy E(vn) := EJ,Hδn (vn) defined in (11) satisfies E(vn) = − Hδn(θ, γp(θ)) dθ + Hδn(θ, γq(θ)) dθ. Symplectic homology for autonomous Hamiltonians 30 Since Hδn → H we infer that E(vn) is uniformly bounded. Floer’s compactness theorem [12, Proposition 3c] applies to our situation and provides a collection of Floer trajectories ui, i = 1, . . . ,m for the pair (H, J) together with holomorphic spheres attached to them, as well as shifts (sni ) such that vn(·+ sni , ·) converges to ui and its associated holomorphic spheres in the sense of nodal curves. Condition (1) implies symplectic asphericity 〈ω, π2(Ŵ )〉 = 0, therefore holomorphic spheres in (Ŵ , J) are constant and the shifted vn converge to ui uniformly on compact sets. Because the action spectrum of ∂W was assumed to be discrete and injective the trajectories ui connect with each other, in the sense that ev(ui) and ev(ui+1) belong to the same family of trajectories Sγi , i = 1, . . . ,m−1. Moreover, ev(um) belongs to Sγ and ev(u1) belongs to Sγ . By Lemma 4.5 there exist shifts sni,± such that vn(· + sni,+, ·) converges to the constant cylinder over ev(ui), and vn(· + sni,−, ·) converges to the constant cylinder over ev(ui). Applying Lemma 4.6 with shifts s i,+ < s i−1,−, i = 2, . . . ,m and n large enough, we obtain broken gradient trajectories ci−1 starting at ev(ui) and ending at ev(ui−1). Let now s + be shifts such that vn(·+sn−, ·) → p and vn(· + sn+, ·) → q. Applying Lemma 4.6 with shifts sn− < snm,− and with shifts sn1,+ < s + we obtain broken gradient trajectories cm starting at p and ending at ev(um) and c0 starting at ev(u1) and ending at q. Since all Sγ are circles of periodic orbits, the broken gradient trajectories ci, i = 0, . . . ,m consist each of a single fragment. The construction of a stable broken Floer trajectory with gradient fragments out of the data ci, ui is straightforward and goes as follows. The collection of points of the form ev(ui+1), ev(ui) which are critical points of fγi determine a partition (cmℓ,ℓ, umℓ,ℓ, cmℓ−1,ℓ, . . . , c1,ℓ, u1,ℓ, c0,ℓ), . . . , (cm1,1, um1,1, . . . , u1,1, c0,1) of the ordered tuple (cm, um, . . . , c1, u1, c0). Note that the cmk,k and c0,k may either be missing or be constant and exactly one of c0,k and cmk−1,k−1 is missing. In such a situation we set cmk,k or c0,k to be a constant trajectory at the relevant critical point, defined on a semi-infinite interval. 4.3 Gluing for Morse-Bott moduli spaces We prove in this subsection the assertions (i-ii) of Theorem 3.7. The following notation was introduced in the previous subsection. For γ ∈ P(H) we choose coordinates (ϑ, z) ∈ S1 × R2n−1 parametrizing a tubular neighbourhood of γ, such that ϑ ◦ γ(θ) = θ and z ◦ γ(θ) = 0. Given a smooth function fγ : Sγ → R, we denote by ϕ s the gradient flow of fγ with respect to the natural metric on S1. The orthogonal projection onto the 1-dimensional kernel of the asymptotic operator at γ ∈ P(H) is denoted by P∞, and we denote Q∞ := 1l− P∞. Let p > 2, d > 0 and δ > 0. Let BAδ = B 1,p,d δ (γp, γq, A;H, {fγ}) be the space of proper maps u : R× S1 → Ŵ which are locally in W 1,p and satisfy Symplectic homology for autonomous Hamiltonians 31 (i) the map u converges uniformly in θ as s → ±∞ to γ , respectively γp, and represents the homology class A ∈ H2(W ;Z); (ii) there exist tubular neighbourhoods U and U of γ and γ respectively, parametrized by (ϑ, z) ∈ S1 × R2n−1 such that ϑ ◦ u(s, θ)− θ − ϕδfγs (θ0) ∈ W 1,p(]−∞,−s0]× S1,R; ed|s|ds dθ), z ◦ u(s, θ) ∈ W 1,p(]−∞,−s0]× S1,R2n−1; ed|s|ds dθ), ϑ ◦ u(s, θ)− θ − ϕδfγs (θ0) ∈ W 1,p([s0,∞[×S1,R; ed|s|ds dθ), z ◦ u(s, θ) ∈ W 1,p([s0,∞[×S1,R2n−1; ed|s|ds dθ), for some s0 > 0 sufficiently large and some θ0, θ0 ∈ S1 satisfying ϕfγs (θ0) = p, lim ϕfγs (θ0) = q. (49) Then BAδ is a Banach manifold and, for d > 0 sufficiently small, it contains the moduli spaces MA(γp, γq;Hδ, J) for all J ∈ J (see Proposition A.2 in the Appendix). Let E → BAδ be the Banach vector bundle with fiber E(u,J) = Lp(R× S1, u∗TŴ ; ed|s|ds dθ). Let ∂̄Hδ,J : BAδ → E be the section defined by ∂̄Hδ,J(u) := ∂su+ Jθ(∂θu−XHδ ). Then MA(γp, γq;Hδ, J) = ∂̄ (0). From now on we fix J ∈ Jreg(H). In order to prove (i) in Theorem 3.7 we need to show that the vertical differential Du : TuBAδ → Eu defined by (12) is surjective for all u ∈ MA(γp, γq;Hδ, J) when δ > 0 is sufficiently small and the expected dimension of the moduli space is zero. We have TuBAδ =W 1,p(R× S1, u∗TŴ ; ed|s|ds dθ)⊕ V u ⊕ V u, where V u, V u are real vector spaces of dimension dim V u = ind(p), dim V u = 1− ind(q). When their dimension is nonzero V u and V u are respectively generated by two sections of u∗TŴ of the form (1− β(s, θ))∇fγ(ϑ ◦ u(s, 0)) and β(s, θ)∇fγ(ϑ ◦ u(s, 0)), with β(s, θ) = β(s) a smooth cutoff function which vanishes for s ≤ 0 and is equal to 1 for s ≥ 1. The fact that V u and V u have varying dimensions is a consequence of condition (49). We shall prove surjectivity of Du by showing that the elements of the moduli spaceMA(γp, γq;Hδ, J) can be approximated, for δ > 0 small enough, by gluing the elements of MA(Sγ , Sγ ;H, J) with fragments of gradient trajectories of the Morse functions fγ . Symplectic homology for autonomous Hamiltonians 32 Given a, b ∈ R, a < b we define intervals I(a, b) = [a, b] if a, b ∈ R, ]−∞, b] if a = −∞, b ∈ R, [a,∞[ if a ∈ R, b = ∞. For b − a > 4 and |ǫ| < 1, we let ha,b,ǫ : R → I(a, b) ⊂ R be a collection of smooth increasing functions such that ha,b,ǫ(s) = a if s ≤ a − ǫ/2, ha,b,ǫ(s) = b if s ≥ b + ǫ/2 and ha,b,ǫ(s) := s if a − ǫ/2 + 1 < s < b + ǫ/2 − 1. We can of course make the family {ha,b,ǫ} depend smoothly on a, b and ǫ. We define ka,b,ǫ(s) := |σ=0h′a−σ,b+σ,ǫ(s). The support of ka,b,ǫ is contained in [a − ǫ/2, a− ǫ/2 + 1] ∪ [b + ǫ/2 − 1, b + ǫ/2]. We may assume without loss of generality that h′a,b,ǫ and ka,b,ǫ are uniformly bounded. Convention. If ǫ = 0 we shall omit it from all subsequent decorations, and we set ǫ = 0 if a = −∞ or b = +∞. a, b ∈ R a ∈ R, b = +∞ ka,b,ǫ ha,b,ǫ h′a,b,ǫ ka,b,ǫ ha,b,ǫ h′a,b,ǫ Figure 2: The reparametrization function ha,b,ǫ and its derivatives. Let γ ∈ Pλ and c : I(a, b) → Sγ ⊂ Ŵ be a fragment of gradient trajectory for the function fγ , i.e. ċ = ∇fγ ◦ c. We define the corresponding gradient cylinder uδ,γ,a,b,ǫ : R× S1 → Sγ ⊂ Ŵ by the equation ϑ ◦ uδ,γ,a,b,ǫ(s, θ) = ϑ ◦ c(δh a (s)) + θ. (50) Then lim ϑ ◦ uδ,γ,a,b,ǫ(s, θ) = ϑ ◦ c(a) + θ and lim ϑ ◦ uδ,γ,a,b,ǫ(s, θ) = ϑ ◦ c(b) + θ. Symplectic homology for autonomous Hamiltonians 33 For γ ∈ Pλ we define Banach manifolds B1,p,dδ (Sγ , Sγ ; fγ), B 1,p,d δ (p, Sγ ; fγ), p ∈ Crit(fγ) and B1,p,dδ (Sγ , q; fγ), q ∈ Crit(fγ) consisting of maps u : R× S1 → Ŵ which are locally of class W 1,p, whose asymptotics are translates of γ, which represent the zero homology class and which satisfy the following asymptotic conditions. (i) for B1,p,dδ (Sγ , Sγ ; fγ): there exists a neighbourhood U of Sγ together with a parametrization (ϑ, z) : U → S1 × R2n−1 such that ϑ ◦ u(s, θ)− θ − θ0 ∈ W 1,p(]−∞,−s0]× S1,R; ed|s|ds dθ), z ◦ u(s, θ) ∈ W 1,p(]−∞,−s0]× S1,R2n−1; ed|s|ds dθ), ϑ ◦ u(s, θ)− θ − θ0 ∈ W 1,p([s0,∞[×S1,R; ed|s|ds dθ), z ◦ u(s, θ) ∈ W 1,p([s0,∞[×S1,R2n−1; ed|s|ds dθ), for some θ0, θ0 ∈ S1 and some s0 > 0. Moreover, there exists T > 0 such T (θ0) = θ0; (51) (ii) for B1,p,dδ (p, Sγ ; fγ): there exists a neighbourhood U of Sγ parametrized by (ϑ, z) ∈ S1 × R2n−1 such that ϑ ◦ u(s, θ)− θ − ϕδfγs (θ0) ∈ W 1,p(]−∞,−s0]× S1,R; ed|s|ds dθ), z ◦ u(s, θ) ∈ W 1,p(]−∞,−s0]× S1,R2n−1; ed|s|ds dθ), ϑ ◦ u(s, θ)− θ − θ0 ∈ W 1,p([s0,∞[×S1,R; ed|s|ds dθ), z ◦ u(s, θ) ∈ W 1,p([s0,∞[×S1,R2n−1; ed|s|ds dθ), for some θ0, θ0 ∈ S1 such that lims→−∞ ϕ s (θ0) = lims→−∞ ϕ s (θ0) = p and some s0 > 0; (iii) for B1,p,dδ (Sγ , q; fγ): there exists a neighbourhood U of Sγ parametrized by (ϑ, z) ∈ S1 × R2n−1 such that ϑ ◦ u(s, θ)− θ − θ0 ∈ W 1,p(]−∞,−s0]× S1,R; ed|s|ds dθ), z ◦ u(s, θ) ∈ W 1,p(]−∞,−s0]× S1,R2n−1; ed|s|ds dθ), ϑ ◦ u(s, θ)− θ − ϕδfγs (θ0) ∈ W 1,p([s0,∞[×S1,R; ed|s|ds dθ), z ◦ u(s, θ) ∈ W 1,p([s0,∞[×S1,R2n−1; ed|s|ds dθ), for some θ0, θ0 ∈ S1 such that lims→∞ ϕ s (θ0) = lims→∞ ϕ s (θ0) = q and some s0 > 0. We will designate one of the above three spaces by B′δ. We define evaluation maps ev and ev on B′δ by ev(u) = lim u(s, ·), ev(u) = lim u(s, ·). Symplectic homology for autonomous Hamiltonians 34 Any map u = uδ,γ,a,b,ǫ belongs to a suitable space B′δ, depending on a, b being finite or not. The tangent space TuB′δ has a natural decomposition TuB′δ =W 1,p,d(R× S1, u∗TŴ )⊕ V u ⊕ V u, (52) where V u are real vector spaces of dimensions 1 if a ∈ R, ind(p) if a = −∞, dim V 1 if b ∈ R, 1− ind(q) if b = +∞. (53) When the dimensions are respectively nonzero the generators of V u are sections given as follows. (i) for B1,p,dδ (Sγ , Sγ ; fγ) the sections are (1− β(s, θ))XH(γ(θ + θ0)) and β(s, θ)XH(γ(θ + θ0)); (ii) for B1,p,dδ (p, Sγ ; fγ) the sections are (1− β(s, θ))∇fγ(ϑ ◦ u(s, 0)) and β(s, θ)XH(γ(θ + θ0)); (iii) for B1,p,dδ (Sγ , q; fγ) the sections are (1− β(s, θ))XH(γ(θ + θ0)) and β(s, θ)∇fγ(ϑ ◦ u(s, 0)). We recall that β(s, θ) = β(s) is a smooth cutoff function which vanishes for s ≤ 0 and is equal to 1 for s ≥ 1. The norm on TuB′δ is chosen such that the norm of the above generators of V u is equal to 1. Let E → B′δ be the Banach vector bundle with fiber Eu = Lp(R× S1, u∗TŴ ; ed|s|ds dθ). We are interested in the family of sections ∂̄a,b,ǫ := ∂̄H′ a,b,ǫ ,J : B′δ → E , with H ′a,b,ǫ = H + h (s)(Hδ −H) = H + δh′a (s)ρfγ(ℓγϑ− ℓγθ). (54) Here we use the definition (25) of Hδ. This is a three-parameter family in case (i) and a two-parameter family in cases (ii) and (iii). Its main feature is that ∂̄a,b,ǫ(uδ,γ,a,b,ǫ) = 0. We note that neither of the operators ∂̄H,J and ∂̄Hδ,J defines a section B′δ → E if a or b is infinite. The vertical differential Du := D a,b,ǫ u : TuB′δ → Eu of each of the sections ∂̄a,b,ǫ is given by formula (12) and is a Fredholm operator whose index has the following values (see also (44)). Symplectic homology for autonomous Hamiltonians 35 (i) for B1,p,dδ (Sγ , Sγ ; fγ) ind(Du) = (µRS(γ)− )− (µRS(γ) + ) + 2 = 1, (ii) for B1,p,dδ (p, Sγ ; fγ) ind(Du) = (µRS(γ)− )− (µRS(γ) + ) + ind(p) + 1 = ind(p), (iii) for B1,p,dδ (Sγ , q; fγ) ind(Du) = (µRS(γ)− )− (µRS(γ) + ) + 1 + 1− ind(q) = 1− ind(q). In formulas (ii) and (iii) the asymptotics of the operator obtained by conjugation with e |s| do not depend on ind(p), ind(q) because, for δ small, the exponential weight d overrides the contribution of the perturbation Hδ −H . Proposition 4.8. Let u = uδ,γ,a,b,ǫ ∈ B′δ. The operator Du :W 1,p(R× S1, u∗TŴ ; ed|s|)⊕ V ′u ⊕ V ′u → Lp(R× S1, u∗TŴ ; ed|s|dsdθ) is surjective for δ > 0 small enough. Proof. In order to compute Du we choose ∇ to be the Levi-Civita connection corresponding to a (split) metric given by (dλ + dt ∧ λ)(·, J ·). It is a general fact that the operator Du can be written in a unitary trivialization of u ∗TŴ as (Duζ)(s, θ) = ∂sζ + J0∂θζ + S(s, θ)ζ(s, θ), where J0 is the standard complex structure on R 2n and S is asymptotically symmetric as s → ±∞. We can choose the trivialization so that XH and ∂/∂t correspond to constant vectors in R2n. We denote S := lims→−∞ S(s, ·). In this situation the matrix S has the following properties: (i) ‖S(s, θ) − S(ϑ ◦ u(s, θ) − ϑ ◦ c(a))‖ is bounded by a constant multiple of δ. This is because, for s ∈ R, the restriction of u to [s − 1, s+ 1] × S1 is δ-close to the constant cylinder over the orbit u(s, ·) ∈ Sγ . (ii) the action of S(s, θ) on the (constant) vector of R2n corresponding to XH is multiplication by δk(s) := δh′a (s)f ′′γ (ϑ(u(s, θ))− θ), and this expression goes to zero with δ. (iii) the matrix S(s, θ) sends the subspace corresponding to ξ to itself and sends ∂/∂t on a multiple of the form (E + δF (s, θ))∂/∂t, with E > 0 and F a bounded function of (s, θ). This follows from (12), (24) and the fact that ∇∂/∂tRλ = 0 and ∇vRλ ∈ ξ, v ∈ ξ; Symplectic homology for autonomous Hamiltonians 36 (iv) there is a constant C > 0 such that ‖S′(s, θ)‖ ≤ Cδ for all s ∈ R and θ ∈ S1. This follows from (50) due to the presence of the factor δ in front of the reparametrization function h a We characterize now the kernel of Du. We first show that each ζ ∈ ker Du is a multiple of the (constant) vector corresponding to XH , or that its component ζ⊥ on the orthogonal complement vanishes. Let F (s) denote the self-adjoint operator J0∂θ + S(s, θ), so that Du = ∂s + F (s). If ζ ∈ ker Du we have (∂s − F (s))(∂s + F (s))ζ = 0, i.e. ∂2sζ − F (s)2ζ + S′(s)ζ = 0. By taking the scalar product in L2(S1,R2n) with ζ⊥ and using property (ii) for S we get 〈ζ⊥, ∂2sζ⊥〉 − ‖F (s)ζ⊥‖2 + 〈ζ⊥, S′(s)ζ⊥〉 = 0. The Morse-Bott assumption and property (i) guarantee that ‖F (s)ζ⊥‖L2 ≥ c‖ζ⊥‖L2 for some c > 0. We obtain ∂2s‖ζ⊥‖2L2 ≥ 2〈ζ⊥, ∂2sζ⊥〉L2 ≥ 2(c2 − Cδ)‖ζ⊥‖2L2 ≥ c2‖ζ⊥‖2L2 if δ > 0 is sufficiently small. In particular ‖ζ⊥‖2L2 can have no local maximum on R. Since ‖ζ⊥‖2 → 0 as s→ ±∞ we deduce that ζ⊥ ≡ 0. We now show that all elements of ker Du are independent of θ. Let ζ ∈ ker Du. Because ζ ⊥ = 0 we have ∂sζ + J0∂θζ + δk(s)ζ = 0, with ∂sζ + δk(s)ζ and ∂θζ pointwise colinear with XH . Hence ∂sζ + δk(s)ζ = 0 and ∂θζ = 0. This shows that the elements of ker Du also belong to the kernel of the linearized Morse operator ζ 7→ ∂sζ + δh′a (s)f ′′γ (ϑ ◦ c(δh a This is a differential equation on R for which the Cauchy problem has a unique solution. Hence the space of solutions is one-dimensional in C∞(R,R) and, in order to determine the dimension of ker Du, we just have to check whether the solutions belong or not to its domain. If a and b are finite the solutions are constant near ±∞, hence belong to the domain of Du and dim ker Du = 1. If a = −∞ (and b is finite) we distinguish two cases: either p is a maximum, in which case f ′′γ (p) < 0, the solutions are unbounded near −∞ and ker Du = 0, or p is a minimum, in which case f ′′γ (p) > 0, the solutions coincide near −∞ with the elements of V ′u and dim ker Du = 1. Hence dim ker Du = ind(p). A similar argument shows that dim ker Du = 1− ind(q) if b = +∞ (and a is finite). In all cases we have dim ker Du = ind(Du), so that Du is surjective. Symplectic homology for autonomous Hamiltonians 37 Up to a translation, the defining interval I(a, b) of a gradient cylinder can be considered to be [−T/2, T/2], T > 0 in case (i), or ]−∞, 1], [−1,∞[ in cases (ii) and (iii) respectively. We shall thus assume in the sequel that the parameters a, b take the values a = −T/2, b = T/2 for T > 0, or a = −∞, b = 1, or a = −1, b = +∞. We consider a tuple (γ, a, b, ǫ) and the gradient cylinder u := uδ := uδ,γ,a,b,ǫ for δ small enough. Let (sδ) be a family of parameters such that sδ ≤ s∗δ and → 1 as δ → 0, where s∗δ := (T + ǫ)/2δ, if a = −T/2, b = T/2, 1/δ, otherwise. In particular we have sδ → ∞ as δ → 0. Our goal now is to define modified norms ‖ · ‖1,δ and ‖ · ‖δ on the domain and target of the operators Du = Duδ such that they admit uniformly bounded right inverses with respect to δ → 0. Let wδ : R → R+ be the weight function defined by wδ(s) = ed||s|−sδ|, if a and b are finite, ed|s−sδ|, if a = −∞ and b is finite, ed|s+sδ|, if a is finite and b = ∞. 0 sδ−sδ Figure 3: Weight function ||s| − sδ| for a, b finite (logarithmic scale). The new norm ‖ · ‖δ on the target of Du is the Lp-norm with weight wδ, and we emphasize it by writing the target as Lp(R× S1, u∗TŴ ;wδ(s)dsdθ). Let V u,δ, V u,δ be vector spaces of the same dimension as V u, given by (53), and which, when their dimension is nonzero, are spanned by the fol- lowing sections. (i) if a, b are both finite the sections are (1− β(s+ sδ, θ))XH(γ(θ + θ0)) and β(s− sδ, θ)XH(γ(θ + θ0)); Symplectic homology for autonomous Hamiltonians 38 (ii) if a = −∞ and b is finite the sections are (1− β(s− sδ, θ))∇fγ(ϑ ◦ u(s, 0)) and β(s− sδ, θ)XH(γ(θ + θ0)); (iii) if a is finite and b = +∞ the sections are (1− β(s+ sδ, θ))XH(γ(θ + θ0)) and β(s+ sδ, θ)∇fγ(ϑ ◦ u(s, 0)). In case a = −∞, b finite or a finite, b = +∞ we define the new norm ‖ · ‖1,δ on the domain of Du by splitting it as domDu =W 1,p(R× S1, u∗TŴ ;wδ(s)dsdθ) ⊕ V u,δ ⊕ V and setting the norm of the above generators of V u,δ, V u,δ to be equal to 1. In case a, b are finite we split the domain of Du as above and further modify the weighted norm on the W 1,p-space. We recall from the proof of Proposi- tion 4.8 that kerDu is 1-dimensional and is spanned by a section ζδ which is constant for |s| ≥ s∗δ . We normalize ζδ by requiring that its value at 0 be equal to the constant vector corresponding to XH . Let 〈·, ·〉 be the scalar product in L2(S1). For an element ζ ∈ W 1,p(R× S1, u∗TŴ ;wδ(s)dsdθ) we denote κδ := 〈ζ(0, ·), ζδ(0, ·)〉 〈ζδ(0, ·), ζδ(0, ·)〉 We denote χδ(s, θ) := β(s+ sδ)β(−s+ sδ)ζδ(s, θ) and define the norm ‖ · ‖1,δ on W 1,p(R× S1, u∗TŴ ;wδ(s)dsdθ) by ‖ζ‖1,δ := ‖ζ − κδχδ‖1,p,δ + |κδ|. Here ‖ · ‖1,p,δ is the weighted norm on W 1,p(R× S1, u∗TŴ ;wδ(s)dsdθ). Proposition 4.9. Let u = uδ = uδ,γ,a,b,ǫ as above. There exists δ2 ∈]0, δ0] such that the operator Du : (domDu, ‖ · ‖1,δ) → (Lp(R× S1, u∗TŴ ;wδ(s)dsdθ), ‖ · ‖δ) is surjective and has a uniformly bounded right inverse Qu = Quδ for δ ∈]0, δ2]. Proof. We choose a unitary trivialization of u∗TŴ as in the proof of Proposi- tion 4.8, so that XH and ∂/∂t correspond to constant vectors in R 2n, and so that the operator Du takes the form (Duζ)(s, θ) = ∂sζ + J0∂θζ + S(s, θ)ζ(s, θ). Here J0 is the standard complex structure on R 2n and S is asymptotically symmetric as s→ ±∞. The matrix S(s, θ) can be written as S′(s, θ)⊕S′′(s, θ) with respect to the splitting ξ ⊕ 〈∂/∂t,XH〉, so that the operator Du is also Symplectic homology for autonomous Hamiltonians 39 split with respect to the decomposition ξ ⊕ L, where L := 〈∂/∂t,XH〉. It is therefore enough to find uniformly bounded right inverses for each of the surjective operators D′u :W 1,p(R× S1, u∗ξ;wδ(s)dsdθ) → Lp(R× S1, u∗ξ;wδ(s)dsdθ), D′′u :W 1,p(R× S1, u∗L; ‖ · ‖1,δ)⊕ V u,δ ⊕ V ′u,δ → Lp(R× S1, u∗L;wδ(s)dsdθ). Here we use the fact that the norm ‖·‖1,δ coincides with the weightedW 1,p-norm on sections with values in the subbundle u∗ξ. Note that D′u is an isomorphism since it has index 0, whereas ind(D′′u) = ind(Du) is either 0 or 1. We treat D′′u and consider first the case of a semi-infinite gradient trajec- tory. The two possible cases are entirely similar, and we assume without loss of generality that a = −∞, b = 1. Let S′′0 := so that limδ→0 S ′′(s, θ) = S′′0 uniformly in (s, θ). Consider the operator D′′0,δ :W 1,p(R×S1, u∗L;wδ(s)dsdθ)⊕V u,δ⊕V u,δ → Lp(R×S1, u∗L;wδ(s)dsdθ) defined by D′′0,δ := ∂s + J0∂θ + S 0 . As in the proof of Proposition 4.8 one sees that D′′0,δ is surjective, and we claim that it admits a right inverse Q 0,δ that is uniformly bounded with respect to δ. Indeed, let Q′′0 be a right inverse of D′′0 := D 0,δ=1 and consider the shift operators (Tδζ)(s) := ζ(s+ sδ) acting from dom(D′′0,δ) → dom(D′′0 ) and from Lp(wδ(s)dsdθ) → Lp(ed|s|dsdθ). It follows from the definitions of ‖ · ‖1,δ and ‖ · ‖δ that the operators Tδ are isometries, and we have D′′0,δ = T 0Tδ since D 0 is independent of s ∈ R. Hence Q′′0,δ = T 0Tδ is a right inverse for D 0,δ such that ‖Q′′0,δ‖ = ‖Q′′0‖, and the claim is proved. Now, if δ is small enough we have ‖S′′(s, θ) − S′′0 ‖ ≤ 1/2‖Q′′0‖, s ∈ R and therefore ‖D′′u −D′′0,δ‖ ≤ 1/2‖Q′′0‖. This implies that ‖D′′uQ′′0,δ − Id‖ = ‖D′′uQ′′0,δ −D′′0,δQ′′0,δ‖ ≤ Thus D′′uQ 0,δ is invertible and the norm of its inverse is ≤ 2. Finally a right inverse for D′′u is given by Q 0,δ(D −1 and has norm ≤ 2‖Q′′0‖. We now treat the case a = −T/2, b = T/2 for T > 0. Let u := uδ,γ,−(T+ǫ)/2,0, u := uδ,γ,0,(T+ǫ)/2 and := D′′u :W 1,p(R× S1, u∗L; ed|s|dsdθ) ⊕ V ′u ⊕ V ′u → Lp(ed|s|dsdθ), D′′ := D′′u :W 1,p(R× S1, u∗L; ed|s|dsdθ) ⊕ V ′u ⊕ V u → Lp(ed|s|dsdθ). Symplectic homology for autonomous Hamiltonians 40 The same argument as above, using the constant operator D′′0 , shows that D and D′′ admit right inverses which are uniformly bounded with respect to δ → 0. Both operators have index 1 and it follows from the description of their kernels given in the proof of Proposition 4.8 that their restrictions toW 1,p⊕V ′u, respectively W 1,p ⊕V ′u are isomorphisms. We choose the right inverses Q , Q′′ to be the inverses of their respective restrictions. Let ζ ∈ kerD′′, ζ ∈ kerD′′ be two sections such that their values at +∞ and respectively −∞ are equal to the (constant) vector corresponding to XH in the chosen trivialization of u∗TŴ . Let V ′, V be the 1-dimensional vector spaces spanned by βζ and (1 − β)ζ respectively. Setting the norm of these generators to be equal to 1 defines a new norm on dom(D ) and dom(D′′), which we emphasize by decomposing the latter as dom(D ) =W 1,p(R× S1, u∗L; ed|s|dsdθ)⊕ V ′u ⊕ V ′, dom(D′′) = D′′u :W 1,p(R× S1, u∗L; ed|s|dsdθ) ⊕ V ′ ⊕ V ′u. It follows from our special choice of the right inverses Q , Q′′ that the latter are also uniformly bounded with respect to this new norm as δ → 0. Let D′′ := D ′′ be the operator obtained by gluing D cut at sδ and D′′ cut at −sδ, with the ‖ · ‖1,δ-norm on its domain and the ‖ · ‖δ-norm on its target. It follows as in [5, Proposition 5] that the right inverses Q , Q′′ give rise to a uniformly bounded right inverse Q′′ for D′′ as δ → 0. On the other hand, we have ‖D′′u − D′′‖ → 0 as δ → 0, and we obtain a uniformly bounded right inverse for D′′u by the previous formula Q u := Q ′′(D′′uQ ′′)−1. We note that, upon gluing, the exponential weights at ±∞ for D′′, D′′ give rise to the peak in the weight function wδ for D ′′, and the fibered sum operation on V ′, V which the norm is fixed, is responsible for the appearance of the distinguished cutoff section ζδ leading to the modified norm ‖ · ‖1,δ. We now treat D′u and start by making a few general remarks. For each s0 ∈ R the operator D′(s0) := ∂s+J0∂θ+S(s0, θ) :W 1,p(R×S1, u∗ξ; dsdθ) → Lp(R×S1, u∗ξ; dsdθ) is δ-close to the R-invariant operator with nondegenerate asymptotics corre- sponding to the constant cylinder over the orbit u(s0, ·). Hence, for δ > 0 small enough, both operators are isomorphisms [24, Lemma 2.4]. Moreover, this property also holds in the presence of weights ed|s|, eds or e−ds. For the weight ed|s| we argue as follows. The operator is still Fredholm between the W 1,p and Lp spaces with weights, of the same index 0. Since the corre- sponding W 1,p space is contained in W 1,p(R × S1, u∗ξ; dsdθ) we infer that the operator is injective, hence an isomorphism. For the weight eds we argue as follows. Multiplication by e s determines linear isomorphisms M : W 1,p(R × S1, u∗ξ; edsdsdθ) →W 1,p(R×S1, u∗ξ; dsdθ) andM : Lp(R×S1, u∗ξ; edsdsdθ) → Symplectic homology for autonomous Hamiltonians 41 Lp(R× S1, u∗ξ; dsdθ). The operator M−1D′(s0)M is an isomorphism and, for ζ ∈ W 1,p(R× S1, u∗ξ; edsdsdθ), we have M−1D′(s0)Mζ = D ′(s0)ζ + Since d > 0 is small as in Proposition A.2 and p > 2, the operatorM−1D′(s0)M is R-invariant and has nondegenerate asymptotics, hence is an isomorphism [24, Lemma 2.4]. An analogous reasoning using the multiplication by e− s proves the claim for the weight e−ds. We now prove that D′u admits a uniformly bounded right inverse in the case a = −T/2, b = T/2, sδ = (T+ǫ)/2δ. We recall the notation u := uδ,γ,−(T+ǫ)/2,0, u := uδ,γ,0,(T+ǫ)/2 and set := D′u :W 1,p(R× S1, u∗ξ; ed|s|dsdθ) → Lp(R× S1, u∗ξ; ed|s|dsdθ), D′ := D′u :W 1,p(R× S1, u∗ξ; ed|s|dsdθ) → Lp(R× S1, u∗ξ; ed|s|dsdθ). We claim that each of the operatorsD , D′ is an isomorphism with uniformly bounded right inverse as δ → 0. We give the proof forD′ since the proof forD′ is entirely analogous. We choose a finite number of points −∞ = s−m < s−m+1 < · · · < s−1 < 0 = s0 < s1 < · · · < sm+1 = +∞ such that ‖S(s, θ) − S(s′, θ)‖ ≤ 1/4C for all θ ∈ S1 and s, s′ ∈ [si, si+1], i = −m, . . . ,m, with C > 0 a constant to be chosen below. Let bi−1 := ai := c −1(u(si, 0)), i = −m, . . . ,m+ 1. We consider the operators D′i := D uδ,γ,ai−1,bi−1 , i = −m+ 1, . . . ,−1, D′0 := D uδ,γ,a−1,b0 D′i := D uδ,γ,ai,bi , i = 1, . . . ,m. For each i = −m + 1, . . . ,m we denote by ui = ui,δ the gradient cylinder corresponding to the operator D′i. The domain and range of the operators D are as follows: D′i :W 1,p(R× S1, u∗i ξ; e−dsdsdθ) → Lp(R× S1, u∗i ξ; e−dsdsdθ), i < 0, D′0 :W 1,p(R× S1, u∗0ξ; ed|s|dsdθ) → Lp(R× S1, u∗0ξ; ed|s|dsdθ), D′i :W 1,p(R× S1, u∗i ξ; edsdsdθ) → Lp(R× S1, u∗i ξ; edsdsdθ), i > 0. We have seen that D′(s0) is an isomorphism for all s0 ∈ R if one uses any of the weights ed|s|, eds, e−ds. Since S(s0, ·) belongs to a compact set of loops of matrices we infer that the norm of the inverse Q′(s0) := D ′(s0) −1 is uniformly bounded with respect to s0 ∈ R for each of these three weights. We choose C := maxweight∈{ed|s|,eds,e−ds} maxs0∈R ‖Q′(s0)‖. Symplectic homology for autonomous Hamiltonians 42 The same argument as forD′′u shows that the inverse of eachD i is bounded by 2C independently of δ. We glue together the operators D′i into D̃ ′ by cutting at ai/δ and bi/δ. Then D̃ ′ is still surjective and the norm of its inverse is bounded by 2CC̃2m−1, with C̃ a universal constant (see [24, Proposition 3.9]). Note that our choice of weights for the operators D′i is such that the resulting weight for the domain and target of D̃′ is still ed|s|. On the other hand we have ‖D̃′ −D′‖ → 0, δ → 0. This is because the two operators coincide outside 2m− 1 intervals of length 2, where the variation of S tends to zero as δ → 0. As a consequence the inverse is also uniformly bounded when δ is small enough. We now glue the operator D cut at sδ with the operator D ′ cut at −sδ, and denote the resulting operator by D′. The argument in [5, Proposition 5] shows that D′ admits a uniformly bounded right inverse Q′, provided one uses the weight wδ(s) on its domain and target. On the other hand ‖D′u −D′‖ → 0, δ → 0 since the two operators differ on a segment of length 2 where the variation of S goes to zero. We infer that D′u also admits a uniformly bounded right inverse. The cases when a = −∞, b = 1 or a = −1, b = ∞ follow now easily by combining the proof of the existence of uniformly bounded right inverses for the operators D with the previous use of a shift operator (Tδζ)(s) = ζ(s± sδ). Remark 4.10. Note that, if a = −T/2, b = T/2, Our construction of a right inverse for D′′ in the proof of Proposition 4.9 is such that its norm is uniformly bounded as δ → 0 even if one uses the “non-compensated” norm ‖·‖1,p,δ instead of ‖ · ‖1,δ. However, our choice of the norm ‖ · ‖1,δ will be essential in the proof of Proposition 4.16. In order to describe the pregluing construction it is convenient to work with a single section over B′δ rather than with a family of sections. We recall that, up to a translation, the defining interval I(a, b) of a gradient cylinder can be considered to be [−T/2, T/2], T > 0 in case (i), or ] − ∞, 1], [−1,∞[ in cases (ii) and (iii) respectively. We are therefore led to consider the section ∂̄ : B′δ → E (56) defined by ∂̄ := ∂̄−∞,1 and ∂̄ := ∂̄−1,∞ in cases (ii) and (iii), and by ∂̄(u) = ∂̄ǫ(u) := ∂̄−Tu/2,Tu/2,ǫ(u) in case (i). Here Tu > 0 is the time needed to flow along the gradient of fγ from the negative limit to the positive limit of u (see (51)). Remark 4.11. In case (i) the section ∂̄ can be described as follows. The one parameter family of sections ∂̄T := ∂̄−T/2,T/2,ǫ gives rise to a section denoted Symplectic homology for autonomous Hamiltonians 43 {∂̄T } of the pull-back bundle pr∗1E → B′δ × R+. There is a codimension one embedding ι : B′δ → B′δ × R+ given by ι(u) = (u, Tu), the composition pr1 ◦ ι is the identity and we have ∂̄ = {∂̄T }|im ι. The situation is summarized in the following commutative diagram. pr∗1E // � ι // B′δ × R+ pr1 // {∂̄T } Given u ∈ B′δ we denote by D′u : TuB′δ → Eu the vertical differential of ∂̄. In cases (ii) and (iii) we have seen that D′u is a Fredholm operator of index ind(p) and 1− ind(q) respectively. In case (i) the vertical differential can be computed explicitly as follows. The vertical differential of {∂̄T }, denoted by D{∂̄T }, is D{∂̄T }(u, T ) · (ζ, τ) = D−T/2,T/2,ǫu ζ − τ(JXHδ−H) h′−T/2δ,T/2δ,ǫ/δ(s) = D−T/2,T/2,ǫu ζ − (JXHδ−H)k−T/2δ,T/2δ,ǫ/δ(s). Let us write a section ζ ∈ TuB′δ as ζ = ζ0 + aζ + bζ, with ζ0 ∈ W 1,p,d, a, b ∈ R and ζ, ζ being the distinguished generators of V u respectively. The vertical differential D′u acts by D′uζ = D{∂̄T}(u, Tu) · (ζ, dTu · ζ) = D−Tu/2,Tu/2,ǫu ζ − dTu · ζ (JXHδ−H)k−Tu/2δ,Tu/2δ,ǫ/δ(s). One can explicitly compute dTu · ζ = dTu · (aζ + bζ) = ċ(−Tu/2)b− ċ(Tu/2)a ċ(−Tu/2) · ċ(Tu/2) where c : R → Sγ is the gradient trajectory satisfying c(−Tu/2) = θ0, c(Tu/2) = θ0 and ċ is the derivative with respect to the XH -parametrization of Sγ . Proposition 4.12. Let T > 0 and u = uδ,γ,−T/2,T/2,ǫ. The index of D equal to 1, its kernel has dimension 2 and a complement of imD′u is spanned by a section supported in [−(T + ǫ)/2δ,−(T + ǫ)/2δ + 1]× S1 [(T + ǫ)/2δ − 1, (T + ǫ)/2δ]× S1. Morever, D′u admits a right inverse defined on its image which is uniformly bounded with respect to δ → 0. Symplectic homology for autonomous Hamiltonians 44 Proof. The first order differential operators D′u and D −Tu/2,Tu/2,ǫ u differ by a term of order zero, hence their indices are equal and indD′u = 1. The operator D{∂̄T }(u, Tu) is surjective and has index 2. As a consequence dimkerD′u ≤ dim kerD{∂̄T }(u, Tu) = 2. Let c : R → Sγ be the gradient curve defining u = uδ,γ,−T/2,T/2,ǫ. For σ close to zero we define c σ(s) := c(σ + s) and denote by uσ1 := u δ,γ,−T/2,T/2,ǫ the gradient cylinder defined by cσ. Then ∂̄(uσ1 ) = ∂̄T (u 1 ) = 0, hence ζ 1 := d |σ=0uσ1 ∈ kerD′u. We also define uσ2 := uδ,γ,−(T+σ)/2,(T+σ)/2,ǫ to be the gradient cylinder associated to c. Then ∂̄(u 2 ) = ∂̄T+σ(u 2 ) = 0, hence ζ 2 := d |σ=0uσ2 ∈ kerD′u. Since ζ1 and ζ2 are linearly independent, we infer that dimkerD′u = 2. We claim that the section η := 1 (JXHδ−H)k−Tu/2δ,Tu/2δ,ǫ/δ(s) spans a com- plement of imD′u. This follows from (i) in Lemma 4.13 below with ℓ := dTu, φ := D −Tu/2,Tu/2,ǫ u , φ̃ := D u, y := η and xy := ζ 2. That D′u admits a uniformly bounded right inverse defined on its image follows from (ii) in Lemma 4.13 and the fact that D −Tu/2,Tu/2,ǫ u has a uniformly bounded right inverse by Proposi- tion 4.9. Lemma 4.13. Let φ : E → F be a surjective map of Banach vector spaces, ℓ : E → R be a nonzero linear functional, y = φ(xy) ∈ F be fixed and φ̃ : E → F be defined by φ̃(x) = φ(x) − ℓ(x)y. We assume that kerφ ⊂ ker ℓ. Then im φ̃ = φ(ker ℓ) if and only if ℓ(xy) = 1, in which case the following hold. (i) The element y spans a complement of im φ̃. (ii) If Q : F → E is a right inverse for φ, then Q|φ(ker ℓ) is a right inverse for φ̃ defined on its image. Proof. We first note that im φ̃ ⊇ φ(ker ℓ). Let us now assume that im φ̃ = φ(ker ℓ). For x /∈ ker ℓ we obtain φ(x)− l(x)φ(xy) ∈ φ(ker ℓ), hence x− l(x)xy ∈ ker ℓ, implying ℓ(x) − ℓ(x)ℓ(xy) = 0 and ℓ(xy) = 1. Conversely, if ℓ(xy) = 1 we obtain x− ℓ(x)xy ∈ ker ℓ for any x ∈ E, hence φ̃(x) = φ(x− ℓ(x)xy) ∈ φ(ker ℓ). The element y does not belong to φ(ker ℓ) because y = φ(xy) with ℓ(xy) = 1 and the preimage xy is well-defined up to an element of kerφ ⊂ ker ℓ. This proves the equivalence in the statement of the Lemma, as well as (i). To prove (ii) we need to show that Q(φ(ker ℓ)) ⊂ ker ℓ. We prove the stronger statement imQ ∩ ker ℓ = Q(φ(ker ℓ)). The inclusion imQ ∩ ker ℓ ⊂ Q(φ(ker ℓ)) follows from the observation that, given x = Qz with ℓ(x) = 0, we have z = φ(Qz) = φ(x) ∈ φ(ker ℓ). On the other hand note that Qφ is the projection to imQ along kerφ. Since kerφ ⊂ ker ℓ, it follows that Qφ(ker ℓ) ⊂ imQ∩ker ℓ. We describe now the pre-gluing construction for elements of the Morse-Bott moduli spaces and gradient cylinders of the form uδ,γ,a,b,ǫ. We define the space B̃δ := B̃1,p,dδ (γp, Sγm−1 , . . . , Sγ1 , γq, A;H, {fγ}) Symplectic homology for autonomous Hamiltonians 45 consisting of tuples w̃ := (u1, . . . , um, v0, . . . , vm) satisfying the following condi- tions. (i) ui ∈ B1,p,d(Sγi , Sγi−1 , Ai;H), i = 1, . . . ,m, with Sγ0 := Sγ , Sγm := Sγ , Sγi 6= Sγi−1 , i = 1, . . . ,m and A1 + . . .+Am = A; (ii) v0 ∈ B1,p,dδ (Sγ , q; fγ), vi ∈ B 1,p,d δ (Sγi , Sγi ; fγi) for i = 1, . . . ,m − 1, and vm ∈ B1,p,dδ (p, Sγ ; fγ); (iii) ev(vi−1) = ev(ui) and ev(vi) = ev(ui) for i = 1, . . . ,m; (iv) ev(v0) belongs to the stable manifold of q, and ev(vm) belongs to the unstable manifold of p. By the definition of the spaces B1,p,dδ (Sγi , Sγi ; fγi) we have ev(vi) 6= ev(vi) for i = 1, . . . ,m − 1. We denote by Ti > 0 the unique positive real number such that ϕ (ev(vi)) = ev(vi), where ϕ s is the gradient flow of fγ . Figure 4: Broken Morse-Bott trajectory w̃. Let us choose a tubular neighbourhood Uγ ⊂ Ŵ for each γ ∈ P(H), parametrized by (ϑ, z) ∈ S1 × R2n−1. Given any subset K ⊂ B̃1,p,dδ (γp, Sγm−1 , . . . , Sγ1 , γq, A;H, {fγ}) for which there exists s0 > 0 such that, for |s| ≥ s0, the components of any w̃ ∈ K belong to the respective tubular neighbourhoods of their asymptotics, Symplectic homology for autonomous Hamiltonians 46 we construct, for δ > 0 small enough and ǫi ∈ R, i = 1, . . . ,m− 1 small enough in absolute value, a pre-gluing map Gδ,ǫ : K → B1,p,dδ (γp, γq, A;H, {fγ}), ǫ := (ǫ1, . . . , ǫm−1). Let β : R → [0, 1] be a smooth increasing cutoff function vanishing on ]−∞, 0] and identically equal to 1 on [1,∞[. Define the gluing profile R = R(δ) by . (57) We define for i = 1, . . . ,m the maps ûi : [−R,R]× S1 → Ŵ by ûi(s, θ) :=   z(s, θ) = β(s+R)z ◦ ui(s, θ), ϑ(s, θ) = θ + β(s+R)(ϑ ◦ ui(s, θ)− θ), s ∈ [−R,−R+ 1], ui(s, θ), s ∈ [−R+ 1, R− 1],{ z(s, θ) = β(−s+R)z ◦ ui(s, θ), ϑ(s, θ) = θ + β(−s+R)(ϑ ◦ ui(s, θ)− θ), s ∈ [R− 1, R]. We define for i = 1, . . . ,m− 1 the maps v̂i : [−(Ti + ǫi)/2δ, (Ti + ǫi)/2δ]× S1 → Ŵ by the analogous formulas in which we replace R by Ti+ǫi . We also define v̂0 : [−1/δ,+∞[×S1 → Ŵ v̂0(s, θ) := z(s, θ) = β(s+ 1 )z ◦ v0(s, θ), ϑ(s, θ) = θ+β(s+ 1 )(ϑ◦v0(s, θ)−θ), s ∈ [− 1 + 1], v0(s, θ), s ∈ [− 1δ + 1,+∞[, as well as v̂m :]−∞, 1/δ]× S1 → Ŵ by the analogous formula with s replaced by −s and v0 replaced by vm. Finally, we define Gδ,ǫ(w̃) as the catenation v̂m, ûm, v̂m−1, . . . , û1, v̂0. The catenation of these maps is performed in the above order and with (obvious) shifts 0 = svm < sum < svm−1 < . . . < su1 < sv0 in the domain defined by suj = svj + ℓj, (58) svj−1 = suj + ℓj−1 Symplectic homology for autonomous Hamiltonians 47 for j = 1, . . . ,m. Here we denote ℓi := R + (Ti + εi)/2δ (59) for i = 0, . . . ,m, with the convention Tm = T0 = 2 and εm = ε0 = 0. We have in particular v̂i(s, θ) = Gδ,ǫ(w̃)(s+ svi , θ), (s, θ) ∈ dom(v̂i), i = 0, . . . ,m, ûj(s, θ) = Gδ,ǫ(w̃)(s+ suj , θ), (s, θ) ∈ dom(ûj), j = 1, . . . ,m. Given u = (cm, um, . . . , u1, c0) ∈ M̂A(p, q;H, {fγ}, J), we denote by Gδ,ǫ(u) the element Gδ,ǫ(w̃) ∈ Bδ, where w̃ := (vm, um, . . . , u1, v0) and vi := uδ,γi,ai,bi,ǫi , i = 0, . . . ,m is the gradient cylinder corresponding to the gradient trajectory ci : I(ai, bi) → Sγi . The section ∂̄Hδ,J(Gδ,ǫ(w̃)) belongs to the space Lp(R× S1, Gδ,ǫ(w̃)∗TŴ ; gδ,ǫ(s)dsdθ), where the continuous function gδ,ǫ(s) is the catenation of the following functions: (i) gδ,ui(s) := e d|s| on the domain [−R,R] of ûi; (ii) gδ,ǫi,vi(s) = e d||s|−si,δ| on the domain [−(Ti + ǫi)/2δ, (Ti + ǫi)/2δ] of v̂i, where si,δ = Ti+ǫi −R ≤ s∗i,δ = Ti+ǫi , i = 1, . . . ,m− 1; (iii) gδ,v0(s) := e d|s+s0,δ| on the domain [−1/δ,+∞[ of v̂0, where s0,δ = 1/δ − R ≤ s∗0,δ = 1/δ; (iv) gδ,vm(s) := e d|s−sm,δ| on the domain ]−∞, 1/δ] of v̂m, with sm,δ = 1/δ − R ≤ s∗m,δ = 1/δ. We denote the norm on the above Lp space with weight gδ,ǫ by ‖ · ‖δ, omitting in the notation the dependence on the numbers Ti + ǫi, i = 1, . . . ,m − 1. We define a norm ‖ · ‖1,δ on the space W 1,p(R× S1, Gδ,ǫ(w̃)∗TŴ ; gδ,ǫ(s)dsdθ) as follows. For j = 1 . . . ,m let 〈ζ(suj −R, ·), XH〉 〈XH , XH〉 , κj = 〈ζ(suj +R, ·), XH〉 〈XH , XH〉 , (60) where 〈·, ·〉 is the inner product in L2(S1). Here suj − R and suj + R are the coordinates of the catenation circles between ûj and v̂j , respectively ûj and v̂j−1. For i = 1, . . . ,m− 1 let 〈ζ(svi , ·), ζi,δ(0, ·)〉 〈ζi,δ(0, ·), ζi,δ(0, ·)〉 Symplectic homology for autonomous Hamiltonians 48 suj +R sujsvj suj − 2R suj −R svj−1suj + 2R Figure 5: The definition of ‖ · ‖1,δ. where the section ζi,δ generates the kernel of the operator Dvi as in Proposition 4.8. The norm ‖ · ‖1,δ is then defined by ‖ζ‖1,δ := ∥∥ζ − κjβ(−s+ suj )β(s − suj + 2R)XH (61) − κjβ(s− suj )β(−s+ suj + 2R)XH κiβ(s− svi + ℓi − 2R)β(−s+ svi + ℓi − 2R)ζi,δ(· − svi , ·) W 1,p(gδ,ǫ) |κj |+ |κj | |κi|. Here ℓj is defined by (59), β : R → [0, 1] is the smooth cutoff function which vanishes on ]−∞, 0] and is equal to 1 on [1,∞[, and ‖ · ‖W 1,p(gδ,ǫ) is the W 1,p- norm with weight gδ,ǫ on W 1,p(ed|s|dsdθ). The graph of the function β(−s+ suj )β(s− suj + 2R) + β(s− suj )β(−s+ suj + 2R) +β(s− svj + ℓj − 2R)β(−s+ svj + ℓj − 2R) +β(s− svj−1 + ℓj−1 − 2R)β(−s+ svj−1 + ℓj−1 − 2R) is depicted in Figure 5. Remark 4.14. The definition of ‖ · ‖1,δ is such that the norm of the gluing map G constructed in the proof of Proposition 4.18 below is uniformly bounded with respect to δ → 0. Proposition 4.15. Let w̃ ∈ B̃δ and ǫ(δ) := (ǫ1(δ), . . . , ǫm−1(δ)) be such that (i) ǫi(δ) → 0, δ → 0 for i = 1, . . . ,m− 1; (ii) ui ∈ MAi(Sγi , Sγi−1 ;H, J), i = 1, . . . ,m; Symplectic homology for autonomous Hamiltonians 49 (iii) the components vi are of the form uδ,γi,ai,bi,ǫi , with bi = −ai = Ti/2 for i = 1, . . . ,m − 1, b0 = +∞, a0 = −1, ǫ0 = 0 and bm = 1, am = −∞, ǫm = 0. ‖∂̄Hδ,J(Gδ,ǫ(w̃))‖δ = 0. Proof. We must check that ‖∂̄Hδ,J(Gδ,ǫ(w̃))|I×S1‖δ → 0 as δ → 0 when I ⊂ R is an interval of the following type. (i) I = [−R + 1, R − 1] is contained in the domain of ûi. Then ∂̄Hδ,J(ûi) = −J(XHδ − XH) ◦ ûi. The norm of this map is pointwise bounded by a constant multiple of δ. Hence its δ-norm is bounded by a constant multiple of δedR → 0, δ → 0; (ii) I = [−R,−R + 1] or I = [R − 1, R] is contained in the domain of ûi. We have ∂̄Hδ,J(ûi) = ∂̄H,J(ûi) − J(XHδ − XH) ◦ ûi. The second term is bounded as in (i). The term ∂̄H,J(ûi) is pointwise bounded by the norms of z ◦ ûi, ϑ ◦ ûi − θ and of their derivatives. By Proposition A.1 their δ-norm is bounded by a constant multiple of e(d−r)R → 0, δ → 0; (iii) I = [−(Ti + ǫi)/2δ + 1, (Ti + ǫi)/2δ − 1] for i = 1, . . .m − 1, or I = [−1/δ + 1,+∞[ or I =] −∞, 1/δ − 1] and is contained in the domain of some v̂i. Since ∂̄H′ −Ti/2,Ti/2,ǫi ,J(v̂i) = 0 and H −Ti/2,Ti/2,ǫi = Hδ for s ∈ I, we already have ‖∂̄Hδ,J(Gδ,ǫ(w̃))|I×S1‖δ = 0; (iv) I = [−(Ti+ ǫi)/2δ,−(Ti+ ǫi)/2δ+1] or I = [(Ti+ ǫi)/2δ− 1, (Ti+ ǫi)/2δ] for i = 1, . . .m − 1, or I = [−1/δ,−1/δ + 1], or I = [1/δ − 1, 1/δ] and is contained in the domain of some v̂i. Then ∂̄Hδ,J(v̂i) involves only ϑ◦ v̂i−θ, its derivative with respect to s and δ∇fγi . By formula (50) the norm of these expressions is pointwise bounded by a constant multiple of δ, therefore their δ-norms are bounded by δedR → 0 as δ → 0. Proposition 4.16. Let [ṽn] ∈ MA(γp, γq;Hδn , J) with δn → 0, n → ∞ and let [u] ∈ MA(p, q;H, {fγ}, J) be a broken Floer trajectory of level ℓ = 1 whose intermediate gradient fragments c1, . . . , cm−1 are nonconstant. Then [ṽn] → [u], n→ ∞ if and only if there exist • representatives vn ∈ [ṽn], v ∈ [u], • real parameters ǫn = (ǫn1 , . . . , ǫnm−1) with ǫni → 0, n→ ∞, • vector fields ζn ∈ TGδn,ǫn (v)Bδ with ζn = (ζ n, ζn, ζn), such that ‖ζn‖1,δn := ‖ζ0n‖1,δn + ‖ζn‖+ ‖ζn‖ → 0, n→ ∞, Symplectic homology for autonomous Hamiltonians 50 satisfying vn := expGδn,ǫn(v) (ζn). Proof. We first prove the converse implication, namely that convergence in norm implies geometric convergence. We define shifts (sni ), i = 1, . . . ,m inductively snm := 1/δn +Rn, s i := s i+1 + 2Rn + (Ti + ǫ i )/δn. We claim that vn(·+sni , ·) → ui, n→ ∞ uniformly on compact sets. Let R0 > 0 be fixed. By assumption ‖ζ0n(·+ sni , ·)|[−R0,R0]×S1‖1,δn → 0, n→ ∞. By the Sobolev embedding theorem this implies ‖ζ0n(·+ sni , ·)|[−R0,R0]×S1‖C0 → 0, n→ ∞. Since Gδn,ǫn(v)(· + sni , ·)|[−R0,R0]×S1 = ui|[−R0,R0]×S1 for n sufficiently large, the conclusion follows. We now prove the direct implication. Let us pick a representative v = (cm, um, cm−1, . . . , u1, c0) ∈ [u] and let Ti, i = 0, . . . ,m be the lengths of the intervals of definition of ci, with the convention T0 = Tm = +∞. We also choose arbitrary representatives vn ∈ [ṽn]. By assumption there exist shifts (sni ) such that vn(· + sni , ·) converges to ui uniformly on compact sets. We define ǫni := δn(s i − sni+1 − 2Rn)− Ti, i = 1, . . . ,m− 1. (62) By Lemma 4.6 we have ǫni → 0, n→ ∞. We define partitions of the real line −∞ = anm ≤ bnm ≤ anm−1 ≤ . . . ≤ an0 ≤ bn0 = +∞ by bnm := 1/δn and ani−1 := b i + 2Rn, b i−1 := a i−1 + (Ti−1 + ǫ i−1)/δn, i = 1, . . . ,m. We define a sequence of shifts (sn) by sn := snm − 1/δn −Rn and we still denote by vn the shifted sequence vn(·+ sn, ·). We first show the existence of a unique vector field ζn satisfying vn = expGδn,ǫn(v) (ζn). For that it is enough to prove s∈In,θ∈S1 dist(vn(s, θ), Gδn,ǫn(v)(s, θ)) = 0, (63) where In is an interval of the following form: Symplectic homology for autonomous Hamiltonians 51 (i) [bni , a i−1], i = 1, . . . ,m; (ii) [ani , b i ], i = 1, . . . ,m− 1; (iii) [bnm −K/δn, bnm] or [an0 , an0 +K/δn], for any K > 0. The asymptotic behaviour of vn and Gδn,ǫn(v) ensures that ζn is an element of the relevant W 1,p-space. We prove case (i) by contradiction. Assume that there exists ǫ > 0 and a sequence (s̃n, θ̃n) ∈ [bmi , ani−1]× S1 such that dist(vn(s̃n, θ̃n), Gδn,ǫn(v)(s̃n, θ̃n)) ≥ ǫ. Since (63) is satisfied if one replaces vn by ui(·−sni , ·) (by definition ofGδn,ǫn(v)), we also have dist(vn(s̃n, θ̃n), ui(s̃n − sni , θ̃n)) ≥ ǫ/2 (64) for n large enough. By the assumption of uniform convergence on compact sets vn(· + sni , ·) → ui(·, ·), up to passing to a subsequence we can assume that s̃n − sni → ±∞. We treat the case s̃n − sni → ∞, the other case being similar. Since s̃n ∈ [bni , ani−1] and δn(ani−1 − bni ) = 2δnRn → 0, we have δn(s̃n − sni ) → 0. By Lemma 4.5 we infer that vn(·+ s̃n, ·) → ev(ui), which means vn(s̃n, ·) = lim ui(s̃n − sni , ·) and this contradicts (64). Note that the above proof shows that vn(· + ani−1, ·) → ev(ui) and vn(· + bni , ·) → ev(ui), i = 1, . . . ,m uniformly on compact sets. We now prove case (ii). Let us fix 1 ≤ i ≤ m − 1. An action argument as the one in the proof of Lemma 4.6 shows that vn(In × S1) is entirely contained in a small neighbourhood of Sγi . We apply Proposition A.3 to vn and In × S1 to obtain (s,θ)∈In×S1 |z ◦ vn(s, θ)| = 0 (s,θ)∈In×S1 |ϑ ◦ vn(s, θ)− θ − ϕ δn(s−a (ev(ui+1))| = 0. The same two equations hold, by definition, if one replaces vn by Gδn,ǫn(v), and the conclusion follows. We now prove (iii). We treat only the case In = [a 0 , a 0 +K/δn], the other case being similar. An action argument as above shows that vn(·+an0 +K/δn, ·) converges uniformly on compact sets to a constant cylinder over some orbit γ ∈ Sγ0 . By Lemma 4.6 we know that γ = ϕ K (ev(u1)), and in particular is not a critical point of fγ0 . Now the conclusion follows in the same way as in case (ii). We now show that ‖ζn|In×S1‖1,δn = 0 (65) Symplectic homology for autonomous Hamiltonians 52 in each of the cases (i)-(iii). We denote in the sequel |ζ(s, θ)|1 := |ζ(s, θ)| + |∇sζ(s, θ)|+ |∇θζ(s, θ)|. We first consider case (i). Let us fix K > 0 large enough. For n large enough we can write In = [s i −Rn, sni −K] ∪ [sni −K, sni +K] ∪ [sni +K, sni +Rn]. We first note that ∫ sni +K |ζ(s, θ)|p1 gδn,ǫn(s)dsdθ = ∫ sni +K |ζ(s, θ)|p1 ed|s−s i |dsdθ ≤ sup s∈[sn −K,sn |ζ(s, θ)|p1 · e Since vn(· + sni , ·) and Gδn,ǫn(v)(· + sni , ·) converge uniformly on compact sets together with their derivatives to ui, the last term goes to zero as n→ ∞. In order to estimate the integral on the interval [sni −Rn, sni −K] we apply Proposition A.3 on [sni+1 +K, s i −K] to vn to obtain |z ◦ vn(s, θ)|1 ≤ C(K) cosh(ρ(s− s i+1+s cosh(ρ( ≤ C1C(K)eρ(s−s i +K) |ϑ ◦ vn(s, θ)− θ − ϕ δn(s−b (pni )|1 ≤ C1C(K)eρ(s−s i +K), where | · |1 stands for the pointwise C1-norm, for some pni ∈ Sγi such that pni → ev(ui), n → ∞. Similar estimates hold, by definition, if one replaces vn by Gδn,ǫn(v) and p i with ev(ui). Hence we obtain |ζn(s, θ)− κni XH |1 ≤ C1C(K)eρ(s−s i +K), (66) where κni → 0 as n→ ∞ and C(K) = Cmax(‖Q∞vn(sni+1 +K)‖, ‖Q∞vn(sni −K)‖, ‖Q∞ṽn(sni+1 + sn +K)‖, ‖Q∞ṽn(sni + sn −K)‖). (67) We obtain ∫ sni −K |ζn(s, θ)− κni XH | 1 gδn(s)dsdθ ∫ sni −K |ζn(s, θ)− κni XH | −d(s−sni )dsdθ ≤ C2C(K)pedK . Symplectic homology for autonomous Hamiltonians 53 A similar estimate holds when the interval of integration is [sni +K, s i + Rn], with C(K) replaced with C′(K). Letting n→ ∞ we obtain In×S1 |ζn(s, θ)− κni β(−s+ sni )XH − κni β(s− sni )XH | 1 gδn(s)dsdθ ≤ C2(C(K)p + C′(K)p)edK . We let now K → ∞. Proposition A.3 implies that, for K > K ′, we have C(K ′) ≤ C3C(K)e−ρ(K ′−K), hence C(K)pedK → 0 as K → ∞ because d < ρp. The equality (65) follows. We now consider case (ii). We fix K > 0 large enough and apply Proposi- tion A.3 on the interval [sni+1+K, s i −K] ⊃ [sni+1+Rn, sni −Rn] = In to obtain as in case (i) |ζn(s, θ)− κni ζi,δ(s, θ)|1 ≤ C(K) cosh(ρ(s− s i+1+s cosh(ρ( where C(K) is given by (67), ζi,δ(s − sni+1+s ) generates the kernel of the lin- earized operator corresponding to gradient trajectory ci as in Proposition 4.8 and κni ζi,δ(b i , ·) = κni XH . In particular, we have κni → 0, n→ ∞. We get ∫ sni −Rn |ζn(s, θ)− κni ζi,δ| 1 gδn(s)dsdθ ≤ C2C(K) pe(d−ρp)(Rn−K). The last term goes to zero as n → ∞. Equality (65) follows now as in case (i). Case (iii) is entirely similar to case (ii). In order to complete the proof of ‖ζn‖1,δn → 0, n→ ∞, it is enough to show that ‖ζn|In×S1‖1,δn → 0 if In =] − ∞, bnm − K/δn] or In = [an0 + K/δn,+∞[, for any K > 1. The two cases are entirely similar and we give the argument only for In =]−∞, bnm −K/δn]. By Proposition A.2, for n sufficiently large we have vn(s, θ) = expuδn,γm,−∞,1(s,θ)(ηn(s, θ)), with ηn = (η n, ηn), η n ∈W 1,p(In× S1, u∗δn,γm,−∞,1TŴ ; e r|s|ds dθ), ηn ∈ V . Since vn(b n , ·) → ev(um) we have ‖ηn‖∞ → 0. Since Gδn,ǫn(v) = uδn,γm,−∞,1 on In, we obtain ζn = ηn, so that ‖ζn‖ → 0. The fact that ‖ζ0n‖1,δn → 0 follows from the fact that d < r. We explain now how to construct a right inverse for DGδ,ǫ( ew) which is uni- formly bounded with respect to δ → 0. The space B̃δ is a Banach manifold whose tangent space at w̃ is T ewB̃δ = TvmB′δ dev ⊕dev TumB dev ⊕dev Tvm−1B′δ dev ⊕ . . .⊕dev Tv0B′δ. (68) Recall that the fibered sum of two vector spaces W1, W2 with respect to linear maps fi :Wi →W is the vector space W1 f1 ⊕f2 W2 := {(w1, w2) ∈W1 ⊕W2 : f1(w1) = f2(w2)}. Symplectic homology for autonomous Hamiltonians 54 If (W1, ‖ · ‖1), (W2, ‖ · ‖2) and W are normed vector spaces, and f1, f2 are continuous linear maps, then W1 f1 ⊕f2 W2 is a closed subspace of W1⊕W2 and inherits the norm ‖ · ‖1 + ‖ · ‖2 from W1 ⊕W2. In our case dev : TvmB′δ =W 1,p,d ⊕ V ′ ⊕ V ′ → Tev(vm)Sγm factors through the projection on V ′, and similarly for the other evaluation maps. Therefore the above fibered sum only affects the summands V , V , V V ′, so that T ewB̃δ is a subspace of codimension 2m in TvmB′δ ⊕ TumB ⊕ Tvm−1B′δ ⊕ . . .⊕ Tv0B′δ. As above, the norm on T ewB̃δ is induced from the ambient space. Recall that the W 1,p-component has weight ed|s| for each TujB, weight ed||s|−si,δ| for each TviB′δ, i = 1, . . . ,m − 1, weight ed|s+s0,δ | for i = 0 and weight ed|s−sm,δ| for i = m, with si,δ as in the definition of gδ,ǫ. The sections ∂̄H,J : B → E and ∂̄ : B′δ → E defined by (41) and (56) give rise to a section over B̃δ. We denote its vertical differential by D ew : T ewB̃δ → Lp,d(v∗mTŴ )⊕ Lp,d(u∗mTŴ )⊕ . . .⊕ Lp,d(v∗0TŴ ), where Lp,d(v∗i TŴ ) := L p(R× S1, v∗i TŴ ; gδ,ǫi,vi(s)dsdθ), Lp,d(u∗iTŴ ) := L p(R× S1, u∗iTŴ ; gδ,ui(s)dsdθ). Lemma 4.17. Let J ∈ Jreg(H) and {fγ} ∈ Freg(H, J). Let ǫ =(ǫ1, . . . , ǫm−1) and let w̃ ∈ B̃δ be as in Proposition 4.15. The image of the operator D ew has codi- mension m−1 and admits a complement spanned by sections ηi ∈ Lp,d(v∗i TŴ ), i = 1, . . . ,m− 1 which are respectively supported in [−(Ti + ǫi)/2δ,−(Ti + ǫi)/2δ + 1]× S1 ∪ [(Ti + ǫi)/2δ − 1, (Ti + ǫi)/2δ]× S1. The operator D ew admits a right inverse Q ew defined on its image and whose norm is uniformly bounded with respect to δ → 0. Proof. We show that imD ew = imD ⊕ imDum ⊕ imD′vm−1 ⊕ . . .⊕ imD =: E. (69) By definition we have imD ew ⊂ E. Let us now choose (xm, ym, . . . , x0) ∈ E and x̃i and ỹj such that D (x̃i) = xi, Duj (ỹj) = yj . We need to modify x̃i and ỹj by elements lying in the kernels of the corresponding operators so that dev(ỹj) = dev(x̃j), dev(ỹj) = dev(x̃j−1), j = 1, . . . ,m. (70) Let us first assume m > 1. We have TvmM′δ,1,−∞(Sγ , Sγ ;H, J)× TumMAm(Sγ , Sγm−1 ;H, J) = kerD′vm × kerDum Symplectic homology for autonomous Hamiltonians 55 and, because {fγ} ∈ Freg(H, J), the map (dev, dev) : kerD′vm × kerDum → Tev(vm)Sγ × Tev(um)Sγ is transverse to the diagonal. We can therefore modify x̃m and ỹm so that dev(ỹm) = dev(x̃m). Similarly the map (dev, dev) : kerDu1 × kerD′v0 → Tev(u1)Sγ × Tev(v0)Sγ is transverse to the diagonal and we can modify ỹ1, x̃0 in order to achieve dev(ỹ1) = dev(x̃1). For i = 1, . . . ,m− 1 the maps (dev, dev) : kerD′vi → Tev(vi)Sγi × Tev(vi)Sγi are surjective and we can modify x̃i so that (70) is satisfied. If m = 1 the regularity hypothesis on fγ ensures that the map (dev, dev, dev, dev) : kerD′v1 × kerDu1 × kerD → Tev(v1)Sγ × Tev(u1)Sγ × Tev(u1)Sγ × Tev(v0)Sγ is transverse to the product of the diagonals in the first two and in the last two factors. We can therefore modify simultaneously x̃1, ỹ1, x̃0 in order to achieve (70). Therefore (69) is proved. It then follows from Proposition 4.12 that the image of D ew has codimension m − 1 and is spanned by sections ηi ∈ Lp,d(v∗i TŴ ) supported in the desired intervals. We now prove that D ew admits a uniformly bounded right inverse defined on its image. We observe that D ew is the restriction to dom(D ew) of the direct sum of operators D := D′vm ⊕Dum ⊕D ⊕· · ·⊕Du1 ⊕D′v0 . Let ζm, ζ0 be generators of kerD′vm , kerD and, for i = 1, . . . ,m− 1, let ζ1i , ζ2i be the basis of kerD′vi constructed in Proposition 4.12. We denote by K the vector space spanned by these 2m sections, viewed as elements of dom(D). Then dim K = 2m and K is a complement of dom(D ew). Let P : dom(D) → dom(D ew) be the projection parallel to K, let Quj , j = 1, . . . ,m be uniformly bounded right inverses forDuj , let Qvi , i = 0, . . . ,m be uniformly bounded right inverses for D defined on their images as in Proposition 4.12, and denote Q := Qvm ⊕Qum⊕Qvm−1 ⊕· · ·⊕ Qu1 ⊕Qu0 . Since K ⊂ kerD the operator P ◦Q : im(D) = im(D ew) → dom(D ew) is a right inverse for D ew defined on its image, and we claim that its norm is uniformly bounded for δ → 0. The norm of Q is uniformly bounded for δ → 0, so that it is enough to prove that the norm of P is uniformly bounded for δ → 0. The sections ζ0, ζm and ζ i , ζ i for i = 1, . . . ,m − 1 have the property that their respective asymptotic values (obtained by applying dev and dev) are not simultaneously zero. Moreover, the same is true for any linear combination of ζ1i and ζ2i for i = 1, . . . ,m− 1. As a consequence, there exists a uniform constant C > 0 such that, for any x = (xm, 0, xm−1, . . . , 0, x0) ∈ K, we have ‖x‖1,δ ≤ C |dev(xm)|+ |dev(x0)|+ |dev(xi)|+ |dev(xi)| . (71) Symplectic homology for autonomous Hamiltonians 56 Given v ∈ dom(D) we have P (v) = v + w for some vector w ∈ K which is uniquely determined by the asymptotic values of the components of v, and it follows from (71) that ‖w‖1,δ ≤ C‖v‖1,δ. We obtain ‖P (v)‖1,δ ‖v‖1,δ ‖v + w‖1,δ ‖v‖1,δ ≤ 1 + C, so that the norm of P is uniformly bounded by 1+C. This proves the Lemma. Proposition 4.18. Let J ∈ Jreg(H) and {fγ} ∈ Freg(H, J). Let w̃ ∈ B̃δ and ǫ(δ) = (ǫ1(δ), . . . , ǫm−1(δ)) be as in Proposition 4.15. The operator DGδ,ǫ( ew) : W 1,p(R× S1, Gδ,ǫ(w̃)∗TŴ ; gδ,ǫ(s)dsdθ)⊕ V ⊕ V ′v0 → Lp(R× S1, Gδ,ǫ(w̃)∗TŴ ; gδ,ǫ(s)dsdθ) is surjective and admits a right inverse Qδ = Qδ,ǫ, ew whose δ-norm is uniformly bounded with respect to δ → 0. Proof. Our proof is modelled on the proof of the gluing theorem for holomorphic spheres by McDuff and Salamon [21, Ch. 10] . Let vδm, u m−1, . . . , u be the extensions of v̂m, ûm, v̂m−1, . . . , û1, v̂0 to R × S1 defined by the same formulas. Note that uδj(s, θ) = uj(s, θ), s ∈ [−R+ 1, R− 1], vδm(s, θ) = vm(s, θ), s /∈ [1/δ − 1, 1/δ], vδ0(s, θ) = v0(s, θ), s /∈ [−1/δ,−1/δ+ 1] and vδi (s, θ) = vi(s, θ) for s outside [−(Ti + ǫi)/2δ,−(Ti + ǫi)/2δ + 1] ∪ [(Ti + ǫi)/2δ − 1, (Ti + ǫi)/2δ] and i = 1, . . . ,m− 1. The difference between vδi and vi on the one hand, and that between uδj and uj on the other hand is exponentially small as δ → 0. This implies that the operatorsDuδ andD′ are surjective for δ small enough and admit uniformly bounded right inverses, while the op- erators D′ , i = 1, . . . ,m− 1 have a codimension one image with a supplement spanned by a smooth section ηi supported in [−(Ti+ ǫi)/2δ,−(Ti+ ǫi)/2δ+1]× S1 ∪ [(Ti + ǫi)/2δ − 1, (Ti + ǫi)/2δ]× S1, and admit uniformly bounded “right inverses” defined on their image. It follows that the vertical differential D ewδ satisfies the conclusions of Lemma 4.17, where w̃δ := (uδ1, . . . , u 0 , . . . , v In particular, it admits a uniformly bounded right inverse defined on its image, which we denote by Q ewδ (see [21, Lemma 10.6.1] for a similar statement in the case of holomorphic spheres). This means that there exists a constant c0 > 0 such that ‖Q ewδx‖W 1,p,d ≤ c0‖x‖Lp,d Symplectic homology for autonomous Hamiltonians 57 for all x ∈ imD ewδ and δ > 0. We define an operator Tδ by the commutative diagram T ewδ B̃δ Lp,d(w̃δ∗TŴ ) dom(DGδ,ǫ( ew)) L p(R× S1, Gδ,ǫ(w̃)∗TŴ ; gδ,ǫ(s)dsdθ) where Lp,d(w̃δ∗TŴ ) := Lp,d(vδm TŴ )⊕ Lp,d(uδm TŴ )⊕ . . .⊕ Lp,d(vδ0 TŴ ). In the rest of the proof we shall omit the subscript ǫ from Gδ,ǫ and gδ,ǫ. An element of Lp,d(w̃δ∗TŴ ) is denoted by x = (xm, ym, . . . , x0). The mixing map P , the splitting map S and the gluing map G are defined below, and we shall prove that P, S,G are uniformly bounded with respect to δ → 0. We shall also prove that Tδ is an approximate right inverse for DGδ( ew), i.e. ‖DGδ( ew)Tδη − η‖δ ≤ ‖η‖δ (72) for δ sufficiently small and η ∈ Lp(R × S1, Gδ(w̃)∗TŴ ; gδ(s)dsdθ). This im- plies that DGδ( ew)Tδ is invertible (with the norm of its inverse bounded by 2), and Tδ(DGδ( ew)Tδ) −1 is a right inverse for DGδ( ew). Since P, S,G are uniformly bounded, the norm of Tδ(DGδ( ew)Tδ) −1 is bounded by a constant multiple of ‖Q ewδ‖, hence is uniformly bounded and the conclusion of the Proposition fol- lows. For every L > 0 we fix a smooth function βL : R → [0, 1] which vanishes for s ≤ 0, which is constant equal to 1 for s ≥ L and whose derivative is bounded by 2/L. We moreover require that, for L large enough, the function βL vanishes for s ≤ 1. We define the mixing map P . Let pi : L p,d(w̃δ∗TŴ ) → imD′ , i = 0, . . . ,m be the projection on Lp,d((vδi ) ∗TŴ ) followed by the projection on imD′ allel to ηi. Recall the definition (59) of ℓi for i = 0, . . . ,m and let qj : L p,d(w̃δ∗TŴ ) → imDuδ qj(x)(s, θ) := yj(s, θ) + β1(s− ℓj) · (1l− pj)(xj) (s− ℓj, θ) + (1− β1(s− ℓj−1)) · (1l− pj−1)(xj−1) (s− ℓj−1, θ) Symplectic homology for autonomous Hamiltonians 58 for j = 1, . . . ,m. We define P : Lp,d(w̃δ∗TŴ ) → imD ewδ P := pm + qm + pm−1 + . . .+ q1 + p0. The norm of P is uniformly bounded with respect to δ → 0 since the norm of each pi is uniformly bounded by 1. We define now the splitting map S(η) := x = (xm, ym, . . . , x0). We recall the definition (58) of the catenation shifts 0 = svm < sum < svm−1 < . . . < su1 < sv0 , and set xm(s, θ) := β1(1/δ − s)η(s, θ), x0(s, θ) := β1(1/δ + s)η(s+ sv0 , θ), and, for i = 1, . . . ,m− 1, j = 1, . . . ,m, yj(s, θ) := (1− β1(−R− s))η(s+ suj , θ), s ≤ 0, (1− β1(−R+ s))η(s+ suj , θ), s ≥ 0, xi(s, θ) := β1((Ti + ǫi)/2δ + s)η(s+ svi , θ), s ≤ 0, β1((Ti + ǫi)/2δ − s)η(s+ svi , θ), s ≥ 0. It follows from the definition that the norm of S is uniformly bounded by 1. We define now the gluing map ζ := G(x̃), x̃ = (x̃m, ỹm, x̃m−1, . . . , x̃0) ∈ T ewδ B̃δ by “slowly interpolating” the components of x̃. For j = 1, . . . ,m, i = 1, . . . ,m− 1 we put ζ(s, θ) := x̃m(s, θ), −∞ < s ≤ 1/δ −R/2, ỹj(s− suj , θ), suj −R/2 ≤ s ≤ suj +R/2, x̃i(s− svi , θ), svi − ℓi + 3R/2 ≤ s ≤ svi + ℓi − 3R/2, x̃0(s− sv0 , θ), sv0 − 1/δ +R/2 ≤ s < +∞. The above formula leaves out two types of intervals, on which the actual inter- polation takes place (see Figure 6). • If svj + ℓj − 3R/2 ≤ s ≤ suj −R/2 (interval of length R), we define ζ(s, θ) := x̃j(+∞, θ) + (1− βR (s− svj − ℓj +R)) x̃j(s− svj , θ)− x̃j(+∞, θ) + (1− βR (−s+ suj −R)) ỹj(s− suj , θ)− ỹj(−∞, θ) Symplectic homology for autonomous Hamiltonians 59 R 0 −R−R x̃j−1 0 −Tj−1+ǫj−12δ −Tj−1+ǫj−1 Tj+ǫj Tj+ǫj Figure 6: The gluing map G. • If suj +R/2 ≤ s ≤ svj−1 − ℓj−1 + 3R/2 (interval of length R), we define ζ(s, θ) := x̃j−1(−∞, θ) + (1−βR (−s+svj−1−ℓj−1 +R)) x̃j−1(s− svj−1 , θ)− x̃j−1(−∞, θ) +(1− βR (s− suj −R)) ỹj(s− suj , θ)− ỹj(+∞, θ) The section ζ is indeed of class W 1,p because ỹj(−∞, θ) = x̃j(+∞, θ), ỹj(+∞, θ) = x̃j−1(−∞, θ). That the norm of G is uniformly bounded with respect to δ → 0 follows directly from the definition (68) of the norm on T ewδ B̃δ, as well as from the definition (61) of the norm ‖ · ‖1,δ on dom(DGδ,ǫ( ew)) (see also Remark 4.14). Let us now prove the estimate (72). On each of the intervals appearing in (73) we have (DGδ( ew)Tδη)(s, θ) = η(s, θ) and we are therefore left to examine intervals of the type [svj + ℓj − 3R/2, suj −R/2] and [suj +R/2, svj−1 − ℓj−1 + 3R/2]. We treat only the first case since the second one is entirely similar. Upon applying the operator DGδ( ew) to ζ we obtain five types of terms as following. • DGδ( ew)x̃j(+∞, θ). Since x̃j(+∞, θ) does not depend on s we can view DGδ( ew) as a family of operators on S 1. Then we have ‖DGδ( ew)x̃j(+∞, θ)‖δ = ‖(DGδ( ew) −Dvj(+∞,θ))x̃j(+∞, θ)‖δ ≤ ‖DGδ( ew) −Dvj(+∞,θ)‖δ‖x̃j(+∞, θ)‖ ≤ C(δ)‖η‖δ. Symplectic homology for autonomous Hamiltonians 60 Here Dvj(+∞,θ) denotes the linearized operator at the constant cylinder vj(+∞, θ), the norm ‖x̃j(+∞, θ)‖ is induced from the (1-dimensional) space V ′vj , and C(δ) → 0, δ → 0. This last statement and the last inequality follow from ‖DGδ( ew)−Dvj(+∞,θ)‖δ ≤ C(‖v̂j−vj(+∞, θ)‖L1,p,d([(Tj+ǫj)/2δ−R/2,(Tj+ǫj)/2δ]×S1) + ‖ûj−uj(−∞, θ)‖L1,p,d([−R,−R/2]×S1)) and the fact that the intervals of integration migrate to ±∞. The above inequality makes crucial use of the fact that the weight gδ on the necks is given by the exponential weight of the ambient spaces Bδ, B′δ. Moreover, we have ‖x̃j(+∞, θ)‖ ≤ ‖x̃‖ ≤ C‖η‖δ because Q ew, P and S are uniformly bounded with respect to δ. • −β′ (s− svj − ℓj +R) x̃j(s− svj , θ)− x̃j(+∞, θ) , as well as β′ suj − R)) ỹj(s − suj , θ) − ỹj(−∞, θ) . The δ-norm of each of these two terms is bounded by C(δ)‖η‖δ, with C(δ) → 0 as δ → 0. To see this we first use that |β′ | ≤ 4/R → 0, δ → 0. Secondly we use that ‖x̃j(s − svj , θ) − x̃j(+∞, θ)‖ ≤ ‖x̃‖ ≤ C‖η‖δ and ‖ỹj(s − suj , θ) − ỹj(−∞, θ)‖ ≤ ‖x̃‖ ≤ C‖η‖δ. • (1 − βR/2(s− svj − ℓj + R))DGδ( ew) x̃j(s− svj , θ)− x̃j(+∞, θ) and (1− βR/2(−s+suj−R))DGδ( ew) ỹj(s−suj , θ)−ỹj(−∞, θ) . The parts involving x̃j(+∞, θ) = ỹj(−∞, θ) are bounded by C(δ)‖η‖δ as above. On the other hand we write DGδ( ew)x̃j = (DGδ( ew) −D ewδ)x̃j +D ewδ x̃j and similarly for DGδ( ew)ỹj. The first term of such a sum is bounded by C(δ)‖η‖δ as above, with C(δ) → 0, δ → 0. We are left with (1− βR/2)D ewδ x̃j(s− svj , θ) + (1− βR/2)D ewδ ỹj(s− suj , θ) (P ◦ S)vjη (s− svj , θ) + (P ◦ S)ujη (s− suj , θ) = η. Here we denote by (P ◦ S)vj , (P ◦ S)uj the components of P ◦ S in Lp,d(vδj TŴ ) and Lp,d(uδj TŴ) respectively. The first equality uses the fact that 1− βR/2 ≡ 1 on the support of (P ◦S)vjη and on the support of (P ◦ S)ujη, as well as D ewδ ◦Q ewδ = 1l. As a conclusion we have ‖DGδ( ew)Tδη − η‖δ ≤ C(δ)‖η‖δ, C(δ) → 0, δ → 0, and the estimate (72) holds for δ small enough. Symplectic homology for autonomous Hamiltonians 61 We shall use the following quantitative form of the implicit function theorem from McDuff and Salamon [21, A.3.4]. Theorem 4.19. Let X and Y be Banach spaces, U ⊂ X be an open set, and f : U → Y be a continuously differentiable map. Let x0 ∈ U be such that D := df(x0) : X → Y is surjective and has a bounded right inverse Q : Y → X. Choose positive constants ε and c such that ‖Q‖ ≤ c, Bε(x0) ⊂ U , and ‖x− x0‖ < ε =⇒ ‖df(x)−D‖ ≤ 1/2c. (74) Then, for any x1 ∈ X satisfying ‖f(x1)‖ < ε/4c, ‖x1 − x0‖ < ε/8, (75) there exists a unique x ∈ X such that f(x) = 0, x− x1 ∈ imQ, ‖x− x0‖ ≤ ε. (76) Moreover, ‖x− x1‖ ≤ 2c‖f(x1)‖. The above theorem will be used within the following setup. Consider an element [u] ∈ MA(p, q;H, {fγ}, J) and denote u0 := Gδ,ǫ(u). Given ε > 0 we denote by Bε(0) the ball of radius ε centered at 0 inW 1,p(R×S1, u∗0TŴ ; ‖·‖1,δ), where ‖ · ‖1,δ is defined by (61). For ζ ∈W 1,p(R× S1, u∗0TŴ ; ‖ · ‖1,δ) we write ζ = ζ1 + κjβ(−s+ suj )β(s− suj + 2R)XH κjβ(s− suj )β(−s+ suj + 2R)XH κiβ(s− svi + ℓi − 2R)β(−s+ svi + ℓi − 2R)ζi,δ(· − svi , ·) with ℓi = R+ (Ti + ǫi)/2δ and ζi,δ the generator of ker Dvi whose value at 0 is the vector field XH along γi. Then ‖ζ‖1,δ = ‖ζ1‖W 1,p(gδ,ǫ) + (|κj |+ |κj |) + |κi|. We denote ζ̃ := ζ1+ κjβ(−s+suj )β(s−suj+2R)XH+κjβ(s−suj )β(−s+suj+2R)XH so that ζ̃(svi , ·) is L2-orthogonal to ζi,δ(0, ·). For each i = 1, . . . ,m − 1 we consider the smooth cutoff function ρi,δ,ǫ(s) := β(s− svi + ℓi − 2R)β(−s+ svi + ℓi − 2R), Symplectic homology for autonomous Hamiltonians 62 so that ρi,δ,ǫ vanishes outside [svi − Ti+ǫi2δ , svi + Ti+ǫi ] and ρi,δ,ǫ ≡ 1 on the interval [svi − Ti+ǫi2δ + 1, svi + Ti+ǫi − 1]. We define ϕζ(u0) : R× S1 → Ŵ by ϕζ(u0)(s, θ) := u0(s, θ), suj −R ≤ s ≤ suj +R, ρi,δ,ǫ(s)κi (u0(s, ·))(θ), svi − Ti+ǫi2δ ≤ s ≤ svi + Ti+ǫi Note that the last formula can also be written in the chart (ϑ, z) around Sγi as ϑ ◦ϕζ(u0)(s, θ) = ϑ ◦ϕ ρi,δ,ǫ(s)κi (u0(s, 0))+ θ. Given a vector field ξ along u0 we define the vector field ϕζ∗ξ along ϕζ(u0) by ϕζ∗ξ(s, θ) := ξ(s, θ), suj −R ≤ s ≤ suj +R, ρi,δ,ǫ(s)κi∗ (u0)ξ(s, θ), svi − Ti+ǫi2δ ≤ s ≤ svi + Ti+ǫi We define a map Φ : Bε(0) → Bδ = B1,p,dδ (γp, γq, A;H, {fγ}) (77) Φ(ζ) := expϕζ(u0)(ϕζ∗ζ̃). Since ρi,δ,ǫ is precisely the coefficient of ζi,δ in our splitting for ζ, it follows that dΦ(0) = Id. Hence, for ε > 0 small enough the map Φ is a diffeomorphism onto its image, i.e. a chart. We denote X := W 1,p(R × S1, u∗0TŴ ; ‖ · ‖1,δ), U := Bε(0) ⊂ X , Y := Lp(R × S1, u∗0TŴ ; gδ,ǫdsdθ), x0 = 0. For ε > 0 small enough the Banach bundle E → Bδ can be trivialized over the image of Φ as Bε(0) × Y , and we denote by f : Bε(0) → Y the section ∂̄Hδ,J ◦Φ read in this trivialization. Then df(0) = Du0 is surjective and has a right inverse Qδ whose δ-norm is uniformly bounded with respect to δ → 0 by Proposition 4.18. In order for the hypotheses of Theorem 4.19 to be satisfied we need to check that (74) holds. Lemma 4.20. There exists a constant C > 0 independent of δ such that, for all x ∈ Bε(0), we have ‖df(x)− df(0)‖ ≤ C‖x‖1,δ. Remark 4.21. The motivation for introducing the chart Φ is that we must use the “compensated” norm ‖ · ‖1,δ. The lemma would fail if one used the usual exponential chart ζ 7→ expu0(ζ) instead of Φ, because the estimate for the expression (82) in the proof below would not hold. Proof. We need to prove the existence of a uniform constant C > 0 such that ‖D(∂̄Hδ,J ◦ Φ)(x) · ζ −D(∂̄Hδ ,J ◦ Φ)(0) · ζ‖δ ≤ C‖x‖1,δ‖ζ‖1,δ (78) Symplectic homology for autonomous Hamiltonians 63 for all ζ ∈ X . We recall the decomposition ζ = ζ̃ + i=1 κiρi,δ,ǫζi,δ, which satisfies ‖ζ‖1,δ = ‖ζ̃‖1,δ + i=1 |κi|. It is therefore enough to prove (78) sep- arately for ζ = ζ̃ and for ζ = ρi,δ,ǫζi,δ, i = 1, . . . ,m− 1. We abbreviate in the following computations ∂̄ = ∂̄Hδ,J . We first assume ζ = ρi,δ,ǫζi,δ. Given x = x̃+ j=1 κjρj,δ,ǫζj,δ we have D(∂̄ ◦ Φ)(x)ζ −D(∂̄ ◦ Φ)(0)ζ = D(∂̄ ◦ Φ)(x)ζ −D(∂̄ ◦ Φ)( κjρj,δ,ǫζj,δ)ζ (79) +D(∂̄ ◦ Φ)( κjρj,δ,ǫζj,δ)ζ −D(∂̄ ◦ Φ)(0)ζ. (80) The term (79) is further equal to ∂̄(expϕx+tζ(u0)(ϕx+tζ∗x̃))− ∂̄(expϕx+tζ(u0)(0)) = Dexpϕx(u0)(ϕx∗ex) ·D2 expϕx(u0)(ϕx∗x̃) · ∇tϕx+tζ∗x̃ + Dexpϕx(u0)(ϕx∗ex) ·D1 expϕx(u0)(ϕx∗x̃) · ρi,δ,ǫ∇fγi(ϕx(u0)) − Dϕx(u0) ·D1 expϕx(u0)(0) · ρi,δ,ǫ∇fγi(ϕx(u0)) = Dexpϕx(u0)(ϕx∗ex) ·D2 expϕx(u0)(ϕx∗x̃) · ∇tϕx+tζ∗x̃ (81) + Dexpϕx(u0)(ϕx∗ex) · (D1 expϕx(u0)(ϕx∗x̃)− T ·D1 expϕx(u0)(0)) ·ρi,δ,ǫ∇fγi(ϕx(u0)) + (Dexpϕx(u0)(ϕx∗ex) · T −Dϕx(u0)) ·D1 expϕx(u0)(0) · ρi,δ,ǫ∇fγi(ϕx(u0)). Here T is the parallel transport in Ŵ along the geodesic τ 7→ expϕx(u0)(τϕx∗x̃), τ ∈ [0, 1], and we have ρi,δ,ǫ∇fγi(ϕx(u0)) = ρi,δ,ǫ(ϕ ρi,δ,ǫκi)∗ζi,δ. We study the first term in (81). We have pointwise bounds |∇tϕx+tζ∗x̃| ≤ C(1 + |κi|)|x̃|, |∇∇tϕx+tζ∗x̃| ≤ C(1 + |κi|)(|x̃|+ |∇x̃|) for some universal constant C > 0. In particular ‖∇tϕx+tζ∗x̃‖W 1,p(gδ,ǫ) ≤ C‖x̃‖1,δ if |κi| ≤ ‖x‖1,δ ≤ ε, with C > 0 a universal constant. On the other hand the operators D2 expϕx(u0)(ϕx∗x̃) : W 1,p(gδ,ǫ) → W 1,p(gδ,ǫ) and Dexpϕx(u0)(ϕx∗ex) : W 1,p(gδ,ǫ) → Lp(gδ,ǫ) are uniformly bounded if ‖x‖∞ ≤ C‖x‖1,δ ≤ Cε (we use here the Sobolev inequality). This implies that the δ-norm of the first term in (81) is bounded by a constant multiple of ‖x̃‖1,δ. We now study the second term in (81). Let ‖| · ‖| be the operator norm for continuous linear maps W 1,p(ϕx(u0) ∗TŴ ; ‖ · ‖1,δ) →W 1,p(expϕx(u0)(ϕx∗x̃) ∗TŴ ; gδ,ǫdsdθ). Symplectic homology for autonomous Hamiltonians 64 We claim that ‖|D1 expϕx(u0)(ϕx∗x̃)− T ·D1 expϕx(u0)(0)‖| ≤ C‖x̃‖1,δ for some uniform constant C > 0, provided ‖x‖1,δ ≤ ε. Indeed, since the metric on Ŵ varies smoothly, for any ξ = ξ̃ + ℓ=1 κ ℓρℓ,δ,ǫ∇fγℓ(ϕx(u0)) we have pointwise bounds ∣∣(D1 expϕx(u0)(ϕx∗x̃)− T ·D1 expϕx(u0)(0))ξ̃ ∣∣ ≤ C|x̃||ξ̃|, ∣∣∇(D1 expϕx(u0)(ϕx∗x̃)− T ·D1 expϕx(u0)(0))ξ̃ ∣∣ ≤ C(|∇x̃||ξ̃|+ |x̃||∇ξ̃|), ∣∣(D1 expϕx(u0)(ϕx∗x̃)− T ·D1 expϕx(u0)(0))ρℓ,δ,ǫ∇fγℓ(ϕx(u0)) ∣∣ ≤ C|x̃|, ∣∣∇(D1 expϕx(u0)(ϕx∗x̃)−T ·D1 expϕx(u0)(0))ρℓ,δ,ǫ∇fγℓ(ϕx(u0)) ∣∣ ≤ C(|x̃|+|∇x̃|). The claim then follows by integration with respect to the weight gδ,ǫ and by using the Sobolev inequalities ‖x̃‖L∞ ≤ C‖x̃‖1,δ and ‖ξ̃‖L∞ ≤ C‖ξ̃‖1,δ. On the other hand, as already seen above, the operator Dexpϕx(u0)(ϕx∗ex) acting from the space W 1,p(gδ,ǫ) to L p(gδ,ǫ) is uniformly bounded for ‖x‖1,δ ≤ ε, since its coefficients are bounded. We infer that the δ-norm of the second term in (81) is bounded by a constant multiple of ‖x̃‖1,δ. We finally study the third term in (81). We claim that ‖Dexpϕx(u0)(ϕx∗ex) · T −Dϕx(u0)‖ ≤ C‖x̃‖1,δ for some uniform constant C > 0, provided ‖x‖1,δ ≤ ε. This follows from the pointwise bounds ∣∣(Dexpϕx(u0)(ϕx∗ex) · T −Dϕx(u0))ξ̃ ∣∣ ≤ C|x̃|(|ξ̃|+ |∇ξ̃|), ∣∣(Dexpϕx(u0)(ϕx∗ex) · T −Dϕx(u0))ρℓ,δ,ǫ∇fγℓ(ϕx(u0)) ∣∣ ≤ C|x̃| by integrating with respect to the weight gδ,ǫ and by using the previous Sobolev inequalities. Since D1 expϕx(u0)(0) = Id, we infer that the δ-norm of the third term in (81) is bounded by a constant multiple of ‖x̃‖1,δ. As a conclusion, the δ-norm of the expression in (79) is bounded by a con- stant multiple of ‖x̃‖1,δ. We now consider the expression in (80), which can be written as D(∂̄ ◦ Φ)(κiρi,δ,ǫζi,δ)ζ −D(∂̄ ◦ Φ)(0)ζ (82) ∂̄(ϕκiζ+tζ(u0))− ∂̄(ϕtζ(u0)) ∂̄(u0(·+ (κi + t)ρi,δ,ǫ, ·))− ∂̄(u0(·+ tρi,δ,ǫ, ·)). Each term in the above difference is supported in the intervals [svi − Ti+ǫi2δ , svi − Ti+ǫi +1] and [svi+ Ti+ǫi −1, svi+ Ti+ǫi2δ ]. Moreover, their difference is pointwise bounded by C|κi| for some uniform constant C > 0. Since the weight gδ,ǫ is uniformly bounded on the above intervals of length 1, we infer that the δ-norm of the expression in (80) is bounded by C|κi|, hence by C‖x‖1,δ for some uniform constant C > 0. Symplectic homology for autonomous Hamiltonians 65 We now assume ζ = ζ̃ and we again decompose D(∂̄ ◦Φ)(x)ζ −D(∂̄ ◦Φ)(0)ζ as the sum of the expressions in (79) and (80). The expression in (79) can be written ∂̄(expϕx(u0)(ϕx∗(x̃ + tζ̃)))− ∂̄(expϕx(u0)(ϕx∗tζ̃)) = Dexpϕx(u0)(ϕx∗ex) ·D2 expϕx(u0)(ϕx∗x̃) · ϕx∗ζ̃ − Dϕx(u0) ·D2 expϕx(u0)(0) · ϕx∗ζ̃ = Dexpϕx(u0)(ϕx∗ex) · (D2 expϕx(u0)(ϕx∗x̃)− T ·D2 expϕx(u0)(0)) · ϕx∗ζ̃ + (Dexpϕx(u0)(ϕx∗ex) · T −Dϕx(u0)) ·D2 expϕx(u0)(0) · ϕx∗ζ̃. (83) Here T denotes the same parallel transport map as above. We claim that the δ-norm of the first term in the expression (83) is bounded by C‖x̃‖1,δ‖ζ̃‖1,δ when ‖x‖1,δ ≤ ε, for some uniform constant C > 0. We have the pointwise estimates |ϕx∗ζ̃| ≤ C(1 + |κj |)|ζ̃|, |∇ϕx∗ζ̃| ≤ C(1 + |κj |)(|ζ̃|+ |∇ζ̃|), which imply ‖ϕx∗ζ̃‖W 1,p(gδ,ǫ) ≤ C‖ζ̃‖1,δ for some uniform constant C > 0, provided ‖x‖1,δ ≤ ε. On the other hand, the pointwise estimates ∣∣(D2 expϕx(u0)(ϕx∗x̃)− T ·D2 expϕx(u0)(0))ξ ∣∣ ≤ C|x̃||ξ|, ∣∣∇(D2 expϕx(u0)(ϕx∗x̃)− T ·D2 expϕx(u0)(0))ξ ∣∣ ≤ C(|∇x̃||ξ|+ |x̃||∇ξ|) show that the norm of the operator D2 expϕx(u0)(ϕx∗x̃) − T · D2 expϕx(u0)(0) acting from W 1,p(gδ,ǫ) to itself is bounded by C‖x̃‖1,δ. Finally, we have already seen that the operator Dexpϕx(u0)(ϕx∗ex) acting between W 1,p(gδ,ǫ) and L p(gδ,ǫ) is uniformly bounded, and the claim follows. We now claim that the δ-norm of the second term in the expression (83) is also bounded by C‖x̃‖1,δ‖ζ̃‖1,δ when ‖x‖1,δ ≤ ε, for some uniform constant C > 0. We have the pointwise estimate ∣∣(Dexpϕx(u0)(ϕx∗ex) · T −Dϕx(u0))ξ ∣∣ ≤ C|x̃|(|ξ|+ |∇ξ|), which implies that the norm of the operator Dexpϕx(u0)(ϕx∗ex) ·T −Dϕx(u0) acting from W 1,p(gδ,ǫ) to L p(gδ,ǫ) is bounded by C‖x̃‖1,δ for some uniform constant C > 0. Since ‖ϕx∗ζ̃‖W 1,p(gδ,ǫ) ≤ C‖ζ̃‖1,δ and D2 expϕx(u0)(0) = Id, the claim follows. Symplectic homology for autonomous Hamiltonians 66 We finally study the term (80) in the decomposition of D(∂̄ ◦Φ)(x)ζ−D(∂̄ ◦ Φ)(0)ζ, which can be written ∂̄(expϕx(u0) ϕx∗tζ̃)− ∂̄(expu0 tζ̃) = Dϕx(u0) ·D2 expϕx(u0)(0) · ϕx∗ζ̃ −Du0 ·D2 expu0(0) · ζ̃ = Dϕx(u0) · ϕx∗ζ̃ −Du0 · ζ̃. This last expression is pointwise bounded by C( j=1 |κj |)(|ζ̃| + |∇ζ̃|), which implies that its δ-norm is bounded by C‖x‖1,δ‖ζ̃‖1,δ for some uniform constant C > 0. This proves the lemma. Proposition 4.22. Let [u] ∈ MA(p, q;H, {fγ}, J). There exists δ1 > 0 and a one-parameter family [uδ] ∈ MA(γp, γq;Hδ, J), 0 < δ < δ1 such that [uδ] → [u], δ → 0. Here convergence is understood in the sense of Definition 4.2. Moreover, if dim MA(p, q;H, {fγ}, J) = 0 then the intermediate gradient fragments in [u] are nonconstant and the above one-parameter family is unique. Remark 4.23. The fact that the intermediate gradient fragments in [u] are nonconstant is the reason why we had to prove the gluing theorem only in the case where the intermediate lengths of gradient trajectories are strictly positive: Ti > 0, i = 1, . . . ,m− 1, where m is the number of sublevels in [u]. Proof. We choose a representative u = (cm, um, . . . , u1, c0) of [u] and we apply Theorem 4.19 in a chart of Bδ as above. By Proposition 4.18 the operator D admits a right inverse Qδ which is uniformly bounded with respect to δ by some constant c. By Lemma 4.20 there exists ε > 0 independent of δ such that condition (74) is satisfied. We set x0 := Gδ(u). By Proposition 4.15 we have ‖f(x0)‖ = 0 and therefore condition (75) is satisfied on some open neighbourhood of x0 if δ is small enough. Taking x1 := x0 in the statement of Theorem 4.19 provides us with an element x ∈ X satisfying (76). We set uδ := x. Then [uδ] ∈ MA(γp, γq;Hδ, J). Because ‖x − x0‖ ≤ 2c‖f(x0)‖ → 0 and x0 = Gδ(u) → u by construction, we infer by Proposition 4.16 that [uδ] → [u], δ → 0. We now assume that the dimension of MA(p, q;H, {fγ}, J) is zero. We have dimMA(γp, γq;Hδ, J) = = µ(γp)− µ(γq) + 2〈c1(TW ), A〉 − 1 = µ(γ)− µ(γ) + 2〈c1(TW ), A〉 − 1 + ind(p)− ind(q), Symplectic homology for autonomous Hamiltonians 67 hence µ(γ)−µ(γ)+2〈c1(TW ), A〉 = 1− ind(p)+ ind(q) ≤ 2. On the other hand µ(γ) − µ(γ) + 2〈c1(TW ), A〉 = i=1 µ(γi) − µ(γi−1) + 2〈c1(TW ), Ai〉, where m ≥ 0 is the number of sublevels of u. Each of the summands is nonnegative by transversality, and the only possibilities occuring are the following: (i) each summand is zero, which means that all the Floer trajectories involved in u are rigid; (ii) one of the summands is 1 and the others vanish. Since [u] is rigid, the only nonrigid summand must be u0 or um, while c0, respectively cm have to be constant.; (iii) two of the summands are 1, and the others vanish. As above, the nonrigid summands must be u0 and um, while c0 and cm are constant; (iv) one of the summands is 2 and the others vanish. Since [u] is rigid we must have m = 1 and c0, c1 have to be constant. In each of the cases (i-iii) the intermediate gradient trajectories have to be nonconstant by transversality of the evaluation maps (34). Let now [ṽδ] → [u], δ → 0. Since the only possible intermediate gradient trajectory in [u] is nonconstant, we can apply Proposition 4.16. We obtain representatives vδ ∈ [ṽδ], v ∈ [u] and functions ǫ = ǫ(δ) = (ǫ1(δ), . . . , ǫm−1(δ)) such that vδ, Gδ,ǫ(v) belong to some ‖ · ‖1,δ,ǫ -chart in Bδ and ‖vδ −Gδ,ǫ(v)‖1,δ,ǫ → 0, δ → 0. (84) We have to prove that uδ and vδ differ by a shift for δ > 0 sufficiently small. Let us choose a continuous path vt ∈ M̂A(p, q;H, {fγ}, J), t ∈ [0, 1] with v0 = u, v1 = v. We denote yt = yt(δ) := Gδ,tǫ(vt) and ‖ · ‖t := ‖ · ‖1,δ,tǫ. Note that, for each δ > 0, there exists a continuous function Cδ : [0, 1]× [0, 1] → R+ such that Cδ(t, t′) ‖ · ‖t′ ≤ ‖ · ‖t ≤ Cδ(t, t′)‖ · ‖t′ , t, t′ ∈ [0, 1] satisfying Cδ(t, t) = 1, t ∈ [0, 1]. As yt(δ) and Dyt(δ) vary continuously with t and δ, we can choose ε > 0 and c > 0 so that the hypotheses of the implicit function theorem 4.19 are satisfied for each yt(δ), t ∈ [0, 1], 0 < δ < δ1 and some suitable constant δ1 > 0. After further shrinking δ1 we can also assume that ‖f(yt)‖t < ε/4c for all t ∈ [0, 1] and 0 < δ < δ1. Finally, in view of (84), for some smaller δ1 we can achieve ‖vδ − y1(δ)‖1 ≤ ε. We define Iδ := {t ∈ [0, 1] : ∃x ∈ [vδ], ‖x− yt(δ)‖t ≤ ε}, 0 < δ < δ1. We prove that Iδ = [0, 1] by showing that it is a nonempty open and closed subset of [0, 1]. Note that Iδ is nonempty since 1 ∈ Iδ. We prove now that Iδ Symplectic homology for autonomous Hamiltonians 68 is closed. Assume that tn ∈ Iδ is such that tn → t. Let xn ∈ [vδ] be such that ‖xn−ytn(δ)‖tn ≤ ε. By the triangular inequality we see that ‖xn−yt(δ)‖t stays bounded, hence the sequence of shifts defining xn is also bounded and, up to a subsequence, we may assume that xn → x ∈ [vδ]. Then ‖xn − yt(δ)‖t ≤ Cδ(t, tn)‖xn − yt(δ)‖tn ≤ Cδ(t, tn)(‖xn − ytn(δ)‖tn + ‖ytn(δ)− yt(δ)‖tn) ≤ Cδ(t, tn)(ε+ ‖ytn(δ)− yt(δ)‖tn). We pass to the limit n→ ∞ and obtain ‖x−yt(δ)‖t ≤ ε, hence t ∈ Iδ. We prove now that Iδ is open. Let t ∈ Iδ and choose an open interval J containing t such that Cδ(t ′, t) < 2 and ‖yt′(δ)−yt(δ)‖t < ε/8 for all t′ ∈ J . Theorem 4.19 applied to x0 := yt(δ) and x1 := yt′(δ) yields x such that f(x) = 0 and ‖x−yt(δ)‖t ≤ ε. The uniqueness statement in the implicit function theorem ensures that the intersection of the space of solutions with the ‖ · ‖t-ball of radius ε centered at x0 is a graph over ker D. Since dim ker D = 1 and since translation in the s-variable already provides a 1-parameter family of solutions, we infer that x ∈ [vδ]. Moreover, the last statement in Theorem 4.19 gives ‖x−yt′(δ)‖t ≤ ε/2. Then ‖x− yt′(δ)‖t′ ≤ Cδ(t′, t)ε/2 < ε, so that t′ ∈ Iδ and Iδ is open. The upshot is that there exists x ∈ [vδ] such that ‖x−y0(δ)‖0 ≤ ε, 0 < δ < δ1. But y0(δ) = Gδ(u) and, again by the uniqueness statement in the implicit function theorem, we get that x and uδ differ by a shift. Hence [uδ] = [vδ]. Proof of Theorem 3.7. We first prove (i) and show the existence of δ1. As- sume by contradiction that there exists a sequence δn → 0 and Floer trajec- tories vn ∈ M̂A(γp, γq;Hδn , J) such that J is not regular for vn. By Propo- sition 4.7 we may assume, up to shifting and passing to a subsequence, that vn → u ∈ M̂A(p, q;H, {fγ}, J). As seen in the proof of Proposition 4.22, the limit u has nonconstant intermediate gradient trajectories since J is regular for u. We can therefore apply Proposition 4.16 and get parameters ǫn and vector fields ζn such that vn = expGδn,ǫn (u) (ζn) and ‖ζn‖1,δn,ǫn → 0. By Proposi- tion 4.18 the operator DGδn,ǫn (u) is surjective and admits a right inverse which is uniformly bounded with respect to δ. We infer that the operator Dvn is also surjective for n large enough, a contradiction. Let us prove (ii). Let (δ, vδ) ∈ M̂A]0,δ1[(γp, γq;H, {fγ}, J) and let I(δ) ⊂]0, δ1[ be a small relatively compact open interval containing δ. Since the norms ‖·‖1,δ′ are equivalent for δ′ ∈ I(δ), the space BI(δ) := δ′∈I(δ){δ′} × Bδ′ is a Banach manifold. Similarly, there is a Banach vector bundle EI(δ) → BI(δ) endowed with an obvious section ∂̄HI(δ),J whose restriction to Bδ′ is ∂̄Hδ′ ,J . The restriction of its linearization D(δ,vδ) at (δ, vδ) to TvδBδ is the surjective operator Dvδ of index 1, hence D(δ,vδ) is surjective and has index 2. Therefore ker D(δ,vδ) projects surjectively onto TδI(δ) = R and the projection in (ii) is a submersion. Symplectic homology for autonomous Hamiltonians 69 We now prove (iii). Let us note that, by Proposition 4.22, we have a map MA(p, q;H, {fγ}, J) → π0(MA]0,δ1[(γp, γp;H, {fγ}, J)), [u] 7→ C[u] := δ∈]0,δ1[ {(δ, [uδ])}, where [uδ] is the uniquely defined one-parameter family of Proposition 4.22 such that [uδ] → [u]. This map is injective because the limit of such a family [uδ] as δ → 0 is unique. In order to prove surjectivity, let C = {(δ, [vδ])} be a connected component of MA ]0,δ1[ (γp, γp;H, {fγ}, J). By Proposition 4.7 there exists a sequence δn → 0 and [u] ∈ MA(p, q;H, {fγ}, J) such that [vδn ] → [u]. By the uniqueness statement in Proposition 4.22 we get that C = C[u]. 4.4 Coherent orientations The structure of this section is as follows. We first present the construction of coherent orientations in the usual Floer setting for (Hδ, J) by adopting the point of view of [5]. We construct coherent orientations on the moduli spaces of Morse-Bott trajectories, out of which we get orientations on the space of Morse- Bott trajectories with gradient fragments. Finally, we prove Proposition 3.9. We denote S1 := R/Z and, for a path of symmetric matrices S : S1 → M2n(R), we denote by ΨS the unique solution of the Cauchy problem Ψ̇(θ) = J0S(θ)Ψ(θ), Ψ(0) = 1l, θ ∈ [0, 1], (85) where J0 is the standard complex structure on R 2n. Then ΨS is a path of symplectic matrices and we denote S := {S : S1 →M2n(R) : tS = S and det(1l−ΨS(1)) 6= 0}. Let us denote by E a symplectic vector bundle of rank 2n over CP 1, or R×S1, or C, with fixed trivializations in neighbourhoods of infinity in the case of R× S1 and C. We denote by O(CP 1, E) the space of linear operators D : W 1,p(CP 1, E) → Lp(CP 1,Λ0,1E) of the form (∂x + J0∂y + S(z))dz̄ in a local trivialization of E, where z = x + iy is a local coordinate on CP 1. Given S, S ∈ S we denote by O(R× S1, E;S, S) the space of linear operators D : W 1,p(R × S1, E) → Lp(R × S1,Λ0,1E) of the form (∂s + J0∂θ + S(s, θ))(ds − idθ) in a local trivialization of E, such that lims→−∞ S(s, ·) = S and lims→∞ S(s, ·) = S in the given trivializations of E. Given S0 ∈ S we denote by O±(C, E;S0) Symplectic homology for autonomous Hamiltonians 70 O+ O+ O(R× S1)O−O+O(CP1) Figure 7: The four possibilities of gluing (O = O(R × S1)). the space of linear operators D :W 1,p(C, E) → Lp(C,Λ0,1E) of the form (∂x + J0∂y + S(z))dz̄ in a local trivialization of E and such that, when expressed in holomorphic cylindrical coordinates (s, θ) with e±2π(s+iθ) = z as (∂s + J0∂θ + S(s, θ))(ds − idθ), we have lims→±∞ S(s, θ) = S0(θ) in the given trivialization of E. Intuitively, the space O+ corresponds to the sphere with one positive puncture, while O− corresponds to the sphere with one negative puncture. It is a standard fact in the literature that each of the above spaces O is contractible and consists of Fredholm operators. Moreover, they each come equipped with a canonical real line bundle Det(O) whose fiber atD is Det(D) := (Λmax ker D)⊗ (ΛmaxcokerD)∗. Each of the bundles Det(O) is trivial since the base is contractible. We now define gluing operations between elements of the above spaces (see Figure 7). Let K ∈ O+(C, E;S0) or K ∈ O(R × S1, E;S, S0), and L ∈ O−(C, F ;S0) or L ∈ O(R × S1, F ;S0, S). Let us choose a cutoff function β : R → [0, 1] such that β(s) = 0 if s ≤ 0 and β(s) = 1 if s ≥ 1. Given R > 0 large we define operators KR and LR by replacing S in the asymptotic expressions ofK and L by S0+β(R−s)(S−S0) and S0+β(R+s)(S−S0) respec- tively. We cut out semi-infinite cylinders {s > R} from the base of E, {s < −R} from the base of F , then identify their boundaries using the coordinate θ. We glue the vector bundles E and F using their given trivializations near infinity and denote the resulting vector bundle by E#F . We define K#RL by concate- nating KR and LR, so that K#RL belongs to one of the spaces O(CP 1, E#F ), O+(C, E#F ;S), O−(C, E#F ;S), or O(R× S1, E#F ;S, S). Following [5, Corollary 7], for R large enough there is a natural isomor- phism Det(K)⊗Det(L) ∼→ Det(K#RL) defined up to homotopy. In particular, given orientations oK of Det(K) and oL of Det(L), we induce a canonical ori- entation oK#oL of Det(K#RL). Moreover, this operation on orientations is associative [13, Theorem 10]. We describe now, following [5], a procedure for constructing orientations on Symplectic homology for autonomous Hamiltonians 71 the spaces O(R × S1, E;S, S) which are coherent with respect to the gluing operation, in the sense of [13, Definition 11]. We denote by θn a trivial sym- plectic vector bundle of rank 2n. We first note that each determinant bundle Det(O(CP 1, E)) is naturally oriented since O(CP 1, E) contains the connected space of complex linear operators and the latter have kernels and cokernels which are canonically oriented as complex vector spaces. We now choose arbi- trary orientations of the determinant bundles Det(O+(C, E;S0)) such that the trivialization of E at infinity extends to C. Remark 4.24. Note that, if S0 commutes with J0, the set of C-linear operators in O+(C, E;S0) forms a nonempty convex set, hence Det(O+(C, E;S0)) has a canonical orientation. We induce orientations on the determinant bundles Det(O−(C, E;S0)) such that the trivialization of E at infinity extends to C by requiring that the orientation induced by gluing on Det(O(CP 1, θn)) is the canonical one. Finally, we induce orientations on Det(O(R×S1, E;S, S)) by requiring that the orientation induced on Det(O(CP 1, θn#E#θn)) by the gluing operation O+(C, θn;S)×O(R× S1, E;S, S)×O−(C, θn;S) → O(CP 1, θn#E#θn) is the canonical one. It is proved in [5] that this defines a system of coherent orientations. The general procedure for inducing orientations of the spaces of Floer trajec- tories M̂A(γp, γq;Hδ, J) out of a system of coherent orientations goes as follows. Let Ψp, Ψq denote the linearizations of the Hamiltonian flow of Hδ along γp, γq in their fixed respective trivializations and let Sp, Sq ∈ S be the corresponding paths of symmetric matrices as in (85). Let E be a symplectic vector bundle over R × S1 with fixed trivializations at infinity and relative first Chern class equal to 〈c1(TŴ ), A〉. For each u ∈ M̂A(γp, γq;Hδ, J) there is an isomorphism of symplectic vector bundles Φu : u ∼→ E, chosen to depend continuously on u. There is a map M̂A(γp, γq;Hδ, J) → O(R× S 1, E;Sp, Sq), u 7→ Φu ◦ D̃u ◦Φ−1u , where D̃u has the same analytical expression as the linearized operator Du : W 1,p(R×S1, u∗TŴ ; ed|s|ds dθ)⊕V u⊕V u → Lp(R×S1, u∗TŴ ; ed|s|ds dθ) con- sidered in Section 4.3. Under the assumption that J is regular and because of elliptic regularity, the operators D̃u and Du have the same kernel, consisting of smooth elements. Hence their determinant lines are naturally isomorphic. It fol- lows that the pull-back of Det(O) under the above map is naturally isomorphic to ΛmaxTM̂ = Λmax ker Du, and we get an orientation on M̂. If the dimension is one, the space M̂ has a canonical orientation given at each point u by the vector field ∂su. Comparing this with the orientation constructed above associates to each connected component [u] of M̂ a sign ǫ(u). Symplectic homology for autonomous Hamiltonians 72 Lemma 4.25. Let S1 ∈ S and define Sm(θ) := S1(mθ). Assume Sm ∈ S and define an automorphism φm of O+(C, E;Sm) by conjugation with the map z 7→ e2iπ/mz. Then φm is orientation reversing for Det(O+(C, E;Sm)) if and only if m is even and the difference of Conley-Zehnder indices µCZ(ΨSm)− µCZ(ΨS1) is odd. Proof. We start by explaining how φm acts on the orientations of the deter- minant bundle. The operators K ∈ O+(C, E;Sm) which are invariant under conjugation by φm, i.e. K(ζ ◦φm) ◦φ−1m = K(ζ) for all ζ, form a convex and, in particular, connected set. Since φm acts on ker K and cokerK, it also acts on Det(K) and the induced action on orientations extends to Det(O+(C, E;Sm)). There is a bijective correspondence between operators K1 ∈ O+(C, E;S1) and operators K ∈ O+(C, E;Sm) which are invariant under conjugation by φm, in which case the pull-back of ker K1 under z 7→ zm is the 1-eigenspace of φm acting on ker K. Since ker K splits as a direct sum of eigenspaces corresponding to them-th roots of unity and since imaginary roots give rise to even-dimensional eigenspaces, we infer that the dimension of the −1-eigenspace has the parity of dimker K − dimker K1. This fact is relevant in our situation since φm reverses the orientation of ker K if and only if this dimension is odd. Similarly, φm reverses the orientation of cokerK if and only if dim cokerK − dim cokerK1 is odd. As a conclusion, φm reverses the orientation of Det(K) if and only if ind(K)− ind(K1) = µCZ(ΨSm)− µCZ(ΨS1) is odd. This can happen of course only if −1 is an m-th root of unity, i.e. m is even. Remark 4.26. The proof of Lemma 4.25 shows that, if m is odd, the difference of Conley-Zehnder indices is automatically even. Lemma 4.27. Let S1 ∈ S, define Sm(θ) := S1(mθ) and assume Sm ∈ S. Let T ∈ O(R× S1, θn;Sm, Sm) be an element of the form T := ∂s + J0∂θ + Sm(θ − β(s)/m), with β : R → [0, 1] a smooth function satisfying β(s) = 0 near −∞, β(s) = 1 near +∞ and with derivative uniformly bounded by some small constant c. We denote by O one of the spaces O+(C, E;Sm) or O(R × S1, E;S, Sm), S ∈ S. The family ψ = {ψR}, R > 0 of automorphisms of O defined by ψR(D) := D#RT induces an action on the orientations of Det(O) which is reversing if and only if m is even and the difference of Conley-Zehnder indices µCZ(ΨSm)− µCZ(ΨS1) is odd. Proof. Note that T is an isomorphism if c is small enough, by the same argument as the one for D′′u in the proof of Proposition 4.9. We now explain what is the action of ψ on the orientations of Det(O). Let D ∈ O and let V ⊂ Lp be a finite dimensional vector space spanned by Symplectic homology for autonomous Hamiltonians 73 smooth sections with compact support, such that V + imD = Lp. We define the stabilization of D by V as DV : V ⊕W 1,p → Lp, (v, ζ) 7→ v +Dζ. ThenDV is a surjective Fredholm operator and there is a canonical isomorphism Det(D) ≃ Λmax ker DV ⊗ ΛmaxV ∗. For R large enough the glued operator DR = D V #RT : V ⊕W 1,p → Lp is surjective with a uniformly bounded right inverse QR, and moreover the projection onto ker DR given by 1l − QRDR is an isomorphism when restricted to ker DV (see [5, Corollary 6], as well as [13, Proposition 9] for a slightly different setup). Since DV #RT = (D#RT ) V , this induces a natural isomorphism between Det(D) and Det(ψR(D)). The gluing of orientations is associative, hence it is enough to prove the statement for O = O+(C, θn;Sm). We claim that the action induced by ψ is the same as the one induced by φm in Lemma 4.25. Let us choose D ∈ O which is s-independent for s large enough, and let DV be a surjective stabilization. We construct a continuous path in O from ψR(DV ) := ψR(D)V to φm(DV ) := φm(D) V as follows. Let DVt be the conjugation of D V by rt : C → C, z 7→ e−2iπt/mz, and let Tt be the operator ∂s + J0∂θ + Sm(θ − (t+ (1− t)β(s))/m). Then DVt #RTt interpolates between ψR(D V ) and φm(D V ) as t varies from 0 to 1. This is a path of surjective operators admitting a continuous family of right inverses Qt. Given a basis (ζ1, . . . , ζk) of ker D V , a basis of ker DVt is given by (ζ1, . . . , ζk) ◦ rt. By projecting along imQt we obtain a basis of ker DVt #RTt. For t = 1, since DV1 = D 1 #RT1, the elements ζi ◦ r1 are preserved by the projection and form a basis of φ−1m (D V ) which is exactly the one giving the action of φ−1m (or φm) on orientations, as explained in Lemma 4.25. Lemma 4.28. Let γ ∈ P≤αλ and γp, γq be the orbits corresponding to the minimum p and maximum q of fγ respectively. For δ > 0 small enough, the moduli space MA(γp, γq;Hδ, J) is empty if A 6= 0, while for A = 0 it consists of exactly two elements u1, u2 corresponding to the two gradient trajectories of fγ running from p to q. Moreover, they satisfy ǫ(u1) + ǫ(u2) = 0, if γ is a good orbit, ±2, if γ is a bad orbit. Proof. Let c1, c2 be the gradient trajectories of fγ running from p to q. By Theorem 3.7, for δ > 0 small enough each element [uδ] ∈ MA(γp, γq;Hδ, J) corresponds to a unique Floer trajectory with gradient fragments [u] whose endpoints are p and q. For energy reasons there can be no nonconstant Floer trajectory involved in [u] and therefore [u] is either c1 or c2. Since the cylinders u1 and u2 of the form uδ,γ,−∞,+∞ associated to c1 and c2 are already Floer trajectories forHδ, we infer that [uδ] equals either [u1] or [u2], and the homology class A is necessarily zero. Let us introduce the notation ǫ(γ) := 1 if γ is a good orbit and ǫ(γ) := −1 if γ is a bad orbit. The conclusion of the Lemma is equivalent to the relation ǫ(u1) = −ǫ(γ)ǫ(u2). (86) Symplectic homology for autonomous Hamiltonians 74 Let us choose a symplectic trivialization Φγ : TŴ |Sγ → Sγ×(R×R2n−1) such that Φγ(XH) = (1, 0). We assume without loss of generality that ċ1 is a positive multiple of XH , so that Φγ(∂su1) = (f1, 0) with f1 > 0 and Φγ(∂su2) = (f2, 0) with f2 < 0. We denote by Du1 , Du2 the elements of O(R× S1, θn;Sp, Sq) ob- tained by conjugation of D̃u1 , D̃u2 with Φγ . The main point is to consider the operator ψ(Du1) = Du1#RT , with T as in Lemma 4.27. A basis of Det(Dui) corresponding to the coherent orientation is by definition ǫ(ui)(fi, 0), i = 1, 2. The image of this basis under the action of ψ is given by ǫ(u1)(f 1 , 0), for some 1 ∈ W 1,p(R× S1,R) with ‖f 1 − f1‖1,p arbitrarily small with R → ∞, hence 1 > 0 for R large enough. By Lemma 4.27, a basis of Det(Du1#RT ) corre- sponding to the coherent orientation is ǫ(γ)ǫ(u1)(f 1 , 0). Finally, the operators Du1#RT and Du2 can be connected by a continuous path of operators Dt, t ∈ [0, 1] satisfying properties (ii)-(iv) in the proof of Proposition 4.9, as well as the following weaker form of property (i) therein. (i’) there exists a smooth path c : R → Sγ with c(±∞) being fixed critical points of fγ , such that ‖S(s, θ)− S(θ+ ϑ ◦ c(s)− ϑ ◦ c(−∞))‖ is bounded by a constant multiple of δ. The connected components of the set of operators satisfying (i’) and (ii)-(iv) are indexed by homotopy classes of paths c as above. Gluing Du1 to T has precisely the effect of concatenating c1 with (γq|[0,1/m])−1, which is homotopic to c2. The proof of Proposition 4.9 works the same with the weaker assumption (i’) and shows that the operators Dt are surjective and that ker Dt is generated by an element of the form (ft, 0), where ft ∈ W 1,p(R×S1,R) has constant sign for t ∈ [0, 1]. We conclude that ǫ(u1)ǫ(γ)f#1 and ǫ(u2)f2 have the same sign, hence (86) is proved. We now generalize the construction of coherent orientations to the moduli spaces of Morse-Bott trajectories with gradient fragments. We define S̃ to be the space of loops of symmetric matrices S : S1 → M2n(R) such that the symplectic matrix ΨS(1) defined by (85) has exactly one eigenvalue equal to 1, corresponding to the eigenspace R⊕0 ⊂ R⊕R2n−1 = R2n. Let β : R → [0, 1] be a smooth function equal to 0 near −∞ and equal to 1 near +∞. We define V , V to be the one-dimensional real vector spaces generated by the vector-valued functions (1−β(s))(1, 0) and β(s)(1, 0) respectively. In the following we denote by W 1,p,d = W 1,p(ed|s|ds dθ), Lp,d = Lp(ed|s|ds dθ). Given S, S ∈ S̃ we denote Õ(R× S1, E;S, S) the space of linear operators D : W 1,p,d(R × S1, E) ⊕ V ⊕ V → Lp,d(R × S1,Λ0,1E) of the form (∂s + J0∂θ + S(s, θ))(ds− idθ) in a local trivialization of E, for which there exist θ, θ ∈ R/Z such that lims→−∞ S(s, θ) = S(θ + θ) and lims→∞ S(s, θ) = S(θ + θ) in the given trivializations at infinity of E. Given S0 ∈ S, S ∈ S̃ we denote by Õu(R× S1, E;S0, S) Symplectic homology for autonomous Hamiltonians 75 the space of linear operators D :W 1,p(R× S1, E; g+(s)ds dθ) ⊕ V → Lp(R× S1,Λ0,1E; g+(s)ds dθ) with g+(s) := max(1, e ds), which are of the form (∂s + J0∂θ + S(s, θ))(ds − idθ) in a local trivialization of E, and for which there exists θ ∈ R/Z such that lims→−∞ S(s, θ) = S0(θ) and lims→∞ S(s, θ) = S(θ + θ) in the given trivializations at infinity of E. Given S ∈ S̃, S0 ∈ S we denote by Õs(R× S1, E;S, S0) the space of linear operators D :W 1,p(R× S1, E; g−(s)ds dθ) ⊕ V → Lp(R× S1,Λ0,1E; g−(s)ds dθ) with g−(s) := max(1, e −ds), which are of the form (∂s + J0∂θ + S(s, θ))(ds − idθ) in a local trivialization of E, and for which there exists θ ∈ R/Z such that lims→−∞ S(s, θ) = S(θ + θ) and lims→∞ S(s, θ) = S0(θ) in the given trivializations at infinity of E. Given S̃ ∈ S̃ we denote by Õ±(C, E; S̃) the space of linear operators D : W 1,p,d(C, E) ⊕ V± → Lp,d(C,Λ0,1E) of the form (∂x + J0∂y + S(z))dz̄ in a local trivialization of E and such that, when expressed in holomorphic cylindrical coordinates (s, θ) with e±2π(s+iθ) = z as (∂s+J0∂θ+S(s, θ))(ds− idθ), there exists θ± ∈ R/Z so that lims→±∞ S(s, θ) = S̃(θ+θ±) in the given trivialization of E near infinity. Here we use the notation V+ := V and V− := V . Due to the exponential weights, each of the above spaces Õ consists of Fred- holm operators and comes equipped with a canonical real line bundle Det(Õ) whose fiber at D is Det(D). Unlike in the nondegenerate case, the spaces Õ are not generally contractible, hence we have to investigate the orientability of Det(Õ). Given S ∈ S̃ we define m = m(S) to be the maximal positive integer such that S(θ + 1/m) = S(θ), θ ∈ R/Z. The number m is infinite if and only if the loop S is constant, in which case the spaces Õ±(C, E;S), Õu(R×S1, E;S0, S), Õs(R×S1, E;S, S0) are contractible. In the following we shall restrict ourselves to nonconstant loops S ∈ S̃, in which case the above spaces have the homotopy type of S1, while Õ(R× S1, E;S, S) has the homotopy type of S1 × S1 (this is because they fiber over S1, respectively S1 × S1 with contractible fibers). We denote by S1 ∈ S̃ the unique loop such that S(θ) = S1(mθ). Lemma 4.29. Let S ∈ S̃ be nonconstant. Then Det(Õ±(C, E;S)) is nonori- entable if and only if m is even and µRS(S)− µRS(S1) is odd. Proof. We prove the statement only for Õ+ := Õ±(C, E;S), the proof of the other case being similar. The following two remarks will allow us to apply Symplectic homology for autonomous Hamiltonians 76 Lemma 4.25. First, Det(D) is naturally isomorphic to Det(D|W 1,p,d) ⊗ V+ and V+ is a trivial bundle over Õ+. Second, the operator D|W 1,p,d is conjugated to an operator D̃ ∈ O+(C, E;S− dp1l). Hence it is enough to study the orientability of the bundle D̃et(Õ+) over Õ+ with fiber Det(D̃). The bundle D̃et(Õ+) is orientable if and only if its restriction to a loop generating π1(Õ) is orientable. After choosing D ∈ Õ+ which is invariant under conjugation with z 7→ e−2iπ/m, the conjugation of D by rt : C → C, z 7→ e−2iπt/mz provides such a loop Dt, t ∈ [0, 1] with D0 = D1 = D. The orientation on Det(D̃1) obtained by continuation along the path Dt from an orientation on Det(D̃0) is the same as the one induced by the action of φ m (or φm) in Lemma 4.25. Since µCZ(S − dp1l) = µRS(S)− 1/2 and µCZ(S1 − 1l) = µRS(S1)− 1/2, the statement follows from Lemma 4.25. The same kind of argument gives the following result. Lemma 4.30. Let S, S, S ∈ S̃ be nonconstant and S0, S0 ∈ S. The line bundles Det(Õ(R× S1, E;S, S)), Det(Õu(R× S1, E;S0, S)), Det(Õs(R× S1, E;S, S0)) are nonorientable if and only if the condition in Lemma 4.29 holds for S and for one of S, S. � The previous results motivate the following definition. We denote S̃good := {S ∈ S̃ : S constant or µRS(S)− µRS(S1) is even}, S̃bad := {S ∈ S̃ : S nonconstant and µRS(S)− µRS(S1) is odd}, so that S̃bad = S̃ \ S̃good. Although the determinant lines over the various spaces Õ are nonorientable if one of the asymptotes is in S̃bad, we can construct covers O of Õ over which the determinant lines become orientable. Let S0 ∈ S and S ∈ S̃bad with S(θ) = S1(mθ), θ ∈ R/Z. We define Os(R× S1, E;S, S0) to consist of pairs (D, θ) such that θ ∈ R/ 2 Z, D = (∂s+J0∂θ+S(s, θ))(ds−idθ) ∈ Õs(R× S1, E;S, S0) with lims→−∞ S(s, θ) = S(θ + θ). The obvious projection Os → Õs is a double cover and the lift of the determinant bundle to Os is orientable. We define in a completely analogous manner double covers O±(S) → Õ±(S), Ou(S0, S) → Õu(S0, S), S ∈ S̃bad and a cover O(S, S) → Õ(S, S) which is double if exactly one of S, S is in S̃bad, and quadruple if both S, S are in S̃bad. We define now gluing operations between elements of the various spaces Õ. Let K in Õ+(C, E;S), Õ(R × S1, E;S, S), or Õu(R × S1, E;S0, S), and L in Õ−(C, F ;S), Õ(R × S1, F ;S, S), or Õs(R × S1, F ;S, S0). We denote by SK , respectively SL the matrix valued functions involved in K and L near infinity. We assume that SK(s, ·) = lim SL(s, ·) = S(·+ θ0) =: Sθ0 Symplectic homology for autonomous Hamiltonians 77 for some θ0 ∈ R/Z. We choose a cutoff function β : R → [0, 1] such that β(s) = 0 if s ≤ 0 and β(s) = 1 if s ≥ 1. Given R > 0 large we define operators KR and LR by replacing SK and SL by Sθ0 +β(R− s)(SK −Sθ0) and Sθ0+β(R+s)(SL−Sθ0) respectively. We cut out semi-infinite cylinders {s > R} from the base of E, {s < −R} from the base of F , then identify their boundaries using the coordinate θ. We glue the vector bundles E and F using their given trivializations near infinity and denote the resulting vector bundle by E#F . We define K#RL by concatenating KR and LR, so that K#RL belongs to one of the spaces O(CP 1), Õ+(C;S), O+(C;S0), or Õ−(R × S1;S), Õ(R × S1;S, S), Õs(R × S1;S, S0), or O−(C;S0), Õu(R × S1;S0, S), O(R × S1;S0, S0), where we have omitted the symbol E#F from the notation. The above gluing operations admit a straightforward extension to the spaces O. For example, two elements (K, θ) ∈ Ou(S0, S), (L, θ) ∈ Os(S, S0) can be glued if θ = θ, in which case they give rise to an element K#RL ∈ O(S0, S0). Recall that the domain of an operator D in some Õ contains a canoni- cally oriented 1-dimensional summand for each asymptote in S̃, together with a canonical isomorphism with R. We denote by VK , VL the summands corre- sponding to the asymptote S of K and L respectively, and we let V := VK⊕RVL be their (canonically oriented) fibered sum. By [5, Corollary 6], for R > 0 large enough there is a natural isomorphism Det(K ⊕R L) ≃ Det(K#RL) defined up to homotopy, where K ⊕R L is the restriction of K ⊕ L to the fibered sum of their domains. Since V is canonically oriented, it follows that Det(K ⊕R L) is canonically isomorphic to Det(K ⊕ L) ≃ Det(K) ⊗ Det(L). Hence we obtain a canonical isomorphism Det(K) ⊗ Det(L) ∼→ Det(K#RL) defined up to ho- motopy, and inducing an associative gluing operation for orientations. Similar considerations apply to the elements of the spaces Remark 4.31. We can construct a system of coherent orientations on the determinant line bundles Det(Õ±(C, E;S)) and Det(Õ(R × S1, E;S, S)) with S, S, S ∈ S̃good by the same procedure as for the spaces O. We can moreover extend this to a system of coherent orientations involving all spaces O, Õ and O. Nevertheless, if we want that certain orientations have a geometric meaning, we have to impose compatibility conditions which seem ad-hoc in such a general setup. This is why we restrict ourselves in the sequel to the spaces O, Õ and which are relevant for our geometric situation. We use now the notations of Section 3. Given γ ∈ P(H) we denote by Ψγ the linearization of the Hamiltonian flow along γ given by (21) and let Sγ : R/Z → M2n(R) be the corresponding loop of symmetric matrices defined by Ψ̇γ = J0SγΨγ . Then Sγ ∈ S̃good if and only if γ is a good orbit. We similarly define Sγq for each γq ∈ P(Hδ), with q ∈ Crit(fγ). For γ ∈ P(H), γq ∈ P(Hδ) we denote Õs(R× S1, E; γ, γ ) := Õs(R× S1, E;Sγ , Sγ ) etc. Convention. In what follows the spaces Õ will be understood to be indexed Symplectic homology for autonomous Hamiltonians 78 only by good orbits, whereas if one of the asymptotic orbits is bad we use the corresponding double or quadruple cover We construct orientations on the determinant bundles over all spaces O, Õ, O indexed by the elements of P(H) and P(Hδ) as follows. We start by choosing arbitrary orientations of Det(Õ+(C, E; γ)), respectively Det( O+(C, E; γ)), γ ∈ P(H) such that the trivialization of E at infinity extends to C. We then choose orientations of Det(Õs(R×S1, E; γ, γq)), respectively Det( Os(R×S1, E; γ, γq)), γ ∈ P(H), q ∈ Crit(fγ) such that the trivializations of E at infinity extend to R × S1, as follows. If γ is good, the space Õs(R × S1, θn; γ, γq) contains a distinguished family of operators of the form Φγ ◦ Du ◦ Φ−1γ , where u = uδ,γ,−1,∞ is the cylinder corresponding to a semi-infinite gradient trajectory ending at q and Φγ : TŴ |Sγ → Sγ × R2n is a fixed trivialization satisfying Φγ(XH) = (1, 0) ∈ R⊕R2n−1. This family is naturally parametrized by W s(q), hence it is connected. As seen in Proposition 4.9 the above Fredholm operators are surjective and have index 1 − ind(q). If the index is zero we choose the orientation sign to be +1. If the index is one the kernel is generated by a nonvanishing section of the form (f, 0), hence is canonically isomorphic to R⊕0 and therefore admits a canonical orientation. If γ is bad, we choose in an arbitrary way a lift of the operator Φγ ◦Du ◦ Φ−1γ , where u = uδ,γ,−1,∞ is the cylinder corresponding to a constant semi-infinite gradient trajectory at q. This determines a lift of the whole path of operators described above, and hence an orientation of Det( Os(R× S1, E; γ, γq)) by the previous rule. We induce orientations on Det(O+(θn)) by gluing orientations on the line bundles Det(Õ+(θn)) and Det(Õs(θn)). The orientations on Det(Õ+(θn)) and Det(O+(θn)) determine orientations on Det(Õ±(E)) and Det(O±(E)) by re- quiring that the glued orientation on Det(O(CP 1, E)) is the canonical one. We get orientations of Det(Õ(R×S1, E)) by requiring that the orientation induced on Det(O(CP 1, θn#E#θn)) by the gluing operation Õ+(C, θn; γ) ev ×ev Õ(R× S1, E; γ, γ)ev ×ev Õ−(C, θn; γ)→ O(CP 1, θn#E#θn) is the canonical one. Here we have denoted by ev, ev the evaluation maps to S1 at −∞ and +∞ respectively. Similarly, we get orientations on Det(O(R × S1, E)), Det(Õu(R×S1, E)) and Det(Õs(R×S1, E)), as well as orientations on O) for the various spaces Lemma 4.32. The above recipe defines a system of coherent orientations. Proof. We have to prove that, given operators K, L that can be glued lying in one of the spaces O, Õ or O, the coherent orientations oK , oL of Det(K) and Det(L) induce an orientation oK#oL that coincides with the coherent ori- entation of Det(K#L). In the case when K, L belong to some O(R × S1), Õ(R × S1) or O(R × S1) this means that, for a suitable choice of operators Symplectic homology for autonomous Hamiltonians 79 O+(C, θn), A ∈ Õ+(C, θn) or A ∈ O+(C, θn), and B ∈ O−(C, θn), B ∈ Õ−(C, θn) or B ∈ O−(C, θn), with oA, oB the coherent orientations on the respective determinant line bundles, oA#(oK#oL)#oB is the canonical ori- entation on Det(O(CP 1)). Let E and F be the symplectic vector bundles corresponding to K and L respectively. If E = θn or F = θn the conclusion is a direct consequence of the definitions and of the associativity of gluing. In the general case E 6= θn and F 6= θn we give the proof whenK ∈ Õu(R×S1, E; γp, γ) and L ∈ Õs(R×S1, F ; γ, γq), the other cases being similar. Let us introduce an auxiliary loop of symmetric matrices S0 ∈ S such that [S0, J0] = 0, and we define the orientations on Det(O±(C, E′;S0)) to be the canonical ones (see Remark 4.24). This determines in turn orientations on Det(Õu(R×S1, E′;S0, Sγ)), γ ∈ P(H) by requiring that gluing induces the coherent orientation on Det(Õ+(C, θn#E′; γ)). Let A1 ∈ O+(C, E1;S0), K1 ∈ Õu(R× S1, θn;S0, Sγ) with E1#θn = θn#E. By the above definition, we have oA1#oK1 = oA#oK . We obtain oA#(oK#oL)#oB = (oA#oK)#oL#oB = (oA1#oK1)#oL#oB = oA1#oK1#L#oB = oA1#oK1#L#B. The operators A1 and K1#L#B are homotopic to C-linear operators with asymptotic condition S0. The main observation now is that the gluing of two C-linear operators is again C-linear, hence the gluing of the above orientations is the canonical one on Det(O(CP 1)). Let γ, γ be good orbits. In this case the procedure for orienting the Morse- Bott spaces of Floer trajectories M̂A(Sγ , Sγ ;H, J) is entirely similar to the corresponding procedure in the nondegenerate case (it is actually simpler since we do not need the intermediate transition from Lp,d to Lp spaces). Namely, we pull back the orientation on Det(Õ) using the natural map M̂ → Õ. This in turn induces orientations on the quotient spaces MA(Sγ , Sγ ;H, J). Recall that, given oriented vector spaces V ⊂W , we define an orientation on W/V by requiring that the isomorphism V ⊕ (W/V ) ≃W is orientation preserving. Since the stable and unstable manifolds of the functions fγ are canoni- cally oriented, one gets orientations (i.e. signs) on all zero-dimensional moduli spaces of Floer trajectories with gradient fragments MA(p, q;H, {fγ}, J) which involve only good orbits. This is done by the following fibered sum rule. Let fi : Wi → W , i = 1, 2 be linear maps of oriented vector spaces such that f : W1 ⊕W2 → W , (w1, w2) 7→ f1(w1) − f2(w2) is surjective. The orientation on the fibered sum W1f1⊕f2W2 := ker f is defined such that the isomorphism of vector spaces (W1 ⊕W2)/ ker f ∼→ W induced by f changes orientations by the sign (−1)dim W2·dim W . Note that this rule is such that the fibered sum operation is associative for oriented vector spaces, and moreover, if f2 is an ori- entation preserving isomorphism, the natural isomorphismW1f1⊕f2W2 ≃W1 is orientation preserving. Similarly, if f1 is an orientation preserving isomorphism, the natural isomorphism W1f1⊕f2W2 ≃W2 is orientation preserving. Symplectic homology for autonomous Hamiltonians 80 The important remark now is that, although the spaces M̂A(Sγ , Sγ ;H, J) with γ or γ being a bad orbit may not be orientable, we can nevertheless define orientations (i.e. signs) on all zero-dimensional moduli spaces of Floer trajec- tories with gradient fragments MA(p, q;H, {fγ}, J). The sign of an (isolated) point [u] = (cm, [um], . . . , c1, [u1], c0) in this moduli space is determined as fol- lows. For each operator Dui , i = 1, . . . ,m with at least one bad asymptote we choose a lift in the corresponding space O(R×S1). For each ci, i = 0, . . . ,m ly- ing on a bad orbit γi the corresponding operatorDuδ,γi,−Ti/2,Ti/2 admits a unique lift to the space O(Sγi , Sγi) such that it can be glued with both Dui+1 and Dui . Since all these operators are surjective, the orientations of the determinant line bundles over the spaces Õ and O induce orientations on TuiM̂Ai(Sγi , Sγi−1), respectively TWu(p), TW s(q) and T(ci(−Ti/2),Ti)(Sγi × R+), i = 1, . . . ,m − 1. By the fibered sum rule we get an orientation on TuM̂A(p, q;H, {fγ}, J) which we call “the coherent orientation”. On the other hand this vector space carries the “geometric orientation” of the basis (∂sum, . . . , ∂su1). We define the sign ǫ(u) = ǫ([u]) (87) to be +1 if these two orientations coincide, and −1 if they are different. We now want to compare the signs ǫ(u) with the signs ǫ(uδ) of the glued trajectories uδ corresponding to u. The situation is expressed by the follow- ing diagram, in which we dropped the decorations A, (H, {fγ}, J) and (Hδ, J) and in which we have indicated on the morphism arrows the way in which the corresponding isomorphisms of vector spaces act on orientations. Coherent orientation // TM̂(p, q) Id ǫ(u) TM̂(γp, γq)⊕ R Id ǫ(uδ) Coherent orientation Geometric orientation 〈∂sum,...,∂su1〉 // TM̂(p, q) ? // TM̂(γp, γq)⊕ R Geometric orientation 〈∂suδ〉⊕ R The map φ is defined from gluing as follows. The tangent space TM̂(p, q) is the kernel of the operator D ew, w̃ = (vm, um, . . . , v1, u1, v0) considered in Lemma 4.17. Moreover, since the cokernel ofD ew is naturally oriented, the coher- ent orientation of Det(D ew) induces a “coherent” orientation on ker D ew. Recall that the analytical expression ofD ew isDvm⊕Dum⊕D′vm−1⊕. . .⊕D ⊕Du1⊕Dv0 , and note that D ew admits a natural stabilization D ew obtained by replacing D′vi , i = 1, . . . ,m− 1 with D{∂̄T }(vi, Tvi) (see Remark 4.11 for the definitions). By [5, Corollary 6] there is a natural isomorphism φ̃ : ker DR ∼→ ker DRm−1 Gδ( ew) which preserves the coherent orientations. We denote by φ : ker DR ker DR the composition of φ̃ with the projection Π on ker DR along the image of the right inverse Qδ of DGδ( ew) given by Proposition 4.18. Since DGδ( ew) Symplectic homology for autonomous Hamiltonians 81 and Duδ are close in the relevant δ-norm, we get that φ is an isomorphism pre- serving coherent orientations. The vertical maps change orientations by ǫ(u), respectively ǫ(uδ) by defi- nition, and the whole work now goes into determining the action of φ on the geometric orientations. Remark 4.33. If γ is a good orbit and p ∈ Crit(fγ), the geometric orientations on Wu(p) and W s(p) coincide with the coherent ones. Indeed, the unstable manifold Wu(p) is naturally identified with the zero set of the section ∂̄−∞,1 defined on B1,p,dδ (p, Sγ ; fγ) by (54), whereas the stable manifold W s(p) is natu- rally identified with the zero set of the section ∂̄−1,∞ defined on B1,p,dδ (Sγ , q; fγ). The assertion for W s(p) is then a direct consequence of the definition of the ori- entation on Det(Õs(R×S1, θn; γ, γq)). As for Wu(p), let us consider the gluing operation Õu(R× S1, θn; γp, γ)ev ×ev Õs(R× S1, θn; γ, γp) → O(R× S1, θn; γp, γp). We choose the surjective operators D1 := Duδ,γ,−∞,1 , D2 := Duδ,γ,−1,∞ corre- sponding to the constant gradient trajectory at p. With these choicesD1#D2 = Duδ,γ,−∞,∞ =: D also corresponds to the constant gradient trajectory at p. The operator D is an isomorphism and, by the coherent choice of the orientations, the determinant line Det(D) ≃ R is positively oriented. If p is the maximum of fγ then ker D2 ≃ TpSγ as oriented vector spaces (by definition), the kernel of D1 is trivial and its determinant line must be positively oriented. If p is the min- imum of fγ then ker D2 is trivial and its determinant line is positively oriented by definition, therefore ker D1 ≃ TpSγ must have the geometric orientation. Lemma 4.34. Assume dim MA(p, q;H, {fγ}, J) = 0 and fix an element [u] ∈ MA(p, q;H, {fγ}, J) with m ≤ 2 sublevels. Then ǫ(u) = ǫ(uδ) if m = 0, 1 and ǫ(u) = −ǫ(uδ) if m = 2. Proof. If m = 0 the statement is obvious since u consists of a single gradient trajectory and u = uδ (see Lemma 4.28). We now have to show that the map φ in the previous diagram preserves the geometric orientation if m = 1, respectively reverses it if m = 2. Since a shift σ on u1 produces a glued trajectory uδ shifted by the same amount σ, we infer that φ(∂su1) = ∂suδ and, for m = 1, the statement follows from the commutativity of the diagram. Let us now examine the case m = 2. We recall that φ = Π ◦ φ̃, where the isomorphism φ̃ is the composition of the gluing map G in the proof of Propo- sition 4.18 with the projection to ker DR Gδ( ew) along the image of Qδ (see [5]). We first show that φ(∂su1 + ∂su2) is close in ‖ · ‖1,δ-norm to ∂suδ. We denote w̃σ,σ := (v2, u2(·+ σ), v1, u1(·+ σ), v0), σ ∈ R. Then G(0⊕ ∂su2 ⊕ 0⊕ ∂su1 ⊕ 0) is ‖ · ‖1,δ-close to ddσ Gδ(w̃ σ,σ), which is ‖ · ‖1,δ-close to ddσ Gδ(w̃)(·+ σ), which is in turn close to d vδ(· + σ) = ∂svδ. Then φ(∂su1 + ∂su2) = Π(G(0 ⊕ ∂su2 ⊕ 0⊕ ∂su1 ⊕ 0)) is ‖ · ‖1,δ-close to Π(∂suδ) = ∂suδ. We now show that φ(∂su1−∂su2) ∈ ker Duδ ⊕R is a vector having a negative component in the R direction and whose component on ker Duδ is ‖ · ‖1,δ-close to −∂su1. Then the conclusion follows. Symplectic homology for autonomous Hamiltonians 82 Let w̃−σ,σ := (v2, u2(· − σ), v1, u1(·+ σ), v0), σ ∈ R. Then G(0⊕ (−∂su2)⊕ 0 ⊕ ∂su1 ⊕ 0) is ‖ · ‖1,δ-close to ddσ Gδ(w̃ −σ,σ). We define ǫ(σ) := 2δσ and the section Gδ,ǫ(σ)(w̃ −σ,σ) = Gδ(w̃ −σ,σ) + 2δ Gδ,ǫ(w̃) (88) is by construction ‖ · ‖1,δ-close to ddσ Gδ(w̃ −σ,−σ), hence ‖ · ‖1,δ-close to −∂suδ. By adapting the arguments in the proof of Proposition 4.15 one sees that the section ∂̄Tv1+ǫGδ,ǫ(w̃) = D∂̄Tv1 Gδ,ǫ(w̃) +D{∂̄T }(Gδ(w̃), Tv1) · (0, 1) is ‖ · ‖δ-small. Here the sections ∂̄T are of the form ∂̄HT ,J , where HT is the s- dependent Hamiltonian given respectively by (54) on the intervals of definition of v2, v1, v0, and equal toH on the intervals of definition of u1, u2. The previous equation shows that ( d Gδ,ǫ(w̃), 1) ∈ dom(DRGδ( ew)) is ‖·‖1,δ-close to ker D On the other hand, equation (88) shows that G(0 ⊕ (−∂su2)⊕ 0⊕ ∂su1 ⊕ 0) is ‖ · ‖1,δ-close to −∂suδ − 2δ ddǫ Gδ,ǫ(w̃). Hence, after projecting to ker D ker Duδ ⊕ R, we get a vector having a negative component in the R direction and whose component on ker Duδ is ‖ · ‖1,δ-close to −∂suδ. Proof of Proposition 3.9. The special statement concerning the case m = 0 was proved in Lemmas 4.28 and 4.34, whereas the equality ǫ(u) = (−1)m−1ǫ(uδ) in case m = 1, 2 was the content of Lemma 4.34. The proof in the case m ≥ 3 is just a more elaborate version of the proof of Lemma 4.34. We consider the basis of TuM̂(p, q) given by e0 := ∂sum + ∂sum−1 + . . .+ ∂su2 + ∂su1, e1 := −∂sum + ∂sum−1 + . . .+ ∂su2 + ∂su1, em−2 := −∂sum − ∂sum−1 − . . .+ ∂su2 + ∂su1, em−1 := −∂sum − ∂sum−1 − . . .− ∂su2 + ∂su1. It is easy to see that the orientation determined by (e0, . . . , em−1) is the same as the geometric orientation determined by (∂sum, . . . , ∂su1). We have to show that the orientation of the basis (φ(e0), . . . , φ(em−1)) differs from the canonical orientation of 〈∂suδ〉 ⊕ Rm−1 by (−1)m−1. As in Lemma 4.34 we see that φ(e0) is ‖ · ‖1,δ-close to ∂suδ. We now show that φ(ek) ∈ kerDuδ ⊕Rm−1, k = 1, . . . ,m− 1 has a negative component which is bounded away from zero along the corresponding factor R ⊂ Rm−1, that the other components in Rm−1 are close to zero, whereas the component along kerDuδ is close to −∂suδ in ‖ · ‖1,δ-norm. Then the conclusion will follow since the orientation defined by (φ(e0), . . . , φ(em−1)) is the same as the orientation defined by (∂suδ, 0, . . . , 0), (−∂suδ,−1, 0, . . . , 0), . . . , (−∂suδ, 0, . . . , 0,−1)). Symplectic homology for autonomous Hamiltonians 83 Let us fix k = 1, . . . ,m−1. We shall freely use the notation ek for the vector 0⊕ (−∂sum)⊕ 0⊕ . . .⊕ (−∂sum−k+1)⊕ 0⊕∂sum−k⊕ . . . ∂su1⊕ 0 in the domain of the gluing map G defined in the proof of Proposition 4.18. For σ > 0 we denote k :=(vm, um(·−σ), . . . , um−k+1(·−σ), vm−k, um−k(·+σ), . . . , u1(·+σ), v0), w̃−σ,−σ :=(vm, um(·−σ), . . . , um−k+1(·−σ), vm−k, um−k(·−σ), . . . , u1(·−σ), v0). Then G(ek) is ‖ · ‖1,δ-close to ddσ Gδ(w̃ k ). We denote ǫk(ǫ) := (0, . . . , ǫ, . . . , 0), where the parameter ǫ > 0 appears on position m− k. The section Gδ,ǫk(2δσ)(w̃ k ) = Gδ(w̃ k ) + 2δ Gδ,ǫk(w̃) (89) is by construction ‖·‖1,δ-close to ddσ Gδ(w̃ −σ,−σ), hence ‖·‖1,δ-close to −∂suδ. As in Lemma 4.34, by adapting the arguments in the proof of Proposition 4.15 one sees that the section ∂̄Tvm−k+ǫGδ,ǫk(ǫ)(w̃) = D∂̄Tvm−k Gδ,ǫk(ǫ)(w̃) + D{∂̄T }(Gδ(w̃), Tvm−k) · (0, 1) is ‖ · ‖δ-small. As before, the sections ∂̄T are of the form ∂̄HT ,J , where HT is the s-dependent Hamiltonian given respectively by (54) on the intervals of definition of vm, vm−1, . . . , v0, and equal to H on the intervals of definition of um, . . . , u1. The previous equation shows that Gδ,ǫk(ǫ)(w̃), 0, . . . , 1, . . . , 0) ∈ dom(D Gδ( ew) is ‖·‖1,δ-close to ker DR . On the other hand, equation (89) shows that G(ek) is ‖·‖1,δ-close to −∂suδ−2δ ddǫ Gδ,ǫk(ǫ)(w̃). After projecting to ker D get a vector whose k-th component in Rm−1 is negative, whose other components in Rm−1 are small, and whose component on ker Duδ is ‖ · ‖1,δ-close to −∂suδ. Remark 4.35. We chose to define the signs ǫ(u) by comparing the orientation induced on TuM̂A(p, q;H, {fγ}, J) by the fiber sum rule from the coherent ori- entations on TM̂Ai(Sγi , Sγi−1 ;H, J), i = 1, . . . ,m with the orientation of the basis (∂sum, . . . , ∂su1). Another possible recipe would have been the following: induce orientations on TMAi(Sγi , Sγi−1 ;H, J) out of the coherent orientations by quotienting out 〈∂s〉, then apply the fiber sum rule in order to get a sign on the zero-dimensional spaces T[u]MA(p, q;H, {fγ}, J). The sign obtained in this way would have differed from the previously defined ǫ(u) by a factor ±1 which Symplectic homology for autonomous Hamiltonians 84 can be explicitly computed and which depends on the combinatorics of the lev- els of u. The curious reader can test this procedure in the case m = 1: it gives a sign equal to ǫ(u) if p, q are both minima, respectively equal to −ǫ(u) if p, q are both maxima. The following two properties of the fibered sum constitute a useful tool for making the verification (here W1 and W2 are oriented vector spaces). • the natural isomorphism W1 0 ∼→ W1 ⊕ ker f2 changes the orien- tation by (−1)dim W1·(dim W2+1); • the natural isomorphism W1 f1 ∼→ ker f1 ⊕W2 preserves the orien- tation. A Appendix: Asymptotic estimates For all γ ∈ P(H), we choose coordinates (ϑ, z) ∈ S1 × R2n−1 parametrizing a tubular neighbourhood of γ, such that ϑ ◦ γ(θ) = θ and z ◦ γ(θ) = 0. Given a smooth function fγ : Sγ → R, we denote by ϕ s the gradient flow of fγ with respect to the natural metric on S1. In a neighbourhood of γ ∈ P(H) the Floer equation ∂su+ J∂θu− JXH = 0 becomes ∂sZ+J∂θZ+J −JXH = 0, where Z(s, θ) := (ϑ◦u(s, θ)−θ, z◦u(s, θ)). Since XH = on {z = 0} this can be rewritten as ∂sZ + J∂θZ + Sz = 0 for some matrix-valued function S = S(ϑ, z). The matrix S∞(θ) := S(θ, 0) is symmetric. Let A∞ : H k(S1,R2n) → Hk−1(S1,R2n) be the operator defined by A∞Z := J Z + S∞(θ)z. The kernel of A∞ has dimension one and is spanned by the constant vector e1 := (1, 0, . . . , 0). We denote by Q∞ the orthogonal projection onto (ker A∞) and we set P∞ := 1l − Q∞. Then A∞ is invertible when restricted to imQ∞ and Q∞A∞ = A∞. Proposition A.1. Let H ∈ H′ be fixed. There exists r > 0 such that for all J ∈ J ℓ and for all u ∈ MA(Sγ , Sγ ;H, J), γ, γ ∈ P(H) we have ϑ ◦ u(s, θ)− θ − θ0 ∈ W 1,p(]−∞,−s0]× S1,R; er|s|ds dθ), z ◦ u(s, θ) ∈ W 1,p(]−∞,−s0]× S1,R2n−1; er|s|ds dθ), ϑ ◦ u(s, θ)− θ − θ0 ∈ W 1,p([s0,∞[×S1,R; er|s|ds dθ), z ◦ u(s, θ) ∈ W 1,p([s0,∞[×S1,R2n−1; er|s|ds dθ), for some θ0, θ0 ∈ S1 and some s0 > 0 sufficiently large. Symplectic homology for autonomous Hamiltonians 85 Proof. We make the proof only at +∞ since the case of −∞ is entirely similar. For s large enough we set S(s, θ) := S(ϑ ◦ u(s, θ), z ◦ u(s, θ)), so that S∞(θ) = lims→∞ S(s, θ) and lims→∞ |∂sS(s, θ)| = 0. Let A(s) : Hk(S1,R2n) → Hk−1(S1,R2n) be the operator defined by A(s)Z := J Z + S(s, θ)z, so that A∞ = lims→∞A(s). We have A(s) = A(s)Q∞, ∂sQ∞ = Q∞∂s. Since A∞ is invertible when restricted to imQ∞ and Q∞A∞ = A∞, the operators A(s) andQ∞A(s) are also invertible when restricted to imQ∞ for s large enough and there exists c > 0 such that ‖A(s)Q∞Z‖2 ≥ ‖Q∞A(s)Q∞Z‖2 ≥ c‖Q∞Z‖2 for all Z ∈ Hk(S1,R2n). For s large enough we define f(s) := ‖Q∞Z(s)‖2. We have f ′′(s) = ‖∂sQ∞Z‖2 + 〈Q∞Z, ∂2sQ∞Z〉 = ‖∂sQ∞Z‖2 − 〈Q∞Z, ∂sQ∞A(s)Q∞Z〉 = ‖Q∞A(s)Q∞Z‖2 − 〈Q∞Z,Q∞(∂sA(s))Q∞Z −Q∞A(s)2Q∞Z〉 ≥ (c− ε)‖Q∞Z‖2+〈(A(s)∗ −A(s))Q∞Z,A(s)Q∞Z〉+‖A(s)Q∞Z‖2 ≥ (2c− 2ε)‖Q∞Z‖2 ≥ 4ρ2f(s). Here A(s)∗ is the adjoint of A(s) and we used the fact that ‖∂sA(s)‖ → 0, A(s)∗ −A(s) → 0 for s→ ∞ and ‖A(s)‖ is uniformly bounded. Let now s0 be large enough and define g(s) := f(s0)e −2ρ(s−s0). Then g′′ = 4ρ2g, (f − g)′′ ≥ 4ρ2(f − g), (f − g)(s0) = 0 and lims→∞ f(s) − g(s) = 0. Then f − g ≤ 0 on [s0,∞[ because it cannot have a strictly positive maximum. Therefore ‖Q∞Z(s)‖ ≤ ‖Q∞Z(s0)‖e−ρ(s−s0). It is important to note that this estimate holds for any Sobolev normHk. By the Sobolev embedding theorem this implies the following pointwise estimate |Q∞Z(s, θ)| ≤ Ce−ρs, |∂θQ∞Z(s, θ)| = |∂θZ(s, θ)| ≤ Ce−ρs, s ≥ s0. Because ∂sZ +A(s)Z = ∂sZ +A(s)Q∞Z = 0 we obtain |∂sZ(s, θ)| ≤ Ce−ρs, s ≥ s0 and, by integration on [s,∞[ and taking into account that Z(s, θ) converges to (θ0, 0, . . . , 0) for s→ ∞, we obtain the pointwise estimate |(ϑ− θ − θ0, z)| ≤ Ce−ρs. This implies the conclusion for any r < ρ. Symplectic homology for autonomous Hamiltonians 86 Proposition A.2. Let H ∈ H′ and {fγ : Sγ → R} be a collection of perfect Morse functions indexed by γ ∈ Pλ. There exist r > 0 and δ0 > 0 such that for all J ∈ J , γ, γ ∈ P(H), p ∈ Crit(fγ), q ∈ Crit(fγ) and for all (δ, u) ∈ ]0,δ0] (γp, γq;H, {fγ}, J), we have ϑ ◦ u(s, θ)− θ − ϕδfγs (θ0) ∈ W 1,p(]−∞,−s0]× S1,R; er|s|ds dθ), z ◦ u(s, θ) ∈ W 1,p(]−∞,−s0]× S1,R2n−1; er|s|ds dθ), ϑ ◦ u(s, θ)− θ − ϕδfγs (θ0) ∈ W 1,p([s0,∞[×S1,R; er|s|ds dθ), z ◦ u(s, θ) ∈ W 1,p([s0,∞[×S1,R2n−1; er|s|ds dθ), for some θ0, θ0 ∈ S1 and some s0 > 0 sufficiently large. Proof. The proof is similar to the one of Proposition A.1. With the same nota- tions as before the Floer equation satisfied by u can be written in local coordi- nates Z = (ϑ− θ, z) as ∂sZ + J∂θZ + Sz − δ∇fγ(Z1) = 0, (90) where Z1 := ϑ−θ. We again show that f(s) = 12‖Q∞Z‖ 2 satisfies an inequality of the form f ′′(s) ≥ 4ρ2f(s). There are two additional terms to estimate in the expression of f ′′(s), namely 〈Q∞Z, δQ∞A(s)∇fγ(Z1)〉 (91) 〈Q∞Z, δQ∞∂s(∇fγ(Z1))〉. (92) Let P∞ := 1l−Q∞ be the orthogonal projection on ker A∞. The main observa- tion is that Q∞∇fγ(P∞Z1) = 0. As a consequence there exists a matrix-valued function L = L(s, θ) such that Q∞∇fγ(Z1) = LQ∞(Z1). The term (91) is then estimated by 〈Q∞Z, δQ∞A(s)∇fγ(Z1)〉 = 〈Q∞Z, δQ∞A(s)Q∞∇fγ(Z1)〉 ≤ Cδ‖Q∞Z‖2 for s ≥ s0, where s0 depends on u, but C depends only on γ and fγ . Similarly, the term (92) is estimated by 〈Q∞Z, δQ∞∂s(∇fγ(Z1))〉 = 〈Q∞Z, δ∂sQ∞∇fγ(Z1)〉 ≤ Cδ‖Q∞Z‖‖∂sQ∞(Z1)‖ ‖Q∞Z‖2 + ‖∂sQ∞Z‖2 The norm of ∂sQ∞Z = Q∞∂sZ satisfies ‖∂sQ∞Z‖ = ‖Q∞A(s)Z − δQ∞∇fγ(Z1)‖ ≤ C‖Q∞Z‖. Symplectic homology for autonomous Hamiltonians 87 As a consequence, there exists δ0 > 0 and ρ > 0 such that f ′′(s) ≥ 4ρ2f(s) for s ≥ s0 and 0 < δ ≤ δ0. As before, we infer the pointwise bounds |Q∞Z(s, θ)| ≤ Ce−ρs, |∂θQ∞Z(s, θ)| = |∂θZ(s, θ)| ≤ Ce−ρs, s ≥ s0. (93) It remains to estimate P∞Z. For that we write ∇fγ(Z1) = ∇fγ(P∞(Z1)) + KQ∞(Z1) for some matrix-valued function K = K(s, θ). Again, for s ≥ s0, the norm ‖K‖ is uniformly bounded by a constant depending only on γ and fγ . By applying P∞ to the equation (90) and using the fact that P∞∇fγ(P∞(Z1)) = ∇fγ(P∞(Z1)) and P∞(Z1) = P∞(Z) we obtain |∂s(P∞Z)− δ∇fγ(P∞Z)| ≤ Ce−ρs. (94) We claim that this implies |P∞Z(s)− ϕ s (θ0)| ≤ Ce−ρs, s ≥ s0 (95) for a suitable θ0. We choose a Morse coordinate x on Sγ around the critical point q of fγ in which the gradient ∇fγ(x) = ±Mx,M > 0. Then equation (94) becomes ∂s(P∞Z)(s)∓ δMP∞Z(s) = G(s) with |G(s)| ≤ Ce−ρs. Then P∞Z(s) = c(s)e±δMs with e±δMs∂sc(s) = G(s). As a consequence, for δ < ρ/M the function c admits a limit c∞ as s→ ∞ and c(s) = c∞ − G(σ)e∓δMσ dσ. Let θ0 be such that ϕ s (θ0) = c∞e ±δMs (note that c∞ = 0 if q is a maximum). Then |P∞Z(s)− ϕ s (θ0)| = |e±δMs G(σ)e∓δMσ dσ| ≤ Ce−ρs. The estimates (93) and (95) imply the conclusion. Proposition A.3. Let δ ∈]0, δ0] and let uδ ∈ M̂A(γp, γq;Hδ, J). Let Iδ = [s0(δ), s1(δ)] ⊂ R be an interval such that uδ(Iδ×S1) is contained in the domain of a coordinate chart Z = (ϑ, z) around Sγ for some γ ∈ P(H). There exist ρ > 0, θ0 ∈ S1, C > 0 and M > 0 such that z ◦ u(s, θ) and its (first order) derivatives are bounded by Cmax(‖Q∞Z(s0)‖, ‖Q∞Z(s1)‖) cosh(ρ(s− s0+s1 cosh(ρ(s1 − s0)/2) for s ∈ Iδ, θ ∈ S1. If P∞Z(s), s ∈ Iδ stays away from all but one of the critical points of fγ, then ϑ ◦ u(s, θ)− θ − ϕ s (θ0) and its (first order) derivatives are bounded by Cmax(‖Q∞Z(s0)‖, ‖Q∞Z(s1)‖)eδM(s1−s0) cosh(ρ(s− s0+s1 cosh(ρ(s1 − s0)/2) Moreover, if P∞Z(s), s ∈ Iδ stays away from all critical points of fγ , the above bound is improved to (96). Symplectic homology for autonomous Hamiltonians 88 Proof. With the notations of Proposition A.2, the Floer equation satisfied by u can be written in local coordinates Z = (ϑ− θ, z) as ∂sZ + J∂θZ + Sz − δ∇fγ(Z1) = 0, (97) where Z1 := ϑ − θ. Let A∞ = J ddθ + S∞(θ) the asymptotic operator at γ, let Q∞ be the orthogonal projection onto (ker A∞) ⊥ and P∞ := 1l − Q∞. Then, as in Proposition A.2, the quantity f(s) = 1 ‖Q∞Z‖2 satisfies an inequality of the form f ′′(s) ≥ 4ρ2f(s). Define g(s) := max(f(s0), f(s1)) cosh(2ρ(s− s0+s1 cosh(ρ(s1 − s0)) Then (f − g)′′ ≥ 4ρ2(f − g) and f − g cannot have a strictly positive maximum. Since f−g ≤ 0 at s0 and s1, we infer that f−g ≤ 0 on Iδ. As in Proposition A.1, we infer the pointwise bounds for s ≥ s0 |Q∞Z(s, θ)| ≤ Cg1(s), (98) |∂θQ∞Z(s, θ)| = |∂θZ(s, θ)| ≤ Cg1(s), |∂s(P∞Z)(s)− δ∇fγ(P∞Z)(s)| ≤ C1g1(s), where g1(s) := max(‖Q∞Z(s0)‖, ‖Q∞Z(s1)‖) cosh(2ρ(s− s0+s1 cosh(ρ(s1 − s0)) If P∞Z(s) stays away from Crit(fγ), we can assume that ∇fγ(P∞Z(s)) = M in some suitable coordinate on S1. Then the last equation becomes ∂s(P∞Z)(s)− δM = G(s), where |G(s)| ≤ C1g1(s). By direct integration we obtain |(P∞Z)(s)− δMs− c0| = s0+s1 G(σ) dσ s0+s1 cosh(2ρ(s− s0 + s1 )) dσ ∣∣ sinh(ρ(s− s0 + s1 cosh(ρ(s− s0 + s1 Here C2 = C1 max(‖Q∞Z(s0)‖, ‖Q∞Z(s1)‖)/ cosh(ρ(s1 − s0)) and we have used the inequality coshx ≤ 2 cosh(x/2). Therefore, there exists a uniquely Symplectic homology for autonomous Hamiltonians 89 determined θ0 ∈ S1 such that |(P∞Z)(s)− ϕδfγs (θ0)| max(‖Q∞Z(s0)‖, ‖Q∞Z(s1)‖) cosh(ρ(s− s0+s1 cosh(ρ(s1 − s0)) max(‖Q∞Z(s0)‖, ‖Q∞Z(s1)‖) cosh(ρ(s− s0+s1 cosh(ρ(s1 − s0)/2) The last inequality follows from cosh(x/2) ≤ coshx. A similar manipulation on (98) gives |Q∞Z(s, θ)| ≤ C 2max(‖Q∞Z(s0)‖, ‖Q∞Z(s1)‖) cosh(ρ(s− s0+s1 cosh(ρ(s1 − s0)/2) The last two inequalities imply the conclusion of the Proposition in the case when P∞Z(s), s ∈ Iδ stays away from Crit(fγ). If P∞Z(s) is allowed to approach one of the critical points of fγ , the estimate on |Q∞Z(s)| stays the same, but the estimate involving P∞Z(s) has to be modified as follows. In a suitable Morse coordinate chart around the critical point we can assume that ∇fγ(x) = ±Mx, M > 0 and we have to study the equation ∂s(P∞Z)(s)∓ δMP∞Z(s) = G(s), with |G(s)| ≤ C1g1(s). As in Proposition A.2 we have P∞Z(s) = c(s)e±δMs with e±δMs∂sc(s) = G(s). 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Asymptotics with degeneracies. in Contact and symplectic geometry (Cambridge, 1994), Ed. C.B. Thomas. Publ. Newton Inst., vol. 8, Cambridge Univ. Press, Cambridge, 1996, 78–117. [20] H. Hofer, K. Wysocki, and E. Zehnder, Polyfolds and Fredholm theory. Part I, Preliminary preprint, 182 pp., 2005. [21] D. McDuff, D. Salamon, J-holomorphic curves and symplectic topology, AMS Colloquium Publications, vol. 52, AMS, 2004. [22] A. Oancea, A survey of Floer homology for manifolds with contact type boundary or sym- plectic homology, Ensaios Mat. 7 (2004), 51–91. [23] J. Robbin, D. Salamon, The Maslov index for paths, Topology 32 (1993), 827–844. [24] D. Salamon, Lectures on Floer Homology, in Symplectic Geometry and Topology, Eds. Y. Eliashberg and L. Traynor. IAS/Park City Math. Series, vol. 7, AMS, 1999, pp. 143–229. [25] D. Salamon, E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992), 1303–1360. Symplectic homology for autonomous Hamiltonians 91 [26] M. Schwarz, Cohomology operations from S1-cobordisms in Floer homology, PhD thesis, ETH Zurich, 1995. [27] I. Ustilovsky, Contact homology and contact structures on S4m+1, Ph.D. dissertation, Stan- ford University, 1999. [28] C. Viterbo, Functors and computations in Floer homology with applications. I. Geom. Funct. Anal. 9 (1999), 985–1033. Index A∞, asymptotic operator, 27, 84 BCa∗ (H), BC ∗(H), Morse-Bott chain groups, 20 Du, linearized operator, 9 D ew, fibered sum linearized operator, 54 G, gluing map, 58 Gδ,ǫ( ew), pre-glued curve, 46 H′a,b,ǫ, 34 Hδ , perturbed Hamiltonian, 14 P , mixing map, 57 P∞, asymptotic operator, 27, 84 Qδ , right inverse for gluing theorem, 56 Q∞, asymptotic operator, 27, 84 R = R(δ), gluing profile, 46 Rλ, Reeb vector field, 6 S, splitting map, 58 SCa∗ (H), symplectic chain group, 8 SHa∗ (W,ω), symplectic homology, 11 Sγ , circle of orbits, 14 S∞, asymptotic matrix, 27, 84 X, Liouville vector field, 6 XH , Hamiltonian vector field, 6 Λω , Novikov ring, 8 ǭ(u), 20 β, cutoff function, 23, 31, 34, 46, 48, 70 βL, cutoff function, 57 AH , action functional, 7 B′δ, 33 BA = B1,p,d(Sγ , Sγ , A;H), 22 BAδ = B 1,p,d (γp, γq , A;H, {fγ}), 30 E(u), energy, 9 Freg(H, J), regular families {fγ}, 18 H, admissible Hamiltonians, 7 H′, autonomous admissible Hamiltonians, 13 J , admissible a.c. structures, 7 J ′, time-indep. admissible a.c. structures, 17 J ′reg(H), 17 Jreg(H), 9, 17 M0(γ, γ;H, J), cM0(γ, γ;H, J), 9 MA(Sγ , Sγ ;H, J), cM A(Sγ , Sγ ;H, J), 16–17 MA(Sγ , eq;H, J), cM A(Sγ , eq;H, J), 16–17 MA(γ, γ;H, J), cMA(γ, γ;H, J), 8–9 MA]0,δ0] (γp, γq ;H, {fγ}, J), 16 MAm(p, q;H, {fγ}, J), M A(p, q;H, {fγ}, J), 18– O, space of CR operators, 69–70 P(H),Pa(H), 7 i−1(a) , 11–12 S, loops of symmetric matrices without degen- erate directions, 69 ∂̄H , 22 ∂̄Hδ,J , 31 ∂̄a,b,ǫ := ∂̄H′ a,b,ǫ ,J , 34 ǫ(u), 21, 80 ǫ(u), ǫ(uδ), 10, 20 γmin , γMax , surviving orbits, 15 ind(p), index of critical point of fγ , 18 µ(γ), index of Reeb orbit, 8, 15 ev, ev, evaluation maps, 17 ∂, Floer differential, 10 ∂, Morse-Bott differential, 20 eO, space of CR operators, 74–75 eS, loops of symmetric matrices with one degen- erate direction, 74 Spec(M,λ), 6 O, cover of space of CR operators, 76 cW , symplectic completion, 6 bω, symplectic form on the completion cW , 6 eBδ , 44 ξ, contact distribution, 6 {∂̄T }, 43 fγ , Morse function on Sγ , 14, 18 gδ,ǫ(s), weight function for gluing, 47 ha,b,ǫ, 32 ka,b,ǫ, 32 uδ,γ,a,b,ǫ, gradient cylinder, 32 Introduction Symplectic homology The Morse-Bott chain complex Morse-Bott moduli spaces Transversality Compactness for Morse-Bott trajectories Gluing for Morse-Bott moduli spaces Coherent orientations Appendix: Asymptotic estimates

0704.1040

Recent Results on Thermal Casimir Force between Dielectrics and Related Problems

October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr International Journal of Modern Physics A c© World Scientific Publishing Company RECENT RESULTS ON THERMAL CASIMIR FORCE BETWEEN DIELECTRICS AND RELATED PROBLEMS B. GEYER, G. L. KLIMCHITSKAYA∗ and V. M. MOSTEPANENKO† Center of Theoretical Studies and Institute for Theoretical Physics, Leipzig University, Augustusplatz 10/11, 04109, Leipzig, Germany Received 26 May 2006 We review recent results obtained in the physics of the thermal Casimir force acting between two dielectrics, dielectric and metal, and between metal and semiconductor. The detailed derivation for the low-temperature behavior of the Casimir free energy, pressure and entropy in the configuration of two real dielectric plates is presented. For dielectrics with finite static dielectric permittivity it is shown that the Nernst heat the- orem is satisfied. Hence, the Lifshitz theory of the van der Waals and Casimir forces is demonstrated to be consistent with thermodynamics. The nonzero dc conductivity of dielectric plates is proved to lead to a violation of the Nernst heat theorem and, thus, is not related to the phenomenon of dispersion forces. The low-temperature asymptotics of the Casimir free energy, pressure and entropy are derived also in the configuration of one metal and one dielectric plate. The results are shown to be consistent with thermo- dynamics if the dielectric plate possesses a finite static dielectric permittivity. If the dc conductivity of a dielectric plate is taken into account this results in the violation of the Nernst heat theorem. We discuss both the experimental and theoretical results related to the Casimir interaction between metal and semiconductor with different charge carrier density. Discussions in the literature on the possible influence of spatial dispersion on the thermal Casimir force are analyzed. In conclusion, the conventional Lifshitz theory taking into account only the frequency dispersion remains the reliable foundation for the interpretation of all present experiments. Keywords: Casimir force; Lifshitz theory; thermal corrections. 1. INTRODUCTION The Casimir effect1 is the force and also the specific polarization of the vacuum arising in restricted quantization volumes and originating from the zero-point oscil- lations of quantized fields. This force acts between two closely spaced macrobodies, between an atom or a molecule and macrobody or between two atoms or molecules. During more than fifty years, passed after the discovery of the Casimir effect, it has attracted much theoretical attention because of numerous applications in quantum field theory, atomic physics, condensed matter physics, gravitation and cosmology, ∗On leave from North-West Technical University, St.Petersburg, Russia. †On leave from Noncommercial Partnership “Scientific Instruments”, Moscow, Russia. http://arxiv.org/abs/0704.1040v1 October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 2 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko mathematical physics, and in nanotechnology (see monographs 2–4 and reviews 5–7). In multidimensional Kaluza-Klein supergravity the Casimir effect was used8 as a mechanism for spontaneous compactification of extra spatial dimensions and for constraining the Yukawa-type corrections to Newtonian gravity.9,10,11,12,13 In quantum chromodynamics the Casimir energy plays an important role in the bag model of hadrons.3 In cavity quantum electrodynamics the Casimir interaction be- tween an isolated atom and a cavity wall leads to the level shifts of atomic electrons depending on the position of the atom near the wall.2 Both the van der Waals and Casimir forces are used14,15 for the theoretical interpretation of recent experiments on quantum reflection and Bose-Einstein condensation of ultracold atoms on or near the cavity wall of different nature. In condensed matter physics the Casimir effect turned out to be important for interaction of thin films, in wetting processes, and in the theory of colloids and lattice defects.16 The Casimir force was used to actuate nanoelectromechanical devices17 and to study the absorption of hydrogen atoms by carbon nanotubes.18 Theoretical work on the calculation of the Casimir ener- gies and forces stimulated important achievements in mathematical physics and in the theory of renormalizations connected with the method of generalized zeta func- tion and heat kernel expansion.6,19 All this made the Casimir effect the subject of general interdisciplinary interest and attracted permanently much attention in the scientific literature. The last ten years were marked by the intensive experimental investigation of the Casimir force between metallic test bodies (see Refs. 20–33). During this time the agreement between experiment and theory on the level of 1-2% of the measured force was achieved. This has become possible due to the use of modern laboratory techniques, in particular, of atomic force microscopes and micromechanical torsional oscillators. Metallic test bodies provide advantage in comparison with dielectrics be- cause their surfaces avoid charging. In Refs. 34, 35, where the importance of the Casimir effect for nanotechnology was pioneered, it was demonstrated that at sepa- rations below 100 nm the Casimir force becomes larger than the typical electrostatic forces acting between the elements of microelectromechanical systems. Bearing in mind that the miniaturization is the main tendency in modern technology, it be- comes clear that the creation of new generation of nanotechnological devices with further decreased elements and separations between them would become impossible without careful account and calculation of the Casimir force. Successful developments of nanotechnologies based on the Casimir effect calls for more sophisticated calculation methods of the Casimir forces. Most of theoreti- cal output produced during the first decades after Casimir’s discovery did not take into account experimental conditions and real material properties of the boundary bodies, such as surface roughness, finite conductivity and nonzero temperature. The basic theory giving the unified description of both the van der Waals and Casimir forces was elaborated by Lifshitz36,37,38 shortly after the publication of Casimir’s paper. It describes the boundary bodies in terms of the frequency dependent dielec- October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 3 tric permittivity ε(ω) at nonzero temperature T . In the applications of the Lifshitz theory to dielectrics it was supposed that the static dielectric permittivity (i.e., the dielectric permittivity at zero frequency) is finite. The case of ideal metals was ob- tained from the Lifshitz theory by using the so-called Schwinger’s prescription,39 i.e., that the limit |ε(ω)| → ∞ should be taken first and the static limit ω → 0 second. For ideal metals the same result, as follows from the Lifshitz theory com- bined with the Schwinger’s prescription, was obtained independently in the frame- work of thermal quantum field theory in Matsubara formulation.40 However, the cases of real dielectrics and metals (which possess some nonzero dc conductivity at T > 0 and finite dielectric permittivity at nonzero frequencies, respectively) re- mained practically unexplored for a long time. The case of semiconductor boundary bodies was also unexplored despite of the crucial role of semiconductor materials in nanotechnology. Starting in 2000, several theoretical groups in different countries attempted to describe the Casimir interaction between real metals at nonzero temperature in the framework of Lifshitz theory. They have used different models of the metal con- ductivity and arrived to controversial conclusions. In Ref. 41, using the dielectric function of the Drude model, quite different results than for ideal metals were ob- tained. According to Ref. 41, at short separations (low temperatures) the thermal correction to the Casimir force acting between real metals is several hundred times larger than between ideal ones. In addition, at large separations of a few microm- eters (high temperatures) a two times smaller magnitude of the thermal Casimir force was found than between ideal metals (the latter is known as “the classical limit”42,43). In Refs. 44, 45 the dielectric permittivity of the plasma model was used to describe real metals and quite different results were obtained. At short sep- arations the thermal correction appeared to be small in qualitative agreement with the case of ideal metals. At large separations for real metals the familiar classical limit was reproduced. Later the approach of Ref. 41 was supported in Refs. 46, 47. The plasma model approach can be used at such separations a that the char- acteristic frequency c/(2a) belongs to the region of infrared optics. Later a more general framework, namely the impedance approach was suggested48,49 which is applicable at any separation larger than the plasma wave length. It was supported in Refs. 7, 50, 51. In the region of the infrared optics, the impedance approach leads to practically the same results as the plasma model approach. As was shown in Refs. 52, 53, the Drude model approach leads to the violation of the Nernst heat theorem when applied to perfect metal crystal lattices with no impurities. This ap- proach was also excluded by experiment at 99% confidence in the separation region from 300 to 500nm and at 95% confidence in the wider separation region from 170 to 700 nm.30,32,33 On the contrary, the plasma model and impedance approaches were shown to be in agreement with thermodynamics and consistent with exper- iment. The polemic between different theoretical approaches to the description of the thermal Casimir force in the case of real metals can be found in Refs. 47, 54–56. October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 4 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko These findings on the application of the Lifshitz theory to real metals have in- spired a renewed interest in the Casimir force between dielectrics. As was mentioned above, at nonzero temperature dielectrics possess an although small but not equal to zero dc conductivity. In Ref. 57, 58 the van der Waals force arising from the dc conductivity of a dielectric plate was shown to lead to large effect in noncontact atomic friction,59 a phenomenon having so far no satisfactory theoretical explana- tion. This brings up the question: Is it necessary or possible to take into account the dc conductivity of dielectrics in the Lifshitz theory? Recall that in the case of a positive answer the static dielectric permittivity of a dielectric material would be infinitely large. It is amply clear that the resolution of the above issue should be in accordance with the fundamentals of thermodynamics. For this reason, it is de- sirable to investigate the low-temperature behavior of the Casimir free energy and entropy for two dielectric plates both with neglected and included effects of the dc conductivity. A major breakthrough in the investigation of this problem was achieved in the year 2005. In Ref. 60 a new variant of perturbation theory was developed in a small parameter proportional to the product of the separation distance between the plates and the temperature. As a result, the behavior of the Casimir free energy, entropy and pressure at low temperatures was found analytically. If the static dielectric permittivity is finite, the thermal correction was demonstrated to be in accordance with thermodynamics. This solves positively the fundamental problem about the agreement between the Lifshitz theory and thermodynamics for the case of two dielectric plates. In Ref. 60 it was shown that, on the contrary, the formal inclusion of a small conductivity of dielectric plates at low frequencies into the model of their dielectric response leads to a violation of the Nernst heat theorem. This result gives an important guidance on how to extrapolate the tabulated optical data for the complex refractive index to low frequencies in numerous applications of the van der Waals and Casimir forces. All these problems and related ones arising for semiconductor materials are discussed in this review. In Sec. 2 we derive the analytical behavior of the Casimir free energy, entropy and pressure in the configuration of two parallel dielectric plates at both low and high temperature. It is demonstrated that if the static dielectric permittivity is finite the Lifshitz theory is in agreement with thermodynamics. Sec. 3 contains the derivation of the low-temperature behavior for the Casimir free energy and entropy between two dielectric plates with included dc conductivity. In this case the Lifshitz theory is found to be in contradiction with the Nernst heat theorem. The conclusion is made that the conductivity properties of a dielectric material at a constant current are unrelated to the van der Waals and Casimir forces and must not be included into the model of dielectric response. In Sec. 4 we consider the thermal Casimir force between metal and dielectric. This problem was first investigated in Ref. 61. It was found that for dielectric plate with finite static dielectric permittivity the Nernst heat theorem is satisfied but the Casimir entropy may take negative values. Here we not only reproduce an analytical proof of the Nernst heat theorem but also October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 5 find the next perturbation orders in the expansion of the Casimir free energy and entropy in powers of a small parameter. The results obtained in high-temperature limit are also provided. Sec. 5 is devoted to the Casimir interaction between metal and dielectric plates with included dc conductivity of the dielectric material. We demonstrate that in this case the Nernst heat theorem is violated. Sec. 6 contains the discussion of semiconductors which present a wide variety of electric properties varying from metallic to dielectric. We consider the Casimir interaction between metal and semiconductor test bodies and formulate the criterion when it is appro- priate to include the dc conductivity of a semiconductor into the model of dielectric response. In doing so, the results of recent experiments62,63 on the measurement of the Casimir force between metal and semiconductor test bodies are taken into account. To this point the assumption has been made that metal, dielectric and semiconductor materials of the Casimir plates possess only temporal dispersion, i.e., can be described by the dielectric permittivity depending only on frequency. In Sec. 7 we discuss recent controversial results by different authors (see, e.g., Refs. 64 – 68) attempting to take into account also spatial dispersion. As is shown in this section, the way of inclusion of spatial dispersion into the Lifshitz theory, used in Refs. 64 – 68, is unjustified. We argue that the account of spatial dispersion cannot influence theoretical results obtained with the help of usual, spatially local, Lifshitz theory within presently used ranges of experimental separations. Sec. 8 contains our conclusions and discussion. 2. NEW ANALYTICAL RESULTS FOR THE THERMAL CASIMIR FORCE BETWEEN DIELECTRICS We consider two thick dielectric plates (semispaces) described by the frequency- dependent dielectric permittivity ε(ω) and restricted by the parallel planes z = ±a/2 with a separation a between them, in thermal equilibrium at temperature T . The Lifshitz formula for the free energy of the van der Waals and Casimir interaction between the plates is given by2,3,4,5,6,36,37,38 F(a, T ) = kBT 1− δl0 k⊥dk⊥ (1) 1− r2‖(ξl, k⊥)e−2aql 1− r2⊥(ξl, k⊥)e−2aql where the reflection coefficients for two independent polarizations of electromagnetic field are defined as r‖(ξl, k⊥) = εlql − kl εlql + kl , r⊥(ξl, k⊥) = kl − ql kl + ql . (2) Here k⊥ is the magnitude of the wave vector in the plane of plates, ξl = 2πkBT l/~ are the Matsubara frequencies, kB is the Boltzmann constant, εl = ε(iξl), and + k2⊥, kl = ε(iξl) + k2⊥. (3) October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 6 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko The problems in the application of the Lifshitz theory to real materials discussed in the Introduction are closely connected with the values of the reflection coefficients at zero Matsubara frequency. For later use we discuss it for the various cases. • For ideal metals40 it holds r‖(0, k⊥) = r⊥(0, k⊥) = 1. (4) • For real metals described by the dielectric function of the Drude model, ε(iξl) = 1 + ξl[ξl + ν(T )] , (5) where ωp is the plasma frequency and ν(T ) is the relaxation parameter, it holds41 r‖(0, k⊥) = 1, r⊥(0, k⊥) = 0. (6) Eq. (6) results in the discontinuity between the cases of ideal and real metals and leads to the violation of the Nernst heat theorem for metallic plates having perfect crystal lattices.52,53,54,56 • For real metals described by the dielectric function of the plasma model, ε(iξl) = 1 + , (7) from Eq. (2) it follows:44,45 r‖(0, k⊥) = 1, r⊥(0, k⊥) = c2k2⊥ + ω p − ck⊥ c2k2⊥ + ω p + ck⊥ . (8) Here, in the limit of ideal metals (ωp → ∞) the continuity is preserved because r⊥(0, k⊥) in Eq. (8) goes to unity. The free energy (1) calculated with the permittivity (8) is also consistent with thermodynamics. • For dielectrics and semiconductors the dielectric permittivities at the imaginary Matsubara frequencies are given by the Ninham-Parsegian representation,69,70 ε(iξl) = 1 + 1 + ξ2l /ω , (9) where the parameters Cj are the absorption strengths satisfying the condi- Cj = ε0 − 1 (10) and ωj are the characteristic absorption frequencies. Here, the static dielec- tric permittivity ε0 ≡ ε(0) is supposed to be finite. Although Eq. (10) is an October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 7 approximate one, it gives a very accurate description for many materials.71 By the substitution of Eq. (9) in Eq. (2) one arrives at r‖(0, k⊥) ≡ r0 = ε0 − 1 ε0 + 1 , r⊥(0, k⊥) = 0. (11) Note that the vanishing of the transverse reflection coefficient for dielectrics at zero frequency in Eq. (11) has another meaning than for the Drude metals in Eq. (6). For Drude metal the parallel reflection coefficient is equal to the physical value for real photons at normal incidence, i.e., to unity, and the transverse one vanishes instead of taking unity, its physical value. This results in the violation of the Nernst heat theorem for perfect crystal lattices. In the case of dielectrics both reflection coefficients at zero frequency in Eq. (11) depart from the physical value for real photons which is equal to ( ε0−1)/( ε0+1). In this case, however, one of them is larger and the other one is smaller than the physical value. As we will see below, this leads to the preservation of Nernst’s heat theorem confirming that Eq. (9), despite being approximate, describes the material properties of dielectric and semiconductor plates in a thermodynamic consistent way. Now we derive the analytic representation for the Casimir free energy in Eq. (1) at low temperatures. For convenience in calculations, we introduce the dimensionless variables = τl, y = 2aql, (12) where ξc = c/(2a) is the characteristic frequency, τ = 4πkBaT/(~c), and ql was defined in Eq. (3). Then the Lifshitz formula (1) takes the form F(a, T ) = ~cτ 32π2a3 1− δl0 dy f(ζl, y) , (13) where f(ζ, y) = f‖(ζ, y) + f⊥(ζ, y), (14) f‖,⊥(ζ, y) = y ln 1− r2‖,⊥(ζ, y)e−y , (15) and reflection coefficients (2), in terms of variables (12), being given by r‖(ζl, y) = εly − y2 + ζ2l (εl − 1) εly + y2 + ζ2l (εl − 1) , r⊥(ζl, y) = y2 + ζ2l (εl − 1)− y y2 + ζ2l (εl − 1) + y . (16) To separate the temperature independent contribution and thermal correction in Eq. (13) we apply the Abel-Plana formula,3,6 1− δl0 F (l) = F (t) dt+ i F (it)− F (−it) e2πt − 1 , (17) where F (z) is an analytic function in the right half-plane. Here, taking it as F (x) = dy f(x, y) (18) October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 8 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko and using Eq. (17), we can identically rearrange Eq. (13) to the form F(a, T ) = E(a) + ∆F(a, T ) , (19) where E(a) is the energy of the van der Waals or Casimir interaction at zero tem- perature, E(a) = 32π2a3 dy f(ζ, y) , (20) and ∆F(a, T ) is the thermal correction to this energy, ∆F(a, T ) = 32π2a3 F (iτt)− F (−iτt) e2πt − 1 . (21) Note that, in fact, Eq. (21) describes the dependence of the free energy on the tem- perature arising from the dependence on temperature of the Matsubara frequencies. Thus, ∆F(a, T ) in (21) coincides with the thermal correction to the energy, defined as F(a, T )− F(a, 0), only for plate materials with temperature independent prop- erties. The asymptotic expressions for the energy E(a) at both short and large sep- arations are well known.6,37,38 Below we find the asymptotic expressions for the thermal correction (21) under the conditions τ ≪ 1 and τ ≫ 1. Taking into account the definition of τ in Eq. (12), the asymptotic expressions at τ ≪ 1 are applicable both at small and large separations if the temperature is sufficiently low. We begin with condition τ ≪ 1. Let us substitute Eq. (9) in Eqs. (14) – (16), expand the function f(x, y) in powers of x = τt, and than integrate the obtained expansion with respect to y from x to infinity in order to find F (x) in Eq. (18) and F (ix)− F (−ix) in Eq. (21). It is easy to check that f⊥(ζ, y) does not contribute to the leading, second, order in the expansion of F (ix)− F (−ix) in powers of x. Thus, we can restrict ourselves by the consideration of the expansion f‖(x, y) = y ln 1− r20e−y 2ε0 r ε0 + 1 y (ey − r20) 4ξ2c r (ε0 + 1)ω ey − r20 +O(x3), (22) where r0 was defined in Eq. (11). Note that for simplicity we consider here only one oscillator in Eq. (9) and put ωj = ω1. The case of several oscillator modes can be considered in an analogous way. As a next step, we integrate Eq. (22) term by term according to Eq. (18), expand the partial results in powers of x and sum up the obtained series. Thereby we obtain the following expressions: Z1(x) ≡ y dy ln 1− r20e−y (1 + nx)e−nx = −Li3(r20)− ln(1− r20) + O(x3) , (23) October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 9 Z2(x) ≡ ε0 + 1 y (ey − r20) = − 2ε0x ε0 + 1 r2n0 Ei(−nx) , (24) Z3(x) ≡ 4r20ξ (ε0 + 1)ω ey − r20 4ξ2cx (ε0 + 1)ω (1 + nx)e−nx (ε0 + 1)ω x2 Li2(r 1− r20 +O(x5) , (25) where Lin(z) is the polylogarithm function and Ei(z) is the exponential integral function. From these equations it follows Z1(ix) − Z1(−ix) = O(x3) , Z3(ix)− Z3(−ix) = O(x5) , (26) Z2(ix) − Z2(−ix) = 2iπ ε0 + 1 1− r20 x2 +O(x3) , (27) and, thus, Z1 and Z3 do not contribute to the leading order in the expansion of F (ix)− F (−ix). The latter is determined by Z2 only. As a result, we arrive at F (ix)− F (−ix) = iπ (ε0 − 1)2 2(ε0 + 1) x2 − iαx3 +O(x4), (28) where r0 was substituted from Eq. (11) and α was introduced for the still unknown real coefficient of the next to leading order resulting from Z1 and Z2 as well as, possibly, from f⊥(ζ, y). At this stage it is difficult to determine the value of this coefficient because all powers in the expansion of f(x, y) contribute to it. Remark- ably, the two leading orders depend only on the static dielectric permittivity ε0 and are not influenced by the dependence of the dielectric permittivity on the frequency contained in Z3. Substituting Eq. (28) in Eq. (21) and using Eq. (19), we obtain F(a, T ) = E(a)− ~c 32π2a3 (ε0 − 1)2 ε0 + 1 τ3 − C4τ4 +O(τ5) , (29) where C4 ≡ α/240 and ζ(z) is the Riemann zeta function. So far we have considered the free energy. The thermal pressure is obtained as P (a, T ) = −∂F(a, T ) = P0(a)− 32π2a4 4 +O(τ5) , (30) where P0(a) = −∂E(a)/∂a is the Casimir pressure at zero temperature. In order to determine the value of the coefficient C4 of the leading term, we express the pressure directly through the Lifshitz formula P (a, T ) = − ~cτ 32π2a4 1− δl0 (ζl, y) ey − r2 (ζl, y) r2⊥(ζl, y) ey − r2⊥(ζl, y) . (31) Again, applying the Abel-Plana formula (17), we represent the pressure as follows, P (a, T ) = P0(a) + ∆P (a, T ), (32) October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 10 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko where the thermal correction to P0(a), the pressure at zero temperature, is ∆P (a, T ) = − i~cτ 32π2a4 Φ(iτt)− Φ(−iτt) e2πt − 1 and the function Φ(x) is given by Φ(x) ≡ Φ‖(x) + Φ⊥(x) , Φ‖,⊥(x) = y2 r2 (x, y) ey − r2 (x, y) . (34) First, we determine the leading term of the expansion of Φ⊥(x) in powers of x. For this purpose, let us introduce the new variable v = y/x and note that the reflection coefficient r⊥(x, v) depends on x only through the frequency dependence of ε given by Eq. (9). Thus, we can rewrite and expand Eq. (34) as follows: Φ⊥(x) = x v2 r2⊥(x, v) evx − r2⊥(x, v) v2 r2⊥(v) 1− r2⊥(v) + O(x4), (35) where, according to Eq. (16), r⊥(v) ≡ r⊥(0, v) = v2 + ε0 − 1− v√ v2 + ε0 − 1 + v . (36) Integration in Eq. (35) with account of Eq. (36) results in Φ⊥(x) = 1− ε0(3− ε0) + O(x4) , (37) from which it follows: Φ⊥(ix)− Φ⊥(−ix) = −i ε0 (3− ε0) + O(x5). (38) The expansion of Φ‖(x) from Eq. (34) in powers of x is somewhat more cumber- some. It can be performed in the following way. As is seen from the second equality in Eq. (26), the dependence of the dielectric permittivity on frequency contributes to F (ix)−F (−ix) starting from only the 5th power in x. Bearing in mind the con- nection between free energy and pressure, we can conclude that the dependence on the frequency contributes to Φ‖(ix)−Φ‖(−ix) starting from the 4th order. We are looking for the lowest (third) order expansion term of Φ‖(ix)−Φ‖(−ix). Because of this, it is permissible to disregard the frequency dependence of ε and describe the dielectric by its static dielectric permittivity. To begin with, we identically rearrange Φ‖(x) in Eq. (34) by subtracting and adding the two first expansion terms of the function under the integral in powers of x, Φ‖(x) = (x, y) ey − r2 (x, y) ey − r20 ε0 + 1 (1− r20e−y) ey − r20 − x2 2ε0 ε0 + 1 (1− r20e−y) , (39) October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 11 and consider these three integrals separately. The first integral in terms of the new variable v = y/x reads Q1(x) ≡ x3 evx − r2 − v2 r evx − r20 ε0 + 1 (1− r20e−vx) , (40) where, in accordance with Eq. (16), r‖(v) ≡ r‖(0, v) = ε0v − v2 + ε0 − 1 ε0v + v2 + ε0 − 1 . (41) Expanding Q1(x) in powers of x and explicitly calculating the remaining inte- grals for the lowest, third, power of x results in Q1(x) = x 1− r2 1− r20 ε0 + 1 (1− r20) +O(x4) (42) 1 + 3ε0 − 2 ε20 2 r20 1− r20 − 12 ε0 ε0 + 1 (1− r20) +O(x4). The second and third integrals on the right-hand side of Eq. (39) are simply deter- mined with the following result: Q2(x) ≡ ey − r20 (2 + 2nx+ n2x2)e−nx = 3Li3(r 1− r20 +O(x4) , (43) Q3(x) ≡ − ε0 + 1 (1− r20e−y) = − 2ε0x ε0 + 1 1− r20e−x = − 2ε0 ε0 + 1 1− r20 − x3 r (1− r20)2 +O(x4) . (44) Substituting Eqs. (42), (43) and (44) into Φ‖(x) = Q1(x) + Q2(x) +Q3(x), we arrive at Φ‖(ix)− Φ‖(−ix) = −i 2ε20 − 3ε0 − 1 + O(x4). (45) Then, by summing Eqs. (38) and (45), the result is obtained Φ(ix)− Φ(−ix) = −i 2 + ε 0 − ε 0 − 2 + O(x4). (46) Now we substitute Eq. (46) in Eq. (33) and perform integration. Finally, from Eq. (32) the desired expression for the Casimir pressure is derived P (a, T ) = P0(a)− 32π2a4 ε0 − 1 ε20 + ε 0 − 2 τ4 +O(τ5) . (47) By comparison with Eq. (30) the explicit form of the coefficient C4 is found as ε0 − 1 ε20 + ε 0 − 2 /720 (48) October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 12 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko and, thus, both two first perturbation orders in the expansion for the free energy (29) are determined. Equations (29), (47) and (48) solve the fundamental problem of the thermody- namic consistency of the Lifshitz theory in the case of two dielectric plates. From Eqs. (29) and (48) the entropy of the van der Waals and Casimir interaction between plates takes the form S(a, T ) = − ∂F(a, T ) 3kBζ(3)(ε0 − 1)2 64π3a2(ε0 + 1) τ2 (49) 2π2(ε0 + 1) 0 + 2ε0 + 2 ε0 + 2 135ζ(3)( ε0 + 1)2 τ + O(τ2) As is seen from Eq. (49), in the limit τ → 0 (T → 0) the lower order contributions to the entropy are of the second and the third powers in the small parameter τ . Thus, the entropy vanishes when the temperature goes to zero as it must be in accordance with the third law of thermodynamics (the Nernst heat theorem). A similar behavior was obtained for ideal metals40,72,73 and for real metals described by the plasma model.45,52 For example, in the case of plates made of ideal metal the entropy at low temperatures is given by40,72,73 S(a, T ) = 3kBζ(3) 32π3a2 1− 2π 135ζ(3) τ + O(τ2) . (50) Note, however, that the expansion coefficients in Eq. (50) cannot be obtained as a straightforward limit |ε0| → ∞ in Eq. (49) and the above equations for the free energy and pressure. The mathematical reason is that it is impermissible to interchange the limiting transitions τ → 0 and |ε0| → ∞ in the power expansions of functions depending on ε0 as a parameter. Remarkably, the low-temperature behavior of the free energy, pressure and en- tropy of nonpolar dielectrics in Eqs. (29), (47) and (49) is universal, i.e., is deter- mined only by the static dielectric permittivity. The absorption bands included in Eq. (9) do not influence the low-temperature behavior. A more simple derivation of the results (29), (47)–(49) for dielectrics with constant ε is contained in Ref. 74. As was demonstrated above, all these results remain unchanged if the dependence of dielectric permittivity on frequency is taken into account. In Ref. 60 more general results were obtained related to two dissimilar dielectric plates with dielectric permittivities ε(1)(ω) and ε(2)(ω). For brevity here we present only the final expressions for the low-temperature behavior of the Casimir free energy, pressure and entropy between dissimilar plates. They are as follows:60 F(a, T ) = E(a)− ~c 32π2a3 0 + ε 0 + 2ε 0 + 1)(ε 0 + 1) 0 − 1)(ε 0 − 1) 0 + ε − C4 τ +O(τ2) October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 13 P (a, T ) = P0(a)− 32π2a4 4 +O(τ5) , (51) S(a, T ) = 3ζ(3) 0 + ε 0 + 2ε 0 + 1)(ε 0 + 1) 0 − 1)(ε 0 − 1) 0 + ε − C4 τ +O(τ2) Here ε (1,2) 0 ≡ ε(1,2)(0) and the coefficient C4 is given by60 0 + ε 0 + ε 0 + 2ε 0 − ε 0 − 3ε 0 − 3ε 0 + 1 0 − ε 0 − ε 0 − 1 0 − 1 0 + ε Artanh 0 + ε 0 − ε 0 − ε It is easily seen that in the limit ε 0 = ε 0 = ε0 equations (51), (52) coincide with equations (29), (47)–(49) having obtained above. Note that in the application region of low-temperature asymptotic expressions the entropy of the Casimir interaction between dielectric plates is nonnegative. The obtained analytic behavior of the free energy, pressure and entropy at low temperatures can be compared with the results of numerical computations using the Lifshitz formula. Dielectric properties of the plates can be described by the static dielectric permittivity or more precisely using the optical tabulated data for the complex index of refraction. As an example, in Fig. 1 we present the thermal corrections to the Casimir energy (a) and pressure (b) at a separation a = 400 nm as functions of temperature in the configuration of two dissimilar plates made of high-resistivity Si and SiO2. The dielectric permittivities of both materials along the imaginary frequency axis were computed in Ref. 75 using the optical data of Ref. 76. The precise thermal corrections computed by taking into account these permittivities are shown by the solid lines and corrections computed by our analyt- ical asymptotic expressions are shown by the long-dashed lines. Short-dashed lines indicate the results computed by the Lifshitz formula with constant dielectric per- mittivities of Si and SiO2 equal to ε 0 = 11.67 and ε 0 = 3.84, respectively. As is seen in Fig. 1a,b, at T < 60K the results obtained using the analytical asymptotic October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 14 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko 25 50 75 100 125 150 175 200 T (K) −∆F (pJ) 25 50 75 100 125 150 175 200 T (K) −∆P (µPa) Fig. 1. Magnitudes of the thermal corrections to the energy (a) and pressure (b) in configuration of two plates, one made of Si and another one of SiO2, at a separation a = 400 nm as a function of temperature calculated by the use of different approaches: by the Lifshitz formula and tabulated optical data (solid lines), by the Lifshitz formula and static dielectric permittivities (short-dashed lines), by the asymptotic expressions in Eqs. (51) and (52) (long-dashed lines). expressions practically coincide with the solid lines computed using the tabulated optical data for the materials of the plates. Now we return to the case of two similar dielectric plates and consider the asymptotic expressions under the condition τ ≫ 1, i.e., at high temperatures (large separations). It is well known37,38,77 that in this case the approximation of static October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 15 dielectric permittivity works good and the main contribution is given by the zero- frequency term of the Lifshitz formula (13) F(a, T ) = ~cτ 64π2a3 ydy ln 1− r20e−y (the other terms being exponentially small). Performing the integration in Eq. (53) we obtain F(a, T ) = − kBT 16πa2 . (54) In a similar manner for the Casimir pressure and entropy at τ ≫ 1 it follows P (a, T ) = − , S(a, T ) = 16πa2 . (55) Equations (54) and (55) are simply generalized60 for the case of two dissimilar di- electric plates by performing the replacement r20 → r 0 , where r (1,2) 0 are defined by Eq. (11) with the static dielectric permittivities of dissimilar plates ε (1,2) 3. IS THE DC CONDUCTIVITY RELATED TO THE CASIMIR INTERACTION BETWEEN DIELECTRICS? As was discussed in the previous section, the zero-frequency term in formula (13), i.e., the contribution with l = 0, is of prime importance and determines many of the basic properties of the Casimir interaction. In the above consideration we have described dielectric materials by Eq. (9) with finite static dielectric permittivity ε0. This resulted in Eq. (11) where one reflection coefficient at zero frequency is larger and the other one is smaller than the physical value for real photons at normal incidence. However, in the Lifshitz theory, the departure of both coefficients from their physical values is coordinated in such a way that the Nernst heat theorem remains valid. It is common knowledge that at nonzero temperatures dielectric materials pos- sess a negligibly small but not equal to zero dc conductivity. From physical intuition it is reasonable to expect that the influence of this conductivity on the van der Waals and Casimir forces should be also negligible. In Ref. 57 it was shown that, on the contrary, the inclusion of small dielectric dc conductivity in the model of dielectric response leads to a large effect in dispersion forces. This raises the question if dc conductivity is related to dispersion forces or if, on the contrary, the zero-frequency contribution should be understood not literally but as an analytic continuation from the region of high frequencies determining the physical phenomenon of dispersion forces. To illustrate this problem in more details, we consider the asymptotic behavior of the free energy and entropy at low temperature with included dc conductivity. What this means is that, instead of the dielectric permittivity ε(iξl) given in Eq. (9), one uses57,58 ε̃(iξl) = ε(iξl) + = εl(iξ) + β(T ) , (56) October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 16 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko where σ0 is the dc conductivity of the plate material and β(T ) = 2~σ0/(kBT ). The conductivity of dielectrics depends on temperature as σ0 ∼ exp(−b/T ) where b is determined by the energy gap ∆which differs for different materials.78 The small- ness of the dc conductivity of dielectrics can be illustrated79 by the example of SiO2 where at T = 300K it holds β ∼ 10−12. Thus, the role of the dc conductivity is really negligible for all l ≥ 1. In addition, β(T ) quickly decreases with decrease of T and, as a consequence, remains negligible at any T . In spite of this, the substi- tution of Eq. (56) into the reflection coefficients (16) leads to different result than in Eq. (11): r̃‖(0, y) = 1, r̃‖(0, y) = 0. (57) Equation (57) is in some analogy to Eq. (6), obtained for metals described by the Drude model, which leads to the violation of the Nernst heat theorem in the case of perfect crystal lattices. Now we substitute the dielectric permittivity ε̃l ≡ ε̃(ξl) in Eqs. (13) – (16) instead of εl and find the Casimir free energy F̃(a, T ) with included dc conductivity. For convenience, we separate the zero-frequency term, subtract and add the usual zero-frequency contribution for dielectric without dc conductivity. The result is F̃(a, T ) = 16πa2 1− e−y 1− r20e−y 16πa2 ydy ln 1− r20e−y 1− r̃2‖(ζl, y)e 1− r̃2⊥(ζl, y)e−y where r0 was defined in Eq. (11). Let us expand the last, third, integral on the right- hand side of Eq. (58) in powers of the small parameters β/l. Then, we combine the zero-order contribution in this expansion with the second integral on the right- hand side of Eq. (58) and obtain the Casimir free energy F(a, T ) calculated with the dielectric permittivities εl. The first integral on the right-hand side of Eq. (58) is calculated explicitly. Then, Eq. (58) can be rewritten as F̃(a, T ) = F(a, T )− kBT 16πa2 ζ(3)− Li3 +R(a, T ). (59) Here, R(a, T ) is of order O(β/l). It represents the first and higher-order contribu- tions in the expansion of the third integral on the right-hand side of Eq. (58) in powers of β/l. Restricting its explicit form to the first order contribution we get R(a, T ) = R1(a, T ) +O (β/l) , (60) R1(a, T ) = dy y2e−y y2 + ζ2l (εl − 1) (2− εl) ζ2l − 2y2 y2 + ζ2l (εl − 1) + εly r‖(ζl, y) 1− r2 (ζl, y)e−y y2 + ζ2l (εl − 1) + y r⊥(ζl, y) 1− r2⊥(ζl, y)e−y . (61) October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 17 Calculating the entropy by the first equality in Eq. (49), we arrive at S̃(a, T ) = S(a, T ) + 16πa2 ζ(3)− Li3 − ∂R(a, T ) , (62) where S(a, T ) is the entropy for the plates with the dielectric permittivity εl given by Eq. (49). Let us now show that the quantity R(a, T ) exponentially goes to zero with the decrease of T . First we consider only the integral in Eq. (61), expand the integrated function in powers of τ (we recall that ζl = τl), restrict ourselves to the main contribution, resulting for τ = 0, and rearrange it appropriately: (ε0 + 1)2 1− r20e−y = − 2 (ε20 − 1) 1− r20e−y (ε20 − 1) dy ye−ny = − (ε20 − 1) 1 + nζl e−nζl . (63) Substituting this in Eq. (61), we find R1(a, T ) = − 4πa2 (ε20 − 1) e−nτl e−nτl = − kBTβ 4πa2 (ε20 − 1) 1− e−nτ enτ − 1 kB Li2 2πa2 (ε20 − 1) Tβ ln τ + TβO(τ0) . (64) Here, the last line is obtained by using the equality 1− e−nτ enτ − 1 = − ln τ + 1− lnn+O(τ2) , (65) substituting only its leading term and observing the definition of the integral log- arithm, Li2 (z) = (1/2) zn/n2. Taking into account that β ∼ (1/T ) exp(−b/T ), we get the conclusion that the temperature dependence of R1(a, T ) is given by R1(a, T ) ∼ e−b/T lnT. (66) Thus, both R1(a, T ) and its derivative with respect to T in Eqs. (59) and (62) go to zero. The terms of the second and higher powers in β in Eq. (60) go to zero even faster than R1 when T → 0. Finally, in the limit T → 0 from Eq. (62) it follows S̃(a, 0) = 16πa2 ζ(3)− Li3 > 0. (67) The right-hand side of this equation depends on the parameter of the system under consideration (the separation distance a) and implies a violation of the Nernst heat theorem. An analogous result was obtained60 in the case of two dissimilar dielectrics. October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 18 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko The violation of the Nernst heat theorem in the Casimir interaction for dielectrics originates from the inclusion of the dc conductivity in the model of dielectric re- sponse. This violation is, however, of different nature than the one discussed above in the case of Drude metals. In the case of dielectrics the entropy at zero tempera- ture is positive but in the case of Drude metals it is negative. In the case of metals the violation is caused by the vanishing contribution from the transverse electric mode at zero frequency whereas the other reflection coefficient takes the physical value 1 [see Eq. (6)]. For dielectrics the situation is quite opposite. In this case the transverse reflection coefficient at zero frequency is always equal to zero [compare Eqs. (11) and (57) in the absence and in the presence of the contribution from dc conductivity]. Here the violation occurs due to the unity value of the parallel reflection coefficient at zero frequency in Eqs. (57) which departs from the value r0 = (ε0 − 1)/(ε0 + 1) coordinated with the zero value of the transverse coefficient in Eq. (11). One can conclude that the dc conductivity of a dielectric is not related to the na- ture of the van der Waals and Casimir forces and must not be included in the model of dielectric response. Ignoring this rule results in a violation of thermodynamics. Physically it is amply clear that there is no fluctuating field of zero frequency and that for such high-frequency phenomena as the van der Waals and Casimir forces the low-frequency behavior should be obtained by analytic continuation from the region of high frequencies. This permits to conclude that the correct procedure consists in the substitution of the finite static dielectric permittivities into the zero-frequency term of the Lifshitz formula, as Lifshitz and his collaborators really did.36,37,38 4. THERMAL CASIMIR FORCE BETWEEN DIELECTRIC AND METAL PLATES The Casimir interaction between metal and dielectric plates suggests the interest- ing opportunity to verify the thermodynamic consistency of the Lifshitz theory with different models of the dielectric response. This configuration was first inves- tigated in Ref. 61 where it was proved that the Casimir entropy is in accordance with the demands of the Nernst heat theorem if the static permittivity of the di- electric plate is finite. In Ref. 61, however, only the first leading terms in the low- temperature asymptotic expressions for the free energy and entropy were obtained and the Casimir pressure was derived only in the dilute approximation. Here we derive the more precise low-temperature behavior for the Casimir free energy, pres- sure and entropy in the configuration of one plate made of ideal metal and another plate made of dielectric with any finite static dielectric permittivity. For the configuration of metal and dielectric plates the Lifshitz formula takes the form80 analogical to Eq. (13) F(a, T ) = ~cτ 32π2a3 1− δl0 ydy (68) October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 19 1− rM‖ (ζl, y) rD‖ (ζl, y) e−y 1− rM⊥ (ζl, y) rD⊥ (ζl, y) e−y Here the reflection coefficients r for metal and dielectric, respectively, are given by Eqs. (16) where εl should be changed for ε l = ε M,D(iξl). For an ideal metal rM‖,⊥(ζl, y) = 1 and Eq. (68) takes the more simple form F(a, T ) = 32π2a3 ydy (69) 1− r‖(ζl, y)e−y 1− r⊥(ζl, y)e−y (here and below we omit the index D near the reflection coefficient and permit- tivity of a dielectric plate). We admit that the dielectric permittivity calculated at Matsubara frequencies εl ≡ ε0, i.e., is equal to its static value and find the asymp- totic behavior of Eq. (69) at small τ . [In analogy with Sec. 2 it is possible to prove that the deviations of ε(iξl) from ε0 at high frequencies do not influence the low- temperature behavior of the Casimir free energy, pressure and entropy. It can be shown also that the results of this section are valid not only for ideal metal plate but for plate made of real metal as well.] The free energy (69) can be represented by Eqs. (18) – (21), with the function f(ζ, y) replaced by f̂(ζ, y) = y ln 1− r‖(ζ, y)e−y + y ln 1− r⊥(ζ, y)e−y ≡ f̂‖(ζ, y) + f̂⊥(ζ, y). (70) In the case of one dielectric and one metal plate both f̂‖ and f̂⊥ contribute to F (ix)− F (−ix). The expansion of f̂(x, y) in powers of x takes the form f̂(x, y) = y ln(1− r0e−y)− ε0 − 1 e−y − ε0 ε0 + 1 x2 +O(x3). (71) Now we integrate Eq. (71) in accordance with Eq. (18) to find the function F (x). The integral of the first term on the right-hand side of Eq. (71) is evaluated using the new variable v = y − x: ydy ln(1− r0e−y) = vdv ln(1− r0e−v) + O(x2), (72) where the coefficient near the first-order contribution in x vanishes. As a result, this term could contribute to F (ix) − F (−ix) only starting from the third expansion order. The integrals of the second-order terms on the right-hand side of Eq. (71) are simply calculated using the formulas = −Ei(−x), = −Ei(−nx). (73) Finally, we obtain F (ix)− F (−ix) = iπ (ε0 − 1)2 4 (ε0 + 1) x2 − iγ x3 +O(x4), (74) October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 20 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko where the unknown third order expansion coefficient is designated as γ. Substituting Eq. (74) in Eq. (21) and using Eq. (19), we find the free energy in the system metal-dielectric in the form F(a, T ) = E(a)− ~c 32π2a3 (ε0 − 1)2 ε0 + 1 τ3 −K4 τ4 +O(τ5) , (75) where K4 ≡ γ/240. The Casimir pressure in the configuration of metal and dielectric plates obtained from Eq. (75) is equal to P (a, T ) = P0(a)− 32π2a4 4 +O(τ5) . (76) The direct application of the Lifshitz formula gives the expression for the pressure analogical to Eq. (31), P (a, T ) = − ~cτ 32π2a4 1− δl0 r‖(ζl, y) ey − r‖(ζl, y) r⊥(ζl, y) ey − r⊥(ζl, y) . (77) Using the Abel-Plana formula (17), Eq. (77) can be represented in the form of Eqs. (32), (33) where Φ‖,⊥(x) = y2 r‖,⊥(x, y) ey − r‖,⊥(x, y) . (78) Again, we deal first with Φ⊥(x). By adding and subtracting the asymptotic behavior of the integrated function at small x, y2r⊥(x, y) ey − r⊥(x, y) (ε0 − 1)x2e−y +O(x3) , (79) and introducing the new variable v = y/x, the function Φ⊥(x) can be identically rearranged and expanded in powers of x as follows: Φ⊥(x) = (ε0 − 1)x2e−x + x3 rn⊥(v)e −nvx − 1 (ε0 − 1)e−vx (ε0 − 1)x2(1− x) + x3 v2r⊥(v) 1− r⊥(v) ε0 − 1 +O(x4). (80) The integral on the right-hand side of Eq. (80) is converging and can be simply calculated with the result Φ⊥(x) = ε0 − 1 x2 − 1 ε0 − 1)x3 +O(x4). (81) To deal with Φ‖(x) we add and subtract in Eq. (78) the two first expansion terms of the integrated function in powers of x, Φ‖(x) = ey − r0 − ε0r0e y2(ε0 + 1) (1− r0e−y)2 r‖(x, y) ey − r‖(x, y) ey − r0 ε0r0e y2(ε0 + 1) (1− r0e−y)2 October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 21 The asymptotic expansion of the first integral on the right-hand side of Eq. (82) is given by 2Li3(r0)− ε0(ε0 − 1) 2(ε0 + 1) (ε0 − 1)(3ε0 − 2)x3 +O(x4), (83) and of the second one by ε0(ε0 − 1)− ε0(ε0 ε0 − 1) + ε0(ε0 − 1) x3 +O(x4). (84) By summing Eqs. (83) and (84) we find Φ‖(x) = 2Li3(r0)− ε0(ε0 − 1) 2(ε0 + 1) x2 (85) (ε0 − 1) + (ε0 ε0 − 1)− 3ε0(ε0 − 1) x3 +O(x4). Finally we add Eq. (85) to Eq. (81) and arrive at Φ(ix) − Φ(−ix) = −i 1− 2ε0 ε0 + ε x3 +O(x4). (86) Substituting this in Eq. (33) and using Eq. (32), the asymptotic expression for the Casimir pressure is obtained P (a, T ) = P0(a)− 32π2a4 1− 2ε0 ε0 + ε τ4 +O(τ5) . (87) Comparing Eqs. (87) and (76), the explicit expression for the coefficient K4 reads 1− 2ε0 ε0 + ε . (88) Now we are in a position to find the asymptotic behavior of the entropy in the limit of low temperatures in the configuration of two parallel plates one of which is metallic and the other one dielectric. By calculating the negative derivative of Eq. (75) with respect to temperature, one arrives at S(a, T ) = 3kBζ(3)(ε0 − 1)2 128π3a2(ε0 + 1) τ2 (89) 8π2(ε0 + 1) 1− 2ε0 ε0 + ε 135ζ(3)(ε0 − 1)2 τ +O(τ2) This equation is in analogy to Eq. (49) obtained for the case of two dielectrics. As is seen from Eq. (89), the entropy of the Casimir and van der Waals interactions between metal and dielectric plates vanishes when the temperature goes to zero, i.e., the Nernst heat theorem is satisfied. Note that the first term of order τ2 on the right-hand side of Eq. (89) was already obtained in Ref. 61. It is notable also that at low temperatures the entropy goes to zero remaining positive. At the same time, as was shown in Ref. 61, at larger temperatures entropy is nonmonotonous and may take negative values. This interesting property distinguishes the configuration of metal and dielectric plates from two dielectric plates. In the latter configuration the October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 22 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko negative entropy appears only for nonphysical dielectrics with anomalously large and frequency independent dielectric permittivities.53 Now let us consider the case τ ≫ 1, i.e., high temperatures (large separations). In the same way as for two dielectric plates, here the main contribution to the free energy is given by the term with l = 0 in Eq. (69), F(a, T ) = ~cτ 64π2a3 ydy ln 1− r0e−y . (90) Performing the integration in Eq. (90), we obtain F(a, T ) = − kBT 16πa2 Li3 (r0) . (91) From this equation, for the Casimir pressure and entropy at τ ≫ 1 it follows P (a, T ) = − kBT Li3 (r0) , S(a, T ) = 16πa2 Li3 (r0) . (92) The results (91) and (92) are analogous to (54) and (55) for two dielectric plates. 5. THE PROBLEM ORIGINATING FROM THE ACCOUNT OF DIELECTRIC DC CONDUCTIVITY In the previous section it was supposed that the static dielectric permittivity ε0 of the dielectric plate is finite. Now we will deal with the configuration of metal and dielectric plates with included dc conductivity of the dielectric material. In doing so the permittivity of the dielectric plate is given by ε̃(iξl) = ε0 + β(T ) , (93) where all notations were introduced in and below Eq. (56). Thus, the reflection coefficients at zero frequency satisfy Eq. (4) for a plate made of ideal metal and Eq. (57) for a plate made of dielectric with included dc conductivity. Let us find the low-temperature behavior of the Casimir entropy and verify the consistency of the Lifshitz theory with thermodynamics in this nonstandard situation. For this purpose we substitute the dielectric permittivity (93) in Eq. (69) in- stead of ε0 and find the Casimir energy F̃(a, T ) with included dc conductivity of a dielectric plate. In the same way as in Sec. 3, it is convenient to separate the zero-frequency term of F̃(a, T ) and subtract and add the usual zero-frequency con- tribution for metal-dielectric plates computed with the dielectric permittivity ε0, F̃(a, T ) = kBT 16πa2 1− e−y 1− r0e−y 16πa2 ydy ln 1− r0e−y 1− r̃‖(ζl, y)e−y 1− r̃⊥(ζl, y)e−y October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 23 Here the reflection coefficients r̃‖,⊥ are calculated with the permittivity (93). We expand the third integral on the right-hand side of Eq. (94) in powers of the small parameter β/l. The zero-order contribution in this expansion together with the second integral of Eq. (94) form the Casimir free energy F(a, T ) calculated with dielectric permittivity ε0. The first integral on the right-hand side of Eq. (94) is calculated explicitly. As a result, Eq. (94) is rearranged to F̃(a, T ) = F(a, T )− kBT 16πa2 [ζ(3)− Li3 (r0)] +Q(a, T ), (95) where Q(a, T ) contains the first and higher-order contributions in the expansion of the third integral on the right-hand side of Eq. (94) in powers of β/l. The explicit form of the main first-order term in Q(a, T ) is the following: Q1(a, T ) = dy y2e−y y2 + ζ2l (ε0 − 1) (2− ε0)ζ2l − 2y2 y2 + ζ2l (ε0 − 1) + ε0y 1− r‖(ζl, y)e−y y2 + ζ2l (ε0 − 1) + y 1− r⊥(ζl, y)e−y . (96) In the same way as in Sec. 3, we expand the integrated function in Eq. (96) in powers of τ (bearing in mind that ζl = τl) and preserve only the main contribution at τ = 0: Q1(a, T ) = − dy yr0e (ε20 − 1) (1− r0e−y) = − kBTβ 4πa2(ε20 − 1) e−nτl e−nτl Dealing with this expression in the same way as with Eq. (64), we arrive at Q1(a, T ) ∼ e−b/T lnT. (98) The Casimir entropy in the configuration of metal and dielectric plates with included dc conductivity of the dielectric plate is obtained as minus derivative of Eq. (95) with respect to temperature, S̃(a, T ) = S(a, T ) + 16πa2 [ζ(3)− Li3 (r0)]− ∂Q(a, T ) . (99) Using Eq. (98), the calculation of the limiting value at T → 0 is straightforward: S̃(a, 0) = 16πa2 [ζ(3)− Li3 (r0)] > 0. (100) From this equation it follows that the inclusion of the dc conductivity of dielectric plate in the configuration metal-dielectric results in a violation of the Nernst heat theorem. In the above this result was obtained for a metallic plate made of ideal metal. It can be shown that it remains valid for a metallic plate made of real metal with finite conductivity. October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 24 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko Thus, both configurations (two dielectric plates or one metal plate and one dielectric) lead to the same conclusion that when the dc conductivity is included in the model of dielectric response of the dielectric plate, the Lifshitz theory loses its consistency with thermodynamics. This confirms the conclusion made in Sec. 3 that the actual properties of dielectric materials at very low frequencies are in fact not related to the van der Waals and Casimir forces. 6. QUALITATIVE DISCUSSION OF CASIMIR INTERACTION BETWEEN METAL AND SEMICONDUCTOR As was mentioned in the Introduction, semiconductors possess a wide variety of electric and optical properties ranging from metallic to dielectric. This opens the possibility to modulate the van der Waals and Casimir forces by changing the charge carrier density. Bearing in mind the discussed above problems on the consistency of the Lifshitz theory with thermodynamics, semiconductors can provide us with a test for the validity of different approaches. Thus, if for good dielectric the dc conductivity does not play any real role in the van der Waals and Casimir forces, the question arises on how much it should be increased in order to become a relevant factor in the description of dispersion forces. In Ref. 81 the Casimir force acting between two Si plates was calculated using the simple analytic expression for Si dielectric permittivity as a function of frequency. The complete tabulated optical data of Si were used in Ref. 75 to calculate the van der Waals interaction of different atoms with a Si wall. The first attempt to measure the van der Waals force between a glass lens and a Si plate and to modify it by light due to the change of carrier density was undertaken in Ref. 82. However, glass is a dielectric and therefore the electric forces due to localized point charges could not be controlled. This might explain that no force change occured on illumination at small separations where the effect should be most pronounced. The first measurements of the Casimir force between a gold coated sphere and a single crystal Si plate were performed in Refs. 62, 63 by means of the atomic force microscope. The experiments used a p-type B doped Si plate of resistivity ρ = 0.0035 Ω cm. The chosen resistivity of the plate is in some sense intermediate between the resistivity of metals (which is usually two or three orders of magnitude lower) and the resistivity of dielectrics (it can be by about a factor of 105 larger; for instance, high-resistivity “dielectric” Si has the resistivity ρ0 = 1000 Ω cm). Thus, the used Si plate had a relatively large absorption typical for semiconductors but it was also enough conductive to avoid the accumulation of charges.63 In Fig. 2, taken from Ref. 63, the differences of the theoretical and mean ex- perimental Casimir forces acting between Au sphere and Si plate are presented as functions of separation. In Fig. 2a the theoretical force F theor is computed using the Lifshitz formula and the dielectric permittivity of a Si plate with the relatively low resistivity ρ used in experiment. This dielectric permittivity goes to infinity as ξ−1 with decreasing frequency (see the solid line in Fig. 3). In Fig. 2b the theo- October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 25 80 100 120 140 (nm) F theor − F̄ expt (pN) 80 100 120 140 (nm) F̃ theor − F̄ expt (pN) Fig. 2. Differences of the theoretical and mean experimental Casimir forces versus separation. Theoretical forces are computed (a) for the Si plate used in experiment and (b) for dielectric Si. Solid and dashed lines indicate 95 and 70% confidence intervals, respectively. retical Casimir force F̃ theor is computed using the dielectric permittivity of the Si plate made of “dielectric” Si with high resistivity ρ0. The dielectric permittivity of high-resistivity Si is shown by the dashed line in Fig. 3. It is characterized by the finite static value εSi(0) = 11.67. The solid and dashed lines in Fig. 2a,b indicate the 95% and 70% confidence intervals, respectively. As is seen from Fig. 2, the the- oretical approach using the dielectric permittivity of high-resistivity “dielectric” Si is excluded by experiment within the separation range from 60 to 110nm at 70% confidence. At the same time, the theory using the dielectric permittivity of Si with a low resistivity ρ is consistent with experiment. October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 26 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko 13.5 14 14.5 15 15.5 16 16.5 17 Log10 [ξ (rad/s)] ε (iξ)D Fig. 3. Dielectric permittivity of Si plate used in experiment along the imaginary frequency axis (solid line). Dashed line shows the dielectric permittivity of dielectric Si. The above results suggest an approach on how to correctly determine the pos- sible role of the low-frequency conductivity properties in dispersion forces. As is seen from Fig. 3, the dielectric permittivity of low-resistivity Si (solid line) sig- nificantly departs from the dielectric permittivity of “dielectric” Si in the region around the important dimensional parameter of the problem, the characteristic fre- quency c/(2a) ∼ 1014 − 1015 rad/s. That is why the low-resistivity sample cannot be described at low frequencies by the static dielectric permittivity of Si equal to εSi(0) = 11.67. To describe it, the term β(T )/l, like in Eq. (93), should be added to εSi(0). Note that in this case β(T )/l > 1 at the first Matsubara frequencies with l = 1, 2, 3, . . . and, thus, this quantity cannot be considered as a small parameter. At the same time, for a high-resistivity sample the inclusion of the dc conduc- tivity would lead to deviations from the dashed line in Fig. 3 only at frequencies ξ < 108 rad/s, which are much less than the characteristic frequency. This compari- son permits to make a conclusion in what experimental situations the conductivity properties of semiconductors at low frequencies should be taken into account and when they should be omitted as being not related to dispersion forces. The future experiments on the modification of semiconductor charge carrier density by laser light83 will bring a more clear understanding of this problem on the connection between the low-frequency material properties and dispersion forces. October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 27 7. DOES SPATIAL DISPERSION LEAD TO AN IMPORTANT IMPACT ON THERMAL CASIMIR FORCE? The presented above new analytic results on the low-temperature behavior of the Casimir free energy, pressure and entropy between dielectrics or between dielectric and metal are based on the conventional Lifshitz theory which describes dielectric materials by means of the frequency dependent dielectric permittivity. In fact, the assumption that the material of the plates possesses only the frequency dispersion means that the components of electric displacement are connected with the compo- nents of electric field by the relation Dk(r, ω) = εkl(r, ω)El(r, ω). (101) This equation is central in all different derivations of the Lifshitz formula (see, e.g., Refs. 2, 6, 36–38, 69, 70, 84). The effects of spatial nonlocality (spatial dispersion) are in fact essential only at shortest separations between the plates comparable with atomic dimensions and also for metals at sufficiently large separations (typically of about 2–3µm) in the frequency region of the anomalous skin effect. The Casimir force in the latter region was described by the Lifshitz theory reformulated in terms of the Leontovich impedance.85 Recently the spatial dispersion came to the attention in connection with the thermal Casimir force.64,65,66,67,68 In particular, it was claimed64,65 that for real metals at any separation the account of spatial dispersion leads to practically the same result (6) for the reflection coefficients at zero frequency as was obtained earlier using the Drude model dielectric function (5). This conclusion, if it is correct, not only returns us to the contradiction with experiment30,32,33 but also casts doubts on all results obtained by means of the conventional Lifshitz theory accounting for only the frequency dispersion. It is natural when the spatial dispersion contributes a small fraction of a percent as it is generally believed in numerous applications of the Lifshitz theory. It is, however, quite another matter when the account of the spatial dispersion results in some “dramatic effects”,64 i.e., in several hundred times larger thermal correction than is obtained in the local case. Below we demonstrate that the conclusions of Refs. 64 – 68 are in fact not reliable because they use the Lifshitz theory of dispersion forces outside of its application range.86 To find the electromagnetic modes associated with an empty gap between the plates, Refs. 64 – 68 use the standard continuity boundary conditions, E1t = E2t, B1n = B2n, D1n = D2n, B1t = B2t, (102) which are commonly applied in the derivation of the Lifshitz formula for spatially local nonmagnetic materials. Here B is the magnetic induction, n is the normal to the boundary directed inside the medium, the subscripts n, t refer to the normal and tangential components, respectively, the subscript 1 refers to the vacuum and subscript 2 to the plate material. In Refs. 64 – 68 the spatial dispersion is described by the longitudinal and transverse dielectric permittivities depending on the wave vector and frequency: εkl = εkl(q, ω). However, as is shown below, in the theory October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 28 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko of the Casimir effect both the boundary conditions (102) and permittivities εkl = εkl(q, ω) are inapplicable. We start from the boundary conditions and recall the set of Maxwell equations in a metal describing the Casimir effect, rotE + = 0, divD = 0, rotB − 1 = 0 divB = 0. (103) Equations (103) do not contain any external, i.e., independent on E, D and B, current or charge densities. The definition of the electric displacement is + 4πi, (104) where the volume current i is induced by E and B and takes into account the conduction electrons. In electrodynamics with spatial dispersion the electric field and magnetic induc- tion are finite at the boundary surfaces, whereas the electric displacement can tend to infinity.87 Then, integrating Eqs. (103) over the thickness of the boundary layer as is done in Ref. 88, we reproduce the first two conditions in Eq. (102) and arrive at the modified third and fourth conditions,87,89 E1t = E2t, B1n = B2n, D2n −D1n = 4πσ, [n× (B2 −B1)] = j, (105) where the induced surface charge and current densities are given by div[n× [D × n]]dl, j = 1 dl. (106) Note that the boundary conditions (105), (106) are obtained from the macroscopic Maxwell equations for physical fields. They should not be mixed with the boundary conditions arising in perturbative theories and for the fictitious fields (see below). In linear electrodynamics for a medium with time-independent properties with- out spatial dispersion the material equation connecting the electric displacement and electric field takes the form Dk(r, t) = ε̂kl(r, t− t′)El(r, t′)dt′. (107) According to this equation, the electric displacement at a point r and moment t is determined by the electric field at the same point r at different moments t′ ≤ t (the spatial dispersion is absent but the temporal may be present). It is easily seen that the substitution of Eq. (107) in Eq. (106) leads to σ = 0, j= 0 and, as a result, the boundary conditions (105) coincide with the standard continuity conditions (102). It is unjustified, however, to use conditions (102) in the presence of spatial dispersion. In Refs. 87, 90 a few examples are presented illustrating that in this case neither σ nor j is equal to zero. We now turn to a discussion of the use of dielectric permittivity εkl(q, ω) in the theory of the Casimir effect with account of spatial dispersion. In the presence of October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 29 only frequency dispersion, it is possible to perform the Fourier transformation of the fields E(r, t) = E(r, ω)e−iωtdω, D(r, t) = D(r, ω)e−iωtdω, (108) in Eq. (107) and arrive at Eq. (101) where εkl(r, ω) = ε̂kl(r, τ)e iωτdτ (109) is the frequency-dependent dielectric permittivity and τ ≡ t− t′. In fact Eqs. (101), (108) and (109) are used in parallel with the boundary conditions (102) in all derivations of the Lifshitz formula.2,6,36,37,38,69,70,84 If the material of the plates is characterized not only by temporal but also spatial dispersion, Eq. (107) is generalized to Dk(r, t) = dr′ε̂kl(r, r ′, t− t′)El(r′, t′). (110) If the material medium were uniform in space (i.e., all points were equivalent), the kernel ε̂ would not depend on r and r′ separately, as in Eq. (110), but on the difference R ≡ r − r′. In this case, by performing the Fourier transformation, E(r, t) = dqE(q, ω)ei(qr−ωt), D(r, t) = dqD(q, ω)ei(qr−ωt), (111) and substituting it in Eq. (110), one could introduce the dielectric permittivities εkl(q, ω) = dR ε̂kl(R, τ)e −i(qR−ωτ), (112) as Refs. 64–68 do, and rearrange Eq. (110) to the form Dk(q, ω) = εkl(q, ω)El(q, ω). (113) In the Casimir effect, however, the material medium is not uniform due to the presence of a macroscopic gap between the two plates (half spaces). Because of this, the assumption that the kernel ε̂ depends only on R and τ is wrong. As a result, it is not possible to introduce the dielectric permittivity εkl(q, ω) depending on the wave vector and frequency. In fact, for systems with spatial dispersion in the presence of boundaries the kernel ε̂ depends not only on R and τ but also on the distance from the boundary.87 In this complicated situation the following approx- imate phenomenological approach is sometimes applicable.87 For electromagnetic waves with a wavelength λ the kernel ε̂(r, r′, τ) in Eq. (110) differs essentially from zero only in a certain vicinity of the point r with characteristic dimensions l ≪ λ (for nonmetallic condensed media l is of the order of the lattice constant). Then it is reasonable to assume that ε̂ is a function of R=r–r′, except for a layer of thickness l adjacent to the boundary surface. If one is mostly interested in bulk phenomena October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 30 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko and neglects the role and influence of a subsurface layer, the quantity εkl(q, ω) may be employed as a reasonable approximation. This approximate phenomenological approach is widely applied in the theoretical investigation of the anomalous skin effect (see, e.g., Ref. 91). Note that in Ref. 91 some kind of fictitious infinite system was introduced and electromagnetic fields in this system are discontinuous on the boundary surface. This discontinuity should not be confused with the discontinuity of physical fields of a real system in the presence of spatial dispersion described by Eqs. (105) and (106). (There is also another approach to the description of the anomalous skin effect in polycrystals using the generalizations of the local Leontovich impedance which takes into account the shape of Fermi surface92). The frequency and wave vector dependent dielectric permittivity in the presence of boundaries is also approximately applied in the theory of radiative heat transfer93 or in the study of electromagnetic interaction of molecules with metal surfaces.94 In all these applications the boundary effects are usually taken into account by the boundary conditions (105) supplemented by so called “additional boundary conditions”. It is unlikely, however, that the approximate phenomenological approach using such quantity as εkl(q, ω) in the presence of boundaries would be applicable in the theory of the Casimir force where the boundary effects on the zero-point electromag- netic oscillations are of prime importance. It is notable also that this approach faces serious theoretical difficulties including the violation of the law of conservation of energy.95 It is then not surprising that the application of this approach in Refs. 64, 65 results in the contribution to the Casimir free energy from the transverse electric mode which is in contradiction with experiment.30,32,33 One more shortcoming of Refs. 64–68 is that they substitute the dielectric per- mittivity εkl(q, ω), depending on both wave vector and frequency, into the conven- tional Lifshitz formula derived in the presence of only temporal dispersion. In the famous review paper96 it has been noticed, however, that with the inclusion of spa- tial dispersion the free energy of a fluctuating field takes the form F = FL +∆F , where FL is given by the conventional Lifshitz expression derived in a spatially local case and written in terms of the Fresnel reflection coefficients, and ∆F is an additional term which can be expressed in terms of the thermal Green’s function of the electromagnetic field and polarization operator. Review96 calls as not reliable the results of, e.g., Ref. 97 obtained by the substitution of dielectric permittivity εkl(q, ω), taking account of spatial dispersion, into the conventional Lifshitz for- mula. It can be true that the Lifshitz formula written in terms of general reflection coefficients is applicable in both spatially local and nonlocal situations. However, as far as the exact reflection coefficients in a spatially nonlocal case are unknown, the use of some approximate phenomenological models, elaborated in literature for applications different than the Casimir effect, may lead to incorrect results for ∆F and create inconsistencies with experiment. October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 31 To conclude this section, the results of Refs. 64–68 on the influence of spatial nonlocality on the Casimir interaction are shown to be not reliable. Although at present there is no fundamental theory of the thermal Casimir force incorporating spatial dispersion, there is no reason to expect that it can play any significant role in the frequency region of infrared optics (experimental separations) or normal skin effect (i.e., at separations between plates greater than 4–5µm). 8. CONCLUSIONS AND DISCUSSION In the foregoing we have presented the derivation of analytic asymptotic expres- sions for the free energy, pressure and entropy of the Casimir interaction between two dielectric plates and between metal and dielectric plates at low and high temper- atures. It was shown that the low-temperature behavior of the Casimir interaction between dielectrics and between dielectric and metal is determined by the static dielectric permittivities of nonpolar dielectrics. The obtained results were shown to be in agreement with thermodynamics when the static dielectric permittivities of dielectrics are finite. In particular, the entropy of the Casimir interaction goes to zero when the temperature vanishes, i.e., the Nernst heat theorem is satisfied. This demonstrates the consistency of the original Lifshitz’s approach to the van der Waals forces between dielectrics which disregards the small conductivity of dielectrics at constant current. The second important result shown above is that the inclusion of the dc con- ductivity of dielectrics into the model of dielectric response leads not to some small corrections to the characteristics of the Casimir interaction, as one could expect, but makes the Lifshitz theory inconsistent with thermodynamics leading to the vi- olation of the Nernst heat theorem. This reveals that real material properties at very low, quasistatic frequencies are in fact not related to the phenomenon of the van der Waals and Casimir forces which is actually determined by sufficiently high frequencies. In this case the zero-frequency contribution to the Casimir force should be understood not literally but as analytic continuation to zero of the material physical behavior in the region around the characteristic frequency. The presented results provide a basis for the calculation of the van der Waals and Casimir forces between real materials. Such calculations are much needed for numerous applications of the Casimir force discussed in the Introduction, in par- ticular for the applications in nanotechnology and for constraining predictions of fundamental physical theories beyond the Standard Model. Bearing in mind that semiconductors are the main constituent materials in nanotechnological devices, it is a subject of high priority to understand the Casimir effect with semiconductor boundaries. In this connection we have discussed new experimental results and the- oretical ideas on the Casimir interaction between a metal sphere and semiconductor plate. It was stressed that by changing the charge carrier density in the semicon- ductor it is possible to bring it in different intermediate states between metallic and dielectric. In this case the problem arises when the conductivity properties of October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 32 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko semiconductor are not related to dispersion forces and when they are becoming rel- evant. A criterion for the resolution of this problem was formulated based on the relation between the typical frequency at which the dc conductivity properties come into play and the characteristic frequency of the Casimir effect. In fact the thermal Casimir interaction between semiconductors remains an open question and much work should be done both in experiment and theory to gain a better insight into this subject. The last major problem discussed in this review is whether or not the spatial dispersion influences essentially the thermal Casimir force between real materials. In the present literature there is no agreement on this subject. We adduced arguments in favor of the statement that in the region of experimental separations the influence of the spatial dispersion on the Casimir force is negligible small. The statements on the opposite, contained in the literature, were shown to be not reliable because they are obtained by the application of the Lifshitz theory outside of its application range. At the same time it was ascertained that at the moment there is no consistent fundamental theory of the van der Waals and Casimir forces taking the spatial dispersion into account. This is the problem to solve in the foreseeable future. ACKNOWLEDGMENTS G.L.K. and V.M.M. are grateful to the Center of Theoretical Studies and the Insti- tute for Theoretical Physics, Leipzig University for their kind hospitality. This work was supported by Deutsche Forschungsgemeinschaft grant 436RUS113/789/0-2. References 1. H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948). 2. P. W. Milonni, The Quantum Vacuum (Academic Press, San Diego, 1994). 3. V. M. Mostepanenko and N. N. Trunov, The Casimir Effect and its Applications (Clarendon Press, Oxford, 1997). 4. K. A. 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0704.1041

Higher dimensional conundra

Higher Dimensional Conundra Steven G. Krantz1 Abstract: In recent years, especially in the subject of harmonic analysis, there has been interest in geometric phenomena of RN as N → +∞. In the present paper we examine several spe- cific geometric phenomena in Euclidean space and calculate the asymptotics as the dimension gets large. 0 Introduction Typically when we do geometry we concentrate on a specific venue in a par- ticular space. Often the context is Euclidean space, and often the work is done in R2 or R3. But in modern work there are many aspects of analysis that are linked to concrete aspects of geometry. And there is often interest in rendering the ideas in Hilbert space or some other infinite dimensional set- ting. Thus one wants to see how the finite-dimensional result in RN changes as N → +∞. In the present paper we study some particular aspects of the geometry of N and their asymptotic behavior as N → ∞. We choose these particular examples because the results are surprising or especially interesting. One may hope that they will lead to further studies. 1 Volume in RN Let us begin by calculating the volume of the unit ball in RN and the surface area of its bounding unit sphere. We let ΩN denote the former and ωN−1 denote the latter. In addition, we let Γ(x) be the celebrated Gamma function of L. Euler. It is a helpful intuition (which is literally true when x is an integer) that Γ(x) ≈ (x− 1)!. We shall also use Stirling’s formula which says k! ≈ kk · e−k · 1We are happy to thank the American Institute of Mathematics for its hospitality and support during this work. http://arxiv.org/abs/0704.1041v1 or, more generally, Γ(x) ≈ (x− 1)x−1e−(x−1) 2π(x− 1) for x ∈ R, x > 0. Lemma 1 We have that e−π‖x‖ dx = 1. Proof: The case N = 1 is familiar from calculus. We write hence e−π‖x‖ (polar coordinates) ‖x‖=1 r ds(x)dr hence S = 1. For the N−dimensional case, write e−π|x| 1dx1 · · · and apply the one-dimensional result. Let σ be the unique rotationally invariant area measure on SN−1 = ∂BN . Lemma 2 We have ωN−1 = 2πN/2 Γ(N/2) Proof: Introducing polar coordinates we have e−π|x| rN−1dr Letting s = r2 in this last integral and doing some obvious manipulations yields the result. Corollary 3 The volume of the unit ball in RN is 2πN/2 Γ(N/2) ·N Proof: We calculate that 1 dV (x) (polar coordinates) ‖x‖=1 1·rN−1 dσ(x)dr = ωN−1· That completes the proof. Now the first nontrivial fact that we wish to observe about the volume of the Euclidean unit ball in N -space is that that volume tends to 0 at N → ∞. More formally, Proposition 4 We have the limit Ω(N) = 0 . Proof: We calculate that (Volume of Unit Ball) = 2πN/2 Γ(N/2) ·N 2πN/2 ((N − 2)/2)(N−2)/2e−(N−2)/2 2π[(N − 2)/2] ·N (2πe)N/2 · 2 N (N−1)/2 · (2πe)N/2 · 2 N (N+1)/2 · This expression clearly tends to 0 as N → +∞. In fact we can actually say something about the rate at which the volume of the ball tends to zero. We have Proposition 5 We have the estimate 0 ≤ ΩN ≤ 2 · 20N/2 N (N+1)/2 Proof: Follows by inspection of the last line of the proof of Proposition 4. In fact something more is true about the volumes of balls in high-dimensional Euclidean space. Proposition 6 Let R > 0 be fixed. Then Vol(B(0, R)) = 0 . In other words, the volume of the ball of radius R tends to 0. Proof: From the formula for the volume of the unit ball we have that Vol(B(0, R)) = lim 2πeR2 This expression clearly tends to 0 as N → +∞. We leave the proof of the next result as an exercise for the reader; simply examine the formula for ωN−1: Proposition 7 Let R > 0. Then the surface area of the sphere of radius R in RN tends to 0 as N → +∞. The following very simple but remarkable fact comes up in considerations of spherical summation of Fourier series. Proposition 8 As N → +∞, the volume of the unit ball in RN is con- centrated more and more out near the boundary sphere. More precisely, let δ > 0. Then volume(B(0, 1) \B(0, 1− δ)) volume(B(0, 1) = 1 . Proof: We have volume(B(0, 1) \B(0, 1− δ)) volume(B(0, 1) = lim [1− (1− δ)N ] · [2πN/2]/[Γ(N/2) ·N ] [2πN/2]/[Γ(N/2) ·N ] = lim 1− (1− δ)N = 1 . That is the desired conclusion. 2 A Case of Leakage The title of this section gives away the punchline of the example. Or so it may seem to some. Consider at first a square box of side two with sides parallel to the coor- dinate axes in the Euclidean plane. We may inscribe in this box four discs of diameter 1, as shown in Figure 1. These discs will be called primary discs. Once those four discs are inscribed, we may inscribe a small, shaded disc in the middle as shown in Figure 2. We set area of shaded disc area of large box The same construction may be performed in Euclidean dimension 3. Ex- amine Figure 3. It suggests a rectangular parallelepiped with all sides equal Figure 1: The configuration in dimension 2. Figure 2: The shaded disc in dimension 2. Figure 3: The configuration in dimension 3. to 2, and 8 unit balls inscribed inside in a canonical fashion. These eight primary balls determine a unique inscribed shaded ball in the center. We set volume of shaded ball volume of large box A similar construction may be performed in any dimension N ≥ 2, with 2N balls inscribed in a rectangular box of side 2. The ratio RN is then calculated in just the same way. The question is then What is the limit limN→∞RN as N → +∞? It is natural to suppose, and most people do suppose, and that this limit (assuming it exists) is between 0 and 1. All other things being equal, it is likely equal to either 0 or 1. Thus it comes as something of a surprise that this limit is in fact equal to +∞. Let us now enunciate this result and prove Proposition 9 The limit RN = +∞ . Figure 4: The disc trapped in dimension 2. Of course this result is counter-intuitive, because we all instinctively be- lieve that the shaded ball, in any dimension, is contained inside the big box. Such is not the case. We are being fooled by the 2-dimensional situation de- picted in Figure 1. In that special situation, any of the two adjacent primary discs actually touch in such a way as to trap the shaded disc in a particular convex subregion of the big box (see Figure 4). So certainly it must be that R2 < 1. But such is not the case in higher dimensions. There is actually a gap on each side of the box through which the shaded ball can leak. And indeed it does. This is what we shall now show. First we shall perform the calculation of RN for each N and confirm that the expression tends to +∞ as N → +∞. Then we shall calculate the first dimension in which the shaded ball actually leaks out of the box. Proof of the Proposition: Notice that the center of one of the primary balls is at the point (1, 1, . . . , 1). It is a simple matter to calculate that a boundary point of this ball that is nearest to the center of the box is located at P ∗ ≡ (1 − 1/ N, 1 − 1/ N, . . . , 1 − 1/ N . Since the shaded ball will osculate the primary ball at that point, we see that the shaded ball has center the origin and radius equal to dist(0, P ∗) = (1− 1/ N)2, 1− 1/ N)2, . . . , 1− 1/ N)2 = N + 1− 2 Thus we see that the volume of the shaded ball is [N + 1− 2 N ]N/2 · ΩN . The ratio RN is then [N + 1− 2 N ]N/2 · ΩN Now we may simplify this last expression to 2 · πN/2 Γ(N/2) ·N · [N + 1− 2 N ]N/2 After some simplification we find that 2(π/4)N/2 · [N + 1− 2 N ]N/2 Γ(N/2) ·N By Stirling’s formula, this last expression is approximately equal to 2 · (π/4)N/2(N + 1− 2 N)N/2 N − 2 )(2−N)/2 · e(N−2)/2 · 1√ π(N − 2) (N + 1− 2 N) · 2 N − 2 · N − 2 π(N − 2) After some manipulation, we finally find that RN = lim N + 1 N − 2 · N − 2 π(N − 2) = lim )N/2 ( N − 2 · N − 2 π(N − 2) Now, in the limit, we may replace expressions like N − 2 by N . And we may reparametrize N as 3N . The result is N − 2 π(N − 2) = lim )3N/2 N − 2 π(N − 2) = lim N − 2 π(N − 2) What we see now is that this last equals Plainly, because πe/2 > 4, this limit is +∞. That proves the result. And now we turn to the question of when the shaded ball starts to leak out of the big box. This is in fact easy to analyze. We need only determine when the radius of the shaded ball exceeds 1. First notice that the radius of the shaded ball is monotone increasing in N . Now we need to solve N + 1− 2 N > 1 . This is a simple algebra problem, and the solution is N > 4. Thus, beginning in dimension 5, the shaded ball will “leak out of” the large box. It may be noted that RichardW. Cottle has made a study of mathematical phenomena that change (in the manner of a catastrophe—see [ZEE]) between dimensions 4 and dimensions 5. The results may be found in [COT]. 3 Centroids This final section of the paper will be more like an invitation to further exploration. We cannot include all the details of the calculations, as they are too recondite and complex. Yet the topic is very much in the spirit of the theme of this paper, and we cannot resist including a few pointers to this new and interesting work (for which see [KRA1] and [KRMP]). Figure 5: Centroids for a triangle. The inspiration for this work is the following somewhat surprising obser- vation. Let T be a triangle in the plane (see Figure 5). There are three ways to calculate the centroid of this figure: (i) average the vertices, (ii) average the edges, or (iii) average the 2-dimensional solid figure. And the question is: are these three versions of the centroid the same? The answer is that (i) and (iii) are always the same. Generically (ii) is different. In fact the three versions of the centroid coincide if and only if the triangle is equilateral [KRMP]. We used this fact as a springing-off point to investigate analogous ques- tions in higher dimensions. Consider the simplex S in RN that is the con- vex hull of the points 0 = (0, 0, . . . , 0), (1, 0, . . . , 0), (0, 1, 0, . . . , 0), . . . , (0, 0, . . . , 0, 1). Refer to Figure 6. Such an N -dimensional geometric figure comes equipped with (N+1) notions of centroid: one can average the vertices (or 1-dimensional skeleton) S0, or one can average over the 1-dimensional skeleton S1, or one can average over the two-dimensional skeleton S2A, or . . . one can average over the (N − 1)-dimensional skeleton SN−1, or one can average over the N -dimensional solid SN . There results the centroids C0,N , C1,N , . . . , CN,N . And the question is: Are these different notions of centroid all the same? And here is the somewhat surprising answer: In dimensions 2 through 12 (for the ambient space), the skeletons S0 and SN for the simplex S have the same centroid. In those Figure 6: A simplex in RN . same dimensions, the skeletons S1, S2, . . . , SN−1 all have different centroids, and the centroids all differ from the common centroid for S0 and SN . But in dimension 13 things are different. In fact in that dimension the skeletons S3 and S8 have the same centroid. Let us say a word about why these facts are true. Let ej denote the j coordinate vector in RN (i.e., the vector with a 1 in the jth position and 0s in all other slots). Then a sophisticated computation with elementary calculus yields that the centroid of the k-skeleton Sk of the simplex which is the convex hull of 0, e1, e2, . . . , eN is Ck,N = k + (N − k) k + 1 (k + 1) + (N − k) k + 1 (e1 + e2 + · · ·+ eN) . From this formula it can immediately be verified that S0 = SN = N + 1 (e1 + e2 + · · ·+ eN ) . It can also be checked that, in dimensions 2 through 12, all the intermediate skeletons have distinct centroids. But, in dimension N = 13, we observe that C3,13 = C8,13 = 13 · 24 (e1 + e2 + · · ·+ eN ) . One may well ask whether dimension N = 13 is the only dimension in which there are two intermediate skeletons with the same centroid. The answer is “no”; there are in fact infinitely many such dimensions (although they are quite sparse—sparser than the prime integers). One may verify this assertion by using the following Diophantine formula. Theorem 10 Fix a dimension N ≥ 2. Consider the simplex S as described above. There are skeletons of dimension k1 and k2, 1 ≤ k1 < k2 ≤ N − 1, of the simplex S which have the same centroid if and only if k1 = a 2 − 1, k2 = b 2 − 1 (for positive integers a and b) and, in addition, N = (b2 + ab+ a2)− (b+ a)− 1 . (⋆) Obviously this theorem gives us a tool for finding dimensions in which the simplex S has two intermediate skeletons with the same centroid. The following table gives some values of the dimension, and of the intermediate dimensions of skeletons which have the same centroid. Of course this data may be confirmed by direct calculation with the formula (⋆). We stress that there are in fact infinitely many dimensions in which this phenomenon occurs. The proof of this statement is a nontrivial exercise in elementary number theory (see [KRMP]). value of N value of k1 value of k2 approx. coord. of centroid 13 3 8 0.0737179487 21 3 15 0.0464285714 29 8 15 0.0340038314 31 3 24 0.0317204301 40 8 24 0.0247619047 43 3 35 0.0229789590 51 15 24 0.0194852941 53 8 35 0.0187368973 57 3 48 0.0173872180 65 15 35 0.0153133903 We conclude this discussion by recording the fact that it is impossible in any dimension for there to be three intermediate skeletons with the same centroid. Proposition 11 For no dimension N can there exists 3 distinct number 1 ≤ k1 < k2 < k3 ≤ N − 1 such that the centroids Ck1,N , Ck2,N , Ck3,N for the simplex S coincide. Proof: We let Q(a, b) = (b2 + ab+ a2)− (b+ a)− 1 . It suffices for us to show that there do not exist natural numbers a < b < c such that Q(a, b) = Q(a, c). Seeking a contradiction, we suppose that such a triple does indeed exist. b2 + ab− b = c2 + ac− c b2 + (a− 1)b = c2 + (a− 1)c . Since a ≥ 1, the function b 7→ b2+(a−1)b is strictly increasing, which yields a contradiction. The exploration of centroids for simplices of high dimension is a new venue of exploration. There are many new phenomena, and more to be discovered. See [KRMP] for more results along these lines. The reference [ZON] is also of interest. References [COT] R. W. Cottle, Quartic barriers, Computational Optimization and Ap- plications 12(1999), 81–105. [KRA1] S. G. Krantz, A Matter of gravity, Amer. Math. Monthly 110(2003), 465–481. [KRMP] S. G. Krantz, J. E. McCarthy, and H. R. Parks, Geometric characteri- zations of centroids of simplices, Journal of Mathematical Analysis and Applications 316(2006), 87–109. [ZEE] E. C. Zeeman, Catastrophe Theory. Selected Papers, 1972–1977, Addison- Wesley, Reading, MA, 1977. [ZON] C. Zong, Strange Phenomena in Convex and Discrete Geometry, Springer- Verlag, New York, 1996. STEVEN G. KRANTZ received his B.A. degree from the University of California at Santa Cruz in 1971. He earned the Ph.D. from Princeton Uni- versity in 1974. He has taught at UCLA, Princeton University, Penn State, and Washington University in St. Louis. Krantz is the holder of the UCLA Alumni Foundation Distinguished Teaching Award, the Chauvenet Prize, and the Beckenbach Book Prize. He is the author of 150 papers and 50 books. His research interests include complex analysis, real analysis, har- monic analysis, and partial differential equations. Krantz is currently the Deputy Director of the American Institute of Mathematics. American Institute of Mathematics, 360 Portage Avenue, Palo Alto, CA 94306 [email protected] Introduction Volume in RN A Case of Leakage Centroids

0704.1042

Entangling and disentangling capacities of nonlocal maps

Entangling and disentangling capacities of nonlocal maps Berry Groisman Centre for Quantum Computation, DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom. Entangling and disentangling capacities are the key manifestation of the nonlocal content of a quantum operation. A lot of effort has been put recently into investigating (dis)entangling capacities of unitary operations, but very little is known about capacities of non-unitary operations. Here we investigate (dis)entangling capacities of unital CPTP maps acting on two qubits. I. INTRODUCTION Entanglement content is one of the fundamental ways to characterize nonlocal quantum resources (nonlocal states and operations). For pure bipartite states the ul- timate measure of entanglement, the von Neumann en- tropy of entanglement, had been recently discovered [1]. A universal measure of entanglement for mixed states had not been found yet and different measures are used depending on the operational context. Nevertheless, the important feature of all entanglement measures of states is that their values are directly inferred using the param- eters of a state itself. Similarly to mixed states, the entanglement content of quantum operations can be characterized in different ways, e.g. via the amount of entanglement necessary to generate that operation or via the amount of entangle- ment the operation is able to produce/destroy (the so called entangling/disentangling capacities). This article is concerned with the two latter measures. Unlike the amount of entanglement in a state, the (dis)entangling capacities of an operation do not have an operational interpretation on their own. They mani- fest themselves via the change of the entanglement of a particular state that the operation acts upon. And the operation has to act on a specific state (the “optimal” state) in order to realize its (dis)entangling capacity in full. Thus, the straightforward way to calculate these quantities is to maximize the change of entanglement over all possible initial states. Substantial progress have been made recently in inves- tigating (dis)entangling capacities of unitary operations. The capacities of two-qubit unitary operations were ex- plicitly calculated [2, 3, 4]. It was also shown that single- shot capacities are equal to asymptotic capacities [3, 5, 6]. Some results for higher dimensions were also obtained [7]. However, extending these techniques to systems of higher dimensionality seems to be a very difficult task. Even in the two-qubit case the capacities of a general unitary op- eration have been calculated numerically, no analytical technique is known. In all real situations an experimentalist never deals with a perfect unitary in the laboratory. And it is need- less to say that calculating capacities of non-unitary op- erations, i.e. nonlocal quantum maps, is even a bigger challenge. In this article we consider nonlocal completely positive trace preserving (CPTP) maps of the form τ(ρ) → pkUkρU , (1) where Uk are unitary transformations. Maps of this type are often called random unitary processes, and they are doubly stochastic. The map (1) may arise, for example, if the desired unitary transformation can be implemented successfully only with certain probability, while another unitary is realized in the case of failure. A continuous version of the map (1) may arise naturally in experiment if parameters of a desired unitary transformation are sub- ject to a noise (the case of Gaussian noise will be analyzed in detail in Section IVC). The scope of this article covers the case of τ that act on two qubits. We calculate single- shot (dis)entangling capacities of τ in some particular cases. The structure of the article is as follows. In Sec. II the definition(s) of (dis)entangling capacities of unitaries are presented and some recent results concerning two- qubit unitaries. Sec. III generalizes the definition of (dis)entangling capacities for non-unitaries. Some nu- merical results for (lower bounds on) (dis)entangling ca- pacities for discreet and continuous mixtures of unitaries are presented in Sec. IV. II. (DIS)ENTANGLING CAPACITY OF A UNITARY: DEFINITIONS AND SOME RELATED RESULTS Consider an unitary operation UAB that acts on a tensor product Hilbert space HA ⊗ HB of two spatially separated particles A and B. If UAB can not be de- composed into a tensor product of local unitaries, i.e. UAB 6= VA⊗WB, then we say that UAB is nonlocal. Un- like local unitaries, nonlocal unitaries have an ability to produce or destroy entanglement. This ability is usually characterized by the entangling, E↑(U), and the disen- tangling, E↓(U), capacities, i.e. by the maximal increase (decrease) of entanglement that can be achieved when U acts on quantum states. To quantify these capacities we have to choose appropriate measures of entanglement. The most sensible choice is to use the entanglement of for- mation [11] as a measure of entanglement of the initial state ρ, and the distillable entanglement [12] as a measure http://arxiv.org/abs/0704.1042v1 of entanglement of the final state UρU †. The reason for this asymmetric choice is purely operational one. What counts is the amount of resources (pure maximally en- tangled states) needed to create ρ (asymptotically) and the amount of pure-state entanglement one will be able to extract from UρU †, again asymptotically. Thus the most general definition is E↑(U) = max [D(UρU †)− EF (ρ)], E↓(U) = max [EF (ρ)−D(UρU †)], where the maximization is over all possible states ρ (mixed and pure) accessible to U . The Hilbert space of an accessible ρ is not necessarily restricted to HA ⊗HB. It turns out to be the case that some U create more en- tanglement if the original particles are entangled with local ancillary particles [2, 3]. It also appears to be the case that the maximization in Eq. (2) can be restricted to pure-states only [3], therefore, the definition (2) can be simplified as E↑(U) = max [E(U |ψ〉) − E(|ψ〉)] E↓(U) = max [E(|ψ〉) − E(U |ψ〉)], where E is an entanglement measure for pure state (Throughout this paper we will use the von Neumann entropy of entanglement as the most appropriate mea- sure). This obviously simplifies the job significantly. Let us briefly recall the main results for A and B being two-level particles, qubits. Any UAB acting on qubits can be decomposed as [2, 10] UAB = [VA ⊗ VB] e α=x,y,z α [WA ⊗WB] , (4) where π/4 ≥ ξx ≥ ξy ≥ |ξz | ≥ 0. The middle term sand- wiched by local unitaries is called the canonical decom- position of U . Any U can be transformed to its canon- ical form by sandwiching it with Hermitian conjugates of corresponding local unitaries. That means that the canonical form is genuinely nonlocal part of U - every- thing else is local. The beauty of this results is that out of 15 real parameters that parameterize a general two-qubit unitary only three are necessary to describe its nonlocal nature. It simplifies considerably the clas- sification of nonlocal unitaries. For the purpose of our discussion three classes can be identified; namely, the Controlled-NOT(CNOT)-class (ξx 6= 0, ξy = ξz = 0), the DoubleCNOT-class (ξx 6= 0, ξy 6= 0, ξz = 0), and the SWAP-class (all three ξα are not equal zero) [15, 16]. The names reflect the fact that the corresponding “mother” unitary transformation (i.e. with ξα = π/4 for α 6= 0) belongs to that class. The main results for qubits are [2, 3]: (a) E↑(U) = E↓(U). (b) For CNOT-class the optimal state, i.e. the state that satisfies definition (3), lives solely in the Hilbert space of particles A and B (no ancillas are needed) and takes the form |ψopt〉 = cosα|0〉A|0〉B ± i sinα|1〉A|1〉B, (5) where ± correspond to E↑ and E↓ respectively. Thus all U from that class achieve their capacity by acting on pure states with the same Schmidt basis (only values of Schmidt coefficients differ depending on the value of ξx). The values α = f(ξx) can be obtained by straightforward numerical optimization. (c) If ξα < π/4 the maximal capacity is achieved when |ψopt〉 is already entangled. (d) Unitaries of the CNOT-class achieve their capaci- ties by acting on optimal states that lie inHA⊗HB. How- ever, unitaries of the DCNOT and SWAP-classes achieve their capacities only if the original particles are entangled with local ancillas. It was conjectured that it is sufficient to take the size of ancillas equal to the size of original particles. This conjecture was supported by numerical simulations for qubits [3]. III. ENTANGLING AND DISENTANGLING CAPACITIES OF A NON-UNITARY For non-unitaries we will use a definition similar to Eq. (2), i.e. we define E↑(τ) = max [D(τ(ρ)) − EF (ρ)] E↓(τ) = max [EF (ρ)−D(τ(ρ))]. However, in general here we cannot justify reducing the search to pure states. This is due to the fact that distillable entanglement is not necessary a convex mea- sure. We can argue, nevertheless, that in the case of mix- tures of unitaries acting on two qubits without ancillas the distillable entanglement can be regarded as a convex measure. Indeed, a mixture of optimal states (5) forms a Bell-diagonal state for which the lower and upper bounds on distillable entanglement [17] S(ρA)− S(ρAB) ≤ D(ρ) ≤ ERE(ρAB) (7) coincide. Here we recall that the relative entropy of en- tanglement ERE(x) is a convex measure. If ancillas are used then the situation is more compli- cated. We leave the question of whether the capacities are attained on pure states as an open question and cal- culate the lower bounds on these capacities using pure states. IV. MIXTURES OF UNITARIES ACTING ON TWO QUBITS The properties of two-qubit unitaries described above in Section II can help us to generalize that approach to -0.75-0.5-0.25 0.25 0.5 0.75 FIG. 1: (color online) E↑(τ ) (solid line) and E↓(τ ) (dashed line) as functions of ∆ for different values of ξ, where p = 0.5. The highest curve corresponds to ξ = π/4. The lowest curve corresponds to ξ = 0. Here ∆ is measured in radians. mixtures of unitaries as in Eq. (1)[18]. Here we use two methods for calculating E↑(τ) and E↓(τ). Method I: We make an assumption about the partic- ular form of the optimal input state, and subsequently find the optimal values of its parameters. Method II: We perform a direct numerical optimiza- tion without making any a priori assumption about the optimal state (except of its purity). A. Example I: Discreet CNOT-mixtures Consider a mixture of unitaries of the CNOT-class, Uk = e x . (8) Here we use Method I. From continuity it follows that the optimal state is expected to lie on the 2-dimensional manifold of (superpositions of) states of the form (5) or their convex mixtures. Moreover, in this special case we can adopt the argument of [3] (see Sec. II) and claim that the search can be restricted to pure states only. Thus, the state optimal for τ is again of the type (5). As a simplest case let us consider only two unitaries U1 and U2: τ(ρ) → pU1ρU †1 + (1 − p)U2ρU For convenience let us define ∆ = ξ2x − ξ1x, and denote ξ1x simply by ξ, so ξ x = ξ +∆. We will fix ξ and analyze E↑ and E↓ for various ∆. For ∆ = 0 the map reduces to a unitary (with an appropriate capacity). As smaller angle means smaller E↑(U), we would expect that if U1 is mixed with U2, where ∆ < 0, then the entangling capac- ity of the resulting map will decrease relative to E↑(U). This intuition is consistent with the results presented on Fig. 1. Similarly, we might expect that the entangling capacity of the map will increase with ∆ > 0, and that -0.4 -0.2 0.2 0.4 FIG. 2: (color online) DCNOT: E↑(τ ) (empty diamonds with solid line fit) and E↓(τ ) (empty triangles with dashed line fit) as functions of ∆ for ξ1x = ξ y = π/8. SWAP: E ↑(τ ) (filled diamonds with solid line fit) and E↓(τ ) (filled triangles with dashed line fit) as functions of ∆ for ξ1x = ξ y = ξ z = π/8. In both cases p = 0.5. Here ∆ is measured in radians. this capacity will reach its maximum for maximal ∆, i.e. maximal E↑(U2). However, Fig. 1 shows that this is not the case. E↑(τ) indeed grows while ∆ is positive and relatively small, reaching its maximum for certain inter- mediate positive value of ∆ and then starting to decrease. In other words, if U1 and p are fixed, then maximal E is achieved for some intermediate U2 with ξ x > ξ x, but not for ξ2x = π/4. This result might seem counterintuitive from the first sight, but it has a clear explanation. Let α1, α2 be the corresponding optimal values of α in Eq. (5) for U1 and U2 respectively, then the optimal value of ατ will lie somewhere in between, i.e. satisfy α1 > ατ > α2. For ξ2x = π/4, U2 can realize its entangling capacity of 1 ebit if α2 = 0. However, when U2 acts on a state with ατ > α2, then it creates less than 1−H [(cosατ )2] ebit. Disentangling capacity, E↓(τ), behaves differently. It is monotonic with ∆. It equals E↑ for ∆ = 0 as expected, and it is strictly larger than E↑ for all other values of ∆. The last observation shows behavior completely opposite to that of unitaries. In a sense, it is easier for non-local map to destroy entanglement rather than create it, while for unitary operations the opposite holds [7]. This approach can be similarly applied to any finite number of unitaries and to continuous distribution of uni- taries, which will be discussed in Section IVC. B. Example II: Discreet DCNOT and SWAP-mixtures For mixtures of unitaries of DCNOT and SWAP-class Method II was used. The details of numerical calcula- tions are presented in Appendix. We conjectured that it is sufficient if local ancillas are qubits. Selected re- sults are shown in Fig. 2. We can see that for DCNOT- mixture the behavior of E↑(τ) and E↓(τ) is qualitatively similar to CNOT-mixture (Fig. 1). However, for SWAP- mixture slightly different behavior is obtained. In partic- ular E↓(τ) exhibits maximum at an intermediate value of ∆. It is also noticeable that for D > 0.357, E↑(τ) is smaller for SWAP-mixture than for DCNOT-mixture. It is a counterintuitive result that SWAP-mixture which is naturally considered as “stronger” that DCNOT-mixture has lower entangling capacity. However, again similar to the line of thought in Example I we can argue that for (relatively) large ∆ the second unitary, U2 is too strong, and therefore when it acts on the optimal state (optimal for the mixture, not for itself) it causes more destruc- tion that corresponding U2 of DCNOT-class would have caused. C. Example III: Entangling capacity of noisy unitary with Gaussian fluctuations So far we analyzed discrete mixtures of unitaries. In this section we analyze a continuous distribution, which is usually what experimentalists deal with. These distribu- tions arise due to uncertainty in one or more parameters. Such uncertainties may be caused by the limits of cali- bration precision of the devices and by high sensitivity of systems used to generate desired interactions. For exam- ple, the strength of exchange coupling between donors in silicon based solid-state architectures for quantum com- puting exhibit significant uncertainty resulting in error in gate operation [19]. In particular, we consider the case when a non-unitary map arises if a unitary from CNOT-class is subject to a Gaussian noise. Recent work [20] analyzed the capability of noisy Hamiltonians to create entanglement. In particular, in- teractions of the form Eq. (8), where ξ is Gaussian dis- tributed with the mean ξ = π/4 and standard deviation Ω, were considered. Without noise this operation (CNOT operation) is able to create a maximally entangled state if it acts on a disentangled pure state. The authors an- alyzed the situation when the noisy operation acts on initially disentangled state, which is by itself subject to a Gaussian noise. Its capability to create entanglement was characterized in terms of the condition for insepara- bility of the resulting mixed state (via Peres-Horodecci separability criterion). The aim of our analysis is different. We consider noisy interactions with ξ ∈ [0, π/4] and calculate their entan- gling and disentangling capacities in terms of ξ and Ω. Thus, we give a comprehensive quantitative characteriza- tion of the non-local content of these noisy maps in terms of their entangling and disentangling capacities. Unlike [20] we do not test the resulting state on inseparability, rather calculate its distillable entanglement explicitly. The action of a unitary U = exp[iξσAx σ x ], where ξ is Gaussian distributed with the mean ξ and the standard deviation Ω, on the state ρAB can be seen as a non- 0.2 0.4 0.6 0.8 FIG. 3: (color online) E↑(τG) (solid line) and E ↓(τG) (dashed line) as a function of ξ for several values of Ω: 0.01, 0.18, 0.35, 0.52, 0.69, and 0.86. The dotted line corresponds to Ω = 0, i.e. a unitary transformation. ξ and Ω are measured in radians. unitary CPTP map τG(ρ) → (ξ−ξ)2 2Ω2 UρU †dξ, (10) which is a continuous mixture of unitaries of the CNOT- class. Similarly to the Sec. IVA we consider pure initial state, i.e. ρ = |ψ〉〈ψ|, where ψ takes the form (5), cal- culate the distillable entanglement of the output mixed state, and maximize it over α. Figure 3 presents nu- merical results for E↑(τ) and E↓(τ) as functions of ξ for several values of Ω. We can see that already for Ω ≈ 0.01 rad we obtain only very small deviation from the (dis)entangling capacity of the unitary, i.e. τG with- out noise - Ω = 0. As Ω increases the disentangling capacity increases and the entangling capacity decreases. The former fact should not be surprizing as it is known that entanglement can be destroyed even by local CPTP unital maps [21]. Thus, the more dispersed the distribu- tion of ξ becomes, the easier for τG to destroy entangle- ment and the harder to create it. Nevertheless, we see that even when Ω is relatively large τG is still able to create considerable amount of entanglement. V. DISCUSSION AND CONCLUSION We have discussed the entangling and disentangled ca- pacities of nonlocal CPTP unital maps, i.e. maps that can be represented as probabilistic mixtures of unitaries, and have calculated these capacities in some particular cases for two qubits. Three classes of unitaries were con- sidered, namely the CNOT, DCNOT, and SWAP classes. We have observed that the disentangling capacity was always larger than corresponding entangling capacity, which contrasts with the unitary case where the both capacities are equal for qubits and for higher dimensions disentangling capacity cannot be greater than entangling capacity [7]. In the case of the CNOT-class our results were obtained via straightforward generalization of the method for CNOT-class unitaries. We argue that the (dis)entangling capacity is achieved when a map acts on the optimal pure state from the same family as in the uni- tary case. Both discrete and continuous mixtures were analyzed. In the case of the DCNOT and SWAP-class direct numerical optimization was performed. We have conjectured that dimensions of the local ancillas are equal to the dimensions of the original particles, i.e. the ancil- las were taken to be qubits. A number of open question can be addressed in a future research. It will be interesting and useful to prove (or disprove) the general conjecture that the sizes of local ancillas can be taken equal to the sizes of original particles. Here we have calculated single-shot capacities. In the case of unitaries it had been shown that in the asymptotic regime one cannot do better [5, 6]. It is important to check whether this result holds in the non-unitary case. In the case of DCNOT and SWAP-mixtures we per- formed maximization over pure states only thereby ob- taining lover bounds on E↑(τ) and E↓(τ), but not their actual values. In our future research we will address the question of whether these bounds are tight. It might be the case that optimal states for these operations are mixed and, consequently, the capacities are higher than we have cal- culated. Acknowledgments This work was funded by the U.K. Engineering and Physical Sciences Research Council, Grant No. EP/C528042/1, and supported by the European Union through the Integrated Project QAP (IST-3-015848) and SECOQC. APPENDIX A We have used two-dimensional ancilla on each side. Consider a general state of four qubits in the tensor- product of the computational bases of the original parti- cles A, B and the ancillary particles A′, B′ |Ψ〉AA′BB′ = i,j,k,l ci,j,k,l|i〉A|j〉A′ |k〉B |l〉B′ . (A1) There are 16 terms in the above superposition with 16 complex amplitudes ci,j,k,l, therefore |Ψ〉 can be parame- terized using 30 real numbers (if we take into account the global phase and normalization). We will parameterize it in the following way [22]. First, to facilitate our analysis it is easier to incorporate four indexes i, j, k and l, each of which runs from 0 to 1, into a single index, x, that runs from 1 to 16. This can be done by using the formula x = 8i + 4j + 2k + l + 1, which is essentially a formula for converting a number from the Boolean representation to the decimal. Second, we present amplitudes cx in the cx = |cx|eiθx , (A2) where |cx|2 = 1 and θ1 = 0. Third, we introduce new parameters φx such that cx = sinφx−1 cosφy, (A3) where φ0 = π/2. Thus the state |Ψ〉 is parameterized by 30 angles. The advantage of this parametrization is that we restrict their values only to the interval [0, 2π] that simplifies numerics. We proceed as follows. A program generates a vec- tor of 30 random numbers in the interval [0, 2π]. This is the initial state. We then apply the non-local map and obtain a final state. We calculate the value of the gain in entanglement ∆S = S(τ(Ψ)BB′ ) − S(τ(Ψ)AA′BB′) − S(TrAA′ |Ψ〉〈Ψ|). After that we vary the values of the ran- dom vector by a small amount and repeat these calcula- tions again, thereby obtaining a gradient of the change in entanglement in that point. We move along the gradient to obtain the next |Ψ〉, and the procedure is repeated. Eventually, the program reaches the maximum where it stops. [1] C.H.Bennett, H.J. Bernstein, S. Popescu, B. Schumacher, Phys. Rev. A , 53 2046 (1996). [2] W. Dür, G. Vidal, J. I. Cirac, N. Linden, and S. Popescu, Phys. Rev. Lett. , 87, 137901 (2001). [3] M.S. Leifer, L. Henderson, and N. Linden, Phys. Rev. A 67, 012306 (2003). [4] D.W.Berry and B.C.Sanders, Phys. Rev. A 71, 022304 (2005); P.Zanardi, C. Zalka, and Lara Faoro, Phys. Rev. A 62, 030301(R) (2000); L. Clarisse, S. Ghosh, S. Sev- erini, A. Sudbery, e-print arXiv:quant-ph/0611075v2. [5] C.H. Bennett, A. Harrow, D.W. Leung, and J.A. Smolin, IEEE Tran. Inf. Theory, 49, 8, 1895 (2003). [6] A.M. Childs, D.W. Leung, F. Verstraete, and G. Vidal, Quant. Inf. Comp. 3, 97 (2003). [7] N. Linden, J.A. Smolin, and A. Winter, e-print quant-ph/0511217. [8] M.A. Nielsen and I.L.Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2004). [9] It was explicitly shown [3] that the optimization can be restricted to pure states. [10] B. Kraus and J.I. Cirac, Phys. Rev. A 63, 062309 (2001). http://arxiv.org/abs/quant-ph/0611075 http://arxiv.org/abs/quant-ph/0511217 [11] W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). [12] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters, Phys. Rev. A 54, 3824 (1996). [13] J. I. Cirac, W. Dür, B. Kraus and M. Lewenstein, Phys. Rev. Lett. , 86, 544 (2001); W. Dür and J. I. Cirac, Phys. Rev. A , 64, 012317 (2001). [14] B. Kraus and J.I. Cirac, Phys. Rev. A 63, 062309 (2001). [15] Strictly speaking DCNOT-class and SWAP-class should be unified under a single class if analyzed according to the criteria of interconvertability under LOCC [16]. In the framework of (dis)entangling capacity it is useful to identify them as separate classes, because their behavior differs qualitatively. [16] W. Dür, G. Vidal, J. I. Cirac, Phys. Rev. Lett. , 89, 057901 (2002). [17] V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 (1998). [18] Here we assume that all unitaries in Eq. (1) belong to the same class, which is typical for real situations where only parameters of the coupling are subject to variations. [19] M. J. Testolin, C. D. Hill, C. J. Wellard, L. C. L. Hollen- berg, e-print quant-ph/0701165. [20] S. Bandyopadhyay and D.A. Lidar, Phys. Rev. A 70, 010301(R) (2004). [21] B.Groisman, S. Popescu, and A. Winter, Phys. Rev. A 72, 032317 (2005). [22] This method is a partial adaptation of the method used in Ref. [17] for calculation the relative entropy of entan- glement. http://arxiv.org/abs/quant-ph/0701165

0704.1043

On the Kolmogorov-Chaitin Complexity for short sequences

October 23, 2018 8:33 World Scientific Review Volume - 9in x 6in LongAbstractDelahaye2 Chapter 1 On the Kolmogorov-Chaitin Complexity for short sequences Jean-Paul Delahaye∗ and Hector Zenil† Laboratoire d’Informatique Fondamentale de Lille Centre National de la Recherche Scientifique (CNRS) Université des Sciences et Technologies de Lille Among the several new ideas and contributions made by Gregory Chaitin to mathematics is his strong belief that mathematicians should transcend the millenary theorem-proof paradigm in favor of a quasi- empirical method based on current and unprecedented access to compu- tational resources.3 In accordance with that dictum, we present in this pa- per an experimental approach for defining and measuring the Kolmogorov- Chaitin complexity, a problem which is known to be quite challenging for short sequences — shorter for example than typical compiler lengths. The Kolmogorov-Chaitin complexity (or algorithmic complexity) of a string s is defined as the length of its shortest description p on a universal Turing machine U , formally K(s) = min{l(p) : U(p) = s}. The major drawback of K, as measure, is its uncomputability. So in practical applica- tions it must always be approximated by compression algorithms. A string is uncompressible if its shorter description is the original string itself. If a string is uncompressible it is said that the string is random since no pat- terns were found. Among the 2n different strings of length n, it is easy to deduce by a combinatoric argument that one of them will be completely random simply because there will be no enough shorter strings so most of them will have a maximal K-C complexity. Therefore many of them will remain equal or very close to their original size after the compression. Most of them will be therefore random. An important property of K is that it is nearly independent of the choice of U . However, when the strings are short in length, the dependence of K on a particular universal Turing machine U ∗[email protected] †[email protected] http://arxiv.org/abs/0704.1043v5 October 23, 2018 8:33 World Scientific Review Volume - 9in x 6in LongAbstractDelahaye2 2 Jean-Paul Delahaye and Hector Zenil is higher producing arbitrary results. In this paper we will suggest an em- pirical approach to overcome this difficulty and to obtain a stable definition of the K-C complexity for short sequences. Using Turing’s model of universal computation, Ray Solomonoff9,10 and Leonid Levin7 developed a theory about a universal prior distribu- tion deeply related to the K-C complexity. This work was later known un- der several titles: universal distribution, algorithmic probability, universal inference, among others.5,6 This algorithmic probability is the probabil- ity m(s) that a universal Turing machine U produces the string s when provided with an arbitrary input tape. m(s) can be used as a universal sequence predictor that outperforms (in a certain sense) all other predic- tors.5 It is easy to see that this distribution is strongly related to the K-C complexity and that once m(s) is determined so is K(s) since the formula m(s) can be written in terms of K as follows m(s) ≈ 1/2K(s). The distri- bution of m(s) predicts that non-random looking strings will appear much more often as the result of a uniform random process, which in our ex- periment is equivalent to running all possible Turing machines and cellular automata of certain small classes according to an acceptable enumeration. By these means, we claim that it might be possible to overcome the problem of defining and measuring the K-C complexity of short sequences. Our pro- posal consists of measuring the K-C complexity by reconstructing it from scratch basically approximating the algorithmic probability of strings to ap- proximate the K-C complexity. Particular simple strings are produced with higher probability (i.e. more often produced by the process we will describe below) than particular complex strings, so they have lower complexity. Our experiment proceeded as follows: We took the Turing machine (TM) and cellular automata enumerations defined by Stephen Wolfram.11 We let run (a) all 2−state 2−symbol Turing machines, and (b) a statistical sample of the 3−state 2−symbol ones, both henceforth denoted as TM(2, 2) and TM(3, 2). Then we examine the frequency distribution of these machines’ outputs performing experiments modifying several parameters: the number of steps, the length of strings, pseudo-random vs. regular inputs, and the sampling sizes. For (a) it turns out that there are 4096 different Turing machines accord- ing to the formula (2sk)sk derived from the traditional 5−tuplet description of a Turing machine: d(s{1,2}, k{1,2}) → (s{1,2}, k{1,2}, {1,−1}) where s{1,2} are the two possible states, k{1,2} are the two possible symbols and the last entry {1,-1} denotes the movement of the head either to the right or to the October 23, 2018 8:33 World Scientific Review Volume - 9in x 6in LongAbstractDelahaye2 On the Kolmogorov-Chaitin Complexity for short sequences 3 left. From the same formula it follows that for (b) there are 2985984 so we proceeded by statistical methods taking representative samples of size 5000, 10000, 20000 and 100000 Turing machines uniformly distributed over TM(3, 2). We then let them run 30, 100 and 500 steps each and we pro- ceeded to feed each one with (1) a (pseudo) random (one per TM) input and (2) with a regular input. We proceeded in the same fashion for all one dimensional binary cellular automata (CA), those (1) which their rule depends only on the left and right neighbors and those considering two left and one right neighbor, henceforth denoted by CA(t, c)‡ where t and c are the neighbor cells in question, to the left and to the right respectively. These CA were fed with a single 1 surrounded by 0s. There are 256 one dimensional nearest-neighbor cellular automata or CA(1, 1), also called Elementary Cellular Automata11) and 65536 CA(2, 1). To determine the output of the Turing machines we look at the string consisting of all parts of the tape reached by the head. We then partition the output in substrings of length k. For instance, if k=3 and the Turing machine head reached positions 1, 2, 3, 4 and 5 and the tape contains the symbols {0,0,0,1,1} then we increment the counter of the substrings 000, 001, 011 by one each one. Similar for CA using the ”light cone” of all positions reachable from the initial 1 in the time run. Then we perform the above for (1) each different TM and (2) each different CA, giving two distributions over strings of a given length k. We then looked at the frequency distribution of the outputs of both classes TM and CA§, (including ECA) performing experiments modifying several parameters: the number of steps, the length of strings, (pseudo) random vs. regular inputs, and the sampling sizes. An important result is that the frequency distribution was very stable under the several variations described above allowing to define a natural distribution m(s) particularly for the top it. We claim that the bottom of the distribution, and therefore all of it, will tend to stabilize by taking bigger samples. By analyzing the following diagram it can be deduced that the output frequency distribution of each of the independent systems of ‡A better notation is the 3− tuplet CA(t, c, j) with j indicating the number of symbols, but because we are only considering 2 − symbol cellular automata we can take it for granted and avoid that complication. §Both enumeration schemes are implemented in Mathematica calling the functions CelullarAutomaton and TuringMachine, the latter implemented in Mathematica version October 23, 2018 8:33 World Scientific Review Volume - 9in x 6in LongAbstractDelahaye2 4 Jean-Paul Delahaye and Hector Zenil computation (TM and CA) follow an output frequency distribution. We conjecture that these systems of computation and others of equivalent com- putational power converge toward a single distribution when bigger samples are taken by allowing a greater number of steps and/or bigger classes con- taining more and increasingly sophisticated computational devices. Such distributions should then match the value of m(s) and therefore K(s) by means of the convergence of what we call their experimental counterparts me(s) and Ke(s). If our method succeeds as we claim it could be possible to give a stable definition of the K-C complexity for short sequences inde- pendent of any constant. Fig. 1.1. The above diagram shows the convergence of the frequency distributions of the outputs of TM and ECA for k = 4. Matching strings are linked by a line. As one can observe, in spite of certain crossings, TM and ECA are strongly correlated and both successfully group equivalent output strings. By taking the six groups — marked with brackets — the distribution frequencies only differ by one. By instance, the strings 0101 and 1010 were grouped in second place, therefore they are the second most complex group after the group com- posed by the strings of a sequence of zeros or ones but before all the other October 23, 2018 8:33 World Scientific Review Volume - 9in x 6in LongAbstractDelahaye2 On the Kolmogorov-Chaitin Complexity for short sequences 5 2n strings. And that is what one would expect since it has a very low K-C complexity as prefix of a highly compressible string 0101 . . .. In fa- vor of our claims about the nature of these distributions as following m(s) and then approaching K(s), notice that all strings were correctly grouped with their equivalent category of complexity under the three possible oper- ations/symmetries preserving their K-C complexity, namely reversion (sy), complementation (co) and composition of the two (syco). This also sup- ports our claim that our procedure is working correctly since it groups all strings by their complexity class. The fact that the method groups all the strings by their complexity category allowed us to apply a well-known lemma used in group theory to enumerate actual different cases, which let us present a single representative string for each complexity category. So instead of presenting a distribution with 1024 strings of length 10 it allows us to compress it to 272 strings. We have also found that the frequency distribution from several real- world data sources also approximates the same distribution, suggesting that they probably come from the same kind of computation, supporting contemporary claims about nature as performing computations.8,11 The paper available online contains more detailed results for strings of length k = 4, 5, 6, 10 as well as two metrics for measuring the convergence of TM(2, 2) and ECA(1, 1) and the real-world data frequency distributions extracted from several sources¶. A paper with mathematical formulations and further precise conjectures is currently in preparation. References 1. G.J. Chaitin, Algorithmic Information Theory, Cambridge University Press, 1987. 2. G.J. Chaitin, Information, Randomness and Incompleteness, World Scien- tific, 1987. 3. G.J. Chaitin, Meta-Math! The Quest for Omega, Pantheon Books NY, 2005. 4. C.S. Calude, Information and Randomness: An Algorithmic Perspective (Texts in Theoretical Computer Science. An EATCS Series), Springer; 2nd. edition, 2002. 5. Kirchherr, W., M. Li, and P. Vitányi. The miraculous universal distribution. Math. Intelligencer 19(4), 7-15, 1997. 6. M. Li and P. Vitányi, An Introduction to Kolmogorov Complexity and Its Applications, Springer, 1997. ¶It can be reached at arXiv: http://arxiv.org/abs/0704.1043. A website with the complete results of the whole experiment is available at http://www.mathrix.org/experimentalAIT/ http://arxiv.org/abs/0704.1043 October 23, 2018 8:33 World Scientific Review Volume - 9in x 6in LongAbstractDelahaye2 6 Jean-Paul Delahaye and Hector Zenil 7. A.K.Zvonkin, L. A. Levin. ”The Complexity of finite objects and the Algo- rithmic Concepts of Information and Randomness.”, UMN = Russian Math. Surveys, 25(6):83-124, 1970. 8. S. Lloyd, Programming the Universe, Knopf, 2006. 9. R. Solomonoff, The Discovery of Algorithmic Probability, Journal of Com- puter and System Sciences, Vol. 55, No. 1, pp. 73-88, August 1997. 10. R. Solomonoff, A Preliminary Report on a General Theory of Inductive In- ference, (Revision of Report V-131), Contract AF 49(639)-376, Report ZTB- 138, Zator Co., Cambridge, Mass., Nov, 1960 11. S. Wolfram, A New Kind of Science, Wolfram Media, 2002. *-14pt 1. On the Kolmogorov-Chaitin Complexity for short sequences Jean-Paul Delahaye and Hector Zenil

0704.1044

The redshift and geometrical aspect of photons

Dark matter field fluid may cause redshift effect on light The redshift and geometrical aspect of photons Hongjun Pan Department of Chemistry University of North Texas, Denton, Texas 76203, U. S. A. Abstract The cosmological redshift phenomenon can be described by the dark matter field fluid model, the results deduced from this model agree very well with the observations. The observed cosmological redshift of light depends on both the speed of the emitter and the distance between the emitter and the observer. If the emitter moves away from us, a redshift is observed. If the emitter moves towards us, whether a redshift, a blueshift or no shift is observed will depend on the speed vs. the distance. If the speed is in the range of c(exp[-βD] – 1) < v < 0, a redshift is observed; if the speed equals c(exp[-βD] – 1), no shift is observed; if the speed v less than c(exp[-βD] – 1), a blueshift is observed. A redshift will be always observed in all directions for any celestial objects as long as their distance from us is large enough. Therefore, many more redshifts than blueshifts should be observed for galaxies and supernovae, etc in the sky. This conclusion agrees with current observations. The estimated value of the redshift constant β of the dark matter field fluid is in the range of 10-3 ~ 10-5 /Mpc. A large redshift value from a distant celestial object may not necessarily indicate that it has a large receding speed. Based on the redshift effect of dark matter field fluid, it is concluded that at least in time average all photons have the same geometry (size and shape) in all inertial reference frames and do not have length contraction effect. 1. Introduction In cosmology, the redshift is one of the most important observations to study the origin and the evolution of the universe and the motion of celestial bodies. Based on the redshift observation, in 1927, Belgian Priest Georges Lemaître was the first to propose that the universe began with the explosion of a primeval atom, such hypothesis is the origin of the Big Bang theory. The hypothesis was supported by Edwin Hubble (Hubble, 1929), he found that distant galaxies in every direction are going away from us with speeds proportional to their distance, which is based on the observation that the redshift is proportional to the distance. According to the Big Bang theory, as light from distant galaxies approaches the Earth there is an increase of space between the Earth and the galaxies due to the expansion of the universe, which leads to wavelengths being stretched, i.e., the light is expanding with the universe. The Big Bang theory becomes a dominant theory at current stage in study of the origin and the evolution of the universe. It can explain many important observations such as the redshifts, the ratio of light elements in the universe, cosmic microwave background radiation, etc. Tired light theory is an alternate explanation of the redshift effect. Tired light was first proposed in 1929 by Fritz Zwicky (Zwicky, 1929) who suggested that photons might slowly lose energy as they travel vast distances. The major problem associated with this theory is that there is no notable machnism causing such energy drop during its journey. In 2004, the author proposed the dark matter field fluid model at the DPF2004 (Pan, 2005). In this model, the interstellar space is assumed to be, for simplicity, more or less uniformly filled with the dark matter field fluid which has fluid property and field property and all “baryonic” matter objects are saturated with such dark matter field fluid. Any motional celestial object will experience the dragging force of the dark matter field fluid. It is demonstrated that the current behavior and past evolution of Earth-Moon system can be described very well by this model (Pan, 2007) and the dragging effect of the dark matter field fluid dominates the evolution of the Earth-Moon system. This paper will extend the application of the dark matter field fluid model to the light traveling through space and compare the results with observations; the geometrical aspects of photons is also discussed based on this model. 2. The possible redshift effect of dark matter field fluid In the proposed dark matter field fluid model (Pan, 2005 and 2007), a spherical body moving through the dark matter field fluid at low Reynolds number condition experiences the following dragging force F, mvrF n−−= 16πη (1) where η is the dark matter field fluid constant which is equivalent to regular fluid viscosity constant, r is the radius of the sphere, n is the constant rising from saturation effect, m is the mass of the sphere and the v is the moving velocity of the sphere. The direction of the force F is opposite to the direction of velocity. The equation of motion of the body is m n−−= 16πη (2) The equation (2) can be written as )( 1 mvr mvd n−−= πη (3) The momentum of the sphere is p=mv, so the equation (3) can be written as pr dp n−−= 16πη (4) The equation (4) shows that under the dragging force of dark matter field fluid at low Reynolds number condition, the decrease rate of momentum of the spherical body is proportional to its momentum. It is further assumed that the general form of Eq. (4) can be applied to all ordinary matter objects (including photons) which move through the dark matter field fluid at low Reynolds number condition, i.e., α−= (5) where α is a parameter depending on the geometrical characteristics of the object (such as the size, shape, etc), the dark matter field fluid constant. Eq. (5) is the law of motion for ordinary objects moving through the dark matter field fluid with low Reynolds number. It is well known that a photon has momentum p=h/λ, where λ is the wavelength of photon and h is the Planck’s constant. For a photon traveling through the dark matter field fluid, if the Eq. (5) is applicable, then, (6) and (7) teαλλ 0= where the λ0 is the wavelength of the photon at time t=0, i.e., the wavelength of the photon which is just emitted by the emitter. The wavelength of the photon exponentially increases with the time it travels. This is the redshift effect of dark matter field fluid. The time t for the photons traveling from the emitter to the observer is t = (8) where the D is the distance from the emitter to the observer and c is the speed of light. Let β =α/c, the redshift constant of dark matter field fluid, the Eq. (7) can be written as . (9) Deβλλ 0= By convention, the redshift z is defined as =z . (10) So the redshift caused by dark matter field fluid is . (11) 1−= Dez β When the βD « 1, Eq, 11 reduces to the regular cosmological redshift formula D β == . (12) Eq. 12 indicates that for a sufficient short distance, the cosmological redshift is directly proportional to the distance. The conventional cosmological redshift formula is D z = (13) where the H is the Hubble constant. One can see that α is equivalent to H. Although the Eq. 12 and Eq. 13 are the same in form, their physical meanings are completely different. In the Eq. 12, the redshift is caused by the dragging effect of dark matter field fluid and the parameter α is a measure of the dragging effect of the dark matter field fluid on the light and has a unique value; but in the Eq. 13, the redshift is caused by universe expansion or the stretching of the space. The emitter, however, may move with speed v relative to the observer, such motion has a Doppler effect on the emitted light (v « c) )1(0 c r += λλ (14) where the λr is the wavelength of the light when the emitter is at rest. Therefore, the actual wavelength of light detected by the observer is Dr ec v βλλ )1( += . (15) The observed redshift is 1)1( −+= . (16) Obviously, the relativistic Doppler formula has to be used when the v is close to c. Eq. 16 indicates that the observed redshift z depends on not only the speed of emitter but also the distance. For sufficiently short distance, the Doppler motion effect may dominate the redshift z; for a large distance D, the contribution from the dragging force of the dark matter field fluid may dominate the redshift z. Fiq. 1 shows how the redshift z varies with the speed of the emitter with parameter βD=0.05. By convention, if z is positive, it is redshift; if z is negative, it is blueshift. Referring to Fig. 1, the line intercepts with the speed axis v/c at (exp[-βD]-1), no at zero. When the emitter (such as galaxies, supernova, pulsars, etc) moves away from the observer (on the Earth), v is positive, both the dragging effect of dark matter field fluid and the effect of motion (Doppler effect) have positive contribution to z, z is positive, so a redshift is observed. When the speed of the emitter is zero, only the effect of dark matter field fluid contributes to z which equals to (exp[βD]-1), and a redshift is observed. When the speed is negative, i.e., the emitter moves toward the observer with the speed in the range between (exp[-βD]-1) and 0, the positive contribution by the dark matter field fluid is greater than the negative contribution by the motion, z is still positive, i.e., a redshift is observed, so the observer can not know which direction the emitter moves based on only the sign of z. When the speed of the emitter equals to (exp[-βD]-1), the positive contribution from the dragging effect of dark matter field fluid equals the negative contribution from the motion effect, the two effects cancel each other, no shift is observed. When the speed of the emitter is at the left side of the (exp[-βD]-1), the negative contribution from motion effect is greater than the positive contribution from the dark matter field fluid, a blueshift is observed. The observer knows that the emitter moves towards to him/her. When the distance D increases, the interception (exp[-βD]-1) moves to the left, covers a larger redshift range, a higher negative speed is needed in order to observe a blueshift. According to this result, due to the redshift effect of dark matter field fluid, much more redshifts than blueshifts should be observed for the celestial objects in the sky and a redshift (z > 0) is always observed for any celestial objects in all directions as long as their distance is sufficiently large. This conclusion is exactly what is observed. So far only few galaxies show blueshifts, the most famous being M31 galaxy (Andromeda). As M31 is our near neighbor and the distance is relatively short, the speed is at the left side of (exp[-βD]-1) on Fig. 1. In this case, the redshift associated with the dark matter field fluid is less than the blueshift from the motional Doppler effect. Most galaxies are much further away from us. It is possible that some of those far away galaxies may move towards us, but the redshift effect of dark matter field fluid is greater than the blueshift of the motional Doppler effect, the final observed shifts are to the red. The significance of the Eq. (16) is that a large value of the redshift z does not necessarily means that the distant emitters(galaxies, supernovae, etc) have a large receding speed. The speed deduced from a redshift z using conventional Doppler formula or conventional cosmological redshift formula (Hubble’s law, v = HD) will be overestimated for v ≥ 0, misleading for c(exp[-βD]-1)≤ v ≤ 0, and underestimated for v < c(exp[-βD]-1). Note, the speed v here is the actual speed of the emitters relative to the observer at the moment it emits the light. The parameter β will be one the important parameters of the nature. Finding the value of β is a challenge. The redshift z of a remote celestial object can be accurately measured; however, accurately measuring the speeds and distances of the distant celestial objects is difficult. We can roughly estimate the range of β. According to the available data, the M31 galaxy has a redshift z = -0.000991 (Huchra et al. 1999) and distance about 2.9 million light years which equals 0.889 Mpc (mega parsec). The speed of M31 would be -297 km/s if only based on the Doppler effect. This speed will be underestimated according to the Eq. 16. If the actual speed is -600 km/s (most likely exaggerated), then, β= 1.14 × 10-3 /Mpc; if the actual speed is -300 km/s, then, β = 1.09 × 10-5 /Mpc. So β is probably in the range of 10-3 to 10-5 /Mpc. According to Eq. 12 and Eq. 13, β = H/C = 2.5×10-4 /Mpc with assumption of H = 75 km/s/Mpc, it is just in the middle of the estimated range. Therefore, the current value of Hubble’s constant can be used as a good guess for the β. With this estimated value of β, the speed of M31 towards us is about -363 km/s. When the sufficiently accurate value of β and the distance of the objects are found, it will be possible to calculate the speed of the objects with the observed value of z. Theoretically, the mechanism of the redshift effect by the dark matter field fluid model can be tested by observing the change of the redshift values of remote celestial objects over the time. As indicated above that when the emitter moves toward the observer with the speed in the range between (exp[-βD]-1) and 0, the positive contribution by the dark matter field fluid is greater than the negative contribution by the motion, z is still positive, i.e., a redshift is observed. However, the value of the redshift z will decrease with time as long as the emitter keeps moving towards the observer, and eventually it will become blueshift when the distance between the emitter and observer is short enough that redshift by dark matter field fluid is less than the blueshift by the motional Doppler effect. However, it will take very long time for any detectable change with reasonable accuracy. For example, it will take about 97 million years for the blueshift of M31 changing from -0.000991 to -0.001000 based on the current data and assuming β = 2.5 x 10-4/Mpc. As indicated in the previous paper (Pan, 2005), the dark matter field fluid may have thermal property with temperature about 2.7 K, the observed cosmic microwave background radiation is the black body radiation of the dark matter field fluid at 2.7K. It is certain that the density distribution and the thermal temperature of the dark matter field fluid are not uniform through the space and not constant in all time scale. The uneven distribution of density and temperature of the dark matter field fluid in the space causes uneven black body radiation which could be the origin of the observed anisotropies of cosmic microwave background radiation. Furthermore, it is very possible that the dark matter field fluid may be converted to other type of matter by certain physical mechanisms, or vise verse, which causes the change of density and temperature distribution, therefore, the current distribution of the density and the temperature of the dark matter field fluid could be different from remote past. It is interesting to notice that the redshift effect of dark matter field fluid is another version of “Tired light” model. But in this model, it is clear that the mechanism to cause the energy loss for photons to pass through the dark matter field fluid is the dragging force, and their wavelength becomes increasingly longer as they do so. No one ever thinks before that there is any relationship between the evolution of the Earth-Moon system and the redshift, but the dark matter field fluid model demonstrates that those two events obey the same law of motion (Eq. 5) and share the same mechanism. Such mechanism should not cause any blurring to the observed light. 3. The geometrical aspects of photons Light is the most mysterious and common phenomenon in the nature, people are always fascinated by the properties of light with all kinds of imagination, such imagination may be more important than the knowledge and can make people thinking in unusual ways without being limited by their knowledge. At 16 years old, Albert Einstein wondered what it would be like to ride on a beam of light. Maxwell described the light as a wave; Einstein described the light as a stream of energy packs which are now called photons, i.e. particles. So the light has wave-particle dual properties. It is quite often that people with different backgrounds and ages wonder what a photon looks like and how big it is. Such question is very interesting, but does not have an answer now, may never have one. However, we can still address some of the geometrical aspects of the photons and interesting information can be extracted out from the above results. We must accept the facts that photons are real matter objects and each photon is created as a whole entity and travels in the space as a whole entity without falling apart during its journey, therefore there must be some kind of an internal force existed inside the photon to hold “all components” of the photon together and keep it stable during the journey, such internal force is equivalent to the internal force to hold “all components” of an electron together, and is equivalent to the internal force to hold all components of an atom together and is equivalent to the internal force to hold all components of solar system together. In general, when we talk about the particle property, we will naturally think about the size and shape which are associated with particles. For wave-like property, on other hand, the size and the shape may lose their meanings according to quantum mechanics. However, this does not mean that those objects which can be perfected treated by quantum mechanics do not have, at least in time average, the geometrical characteristics such as size and shape. The instant geometries of photons, electrons and other particles may vary rapidly with time, however, in time average, they should have certain stable sizes and shapes, although it is hard to know what such sizes and shapes are. For example, the instant size and shape of hydrogen atom rapidly changes with time due to the fact that the distance and orientation of electron to the proton rapidly changes because of the fast motion of electron around the proton. However, in time average, the electron cloud is distributed around the proton which makes the hydrogen atom have a “ball” shape with the stable average size well represented by the Bohr radius (0.53 Å). From above results, one can see that the observed redshifts agree with the description of the model very well and photons with all wavelength-band follow the law of motion Eq. 5. As indicated above, the parameter α in Eq. 5 depends on the geometrical characteristics of the object (such as the size, shape, etc) and the α is the same for all photons; therefore, based on such information, we can conclude that all photons have the same geometrical characteristics (size and shape, at least in time average). This means that when a photon travels through the dark matter field fluid, it gradually loses its energy due to the dragging effect of the dark matter field fluid, but its geometry remains the same all the way in its journey. We can further conclude that at least in time average all photons in all inertial reference frames have the same geometry; therefore, photons do not have the length contraction effect. This is a very important property of photons in addition to that all photons have the same speed in all inertial reference frames. In contrast, an object with rest mass has length contraction effect when it is observed in different inertial reference frames with relative motions according to the special relativity. 4. Conclusion The dark matter field fluid model has been successfully applied to the cosmological redshift, the results deduced from this model agree very well with the observations. The observed cosmological redshift of light depends on both the speed of the emitter and the distance between the emitter and the observer. If the emitter moves away from us, a redshift is observed. If the emitter moves towards us, whether a redshift, a blueshift or no shift is observed will depend on the speed vs. the distance. If the speed is in the range of c(exp[-βD] – 1) < v < 0, a redshift is observed; if the speed equals c(exp[- βD] – 1), no shift is observed; if the speed v < c(exp[-βD] – 1), a blueshift is observed. A redshift will be always observed in all directions for any celestial objects as long as their distance from us is large enough. Therefore, many more redshifts than blueshifts should be observed for galaxies and supernovae, etc in the sky. This conclusion agrees with current observations. The estimated value of the redshift constant β of the dark matter field fluid is in the range of 10-3 ~ 10-5 /Mpc. A large redshift value from a distant celestial object may not necessarily indicate that it has a large receding speed. At least in time average, all photons have the same geometry in any inertial reference frames and do not have length contraction effect. 5. References 1. Hubble, Edwin, "A Relation between Distance and Radial Velocity among Extra- Galactic Nebulae" (1929) Proceedings of the National Academy of Sciences of the United States of America, Volume 15, Issue 3, pp. 168-173 2. Zwicky, F. On the Red Shift of Spectral Lines through Interstellar Space. PNAS 15:773-779, 1929. 3. Pan, H. Application of fluid mechanics to the dark matter, Internat. J. Modern Phys. A, 20(14), 3135 (2005). 4. Pan, H. The evolution of the Earth-Moon system based on the dark matter field fluid model, arXiv:0704.0003 (2007). 5. Huchra, J. P., Vogeley, M. S., Geller, M. J., Astrophys. J. Suppl. Ser, 121(2), 287(1999). http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1929PNAS...15..168H&db_key=AST&data_type=HTML&format=&high=42ca922c9c30954 http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1929PNAS...15..168H&db_key=AST&data_type=HTML&format=&high=42ca922c9c30954 http://arxiv.org/abs/0704.0003 The dependence of observed redshift z on the speed v of the emitter with βD = 0.05. The speed v of the emitter is in unit of speed of light c.

0704.1045

Cool Stars in Hot Places

Cool Stars in Hot Places ASP Conference Series, Vol. To appear in proceedings of Cool Stars 14, Ed. Gerard Van Belle Cool Stars in Hot Places S. T. Megeath Ritter Observatory, Department of Physics and Astronomy, University of Toledo, Toledo, OH 43606 E. Gaidos Department of Geology & Geophysics, University of Hawaii, Honolulu, HI 96822 J. J. Hester Arizona State University, Department of Physics & Astronomy, Tempe, AZ 85287 F. C. Adams Physics Department, University of Michigan, Ann Arbor, MI 48109 J. Bally Center for Astrophysics and Space Astronomy, University of Colorado, Boulder, CO 80309 J.-E. Lee Physics and Astronomy Department, The University of California at Los Angeles, Los Angeles, CA 90095 S. Wolk Harvard Smithsonian Center for Astrophysics, Cambridge, MA 02138 Abstract. During the last three decades, evidence has mounted that star and planet formation is not an isolated process, but is influenced by current and previous generations of stars. Although cool stars form in a range of environments, from isolated globules to rich embedded clusters, the influences of other stars on cool star and planet formation may be most significant in embedded clusters, where hundreds to thousands of cool stars form in close proximity to OB stars. At the cool stars 14 meeting, a splinter session was convened to discuss the role of environment in the formation of cool stars and planetary systems; with an emphasis on the “hot” environment found in rich clusters. We review here the basic results, ideas and questions presented at the session. We have organized this contribution into five basic questions: what is the typical environment of cool star formation, what role do hot star play in cool star formation, what role does environment play in planet formation, what is the role of hot star winds and supernovae, and what was the formation environment of the Sun? The intention is to review progress made in addressing each question, and to underscore areas of agreement and contention. http://arxiv.org/abs/0704.1045v1 2 Megeath et al. 1. What is the Typical Environment of Cool Star Formation? Cool stars form in a range of environments, from isolated Bok globules, to modest sized clusters containing 100-200 stars, and finally to large, dense clusters with thousands of cool stars and several to tens of OB stars. This is in sharp contrast to OB stars, which form almost entirely in large clusters. This motivates the question: in what environment do most cool stars form? Surveys of the molecular gas in our Galaxy indicate that most of the cold molecular gas is in giant molecular clouds (GMCs) with masses of 105 to 106 M⊙ (Heyer & Terebey 1998). These massive molecular clouds are thought to form entire associations of hot OB stars as well thousands of low mass stars. Coupled with analyses indicating that 80-90% of cool stars form in large clusters (Porras 2003; Lada & Lada 2003; Carpenter 2000); these results seemed to point to a galaxy in which the vast majority of cool star formation takes place in rich crowded clusters in close proximity to hot stars. However, since there was little information on the numbers of isolated stars, the analyses of Porras (2003) and Lada & Lada (2003) considered only stars in groups and clusters. In an analysis of the 2MASS point source catalog toward several molecular clouds, Carpenter (2000) found evidence for substantial numbers of isolated stars, but the estimates contained significant uncertainties. More recently, surveys of giant molecular clouds with the Spitzer space telescope provided the means to identify isolated young stars and protostars through the infrared excesses from their disks and envelopes (Allen et al. 2007). Spitzer surveys of four giant molecular clouds containing young massive hot stars, the Orion A cloud, Orion B cloud, Cep OB3 cloud and Mon R2 cloud, show that in addition to clusters associated with regions of massive star formation, there are large number of stars in small groups or isolation. In these clouds, 46% of the young stars with excesses are found in clusters with over 90 sources, 11% are found in small clusters of 90-30 stars, 8% in groups of 30-10 stars, and the remaining 35% in groups with less than 10 members or isolation (Megeath et al. 2007; Gutermuth et al. 2007). About 33% of the stars are found in the two largest clusters with over 700 members each. Thus, although most cool stars may form in OB associations, young cool stars in OB associations are not found primarily in large clusters. Instead, they are found in a range of environments, with a significant fraction of stars forming in relative isolation several to tens of parsecs away from the nearest OB stars (Fig. 1). 2. What Role do Hot Stars play in Cool Star Formation? Although cool stars dominate star-forming regions in both number and total stellar mass, hot stars are thought to be the primary agents of molecular cloud evolution. The extreme-UV radiation from young O and early B-type stars pho- toionizes the surfaces of molecular clouds, resulting in flows of ionized gas which erode the clouds. Far-UV radiation may play a similar role in regions where only B-stars are present by heating and photodissociating the molecular gas. Clus- ters with O and/or B stars and ages of only a few million years appear to have partially or fully dispersed their molecular clouds. An example is the 2.5 Myr old σ Ori cluster (Sherry et al. 2004); this cluster sits outside the Orion B cloud Cool Stars in Hot Places 3 Figure 1. The Orion OB association. The contours show an AV map of the Orion region made from the 2MASS database (Gutermuth, p. com.), the gray circles are the O stars (large circles) and B stars (small circles) from Brown et al. (1994) and the dots are Spitzer identified young cool stars and protostars (Megeath et al. 2007; Hernandez et al. 2007). Only regions of high molecular column density and the σ Ori cluster have been surveyed by Spitzer, and many more young cool stars certainly exist in the OB association (Fig. 1). Other examples are discussed in Allen et al. (2007). It is estimated that only 10% of embedded clusters survive gas dispersal and presist as clusters for more than 10 Myr (Lada & Lada 2003). The detection of Evaporating Gaseous Globules (EGGs), 1000 AU diameter photoevaporating dark globules, demonstrated that hot stars may directly im- pact protostellar evolution (Hester et al. 1996). EGGs appear to be protostellar or prestellar cores which emerge from their parental clouds as the surrounding lower density gas is ionized (Hester & Desch 2005). In M16, 15% of the EGGs contain embedded stars, indicating that they are the sites of recent or ongoing star formation (McCaughrean & Anderson 2002). This suggests that hot stars can directly affect protostellar evolution by photoevaporating the infalling gas 4 Megeath et al. and limiting the ultimate mass of the nascent star. However, it is not known what fraction of stars emerge from their clouds in EGGs. A more controversial issue is whether OB stars trigger cool star forma- tion. This possibility has been discussed in the literature for decades (e.g. Elmegreen & Lada 1977). Hester & Desch (2005) proposed that in regions with hot stars, cool star formation is driven primarily by shock fronts preceeding ad- vancing ionization fronts. The shock fronts overtake and compress pre-existing density enhancements, inducing collapse and the formation of clusters of low mass stars. Evidence for this is found in the detection of clusters of young stars at the surfaces of molecular faces being eroded by hot stars (Sugitani et al. 1995; Megeath et al. 2004; Allen et al. 2006). However, additional evidence, such as the detection of the shock fronts, is needed to determine whether the clusters have been triggered, or whether they are regions of ongoing star formation which have been overtaken by ionization fronts (Megeath & Wilson 1997). Although there is growing evidence that triggering does happen, it is not clear what fraction of cool star formation is triggered. Assessing the overall importance of triggered star formation can be difficult due to the rapid evolution and even rapid motions of OB stars. For example, Hoogerwerf et al. (2001) argued that the interaction of the ι Ori binary system with a second system led to the ejection of the runaway stars AE Aur and µ Col 2.5 Myr ago (both are 09.5 stars). Although they suggested that these stars originated in the Orion Nebula Cluster, the lack of a visible HII region surrounding ι Or, an O9 III star which in projection appears conicident with the Orion A cloud, suggests that it is several to tens of parsecs away from the Orion A molecular cloud and is part of the 5 Myr OB1c association (Brown et al. 1994). At the time of their ejection, these three O–stars may have had a significant impact on the Orion A cloud, and could have been responsible for triggering star formation in the Orion Nebula Cluster. Another possible example is the LDN 1551 dark cloud in the Taurus dark cloud complex. This cloud has a cometary morphology with the “head” of the comet pointing toward the Orion constellation. Moriarty-Schieven et al. (2005) argued that the cometary shape may be due to the interaction of LDN 1551 (149 pc from the Sun) with the B8I star Rigel (Hipparcos distance is 240 pc) and the M2I star Betelgeuse (Hipparcos distance 130 pc). The high proper motion of Betelgeuse would place it southeast of LDN 1551 several million years ago; hence, both Betelgeuse and Rigel could have plausibly interacted with LDN 1551, creating the cometary morphology. These observations demonstrate the difficulties in determining causal rela- tionships between subsequent generations of star formation and establishing the importance of triggering. Although ongoing triggering can be identified by the detection of clusters near ionization fronts, in many cases, evidence of triggering may be erased by the evolution and motion of massive stars. 3. What Role does Environment Play in Planet Formation? Environment may also play a role in planet formation by altering the properties of protoplanetary disks. We discuss here two mechanisms: tidal interactions between stars in clusters and the photo-ablation of disks by UV photons from nearby OB stars. Cool Stars in Hot Places 5 Tidal interactions occur when a disk around a star in a cluster is distorted or stripped during a close encounter with another cluster member. Such inter- actions appear to be unimportant. Adopting a stellar density of 104 pc−3 (the peak density for many embedded clusters) and assuming virialized velocities, Gutermuth et al. (2005) used a simple mean free path argument to estimate the frequency of close approaches. They estimated that even in the dense, central cores of clusters, close approaches at distances of 100 AU would occur once in a 10 Myr interval. However, the high stellar densities assumed by Guter- muth et al. may only persist for a few million years before the clusters begin to expand. This result is supported by N-body simulations of bound clusters which show that such interactions are rare over the lifetime of an embedded cluster (Adams et al. 2006; Throop & Bally in prep). Adams et al. (2006) find that each star in a 1000-member (initially) embedded cluster will experience one close-approach within 700-4000 AU over a 10 Myr interval. This distance is more than three times the typical radius of observed circumstellar disks in nearby dark clouds (Andrews et al. 2007) and much larger than the size of the Solar System. Since the adopted timescale for gas removal in these simulations was 5 Myr, longer than the observed timescale (Sec. 2), the close-approach dis- tances should be considered lower limits. In summary, the results from three independent investigations are in agreement; unless embedded clusters exist in our galaxy with much higher stellar densities than observed in nearby regions such as the Orion Nebula Cluster, tidal interactions in clusters rarely influence disk evolution and planet formation. In contrast, photoevaporation of disks by nearby OB stars appears to be a much more influential process. The UV radiation from the OB stars heats the gas in disks through photoionization and photodissociation, resulting in flows of gas off the disks. This process was discovered in VLA and HST observations of young stars in the Orion Nebula (Churchwell et al. 1987; O’Dell & Zheng 1994). The inferred mass loss rates were 10−7 M⊙, suggesting disk lifetimes of only a few hundred thousand years (Bally et al. 1998). However, the mass loss occurs in the outer disk where the thermal velocity of ionized gas exceeds the escape velocity from the star, and the gas in the inner disk may not be strongly affected. More recent calculations include the effect of the far-UV radiation and the time dependent nature of the UV-field as the stars orbit within the cluster potential. Adams et al. (2004) calculate the mass loss from a disk as a function of the intensity of the far-UV radiation field. They find the radiation field can truncate a disk to the size of our solar system in several million years; the exact radius depends on the duration of the exposure to UV radiation, the intensity of the UV radiation, and the mass of the central star (Adams et al. 2004). Throop & Bally (in prep) use N-body simulations to calculate the time dependent flux of UV radiation incident on a young star with disk as it orbits in a cluster which contains OB stars in its center. They find that typical stars experience only a brief exposure to intense UV as they pass within 10,000 AU of the central OB stars. Consequently, the UV flux incident on a disk varies in an stochastic manner over the lifetime of the cluster. Recently, Throop & Bally (2005) proposed that the photoevaporation of disks may in fact trigger the formation of planets. In their model, grain growth and dust settling concentrates dust grains in the midplane of the disk. Conse- quently, the ablation of the gas from the disk surface (as well as the remaining 6 Megeath et al. dust grains entrained in the gas) reduces the ratio of the gas surface density to dust surface density. If the surface density of gas is reduced to less then 10 times the dust density, the disk becomes unstable to gravitational collapse (Sekiya 1998; Youdin & Shu 2002). Although photoevaporation may be important in rich embedded clusters with OB stars, many young cool stars in OB associations are not found in such clusters. Young cool stars with disks identified in the Spitzer survey of the Orion A cloud have a median projected distance of 4.1 pc to the nearest O to B0 star, and a median projected distance of 2.1 pc to the nearest B1-B3 stars (Megeath et al. 2007). Hence, in OB associations, most cool stars may form at large distances from the central OB stars and are unaffected by their UV radiation. 4. What is Role of Hot Star Winds and Supernovae? Chandra X-ray observations of young stellar clusters have detected diffuse X- ray emission in nine regions. The total luminosities of this gas range from 1− 200× 1033 erg s−1 (Wolk et al. 2002; Townsley 2006). Although supernovae could generate this gas, in most cases the diffuse gas appears to be generated by stellar winds from massive stars colliding with other winds or the surrounding HII region. However, in the Carina region, a component of hot gas enriched in Fe was likely created by a supernova (Townsley 2006). The impact of the extremely hot gas on star and planet formation is not well understood. In addition to destroying the surrounding the cloud, the blast waves from a supernova could compress surrounding cores of gas causing them to collapse into stars (Boss 1995; Melioli et al. 2006). Disks can survive at distances of ≤ 1 pc from a supernova (Chevalier 2000); however, these disks will be heated by the radiation and blast wave, and may also be stripped by the blast wave when the disks are only 0.25 pc from the supernova (Chevalier 2000). The hot X-ray gas created by winds may fill bubbles within the larger HII region. This hot, low density gas would be transparent to UV photons, and hence any young stars within the bubble may be exposed to a more intense UV field than those in the surrounding HII region. 5. What Was the Formation Environment of the Sun? Did our Sun also form in the “hot” environment of a large embedded cluster? Tremaine (1991) and Gaidos (1995) proposed that our Solar System might pre- serve dynamical evidence of its birth environment. Gaidos (1995) and Adams & Laughlin (2001) used the low inclination and eccentricity of Neptune to place constraints on the time-integrated tidal field of a cluster and the closest stellar passage. However, such reasoning must now be re-examined in light of the expectation that most embedded clusters expand and disperse in a few Myr (although some clusters would form bound open clusters, Sec. 2) and the realization that Nep- tune (and Uranus) migrated outward to its present orbit by scattering in a residual planetesimal disk, a process that was probably not completed until after a parental cluster dispersed (Hahn & Malhotra 2005). Scattering inside the disk itself, which dampens any non-circular motion, could have produced the low eccentricity and inclination observed today. Similar arguments can Cool Stars in Hot Places 7 be made that other parts of the outer Solar System (the Edgewood-Kuiper belt, Oort Cloud) formed after the cluster evaporated (Levison & Morbidelli 2003). Kenyon & Bromley (2004) and Morbidelli & Levison (2004) proposed that Sedna, a member of the scattered Kuiper Belt, was produced by the close passage of a star, but there are other explanations (Barucci et al. 2005; Gladman & Chan 2006). Thus, it is likely that the structure of the outer Solar System post-dates an embedded cluster phase. The strongest evidence for an early cluster environment is the inferred pres- ence of short-lived radionuclides (SLRs) during the formation of solids now found in meteorites. There are at least three possible sources of SLRs: particle irradi- ation within the primordial solar nebula, the wind from a nearby AGB star, and the wind and/or supernova ejecta from a nearby massive star. The discovery of 60Fe in the early Solar System (Tachibana & Huss 2003) firmly establishes that the Sun formed in a rich cluster containing massive stars (Hester 2004; Hester & Desch 2005). Neutron-rich isotopes such as 60Fe cannot be produced by particle irradiation. The uniform distribution of the SLR 26Al makes it unlikely it was produced by irradiation (Thrane et al. 2006). Finally, it is sta- tistically unlikely that the SLRs originated in an AGB star (Kastner & Myers 1994). Further evidence is found in the mass-independent fractionation of the oxy- gen isotopes (17O and 18O) in meteorites. Following a proposal by Clayton, Grossman & Mayeda (1973), Lee et al. (2007) have made a theoretical analysis of the time-dependent chemistry in a collapsing envelope subjected to an external UV field. Due to “self-shielding” of the much more abundant C16O, the UV field preferentially dissociates C18O and C17O, producing an enhancement of 18O and 17O in the gaseous envelope. These heavier isotopes are then incorporated (as water) into ice grains and transported into the inner region of the solar nebula. This pro- cess depends on the intensity of the external UV radiation field (from OB stars) so that the measured fractionation can constrain the formation environment of the Sun. Lee et al. (2007) conclude that the observed isotopic ratios are best explained by a radiation field 105 greater than the interstellar field, again sup- porting the presence of nearby massive stars. The current evidence firmly indicates that the Sun formed in a hot environ- ment enriched by the ejecta of one or more nearby supernova; however, there is a continuing debate over how the solar nebula was enriched. Cameron et al. (1995) argued that the enrichment occurred when the collapse of the proto- solar molecular cloud was triggered by the blast wave of a supernova (also see Vanhala & Boss 2002). Hester & Desch (2005) question whether this process could enrich the collapsing molecular gas. Alternatively, the protostellar en- velope of the Sun may have been directly enriched while collapsing onto the proto-Sun (Looney et al. 2006). For example, if the solar system formed in an EGG, then it may have been subjected to a blast wave from a supernova. Finally, the SLRs may have been injected directly into the disk of the solar nebula when the Sun was in its T-Tauri phase; a possible mechanism for this is the “aerogel” model, in which grains in SN ejecta are deaccelerated and vaporized within the gaseous primordial disk (Ouellette et al. 2005). This scenario is supported by observations showing that 40% of disks may persist for 4 Myr (Hernandez 2007), the lifetime of a 60 M⊙ star. 8 Megeath et al. Recent quantitative analyses have constrained the distance between the Sun and the supernova from which the SLRs presumably originated. If the en- richment occurred while the Sun was in a T Tauri phase with a 200 AU disk (Andrews et al. 2007), the estimated distance is between 0.04-0.4 pc (Looney et al. 2006; Ellinger et al. in prep). If the enrichment occurred in the protostellar phase (5000 AU diameter), the estimated distance is between 0.12- 1.6 pc (Looney et al. 2006). The question has been raised whether these distances are consistent with observations showing that embedded clusters largely disrupt their parental cloud and disperse in a few million years (see Sec. 2). The dispersal of the molecular gas makes the presence of nearby protostars unlikely, and the subsequent expan- sion of the cluster make the presence of young stars with disks less likely. There are possible solutions to this problem. The Sun may have remained close to a hot star as the cluster dispersed. Only one low mass star with a disk is found within a projected distance of ∼ 0.3 pc of the O6 star HD206267 in the 4 Myr old IC 1396 association (Sicilia-Aguilar et al. 2006), suggesting that this may be rare occurrence. The Sun may have been a bound companion to a massive star, such as the companions with disks found around the OB stars comprising the Orion Trapezium (Schertl et al. 2003); however, it unclear how long such a disk may survive. The Sun could have formed in a massive embedded cluster which evolved into a bound open cluster. In this case, the solar system would have to survive photoevaporation and perturbations from tidal interactions as it orbited within the cluster (Adams & Laughlin 2001). Finally, the solar system may have been enriched by the combined ejecta of many supernova (Hester & Desch 2005; Williams & Gaidos in prep). Additional data on SLRs in meteorites, de- tailed modeling of the evolution and dispersal of embedded clusters, and the study of other planetary systems in hot environments should bring a more de- tailed understanding of our Sun’s formation environment. The presence of SLRs may have had a significant impact on planet formation in the solar nebula. Radioactive decay of 26Al and 60Fe provides by far the largest source of energy for melting and differentiating planetesimals in the early Solar System (Bizzarro et al. 2005; Hevey & Sanders 2006). In summary, it has been amply demonstrated by observation and theory that environment plays a significant role in the formation of cool stars and planets. A comprehensive understanding of star and planet formation must not treat young stars and protoplanetary solely as isolated objects, but as parts of larger associations and clusters in which the formation of cool and hot stars are inextricably linked. References Adams,F. 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0704.1046

Heights and metrics with logarithmic singularities

Heights and metrics with logarithmic singularities Gerard Freixas i Montplet Abstract We prove lower bound and finiteness properties for arakelovian heights with respect to pre-log-log hermitian ample line bundles. These heights were introduced by Burgos, Kramer and Kühn in [2], in their extension of the arithmetic intersection theory of Gillet and Soulé [8], aimed to deal with hermitian vector bundles equipped with metrics admitting suit- able logarithmic singularities. Our results generalize the corresponding properties for the heights of cycles in Bost-Gillet-Soulé [1], as well as the properties established by Faltings [7] for heights of points attached to her- mitian line bundles whose metrics have logarithmic singularities. We also discuss various geometric constructions where such pre-log-log hermitian ample line bundles naturally arise. MSC: 14G40 (Primary) 11J25, 11J97 (Secondary). Contents 1 Introduction 2 2 Differential forms with logarithmic singularities 5 3 Logarithmically singular hermitian vector bundles 13 4 Global bounds for real log-log growth (1,1)-forms 17 5 Bounding height integrals 25 6 Arakelovian heights 39 7 Examples 46 8 Appendix 52 Conventions We fix some conventions and notations to be followed throughout this paper. The open disc of C centered at 0 and of radius ε > 0 will be denoted by ∆ε. If f, g : E → R are two functions on a set E 6= ∅, we write f ≺ g to mean that there exists a constant C such that f(x) ≤ Cg(x) for all x ∈ E. If the involved constant depends on some data D that we want to specify, we write f ≺D g. http://arxiv.org/abs/0704.1046v1 If X is a complex analytic manifold we decompose the exterior differential operator as d = ∂+ ∂̄ and we define dc = (4πi)−1(∂− ∂̄), so that ddc = i ∂ ∂̄ /2π. Let k be a field. By an algebraic variety X over k we mean a separated and reduced scheme of finite type over k. In particular, X is noetherian and it has a finite number of irreducible components. When k = C, for every separated scheme of finite type X over C there is an associated complex analytic space Xan, whose underlying topological space equals the set of complex points X(C). If F is a closed subscheme of X , then F an is an analytic subspace of Xan. The scheme X is proper over C if, and only if, Xan is compact. Also, X is a nonsingular variety over C if, and only if, Xan is a complex analytic manifold. To simplify notations we will write X instead of Xan or X(C) (it will be clear from the context the category we are working on). Let K be a number field. Its ring of integers is denoted by OK . Let S = SpecOK . An arithmetic variety over S will be a flat and projective scheme π : X → S , with regular generic fiber XK = X ×S SpecK of pure dimension n. The set of complex points X (C) has a natural structure of complex analytic manifold, and it can be partitioned as X (C) = σ:K →֒C Xσ(C). The complex conjugation induces an antiholomorphic involution F∞ : X (C) → X (C). 1 Introduction In this paper we establish a common generalization of the following two state- ments. Theorem 1.1 (Faltings [7], Lemma 3). Let X be a projective arithmetic variety and Y ⊆ X a Zariski closed subset. Let L = (L , ‖ ·‖) be an ample line bundle on X endowed with a smooth hermitian metric on L |X (C)\Y (C). Suppose that ‖ · ‖ has logarithmic singularities along Y (C). Fix a number field K. For any real constant C, there are only finitely many points P ∈ X (K) \ Y (K) with (P ) ≤ C. The notions of function and metric with logarithmic singularities are recalled in §2 and §3 below. Theorem 1.2 (Bost-Gillet-Soulé [1], Proposition 3.2.4 and Theorem 3.2.5). Let X be a projective arithmetic variety and L a smooth hermitian ample line bundle on X . 1. For any real constant A, there are only finitely many effective cycles z ∈ Zp(X ) such that deg z ≤ A and h (z) ≤ A. 2. There exists a positive constant κ such that h (z) ≥ −κ deg z for every effective cycle z ∈ Zp(X ). Let us place under the hypothesis of Theorem 1.1. Let P ∈ X (K) \ Y (K), extended to P : SpecOK → X by properness of X . Since L is ample, there exists some positive power L ⊗N admitting a global section s non-vanishing at P . Then (P ) = log ♯ (L ⊗N ) (sOK) σ:K →֒C log ‖s‖σ(Pσ), where Pσ is the point in Xσ(C) induced by P . Therefore, in Faltings’ result, only the metric as a function is required. The definition of the height of a cycle of positive relative dimension, with respect to a smooth hermitian line bundle, involves the derivatives of the metric up to second order (see [1] §3 and also §6 below). Therefore, to extend both theorems, we need to describe the kind of logarithmic singularities allowed to the derivatives of the metric, up to second order. The arithmetic intersection theory of Gillet and Soulé needs to be reinterpreted so we can deal with such metrics. This has been done in [2], where Burgos, Kramer and Kühn develop a theory of abstract arithmetic Chow groups, and apply it to the case of logarithmic singularities. Before the statement of our main theorem we introduce some notations. Let K be a number field, OK its ring of integers and X a projective arithmetic variety over SpecOK . Suppose that D ⊆ XK is a divisor such that D(C) has normal crossings. Write U = X (C) \D(C). By Z U (X ) we denote the group of codimension p cycles z on X , such that z(C) intersects D(C) properly. A cycle on X is called horizontal if it is the Zariski closure of a cycle on XK . We write U (XK) for the subgroup of Z U (X ) consisting of horizontal cycles. Now let L be a line bundle on X , endowed with a pre-log-log hermitian metric ‖ · ‖, with singularities along D (see §3 below for the definition). According to Burgos, Kramer and Kühn, this is the notion of metric with logarithmic singularities well suited to define heights on Z U (X ) (see [2] §7). If LK is ample there is a well defined normalized height h̃ U (XK) with respect to L = (L , ‖ · ‖). We refer to §6 for a summary of the theory of heights in Arakelov theory. Theorem 1.3 (Main Theorem). Let X be a projective arithmetic variety over SpecOK , D ⊆ XK a reduced divisor such that D(C) ⊆ X (C) has simple normal crossings and U = X (C) \ D(C). Let L be a line bundle on X with LK ample. Let ‖ · ‖ be a pre-log-log hermitian metric on L , with singularities along D, and ‖ · ‖0 a smooth metric on L . Then there exist constants α, β, γ > 0 and R ∈ Z≥0 such that for every effective cycle z ∈ Z U (XK) we have (z) + γ ≥ 1 and (1.1) ∣∣∣h̃ (z)− h̃ ∣∣∣ ≤ α+ β logR (z) + γ If moreover ‖ · ‖ is good along D, then we can take R = 1. In conjunction with Theorem 1.2, Theorem 1.3 immediately yields the de- sired finiteness property as well as the existence of a universal lower bound. Corollary 1.4. Let X be a projective arithmetic variety over SpecOK , D ⊆ XK a reduced divisor such that D(C) has simple normal crossings and U = X (C) \D(C). Let L be a pre-log-log hermitian ample line bundle on X , with singularities along D. 1. For any real constant A, there are only finitely many effective cycles z ∈ U (X ) such that degLK z ≤ A and hL (z) ≤ A. 2. There exists a positive constant κ such that h (z) ≥ −κ deg z for every effective cycle z ∈ Z U (X ). The techniques employed for the proof of the main theorem were initially in- spired by the work of Carlson and Griffiths on the defect relation for equidimen- sional holomorphic maps, in higher dimensional Nevanlinna theory [4]. Moreover (1.1) may be interpreted as a vast generalization of the naive Liouville’s inequal- ity for the distance between an algebraic number and a rational number. This point of view is conceptually interesting, since it opens the natural question of finding an analogue to Roth’s theorem for the distance(1) between a divisor with simple normal crossings and effective cycles of arbitrary dimension (all defined over a number field)(2). This paper is organized as follows. In Section 2 we review the theory of differential forms with logarithmic sin- gularities necessary in the rest of the paper. In Section 3 we study in detail several notions of logarithmically singular hermitian vector bundles. The re- sults we recall provide a wealth of constructions to which Theorem 1.1 and Theorem 1.3 apply. Both sections are complemented with examples for a bet- ter understanding of the theory. In Section 4 we establish global bounds for real log-log growth (1,1)-forms. An outstanding consequence is a decomposition theorem (Theorem 4.3) for pre-log-log functions, which plays a crucial role in the proof of the main theorem. In Section 5 we prove estimates for integrals of pre-log-log forms appearing in the archimedian part of the definition of height. This leads to the proof of the main theorem in Section 6, where the reader will find an overview of the theory of heights in Arakelov geometry. In Section 7 we present some examples of good hermitian line bundles interesting for arith- metic purposes (for instance in relation with Theorem 1.3). We treat the case of fully decomposed automorphic vector bundles on locally symmetric varieties, the relative dualizing sheaf of the universal curve over the moduli space of stable curves, equipped with the family hyperbolic metric, the Weil-Petersson metric on the moduli space of curves and the Kähler-Einstein metric on quasi-projective varieties. Finally, in the Appendix we prove a Bertini’s type theorem needed for the preliminary reductions in the proof of the main theorem. Acknowledgements. I am deeply indebted to J.-B. Bost and J. I. Burgos Gil for proposing me to work on this subject as the starting point of my PhD. thesis, and for their guidance and constant encouragement. During the preparation of this paper I benefited from stimulating discussions with several people: I warmly thank S. Fischler, U. Kühn, V. Maillot, C. Mourougane and D. Roessler for the time they devoted to me. Their useful advice is reflected throughout the paper. This work was presented, in a preliminary form, at the Internationales Graduiertenkolleg Arithmetic and Geometry of the Humboldt-Universität zu Berlin and the ETH of Zürich, in May 2005. I am grateful to J. Kramer for inviting me to talk to their summer school. (1)Suitable candidates for the logarithm of the distance are provided by the height integrals introduced in §5 below. (2)The height morphism h is defined for any pre-log-log hermitian line bundle L , with singularities along a divisor in X (C). However, for the main theorem to hold, the divisor needs to be defined over a number field. This essentially goes back to the construction of transcendental numbers due to Liouville. 2 Differential forms with logarithmic singulari- Let X be a complex analytic manifold and F a closed analytic subspace. In this section we introduce several notions of differential forms on X with logarithmic singularities along F , relevant to our work. We distinguish the case of functions from the case of differential forms, since the former can be presented in a more general geometric frame. Indeed, while we can define functions with logarithmic singularities along an arbitrary closed analytic subspace F , the appropriate analogues for differential forms of any order require F to be a divisor with normal crossings. 2.1 Functions with logarithmic singularities Let X be a complex analytic manifold and F a closed analytic subspace of X . We denote by IF ⊆ OX the ideal sheaf defining F and supp(F ) for the support of F . For every x ∈ X , there exist an open neighborhood V and holomorphic functions s1, . . . , sm ∈ OX(V ) such that i. the germ of ideal sheaf IF,x is generated by s1, . . . , sm; ii. the trace of the support of F on V is supp(F ) ∩ V = {z ∈ V | s1(z) = . . . = sm(z) = 0}. Since OX is a coherent sheaf, so is IF . Then, given s1, . . . , sm as above, after possibly shrinking V , s1, . . . , sm generate all the germs IF,z for z ∈ V . In this case we say that s1, . . . , sm generate IF |V and we write IF |V = (s1, . . . , sm) as a shortcut. The reader is referred to [6] for further details on analytic spaces. Definition 2.1. Let X be a complex analytic manifold and F ⊆ X a closed analytic subspace. A smooth function f : X \ supp(F ) → C has logarithmic sin- gularities along F if for every open subset V of X such that IF |V = (s1, . . . , sm) and every relatively compact open subset V ′ ⊂⊂ V , there exists an integer N ≥ 0 such that (2.1) |f|V ′\supp(F ) | ≺ ∣∣∣∣log( maxi=1,...,m |si|) For a function f : X \ supp(F ) → C to be with logarithmic singularities along F it is enough that (2.1) be satisfied for a given open covering of X and given local generators of IF . This is the content of the next lemma. Lemma 2.2. Let X be a complex analytic manifold and F a closed analytic subspace. Fix an open covering {Vα}α of X such that IF |Vα is generated by sα1 , . . . , s . Suppose given V ′α ⊂⊂ Vα still forming an open covering of X. Then a smooth function f : X \ supp(F ) → C has logarithmic singularities along F if, and only if, for every α there exists an integer N ≥ 0 such that |f|V ′α\supp(F ) ∣∣∣∣log( maxi=1,...,mα |sαi |) Proof. Left as an easy exercise. Lemma 2.3. Let X be a complex analytic manifold and F , G closed analytic subspaces with supp(F ) = supp(G). A smooth function f : X\supp(F ) → C has logarithmic singularities along F if, and only if, it has logarithmic singularities along G. Proof. Write I, J for the ideal sheaves of F , G, respectively. Let V be an open neighborhood of x ∈ X such that I|V = (s1, . . . , sl) and J|V = (t1, . . . , tm). By Hilbert’s Nullstellensatz (see (4.22) in [6]), after possibly shrinking V , there exists an integer N ≥ 0 such that sNi = i tj , t µijsi. for some holomorphic functions λ i , µ j . Hence, if V ′ ⊆ V is a relatively compact open subset, there exists a constant C > 0 such that N ≤ C max j=1,...,m |tj |, |tj | N ≤ C max i=1,...,l |si|. The lemma follows. The meaning of the lemma is that the notion of function with logarithmic sin- gularities along F depends only on the support of F . Proposition 2.4. i. Let X be a complex analytic manifold and F,G closed an- alytic subspaces with supp(F ) ⊆ supp(G). A smooth function f : X\supp(F ) → C with logarithmic singularities along F has logarithmic singularities along G. ii. Let ϕ : X → Y be a morphism of complex analytic manifolds and F ⊆ Y a closed analytic subspace. If f : Y \ supp(F ) → C has logarithmic singularities along F , then ϕ∗f = f ◦ ϕ has logarithmic singularities along ϕ−1(F ). If ϕ is surjective and proper, the converse holds. Proof. The first item i is straightforward. We shall prove the second part ii. Let V be an open subset of Y such that IF |V = (s1, . . . , sm). The ideal sheaf of ϕ−1(F ∩ V ) is (ϕ∗s1, . . . , ϕ ∗sm). Let {Vα}α be an open covering of V by relatively compact subsets. For every open U ⊂⊂ ϕ−1(V ) define Uα = U ∩ ϕ−1(Vα). Then Uα is relatively compact in ϕ −1(V ) and ϕ(Uα) ⊆ Vα. The estimate |f|Vα\supp(F )| ≺ ∣∣∣∣log( maxi=1,...,m |si|) implies the corresponding inequality for ϕ∗(f) on Uα. We now prove the converse under the surjectivity and properness assumption. Let V be as above and V ′ ⊂⊂ V an open subset. Since ϕ is proper, ϕ−1(V ′) is relatively compact in ϕ−1(V ). The hypothesis of logarithmic singularities of ϕ∗f asserts the existence of an integer N ≥ 0 such that |ϕ∗f|ϕ−1(V ′\supp(F ))| ≺ ∣∣∣∣log( maxi=1...,m |ϕ ∗si|) We come up with the conclusion by the surjectivity of ϕ. 2.2 Differential forms with logarithmic singularities Definition 2.5 (Divisor with normal crossings). Let X be a complex analytic manifold of dimension n. A reduced analytic subspace D of X is a divisor with normal crossings if X can be covered by open subsets V with coordinates z1, . . . , zn such that D ∩ V is defined by z1 · . . . · zm = 0, for some 0 ≤ m ≤ n. We say that D has simple normal crossings if it can be written as a finite union of smooth analytic hypersurfaces of X . Definition 2.6 (Adapted analytic chart [2]). Let X be a complex analytic manifold of dimension n and D a divisor with normal crossings in X . An analytic chart (V ; {zi} i=1) is said to be adapted to D if |zi| < 1/e, i = 1, . . . , n and D ∩ V is defined by z1 · . . . · zm = 0, for some 0 ≤ m ≤ n. The integer m will be understood when no confusion can arise. Notation 2.7. Let X be a complex analytic manifold and D ⊆ X a divisor with normal crossings. Let (V ; {zi}i) be an analytic chart such that D ∩ V is defined by z1 · . . . · zm = 0. We define dζk = zk log |zk|−1 , if 1 ≤ k ≤ m dzk, if k > m and similarly for dζ̄k. Given I, J ordered subsets of {1, . . . , n}, we abbreviate dζI ∧ dζ̄J = dζi ∧ dζ̄j . In the following definitions we write X for a complex analytic manifold of dimension n, D a divisor with normal crossings in X , U = X \D and ι : U →֒ X for the natural open immersion. The sheaf of C∞ complex differential forms on U is denoted by E∗U . Definition 2.8 (Poincaré growth forms [14]). The sheaf of Poincaré growth forms on X , with singularities along D, is the subalgebra PD of ι∗E U gener- ated, on every open analytic chart (V ; {zi}i) adapted to D, by C ∞(V \D,C) ∩ L∞loc(V,C) and the differential forms dζk, dζ̄k, k = 1, . . . , n. Namely, for every analytic chart (V ; {zi}i) adapted to D, ι∗E U (V ) is the C-vector space generated by differential forms ∑ αI,JdζI ∧ dζ̄J where αI,J ∈ C ∞(V \D) are locally bounded on V . Definition 2.9 (Good forms [14]). A good differential form on an open subset V of X , with singularities along D, is a section ω ∈ Γ(V,PD) such that dω ∈ Γ(V,PD). Definition 2.10 (log-log growth forms [2]). The sheaf of log-log growth forms on X , with singularities along D, is the subalgebra LD of ι∗E U generated, on every analytic chart (V ; {zi}i) adapted to D, by the functions f ∈ C ∞(V \D,C) such that on every open V ′ ⊂⊂ V (2.2) |f(z1, . . . , zn)| ≺ (log log |zk| for some integer N ≥ 0, together with the differential forms dζk, dζ̄k, k = 1, . . . , n. Remark 2.11. Observe that the following inequalities hold: (log log |zk| −1)N ≤ (log log |zk| −1)Nm ≤ 2m−1 (log log |zk| −1)Nm. Therefore, a smooth function f on V \ D has log-log growth along D if, and only if, for every open V ′ ⊂⊂ V there is an estimate (2.3) |f|V ′\D| ≺ (log log |zk| −1)N . In concrete computations involving functions with log-log growth, we may use either formulations (2.2) or (2.3). Proposition 2.12. Let f : X \D → C be a smooth function. Suppose that df has log-log growth with singularities along D. Then f has log-log growth with singularities along D. More precisely, for every analytic chart (V ; {zi}i) adapted to D and every open V ′ ⊂⊂ V , if df = j gjdζj + j hjdζ̄j with |gj|V ′\D |, |hj|V ′\D | ≺ (log log |zk| −1)N , then we have |f|V ′\D| ≺ (log log |zi| −1)N+1 i<j≤m (log log |zj | −1)N . Proof. After localizing to an analytic chart adapted to D we reduce to V = with V \D = ∆∗r . Let 0 < ε < 1 and Uε = ∆ be contained in V . Let (w1, . . . , wn) ∈ Uε \D. Fix 1 ≤ i ≤ r. Define the curve γ : [0, 1] −→ V t 7−→ (tw1 + (1− t) e|w1| , w2, . . . , wn). Then we have (f ◦ γ)(1)− (f ◦ γ)(0) = γ∗(df). Now we write γ∗(df|Uε\D) = γ z1 log |z1|−1 z1 log |z1|−1 where g, h ∈ C∞(Uε \D,C) satisfy |g|, |h| ≺ (log log |zk| −1)N on Uε \D. A straightforward computation yields df|Uε\D ) ∣∣∣∣ ≺f,Uε 1<j≤r (log log |wj | −1)N · (log log γ∗|z1| log γ∗|z1| log γ∗|z1|−1 ≤ (log log |w1| −1)N+1 1<j≤r (log log |wj | −1)N . It follows that |f(w1, . . . , wn)| ≺f,Uε ∣∣∣∣f e|w1| , w2, . . . , wn )∣∣∣∣ +(log log |w1| −1)N+1 1<j≤r (log log |wj | −1)N . By induction we find |f(w1, . . . , wn)| ≺f,Uε sup |f(z1, . . . zn)| (log log |wi| −1)N+1 i<j≤r (log log |wj | Observe that the sup is finite because f is smooth on X \ D. The proof is complete. Definition 2.13 (pre-log-log forms [2]). The sheaf of pre-log-log forms on X , with singularities along D, is the subalgebra E∗X〈〈D〉〉pre of ι∗E U generated by log-log growth forms ω in LD such that ∂ ω, ∂̄ ω and ∂ ∂̄ ω are also log-log growth forms in LD. A particular case of pre-log-log forms that deserves to be distinguished is that of P-singular functions. Definition 2.14 (P-singular function). Let f : X\D → C be a smooth function. We say that f is a P-singular function, with singularities along D, if, and only if, df and ddcf have Poincaré growth along D. Corollary 2.15. Let f : X \D → C be a P-singular function. Then, for every adapted analytic chart (V ; {zi}i) and every open V ′ ⊂⊂ V , we have |fV ′\D| ≺ (log log |zk| Consequently, f is a pre-log-log function. Proof. This is a straightforward application of Proposition 2.12. For later computations it will be worth having at our disposal the following basic properties of log-log growth forms. Proposition 2.16. i. Any log-log growth form is locally integrable. Moreover, log-log growth functions and log-log growth 1-forms are locally L2. ii. (Stokes’ theorem for pre-log-log forms.) If ω ∈ Γ(X, E∗X〈〈D〉〉pre) and [ω] denotes its associated current, then d[ω] = [dω] and similarly for ∂, ∂̄ and ∂ ∂̄. iii. If f : X → Y is a morphism of complex analytic manifolds and DX ⊆ X, DY ⊆ Y normal crossing divisors with f −1(DY ) ⊆ DX , then f ∗PDY ⊆ PDX and f∗LDY ⊆ LDX . Therefore f ∗E∗Y 〈〈DY 〉〉pre ⊆ E X〈〈DX〉〉pre. In particular, this is true for f being the natural inclusion of a complex analytic submanifold X ⊆ Y intersecting DY transversally and DX = DY ∩X. Proof. The proposition quotes [2], Proposition 7.5 and Proposition 7.6. How- ever, for later use, we may comment on the proof of i. After changing to polar coordinates, it is enough to observe that for every 0 < δ < 1 we have an estimate ∫ ε/e (log log t−1)N t(log t)2 ∫ ε/e t(log t−1)1+δ < +∞. We finally give an example showing that the notion of pre-log-log function depends on the compactification X of U . Example 2.17. Let X = P2 with projective coordinates (w0 : w1 : w2). As divisor with normal crossings set D = (w0 = 0) ∪ (w1 = 0). Define the smooth function on U = X \D g(w0 : w1 : w2) = |w0|2 + |w1|2 Denote by X̃ the blowing-up of X at (0 : 0 : 1). X̃ admits the following description: ((w0 : w1 : w2), (z0 : z1)) ∈ X × P | w0z1 = w1z0 The map realizing the blowing-up is the projection onto the first factor π : X̃ → X . In particular, since (0 : 0 : 1) ∈ D, we have an isomorphism π−1(U) → U , and π−1(D) is a divisor with normal crossings. Observe that the pullback of g by π is f((w0 : w1 : w2), (z0 : z1)) = |z0|2 + |z1|2 The function f extends to a smooth function on the whole X̃, in particular pre- log-log along π−1(D). However, we claim that g is not a pre-log-log function alongD. To see this we compute ∂ g. We may localize at the affine neighborhood w2 6= 0 of (0 : 0 : 1) and write u = w0/w2, v = w1/w2. In coordinates u, v we ∂ g = |u|2|v|2 (|u|2 + |v|2)2 But the function |u|2|v|2 log |u|−1/(|u|2 + |v|2)2 does not have log-log growth along D, as we see after restriction to |u| = |v|. This proves the claim. 2.3 Variants: log-log forms Following [3] we introduce a variant of the sheaf of pre-log-log forms, by imposing bounds on all the derivatives of the component functions of the differential forms. There are also corresponding variants for Poincaré growth forms and good forms, for which we refer to loc. cit. We fix a complex analytic manifold X and D ⊆ X a divisor with normal crossings. Write U = X \D and ι : U →֒ X for the natural open immersion. Definition 2.18 (log-log functions of infinite order [3]). A smooth function f : X \D → C is said to be a log-log function of infinite order, with singularities along D, if for every analytic chart (V ; {zi}i) adapted toD, every open V ′ ⊂⊂ V and multi-indices α = (α1, . . . , αn), β = (β1, . . . , βn), there is a bound on V ∣∣∣∣∣ f(z1, . . . , zn) ∣∣∣∣∣ ≺ k=1(log log |zk| where zα = zα11 . . . z m (similarly for z β≤m) and N depends on V ′, α, β. Definition 2.19 (log-log growth forms of infinite order [3]). The sheaf of log-log growth forms of infinite order onX , with singularities along D, is the subalgebra of ι∗E U generated, on every analytic chart V adapted to D, by log-log growth functions of infinite order and the differential forms dζk, dζ̄k, k = 1, . . . , n (see Notation 2.7). Remark 2.20. Let ω be a log-log growth form of infinite order alongD, defined on some analytic open subset of X . Then the complex conjugate ω̄ is also a log-log growth form of infinite order along D. Definition 2.21 (log-log forms [3]). The sheaf of log-log forms on X , with singularities along D, is the subalgebra E∗X〈〈D〉〉 of ι∗E U generated by log-log growth forms ω of infinite order, such that ∂ ω, ∂̄ ω and ∂ ∂̄ ω are also log-log growth forms of infinite order. Remark 2.22. There is an obvious inclusion E∗X〈〈D〉〉 ⊆ E X〈〈D〉〉pre. Log-log growth forms of infinite order enjoy of analogous properties to the log-log growth forms introduced before. We refer to [3] for details. An advan- tage of the sheaf of log-log forms over the sheaf of pre-log-log forms is that a Poincaré’s type lemma holds for the former. The next essential property follows. Theorem 2.23 ([3]). The natural inclusion Ω∗X −→ E X〈〈D〉〉 is a filtered quasi-isomorphism with respect to the Hodge filtration. Proposition 2.24. Let f : X \ D → C be a smooth function. Then f is a log-log form, with singularities along D, if, and only if, df is locally L2 on X and ddcf is a log-log growth form of infinite order along D. Proof. The direct implication is an easy exercise. Let us see the converse. By hypothesis, ∂̄ ∂ f is a log-log form, with singularities along D. Let x ∈ X . By Theorem 2.23, there exists an open neighborhood V of x and a log-log form ω on V such that ∂̄ ∂ f = ∂̄ ω. Therefore, we can write ∂ f = ω + θ. for some holomorphic form θ on V \D. Observe that θ is locally L2, because ∂ f is locally L2 by hypothesis and ω is a log-log 1-form (see Proposition 2.16). By Lemma 2.25 below, θ must be holomorphic on V . This proves that ∂ f has log-log growth of infinite order along D. The same reasoning applied to ∂̄ ∂ f̄ (the complex conjugate of ∂ ∂̄ f) proves that ∂̄ f has log-log growth of infinite order along D. Therefore df has log-log growth of infinite order. Again by Theorem 2.23, after possibly shrinking V , there exists a log-log func- tion of infinite order g and a holomorphic function h on V \D such that f = g+h. We claim that h is locally L2. Since this is true for g, we are reduced to prove it for f . But we have already shown that df has log-log growth of infinite order along D, so that Proposition 2.12 implies that f has log-log growth along D. In particular, f is locally L2 (Proposition 2.16). By Lemma 2.25, g is holomorphic on V and hence f has log-log growth of infinite order along D. This finishes the proof. Lemma 2.25. Let X be a complex analytic manifold and D ⊆ X a divisor with normal crossings. Let θ be a holomorphic function on X \D. If θ is locally L2 on X, then θ extends to a holomorphic function on X. Proof. The lemma is well-known, but we include the proof for lack of reference. It is enough to treat the case whenX = ∆nε ⊆ C n andD is defined by z1·. . .·zr = 0, so that X \D = ∆∗rε ×∆ ε. We write δ = (δ1, . . . , δr) ∈ R >0. Since θ is locally (2.4) ‖θ‖2ε := lim Iδ < +∞ where ∆ε/2\∆δk )×∆ ∣∣∣∣∣ dzk ∧ dz̄k ∣∣∣∣∣ . The Laurent series development (2.5) θ(z1, . . . , zn) = is absolutely and uniformly convergent on any ( k=1 ∆ε/2\∆δk)×∆ . There- fore, the integral Iδ can be computed term by term: ν,µ∈Zn ∆ε/2\∆δk k |dzk ∧ dz̄k| zνkk z k |dzk ∧ dz̄k| Recall that given integers a, b we have eaiθebiθdθ = 2πδa,b, so that |aν | ∆ε/2\∆δk 2νk |dzk ∧ dz̄k| 2νk |dzk ∧ dz̄k| (2.6) We reason by contradiction and assume that θ does not extend to a holomorphic function on ∆n . We can suppose that in (2.5) there appears a term aνz ν 6= 0, with ν = (ν1, . . . , νn) ∈ Z <0 × Z ≥0 , 1 ≤ l ≤ r. From (2.6) and by direct computation we find (2.7) Iδ ≥ (4π) n|aν | k=l+1 (ε/2)2νk+2 2νk + 2 δ2νk+2k 2νk + 2 (ε/2)2νk+2 2νk + 2 where Jδk = if νk = −1, (ε/2)2νk+2 2νk+2 2νk+2 2νk+2 if νk < −1. Since aν 6= 0 and Jδk → +∞ as δ → 0, we see from (2.7) that Iδ → +∞ as δ → 0. This contradicts (2.4). The proof is complete. 3 Logarithmically singular hermitian vector bun- Let X be a complex analytic manifold and D ⊆ X a divisor with normal cross- ings. Write U = X \D and ι : U →֒ X for the natural open immersion. In this section we study vector bundles endowed with hermitian metrics with singular- ities of logarithmic type along D. The reader is referred to §2 for the several definitions and properties of differential forms with singularities of logarithmic type along D. Definition 3.1 ([3] and [14]). Let E be a vector bundle of rank r on X . A smooth hermitian metric h on E|U is said to have logarithmic singularities along D if, for every trivializing open subset V and holomorphic frame e1, . . . , er of E|V , putting hij = h(ei, ej) and H = (hij) on V \D, the following condition is fulfilled: (L(E, h)) the functions |hij |, detH −1 have logarithmic singularities along D ∩ V (see Definition 2.1). We say that h is (pre-)log-log (resp. good) along D if moreover, for every such data V and e1, . . . , er, the following property holds: (G(E, h)) the entries of the matrix (∂ H)H−1 are (pre-)log-log (resp. good) forms on V , with singularities along D (see Definition 2.9). We will usually write E = (E, h) when no confusion on the metric can arise. Sometimes we use some variants of the definition, and we say for instance “E has logarithmic singularities along D” or “E is (pre-)log-log (resp. good) along In the case of line bundles, the notions of (pre-)log-log and good hermitian metrics can be characterized by slightly simpler properties. Proposition 3.2. Let L be a hermitian line bundle on X and h a smooth hermitian metric on L|U . Write ‖ · ‖ for the norm associated to h. i. The metric h is (pre-)log-log (resp. good) with singularities along D if, and only if, for every trivializing open subset V and holomorphic frame e of L|V , the function log h(e, e) is (pre-)log-log (resp. P-singular) on V , with singularities along D. ii. The metric h is log-log with singularities along D if, and only if, for every trivializing open subset V and holomorphic frame e of L|V , the form ∂ log h(e, e) is locally L2 on V and ∂̄ ∂ log h(e, e) has log-log growth of infinite order, with singularities along D. Proof. This follows from the definitions and Proposition 2.12, Proposition 2.15 and Proposition 2.24 applied to the smooth real function log h(e, e) on V \D. An essential extension property of hermitian vector bundles with logarithmic singularities is the following observation due to Mumford. Proposition 3.3. Let (E◦, h) be a smooth hermitian vector bundle on U . Then there exists at most one extension of (E◦, h) to a hermitian vector bundle (E, h) on X, with logarithmic singularities along D. More precisely, if (E, h) is such an extension, then for every open subset V in X Γ(V,E) = s ∈ Γ(V, ι∗E ◦) | h(s, s) has log. sing. along D ∩ V Proof. This is Proposition 1.3 in [14]. Hermitian vector bundles with logarithmic singularities along D admit the following characterization. Proposition 3.4. Let E be a vector bundle on X and hE a smooth hermitian metric on E|U . Denote by hE∨ the dual metric. Then hE has logarithmic singularities along D if, and only if, the following condition is satisfied with F = E and F = E (L̃(F, hF )) for every open subset V and any holomorphic section s of F|V , the function hF (s, s) has logarithmic singularities along D ∩ V . Proof. For the direct implication, first take a holomorphic section s of E over an open subset V . We may assume that s does not vanish on V . After pos- sibly shrinking V , we can complete s to a holomorphic frame e1 = s, . . . , er of E|V . By the definition of metric with logarithmic singularities, the function hE(s, s) = hE(e1, e1) has logarithmic singularities along D. Secondly, let V be a trivializing open subset of E and e1, . . . , er a holomor- phic frame of E|V . Write H = (hij) for the matrix of hE in base {ei}i and H−1 = (gij) for the inverse matrix. From the very construction of H −1 and the logarithmic singularities of the functions hij and detH −1, it is immedi- ate to check that the functions gij have logarithmic singularities along D. If B is the matrix of hE∨ in any holomorphic frame of E , then there exists A ∈ GLr(Γ(V,OX)) such that B = At ·H−1 · A. Since the entries of A are holomorphic, the entries of B inherit from H−1 the logarithmic singularities along D ∩ V . Let now s be a holomorphic section of E∨ over an open subset V . Replacing V by a smaller open subset, we can complete s to a holomorphic frame of E∨ v1 = s, . . . , vr. As we have just proven the functions hE∨(vi, vj) have logarithmic singularities along D, in particular so does hE∨(s, s) = h(v1, v1). Now for the converse. Let V be a trivializing open subset, adapted to D. Let e1, . . . , er be a frame for E|V . Write H = (hij) for the matrix of the hermitian metric hE in base {ei}i. By hypothesis, for every open subset V ′ ⊂⊂ V , there exists an integer N ≥ 0 such that on V ′ hE(ei, ei) ≺ (log |z1 · . . . · zm| −1)N . Applying Schwarz’s inequality we get |hij | 2 ≺ (log |z1 · . . . · zm| −1)2N . The same argument provides similar bounds for the entries of the matrix of hE∨ in the dual basis, namely H−1. Since the determinant of H−1 is a polynomial in the entries of this matrix, we derive a bound detH−1 ≺ (log |z1 · . . . · zm| for some integer M . This concludes the proof. As an immediate consequence of the proposition we establish the following corollary. Corollary 3.5. Let E = (E, hE) be a hermitian vector bundle with logarithmic singularities along D. For every exact sequence of vector bundles 0 −→ F −→ E −→ Q −→ 0, the induced hermitian vector bundles F = (F, hF ) (restricted metric) and Q = (Q, hQ) (quotient metric) have logarithmic singularities along D. Proof. It is enough to prove that for every exact sequence as in the statement, conditions (L̃(F )) and (L̃(Q)) hold. Indeed, since E has logarithmic singular- ities along D, conditions (L̃(F )) and (L̃(Q )) automatically follow by duality. Then we conclude applying Proposition 3.4. The validity of L̃(F ) is clear. For L̃(Q), we just observe that if s is a holomorphic section of Q|V and s̃ is a holomorphic section of E|V lifting s, then hQ(s, s) ≤ hE(s̃, s̃). Thus we see that L̃(E) implies L̃(Q). We next state the main formal properties of logarithmically singular (resp. (pre-)log-log, resp. good) hermitian vector bundles. Proposition 3.6. Let E, F be two vector bundles on X and hE and hF smooth hermitian metrics on E|U and F|U , respectively. If hE and hF have logarithmic (resp. (pre-)log-log, resp. good) singularities along D, then E , E ⊗ F , SkE and ∧kE have logarithmic (resp. (pre-)log-log, resp. good) singularities along Proof. Left as an elementary exercise. Proposition 3.7. Let X, Y be complex analytic manifolds and DX ⊆ X, DY ⊆ Y normal crossing divisors. Let f : X → Y be a morphism of com- plex analytic manifolds. Let E = (E, h) be a hermitian vector bundle on Y whose metric is defined and smooth on Y \DY . i. If f−1(DY ) ⊆ DX and h has logarithmic (resp. (pre-)log-log, resp. good) singularities along DY , then the metric f ∗(h) on f∗(E) has logarithmic (resp. (pre-)log-log, resp. good) singularities along DX . ii. Suppose that f is surjective, proper and f−1(DY ) = DX . Then h has logarithmic singularities along DY if, and only if, f ∗(E) has logarithmic singu- larities along DX . Proof. The first item i follows from Proposition 2.4 i and Proposition 2.16 iii. The second item ii is automatically deduced from Proposition 2.4 ii. Corollary 3.8. Let (E, h) be a hermitian vector bundle on X, with singularities along D. Let OE(1) be the dual of the trivial vector bundle of P(E), the projective space of lines in E∨. Denote by π : P(E) → X the natural projection. Then the metric on OE(1) induced by π ∗(h) has logarithmic singularities along π−1(D). Proof. By definition, the line bundle OE(1) is a quotient of π ∗(E). The hermi- tian metric on OE(1) is the quotient metric from π ∗(E). By Proposition 3.7, π∗(E) has logarithmic singularities along π−1(D). Then, by Corollary 3.5, so does the induced metric on OE(1). The end of this section is devoted to some counter-examples. Example 3.9. i. Counter-example to Corollary 3.5 and Corollary 3.8 for pre- log-log hermitian vector bundles. Let X = A1 be the complex line, with analytic coordinate z. Let D be the divisor with normal crossings z = 0. As vector bundle we take E = O⊕2X . We consider a hermitian metric h on E such that, in the standard basis e1, e2 and near the origin, its matrix H looks like log |z|−1 0 It is easily seen that (E, h) is pre-log-log along D (actually good). However, the induced metric on the line bundle OE(−1) ⊆ π ∗(E∨) on P(E) = P1 is not pre-log-log. Indeed, identifiy A1 as an open subset of P1 via t 7→ (1 : t). If e∨1 , e∨2 is the dual basis, define the section s of OE(−1)|A1 s = e∨1 + te Then h∨(s, s) = (log |z|−1)−1 + |t|2 and log h∨(s, s) = (log |z|−1)−1 + |t|2 If we restrict ∂ log h∨(s, s)/ ∂ t to the set C : t = (log |z|−1)−1/2, we find log h∨(s, s) (log |z|−1)1/2 which does not have log-log growth near z = 0. ii. The notion of hermitian vector bundle with logarithmic singularities depends on the compactification. Let Y be a smooth complex projective surface. Let p be a closed point in Y and π : X → Y the blowing-up of Y at p. Let D be a divisor with normal crossings in Y with p ∈ D. Then π−1(D) is a divisor with normal crossings. Define U = X \ π−1(D) and V = Y \D. Then π induces an isomorphism between U and V . Let h be a smooth hermitian metric on ωY |V , and endow ωX |U with the induced metric π ∗(h). Assume that (ωX , π ∗(h)) has logarithmic singularities along π−1(D). Then we claim that h does not define a metric on ωY with logarithmic singularities along D. Indeed, suppose that ωY = (ωY , h) had logarithmic singularities along D. Then, by Proposition 3.7, π∗(ωY ) would have logarithmic singularities along π −1(D). Observe that π∗(ωY )|U = ωX |U . By Proposition 3.3 we would derive the equality π∗(ωY ) = ωX . However we know that π∗(ωY ) = ωX ⊗O(−E) where E is the exceptional divisor π−1(p). Since the self-intersection (E2) = −1, O(−E) is not trivial. We thus arrive to a contradiction and the claim is proved. We remark that we can produce such examples just endowing ωX with a smooth hermitian metric and then restricting it to U . 4 Global bounds for real log-log growth (1,1)- forms 4.1 Statement of the theorem and consequences Let X be a complex analytic manifold and D ⊆ X a divisor with simple normal crossings. DecomposeD into smooth irreducible components,D = D1∪. . .∪Dm. For every Dk we fix a global section sk of O(Dk) with divisor div sk = Dk. We endow O(Dk) with a smooth hermitian metric ‖ · ‖k such that ‖sk‖ k ≤ e Therefore 1 ≤ log log ‖sk‖ −2 ≤ +∞ on X . Notation 4.1. For every integer N ≥ 0 we define the real positive smooth function on X \D (log log ‖sk‖ The purpose of this section is the proof of the following global bounds for real log-log growth (1,1)-forms on a compact complex analytic manifold. Theorem 4.2. Suppose that X is compact and let ω be a smooth positive (1,1)- form on X. Let η be a real log-log growth (1,1)-form on X, with singularities along D. Then there exist constants A,B > 0 and an integer N ≥ 0 such that on X \D (4.1) η +BΘN (dd c(−Θ1) +Aω) ≥ 0. If moreover η has Poincaré growth along D, then N can be chosen to be 0. The proof of the theorem is postponed until §4.3. Now we may discuss a result appearing as a particular instance of Theorem 4.2. Theorem 4.3. Suppose that X is compact and let ω be a smooth positive (1,1)- form on X. Let f : X \D → R be a pre-log-log function, with singularities along D. Then there exist positive pre-log-log functions, with singularities along D, ϕ, ψ : X \D −→ R≥0 and constants A,B ≥ 0, N ∈ Z≥0 with the properties i. f is the difference of ϕ and ψ: f = ϕ− ψ; ii. the following inequalities hold on X \D: ωϕ :=dd c(−ϕ) +BΘN (dd c(−Θ1) +Aω) ≥ 0, ωψ :=dd c(−ψ) +BΘN (dd c(−Θ1) +Aω) ≥ 0. If f is P-singular, then N can be chosen to be 0; iii. if f is P-singular, one can take ϕ, ψ to be P-singular with ddc(−ϕ) +Aω ≥ 0, ddc(−ψ) +Aω ≥ 0 on X \D. Proof. Since X is compact, from the log-log growth of f it is easily seen that for some constant C > 0 and integer M ≥ 0, f + CΘM ≥ 0 on X \D. If f is P-singular, then by Corollary 2.15 we can take (as we do) M ≤ 1. We define ϕ̃ = f + CΘM and ψ̃ = CΘM . These are positive pre-log-log functions, with singularities along D (see Lemma 4.7 below). If f is P-singular (hence M ≤ 1), then ϕ̃, ψ̃ are P-singular (again by Lemma 4.7). By Theorem 4.2 there exist constants A,B ≥ 0 and N ∈ Z≥0 such that eϕ := dd c(−ϕ̃) +BΘN (dd c(−Θ1) +Aω) ≥ 0 := ddc(−ψ̃) +BΘN (dd c(−Θ1) +Aω) ≥ 0 hold on X \D. Hence ϕ = ϕ̃ and ψ = ψ̃ satisfy the requirements of i and ii. If f is P-singular, then ddc(−ϕ̃), ddc(−ψ̃) have Poincaré growth along D and we may take N = 0. In this case we have eϕ =dd c(−ϕ̃) +mBddc(−Θ1) +mABω =ddc(−(ϕ̃+mBΘ1)) +mABω and similarly = ddc(−(ψ̃ +mBΘ1)) +mABω In view of these equalities, ϕ = ϕ̃ + mBΘ1 and ψ = ψ̃ + mBΘ1 fulfill the requirement of iii. We include the next corollary for its own interest, but we will not need it in the sequel. Corollary 4.4. Suppose that X is compact and Kähler. Let ω be a Kähler form on X. Let f : X \ D → R be a P-singular function and f = ϕ − ψ a decomposition as in Theorem 4.3 iii. Then the functions −ϕ,−ψ : X \D −→ R≤0 uniquely extend to quasiplurisubharmonic functions(3) on X. Proof. First of all, since −ϕ and −ψ are pre-log-log alongD, by Proposition 2.16 we have the equality of currents ddc[−ϕ] = [ddc(−ϕ)] and ddc[−ψ] = [ddc(−ψ)] on X . The inequalities ddc[−ϕ] +Aω ≥ 0, ddc[−ψ] +Aω ≥ 0 (4.2) then hold on X in the sense of currents. Let U ⊂ X be an open subset diffeo- morphic to a complex euclidian ball. Because ω is d-closed (Kähler assumption), by Poincaré’s lemma ω|U is d-exact. Since U itself is Kähler, ω|U is in fact dd exact. Write ω|U = dd ch for some smooth function h on U . Then the currents ddc[−ϕ|U +Ah] and dd c[−ψ|U +Ah] are positive on U , by (4.2). Since −ϕ and −ψ are bounded above and D is polar, ϕ̃ := −ϕ|U + Ah, ψ̃ := −ψ|U + Ah uniquely extend to plurisubharmonic functions on U (see [6], Theorem 5.24). Since h is smooth, these extensions determine extensions of −ϕ|U and −ψ|U to quasiplurisubharmonic functions on U , clearly unique. The corollary fol- lows. 4.2 Construction of pre-log-log functions 4.2.1 Preliminaries Lemma 4.5. Let M be a complex analytic manifold and α, β, C∞ differential forms of type (1,0) on M . For every function K > 0 on M , the following inequality holds: 2Re(iα ∧ β) ≥ − α ∧ α− iKβ ∧ β. (3)A quasiplurisubharmonic function on a complex analytic manifold M is an upper semi- continuous function h : M → [−∞,+∞[ which is locally the sum of a smooth function and a plurisubharmonic function. Proof. On one hand, the (1,1)-form µ = i(α/K1/2 +K1/2β) ∧ (α/K1/2 +K1/2β) is semi-positive. On the other hand, there is an equality α ∧ α+ 2Re(iα ∧ β) + iKβ ∧ β. The lemma follows. Lemma 4.6. Let L be a line bundle on X. Suppose that s ∈ Γ(X,L) is a global section such that div s is a divisor with normal crossings. Let ‖ · ‖ be a smooth hermitian metric on L with ‖s‖2 ≤ e−e. For every integer N ≥ 0, the smooth function θN = (log log ‖s‖ −2)N : X \ div s −→ R≥1 is pre-log-log, with singularities along div s. On X \div s the following identities hold: ∂ θN = NθN−1 ∂ log ‖s‖−2 log ‖s‖−2 ddc(−θN ) = θ1 −N + 1 ∂ θN ∧ ∂̄ θN −NθN−1 c1(L) log ‖s‖−2 provided N ≥ 1. If N = 1, θ1 is P-singular along div s. Proof. The lemma is trivial for N = 0. We suppose N ≥ 1. First of all we remark that θN has log-log growth along div s. Next we compute ∂ θN and ddc(−θN ) on X \ div s. We find (4.3) ∂ θN = NθN−1 ∂ log ‖s‖−2 log ‖s‖−2 ddc(−θN ) = − ∂ ∂̄ θN = − N ∂ θN−1 ∧ ∂̄ log ‖s‖−2 log ‖s‖−2 NθN−1 ∂ log ‖s‖−2 ∧ ∂̄ log ‖s‖−2 (log ‖s‖−2)2 NθN−1 ∂ ∂̄ log ‖s‖−2 log ‖s‖−2 To simplify the last equality, rearrange terms, use (4.3) and the trivial fact θN+M = θNθM , and recall that on X \ div s we have c1(L) = dd c log ‖s‖−2. We (4.4) ddc(−θN ) = θ1 −N + 1 ∂ θN ∧ ∂̄ θN −NθN−1 c1(L) log ‖s‖−2 Observe that the quotient (θ1−N+1)/NθN is bounded, so it has log-log growth along div s. Also the function 1/ log ‖s‖−2 is bounded and c1(L) is smooth, so that c1(L)/ log ‖s‖ −2 has log-log growth along div s. Hence from (4.3) and (4.4) we see that it is enough to prove that ∂ θN has log-log growth along div s or, still, that ∂ log ‖s‖−2/ log ‖s‖−2 has log-log growth along div s. Let V be an open analytic chart adapted to div s such that L|V can be trivialized and s = z1 . . . zme, where e is a holomorphic frame of L|V . We can write ∂ log ‖s‖−2 log ‖s‖−2 ∂ log ‖e‖−2 log ‖s‖−2 log |zk| log ‖s‖−2 zk log |zk|−1 The differential form ∂ log ‖e‖−2 is smooth on V and (log ‖s‖−2)−1 is bounded on V , so that the first term is bounded on any small enough open V ′ ⊂⊂ V . As for the sum, we observe that log |zk|/ log ‖s‖ −2 is bounded on any small enough open V ′ ⊂⊂ V , because log ‖s‖−2 = log ‖e‖−2 + j=1 log |zj| −2 and log ‖e‖−2 is smooth. Hence ∂ log ‖s‖−2/ log ‖s‖−2 has log-log growth along div s. The proof is complete. Lemma 4.7. Under the hypothesis of §4.1 and with the notations therein, the functions ΘN are pre-log-log, with singularities along D. If N = 1, then Θ1 is P-singular. Proof. Write θ N = (log log ‖sk‖ N , ΘN = k=1 θ N and apply Lemma 4.6. 4.2.2 Local results Let L be a line bundle on X admitting a global section s ∈ Γ(X,L) whose associated divisor div s is smooth and irreducible. Let ‖·‖ be a smooth hermitian metric on L with ‖s‖2 ≤ e−e on X . Define, as before, the function θ1 = log log ‖s‖ −2 : X \ div s −→ R≥1. By Lemma 4.6 we can write (4.5) ddc(−θ1) = ∂ log ‖s‖−2 ∧ ∂̄ log ‖s‖−2 (log ‖s‖−2)2 c1(L) log ‖s‖−2 Let (V ; z1, . . . , zn) be an analytic chart adapted to div s with V ∩div s 6= ∅. We suppose that L|V can be trivialized and we denote u for a holomorphic frame such that s = z1u. In local coordinates equality (4.5) becomes dd c(−θ1) = α+ β + γ, where ∂ log |z1| −2 ∧ ∂̄ log |z1| (log ‖s‖−2)2 dz1 ∧ dz̄1 |z1|2(log |z1|−2)2 β :=− 2Re ∂ log |z1| −2 ∧ ∂̄ log ‖u‖−2 (log ‖s‖−2)2 =2aRe dz1 ∧ ∂̄ log ‖u‖ z1(log |z1|−2)2 ∂ log ‖u‖2 ∧ ∂̄ log ‖u‖2 (log ‖s‖−2)2 c1(L) log ‖s‖−2 and a = (log |z1|/ log ‖s‖) 2. Decompose ∂̄ log ‖u‖2 = j=1 qjdz̄j , where the functions qj are smooth on V . Then β can be expanded as a sum β = j=1 βj , βj := 2aRe dz1 ∧ dz̄j z1(log |z1|−2)2 Proposition 4.8. Let L be a line bundle on X admitting a global section s ∈ Γ(X,L) such that div s is smooth and irreducible. Let ‖·‖ be a smooth hermitian metric on L with ‖s‖2 ≤ e−e and define θ1 = log log ‖s‖ −2. Let ω be a positive (1,1)-form on X. For every analytic chart (V ; z1, . . . , zn) adapted to div s and any open V ′ ⊂⊂ V intersecting div s, there exists A > 0 such that on V ′ \ div s ddc(−θ1) +Aω ≥ dz1 ∧ dz̄1 |z1|2(log |z1|−2)2 Proof. Without loss of generality, suppose supV | log ‖u‖| and supV |qj | finite for all j (otherwise, replace V by a relatively compact open subset containing V ′). We divide the proof into three steps. Step 1. Observe that because log ‖s‖ = log |z1|+log ‖u‖ and log ‖u‖ is bounded on V , the function a uniformly tends to 1 as z1 tends to 0. Therefore there exists an open V ′′ ⊂ V ′ such that 1/2 ≤ a ≤ 2 on V ′′ \ div s. For later need, we take V ′′ so that V ′ \ V ′′ does not intersect div s. On V ′′ \div s the following inequality holds: (4.6) α ≥ dz1 ∧ dz̄1 |z1|2(log |z1|−2)2 Step 2. Define C = maxj supV |qj |. Shrinking V ′′ if necessary, we assume that |z1| ≤ 1/16n(C + 1) on V ′′. Since a ≤ 2 on V ′′, we have β1 = 2aRe dz1 ∧ dz̄1 |z1|2(log |z1|−2)2 ≥ −4C|z1| dz1 ∧ dz̄1 |z1|2(log |z1|−2)2 dz1 ∧ dz̄1 |z1|2(log |z1|−2)2 (4.7) on V ′′ \ div s. Fix a constant K ≥ 8nC. Applying Lemma 4.5, for every j > 1 we find (4.8) βj ≥ −2 dz1 ∧ dz̄1 |z1|2(log |z1|−2)2 − 2CK dzj ∧ dz̄j (log |z1|−2)2 also on V ′′ \ div s. From inequalities (4.7) and (4.8) we derive that (4.9) β ≥ − dz1 ∧ dz̄1 |z1|2(log |z1|−2)2 − 2CK dzj ∧ dz̄j (log |z1|−2)2 holds on V ′′ \ div s. Step 3. To conclude, add up (4.6) and (4.9) to find (4.10) ddc(−θ1) ≥ dz1 ∧ dz̄1 |z1|2(log |z1|−2)2 − 2CK dzj ∧ dz̄j (log |z1|−2)2 on V ′′ \ div s. The last two terms in (4.10) define a smooth differential form on V \div s, bounded on V ′′ \div s. Hence, there exists a constant A > 0 such that (4.11) ddc(−θ1) +Aω ≥ dz1 ∧ dz̄1 |z1|2(log |z1|−2)2 holds on V ′′\div s. Because θ1 is smooth away from div s and V ′ \ V ′′ is compact and disjoint from div s, after possibly increasing A inequality (4.11) holds on V ′ \ div s as well, as was to be shown. 4.2.3 Global results We keep the hypothesis and notations of §4.1. Proposition 4.9. Suppose that X is compact and let ω be a smooth positive (1,1)-form on X. Let L be a line bundle on X admitting a global section s ∈ Γ(X,L) such that div s is a divisor with normal crossings. Let ‖ · ‖ be a smooth hermitian metric on L such that ‖s‖2 ≤ e−e. Define θN = (log log ‖s‖ for any N ∈ Z≥0. Then there exists a constant A = A(N) > 0 such that ddc(−θN ) +Aω ≥ 0 on X \ div s. Proof. The case N = 0 is trivial. We treat the case N ≥ 1. By Lemma 4.6 the following identity holds: ddc(−θN ) = θ1 −N + 1 ∂ θN ∧ ∂̄ θN −NθN−1 c1(L) log ‖s‖−2 First of all, the function θN−1/ log ‖s‖ −2 is bounded and the differential form c1(L) is smooth on X . Since X is compact, there exists a constant A > 0 such (4.12) −NθN−1 c1(L) log ‖s‖−2 ω ≥ 0 on X \ div s Still by the compactness hypothesis and by the very definition of θ1, there exists an open neighborhood V of div s such that θ1|V ≥ N − 1. Moreover i ∂ θN ∧ ∂̄ θN ≥ 0, so that (4.13) θ1 −N + 1 ∂ θN ∧ ∂̄ θN ≥ 0 on V \ div s. Finally, since θM ≥ 1 is smoooth on X \div s for every integerM ≥ 0 and X \V is compact, after possibly increasing A we have (4.14) θ1 −N + 1 ∂ θN ∧ ∂̄ θN + ω ≥ 0 on X \ V. Equations (4.12), (4.13) and (4.14) together give the desired positivity property. Corollary 4.10. Suppose that X is compact and let ω be a smooth positive (1,1)-form on X. For every integer N ≥ 0 there exists A = A(N) > 0 such that ddc(−ΘN ) +Aω ≥ 0 holds on X \D. Proof. This follows from Proposition 4.9 by writing ΘN = k=1 θ N , with the notation θ N = (log log ‖sk‖ The last proposition of this subsection provides a first approach to Theorem 4.2. We may thus place under the hypothesis and notations therein. Proposition 4.11. Suppose that X is compact and let ω be a smooth positive (1,1)-form on X. For every finite covering {(Vα; z 1 , . . . , z n )}α of X by analytic charts adapted to D, together with relatively compact open subsets V ′α ⊂⊂ Vα still forming a covering, there exists a constant A > 0 such that ddc(−Θ1) +Aω ≥ dzαk ∧ dz̄ |zαk | 2(log |zαk | holds on V ′α \D for every α. Proof. This follows immediately from Proposition 4.8. 4.3 Proof of Theorem 4.2 We now complete the proof of Theorem 4.2. Since X is compact, we can choose a finite open covering Vα of X as in Proposition 4.11. It is enough to prove the existence of constants A,B,N ful- filling (4.1) on a single V ′α ⊂⊂ Vα. We write {zi}i for the coordinates on Vα, instead of {zαi }i. Following Notation 2.7, we develop dζj ∧ dζ̄j + 2Re(hjk dζj ∧ dζ̄k), where the functions hjk, j ≤ k, have log-log growth along D ∩ Vα. There exist a constant C > 0 and an integer N ≥ 0 such that |hjk|V ′α\D| ≤ CΘN . Therefore, by Lemma 4.5, on V ′α \D there is a lower bound η ≥ −CΘN dζj ∧ dζ̄j − CΘN dζj ∧ dζ̄j + dζk ∧ dζ̄k From this inequality and Proposition 4.11 we see that there exist B ≥ 1 and a smooth differential form σ on Vα such that η +BΘN (dd c(−Θ1) +Aω) ≥ ΘNσ holds on V ′α \D. After possibly increasing A, we also have σ +Aω ≥ 0 on V Since B ≥ 1, we finally find η +BΘN(dd c(−Θ1) + 2Aω) ≥ 0 on V ′α \D, as was to be shown. Observe that if η has Poincaré growth, then we can choose N = 0. 5 Bounding height integrals 5.1 Geometric assumptions and statement of the theorem Let X be a complex analytic manifold and D ⊂ X a divisor with normal cross- ings. Let L be a line bundle on X and ‖ · ‖ a pre-log-log hermitian metric on L, with singularities along D (see Definition 3.1). If ‖ · ‖0 is any smooth hermitian metric on L, then we can write ‖ · ‖ = e−f/2‖ · ‖0, where f : X \D → R is a pre-log-log function. If ‖ · ‖ is good along D, then f is P-singular along D. As usual, abbreviate L = (L, ‖ · ‖) and L0 = (L, ‖ · ‖0). Suppose now that Y ⊂ X is a compact complex analytic submanifold of pure dimension d. We assume the following conditions are fulfilled: i. the submanifold Y meets D in a divisor with normal crossings E in Y ; ii. the restriction ω := c1(L0)|Y is semi-positive and degL Y := ωd > 0; iii. there exists a global section s ∈ Γ(Y, L) such that ‖s‖20 ≤ e −e on Y and div s is a divisor with normal crossings containing E. In particular div s is reduced, so that we may indistinctly treat div s as a reduced Weil divisor or a reduced scheme. For every integer N ≥ 0, we define the pre-log-log function, with singularities along div s, ℓN = (log log ‖s‖ N : Y \ div s −→ R≥1; iv. there exist pre-log-log functions, with singularities along E, Θ1,ΘN , ϕ, ψ : Y \ E −→ R≥0 (N ∈ Z≥0 depending on f) with f = ϕ− ψ, and bounds ϕ ≤ CℓM , ψ ≤ CℓM ,Θ1 ≤ Cℓ1,ΘN ≤ CℓN , for some constant C ≥ 0 and integer M ≥ 0. Moreover, if f is P-singular, we suppose that M = 1 and N = 0; v. there exists A > 0 such that τ := ddc(−Θ1) + Aω ≥ 0, and for every integer Q ≥ 0 there exists AQ > 0 such that τQ := dd c(−ℓQ) + AQω ≥ 0. For Q = 0, we can choose A0 = 1, so that τ0 = ω; vi. there exists B > 0 such that ddc(−ϕ) +BΘNτ ≥ 0, ddc(−ψ) +BΘNτ ≥ 0 hold on Y \E (and so on Y \div s). Observe that by the bounds in iv, we then have ωϕ := dd c(−ϕ) +BCℓNτ ≥ 0, ωψ := dd c(−ψ) +BCℓNτ ≥ 0 on Y \ div s. The aim of this section is to find bounds for the height integrals Jp := f c1(L) p c1(L0) d−p, 0 ≤ p ≤ d. We observe that f c1(L) p c1(L0) is a pre-log-log differential form on Y with singularities along E, hence locally integrable (see Proposition 2.16). In partic- ular, since E ⊆ div s and div s is Lebesegue negligible, the integrals Jp can be computed on Y \ div s. Theorem 5.1. There exist constants α, β > 0, R ∈ Z≥0, depending only on A, {AN}N , B, C, M and N , such that, for any p ∈ {0, . . . , d}, |Jp| ≤ α degL Y + β · (degL Y ) · log log ‖s‖−20 c1(L0) degL Y If ‖ · ‖ is good along E, then we can take R = 1, so that |Jp| ≤ α degL Y + β · (degL Y ) · log log ‖s‖−20 c1(L0) degL Y The theorem will be reduced to the bounds claimed by the following two propositions. Proposition 5.2. Let σ be a closed, real and semi-positive pre-log-log (t,t)-form on Y , with singularities along div s. Let a, b be integers such that a+ b+ t = d. If a, b ≥ 0, define the integral I(M,a, b, σ) = Otherwise set I(M,a, b, σ) = 0. Then, if a > 0, the following bound holds I(M,a, b, σ) ≤ BCI(M +N, a− 1, b, στ) + CI(M,a− 1, b, στM ) + bBC2I(2M,a− 1, b− 1, σττN ) + (a− 1)BC2I(2M,a− 2, b, σττN ). If f is P-singular (in which case N = 0 and M = 1), then I(1, a, b, σ) ≤ ABCI(1, a− 1, b, στ0) + (B + 1)CI(1, a− 1, b, στ1). Similar bounds are true if b > 0, exchanging the role of a and b. Proposition 5.3. Let σ be a (d, d)-form which is a product of (1,1)-forms of the kind τ or τQ, Q ≥ 0. Let W be the set of integers Q ≥ 0 such that τQ appears in σ. Fix an integer K ≥ 0. Then there exist constants α, β > 0 and an integer R ≥ 0, depending only on A, {AN}N , C, K and W , such that ℓKσ ≤ α degL Y + β(degL Y ) log log ‖s‖−20 c1(L0) degL Y If K = 1 and σ is a product of differential forms of the kind τ0, τ1 (i.e. W ⊆ {0, 1}), then we can take R = 1. Assuming for the moment the propositions, we prove Theorem 5.1. Proof of Theorem 5.1. We first observe that c1(L) = dd cf + c1(L0), so that (5.1) Jp = f(ddcf)jωd−j. Next, on Y \ div s we write f = ϕ− ψ and ddcf = ωψ − ωϕ. We get (−1)j−k ϕωkψω d−j − ψωkψω The coefficients can be bounded in terms of dimX (hence independently of Y and the hermitian line bundles). Therefore we are reduced to bound integrals of the kind ∫ ϕωaϕω ψωaϕω for integers a, b, c ≥ 0 such that a + b + c = d. Since the differential forms c are semi-positive and 0 ≤ ϕ, ψ ≤ CℓM , we have to find upper bounds for the integrals I(M,a, b, ωc) = Successively applying Proposition 5.2, we reduce our problem to find bounds for integrals ℓKσ, where σ is a product of (1,1)-forms of type τ or τQ, for some integers K, Q ≥ 0. If f is P-singular, then K = 1 and σ is a product of forms of type τ0 and τ1. We conclude by Proposition 5.3. 5.2 Proofs of Proposition 5.2 and Proposition 5.3 We proceed to prove Proposition 5.2 and Proposition 5.3. The proofs make an extensive use of Stokes’ theorem for pre-log-log differential forms. We refer to Proposition 2.16 for the statement and references. To simplify the exposition, it will be worth having at our disposal the computations summarized in the next lemma. Lemma 5.4. Let a, b ≥ 0 be integers. On Y \ E the following equalities hold: ∂(ωaϕω ψ) = aBC(∂ ℓN)ω ψτ + bBC(∂ ℓN )ω ψ τ ; ∂ ∂̄(ωaϕω ψ) =aBC(∂ ∂̄ ℓN)ω −aBC(∂̄ ℓN ) ∂(ω +bBC(∂ ∂̄ ℓN )ω −bBC(∂̄ ℓN ) ∂(ω ψ )τ. Proof. It is enough to apply the definition of ωϕ, ωψ, Leibniz’ rule and the fact that ddc(−ϕ), ddc(−ψ) and τ are ∂ and ∂̄-closed. Proof of Proposition 5.2. Write I = I(M,a, b, σ). We suppose that a > 0 and b ≥ 0. We decompose ωaϕ = dd c(−ϕ)ωa−1ϕ +BCℓNτω Accordingly, the integral I decomposes as I = I1 +BCI2, with c(−ϕ)ωa−1ϕ ω ℓM+Nω Bounding I1. To get bounds on I1 we apply Stokes’ theorem for pre-log-log forms. For this, we first recall that σ is closed of degree (t, t) and ωϕ, ωψ are of degree (1, 1). Then, by Leibniz’ rule, we compute ∂̄(−ϕ)ωa−1ϕ ω =ℓMdd c(−ϕ)ωa−1ϕ ω +(∂ ℓM ) ∂̄(−ϕ)ωa−1ϕ ω ∂̄(−ϕ) ∂(ωa−1ϕ ω where we used that ddc = i ∂ ∂̄ /2π. By Stokes’ theorem, we find I1 = I1,1+I1,2, where I1,1 = − (∂ ℓM ) ∂̄(−ϕ)ω I1,2 = ℓM ∂̄(−ϕ) ∂(ω Bounding I1,1. Again we apply Stokes’ theorem. By Lebniz’ rule we have (−ϕ)(∂ ℓM )ω (∂ ℓM ) ∂̄(−ϕ)ω (−ϕ)(∂ ∂̄ ℓM )ω (−ϕ)(∂ ℓM ) ∂̄(ω Therefore, by Stokes’ theorem, we get I1,1 = I1,1,1 + I1,1,2, where I1,1,1 = ϕddc(−ℓM )ω I1,1,2 = (−ϕ)(∂ ℓM ) ∂̄(ω Bounding I1,1,1. Recall that, by assumption, there exists a constant AM > 0 such that τM = dd c(−ℓM )+AMω ≥ 0. Then, since ω ψωσ is a semi-positive form and 0 ≤ ϕ ≤ CℓM , we have a bound I1,1,1 ≤ ϕ(ddc(−ℓM ) +AMω)ω =CI(M,a− 1, b, στM ). (5.2) Bounding I1,1,2. To bound the integral I1,1,2 we first appeal to Lemma 5.4 to develop ∂̄(ωa−1ϕ ω ψ) = (a− 1)BC(∂̄ ℓN )ω ψτ + bBC(∂̄ ℓN)ω Then we write I1,1,2 = (a− 1)BCI 1,1,2 + bBCI 1,1,2, where 1,1,2 = (−ϕ)(∂ ℓM )(∂̄ ℓN )ω 1,1,2 = (−ϕ)(∂ ℓM )(∂̄ ℓN )ω ψ στ. In these integrals we observe that (5.3) i ∂ ℓM ∧ ∂̄ ℓN = NMℓN+M−2i ∂ ℓ1 ∧ ∂̄ ℓ1, which is a semi-positive differential form. Since ϕ ≥ 0 and ωϕ, ωψ, σ and τ are also semi-positive, we find I 1,1,2 ≤ 0 and I 1,1,2 ≤ 0. This shows (5.4) I1,1,2 = (a− 1)BCI 1,1,2 + bBCI 1,1,2 ≤ 0. From (5.2) and (5.4) we conclude with the bound (5.5) I1,1 = I1,1,1 + I1,1,2 ≤ CI(M,a− 1, b, στM ). Bounding I1,2. As before we proceed by successive applications of Stokes’ the- orem. First of all we compute ℓM (−ϕ) ∂(ω ℓM ∂̄(−ϕ) ∂(ω (−ϕ)(∂̄ ℓM ) ∂(ω (−ϕ)ℓM ∂̄ ∂(ω By Stokes’ theorem we find I1,2 = I1,2,1 + I1,2,2, with I1,2,1 = ϕ(∂̄ ℓM ) ∂(ω I1,2,2 = (−ϕ)ℓM ∂ ∂̄(ω Bounding I1,2,1. By Lemma 5.4 we get the expansion ∂(ωa−1ϕ ω ψ) = (a− 1)BC(∂ ℓN )ω ψτ + bBC(∂ ℓN)ω Accordingly, we decompose I1,2,1 = (a− 1)BCI 1,2,1 + bBCI 1,2,1, where 1,2,1 = − ϕ(∂ ℓN )(∂̄ ℓM )ω 1,2,1 = − ϕ(∂ ℓN )(∂̄ ℓM )ω ψ στ. By (5.3) the differential form i ∂ ℓN ∧ ∂̄ ℓM is semi-positive. Since ϕ ≥ 0 and ωϕ, ωψ, σ and τ are also semi-positive, we have I 1,2,1, I 1,2,1 ≤ 0. This proves (5.6) I1,2,1 = (a− 1)BCI 1,2,1 + bBCI 1,2,1 ≤ 0. Bounding I1,2,2. To bound the integral I1,2,2 we first recall from Lemma 5.4 ∂ ∂̄(ωa−1ϕ ω ψ) = (a− 1)BC(∂ ∂̄ ℓN)ω − (a− 1)BC(∂̄ ℓN) ∂(ω + bBC(∂ ∂̄ ℓN)ω − bBC(∂̄ ℓN ) ∂(ω ψ )τ. Corresponding to this expansion, we write I1,2,2 = (a − 1)BCI 1,2,2 + (a − 1)BCI 1,2,2 + bBCI 1,2,2 + bBCI 1,2,2, with the obvious notations for the inte- grals I 1,2,2 (see below). Bounding I 1,2,2. We have 1,2,2 = ϕℓMdd c(−ℓN)ω Recall that for some constant AN > 0 the differential form τN = dd c(−ℓN) + ANω is semi-positive. Moreover 0 ≤ ϕ ≤ CℓM . Thus we find the bound 1,2,2 ≤C ψσττN =CI(2M,a− 2, b, σττN ). (5.7) Bounding I 1,2,2. We write 1,2,2 = ϕℓM (∂̄ ℓN) ∂(ω ψ)στ. By (5.3) and Lemma 5.4, i ∂(ωa−1ϕ ω ψ) ∧ ∂̄ ℓN is semi-positive, so that (5.8) I 1,2,2 ≤ 0. Bounding I 1,2,2. We have 1,2,2 = ϕℓMdd c(−ℓN)ω Reasoning as for I 1,2,2, we arrive to 1,2,2 ≤C ψ σττN =CI(2M,a− 1, b− 1, σττN ). (5.9) Bounding I 1,2,2. We finally bound the integral 1,2,2 = ϕℓM (∂̄ ℓN ) ∂(ω ψ )στ. Again by (5.3) and Lemma 5.4, the differential form i ∂(ωa−1ϕ ω ψ ) ∧ (∂̄ ℓN ) is semi-positive, so that (5.10) I 1,2,2 ≤ 0. We now conclude with a bound for I1,2, since the inequalities (5.6)–(5.10) yield I1,2 ≤ I1,2,2 ≤(a− 1)BC 2I(2M,a− 2, b, σττN ) +bBC2I(2M,a− 1, b− 1, σττN ). (5.11) As for I1, the bounds (5.5) and (5.11) lead to I1 = I1,1 + I1,2 ≤CI(M,a− 1, b, στM ) +bBC2I(2M,a− 1, b− 1, σττN ) +(a− 1)BC2I(2M,a− 2, b, σττN ). (5.12) Bounding I2. We have (5.13) I2 = ℓM+Nω ψστ = I(M +N, a− 1, b, στ) To conclude we put (5.12) and (5.13) together and we get I = I1 +BCI2 ≤BCI(M +N, a− 1, b, στ) +CI(M,a− 1, b, στM ) +bBC2I(2M,a− 1, b− 1, σττN ) +(a− 1)BC2I(2M,a− 2, b, σττN ), as was to be shown. Suppose now that f is P-singular (so that N = 0 andM = 1). We can write ωϕ = dd c(−ϕ̃) +ABCω, ωψ = dd c(−ψ̃) +ABCω, where ϕ̃ = ϕ + BCΘ1 and ψ̃ = ψ + BCΘ1. The same method followed above allows to establish the bound I(1, a, b, σ) ≤ ABCI(1, a− 1, b, στ0) + (B + 1)CI(1, a− 1, b, στ1). The details are left to the reader. Proof of Proposition 5.3. Let σ be a (d, d)-form which is a product of (1,1)- forms of type τ or τQ, Q ≥ 0. We write σ = τ sσ1, for some s ≥ 0 and σ1 a product of forms of type τQ. Define (5.14) J(K, s, σ1) = First of all we show how to reduce s to 0. The argument is by induction. If s > 0, recalling that τ = ddc(−Θ1) +Aω we write (5.15) τs = ddc(−Θ1)τ s−1 +Aωτs−1. Since τ0 = ω, we get from the definition of J in (5.14) and from (5.15) (5.16) J(K, s, σ1) = AJ(K, s− 1, σ1τ0) + c(−Θ1)τ s−1σ1. We next bound the integral on the right hand side of (5.16). Since τ and σ1 are ∂ and ∂̄-closed, applying Stokes’ theorem for pre-log-log forms we get (5.17) c(−Θ1)τ s−1σ1 = c(−ℓK)τ s−1σ1. Now ddc(−ℓK) = τK −AKτ0, the forms τ0, τK are semi-positive and 0 < Θ1 ≤ Cℓ1, so that from (5.17) and the definition of J we derive (5.18) c(−Θ1)τ s−1σ1 ≤ CJ(1, s− 1, σ1τK). Observe that because ℓK = ℓ 1 and ℓ1 ≥ 1, the inequality ℓ1 ≤ ℓK holds. Therefore (5.19) J(1, s− 1, σ1τK) ≤ J(K, s− 1, σ1τK). From (5.16)–(5.19) we arrive to J(K, s, σ1) ≤ AJ(K, s− 1, σ1τ0) + CJ(K, s− 1, σ1τK). Hence we may suppose that s = 0, so that σ is a product τQ1 . . . τQd . We have to deal with L(K,Q1, . . . , Qd) = ℓKτQ1 . . . τQd . Again by an inductive argument we show how to reduce all the integers Qi to 0. Suppose that Q1 > 0. Then we write τQ1 = dd c(−ℓQ1) +AQ1τ0, so that (5.20) L(K,Q1, . . . , Qd) = AQ1L(K, 0, Q2, . . . , Qd) + c(−ℓQ1)σ1 where σ1 = τQ2 . . . τQd . We study the integral on the right hand side of (5.20). Because σ1 is ∂ and ∂̄-closed, by Stokes’ theorem for pre-log-log forms we find L2 := c(−ℓQ1)σ1 = (∂ ℓK)(∂̄ ℓQ1)σ1 KQ1ℓK+Q1−2(∂ ℓ1)(∂̄ ℓ1)σ1. (5.21) By the very definition of ℓ1, we have (5.22) ∂ ℓ1 ∧ ∂̄ ℓ1 = ∂ log ‖s‖−20 ∧ ∂̄ log ‖s‖ (log ‖s‖−20 ) On the other hand, since ‖s‖20 ≤ e −e, there exists a constant D depending only on K +Q1 − 2 such that (5.23) ℓK+Q1−2 (log ‖s‖−20 ) (log log ‖s‖−20 ) K+Q1−2 (log ‖s‖−20 ) (log ‖s‖−20 ) Moreover observe that (5.24) (log ‖s‖−20 ) ∂ log ‖s‖−20 = −2 ∂(log ‖s‖ −1/2. Because i ∂ log ‖s‖−20 ∧ ∂̄ log ‖s‖ 0 and σ1 are semi-positive, combining (5.21)– (5.24) we get (5.25) L2 ≤ −2 KQ1D(∂(log ‖s‖ −1/2)(∂̄ log ‖s‖−20 )σ1. By Lemma 5.5 below, we can apply Stokes’ theorem to the right hand side of (5.25) and obtain (5.26) L2 ≤ 2 KQ1D(log ‖s‖ −1/2ωσ1. By the hypothesis on ‖ · ‖0, (log ‖s‖ −1/2 ≤ 1. Using the positivity of ωσ1, from (5.26) we derive (5.27) L2 ≤ 2KQ1D Now recall that ω is ∂ and ∂̄-closed, and τQ = dd c(−ℓQ) +AQω, so that ωσ1 = AQ2 . . . AQdω d + ddcσ2 for some pre-log-log form σ2. Applied to (5.27) this provides (5.28) L2 ≤ 2Q1AQ2 . . . AQdKD ωd = 2Q1AQ2 · . . . · AQdKD degL Y. The identity (5.20) together with (5.28) imply the inequality L(K,Q1, . . . , Qd) ≤ AQ1L(K, 0, Q2, . . . , Qd) + 2Q1AQ2 · . . . · AQdKD degL Y. Successively repeating this argument, we reduce Q2, . . . , Qd to 0. Hence it remains to treat the integrals M(K) = We want to apply Jensen’s inequality. First of all, rewrite M(K) = (degL Y ) degL Y so that ωd/ degL Y defines a probability measure on Y . Secondly, the function x 7→ (log x)K is concave on ]eK−1,+∞[, because (log x)K = (log x)K−2(K − 1− log x). Since ‖s‖−20 ≥ e e, in particular eK+1 log ‖s‖−20 > e K−1. We use that logK is an increasing function and apply Jensen’s inequality: M(K) ≤(degL Y ) logK(eK+1 log ‖s‖−20 ) degL Y ≤(degL Y ) log log ‖s‖−20 degL Y By the trivial inequality x+ y ≤ 2xy for real x, y ≥ 1, we finally arrive to M(K) ≤ (degL Y )(2K + 2) K logK log ‖s‖−20 degL Y This concludes the proof of the proposition, except for the fact that we can take R = 1 when K = 1 and σ is a product of forms τ0 and τ1. This last case is similarly treated and left to the reader. Lemma 5.5. Let µ be a closed pre-log-log (d-1,d-1)-form on Y , with singulari- ties along div s. Then we have (log ‖s‖−20 ) ∧ ∂̄ log ‖s‖−20 µ = (log ‖s‖−20 ) ∧ ωµ. Proof. The proof of the lemma follows the ideas of Lemma 7.36 in [2]. First of all, as for log-log growth differential forms, the form (log ‖s‖−20 ) ∧ ∂̄ log ‖s‖−20 µ = − (log ‖s‖−20 ) 1/2 ∂ ℓ1 ∧ ∂̄ ℓ1µ is locally integrable on Y . Indeed, after localizing to an analytic chart adapted to div s and changing to polar coordinates, we are reduced to point out that, for every 0 < δ < 1/2, we have an estimate (5.29) ∫ ε/e (log log t−1)N (log t−1)1/2 t(log t)2 ∫ ε/e t(log t−1)3/2−δ < +∞. Define I = − (log ‖s‖−20 ) ∧ ∂̄ log ‖s‖−20 µ. We construct a finite open covering {(V ′α; {z i }i)}α of div s, by adapted analytic open charts. Suppose that via the coordinates {zαi }α, V α is identified with (r = r(α)), so that V ′α \ D corresponds to ∆ further assume that in these coordinates we can write s = zα1 · . . . · z r uα, where uα is a holomorphic unit. After possibly adding a finite number of adapted analytic charts to {(V ′α; {z i }i)}α, the open subsets Vα ⊂⊂ V α identified with via the coordinates {zαi }i still cover div s. Write Ω = ∪αVα. Take a finite open covering {Vβ}β of X \ Ω, so that Vβ ∩ (div s) = ∅ for all β. Let {χα}α ∪ {χβ}β be a partition of unity subordinate to {Vα}α ∪ {Vβ}β, with χγ vanishing outside Vγ for all γ = α, β. We can expand where Iγ := − (log ‖s‖−20 ) ∧ ∂̄ log ‖s‖−20 χγµ. We first treat the integrals Iβ . Observe that for any C ∞ differential form ν on Y \ div s, the differential form χβν is C ∞ on Y , because χβ vanishes on Y \ Vβ and Vβ ∩ (div s) = ∅. Moreover the equality d(χβν) = dχβ ∧ ν + χβ ∧ dν holds on Y . For ν = i ∂ ∂̄ log ‖s‖−20 /2π we find χβν = χβω on Y . These observations, together with dµ = 0, yield, by Stokes’ theorem, (log ‖s‖−20 ) (log ‖s‖−20 ) ∂̄ log ‖s‖−20 (dχβ)µ, (5.30) On the other hand, for every α define Bαε (div s) to be the ε-neighborhood of div s given by Bαε (div s) = Bαε (Tk), where Tk is the divisor z k = 0 in Vα and Bαε (Tk) = ∆ ×∆ε ×∆ ×∆s1/2e ⊂ Vα. Then we write Iα = limε→0 Iα,ε, where Iα,ε = I α,ε + I α,ε + I α,ε and I(1)α,ε :=− Y \Bαε (div s) (log ‖s‖−20 ) ∂̄ log ‖s‖−20 χαµ I(2)α,ε := Y \Bαε (div s) (log ‖s‖−20 ) I(3)α,ε :=− Y \Bαε (div s) (log ‖s‖−20 ) ∂̄ log ‖s‖−20 (dχα)µ. The differential form in I α,ε is integrable on Y , since it has log-log growth along div s (Proposition 2.16). The differential form in I α,ε is integrable on Y , too. Indeed, after localizing to an analytic chart adapted to div s and changing to polar coordinates, we are reduced to prove the convergence of the integrals (5.31) ∫ 1/e (log log t−1)N (log t−1)1/2 (5.32)∫ 1/e (log log t−1)N (log t−1)1/2 t log t−1 ∫ 1/e (log log t−1)N t(log t−1)3/2 From the boundedness of (log log t−1)N/(log t−1)1/2 the convergence of (5.31) follows. The second one (5.32) has already been treated (5.29). Therefore limε→0 I α,ε = I α and limε→0 I α,ε = I α , where I(2)α := (log ‖s‖−20 ) ωχαµ, I(3)α :=− (log ‖s‖−20 ) ∂̄ log ‖s‖−20 (dχα)µ. As for the integral I α,ε, after applying Stokes’ theorem we find I(1)α,ε = ∂ Bαε (div s) (log ‖s‖−20 ) ∂̄ log ‖s‖−20 χαµ. Taking into account that χα vanishes on ∂ Vα, we easily see that (5.33) |I(1)α,ε| ≤ ∂∗ Bαε (Tk) (log ‖s‖−20 ) ∣∣∂̄ log ‖s‖−20 χαµ with the notation ∂∗Bαε (Tk) = ∆ × ∂∆ε ×∆ Observe that ∂∗Bαε (Tk) is fibered in circles over Tk, via the projection pαk,ε : ∂ ∗Bε(Tk) −→ Tk (zα1 , . . . , z d ) 7−→ (z 1 , . . . , z k−1, 0, z k+1 . . . , z we will mean integral along the fibers of pαk,ε. We claim that, for every k = 1, . . . , r, (5.34) lim (log ‖s‖−20 ) ∣∣∂̄ log ‖s‖−20 χαµ ∣∣ = 0. Indeed, write s = zα1 . . . z r uα on V α, where uα is a holomorphic unit. Since χαµ is a (d− 1, d− 1) pre-log-log form, the differential form to integrate under has the shape f(zα1 , . . . , z (log log |zαk | (log ‖s‖−20 ) ∣∣∣∣∣∣ dz̄αk j 6=k dzαj ∧ dz̄ ∣∣∣∣∣∣ where f behaves as follows: 0 ≤ f(zα1 , . . . , z d ) ≺ j 6=k (log log |zαj | |zαj | 2(log |zαj | We point out that |dz̄αk /z̄ k | is bounded along the fibers of p k,ε, that the form j 6=k dz j ∧ dz̄ ∣∣∣ is integrable and (log log |zαk |−1)M/(log ‖s‖ 1/2 vanishes along Tk. This is enough to prove the claim (5.34). Therefore, from (5.33) limε→0 I α,ε = 0 and consequently Iα =I α + I (log ‖s‖−20 ) (log ‖s‖−20 ) ∂̄ log ‖s‖−20 (dχα)µ. (5.35) Finally, since the open covering {Vα}α ∪ {Vβ}β is finite, from (5.30) and (5.35) we derive (log ‖s‖−20 ) χγ)ωµ (log ‖s‖−20 ) ∂̄ log ‖s‖−20 d( χγ)µ. (5.36) The assertion of the lemma follows from (5.36) once we note that γ χγ = 1 and d( γ χγ)= 0. 5.3 Application to the projective case We suppose that X is a nonsingular complex projective variety (non necessarily connected) and D ⊆ X a divisor with simple normal crossings. Consider the following elements: i. as in section 4, for every integer N ≥ 0 introduce a function ΘN (see Notation 4.1). By Lemma 4.7, the functions ΘN are pre-log-log, with singularities along D; ii. an ample line bundle L on X , admitting a global section s ∈ Γ(X,L) such that div s is a divisor with simple normal crossings containing D. In particular div s is reduced and may be seen as a reduced Weil divisor or a reduced scheme; iii. a pre-log-log metric ‖ · ‖ on L, with singularities along D; iv. a smooth hermitian metric ‖·‖0 on L with ω := c1(L0) > 0 and ‖s‖ 0 ≤ e v. as in §5.1 we introduce ℓQ = (log log ‖s‖ Q, Q ∈ Z≥0. By Lemma 4.6, the function ℓQ is pre-log-log, with singularities along div s. Moreover, Proposition 4.9 asserts the existence of a constant AQ > 0 such that τQ := dd c(−ℓQ) +AQω ≥ 0. We can take A0 = 1. We write ‖ · ‖ = e−f/2‖ · ‖0, where f : X \D → R is a pre-log-log function, with singularities along D. If ‖ · ‖ is good, then f is P-singular. Since X is compact, associated to ω = c1(L0) there is a decomposition f = ϕ − ψ as in Theorem 4.3. Moreover, because D ⊆ div s, there exist a constant C ≥ 0 and an integer M ≥ 0 such that ϕ ≤ CℓM , ψ ≤ CℓM ,Θ1 ≤ Cℓ1,ΘN ≤ CℓN hold on X \div s. If ‖ · ‖ is good, then we can take M = 1. Finally, by Theorem 4.3 and Proposition 4.9, there exist constants A,B > 0 and an integer N ≥ 0 such that τ := ddc(−Θ1) +Aω ≥ 0 and ωϕ :=dd c(−ϕ) +BΘN(dd c(−Θ1) +Aω) ≥ 0, ωψ :=dd c(−ψ) +BΘN (dd c(−Θ1) +Aω) ≥ 0. If ‖ · ‖ is good along D, then we can take N = 0. Therefore the assumptions of §5.1 are fulfilled for X . Let now π : X ′ → X be a morphism of complex analytic manifolds such that the inverse image schemes π−1(D) ⊆ π−1(div s) are divisors with normal crossings. Let Y ⊆ X ′ be a compact complex analytic submanifold of pure dimension d. Suppose that Y meets π−1(div s) in a divisor with normal crossings in Y . Then Y ∩D is a divisor with normal crossings in Y , too. We pull-back by π all the objects introduced above (π−1(D), π∗ΘN , π ∗L, π∗s, π∗‖ · ‖, etc.) and then we restrict them to Y . We obtain corresponding objects on Y (π−1(D) ∩ Y , (π∗ΘN)|Y , (π ∗L)|Y , (π ∗s)|Y , (π ∗‖ · ‖)|Y , etc.). Provided that degπ∗L Y > 0, the requirements of §5.1 are fulfilled on Y . It is important to point out that the involved constants A, {AQ}Q, B, C, M , N don’t depend on the data X ′, π, Y . For every integer 0 ≤ p ≤ d define J∗p = (π∗f) c1(π ∗L)p c1(π As a consequence of Theorem 5.1 we get the following corollary. Corollary 5.6. Let π : X ′ → X be a morphism of complex analytic manifolds such that π−1(D) and π−1(div s) are divisors with normal crossings. Let Y ⊆ X ′ be a compact complex analytic submanifold of pure dimension d, intersecting div s in a divisor with normal crossings in Y . Suppose that degπ∗L Y > 0. There exist constants α, β > 0 and an integer R ≥ 0, depending only on A, {AQ}Q, B, C, M and N such that, for all p ∈ {0, . . . , d}, |J∗p | ≤ α degπ∗L Y + β · (degπ∗L Y ) · log log π∗‖s‖−20 degπ∗L Y If ‖ · ‖ is good along D, then we can take R = 1: |J∗p | ≤ α degπ∗L Y + β · (degπ∗L Y ) · log log π∗‖s‖−20 degπ∗L Y Let Z be a reduced closed subscheme of X of pure dimension d, intersecting div s properly. Denote by Z1, . . . , Zr the irreducible components of Z. For every i = 1, . . . , r, let πi : Xi → X be an imbedded resolution of singularities of Zi, such that π−1i (div s) is a divisor with normal crossings intersecting the strict transform Z̃i of Zi in a divisor with normal crossings in Z̃i (see [2], Theorem 7.27). Then Corollary 5.6 applies to πi, Zi, for every i = 1, . . . , r. If θ is a smooth differential form of degree 2d on X \div s, locally integrable on X , then we adopt the convention ∫ π∗i θ. This definition intrinsically depends on Z, and not on the choice of the resolu- tions πi. With this convention, define f c1(L) p c1(L0) In this situation Corollary 5.6 reads as follows. Corollary 5.7. Let Z be a reduced closed subscheme of X of pure dimension d, intersecting div s properly. There exist positive constants α, β, and an integer R ≥ 0, depending only on A, {AQ}Q, B, C, M and N such that, for all p ∈ {0, . . . , d}, |Jp| ≤ α degL Z + β · (degL Z) · log log ‖s‖−20 c1(L0) degL Z with R = 1 whenever ‖ · ‖ is good along D. Proof. Decompose Z into irreducible components: Z = Z1∪ . . .∪Zr. Following the convention above, define J (i)p = f c1(L) p c1(L0) so that (5.37) Jp = J (i)p . By Corollary 5.6, there exist constants α, β > 0 and an integer R ≥ 0, depending only on A, {AQ}Q, B, C, M and N such that, for all p ∈ {0, . . . , d}, (5.38) |J (i)p | ≤ α degL Zi + β · (degL Zi) · log log ‖s‖−20 c1(L0) degL Zi If ‖ · ‖ is good along D, then we can take R = 1. Now recall that the function logR is increasing and concave on ]eR−1,+∞[, so that degL Zi degL Z log ‖s‖−20 c1(L0) degL Zi ≤ logR degL Zi degL Z log ‖s‖−20 c1(L0) degL Zi = logR log ‖s‖−20 c1(L0) degL Z ≤(2R+ 2)R logR log ‖s‖−20 c1(L0) degL Z (5.39) For the last inequality we used that x + y ≤ 2xy for real x, y ≥ 1. The lemma follows combining (5.37)–(5.39). 6 Arakelovian heights In this section we turn to an arithmetic situation and deal with arakelovian heights on arithmetic varieties. We prove Theorem 1.3, which can be seen as an arithmetic counterpart of the bounds in §5. A remarkable and straightforward outcome is the finiteness property of arakelovian heights with respect to pre-log- log hermitian ample line bundles, as well as the existence of a universal lower bound (Corollary 1.4). 6.1 Heights attached to pre-log-log hermitian line bundles Let K be a number field and OK its ring of integers. Write S = SpecOK . Throughout this section we work with a fixed arithmetic variety π : X → S of relative dimension n. We recall this means that X is a flat and projective scheme over S , with regular generic fiber XK = X ×S SpecK of pure di- mension n. The set of complex points X (C) of X has a natural structure of complex analytic manifold, and it can be decomposed as X (C) = σ:K →֒C Xσ(C). Complex conjugation induces an antiholomorphic involution F∞ : X (C) −→ X (C). We fixD ⊂ XK a divisor, such that D(C) ⊂ X (C) has simple normal crossings. Write U = X (C) \D(C). Notation 6.1 ([2]). We define Z U (X ) to be the free abelian group generated by the irreducible reduced subschemes Z ⊆ X of codimension p, such that Z(C) intersects D(C) properly. We call Z U (X ) the group of cycles of codimension p, intersecting D(C) properly. A cycle z is said to be vertical if its components are supported on closed fibers X℘, ℘ ∈ S \ {(0)}. A cycle z is said to be horizontal if its irreducible components are flat over S . We denote by Z U (XK) the subgroup of Z U (X ) of horizontal cycles. Definition 6.2. A pre-log-log hermitian line bundle on X , with singularities along D, is a couple L = (L , ‖ · ‖) formed by i. a line bundle (invertible sheaf) L on X ; ii. a pre-log-log hermitian metric ‖·‖ on the line LC on X (C), with singular- ities along D(C), and invariant under the action of complex conjugation F∞: F ∞‖ · ‖ = ‖ · ‖. In [2], [3], Burgos, Kramer and Kühn attach a height morphism to a pre- log-log hermitian line bundle L , U (X ) −→ R, generalizing the height morphism for smooth hermitian line bundles introduced by Bost, Gillet and Soulé in [1]. We refer the reader to the cited bibliography for the precise definition and basic properties of h , both in the smooth and pre-log-log case. For our purposes, it will be enough to state the following propositions summarizing the main features of h Proposition 6.3. Let L be a pre-log-log hermitian line bundle on X , with singularities along D. The height h satisfies the following three properties: H1. if PK : SpecK → X is a K-valued point whose image does not belong to D, and P : S → X denotes its extension to S , then (P∗S ) = d̂eg(P L )(4) (4)The arithmetic degree ddeg of a hermitian line bundle M = (M, ‖ · ‖) over SpecOK is defined as follows: if s is a non zero global section of M, then ddeg(M) = log ♯ σ:K →֒C log ‖s‖σ . H2. if z is a vertical cycle supported on a closed fiber X℘, then (z) = log(N℘) deg where N℘ denote the norm of the ideal ℘; H3. let z ∈ Z n+1−p U (XK) be irreducible and reduced. Let s be a rational section of L ⊗N which does not identically vanish on z. If (div(s).z)(C) intersects D(C) properly, then (z) = h (div(s).z)− log(‖s‖L⊗N ) c1(L ) Proof. This follows from the definition of h and the extended arithmetic in- tersection theory in [2]. Remark 6.4. i. The convergence of the integral in H3 is implicit in the state- ment. ii. If LK is an ample line bundle, then an easy inductive argument shows that the properties H1, H2 and H3 actually characterize h Proposition 6.5. Let L be a line bundle on X , ‖ · ‖ a pre-log-log hermitian metric on L , with singularities along D, and ‖ · ‖0 a smooth hermitian metric on L . Write L = (L , ‖ · ‖), L 0 = (L , ‖ · ‖0) and ‖ · ‖ = e −f/2‖ · ‖0, where f : X (C) \D(C) → R is a pre-log-log function, with singularities along D(C). For any cycle z ∈ Z n+1−p U (X ) we have (z) = h (z) + f c1(L ) k c1(L 0) p−1−k. Proof. This is contained in [3], Theorem 4.1. Remark 6.6. Proposition 6.5 allows to recover Proposition 6.3 once it is known for smooth hermitian line bundles. In this case the properties H1, H2 and H3 are already established in [1]. Let F |K be a finite extension of fields and write T = SpecOF . Base changing by T → S , we get an arithmetic variety XT → T , together with a finite flat morphism g : XT → X of degree [F : K]. If D ⊆ XK is an effective divisor such that D(C) ⊆ X (C) has simple normal crossings, then DF ⊆ XF is an effective divisor and DF (C) ⊆ XT (C) has simple normal crossings as well. Let L be a pre-log-log hermitian line bundle on X , with singularities along D. The pull-back g∗L of L to XT is a pre-log-log hermitian line bundle, with singularities along DF . Since g is flat, for every cycle z on X there is a well defined pull-back cycle g∗(z). Observe that if z(C) intersects D(C) properly, then g∗(z)(C) intersects DF (C) properly. Namely, the correspondence z 7→ g∗(z) induces a morphism U (X ) −→ Z V (XT ) z 7−→ g∗(z), (6.1) where V = XT (C) \DF (C). This morphism maps Z U (XK) into Z V (XF ). Lemma 6.7. Let L be a pre-log-log hermitian line bundle on X , with singu- larities along D. Let F |K be a finite extension of fields and T = SpecOF . Write g : XT → X for the finite flat projection induced by T → S . Let w ∈ Z V (XF ) be an irreducible and reduced cycle and set z = g(w)red. Let δ be the degree of g |w. Then we have the equality (w) = δh Consequently, for the morphism g∗ of (6.1), we have hg∗L (g ∗(z)) = [F : K]h for every z ∈ Z U (X ) and every p. Proof. This follows for instance from the case of smooth metrics (see [1], §3.1.4 and Proposition 3.2.1) and Proposition 6.5, since g|w(C) : w(C) → z(C) is gener- ically smooth and finite of degree δ, so that f c1(L ) k c1(L 0) p−1−k f c1(L ) k c1(L 0) p−1−k. Notation 6.8. Let L be a pre-log-log hermitian line bundle, with singularities along D. Let z ∈ Z U (XK) with degLK z 6= 0. We define its normalized height to be (z) = [K : Q] deg Lemma 6.9. Let g : XT → X be as before. Let w ∈ Z V (XF ) be an irreducible and reduced cycle and set z = g(w)red. For the normalized height we have (w) = h̃ (z) and h̃ (g∗z) = h̃ Proof. For every w ∈ Z n+1−p V (XF ) irreducible and reduced and z = g(w)red, we have the equalities deg(g∗L )F w = [F : Q] L )p−1 [F : Q] c1(L ) [F : K] where δ is the degree of g|w. It follows that deg(g∗L )F (g ∗z) = deg z. Com- bined with Lemma 6.7, we get h̃g∗L (w) = h̃L (z) and h̃g∗L (g ∗z) = h̃ Remark 6.10. The normalization h̃ just introduced is not the standard one. However it appears naturally in the statement and proof of Theorem 1.3. 6.2 Proof of the main theorem We now proceed to prove Theorem 1.3. The argument mainly relies on the bounds established in §5 , and more concretely the situation studied in §5.3. However, in the reduction steps we will need the following Bertini’s type theo- rem. The proof is essentially well known, but we include it in the Appendix for lack of reference. Proposition 6.11. Let X be a nonsingular projective scheme over an alge- braically closed field k. Let D ⊆ X be a divisor with simple normal crossings. Let L be an ample line bundle on X. Then there exist an integer N > 0 and global sections s1, . . . , sr ∈ H 0(X,L⊗N ) such that supp(div s1),. . . , supp(div sr) are divisors with simple normal crossings and the following equality of schemes holds: D = (supp(div s1) ∩ . . . ∩ supp(div sr))red. Under the hypothesis of Theorem 1.3, since D(C) has simple normal cross- ings, DK has also simple normal crossings in XK . We will apply Proposition 6.11 through the following straightforward corollary. Corollary 6.12. There exist a finite extension K ′|K, a positive integer N and global sections s1, . . . , sr ∈ H 0(XK′ ,L K′ ) such that B1. supp(div sj)(C) is a divisor with simple normal crossings in X (C), for every j = 1, . . . , r; B2. DK′ = (supp(div s1) ∩ . . . ∩ supp(div sr))red. The next two lemmas provide the final reductions before the proof of Theo- rem 1.3. Lemma 6.13. It is enough to proof Theorem 1.3 in the following situations: i. after some finite extension K ′ | K; ii. L is very ample and z ∈ Z U (XK) is irreducible and reduced. Proof. The first claim i is clear. For the proof of ii, we first note that the statement of Theorem 1.3 for L ⊗N already implies the statement of the theorem for L , since h̃ ⊗N (z) = Nh̃ (z) and h̃ (z) = Nh̃ (z). Hence we assume LK is very ample. We then proceed in two steps. Step 1. We can suppose that L is very ample. Indeed, there exists some model (Y ,A ) of (XK ,LK) with A very ample. The metrics ‖ · ‖, ‖ · ‖0 on L induce metrics on A , and we write A and A 0 for the corresponding hermitian line bundles. For every effective cycle z ∈ Z U (XK), we write z̃ for the corresponding effective and horizontal cycle on Y . By Proposition 3.2.2 in [1], there exists a positive constant C, independent of z, such that (6.2) (z̃)− h ∣∣ ≤ C deg Since A C and L C are isometric, Proposition 6.5 and (6.2) together give (z̃)− h (z̃)− h ∣∣ ≤ C deg Step 2. We can suppose that z is irreducible and reduced. Indeed, suppose that we have shown the existence of constants α, β, γ > 0 and R ∈ Z≥0 such that, for every w ∈ Z U (XK) irreducible and reduced, we have h̃L 0(w) + γ ≥ 1 and (6.3) ∣∣∣h̃ (w)− h̃ ∣∣∣ ≤ α+ β logR (w) + γ After possibly increasing γ we can suppose that h̃ (w) + γ > eR−1, for every irreducible and reduced w ∈ Z U (XK). Let us now consider w = i∈I wi ∈ U (XK) \ {0} where the wi ∈ Z U (XK) are irreducible and reduced. Then (6.3) yields ∣∣∣h̃ (w) − h̃ ∣∣∣ ≤ ∣∣∣h̃ (wi)− h̃L 0(wi) ≤ α+ β (wi) + γ Since h̃ (wi) + γ > e R−1 for all i and logR is concave on ]eR−1,+∞[, we conclude (wi) + γ ≤ logR (wi) + γ) = logR (w) + γ This completes the proof. By Corollary 6.12 and Lemma 6.13, after possibly extending K and choosing a suitable model of (XK ,LK), we can suppose that L is very ample and there exist sections s1, . . . , sr ∈ H 0(XK ,LK) with the properties B1, B2 above (with K ′ = K). After possibly multiplying the sections sj by a sufficiently divisible integer, we can even suppose that s1, . . . , sr ∈ H 0(X ,L ). We denote Ej = supp(div sj)K . We fix these data until the end of the proof. Lemma 6.14. Let z ∈ Z U (XK) be irreducible and reduced. Let F be a finite extension of K over which all the irreducible components of zK are defined. Let T = SpecOF and g : XT → X be the finite flat projection induced by T → S . Write g∗(z) = i∈I zi, zi irreducible, reduced and flat over T . Then z(C) intersects D(C) properly if, and only if, every zi(C) intersects one of the Ej,F (C) properly. Proof. Straightforward. Now we can complete the proof of Theorem 1.3. Proof of Theorem 1.3. First of all, for every integer M ≥ 0 we construct a function ΘM , as in §4, for the complex analytic variety X (C) and the divisor with simple normal crossings D(C). Observe that we can suppose that the metric ‖ · ‖0 satisfies c1(L 0) > 0 and 0 ≤ e −e for every j = 1, . . . , r. Indeed, by [1], Proposition 3.2.2 (or also Proposition 6.5 for smooth metrics), a change of smooth metric causes only bounded variations of the normalized height. We introduce the pre-log-log functions Q = (log log ‖sj‖ Q : X (C) \ Ej(C) −→ R. For everyQ we fix a positive constantAQ > 0 such that dd Q )+AQ c1(L 0) ≥ 0, A0 = 1, for all j = 1, . . . , r (see Proposition 4.9). Write ‖ · ‖ = e−f/2‖ · ‖0, where f : X (C) \ D(C) → R is a pre-log-log function (resp. P-singular if ‖ · ‖ is good). Attached to c1(L 0) we perform a decomposition f = ϕ − ψ as in Theorem 4.3. Recall that ϕ, ψ are positive pre-log-log (resp. P-singular) functions along D(C), with ωϕ :=dd c(−ϕ) +BΘN(dd c(−Θ1) +A c1(L 0)) ≥ 0, ωψ :=dd c(−ψ) + BΘN(dd c(−Θ1) +A c1(L 0)) ≥ 0, for some A,B > 0 and N ∈ Z≥0. If ‖ · ‖ is good, then we can take N = 1. For every j = 1, . . . , r, D ⊆ Ej . By compactness of X (C) there exist constants C > 0, M ∈ Z≥0 such that ϕ ≤ Cℓ M , ψ ≤ Cℓ M ,Θ1 ≤ Cℓ 1 ,ΘN ≤ Cℓ for all j ∈ {1, . . . , r}. Let z ∈ Z n+1−p U (XK) be irreducible and reduced. Denote by F an extension of K such that all the irreducible components of zK are defined over F . Let T = SpecOF , g : XT → X be as before. Decompose g ∗(z) = i∈I zi, with the zi irreducible and flat over T . By Lemma 6.14, for every zi there exists j = j(i) ∈ {0, . . . , r} such that zi(C) intersects Ej,F (C) properly. By Corollary 5.7 and Proposition 6.5 we have (6.4) ∣∣∣h̃g∗L (zi)− h̃g∗L 0(zi) ∣∣∣ ≤ α+β logR zi(C) log g∗‖sj‖ [K : Q] deg(g∗L )F zi for α, β > 0, R ∈ Z≥0 depending only on A, {AQ}Q, B, C,M and N . Moreover, if ‖ · ‖ is good, then we can take R = 1. Applying the property H3 of heights (see Proposition 6.3), we rewrite (6.4) as (6.5)∣∣∣h̃g∗L (zi)− h̃g∗L 0(zi) ∣∣∣ ≤ α+ β logR g∗L 0 (zi)− 2h̃g∗L 0(div(g ∗sj).zi) To derive this inequality we point out that deg(g∗L )F (div(g ∗sj).zi) = deg(g∗L )F zi, so that hg∗L 0(div(g ∗sj).zi) [F : Q] deg(g∗L )F zi = h̃g∗L 0(div(g ∗sj).zi). By Theorem 1.2 there exists a positive constant κ > 0 such that, for every effec- tive cycle w 6= 0 on X , we have h̃ (w) > −κ. Combined with Lemma 6.9 this yields h̃g∗L 0(div(g ∗sj).zi) > −κ, because div(g ∗sj).zi is effective. Therefore, from (6.5) we deduce ∣∣∣h̃g∗L (zi)− h̃g∗L 0(zi) ∣∣∣ ≤α+ β logR 2h̃g∗L 0(zi) + 2κ+ 2e ≤α+ 2Rβ logR h̃g∗L 0(zi) + κ+ e (6.6) where we applied the trivial inequalities log 2 ≤ 1 and x + y ≤ 2xy for real x, y ≥ 1. Now h̃g∗L 0(zi)+κ+ e R+1 > eR−1 and logR is concave on ]eR−1,+∞[. From (6.6) we derive ∣∣∣h̃g∗L (g ∗z)−h̃g∗L 0(g deg(g∗L )F (zi) deg(g∗L )F (g ∣∣∣h̃g∗L (zi)− h̃g∗L 0(zi) ≤α+ 2Rβ logR deg(g∗L )F (zi) deg(g∗L )F (g h̃g∗L 0(zi) + κ+ e =α+ 2Rβ logR h̃g∗L 0(g ∗z) + κ+ eR+1 (6.7) By Lemma 6.9, h̃ (g∗z) = h̃ (z) and h̃ g∗L 0 (g∗z) = h̃ (z). Hence (6.7) is equivalent to (6.8) ∣∣∣h̃ (z)− h̃ ∣∣∣ ≤ α+ 2Rβ logR (z) + κ+ eR+1 The constants α, 2Rβ, γ := κ + eR+1 > 0, R ∈ Z≥0 (R = 1 if ‖ · ‖ is good) in (6.8) depend only on L and L 0, and not on z. This concludes the proof of the theorem. 7 Examples 7.1 Automorphic vector bundles on toroidal compactifi- cations The first natural examples of good hermitian vector bundles are provided by the theory of (fully decomposed) automorphic vector bundles on locally symmetric varieties, and their extensions to smooth toroidal compactifications. These have been firstly worked by Mumford in his proof of Hirzebruch’s proportionality principle in the non-compact case [14]. In this section we quote from loc. cit. the main construction and Mumford’s theorem. As an application, we briefly discuss the example of modular forms. Let B be a bounded symmetric domain. We can write B = G/K, where G is a semi-simple adjoint group and K is a maximal compact subgroup. De- note KC, GC the complexifications of K and G. Inside GC there is a parabolic subgroup of the form P+ · KC (P+ being its unipotent radical), such that K = G∩ (P+ ·KC) and G · (P+ ·KC) is open in GC. Then B̌ := GC/G · (P+ ·KC) is a rational projective variety and there is a G-equivariant immersion B →֒ B̌ compatible with the complex structure of B. Let E0 be a G-equivariant vector bundle on B attached to a representation σ : K → GLr(C). We complexify σ and extend it to P+ · KC, by letting it act trivially on P+. This extension induces a GC equivariant analytic vector bundle Ě0 on B̌, with ι ∗(Ě0) = E0. This way we get a holomorphic structure on E0 (which depends on the chosen extension of σ to P+ ·KC). Let Γ be a neat arithmetic subgroup of G acting on B. Then X = Γ\B is a smooth quasi-projective complex variety. The vector bundle E0 descends to a holomorphic vector bundle E on X . Such a vector bundle is called fully decom- posed automorphic vector bundle. Since K is compact, there is a G-invariant hermitian metric h0 on E0, thus inducing a hermitian metric h on E. Theorem 7.1. Let X be a smooth toroidal compactification of X with D = X \X a divisor with normal crossings. Then the automorphic vector bundle E extends to a vector bundle E1 over X, such that h is good along D. The following proposition may be interesting for some arithmetic purposes. Proposition 7.2. Suppose that E0 = ωB is the canonical bundle of B. Then E1 = ωX(D) and coincides with the pull-back of an ample line bundle O(1) on the Baily-Borel compactification X∗ of X. The global sections of O(n) naturally correspond to the modular forms whose automorphy factor is the nth power of the jacobian. Remark 7.3. Under the hypothesis of the proposition, the line bundle ωX(D) is not ample in general. Equip the line bundle E0 = ωB with the hermitian metric h0 induced by the Kähler-Einstein metric on B, say with Einstein constant −1. The existence and uniqueness is guaranteed by a result of Mok and Yau [13]. Since the Kähler- Einstein metric is invariant under automorphisms, h0 is G-equivariant. By The- orem 7.1 h0 induces a good hermitian metric h on E1 = ωX(D), with singulari- ties along D. Observe that this metric is induced by the Kähler-Einstein metric on X with Einstein constant −1, by uniqueness. Since the Kähler-Einstein met- ric has negative Ricci curvature, c1(ωX(D), h) ≥ 0 on X . Together with the fact that ωX(D) is the pull-back of an ample line bundle O(1) on X ∗, this can be shown to be enough for the main theorem to hold, as soon as X and X are defined over a number field K and we have chosen suitable models X of X, X ∗ of X∗, etc. over SpecOK . Suppose that O(1) extends to an ample line bundle A on X ∗, that there is a morphism π : X → X ∗ extending the natural projection X → X∗ and put L = π∗(A). The line bundle L is a model of ωX(D) and it can be endowed with the good hermitian metric induced by h. Then Corollary 1.4 hold for L , provided we restrict to effective horizontal cycles. The proof follows the same lines as for pre-log-log hermitian ample line bundles, and it will be detailed elsewhere. 7.2 Some natural hermitian vector bundles on the moduli space of curves Let g ≥ 2 be an integer and Mg the moduli space of complex stable curves of genus g. We denote by π : Cg → Mg the universal curve. By Mg we mean the open subset of Mg parametrizing smooth curves, and we write Cg = π −1(Mg). The boundary ∂Mg = Mg \ Mg, which classifies singular stable curves, is a divisor with normal crossings. We write ∂ Cg = π −1(∂Mg), which is a divisor with normal crossings, too. For the sake of simplicity we neglect that Mg and Cg are actually V -manifolds, and we work as if they were complex analytic manifolds (see [17] for the definition of V -manifold and the description of the moduli space of curves as a V -manifold). The first example concerns ωCg/Mg , the relative dualizing sheaf of π. Every fiber of π|Cg admits a unique complete hyperbolic metric of constant negative curvature −1. By Teichmüller’s theory these metrics glue together and define a smooth hermitian metric on ω∨ Cg/Mg . We get a smooth hermitian metric on ωCg/Mg . By a theorem of Wolpert [18] this metric extends to a good hermitian metric on ωCg/Mg , with singularities along ∂ Cg. It is well known that ωCg/Mg is relatively ample ([5], Corollary to Theorem 1.2). Let us now consider (ΩMg , hWP ) the cotangent bundle of Mg with the Weil- Petersson metric. Recall that if p is a point of Mg representing a Riemann surface R, then ΩMg ,p is isomorphic to the space of holomorphic quadratic differentials on R. If R is written as H/Γ (H Poincaré’s upper half plane and Γ ⊆ PSL2(R) a discrete subgroup), then the metric hWP on the stalk ΩMg ,p is the usual scalar product 〈ϕ, ψ〉 = ϕ(z)ψ(z)y2dxdy on automorphic forms of weight (2,0) for the group Γ. By a result of Trapani [16], (detΩMg , dethWP ) extends to a good hermitian line bundle ωMg (log ∂Mg), with singularities along ∂Mg. Moreover Trapani shows that ωMg (log ∂Mg) admits a smooth hermitian metric with positive curvature form. Therefore, its pull-back to the moduli space of curves of genus g with level n structure (n ≥ 3) is ample. In an ambitious program pioneered by [11], [12], Liu, Sun and Yau study the goodness and bounded geometry of several natural Kähler metrics on the moduli space of curves. The interested reader is referred to loc. cit. for precise statements. 7.3 Kähler-Einstein metrics on quasi-projective varieties 7.3.1 Complex theory The main references we follow are [10], [15], and [19]. Let M be a complex analytic manifold of dimension n and Ω a smooth positive (n, n)-form on M , namely a volume form. For every analytic chart (V ; z1, . . . , zn) of M write Ω|V = ξV dzk ∧ dz̄k The functions {ξV }V define a smooth hermitian metric ‖ · ‖Ω on the canonical line bundle ωM . By definition, the Ricci form of Ω is the real (1,1)-form RicΩ = c1(ωM , ‖ · ‖Ω) which is locally given by RicΩ |V= dd c log ξV . The generalized Fefferman operator J acting on volume forms is defined as J : Ω 7−→ (RicΩ)n Theorem 7.4 (Kobayashi [10]). Let X be a compact complex analytic manifold and D ⊆ X a divisor with simple normal crossings. Suppose that the line bundle ωX(D) is ample on X. Then there exists a unique complete Kähler-Einstein metric gKE on X \D with constant negative Ricci curvature -1. If ΩKE is the volume form of gKE, then gKE is characterized by being complete on X \D and by the equation J(ΩKE) = 1. Proposition 7.5. Let X be a compact complex analytic manifold and D ⊆ X a divisor with simple normal crossings. Suppose that ωX(D) is ample on X. Let U = X \ D and endow ωU with the smooth hermitian metric ‖ · ‖KE induced by ΩKE. Then (ωU , ‖ · ‖KE) extends to a good hermitian line bundle (ωX(D), ‖ · ‖KE), with singularities along D. The proof of Proposition 7.5 follows easily from the more precise growth properties established in the proof of Theorem 7.4. For the sake of completeness we now deepen some of the details involved. In the sequel we fix X and D as in the proposition. Definition 7.6. Let M be a complex analytic manifold of dimension n. Let V ⊆ Cn be an open subset. A holomorphic map φ : V → M is called a quasicoordinate if rank(dvφ) = n for every v ∈ V . Then (V, φ) is called a local quasicoordinate of M . Let x ∈ D and (V = V (x); z1, . . . , zn) be an analytic chart of X centered at x, by means of which V gets identified with ∆r1 ×∆ 1 and V \D with ∆ (r = r(x)). For every η ∈ (0, 1)r, define the quasicoordinate φη : Vη = (∆3/4) r ×∆s1 −→ V \D v = (v1, . . . , vn) 7−→ (φη,1(v), . . . , φη,n(v)) where φη,k(v) = if 1 ≤ k ≤ r, vk if k > r. Observe that η∈(0,1)r φη(Vη). We now construct a quasicoordinate covering V = {(Vβ , φβ)}β , containing ex- actly: • all the quasicoordinates {(Vη, φη)}η∈(0,1)r , for (V = V (x); z1, . . . , zn), x ∈ D, as above. Denote by W the union of the images of all these quasicoordinates. This is an open neighborhood of D; • a finite coordinate covering of the compact subset X \W . We introduce the Hölder space of Ck,α-functions on U = X \D, with respect to the quasicoordinate covering V . Definition 7.7 (Hölder spaces). Let k ≥ 0 be an integer and α ∈ (0, 1). The Ck,α-norm with respect to V of a function u ∈ Ck(U) is |u‖k,α,V = sup (V,φ)∈V ‖φ∗(u)‖k,α = sup (V,φ)∈V |p|+|q|≤k ∣∣∣∣∣ ∂|p|+|q| ∂ vp ∂ vq φ∗(u)(v) ∣∣∣∣∣ + sup v,v′∈V |p|+|q|=k |v − v′|−α ∣∣∣∣∣ ∂|p|+|q| ∂ vp ∂ vq φ∗(u)(v) |p|+|q| ∂ vp ∂ vq φ∗(u)(v′) ∣∣∣∣∣ We define the space of Ck,α-functions on U as Ck,α(U) = {u ∈ Ck(U) | ‖u‖k,α,V < +∞}, which is seen to be a Banach space with respect to the norm ‖ · ‖k,α,V . Definition 7.8. We define Rr,s(U) as the space of (r, s)-differential forms ω on U such that, for all quasicoordinate (V, φ) ∈ V , φ∗(ω) = |p|=r |q|=s (apdv p + bqdv ‖ap‖k,α, ‖bq‖k,α < +∞ for all multi-indices p, q with |p| = r, |q| = s and all k ≥ 0, α ∈ (0, 1). If (v1, . . . , vn) are the standard coordinates on V ⊂ C n and p = (i1, . . . , ir), q = (j1, . . . , js) are ordered multi-indices, we wrote dvp = dvi1 ∧ . . . ∧ dvir , dv q = dvj1 ∧ . . . ∧ dvjs . Lemma 7.9. i. Rr,s(U) is a C-vector space; ii. Rr,s(U) ∧Rr ′,s′(U) ⊆ Rr+r ′,s+s′(U); iii. ∂ Rr,s(U) ⊆ Rr+1,s(U) and ∂̄ Rr,s(U) ⊆ Rr,s+1(U). Proof. Immediate to check from the definition of Rr,s(U). Recall that PD denotes the sheaf of Poincaré forms on X with singularities along D (Definition 2.8). Lemma 7.10. We have an inclusion Rr,s(U) ⊆ Γ(X,PD) (r,s), where the su- perscript stands for the (r, s) part with respect to the usual bigrading of complex differential forms. Proof. We localize near the divisor D and consider a quasicoordinate φη : Vη → V \ D in V . Hence V has coordinates z1, . . . , zn and V \ D is the divisor z1 . . . zr = 0. For simplicity we consider the differential form ω = h z1 log(|z1|−1) ∧ . . . ∧ zr log(|zr|−1) ∧ dzr+1 ∧ . . . dzn and suppose that φ∗η(ω) has finite C k,α-norm for all k ≥ 0 and α ∈ (0, 1). We have to prove that h is bounded on the image of φη. From the definition of φη, a straightforward computation shows that φ∗η(ω) = φ 1− |vi|2 |vi − 1| (vi − 1)2 dv1 ∧ . . . ∧ dvn. The hypothesis implies the coefficient of dv1 ∧ . . .∧ dvn has bounded sup-norm. Since |vi| ≤ 3/4 for i = 1, . . . , r, this immediately yields the boundedness of φ∗(h). Corollary 7.11. Let u : U = X \D → C be a smooth function. If u ∈ R0,0(U), then u is a P-singular function with singularities along D. Proof. By Lemma 7.9, du ∈ R1,0(U)⊕R0,1(U) and ddcu ∈ R1,1(U). By Lemma 7.10, du and ddcu have Poincaré growth with singularities along D. Hence u is a P-singular function with singularities along D. Theorem 7.12 (Kobayashi [10]). Let X be a compact complex analytic manifold and D ⊆ X a divisor with simple normal crossings. Suppose that ωX(D) is ample on X. Let D1, . . . , Dm be the irreducible components of D and si ∈ Γ(X,O(Di)) sections with div si = Di, for all i = 1, . . . ,m. Let Ω be a volume form on X. There exist suitable smooth hermitian metrics on the line bundles O(Di), that we write ‖ · ‖ for simplicity, with ‖si‖ small enough, and a function u ∈ R0,0(U) such that the volume form ΩKE of the Kähler-Einstein metric on ΩKE = e u Ω∏m i=1 ‖si‖ 2 log(‖si‖2)2 With Theorem 7.12 at hand, we can prove Proposition 7.5. Proof of Proposition 7.5. We may localize at an analytic chart (V ; z1, . . . , zn) of X by means of which V gets identified with ∆r1 ×∆ 1 and V \D = ∆ For simplicity suppose that Di ∩ V gets identified with zi = 0, for i = 1, . . . , r and Di ∩ V = ∅ for i > r. A local analytic frame of ωX(D)|V is , . . . , , dzr+1, . . . , dzn. Write ‖si‖ 2 = |zi| 2hi for i = 1, . . . , r. Then by Theorem 7.12 we can write ΩKE|V = e hk log(‖sk‖2)2 ‖sk‖2 log(‖sk‖2)2 dzk ∧ dz̄k where γ is a smooth positive function. Observe also that the functions hk as well as the second product are smooth positive functions. We are thus reduced to prove that u and log(log(‖si‖ 2)2) are P-singular functions with singularities along D. On one hand, Lemma 7.11 proves that u is P-singular. On the other hand, log(log(‖si‖ 2)2) is P-singular by Lemma 4.6. 7.3.2 Arithmetic theory Let K be a number field and X a nonsingular projective variety over K. Let D ⊆ X be a reduced effective divisor such that D(C) ⊆ X(C) has simple normal crossings. Suppose that ωX(D) is an ample divisor on X . Then, for every σ : K →֒ C, ωXσ,C(Dσ,C) is ample and there exists a unique Kähler- Einstein metric on Xσ(C) \ Dσ(C) with constant negative Ricci curvature -1 (see Theorem 7.4). By Proposition 7.5, these metrics induce good hermitian metrics on the lines ωXσ,C(Dσ,C), with singularities along Dσ(C), respectively. The collection of these metrics, for varying σ : K →֒ C, is invariant under the action of complex conjugation F∞. Indeed, let gKE,σ be the Kähler-Einstein metric on Xσ(C) \Dσ(C). Then F ∞(gKE,σ) defines a complete Kähler metric on Xσ(C) \Dσ(C). Let ΩKE,σ be the volume form of gKE,σ, so that F ∞ΩKE,σ is the volume form of F ∗∞(gKE,σ). Since RicF ∞ΩKE,σ = F ∞ RicΩKE,σ, we find J(F ∗∞ΩKEσ) = F ∞J(ΩKE,σ) = 1. By uniqueness we derive F ∗∞(gKE,σ) = gKE,σ. We write ωX(D)KE for the resulting good hermitian line bundle, with singularities along D. Let now (X ,L ) be any model of (X,ωX(D)KE) over S = SpecOK . Then Theorem 1.3 can be applied for any choice of smooth metric ‖ · ‖0 on L . If L is ample over X , then Corollary 1.4 applies to (X ,L ). In this situation L verifies the finiteness and the universal lower bound properties. 8 Appendix The appendix is aimed to prove Proposition 6.11. Proof of Proposition 6.11. Decompose D into smooth irreducible components, D = D1 ∪ . . . ∪ Dm. Let D ∗ be the Weil divisor D1 + . . . + Dm. Denote by F the family of nonsingular subschemes of X consisting of X itself and all the irreducible components of the intersections where I runs over all the subsets of {1, . . . ,m}. Since L is ample, there ex- ists some positive integer N such that L⊗N and L⊗N(−D∗) are very ample. Consider the exact sequence 0 → L⊗N (−D∗) → L⊗N → L⊗ND → 0. Taking global sections, we find the exact sequence (8.1) 0 → Γ(X,L⊗N(−D∗)) → Γ(X,L⊗N ) → Γ(X,L⊗ND ). For every Y ∈ F , Bertini’s theorem ([9], Chapter II, Theorem 8.18) provides us with a dense open subset UY of the projective space P = P(Γ(X,L ⊗N(−D∗))) such that, for any t ∈ UY , supp(div t) intersects Y transversally. Since F is finite and the open subsets UY are dense, the intersection is a non-empty open subset of P. Therefore we can take t1, . . . , tr ∈ U such that supp(div t1) ∩ . . . ∩ supp(div tr)=∅. Let t1, . . . , tr ∈ Γ(X,L ⊗N(−D∗)) be representatives of t1, . . . , tr, respectively. Let s1 = j(t1), . . . , sr = j(tr) be their images by the morphism j of (8.1). Since p(si) = 0, D ⊆ supp(div si) for all i = 1, . . . , r. Actually, for every i = 1, . . . , r, we have div si = div ti +D Therefore, for the support of div si we find supp(div si) = supp(div ti) ∪D. By the choice of the sections ti (ti ∈ U), supp(div si) is a divisor with simple normal crossings. Finally, since supp(div t1) ∩ . . . ∩ supp(div tr) = ∅, we have an equality of reduced closed subschemes of X D = (supp(div s1) ∩ . . . ∩ supp(div sr))red. The proof of the proposition is complete. References [1] J.-B. Bost, H. Gillet and C. Soulé, Heights of projective varieties and pos- itive Green forms, J. Amer. Math. Soc. 7 (1994), 903–1027. [2] J.I. Burgos, J. Kramer and U. Kühn, Cohomological arithmetic Chow rings, to appear in J. Inst. Math. Jussieu (2007). [3] , Arithmetic characteristic classes of automorphic vector bundles, Documenta Math. 10 (2005), 619–716. [4] J. Carlson and P.A. Griffiths, A defect relation for equidimensional holo- morphic mappings between algebraic varieties, Ann. Math. 95 (1972), 557– [5] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Publ. Math. IHES 36 (1969), 75–109. [6] J.-P. Demailly, Complex Analytic and Differential Geometry, unpublished book. [7] G. Faltings, Finiteness theorems for abelian varieties over number fields, Arithmetic Geometry (G. Cornell and J.H. Silverman, eds.), Springer- Verlag, 1986, 9–27. [8] H. Gillet and C. Soulé, Arithmetic intersection theory, Publ. Math. IHES 72 (1990), 94–174. [9] R. Hartshorne, Algebraic Geometry, Graduate Texts in Math., vol. 52, Springer-Verlag, 1977. [10] R. Kobayashi,Kähler-Einstein metric on an open algebraic manifold, Osaka J. Math 21 (1984), 399–418. [11] K. Liu, X. Sun and S.-T. Yau, Canonical metrics on the moduli space of Riemann surfaces I, J. Differential Geometry 68 (2004), 571–637. [12] , Canonical metrics on the moduli space of Riemann surfaces II, J. Differential Geometry 69 (2005), 163–216. [13] N. Mok and S.T. Yau, Completeness of the Kähler-Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions, Proc. Symp. in Pure Math. 39 (1983), 41–59. [14] D. Mumford, Hirzebruch’s proportionality theorem in the non-compact case, Invent. Math. 42 (1977), 239–272. [15] G. Tian and S.T. Yau, Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, Math. As- pects of String Theory (San Diego, California, 1986), Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore, 1987. [16] S. Trapani, On the determinant of the bundle of meromorphic quadratic differentials on the Deligne-Mumford compactification of the moduli space of Riemann surfaces, Math. Ann. 293 (1992), 681–705. [17] S.A. Wolpert, On obtaining a positive line bundle from the Weil-Petersson class, Amer. J. of Math. 107 No. 6 (1985), 1485–1507. [18] S.A. Wolpert, The hyperbolic metric and the geometry of the universal curve, J. Differential Geometry 31 (1990) 417–472. [19] D. Wu, Higher canonical asymptotics of Kähler-Einstein metrics on quasi- projective manifolds, Comm. Anal. and Geom., 14 No. 4 (2006), 795–845. G. Freixas i Montplet, Département de Mathématiques, Univer- sité Paris-Sud, Bâtiment 425, 91405 Orsay cedex, France E-mail address : [email protected] Introduction Differential forms with logarithmic singularities Logarithmically singular hermitian vector bundles Global bounds for real log-log growth (1,1)-forms Bounding height integrals Arakelovian heights Examples Appendix

0704.1048

Anisotropic quasiparticle renormalization in Na0.73CoO2: role of inter-orbital interactions and magnetic correlations

Microsoft PowerPoint - figures Anisotropic quasiparticle renormalization in Na0.73CoO2: role of inter-orbital interactions and magnetic correlations J. Geck1,2,∗, S.V. Borisenko1, H. Berger3, H. Eschrig1, J. Fink1, M. Knupfer1, K. Koepernik1, A. Koitzsch1, A.A. Kordyuk1,4, V.B. Zabolotnyy1, and B. Büchner1 1IFW Dresden, P.O. Box 270116,D-01171 Dresden, Germany 2Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, V6T 1Z1, Canada 3Institut de physique de la matiére complex, EPF Lausannne, 1015 Lausanne, Switzerland and 4Institute of Metal Physics of the National Academy of Sciences of Ukraine, 03142 Kyiv, Ukraine (Dated: Received: November 8, 2018) Abstract We report an angular resolved photoemission study of NaxCoO2 with x≃0.73 where it is found that the renormalization of the quasiparticle dispersion changes dramatically upon a rotation from ΓM to ΓK. The comparison of the experimental data to the calculated band structure reveals that the quasiparticle renormalization is most pronounced along the ΓK-direction, while it is significantly weaker along the ΓM-direction. We discuss the observed anisotropy in terms of multiorbital effects and point out the relevance of magnetic correlations for the band structure of NaxCoO2 with x ≃ 0.75. PACS numbers: 71.27.+a, 71.18.+y,74.25.Jb,74.70.-b ∗ Email: [email protected] http://arxiv.org/abs/0704.1048v2 The unconventional behavior of correlated electrons in materials comprised of square lat- tices has attracted a vast amount of attention [1, 2]. However, besides strong electronic correlations, the topology of the underlying lattice structure itself constitutes another im- portant ingredient that can produce exotic electronic ground states. The CoO2-layers in the NaxCoO2 materials, which are built up from edge-sharing CoO6 octahedra, constitute a realization of a correlated electron system based on a triangular lattice. More specifically, these compounds possess a layered hexagonal structure, where strongly covalent CoO2- and ionic Na-layers (ab-planes) alternate along the perpendicular c-axis [3]. Upon changing the Na content x, these materials can be doped with electrons and, in addition, water molecules can be intercalated. Both changing x and water intercalation drastically alter the electronic properties of NaxCoO2, leading most notably to the emergence of superconductivity upon hydration [4, 5]. We will focus on the non-hydrated compounds with x ≃ 0.7, where an anomalous metallic state with an extremely large and field dependent thermoelectric power as well as a giant field dependent scattering rate was observed [6, 7]. There is also evidence for the seemingly paradoxical coexistence of electron itinerancy and localized magnetic moments, as well as for unusual charge order CDW phenomena in these macroscopically metallic compounds [8, 9, 10]. The unconventional electronic properties described above strongly motivate the study of the electronic structure of NaxCoO2 by means of angular resolved photoemission spec- troscopy (ARPES), which provides direct and unique experimental access to single-particle excitations and, thus, to the many-body effects in these materials. Previous ARPES stud- ies on NaxCoO2 [12, 13, 14, 15, 16, 17] showed pronounced deviations from the electronic structure predicted by LDA calculations [11]. In addition, a strongly renormalized heavy QP band, displaying a kink feature was reported [13, 14, 15]. In this ARPES study we focus on the momentum (k) dependent renormalization effects in Na0.73CoO2, which is essential to unravel the relevant couplings that govern the low energy physics. ARPES experiments were carried out using a lab-based system equipped with a SCIENTA SES 200 analyser and a Gammadata He discharge lamp at an excitation energy hν = 21.2 eV (energy resolution 30meV, angular resolution 0.3◦). The high-quality Na0.73CoO2 single crystals examined in this study were grown by the sodium chloride flux methods as described in Ref. [18]. A typical Fermi level (EF ) crossing observed at T=25K is shown in Fig. 1, where the image on the left hand side shows the photoelectron intensity as a function of momentum k and binding energy EB. A single and well defined QP band, which crosses the Fermi energy at EB = 0 eV can be observed. On the right hand side of Fig. 1 an energy distribution curve (EDC) and a momentum distribution curve (MDC) are shown. By mapping the EF crossings over a large area of k-space, a cut through the Fermi surface (FS) of Na0.73CoO2 parallel to the CoO2 planes was measured. The data allows for a precise calibration of the k-scale, because dispersions from the first into the second Brillouin zone were captured. In Figs. 2 (a),(b) the k-dependent photoelectron intensity for two fixed binding energies of EB = 0 eV and 0.132 eV are shown. Focussing on the cut at EB = 0 eV in Fig. 2 (a), a large hole-pocket centered around the Brillouin zone center Γ is observed. The FS displays a clear hexagonal topology. The intensity variations along the FS, leading to a seemingly lower symmetry, are caused by matrix element effects. The size of the observed FS agrees well with the doping level, provided that only one of the two a1g-bands (cf. Fig. 2 (c)) crosses EF . This agrees with the data shown in Fig. 1, where a single QP-band crosses EF . In accordance with previous studies, the hole-pockets along ΓK are not observed for EB = 0 eV. It can be seen in Fig. 2 (b) that the corresponding e g-band (cf. Fig. 2 (c)) lies well below EF , as reported earlier [15]. This can also be observed in Fig. 2 (c), where the measured band structure along ΓK and ΓM is compared to the band structure obtained by LDA [11]. There is a fair agreement of the ARPES data and LDA at higher binding energies above 1.5 eV. However, close to EF deviations occur which are particularly pronounced along In Fig. 3 two cuts of the data set shown in Fig. 2 along the ΓK- (left) and ΓM-direction (right) are shown. Focussing on the cut along ΓK, two prominent features can be observed. First, there is a band crossing EF that forms the FS. Second, there is another band at higher binding energies, which can be identified as the e′g-band. The data indicates that the top of the e′g-band is at about 85meV. By tracking the maximum of the MDCs for different EB, the QP-dispersion (E(k)) in- dicated by black symbols is obtained. Close to EF the dispersion can be well described by a linear behavior, yielding the Fermi velocity vF = (0.3 ± 0.05) eV Å along ΓK. However, at slightly higher EB, the measured dispersion is bent and thus deviates from the linear behavior close to EF . We define the energy where this deviation becomes significant as Ed, resulting in Ed = (26 ± 8)meV for the ΓK-direction. The second bend in the dispersion around 80meV is related to the crossing of the two bands [17]. Applying the same analysis to the cut along ΓM leads to vF = (0.6± 0.08) eV Å and Ed = (66 ± 5)meV. Clearly, both vF and Ed depend on the direction in k-space. The observed bending of the dispersion that sets in around Ed could be due to a coupling of the QP to bosonic excitations. In fact, consistent with this interpretation, we observe an enhanced increase of the scattering rate around Ed. It has to be noted, however, that the feature in the dispersion observed here is not as well defined as the kink in the cuprates, for example. This means that the values of Ed cannot be identified straightforwardly with a specific mode energy and, moreover, a coupling to several different bosonic excitations is possible. In the following we will focus on vF , which is a well defined quantity: Cuts for different ψ-values (cf. Fig. 1 (b)) were systematically analyzed in the same way as described above. The obtained variation of vF as determined from the ARPES data is shown in Fig. 4 (a). Upon rotating the direction of the cut from ΓM to ΓK, vF decreases by about a factor of 2. Using the function kF (ψ) that was determined from the data in Fig. 2 (a) together with the obtained values of vF , the variation of the effective mass m ∗ with ψ can be calculated. The result is given in Fig. 4 (c). Remarkably, the QP along the ΓK direction is about twice as heavy as the QP along ΓM. This pronounced anisotropy of m∗ is expected to have a strong impact on the in-plane electronic properties of Na0.73CoO2. To determine the renormalization effects in Na0.73CoO2, we use the LDA band structure as a reference and compare the ARPES Fermi velocities (vPESF ) to the corresponding LDA values (vLDAF ). As it can be observed in Figs. 4 (a),(b), these two quantities show exactly the opposite behavior: vPESF decreases and v F increases upon rotation from ΓM to ΓK. In order to check whether the deviation between ARPES and LDA is related to a lattice distortion at the surface, LDA calculations were performed for structures where the distance between the oxygen and the cobalt layers, i.e. the Co–O–Co bond angle, was changed. In agreement with previous calculations, we observe that the top of the e′g band is shifted to higher binding energies upon increasing the Co–O–Co bond angle. At the same time the anisotropy of vLDAF is slightly reduced, but unchanged qualitatively. This strongly suggests that the measured anisotropy of vPESF is not caused by a lattice distortion at the surface. The comparison of LDA and ARPES therefore shows that the deviation of the LDA band- structure from the measured QP-dispersion increases dramatically close to the ΓK-direction (cf. inset of Fig. 3). In the following we will refer to this deviation as QP-renormalization (QPR). This QPR can be characterized using a constant κ defined by (1+ κ) vPESF = v The ψ- or, in other words, k-dependence of κ is shown in Fig. 4 (d), revealing the strong anisotropy of the QPR in Na0.73CoO2. We note that, although the photoelectron intensity in Fig. 2 (a) at kF is also influenced by matrix element effects, it is always significantly lower in the ΓK- than in the ΓM-direction, in agreement with enhanced renormalization effects along ΓK. We find ∂kE ≃ ∂kE LDA/2 along ΓK in the whole energy range up to EB = 85meV (inset of Fig. 3). At the same time it is remarkable that the QPR gets stronger the closer the so-called e′g-band gets to the Fermi level (c.f. Fig.2 (b)). This points to an effect related to coupling between the a1g- and the e g-bands. In fact, a strong interaction between these bands is manifested by a large hybridization gap at higher binding energies and the polar- ization dependence along ΓK found in a recent ARPES study [17]. Hence, the k-dependent QPR at EF is most likely caused by multiorbital effects, i.e. interactions between the states of e′g and a1g symmetry. In this case, the QP-states along ΓM and ΓK display different prop- erties: Along ΓM the QP-states have largely a1g symmetry, while they display pronounced multiorbital properties along ΓK. This is of crucial importance for the many-body effects in these materials, since the coupling of the QP to bosonic excitations depends critically on the symmetry of the QP-states [21]. To conclude so far, the observed anisotropies clearly indicate that multiorbital effects play an important role for the QP-dynamics at EF . Furthermore, our DFT studies –details will be provided in a forthcoming publication– show that magnetic correlations play an important role for the QP-dynamics as well: Accord- ing to non-magnetic LDA calculations the band structure of Na0.75CoO2 displays a strong 3D character. In agreement with a previous DFT study [22], we obtain a sizeable kz-dispersion parallel to the c-axis that leads to additional caps of the FS as shown in Figs. 5 (a),(b). Such a strong 3D character is not in agreement with ARPES data: (i) in general, the QP peaks at EF are expected to be considerably broadened in a 3D system, in particular because the short life time of the final states becomes important [19]. This is not the case (Fig. 1). (ii) ARPES measurements at various excitation energies do not show any evidence for a strong dispersion along c [17]. However, magnetic LSDA calculations yield an AFM ground state, where ferromagnetic ab-planes are coupled antiferromagnetically along c. This agrees well with neutron data [23]. In the AFM state, the kz-dispersion is strongly reduced, which removes the aforementioned FS-caps and yields the FS shown in Fig. 5 (c). In other words, according to LSDA, 3D AFM correlations render the electronic structure of Na0.75CoO2 effectively 2D. The top of the e′g-band at EB ≃ 70meV as well as the topology and size of the FS obtained in LSDA are in good agreement with the ARPES data as demonstrated in 5 (d). The above results together with the neutron data indicate that AFM correlations have a strong influence on the electronic structure of NaxCoO2 with x ≃ 0.75. In conclusion, we have shown that the QPR in Na0.73CoO2 is strongly anisotropic and provided clear evidence for the relevance of multiorbital effects for the QP dynamics in this material. In addition, detailed DFT studies highlight the impact of magnetic correlations on the QP-states near EF , which is expected to be directly related to the unusual temperature as well as the field dependencies of the thermopower and the QP scattering rates [6, 7, 14]. Hence, both the interactions between the a1g and e g states as well as magnetic correlations have to be taken into account in order to obtain a realistic description of these materials. Acknowledgements: We thank Dr. Bussy (Univ. of Lausanne) for the micro probe analysis and I. Elfimov, K.M. Shen, D.G. Hawthorn and G.A. Sawatzky for helpful discus- sions. This work was supported by the Swiss NCCR research pool MaNEP of the Swiss NSF, the DFG (FOR 538 research unit, Grant 51195121) and the BMBF (Grant 05KS4OD2/8). J.G. gratefully acknowledges the support by the DFG. Figure captions FIG. 1: Left: Typical Fermi level crossing observed along a cut close to ΓM (ψ = 173◦, cf. Fig. 2 (b)) Right: Corresponding energy distribution curve (EDC) and momentum distribution curve (MDC) at k = kF and EB = 0 eV, respectively. FIG. 2: ARPES data for Na0.73CoO2 (excitation energy hν = 21.2 eV). (a), (b): Momen- tum distribution maps of the photoelectron intensity integrated over a small energy interval (EB ± 3meV) at EB = 0 eV and 0.132 eV measured at T=25K. The measured k-region is indicated by the black dotted line in (a). The other regions in k-space have be obtained by rotating this data set by 120◦ and 240◦. The broken white lines show the two-dimensional Brillouin zone. High-symmetry points Γ, K, and M are indicated in (a) and the definition of ψ is given in (b). A fit to kF = kF (ψ) is shown as a solid black line. (c): Comparison of the measured band structure and the LDA calculation by Singh (black lines) [11]. The crystal field split e′g- and a1g-manifolds are indicated. FIG. 3: Cuts through the map data shown in Fig. 2 (a) and (b). The data is normalized to binding energies above 0.25 eV. Black symbols: QP-dispersion determined by fitting MDCs at different EB. Broken lines indicate the fitted linear dispersions (see text). The inset shows the ARPES- (symbols) and LDA- (lines) dispersions as a function of k−kF (ψ). LDA for x = 0.73 in the rigid band approximation (cf. Fig. 4). FIG. 4: (a),(b): vPESF and v F as a function of ψ. The experimental v F values at a given value of ψ were obtained by averaging over two equivalent cuts (e.g. ψ = 150◦, 210◦). The LDA calculations were performed in the rigid band approximation for the low temper- ature lattice structure using Wien2K. The same behavior was also found by LDA/LSDA calculations in the virtual crystal approximation (cf. Fig. 5). (c): Effective mass of the QP. (a)-(c): hν = 21.2 eV. Solid curves are fits to a sinus-function intended to serve as guides to the eye. (d): κ = vLDAF /v F − 1 characterizing the QPR. FIG. 5: FS obtained for x = 0.75 by LDA in the virtual crystal approximation (VCA), revealing a three-dimensional band structure. (c): FS for the AFM ground state obtained by LSDA in the VCA where the band structure retains its pronounced two-dimensionality. The color scale in (a)-(c) indicates vF . (d) Comparison of the measured and LSDA FS. The DFT calculations have been performed using the FPLO code [20]. [1] J. Orenstein and A. Millis, Science 288, 468 (2000). [2] Y. Tokura and N. Nagaosa, Science 288, 462 (2000). [3] Q. Huang, M. L. Foo, R. A. Pascal, J. W. Lynn, B. H. Toby, T. He, H. W. Zandbergen, and R. J. Cava, Phys. Rev. B 70, 184110 (2004). [4] M. L. Foo, Y. Wang, S. Watauchi, H. Zandbergen, T. He, R. J. Cava, and N. P. Ong, Phys. Rev. Lett. 92, 247001 (2004). [5] K. Takada, H. Sakurai, E. Takayama-Muromachi, F. Izumi, R. Dilanian, and T. Sasaki, Nature 422, 53 (2003). [6] Y. Wang, N. Rogado, R. J. Cava, and N. Ong, Nature 423, 425 (2003). [7] S. Li, L. Taillefer, D. 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0704.1049

Charmless Three-body Decays of B Mesons

July, 2007 Charmless Three-body Decays of B Mesons Hai-Yang Cheng1, Chun-Khiang Chua2 and Amarjit Soni3 1 Institute of Physics, Academia Sinica Taipei, Taiwan 115, Republic of China 2 Department of Physics, Chung Yuan Christian University Chung-Li, Taiwan 320, Republic of China 3 Physics Department, Brookhaven National Laboratory Upton, New York 11973 Abstract An exploratory study of charmless 3-body decays of B mesons is presented using a simple model based on the framework of the factorization approach. The nonresonant contributions arising from B → P1P2 transitions are evaluated using heavy meson chiral perturbation theory (HMChPT). The momentum dependence of nonresonant amplitudes is assumed to be in the exponential form e−αNRpB ·(pi+pj) so that the HMChPT results are recovered in the soft meson limit pi, pj → 0. In addition, we have identified another large source of the nonresonant signal in the matrix elements of scalar densities, e.g. 〈KK|s̄s|0〉, which can be constrained from the decay B0 → KSKSKS or B− → K−KSKS . The intermediate vector meson contributions to 3-body decays are identified through the vector current, while the scalar meson resonances are mainly associated with the scalar density. Their effects are described in terms of the Breit-Wigner formalism. Our main results are: (i) All KKK modes are dominated by the nonresonant background. The predicted branching ratios of K+K−KS(L), K +K−K− and K−KSKS modes are consistent with the data within errors. (ii) Although the penguin-dominated B0 → K+K−KS decay is subject to a potentially significant tree pollution, its effective sin 2β is very similar to that of the KSKSKS mode. However, direct CP asymmetry of the former, being of order −4%, is more prominent than the latter. (iii) For B → Kππ decays, we found sizable nonresonant contributions in K−π+π− and K0π+π− modes, in agreement with the Belle measurements but larger than the BaBar result. (iv) Time-dependent CP asymmetries in KSπ 0π0, a purely CP -even state, and KSπ +π−, an admixture of CP -even and CP -odd components, are studied. (v) The π+π−π0 mode is found to have a rate larger than π+π−π− even though the former involves a π0 in the final state. They are both dominated by resonant ρ contributions. (vi) We have computed the resonant contributions to 3-body decays and determined the rates for the quasi-two-body decays B → V P and B → SP . The predicted ρπ, f0(980)K and f0(980)π rates are in agreement with the data, while the calculated φK, K ∗π, ρK andK∗0 (1430)π are in general too small compared to experiment. (vii) Sizable direct CP asymmetry is found in K+K−K− and K+K−π− modes. http://arxiv.org/abs/0704.1049v2 I. INTRODUCTION Three-body decays of heavy mesons are more complicated than the two-body case as they receive resonant and nonresonant contributions and involve 3-body matrix elements. The three- body meson decays are generally dominated by intermediate vector and scalar resonances, namely, they proceed via quasi-two-body decays containing a resonance state and a pseudoscalar meson. The analysis of these decays using the Dalitz plot technique enables one to study the properties of various resonances. The nonresonant background is usually believed to be a small fraction of the total 3-body decay rate. Experimentally, it is hard to measure the direct 3-body decays as the interference between nonresonant and quasi-two-body amplitudes makes it difficult to disentangle these two distinct contributions and extract the nonresonant one. The Dalitz plot analysis of 3-body B decays provides a nice methodology for extracting infor- mation on the unitarity triangle in the standard model (SM). For example, the Dalitz analysis combined with isospin symmetry allows one to extract the angle α from B → ρπ → πππ [1]. Recently, a method has been proposed in [2] for determining CKM parameters in 3-body decays B → Kππ and Bs → Kππ. This method was extended further in [3] to ∆I = 1, I(K∗π) = 1/2 amplitudes in the above decays and to I = 1 amplitudes in Bs → K∗K and Bs → K Nonresonant 3-body decays of charmed mesons have been measured in several channels and the nonresonant signal in charm decays are found to be less than 10% [4]. In the past few years, some of the charmless B to 3-body decay modes have also been measured at B factories and studied using the Dalitz plot analysis. The measured fractions and the corresponding branching ratios of nonresonant components for some of 3-body B decay modes are listed in Table I. We see that the nonresonant 3-body decays could play an essential role in B decays. It is now well established that the B → KKK modes are dominated by the nonresonant background. For example, the nonresonant fraction is about 90% in B 0 → K+K−K0 decay. While this is a surprise in view of the rather small nonresonant contributions in 3-body charm decays, it is not entirely unexpected because the energy release scale in weak B decays is of order 5 GeV, whereas the major resonances lie in the energy region of 0.77 to 1.6 GeV. Consequently, it is likely that 3-body B decays may receive sizable nonresonant contributions. At any rate, it is important to understand and identify the underlying mechanism for nonresonant decays. The direct nonresonant three-body decays of mesons in general receive two distinct contributions: one from the point-like weak transition and the other from the pole diagrams that involve three- point or four-point strong vertices. For D decays, attempts of applying the effective SU(4)×SU(4) chiral Lagrangian to describe the DP → DP and PP → PP scattering at energies ∼ mD have been made by several authors [18, 19, 20, 21, 22] to calculate the nonresonant D decays, though in principle it is not justified to employ the SU(4) chiral symmetry. As shown in [21, 22], the predictions of the nonresonant decay rates in chiral perturbation theory are in general too small when compared with experiment. With the advent of heavy quark symmetry and its combination with chiral symmetry [23, 24, 25], the nonresonant D decays can be studied reliably at least in the kinematic region where the final pseuodscalar mesons are soft. Some of the direct 3-body D decays were studied based on this approach [26, 27]. TABLE I: Branching ratios of various charmless three-body decays of B mesons. The fractions and the corresponding branching ratios of nonresonant (NR) components are included whenever available. The first and second entries are BaBar and Belle results, respectively. Decay BR(10−6) BRNR(10 −6) NR fraction(%) Ref. B− → π+π−π− 16.2 ± 1.2± 0.9 2.3± 0.9± 0.5 < 4.6 13.6 ± 6.1 [5] – – – B− → K−π+π− 64.1 ± 2.4± 4.0 2.87 ± 0.65 ± 0.43+0.63−0.25 4.5± 1.5 [6] 48.8 ± 1.1± 3.6 16.9 ± 1.3± 1.3+1.1−0.9 34.0 ± 2.2 −1.8 [7] B− → K+K−K− 35.2 ± 0.9± 1.6 a 50± 6± 4 141± 16± 9 [8] 32.1 ± 1.3 ± 2.4 b 24.0 ± 1.5± 1.5 c 74.8 ± 3.6 c [9] B− → K−KSKS 10.7 ± 1.2± 1.0 [10] 13.4 ± 1.9± 1.5 [11] 0 → K0π+π− 43.0 ± 2.3± 2.3 [12] 47.5 ± 2.4± 3.7 19.9 ± 2.5± 1.6+0.7−1.2 41.9± 5.1 ± 0.6 −2.5 [13] 0 → K−π+π0 34.9 ± 2.1± 3.9 < 4.6 [14] 36.6+4.2−4.3 ± 3.0 5.7 +2.7+0.5 −2.5−0.4 < 9.4 [15] 0 → K+K−K0 23.8 ± 2.0± 1.6 26.7 ± 4.6 112.0 ± 14.9 [16] 28.3 ± 3.3± 4.0 [11] 0 → KSKSKS 6.9+0.9−0.8 ± 0.6 [17] 4.2+1.6−1.3 ± 0.8 [11] aWhen the intrinsic charm contribution is excluded, the charmless branching ratio will become (33.5±0.9± 1.6)× 10−6. bWhen the contribution from B+ → χc0K+ is excluded, the charmless branching ratio will become (30.6± 1.2± 2.3)× 10−6. cBelle found two solutions for the fractions and branching ratios. We follow Belle to use the large solution. For the case of B mesons, consider the three-bodyB decay B → P1P2P3. Under the factorization hypothesis, one of the nonresonant contributions arises from the transitions B → P1P2. The nonresonant background in charmless three-body B decays due to the transition B → P1P2 has been studied extensively [28, 29, 30, 31, 32, 33] based on heavy meson chiral perturbation theory (HMChPT) [23, 24, 25]. However, the predicted decay rates are, in general, unexpectedly large. For example, the branching ratio of the nonresonant decay B− → π+π−π− is predicted to be of order 10−5 in [28, 29], which is too large compared to the limit 4.6 × 10−6 set by BaBar [5]. Therefore, it is important to reexamine and clarify the existing calculations. The issue has to do with the applicability of HMChPT. In order to apply this approach, two of the final-state pseudoscalars in B → P1P2 transition have to be soft. The momentum of the soft pseudoscalar should be smaller than the chiral symmetry breaking scale of order 1 GeV. For 3-body charmless B decays, the available phase space where chiral perturbation theory is applicable is only a small fraction of the whole Dalitz plot. Therefore, it is not justified to apply chiral and heavy quark symmetries to a certain kinematic region and then generalize it to the region beyond its validity. In this work we shall assume the momentum dependence of nonresonant amplitudes in the exponential form e−αNRpB ·(pi+pj) so that the HMChPT results are recovered in the soft meson limit pi, pj → 0. We shall see that the parameter αNR can be fixed from the tree-dominated decay B− → π+π−π−. However, the nonresonant background in B → P1P2 transition does not suffice to account for the experimental observation that the penguin-dominated decay B → KKK is dominated by the nonresonant contributions. This implies that the two-body matrix element e.g. 〈KK|s̄s|0〉 induced by the scalar density should have a large nonresonant component. In the absence of first-principles calculation, we will use the B 0 → KSKSKS mode in conjunction with the mass spectrum in 0 → K+K−K0 to fix the nonresonant contribution to 〈KK|s̄s|0〉. In this work, we shall study the charmless 3-body decays of B mesons using the factorization approach. Besides the nonresonant background as discussed above, we will also study resonant contributions to 3-body decays. Vector meson and scalar resonances contribute to the two-body matrix elements 〈P1P2|Vµ|0〉 and 〈P1P2|S|0〉, respectively. They can also contribute to the three- body matrix element 〈P1P2|Vµ − Aµ|B〉. Resonant effects are described in terms of the usual Breit-Wigner formalism. In this manner we are able to figure out the relevant resonances which contribute to the 3-body decays of interest and compute the rates of B → V P and B → SP . In conjunction with the nonresonant contribution, we are ready to calculate the total rates for three-body decays. It should be stressed from the outset that in this work we take the factorization approximation as a working hypothesis rather than a first-principles starting point. If we start with theories such as QCD factorization [34], or pQCD [35] or soft-collinear effective theory [36], then we can take power corrections seriously and make an estimation. Since factorization has not been proved for three-body B decays, we shall work in the phenomenological factorization model rather than in the established theories such as QCDF. That is, we start with the simple idea of factorization and see if it works for three-body decays, in the hope that it will provide a useful zeroth step for others to try to improve. The penguin-induced three-body decays B0 → K+K−KS and KSKSKS deserve special atten- tion as the current measurements of the deviation of sin 2βeff in KKK modes from sin 2βJ/ψKS may indicate New Physics in b → s penguin-induced modes. It is of great importance to examine and estimate how much of the deviation of sin 2βeff is allowed in the SM. Owing to the presence of color-allowed tree contributions in B0 → K+K−KS , this mode is subject to a potentially signif- icant tree pollution and the deviation of the mixing-induced CP asymmetry from that measured in B → J/ψKS could be as large as O(0.10). Since the tree amplitude is tied to the nonresonant background, it is very important to understand the nonresonant contributions in order to have a reliable estimate of sin 2βeff in KKK modes. The layout of the present paper is as follows. In Sec. II we shall apply the factorization approach to study B0 → K+K−KS and KSKSKS decays and discuss resonant and nonresonant contributions. In order to set up the framework for calculations we will discuss B → KKK modes in most details. We then turn to Kππ modes in Sec. III. The tree-dominated modes KKπ in Sec. IV, and πππ in Sec. V. In Sec. VI, we determine the rates for B → V P and B → SP and compare our results with the approach of QCD factorization. Sec. VII contains our conclusions. The factorizable amplitudes of various B → P1P2P3 decays are summarized in Appendix A. The relevant input parameters such as decay constants, form factors, etc. are collected in Appendix B. II. B → KKK DECAYS For 3-body B decays, the b → sqq̄ penguin transitions contribute to the final states with odd number of kaons, namely, KKK and Kππ, while b → uqq̄ tree and b → dqq̄ penguin transitions contribute to final states with even number of kaons, e.g. KKπ and πππ. We shall first discuss the b→ s penguin dominated 3-body decays in details and then turn to b→ u tree dominated modes. For B → KKK modes, we shall first consider the neutral B decays as they involve mixing-induced CP asymmetries. 0 → KKK decays We consider the decay B 0 → K+K−K0 as an illustration. Under the factorization approach, the B0 → K+K−K0 decay amplitude consists of three distinct factorizable terms: (i) the current- induced process with a meson emission, 〈B0 → K+K0〉 × 〈0 → K−〉, (ii) the transition process, 〈B0 → K0〉 × 〈0 → K+K−〉, and (iii) the annihilation process 〈B0 → 0〉 × 〈0 → K+K−K0〉, where 〈A → B〉 denotes a A → B transition matrix element. In the factorization approach, the matrix element of the B → KKK decay amplitude is given by 〈KKK|Heff |B〉 = p=u,c λ(s)p 〈KKK|Tp|B〉, (2.1) where λ p ≡ VpbV ∗ps and the explicit expression of Tp in terms of four-quark operators is given in Eq. (A2). The factorizable B 0 → K+K−K0 decay amplitude is given in Eq. (A4). Note that the OZI suppressed matrix element 〈K+K−|(d̄d)V −A|0〉 is included in the factorizable amplitude since it could be enhanced through the long-distance pole contributions via the intermediate vector mesons such as ρ0 and ω. Likewise, the OZI-suppressed matrix elements 〈K+K−|(d̄b) |B0〉 and 〈K+K−|d̄(1 − γ5)b|B 0〉 are included as they receive contributions from the scalar resonances like f0(980). For the current-induced process, the two-meson transition matrix element 〈K0K+|(ūb)V −A|B0〉 has the general expression [37] 〈K0(p1)K+(p2)|(ūb)V−A|B0〉 = ir(pB − p1 − p2)µ + iω+(p2 + p1)µ + iω−(p2 − p1)µ +h ǫµναβp B(p2 + p1) α(p2 − p1)β, (2.2) where (q̄1q2)V −A ≡ q̄1γµ(1− γ5)q2. This leads to AHMChPTcurrent−ind ≡ 〈K−(p3)|(s̄u)V−A|0〉〈K0(p1)K+(p2)|(ūb)V −A|B0〉 2m23r + (m B − s12 −m23)ω+ + (s23 − s13 −m22 +m21)ω− , (2.3) (c) (d) FIG. 1: Point-like and pole diagrams responsible for the B 0 → K+K0 matrix element induced by the current ūγµ(1− γ5)b, where the symbol • denotes an insertion of the current. where sij ≡ (pi + pj)2. To compute the form factors r, ω± and h, one needs to consider not only the point-like contact diagram, Fig. 1(a), but also various pole diagrams depicted in Fig. 1. In principle, one can apply HMChPT to evaluate the form factors r, ω+ and ω− [37]. However, this will lead to too large decay rates in disagreement with experiment [38]. The heavy meson chiral Lagrangian given in [23, 24, 25] is needed to compute the strong B∗BP , B∗B∗P and BBPP vertices. The results for the form factors are [29, 37] ω+ = − fB∗smB∗s mBmB∗s s23 −m2B∗s 1− (pB − p1) · p1 m2B∗s fB∗smB∗s mBmB∗s s23 −m2B∗s (pB − p1) · p1 m2B∗s pB · (p2 − p1) (pB − p1 − p2)2 −m2B 2gfB∗s (pB − p1) · p1 s23 −m2B∗s 4g2fB mBmB∗s (pB − p1 − p2)2 −m2B p1 ·p2 − p1 ·(pB − p1) p2 ·(pB − p1)/m2B∗s s23 −m2B∗s , (2.4) where fπ = 132 MeV, g is a heavy-flavor independent strong coupling which can be extracted from the CLEO measurement of the D∗+ decay width, |g| = 0.59± 0.01± 0.07 [39]. We shall follow [23] to fix its sign to be negative. The point-like diagram Fig. 1(a) characterized by the term fB/(2f contributes to the form factors ω+ and r, while Figs. 1(b) and 1(d) contribute to r and Fig. 1(c) contributes to all the form factors. A direct calculation indicates that the branching ratio of B 0 → K+K−K0 arising from the current-induced process alone is already at the level of 77 × 10−6 which exceeds the measured total branching ratio of 25 × 10−6 (see Table I). The issue has to do with the applicability of HMChPT. In order to apply this approach, two of the final-state pseudoscalars (K+ and K this example) have to be soft. The momentum of the soft pseudoscalar should be smaller than the chiral symmetry breaking scale Λχ of order 0.83 − 1.0 GeV. For 3-body charmless B decays, the available phase space where chiral perturbation theory is applicable is only a small fraction of the whole Dalitz plot. Therefore, it is not justified to apply chiral and heavy quark symmetries to a certain kinematic region and then generalize it to the region beyond its validity. If the soft meson result is assumed to be the same in the whole Dalitz plot, the decay rate will be greatly overestimated. In [38, 40] we have tried to circumvent the aforementioned problem by applying HMChPT only to the strong vertex and use the form factors to describe the weak vertex. Moreover, we introduced a form factor to take care of the off-shell effect. For example, Fig. 1(c) can be evaluated by considering the strong interaction B0 → K0B∗s followed by the weak transition B s → K+ and the result is [38] AF ig.1(c) = mBmB∗s s23 −m2B∗s F (s23,mB∗s )F m2B − s23 FBsK1 (m m21 +m 3 − s13 + (s23 −m22 +m23)(m2B − s23 −m21) 2m2B∗s , (2.5) where FBsK0,1 are the Bs → K weak transition from factors in the standard convention [41] and we have introduced a form factor F (s23,mB∗s ) to take into account the off-shell effect of the B s pole [40]. It is parameterized as F (s23,mB∗s ) = (Λ 2−m2B∗s )/(Λ 2−s23) with the cut-off parameter Λ chosen to be Λ = mB∗s +ΛQCD. Needless to say, this parametrization of the form factor is somewhat arbitrary. Moreover, the nonresonant contribution thus calculated is too small compared to experiment. The Dalitz plot analysis of B 0 → K+K−K0 has been recently performed by BaBar [16]. In the BaBar analysis, a phenomenological parametrization of the non-resonant amplitudes is described ANR = (c12e iφ12e−αs 12 + c13e iφ13e−αs 13 + c23e iφ23e−αs 23)(1 + bNRe i(β+δNR)), (2.6) and resonant terms are described by cr(1 + br)fre i(φr+δr+β), ĀR = cr(1 − br)frei(φr−δr+β). (2.7) The BaBar results for isobar amplitudes, phases and fractions from the fit to the B0 → K+K−K0 are summarized in Table II. It is evident that this decay is dominated by the nonresonant back- ground. For our purpose, we will parametrize the current-induced nonresonant amplitude Eq. (2.3) Acurrent−ind = A HMChPT current−ind e pB·(p1+p2)eiφ12 , (2.8) so that the HMChPT results are recovered in the chiral limit p1, p2 → 0. That is, the nonresonant amplitude in the soft meson region is described by HMChPT, but its energy dependence beyond the chiral limit is governed by the exponential term e−αNRpB·(p1+p2). In what follows, we shall use the tree-dominated B− → π+π−π− decay data to fix α , which turns out to be = 0.103+0.018−0.011 GeV −2. (2.9) This is very close to the naive expectation of α ∼ O(1/(2mBΛχ)) based on the dimensional argument. The phase φ12 of the nonresonant amplitude in the (K ) system will be set to zero for simplicity. TABLE II: BaBar results for isobar amplitudes, phases, and fractions from the fit to the B0 → K+K−K0 [16]. Three rows for non-resonant contribution correspond to coefficients of exponential functions in Eq. (2.6), while the fraction is given for the combined amplitude. For the nonresonant decay mode in K+K−, the amplitude c12 and the phase φ12 in Eq. (2.6) are fixed to be one and zero, respectively. Errors are statistical only. Decay Amplitude cr Phase φr Fraction (%) φ(1020)K0 0.0085 ± 0.0010 −0.016 ± 0.234 12.5 ± 1.3 f0(980)K 0 0.622 ± 0.046 −0.14 ± 0.14 40.2 ± 9.6 X0(1550)K 0 0.114 ± 0.018 −0.47 ± 0.20 4.1± 1.3 (K+K−)NRK 0 1 (fixed) 0 (fixed) (K+K0)NRK − 0.33± 0.07 1.95 ± 0.27 112.0 ± 14.9 (K−K0)NRK + 0.31± 0.08 −1.34 ± 0.37 χc0(1P )K 0 0.0306 ± 0.00649 0.81−2.33 ± 0.54 3.0± 1.2 D+K− 1.11± 0.17 – 3.6± 1.5 D+s K − 0.76± 0.14 – 1.8± 0.6 For the transition amplitude, we need to evaluate the 2-kaon creation matrix element which can be expressed in terms of time-like kaon current form factors as 〈K+(pK+)K−(pK−)|q̄γµq|0〉 = (pK+ − pK−)µFK 〈K0(pK0)K (pK̄0)|q̄γµq|0〉 = (pK0 − pK̄0)µFK q . (2.10) The weak vector form factors FK q and F K0K̄0 q can be related to the kaon electromagnetic (e.m.) form factors FK em and F K0K̄0 em for the charged and neutral kaons, respectively. Phenomenologi- cally, the e.m. form factors receive resonant and nonresonant contributions and can be expressed em = Fρ + Fω + Fφ + FNR, F K0K̄0 em = −Fρ + Fω + Fφ + F ′NR. (2.11) It follows from Eqs. (2.10) and (2.11) that u = F K0K̄0 d = Fρ + 3Fω + (3FNR − F ′NR), d = F K0K̄0 u = −Fρ + 3Fω, s = F K0K̄0 s = −3Fφ − (3FNR + 2F NR), (2.12) where use of isospin symmetry has been made. The resonant and nonresonant terms in Eq. (2.11) can be parametrized as Fh(s23) = m2h − s23 − imhΓh NR(s23) = , (2.13) with Λ̃ ≈ 0.3 GeV. The expression for the nonresonant form factor is motivated by the asymptotic constraint from pQCD, namely, F (t) → (1/t)[ln(t/Λ̃2)]−1 in the large t limit [42]. The unknown parameters ch, xi and x i are fitted from the kaon e.m. data, giving the best fit values (in units of GeV2 for ch) [43]: cρ = 3cω = cφ = 0.363, cρ(1450) = 7.98 × 10−3, cρ(1700) = 1.71 × 10−3, cω(1420) = −7.64× 10−2, cω(1650) = −0.116, cφ(1680) = −2.0× 10−2, (2.14) x1 = −3.26 GeV2, x2 = 5.02 GeV4, x′1 = 0.47 GeV2, x′2 = 0. (2.15) Note that the form factors Fρ,ω,φ in Eqs. (2.11) and (2.12) include the contributions from the vector mesons ρ(770), ρ(1450), ρ(1700), ω(782), ω(1420), ω(1650), φ(1020) and φ(1680). It is interesting to note that (i) the fitted values of cV are very close to the vector meson dominance expression gV KK for V = ρ, ω, φ [4, 44], where gV γ is the e.m. coupling of the vector meson defined by 〈V |jem|0〉 = gV γε∗V and gV KK is the V → KK strong coupling with −gφK+K− ≃ gρK+K−/ gωK+K−/ 2 ≃ 3.03, and (ii) the vector-meson pole contributions alone yield FK+K−u,s (0) ≈ 1,−1 and FK d (0) ≈ 0 as the charged kaon does not contain the valence d quark. The matrix element for the current-induced decay process then has the expression 〈K0(p1)|(s̄b)V −A|B0〉〈K+(p2)K−(p3)|(q̄q)V−A|0〉 = (s12 − s13)FBK1 (s23)FK q (s23). (2.16) We also need to specify the 2-body matrix elements 〈K+K−|s̄s|0〉〈K0|s̄b|B0〉 induced from the scalar densities. The use of the equation of motion leads to 〈K0(p1)|s̄b|B (pB)〉 = m2B −m2K mb −ms FBK0 (s23). (2.17) The matrix element 〈K+K−|s̄s|0〉 receives resonant and non-resonant contributions: 〈K+(p2)K−(p3)|s̄s|0〉 ≡ fK s (s23) = mf0i f̄ gf0i→K m2f0i − s23 − imf0iΓf0i + fNRs , fNRs = (3FNR + 2F NR) + σNRe −α s23 , (2.18) where f0i denote the generic f0-type scalar mesons, f0i = f0(980), f0(1370), f0(1500),X0(1550), · · ·, the scalar decay constant f̄ sf0i is defined by 〈f0i|s̄s|0〉 = mf0i f̄ [see Eq. (B1)], gf0i→K +K− is the f0i → K+K− strong coupling, and the nonresonant terms are related to those in FK s through the equation of motion. The presence of the nonresonant σ term will be explained shortly. The main scalar meson pole contributions are those that have dominant ss̄ content and large coupling to KK. We consider the scalar mesons f0(980) and X0(1550) (denoted as fX(1500) by Belle) which are supposed to have the largest couplings with the KK pair. Note that the nature of the broad state X0(1550) observed by BaBar and Belle, for example, what is its relation with f0(1500), is not clear. To proceed with the numerical calculations, we shall use gf0(980)→K +K− = 4.3 GeV,1 1 This is different from the coupling gf0(980)→K +K− = 1.5 GeV originally employed in [40]. The coupling gf0(980)→π +π− ∼ 1.33 GeV can be fixed from a recent Belle measurement of Γ(f0(980) → π+π−) [see Eq. (3.18)]. Using the BES result (gf0(980)→KK/gf0(980)→ππ)2 = 4.21± 0.25± 0.21 [45], one can deduce that gf0(980)→KK = 2.7± 0.6 GeV. In this work, we found that a slightly large coupling gf0(980)→KK will give better numerical results. gX0(1550)→K +K− = 1.4 GeV, Γf0(980) = 80 MeV, ΓX0(1550) = 0.257 GeV [8], f̄f0(980)(µ = mb/2) ≃ 0.46 GeV [46] and f̄f0(1530) ≃ 0.30 GeV. The sign of the resonant terms is fixed by fK s (0) = v from a chiral perturbation theory calculation (see, for example, [47]). It should be stressed that although the nonresonant contributions to fKKs and F s are related through the equation of motion, the resonant ones are different and not related a priori. As stressed in [40], to apply the equation of motion, the form factors should be away from the resonant region. In the presence of the resonances, we thus need to introduce a nonresonant term characterized by the parameter σNR in Eq. (2.18) which will be specified later. The parameter α appearing in the same equation should be close to the value of α given in Eq. (2.9). We will use the experimental measurement α = (0.14 ± 0.02)GeV−2 [16]. As noticed before, the matrix elements 〈K+K−|(d̄b) |B0〉 and 〈K+K−|d̄(1 − γ5)b|B 0〉 are included in Eq. (A4) as they receive intermediate scalar pole contributions. More explicitly, 〈K+(p2)K−(p3)|(d̄b)V −A |B 0〉R = gf0i→K m2f0i − s23 − imf0iΓf0i 〈f0i|(d̄b)V −A |B 0〉. (2.19) Hence, 〈K0(p1)|(s̄d)V −A |0〉〈K +(p2)K −(p3)|(d̄b)V −A |B gf0i→K m2f0i − s23 − imf0iΓf0i B −m2f0i). (2.20) The superscript u of the form factor F 0 reminds us that it is the uū quark content that gets involved in the B to f0i form factor transition. In short, the relevant f0(980) pole contributions to 0 → K+K−K0 are 〈K0K+K−|Tp|B 0〉f0 = gf0(980)→K m2f0 − s23 − imf0Γf0 f̄ sf0F )(m2B −m2K) + fKF B −m2f0) 10 − (a , (2.21) where we have employed Eq. (2.18) and applied equations of motion to the matrix elements 〈K0|s̄γ5d|0〉〈K+K−|d̄γ5b|B 0〉. Comparing this equation with Eq. (A6) of [48], we see that the expression inside {· · ·} is identical to that of B0 → f0(980)K , as it should be. We digress for a moment to discuss the wave function of the f0(980). What is the quark structure of the light scalar mesons below or near 1 GeV has been quite controversial. In this work we shall consider the conventional qq̄ assignment for the f0(980). In the naive quark model, the flavor wave functions of the f0(980) and σ(600) read (uū+ dd̄), f0 = ss̄, (2.22) where the ideal mixing for f0 and σ has been assumed. In this picture, f0(980) is purely an ss̄ state. However, there also exist some experimental evidences indicating that f0(980) is not purely an ss̄ state. First, the observation of Γ(J/ψ → f0ω) ≈ 12Γ(J/ψ → f0φ) [4] clearly indicates the existence of the non-strange and strange quark content in f0(980). Second, the fact that f0(980) and a0(980) have similar widths and that the f0 width is dominated by ππ also suggests the composition of uū and dd̄ pairs in f0(980); that is, f0(980) → ππ should not be OZI suppressed relative to a0(980) → πη. Therefore, isoscalars σ(600) and f0 must have a mixing |f0(980)〉 = |ss̄〉 cos θ + |nn̄〉 sin θ, |σ(600)〉 = −|ss̄〉 sin θ + |nn̄〉 cos θ, (2.23) with nn̄ ≡ (ūu+ d̄d)/ 2. Experimental implications for the f0−σ mixing angle have been discussed in detail in [49]. It is found that θ lies in the ranges of 25◦ < θ < 40◦ and −40◦ < θ < −15◦ (or 140◦ < θ < 165◦). Note that the phenomenological analysis of the radiative decays φ → f0(980)γ and f0(980) → γγ favors a solution of the θ to be negative (or in the second quadrant). In this work, we shall use θ = −25◦. Finally, the matrix elements involving 3-kaon creation are given by [38] 〈K0(p1)K+(p2)K−(p3)|(s̄d)V −A|0〉〈0|(d̄b)V −A|B0〉 ≈ 0, (2.24) 〈K0(p1)K+(p2)K−(p3)|s̄γ5d|0〉〈0|d̄γ5b|B0〉 = v s13 −m21 −m23 m2B −m2K FKKK(m2B), where mu +ms m2K −m2π ms −md , (2.25) characterizes the quark-order parameter 〈q̄q〉 which spontaneously breaks the chiral symmetry. Both relations in Eq. (2.24) are originally derived in the chiral limit [38] and hence the quark masses appearing in Eq. (2.25) are referred to the scale ∼ 1 GeV . The first relation reflects helicity suppression which is expected to be even more effective for energetic kaons. For the second relation, we introduce the form factor FKKK to extrapolate the chiral result to the physical region. Following [38] we shall take FKKK(q2) = 1/[1 − (q2/Λ2χ)] with Λχ = 0.83 GeV being a chiral symmetry breaking scale. To proceed with the numerical calculations, we need to specify the input parameters. The relevant CKM matrix elements, decay constants, form factors, the effective Wilson coefficients i and the running quark masses are collected in Appendix B. As for the parameter σNR in Eq. (2.18), in principle we can set its phase φσ to zero and use the measured KSKSKS rate, namely, B(B0 → KSKSKS) = (6.2±0.9)×10−6 [50], to fix the parameter σNR and then use the data obtained from the Dalitz plot analysis to determine the strong phases φr for resonant amplitudes. However, in doing so one needs the data of invariant mass spectra. In the absence of such information, instead we will treat φσ as a free parameter and do not assign any other strong phases to the resonant amplitudes except for those arising from the Breit-Wigner formalism. It turns out that if φσ is small, the K+K− mass spectrum in B 0 → K+K−KS will have a prominent hump at the invariant mass mK+K− = 3 GeV, which is not seen experimentally (see Fig. 2(c)). We found that φσ ≈ π/4 will yield K+K− mass spectrum consistent with the data = eiπ/4 3.36+1.12−0.96 GeV. (2.26) Note that the phase of σ is consistent with the BaBar measurement shown in Table II, namely, φBaBarσ = 1.19± 0.37. The calculated branching ratios of resonant and nonresonant contributions to B 0 → K+K−K0 are summarized in Table III. The theoretical errors shown there are from the uncertainties in (i) TABLE III: Branching ratios (in units of 10−6) of resonant and nonresonant (NR) contributions to 0 → K+K−K0. Theoretical errors correspond to the uncertainties in (i) α , (ii) ms, F 0 and , and (iii) γ = (59 ± 7)◦. We do not have 1/mb power corrections within this model. However, systematic errors due to model dependent assumptions may be sizable and are not included in the error estimates that we give. Experimental results are taken from Table II. Decay mode BaBar [16] Theory 2.98 ± 0.45 2.6+0.0+0.5+0.0−0.0−0.4−0.0 f0(980)K 9.57 ± 2.51 5.8+0.0+0.1+0.0−0.0−0.5−0.0 X0(1550)K − 0.98 ± 0.33 0.93+0.00+0.16+0.00−0.00−0.15−0.00 NR 26.7 ± 4.6 18.1+0.6+5.1+0.2−0.7−3.8−0.2 total 23.8 ± 2.0± 1.6 19.8+0.4+0.5+0.1−0.4−0.4−0.2 the parameter αNR which governs the momentum dependence of the nonresonant amplitude, (ii) the strange quark mass ms, the form factor F 0 and the nonresonant parameter σNR , and (iii) the unitarity angle γ. In QCD calculations based on a heavy quark expansion, one faces uncertainties arising from power corrections such as annihilation and hard-scattering contributions. For example, in QCD factorization, there are large theoretical uncertainties related to the modelling of power corrections corresponding to weak annihilation effects and the chirally-enhanced power corrections to hard spectator scattering. Even for two-body B decays, power corrections are of order (10-20)% for tree-dominated modes, but they are usually bigger than the central values for penguin-dominated decays. Needless to say, 1/mb power corrections for three-body decays may well be larger. However, as stressed in Introduction, in this exploratory work we use the phenomenological factorization model rather than in the established theories based on a heavy quark expansion. Consequently, uncertainties due to power corrections, at this stage, are not included in our calculations, by assumption. In view of such shortcomings we must emphasize that the additional errors due to such model dependent assumptions may be sizable. From Table III we see that the predicted rates for resonant and nonresonant components are consistent with experiment within errors. The nonresonant contribution arises dominantly from the transition process (88%) via the scalar-density-induced vacuum to KK̄ transition, namely, 〈K+K−|s̄s|0〉, and slightly from the current-induced process (3%). Therefore, it is natural to conjecture that nonresonant decays could also play a prominent role in other penguin dominated 3-body B decays. The K+K−KS mode is an admixture of CP -even and CP -odd components. By excluding the major CP -odd contribution from φKS , the 3-bodyK +K−KS final state is primarily CP -even. The K+K− mass spectra of the B0 → K+K−KS decay from CP -even and CP -odd contributions are shown in Fig. 2. For the CP -even spectrum, there are peaks at the threshold and mK+K− = 1.5 GeV region. The threshold enhancement arises from the f0(980)KS and the nonresonant f 1 1.5 2 2.5 3 3.5 4 4.5 m K+ K- HGeVL 1 1.5 2 2.5 3 m K+ K- HGeVL 1 1.02 1.04 (GeV)K+K-m 1 1.5 2 2.5 3 3.5 4 4.5 (GeV)K+K-m 1 1.5 2 2.5 3 3.5 4 4.5 BABAR preliminary (a) (b) (c) FIG. 2: The K+K− mass spectra for B0 → K+K−KS decay from (a) CP -even and (b) CP -odd contributions. The insert in (b) is for the φ region. The full K+K−KS spectrum, which is the sum of CP -even and CP -odd parts, measured by BaBar [16] is depicted in (c). contributions [see Eq. (2.18)]. 2 For the CP -odd spectrum, the peak on the lower end corresponds to the φKS contribution, which is also shown in the insert. The b → u transition is governed by the current-induced process 〈B0 → K+K0〉 × 〈0 → K−〉 [see Eq. (A4)]. From Eq. (2.8) it is clear that the b→ u amplitude prefers a small invariant mass of K+ and K0 and hence a large invariant mass of K+ and K−. In contrast, the b → c amplitude prefers a small s23. Consequently, their interference is largely suppressed. The full K+K−KS spectrum, which is the sum of the CP -even and the CP -odd parts, has been measured by BaBar [Fig. 2(c)]. It clearly shows the phenomenon of threshold enhancement and the scalar resonances X0(1550) and χc0. The decay B 0 → KSKSKS is a pure penguin-induced mode [cf. Eq. (A7)] and it receives intermediate pole contributions only from the iso-singlet scalar mesons such as f0(980). Just like other KKK modes, this decay is governed by the nonresonant background dominated by the σ term defined in Eq. (2.18). Hence, this mode is ideal for determining the unknown parameter σ which is given in Eq. (2.26). Time-dependent CP violation in neutral 3-body decay modes with fixed CP parity was first discussed by Gershon and Hazumi [51]. Results for the decay rates and CP asymmetries in B0 → K+K−KS(L), KSKSKS(L) are dis- played in Table IV and Table V, respectively. (For the decay amplitudes of B0 → KSKSKS(L), see [40] for details.) The mixing-induced CP violations are defined by SKKK,CP± = Im(e−2iβACP±Ā CP±)ds12ds23 |ACP±|2ds12ds23 + |ĀCP±|2ds12ds23 SKKK = Im(e−2iβAĀ∗)ds12ds23 |A|2ds12ds23 + |Ā|2ds12ds23 = f+ SKKK,CP+ + (1− f+)SKKK,CP−, (2.27) 2 In our previous work [40] we have argued that the spectrum should have a peak at the large mK+K− end. This is because we have introduced an additional nonresonant contribution to the ω− parameter parametrized as ωNR− = κ 2pB ·p2 and employed the B− → D0K0K− data and applied isospin symmetry to the B → KK matrix elements to determine the unknown parameter κ. Since this nonresonant term favors a small mK+KS region, a peak of the spectrum at large mK+K− is thus expected. However, such a bump is not seen experimentally [16]. In this work we will no longer consider this term. TABLE IV: Branching ratios for B0 → K+K−KS , KSKSKS , KSKSKL decays and the fraction of CP-even contribution to B 0 → K+K−KS , f+. The branching ratio of CP-odd K+K−KS with φKS excluded is shown in parentheses. Results for (K +K−KL)CP± are identical to those for (K+K−KS)CP∓. For theoretical errors, see Table III. Experimental results are taken from [50]. Final State B(10−6)theory B(10−6)expt K+K−KS 9.89 +0.19+2.28+0.07 −0.21−1.81−0.08 12.4± 1.2 (K+K−KS)CP+ 8.33 +0.10+1.82+0.05 −0.12−1.49−0.06 (K+K−KS)CP− 1.57 +0.09+0.46+0.02 −0.10−0.32−0.02 (0.14+0.06+0.14+0.01−0.06−0.06−0.01) KSKSKS input 6.2± 0.9 KSKSKL 7.63 +0.01+1.37+0.03 −0.01−1.19−0.03 < 14 theory K+K−KS 0.98 +0.01+0.01+0.00 −0.01−0.02−0.00 0.91 ± 0.07 theory K+K−KL 0.98 +0.01+0.01+0.00 −0.01−0.02−0.00 where A is the decay amplitude of B 0 → K+K−KS(L) or KSKSKS(L) and Ā is the conjugated B0 decay amplitude, and f+ is the CP even fraction defined by ΓCP+ + ΓCP+ Γ + Γ φKS excluded. (2.28) Generally, it is more convenient to define an effective sin 2β via Sf ≡ −ηf sin 2βeff with ηf = 2f+−1 for K+K−KS . The predicted value of f+ is consistent with the data but it is on the higher end of the experimental measurement because the CP -odd contributions from the vector mesons ρ, ω, · · · , are OZI suppressed and the CP -odd nonresonant contribution is constrained by the π+π−π− rate. The deviation of the mixing-induced CP asymmetry in B0 → K+K−KS and KSKSKS from that measured in B → φcc̄KS , i.e. sin 2βφcc̄KS = 0.681± 0.025 [50], namely, ∆ sin 2βeff ≡ sin 2βeff − sin 2βφcc̄KS , is calculated from Table V to be ∆ sin 2βK+K−KS = 0.047 +0.028 −0.033 , ∆sin 2βKSKSKS = 0.038 +0.027 −0.032 . (2.29) The corresponding experimental values are 0.049 ± 0.10 and −0.101 ± 0.20, respectively. Due to the presence of color-allowed tree contributions in B 0 → K+K−KS , it is naively expected that this penguin-dominated mode is subject to a potentially significant tree pollution and hence ∆ sin 2βeff can be as large as O(10%). However, our calculation indicates the deviation of the mixing-induced CP asymmetry in B 0 → K+K−KS from that measured in B 0 → φcc̄KS is very similar to that of the KSKSKS mode as the tree pollution effect in the former is somewhat washed out. Nevertheless, TABLE V: Mixing-induced and direct CP asymmetries sin 2βeff (top) and Af (in %, bottom), respectively, in B 0 → K+K−KS and KSKSKS decays. Experimental results for K+K−KS and K+K−KL modes are obtained from the data of B 0 → K+K−K0. Results for (K+K−KL)CP± are identical to those for (K+K−KS)CP∓. For theoretical errors, see Table III. Experimental results are taken from [50]. Final state sin 2βeff Expt. (K+K−KS)φKS excluded 0.728 +0.001+0.002+0.009 −0.002−0.001−0.020 0.73 ± 0.10 (K+K−KS)CP+ 0.732 +0.003+0.006+0.009 −0.004−0.004−0.020 (K+K−KL)φKL excluded 0.728 +0.001+0.002+0.009 −0.002−0.001−0.020 0.73 ± 0.10 KSKSKS 0.719 +0.000+0.000+0.008 −0.000−0.000−0.019 0.58 ± 0.20 KSKSKL 0.718 +0.000+0.000+0.008 −0.000−0.000−0.019 Af (%) Expt. (K+K−KS)φKS excluded −4.63 +1.35+0.53+0.40 −1.01−0.54−0.34 −7± 8 (K+K−KS)CP+ −4.86+1.43+0.52+0.42−1.09−0.55−0.35 (K+K−KL)φKL excluded −4.63 +1.35+0.53+0.40 −1.01−0.54−0.34 −7± 8 KSKSKS 0.69 +0.01+0.01+0.05 −0.01−0.01−0.06 14 ± 15 KSKSKL 0.77 +0.01+0.01+0.05 −0.01−0.03−0.07 direct CP asymmetry of the former, being of order −4%, is more prominent than the latter.3 B. B− → KKK decays The B− → K+K−K− decay amplitude has a similar expression as Eq. (A4) except that one also needs to add the contributions from the interchange s23 → s12 and put a factor of 1/2 in the decay rate to account for the identical particle effect. Branching ratios of resonant and nonresonant contributions to B− → K+K−K− are shown in Table VI. It is clear that the predicted rates of resonant and nonresonant components are consistent with the data except for the broad scalar resonance X0(1550). Both BaBar and Belle have seen a large fraction from X0(1550), (121 ± 19 ± 6)% by BaBar [8] and (63.4 ± 6.9)% by Belle [9], 4 while our prediction is similar to that in B 0 → K+K−K0. It is not clear why there is a huge 3 In our previous work [40], ∆ sin 2βeff is found to be ∆ sin 2βK+K−KS = 0.06 +0.09 −0.04 , ∆sin 2βKSKSKS = 0.06 +0.03 −0.04 , for sin 2βJ/ψKS = 0.687 ± 0.032, while direct CP asymmetry is less than 1% in both modes. Note that due to an oversight the experimental error bars were not included in our previous paper for the theoretical calculation of ∆ sin 2βeff . 4 Belle [9] actually found two solutions for the fraction of X0(1550)K −: (63.4± 6.9)% and (8.21± 1.94)%. The first solution is preferred by Belle. disparity between B− → K+K−K− and B0 → K+K−K0 as far as the X0(1550) contribution is concerned. Obviously, a refined measurement of the X0(1550) contribution to the K +K−K− mode is urgently needed in order to clarify this issue. Our result for the nonresonant contribution is in good agreement with Belle, but disagrees with BaBar. Notice that Belle did not see the scalar resonance f0(980) as Belle employed the E791 result [53] for g f0→KK̄ which is smaller than gf0→ππ. In contrast to E791, the ratio gf0→KK̄/gf0→ππ is measured to be larger than 4 in the existing e+e− experiments [45, 54] TABLE VI: Branching ratios (in units of 10−6) of resonant and nonresonant (NR) contributions to B− → K+K−K−. For theoretical errors, see Table III. Decay mode BaBar [8] Belle [9] Theory φK− 4.14 ± 0.32± 0.33 4.72 ± 0.45 ± 0.35+0.39−0.22 2.9 +0.0+0.5+0.0 −0.0−0.5−0.0 f0(980)K − 6.5± 2.5± 1.6 < 2.9 7.0+0.0+0.4+0.1−0.0−0.7−0.1 X0(1550)K − 43± 6± 3 1.1+0.0+0.2+0.0−0.0−0.2−0.0 f0(1710)K − 1.7± 1.0± 0.3 NR 50± 6± 4 24.0 ± 1.5± 1.8+1.9−5.7 25.3 +0.9+4.8+0.3 −1.0−4.4−0.3 Total 35.2 ± 0.9± 1.6 32.1 ± 1.3± 2.4 25.5+0.5+4.4+0.2−0.6−4.1−0.2 We next turn to the decay B− → K−KSKS . Following [55], let us consider the symmetric state of K0K |K0K0〉sym ≡ |K0(p1)K (p2)〉+ |K (p1)K 0(p2)〉 = [|KS(p1)KS(p2)〉 − |KL(p1)KL(p2)〉] / 2. (2.30) Hence, B(B− → K−KSKS) = [B(B− → K−KSKS) + B(B− → K−KLKL)] B(B− → K−(K0K0)sym). (2.31) The factorizable amplitude of B− → K−K0K0 is given by Eq. (A8). Just as other KKK modes, this decay is also expected to be dominated by the nonresonant contribution (see Table VII). The calculated total rate is in good agreement with experiment. Just as the pure penguin mode KSKSKS , the decay B − → K−KSKS also can be used to constrain the nonresonant parameter As pointed out in [55], isospin symmetry implies the relation A(B− → K−K0K0) = −A(B0 → K0K+K−). (2.32) This leads to B(B− → K−(K0K0)sym) = τ(B−) τ(B0) B(B0 → K+K−K0)φK excluded. (2.33) Experimentally, this relation is well satisfied: LHS=(23.0±2.6)×10−6 and RHS=(22.1±2.1)×10−6 . Hence, the isospin relation Eq. (2.32) is well respected. TABLE VII: Branching ratios (in units of 10−6) of resonant and nonresonant (NR) contributions to B− → K−KSKS . For theoretical errors, see Table III. Decay mode f0(980)K − X0(1550)K − NR total Theory 5.2+0.0+0.3+0.1−0.0−0.5−0.1 0.92 +0.00+0.16+0.00 −0.00−0.15−0.00 12.4 +0.2+2.1+0.1 −0.3−2.0−0.1 12.2 +0.0+1.5+0.0 −0.0−1.7−0.0 Expt. 11.5 ± 1.3 III. B → Kππ DECAYS In this section we shall consider five B → Kππ decays, namely, B− → K−π+π−, K0π−π0, 0 → K−π+π0, K0π+π− and K0π0π0. They are dominated by b → s penguin transition and consist of three decay processes: (i) the current-induced process, 〈B → ππ〉 × 〈0 → K〉, (ii) the transition processes, 〈B → π〉 × 〈0 → πK〉, and 〈B → K〉 × 〈0 → ππ〉, and (iii) the annihilation process 〈B → 0〉 × 〈0 → Kππ〉. The factorizable amplitudes for B− → K−π+π−, K0π−π0, B0 → K−π+π0, K0π+π− and π0π0 are given in Eqs. (A10-A14), respectively. All five channels have the three-body matrix element 〈ππ|(q̄b)V−A|B〉 which has the similar expression as Eqs. (2.3) and (2.4) except that the pole B∗s is replaced by B ∗ and the kaon is replaced by the pion. However, there are additional resonant contributions to this three-body matrix element due to the intermediate vector ρ and scalar f0 mesons 〈π+(p2)π−(p3)|(ūb)V −A |B −〉R = →π+π− m2ρi − s23 − imρiΓρi ε∗ · (p2 − p3)〈ρ0i |(ūb)V −A |B gf0i→π m2f0i − s23 − imf0iΓf0i 〈f0i|(ūb)V −A |B −〉, (3.1) where ρi denote generic ρ-type vector mesons, e.g. ρ = ρ(770), ρ(1450), ρ(1700), · · ·. Applying Eqs. (B1) and (B6) we are led to 〈π+(p2)π−(p3)|(ūb)V−A|B−〉R 〈K−(p1)|(s̄u)V −A |0〉 →π+π− − s23 − imρiΓρi (s12 − s13) (mB +mρi)A mB +mρi (s12 + s13 − 3m2π)− 2mρi [A 2)−ABρi0 (q f0i→π m2f0i − s23 − imf0iΓf0i (m2B −m2f0i)F 2). (3.2) Likewise, the 3-body matrix element 〈K−π+|(s̄b) |B0〉 appearing in B0 → K−π+π0 also receives the following resonant contributions 〈K−(p1)π+(p2)|(s̄b)V −A |B 0〉R = →K−π+ − s12 − imK∗ ε∗ · (p1 − p2)〈K i |(s̄b)V −A |B (3.3) with K∗i = K ∗(892),K∗(1410),K∗(1680), · · ·. For the two-body matrix elements 〈π+K−|(s̄d)V−A|0〉, 〈π+π−|(ūu)V−A|0〉 and 〈π+π−|s̄s|0〉, we note that 〈K−(p1)π+(p2)|(s̄d)V −A |0〉 = 〈π +(p2)|(s̄d)V −A |K +(−p1)〉 = (p1 − p2)µFKπ1 (s12) m2K −m2π (p1 + p2)µ −FKπ1 (s12) + FKπ0 (s12) , (3.4) where we have taken into account the sign flip arising from interchanging the operators s ↔ d. Hence, 〈K−(p1)π+(p2)|(s̄d)V−A|0〉〈π−(p3)|(d̄b)V−A|B−〉 = FBπ1 (s12)F 1 (s12) s23 − s13 − (m2B −m2π)(m2K −m2π) + FBπ0 (s12)F 0 (s12) (m2B −m2π)(m2K −m2π) . (3.5) However, the form factor F1 also receives resonant contributions − s12 − imK∗ − s12 − imK∗ (p1 − p2)µ , (3.6) K∗πK = 〈K −(p1)π +(p2)|K∗〉 = gK ∗→πK ε∗ · (p1 − p2), (3.7) where K∗0 i = K 0 (1430), · · ·. Hence, the resonant contributions to the form factor FKπ1 are FKπ1,R (s) = − s− imK∗ − s− imK∗ . (3.8) In principle, the weak vector form factor F π +π− defined by 〈π+(pπ+)π−(pπ−)|ūγµu|0〉 = (pπ+ − pπ−)µF π +π− , (3.9) can be related to the time-like pion electromagnetic form factors. However, unlike the kaon case, the time-like e.m. form factors of the pions are not well measured enough allowing us to determine the resonant and nonresonant parts. Therefore, we shall only consider the resonant part which has the expression F ππR (s) = mρifρig ρi→ππ m2ρi − s− imρiΓρi . (3.10) Following Eq. (2.18), the relevant matrix elements of scalar densities read 〈π+(p2)π−(p3)|s̄s|0〉 = mf0i f̄ gf0i→π m2f0i − s23 − imf0iΓf0i + 〈π+(p2)π−(p3)|s̄s|0〉NR, (3.11) 〈K−(p1)π+(p2)|s̄d|0〉 = →K−π+ − s12 − imK∗ + 〈K−(p1)π+(p2)|s̄d|0〉NR. (3.12) Note that for the scalar meson, the decay constants fS and f̄S are defined in Eq. (B1) and they are related via Eq. (B2). The nonresonant contribution 〈π+(p2)π−(p3)|s̄s|0〉NR vanishes under the OZI rule, while under SU(3) symmetry5 〈K−(p1)π+(p2)|s̄d|0〉NR = 〈K+(p1)K−(p2)|s̄s|0〉NR = fNRs (s12), (3.13) with the expression of fNRs given in Eq. (2.18). It is known that in the narrow width approximation, the 3-body decay rate obeys the factoriza- tion relation Γ(B → RP → P1P2P ) = Γ(B → RP )B(R→ P1P2), (3.14) with R being a resonance. This means that the amplitudes A(B → RP → P1P2P ) and A(B → RP ) should have the same expressions apart from some factors. Hence, using the known results for quasi-two-body decay amplitude A(B → RP ), one can have a cross check on the three-body decay amplitude of B → RP → P1P2P . For example, from Eq. (A12) we obtain the factorizable amplitude A(B 0 → K∗00 (1430)π0;K∗00 (1430) → K−π+) as 〈K−(p1)π+(p2)π0(p3)|Tp|B 0〉K∗0 (1430) = (1430)→K−π+ − s12 − imK∗ −ap4 + r 10 − r FBπ0 (m )(m2B −m2π) a2δpu + (a9 − a7) B −m2K∗ , (3.15) where χ (µ) = 2m2K∗ mb(µ)(ms(µ)−mq(µ)) . (3.16) The expression inside {· · ·} is indeed the amplitude of B0 → K∗00 (1430)π0 given in Eq. (A6) of [48]. The strong coupling constants such as gρ→π +π− and gf0(980)→π +π− are determined from the measured partial widths through the relations 8πm2S g2S→P1P2 , ΓV = 4πm2V g2V→P1P2 , (3.17) for scalar and vector mesons, respectively, where pc is the c.m. momentum. The numerical results +π− = 6.0, gK ∗→K+π− = 4.59, gf0(980)→π +π− = 1.33+0.29−0.26 GeV, g →K+π− = 3.84GeV. (3.18) In determining the coupling of f0 → π+π−, we have used the partial width Γ(f0(980) → π+π−) = (34.2+13.9+8.8−11.8−2.5)MeV (3.19) 5 The matrix elements of scalar densities can be generally decomposed into D-, F - and S(singlet)-type components. Assuming that the singlet component is OZI suppressed, SU(3) symmetry leads to, for example, the relation 〈Kπ|s̄q|0〉NR = 〈KK̄|s̄s|0〉NR. measured by Belle [56]. The momentum dependence of the weak form factor FKπ(q2) is parametrized as FKπ(q2) = FKπ(0) 1− q2/Λχ2 + iΓR/Λχ , (3.20) where Λχ ≈ 830 MeV is the chiral-symmetry breaking scale [57] and ΓR is the width of the relevant resonance, which is taken to be 200 MeV [38]. The results of the calculation are summarized in Tables VIII-XII. We see that except for f0(980)K, the predicted rates for K ∗π, K∗0 (1430)π and ρK are smaller than the data. Indeed, the predictions based on QCD factorization for these decays are also generally smaller than experiment by a factor of 2∼5. This will be discussed in more details in Sec. VI. TABLE VIII: Branching ratios (in units of 10−6) of resonant and nonresonant (NR) contributions to B− → K−π+π−. For theoretical errors, see Table III. Decay mode BaBar [6] Belle [7] Theory π− 9.04± 0.77 ± 0.53+0.21−0.37 6.45 ± 0.43 ± 0.48 +0.25 −0.35 3.0 +0.0+0.8+0.0 −0.0−0.7−0.0 0 (1430)π − 34.4± 1.7 ± 1.8+0.1−1.4 32.0 ± 1.0± 2.4 −1.9 10.5 +0.0+3.2+0.0 −0.0−2.7−0.1 ρ0K− 5.08± 0.78 ± 0.39+0.22−0.66 3.89 ± 0.47 ± 0.29 +0.32 −0.29 1.3 +0.0+1.9+0.1 −0.0−0.7−0.1 f0(980)K − 9.30± 0.98 ± 0.51+0.27−0.72 8.78 ± 0.82 ± 0.65 +0.55 −1.64 7.7 +0.0+0.4+0.1 −0.0−0.8−0.1 NR 2.87± 0.65 ± 0.43+0.63−0.25 16.9 ± 1.3± 1.3 −0.9 18.7 +0.5+11.0+0.2 −0.6− 6.3−0.2 Total 64.4± 2.5 ± 4.6 48.8 ± 1.1± 3.6 45.0+0.3+16.4+0.1−0.4−10.5−0.1 TABLE IX: Same as Table VIII except for the decay B− → K0π−π0. Decay mode Theory Decay mode Theory K∗−π0 1.5+0.0+0.3+0.2−0.0−0.3−0.2 K π− 1.5+0.0+0.4+0.0−0.0−0.3−0.0 K∗−0 (1430)π 0 5.5+0.0+1.6+0.1−0.0−1.4−0.1 K 0 (1430)π − 5.2+0.0+1.6+0.0−0.0−1.4−0.0 1.3+0.0+3.0+0.0−0.0−0.9−0.0 NR 10.0 +0.2+7.1+0.0 −0.2−3.7−0.0 Total 27.0+0.3+15.4+0.2−0.2− 8.8−0.2 While Belle has found a sizable fraction of order (35 ∼ 40)% for the nonresonant signal in K−π+π− and K π+π− modes (see Table I), BaBar reported a small fraction of order 4.5% in K−π+π−. The huge disparity between BaBar and Belle is ascribed to the different parameteri- zations adopted by both groups. BaBar [6] used the LASS parametrization to describe the Kπ S-wave and the nonresonant component by a single amplitude suggested by the LASS collaboration to describe the scalar amplitude in elastic Kπ scattering. As commented in [7], while this approach is experimentally motivated, the use of the LASS parametrization is limited to the elastic region of M(Kπ) <∼ 2.0 GeV, and an additional amplitude is still required for a satisfactory description of the data. In our calculations we have taken into account the nonresonant contributions to the TABLE X: Same as Table VIII except for the decay B 0 → K0π+π−. Decay mode Belle [13] Theory K∗−π+ 5.6± 0.7± 0.5+0.4−0.3 2.1 +0.0+0.5+0.3 −0.0−0.5−0.3 K∗−0 (1430)π + 30.8± 2.4 ± 2.4+0.8−3.0 10.1 +0.0+2.9+0.1 −0.0−2.5−0.2 6.1± 1.0± 0.5+1.0−1.1 2.0 +0.0+1.9+0.1 −0.0−0.9−0.1 f0(980)K 7.6± 1.7± 0.7+0.5−0.7 7.7 +0.0+0.4+0.0 −0.0−0.7−0.0 NR 19.9± 2.5 ± 1.6+0.7−1.2 15.6 +0.1+8.3+0.0 −0.1−4.9−0.0 Total 47.5± 2.4 ± 3.7 42.0+0.3+15.7+0.0−0.2−10.8−0.0 TABLE XI: Branching ratios (in units of 10−6) of resonant and nonresonant (NR) contributions 0 → K−π+π0. Note that the branching ratios for K∗−π+ and K∗0π0 given in [14] and [15] are their absolute ones. We have converted them into the product branching ratios, namely, B(B → Rh)× B(R→ hh). For theoretical errors, see Table III. Decay mode BaBar [14] Belle [15] Theory K∗−π+ 3.6± 0.8± 0.5 4.9+1.5+0.5+0.8−1.5−0.3−0.3 1.0 +0.0+0.3+0.1 −0.0−0.3−0.1 π0 2.0± 0.6± 0.3 < 2.3 1.0+0.0+0.3+0.2−0.0−0.2−0.1 K∗−0 (1430)π + 11.2± 1.5 ± 3.5 5.1± 1.5+0.6−0.7 5.0 +0.0+1.5+0.1 −0.0−1.3−0.1 0 (1430)π 0 7.9± 1.5± 2.7 6.1+1.6+0.5−1.5−0.6 4.2 +0.0+1.4+0.0 −0.0−1.2−0.0 ρ+K− 8.6± 1.4± 1.0 15.1+3.4+1.4+2.0−3.3−1.5−2.1 2.5 +0.0+3.6+0.2 −0.0−1.4−0.2 NR < 4.6 5.7+2.7+0.5−2.5−0.4 < 9.4 9.6 +0.3+6.6+0.0 −0.2−3.5−0.0 Total 34.9± 2.1 ± 3.9 36.6+4.2−4.1 ± 3.0 28.9 +0.2+16.1+0.2 −0.2− 9.4−0.2 two-body matrix elements of scalar densities, 〈Kπ|s̄q|0〉. Recall that a large nonresonant contribu- tion from 〈KK|s̄s|0〉 is needed in order to explain the observed decay rates of B0 → KSKSKS and B− → K−KSKS . From Tables VIII-XII we see that our predicted nonresonant rates are in agree- ment with the Belle measurements. The reason why the nonresonant fraction is as large as 90% in KKK decays, but becomes only (35 ∼ 40)% in Kππ channels (see Table I) can be explained as follows. Under SU(3) flavor symmetry, we have the relation 〈Kπ|s̄q|0〉NR = 〈KK̄|s̄s|0〉NR. Hence, the nonresonant rates in the K−π+π− and K π+π− modes should be similar to that in K+K−K or K+K−K−. Since theKKK channel receives resonant contributions only from φ and f0i mesons, while K∗i ,K 0i, ρi, f0i resonances contribute to Kππ modes, this explains why the nonresonant frac- tion is of order 90% in the former and becomes of order 40% in the latter. Note that the predicted nonresonant contribution in the K−π+π0 mode is larger than the BaBar’s upper bound and barely consistent with the Belle limit. It is conceivable that the SU(3) breaking effect in 〈Kπ|s̄q|0〉NR may lead to a result consistent with the Belle limit. It is interesting to notice that, based on a simple fragmentation model and SU(3) symmetry, Gronau and Rosner [55] found the relations Γ(B− → K+K−K−)NR = 2Γ(B 0 → K+K−K0)NR = 2Γ(B− → K−π+π−)NR TABLE XII: Same as Table VIII except for the decay B 0 → K0π0π0. Decay mode f0(980)K 0 (1430)π 0 NR Total Theory 3.8+0.0+2.0+0.0−0.0−0.4−0.0 0.55 +0.00+0.16+0.00 −0.00−0.13−0.00 2.3 +0.0+0.8+0.0 −0.0−0.6−0.0 5.3 +0.0+1.8+0.0 −0.0−1.1−0.0 12.9 +0.0+4.0+0.1 −0.0−3.0−0.1 TABLE XIII: Branching ratios, mixing-induced and direct CP asymmetries for B 0 → KSπ+π− decays. Results for (KLππ)CP± are identical to those for (KSππ)CP∓. For theoretical errors, see Table III. Final state Branching ratio +π−)CP+ 13.52 +0.02+4.03+0.01 −0.03−3.06−0.01 +π−)CP− 7.45 +0.10+3.79+0.02 −0.08−2.32−0.02 f+ 0.65 +0.00+0.03+0.00 −0.00−0.04−0.00 Final state sin 2βeff +π−)CP+ 0.693 +0.000+0.003+0.003 −0.000−0.002−0.014 +π−)full 0.718 +0.001+0.017+0.008 −0.001−0.007−0.018 Final state Af (%) +π−)CP+ 4.27 +0.00+0.19+0.28 −0.00−0.12−0.35 +π−)full 4.94 +0.03+0.03+0.32 −0.02−0.05−0.40 = 2Γ(B 0 → K0π+π−)NR = 4Γ(B 0 → K−π+π0)NR. (3.21) Again, a large nonresonant background in K−π+π− and K π+π− is favored by this model. Although the B 0 → KSπ0π0 rate has not been measured, its time-dependent CP asymmetries have been studied by BaBar [58] with the results sin 2βeff = −0.72± 0.71 ± 0.08, ACP = −0.23 ± 0.52 ± 0.13 . (3.22) Note that this mode is a CP-even eigenstate. We found that its branching ratio is not so small, of order 6 × 10−6, in spite of the presence of two neutral pions in the final state (see Table XII). Theoretically, we obtain sin 2βeff = 0.729 +0.000+0.001+0.009 −0.000−0.001−0.020 , ACP = 0.28+0.09+0.07+0.02−0.06−0.06−0.02 %. (3.23) Finally, we consider the mode KSπ +π− which is an admixture of CP-even and CP-odd compo- nents. Results for the decay rates and CP asymmetries are displayed in Table XIII. We see that the effective sin 2β is of order 0.718 and direct CP asymmetry of order 4.9% for KSπ IV. B → KKπ DECAYS We now turn to the three-body decay modes dominated by b → u tree and b → d penguin transitions, namely, KKπ and πππ. We first consider the decay B− → K+K−π− whose factorizable TABLE XIV: Same as Table VIII except for the decay B− → K+K−π−. Decay mode f0(980)π − K∗0K− K∗00 (1430)K − NR Total Theory 0.50+0.00+0.06+0.02−0.00−0.04−0.02 0.23 +0.00+0.04+0.02 −0.00−0.04−0.02 0.82 +0.00+0.18+0.09 −0.00−0.16−0.08 1.8 +0.5+0.4+0.2 −0.5−0.2−0.2 4.0 +0.5+0.7+0.3 −0.6−0.5−0.3 Expt. < 6.3 (BaBar)[59] < 13 (Belle) [11] TABLE XV: Same as Table VIII except for B− → π+π−π−. The nonresonant background is used as an input to fix the parameter α defined in Eq. (2.8). Decay mode BaBar [5] Theory ρ0π− 8.8 ± 1.0± 0.6+0.1−0.7 7.7 +0.0+1.7+0.3 −0.0−1.6−0.2 f0(980)π − 1.2 ± 0.6± 0.1± 0.4 < 3.0 0.39+0.00+0.01+0.03−0.00−0.01−0.02 NR 2.3 ± 0.9± 0.3± 0.4 < 4.6 input Total 16.2 ± 1.2± 0.9 12.0+1.1+2.0+0.4−1.2−1.8−0.3 amplitude is given by Eq. (A9). Note that we have included the matrix element 〈K+K−|d̄d|0〉. Although its nonresonant contribution vanishes as K+ and K− do not contain the valence d or d̄ quark, this matrix element does receive a contribution from the scalar f0 pole 〈K+(p2)K−(p3)|d̄d|0〉R = mf0i f̄ gf0i→π m2f0i − s23 − imf0iΓf0i , (4.1) where 〈f0|d̄d|0〉 = mf0 f̄df0 . In the 2-quark model for f0(980), f̄ f0(980) = f̄f0(980) sin θ/ 2. Also note that the matrix element 〈K−(p3)|(s̄b)V −A|B−〉〈π−(p1)K+(p2)|(d̄s)V−A|0〉 has a similar expression as Eq. (3.5) except for a sign difference 〈K−(p3)|(s̄b)V−A|B−〉〈π−(p1)K+(p2)|(d̄s)V−A|0〉 = −FBK1 (s12)FKπ1 (s12) s23 − s13 − (m2B −m2K)(m2K −m2π) −FBK0 (s12)FKπ0 (s12) (m2B −m2K)(m2K −m2π) . (4.2) As in Eq. (3.8), the form factor FKπ1 receives a resonant contribution for the K ∗ pole. The nonresonant and various resonant contributions to B− → K+K−π− are shown in Table XVI. The predicted total rate is consistent with upper limits set by BaBar and Belle. V. B → πππ DECAYS The factorizable amplitudes of the tree-dominated decay B− → π+π−π− and B0 → π+π−π0 are given by Eqs. (A15) and (A16), respectively. We see that the former is dominated by the ρ0 TABLE XVI: Same as Table VIII except for the decay B 0 → π+π−π0. Decay mode ρ+π− ρ−π+ ρ0π0 f0(980)π 0 NR Total Theory 8.5 +0.0+1.1+0.2 −0.0−1.0−0.1 +0.0+4.0+0.3 −0.0−3.5−0.3 +0.0+0.3+0.0 −0.0−0.2−0.0 0.010 +0.000+0.003+0.000 −0.000−0.002−0.000 +0.02+0.01+0.00 −0.02−0.01−0.00 +0.0+5.6+0.2 −0.0−5.0−0.2 pole, while the latter receives ρ± and ρ0 contributions. As a consequence, the π+π−π0 mode has a rate larger than π+π−π− even though the former involves a π0 in the final state. The π+π−π− mode receives nonresonant contributions mostly from the b → u transition as the nonresonant contribution in the matrix element 〈π+π−|d̄d|0〉 is suppressed by the smallness of penguin Wilson coefficients a6 and a8. Therefore, the measurement of the nonresonant contribution in this decay can be used to constrain the nonresonant parameter α in Eq. (2.8) VI. DIRECT CP ASYMMETRIES Direct CP asymmetries for various charmless three-body B decays are collected in Table XVII. Mixing-induced and direct CP asymmetries in B0 → K+K−KS,L and KSKSKS,L decays are al- ready shown in Table V. It appears that direct CP violation is sizable in K+K−K− and K+K−π− modes. The major uncertainty with direct CP violation comes from the strong phases which are needed to induce partial rate CP asymmetries. In this work, the strong phases arise from the effective Wilson coefficients a i listed in (A3) and from the Breit-Wigner formalism for resonances. Since direct CP violation in charmless two-body B decays can be significantly affected by final-state rescattering [60], it is natural to extend the study of final-state rescattering effects to the case of three-body B decays. We will leave this to a future investigation. TABLE XVII: Direct CP asymmetries (in %) for various charmless three-body B decays. For theoretical errors, see Table III. Experimental results are taken from [50]. Final state BaBar Belle Theory K+K−K− −2± 3± 2 −10.4+1.7+0.9+0.9−1.3−1.0−0.8 K−KSKS −4± 11± 2 −3.9+0.0+0.6+0.3−0.0−0.8−0.3 K+K−π− 0± 10± 3 17.5+1.9+2.2+0.0−3.8−3.4−0.2 K−π+π− −1.3± 3.7± 1.1 4.9± 2.6± 2.0 −3.3+0.7+0.4+0.3−0.5−0.4−0.2 K−π+π0 7± 11± 1 6.3+0.6+1.4+0.5−0.7−1.4−0.5 π+π− 4.9+0.0+0.0+0.3−0.0−0.1−0.4 π0π0 −23± 52± 13 −17± 24± 6 0.28+0.09+0.07+0.02−0.06−0.06−0.02 π−π0 0.4+0.0+0.4+0.0−0.0−0.4−0.0 π+π−π− −1± 8± 3 4.4+0.8+1.2+0.0−0.6−0.9−0.2 π+π−π0 −3.0+0.1+0.2+0.3−0.1−0.3−0.2 VII. TWO-BODY B → V P AND B → SP DECAYS Thus far we have considered the branching ratio products B(B → Rh1)B(R → h2h3) with the resonance R being a vector meson or a scalar meson. Using the experimental information on B(R→ h2h3) [4] B(K∗0 → K+π−) = B(K∗+ → K0π+) = 2B(K∗+ → K+π0) = B(K∗00 (1430) → K+π−) = 2B(K∗+0 (1430) → K +π0) = (0.93 ± 0.10), B(φ→ K+K−) = 0.492 ± 0.006 . (7.1) one can extract the branching ratios of B → V P and B → SP . The results are summarized in Table XVIII. Two remarks about the experimental branching ratios are in order: (i) The BaBar results for the branching ratios of B 0 → K∗−π+, K∗0π0, K∗−0 (1430)π+ are inferred from the three-body decays 0 → K0π+π− (see Table XI) and Belle results are taken from B0 → K−π+π0 (see Table X). (ii) Branching ratios of B 0 → φK0 shown in Table XVIII are not inferred from the Dalitz plot analysis of B → KKK decays. For comparison, the predictions of the QCD factorization approach for B → V P [61] and B → SP [48] are also exhibited in Table XVIII. In order to compare theory with experiment for B → f0(980)K → π+π−K, we need an input for B(f0(980) → π+π−). To do this, we shall use the BES measurement [45] Γ(f0(980) → ππ) Γ(f0(980) → ππ) + Γ(f0(980) → KK) = 0.75+0.11−0.13 . (7.2) Assuming that the dominance of the f0(980) width by ππ and KK and applying isospin relation, we obtain B(f0(980) → π+π−) = 0.50+0.07−0.09 , B(f0(980) → K +K−) = 0.125+0.018−0.022 . (7.3) At first sight, it appears that the ratio defined by R ≡ B(f0(980) → K B(f0(980) → π+π−) = 0.25± 0.06 (7.4) is not consistent with the value of 0.69 ± 0.32 inferred from the BaBar data (see Tables VI and VIII) Γ(B− → f0(980)K−; f0(980) → K+K−) Γ(B− → f0(980)K−; f0(980) → π+π−) 6.5± 2.5 ± 1.6 9.3± 1.0+0.6−0.9 , (7.5) where we have applied the narrow width approximation Eq. (3.14). The above-mentioned discrepancy can be resolved by noting that the factorization relation Eq. (3.14) for the resonant three-body decay is applicable only when the two-body decays B → RP and R → P1P2 are kinematically allowed and the resonance is narrow, the so-called narrow width approximation. However, as the decay f0(980) → K+K− is kinematically barely or even not allowed, the off resonance peak effect of the intermediate resonant state will become important. TABLE XVIII: Branching ratios of quasi-two-body decays B → V P and B → SP obtained from the studies of three-body decays based on the factorization approach. Unless specified, the experimental results are obtained from the 3-body Dalitz plot analyses given in previous Tables. Theoretical uncertainties have been added in quadrature. QCD factorization (QCDF) predictions taken from [61] for V P modes and from [48] for SP channels are shown here for comparison. Decay mode BaBar Belle QCDF This work φK0 8.4+1.5−1.3 ± 0.5 a 9.0 −1.8 ± 0.7 b 4.1 +0.4+1.7+1.8+10.6 −0.4−1.6−1.9− 3.0 5.3 φK− 8.4± 0.7 ± 0.7 9.60 ± 0.92+1.05−0.84 4.5 +0.5+1.8+1.9+11.8 −0.4−1.7−2.1− 3.3 5.9 π− 13.5 ± 1.2+0.8−0.9 9.8± 0.9 −1.2 3.6 +0.4+1.5+1.2+7.7 −0.3−1.4−1.2−2.3 4.4 π0 3.0± 0.9 ± 0.5 < 3.5 0.7+0.1+0.5+0.3+2.6−0.1−0.4−0.3−0.5 1.5 K∗−π+ 11.0 ± 1.5± 0.7 8.4± 1.1+0.9−0.8 3.3 +1.4+1.3+0.8+6.2 −1.2−1.2−0.8−1.6 3.1 K∗−π0 6.9 ± 2.0± 1.3 b 3.3+1.1+1.0+0.6+4.4−1.0−0.9−0.6−1.4 2.2 K∗0K− 0.30+0.11+0.12+0.09+0.57−0.09−0.10−0.09−0.19 0.35 +0.06 −0.06 ρ0K− 5.1± 0.8+0.6−0.9 3.89 ± 0.47 +0.43 −0.41 2.6 +0.9+3.1+0.8+4.3 −0.9−1.4−0.6−1.2 1.3 4.9± 0.8 ± 0.9 6.1± 1.0 ± 1.1 4.6+0.5+4.0+0.7+6.1−0.5−2.1−0.7−2.1 2.0 ρ+K− 8.6± 1.4 ± 1.0 15.1+3.4+2.4−3.3−2.6 7.4 +1.8+7.1+1.2+10.7 −1.9−3.6−1.1− 3.5 2.5 8.0+1.4−1.3 ± 0.5 b 5.8 +0.6+7.0+1.5+10.3 −0.6−3.3−1.3− 3.2 1.3 ρ0π− 8.8± 1.0+0.6−0.9 8.0 −2.0 ± 0.7 b 11.9 +6.3+3.6+2.5+1.3 −5.0−3.1−1.2−1.1 7.7 ρ−π+ 21.2+10.3+8.7+1.3+2.0− 8.4−7.2−2.3−1.6 15.5 ρ+π− 15.4+8.0+5.5+0.7+1.9−6.4−4.7−1.3−1.3 8.5 ρ0π0 1.4± 0.6 ± 0.3 3.1+0.9+0.6−0.8−0.8 0.4 +0.2+0.2+0.9+0.5 −0.2−0.1−0.3−0.3 1.0 f0(980)K 0; f0 → π+π− 5.5± 0.7 ± 0.6 7.6± 1.7+0.8−0.9 6.7 +0.1+2.1+2.3 −0.1−1.5−1.1 c 7.7+0.4−0.7 f0(980)K −; f0 → π+π− 9.3± 1.0+0.6−0.9 8.8± 0.8 −1.8 7.8 +0.2+2.3+2.7 −0.2−1.6−1.2 c 7.7+0.4−0.8 f0(980)K 0; f0 → K+K− 5.3± 2.2 5.8+0.1−0.5 f0(980)K −; f0 → K+K− 6.5± 2.5 ± 1.6 < 2.9 7.0+0.4−0.7 f0(980)π −; f0 → π+π− < 3.0 0.5+0.0+0.2+0.1−0.0−0.1−0.0 c 0.39 +0.03 −0.02 f0(980)π −; f0 → K+K− 0.50+0.06−0.04 f0(980)π 0; f0 → π+π− 0.02+0.01+0.02+0.04−0.01−0.00−0.01 c 0.010 +0.003 −0.002 0 (1430)π − 36.6 ± 1.8± 4.7 51.6 ± 1.7+7.0−7.4 11.0 +10.3+7.5+49.9 − 6.0−3.5−10.1 16.9 0 (1430)π 0 12.7 ± 2.4± 4.4 9.8± 2.5 ± 0.9 6.4+5.4+2.2+26.1−3.3−2.1− 5.7 6.8 K∗−0 (1430)π + 36.1 ± 4.8 ± 11.3 49.7 ± 3.8+4.0−6.1 11.3 +9.4+3.7+45.8 −5.8−3.7− 9.9 16.2 K∗−0 (1430)π 0 5.3+4.7+1.6+22.3−2.8−1.7− 4.7 8.9 K∗00 (1430)K − < 2.2 b 1.3+0.3−0.3 aFrom the Dalitz plot analysis of B0 → K+K−K0 decay measured by BaBar (see Table III), we obtain B(B0 → φK0) = (6.2± 0.9)× 10−6. The experimental value of BaBar cited in the Table is obtained from a direct measurement of B0 → φK0. bnot determined directly from the Dalitz plot analysis of three-body decays. cWe have assumed B(f0(980) → π+π−) = 0.50 for the QCDF calculation. Therefore, it is necessary to take into account the finite width effect of the f0(980) which has a width of order 40-100 MeV [4]. In short, one cannot determine the ratio R by applying the narrow width approximation to the three-body decays. That is, one should employ the decays B → Kππ rather than B → KKK to extract the experimental branching ratio for B → f0(980)K provided B(f0(980) → ππ) is available. We now compare the present work for B → V P and B → SP with the approach of QCD factorization [34, 48]. In this work, our calculation of 3-body B decays is similar to the simple generalized factorization approach [62, 63] by assuming a set of universal and process independent effective Wilson coefficients a i with p = u, c in Eq. (A3). In QCDF, the calculation of a rather sophisticated. They are basically the Wilson coefficients in conjunction with short-distance nonfactorizable corrections such as vertex corrections and hard spectator interactions. In general, they have the expressions [34, 61] i (M1M2) = Ni(M2) + Vi(M2) + Hi(M1M2) i (M2), (7.6) where i = 1, · · · , 10, the upper (lower) signs apply when i is odd (even), ci are the Wilson coefficients, CF = (N c − 1)/(2Nc) with Nc = 3, M2 is the emitted meson and M1 shares the same spectator quark with the B meson. The quantities Vi(M2) account for vertex corrections, Hi(M1M2) for hard spectator interactions with a hard gluon exchange between the emitted meson and the spectator quark of the B meson and Pi(M2) for penguin contractions. Hence, the effective Wilson coefficients i (M1M2) depend on the nature ofM1 andM2; that is, they are process dependent. Moreover, they depend on the order of the argument, namely, a i (M2M1) 6= a i (M1M2) in general. In the above equation, Ni(M2) vanishes for i = 6, 8 and M2 = V , and equals to unity otherwise. For three- body decays, in principle one should also compute the vertex, gluon and hard spectator-interaction corrections. Of course, these corrections for the three-body case will be more complicated than the two-body decay one. One possible improvement of the present work is to utilize the QCDF results for the effective parameters a i (M1M2) in the vicinity of the resonance region. We next proceed to the comparison of numerical results. For φK, K∗π and K∗K modes, the QCDF and the present work have similar predictions. For the ρ meson in the final states, QCDF predicts slightly small ρK and too large ρπ compared to experiment. 6 In contrast, in the present work we obtain reasonable ρπ but too small ρK. This is ascribed to the form factor A 0 (0) = 0.37 ± 0.06 employed in [61] that is too large compared to ours ABρ0 (0) = 0.28 ± 0.03 (see Table XIX). Recall that the recent QCD sum rule calculation also yields a smaller one A 0 (0) = 0.30 +0.07 −0.03 [64]. For B → f0(980)K and B → f0(980)π, QCDF [48] and this work are in agreement with experiment. The large rate of the f0(980)K mode is ascribed to the large f0(980) decay constant, f̄f0(980) ≈ 460 MeV at the renormalization scale µ = 2.1 GeV [48]. In contrast, the predicted 0 (1430)π − and K∗−0 (1430)π + are too small compared to the data. The fact that QCDF leads to too small rates for φK, K∗π, ρK and K∗0 (1430)π may imply the importance of power corrections 6 Recall that the world average of the branching ratio of B0 → ρ±π∓ is (24.0±2.5)×10−6 [50], while QCDF predicts it to be ∼ 36.6× 10−6 [61]. due to the non-vanishing ρA and ρH parameters arising from weak annihilation and hard spectator interactions, respectively, which are used to parametrize the endpoint divergences, or due to possible final-state rescattering effects from charm intermediate states [60]. However, this is beyond the scope of the present work. VIII. CONCLUSIONS In this work, an exploratory study of charmless 3-body decays of B mesons is presented using a simple model based on the framework of the factorization approach. The 3-body decay process consists of resonant contributions and the nonresonant signal. Since factorization has not been proved for three-body B decays, we shall work in the phenomenological factorization model rather than in the established theories such as QCD factorization. That is, we start with the simple idea of factorization and see if it works for three-body decays. Our main results are as follows: • If heavy meson chiral perturbation theory (HMChPT) is applied to the three-body matrix elements for B → P1P2 transitions and assumed to be valid over the whole kinematic region, then the predicted decay rates for nonresonant 3-body B decays will be too large and even exceed the measured total rate. This can be understood because chiral symmetry has been applied beyond its region of validity. We assume the momentum dependence of nonresonant amplitudes in the exponential form e−αNRpB ·(pi+pj) so that the HMChPT results are recovered in the soft meson limit pi, pj → 0. The parameter αNR can be fixed from the tree-dominated decay B− → π+π−π−. • Besides the nonresonant contributions arising from B → P1P2 transitions, we have identified another large source of the nonresonant background in the matrix elements of scalar densi- ties, e.g. 〈KK|s̄s|0〉 which can be constrained from the KSKSKS (or K−KSKS) mode in conjunction with the mass spectrum in the decay B 0 → K+K−K0 . • All KKK modes are dominated by the nonresonant background. The predicted branching ratios of K+K−KS(L), K +K−K− and K−KSKS modes are consistent with the data within the theoretical and experimental errors. • Although the penguin-dominated B0 → K+K−KS decay is subject to a potentially signifi- cant tree pollution, its effective sin 2β is very similar to that of the KSKSKS mode. However, direct CP asymmetry of the former, being of order −4%, is more prominent than the latter, • The role played by the unknown scalar resonance X0(1550) in the decay B− → K+K−K− should be clarified in order to see if it behaves in the same way as in the K+K−K mode. • Applying SU(3) symmetry to relate the nonresonant component in the matrix element 〈Kπ|s̄q|0〉 to that in 〈KK|s̄s|0〉, we found sizable nonresonant contributions in K−π+π− and K π+π− modes, in agreement with the Belle measurements but larger than the BaBar results. In particular, the predicted nonresonant contribution in the K−π+π0 mode is consis- tent with the Belle limit and larger than the BaBar’s upper bound. It will be interesting to have a refined measurement of the nonresonant contribution to this mode to test our model. • The π+π−π0 mode is predicted to have a rate larger than π+π−π− even though the former involves a π0 in the final state. This is because the latter is dominated by the ρ0 pole, while the former receives ρ± and ρ0 resonant contributions. • Among the 3-body decays we have studied, the decay B− → K+K−π− dominated by b→ u tree transition and b→ d penguin transition has the smallest branching ratio of order 4×10−6. It is consistent with the current bound set by BaBar and Belle. • Decay rates and time-dependent CP asymmetries in the decays KSπ0π0, a purely CP -even state, and KSπ +π−, an admixture of CP -even and CP -odd components, are studied. The corresponding mixing-induced CP violation is found to be of order 0.729 and 0.718, respec- tively. • Since the decay f0(980) → K+K− is kinematically barely or even not allowed, it is crucial to take into account the finite width effect of the f0(980) when computing the decay B → f0(980)K → KKK. Consequently, one should employ the Dalitz plot analysis of Kππ mode to extract the experimental branching ratio for B → f0(980)K provided B(f0(980) → ππ) is available. The large rate of B → f0(980)K is ascribed to the large f0(980) decay constant, f̄f0(980) ≈ 460 MeV. • The intermediate vector meson contributions to 3-body decays e.g. ρ, φ, K∗ are identified through the vector current, while the scalar meson resonances e.g. f0(980), X0(1550), K 0 (1430) are mainly associated with the scalar density. Their effects are described in terms of the Breit-Wigner formalism. • Based on the factorization approach, we have computed the resonant contributions to 3- body decays and determined the rates for the quasi-two-body decays B → V P and B → SP . The predicted ρπ, f0(980)K and f0(980)π rates are consistent with experiment, while the calculated φK, K∗π, ρK and K∗0 (1430)π are too small compared to the data. • Direct CP asymmetries have been computed for the charmless 3-body B decays. We found sizable direct CP violation in K+K−K− and K+K−π− modes. • In this exploratory work we use the phenomenological factorization model rather than in the established theories based on a heavy quark expansion. Consequently, we don’t have 1/mb power corrections within this model. However, systematic errors due to such model dependent assumptions may be sizable and are not included in the error estimates that we give. Note added: After the paper was submitted for publication, BaBar (arXiv:0708.0367 [hep-ex]) has reported the observation of the decay B+ → K+K−π+ with the branching ratio (5.0± 0.5± 0.5)× 10−6. Our prediction for this mode (see Table XIV) is consistent with experiment. Acknowledgments This research was supported in part by the National Science Council of R.O.C. under Grant Nos. NSC95-2112-M-001-013, NSC95-2112-M-033-013, and by the U.S. DOE contract No. DE- AC02-98CH10886(BNL). http://arxiv.org/abs/0708.0367 APPENDIX A: DECAY AMPLITUDES OF THREE-BODY B DECAYS In this appendix we list the factorizable amplitudes of the 3-body decays B → KKK,KKπ,Kππ, πππ. Under the factorization hypothesis, the decay amplitudes are given by 〈P1P2P3|Heff |B〉 = p=u,c λ(r)p 〈P1P2P3|Tp|B〉, (A1) where λ p ≡ VpbV ∗pr with r = d, s. For KKK and Kππ modes, r = s and for KKπ and πππ channles, r = d. The Hamiltonian Tp has the expression [34] Tp = a1δpu(ūb)V−A ⊗ (s̄u)V−A + a2δpu(s̄b)V−A ⊗ (ūu)V−A + a3(s̄b)V−A ⊗ (q̄q)V−A (q̄b)V−A ⊗ (s̄q)V−A + a5(s̄b)V −A ⊗ (q̄q)V+A −2ap6 (q̄b)S−P ⊗ (s̄q)S+P + a7(s̄b)V−A ⊗ eq(q̄q)V+A −2ap8 (q̄b)S−P ⊗ eq(s̄q)S+P + a9(s̄b)V−A ⊗ eq(q̄q)V−A (q̄b)V−A ⊗ eq(s̄q)V−A, (A2) with (q̄q′)V±A ≡ q̄γµ(1 ± γ5)q′, (q̄q′)S±P ≡ q̄(1 ± γ5)q′ and a summation over q = u, d, s being implied. For the effective Wilson coefficients, we use a1 ≈ 0.99 ± 0.037i, a2 ≈ 0.19 − 0.11i, a3 ≈ −0.002 + 0.004i, a5 ≈ 0.0054 − 0.005i, au4 ≈ −0.03− 0.02i, ac4 ≈ −0.04 − 0.008i, au6 ≈ −0.06− 0.02i, ac6 ≈ −0.06− 0.006i, a7 ≈ 0.54 × 10−4i, au8 ≈ (4.5 − 0.5i) × 10−4, ac8 ≈ (4.4 − 0.3i) × 10−4, (A3) a9 ≈ −0.010 − 0.0002i, au10 ≈ (−58.3 + 86.1i) × 10−5, ac10 ≈ (−60.3 + 88.8i) × 10−5, for typical ai at the renormalization scale µ = mb/2 = 2.1 GeV which we are working on. Various three-body B decay amplitudes are collected below. B → KKK 〈K0K+K−|Tp|B 0〉 = 〈K+K0|(ūb)V−A|B0〉〈K−|(s̄u)V−A|0〉 a1δpu + a 4 + a 10 − (a 6 + a +〈K+K−|(d̄b)V−A|B0〉〈K 0|(s̄d)V−A|0〉 +〈K0|(s̄b)V−A|B0〉〈K+K−|(ūu)V−A|0〉(a2δpu + a3 + a5 + a7 + a9) +〈K0|(s̄b)V−A|B0〉〈K+K−|(d̄d)V−A|0〉 a3 + a5 − (a7 + a9) +〈K0|(s̄b)V−A|B0〉〈K+K−|(s̄s)V−A|0〉 a3 + a 4 + a5 − (a7 + a9 + a +〈K0|s̄b|B0〉〈K+K−|s̄s|0〉(−2ap6 + a +〈K+K−|d̄(1− γ5)b|B0〉〈K 0|s̄(1 + γ5)d|0〉 (−2ap6 + a +〈K+K−K0|(s̄d)V −A|0〉〈0|(d̄b)V−A|B0〉 +〈K+K−K0|s̄γ5d|0〉〈0|d̄γ5b|B0〉(−2ap6 + a 8), (A4) with rPχ = mb(µ)(m2+m1)(µ) 〈K+K−K−|Tp|B−〉 = 〈K+K−|(ūb)V−A|B−〉〈K−|(s̄u)V−A|0〉 a1δpu + a 4 + a 10 − (a 6 + a +〈K−|(s̄b)V−A|B−〉〈K+K−|(ūu)V−A|0〉(a2δpu + a3 + a5 + a7 + a9) +〈K−|(s̄b)V−A|B−〉〈K+K−|(d̄d)V −A|0〉 a3 + a5 − (a7 + a9) +〈K−|(s̄b)V−A|B−〉〈K+K−|(s̄s)V−A|0〉 a3 + a 4 + a5 − (a7 + a9 + a +〈K−|s̄b|B−〉〈K+K−|s̄s|0〉(−2ap6 + a +〈K+K−|ū(1− γ5)b|B0〉〈K−|s̄(1 + γ5)u|0〉 (−2ap6 + a +〈K+K−K−|(s̄u)V−A|0〉〈0|(ūb)V−A|B−〉 +〈K+K−K−|s̄γ5u|0〉〈0|ūγ5b|B−〉(−2ap6 + a 8). (A5) Since there are two identical K− mesons in this decay, one should take into account the identical particle effects. For example, 〈K+K−|(ūb)V−A|B−〉〈K−|(s̄u)V−A|0〉 = 〈K+(p1)K−(p2)|(ūb)V −A|B−〉〈K−(p3)|(s̄u)V−A|0〉 + 〈K+(p1)K−(p3)|(ūb)V −A|B−〉〈K−(p2)|(s̄u)V−A|0〉, and a factor of 1 should be put in the decay rate. 〈K0K0K0|Tp|B 0〉 = 〈K0K0|(d̄b)V−A|B0〉〈K0|(s̄d)V−A|0〉 10 − (a +〈K0|(s̄b)V −A|B0〉〈K0K0|(d̄d)V−A|0〉 a3 + a5 − (a7 + a9) +〈K0|(s̄b)V −A|B0〉〈K0K0|(s̄s)V−A|0〉 a3 + a 4 + a5 − (a7 + a9 + a +〈K0|s̄b|B0〉〈K0K0|s̄s|0〉(−2ap6 + a +〈K0K0K0|(s̄d)V−A|0〉〈0|(d̄b)V−A|B0〉 (a7 + a9 + a +〈K0K0K0|s̄γ5d|0〉〈0|d̄γ5b|B0〉(−2ap6 + a 8). (A7) The second and third terms do not contribute to the purely CP -even decay B 0 → KSKSKS . 〈K−K0K0|Tp|B−〉 = 〈K0K 0|(ūb)V−A|B−〉〈K−|(s̄u)V−A|0〉 a1δpu + a 4 + a 10 − (a 6 + a +〈K0K−|(d̄b)V−A|B−〉〈K0|(s̄d)V−A|0〉 10 − (a +〈K−|(s̄b)V−A|B−〉〈K0K0|(d̄d)V−A|0〉 a3 + a5 − (a7 + a9) +〈K−|(s̄b)V−A|B−〉〈K0K0|(s̄s)V−A|0〉 a3 + a 4 + a5 − (a7 + a9 + a +〈K−|s̄b|B−〉〈K0K0|s̄s|0〉(−2ap6 + a +〈K−K0K0|(s̄u)V −A|0〉〈0|(ūb)V−A|B0〉(a1δpu + ap4 + a +〈K−K0K0|s̄γ5u|0〉〈0|ū(1− γ5)b|B−〉(2ap6 + 2a 8). (A8) The third and fourth terms do not contribute to the decay B− → K−KSKS . B → KKπ 〈π−K+K−|Tp|B−〉 = 〈K+K−|(ūb)V−A|B−〉〈π−|(d̄u)V −A|0〉 a1δpu + a 4 + a 10 − (a 6 + a +〈π−|(d̄b)V−A|B−〉〈K+K−|(ūu)V−A|0〉(a2δpu + a3 + a5 + a7 + a9) +〈π−|d̄b|B−〉〈K+K−|d̄d|0〉(−2ap6 + a +〈π−|(d̄b)V−A|B−〉〈K+K−|(s̄s)V−A|0〉 a3 + a5 − (a7 + a9) +〈K−|(s̄b)V−A|B−〉〈K+π−|(d̄s)V−A|0〉(ap4 − +〈K−|s̄b|B−〉〈K+π−|d̄s|0〉(−2ap6 + a +〈K+K−π−|(d̄u)V−A|0〉〈0|(ūb)V −A|B−〉 a1δpu + a 4 + a +〈K+K−π−|d̄γ5u|0〉〈0|ūγ5b|B−〉(2ap6 − a 8). (A9) B → Kππ 〈K−π+π−|Tp|B−〉 = 〈π+π−|(ūb)V−A|B−〉〈K−|(s̄u)V−A|0〉 a1δpu + a 4 + a 10 − (a 6 + a +〈K−|(s̄b)V−A|B−〉〈π+π−|(ūu)V −A|0〉 a2δpu + (a7 + a9) +〈K−|s̄b|B−〉〈π+π−|s̄s|0〉(−2ap6 + a +〈π−|(d̄b)V−A|B−〉〈K−π+|(s̄d)V−A|0〉(ap4 − +〈π−|d̄b|B−〉〈K−π+|s̄d|0〉(−2ap6 + a +〈K−π+π−|(s̄u)V−A|0〉〈0|(ūb)V−A|B−〉(a1δpu + ap4 + a +〈K−π+π−|s̄γ5u|0〉〈0|ūγ5b|B−〉(2ap6 + 2a 8). (A10) 〈K0π+π−|Tp|B 0〉 = 〈π+π−|(d̄b)V−A|B 0〉〈K0|(s̄d)V −A|0〉 10 − (a +〈K0|(s̄b)V−A|B 0〉〈π+π−|(ūu)V−A|0〉 a2δpu + (a7 + a9) +〈K0|s̄b|B0〉〈π+π−|s̄s|0〉(−2ap6 + a +〈π+|(ūb)V−A|B 0〉〈K0π−|(s̄u)V−A|0〉(a1 + ap4 + a +〈π+|ūb|B0〉〈K0π−|s̄u|0〉(−2ap6 − 2a +〈K0π+π−|(s̄d)V −A|0〉〈0|(d̄b)V−A|B 0〉(a1δpu + ap4 + a +〈K0π+π−|s̄(1 + γ5)d|0〉〈0|d̄γ5b|B 0〉(2ap6 − a 8). (A11) 〈K−π+π0|Tp|B 0〉 = 〈π+π0|(ūb)V−A|B 0〉〈K−|(s̄u)V −A|0〉 a1δpu + a 4 + a 10 − (a 6 + a +〈K−π+|(s̄b)V−A|B 0〉〈π0|(ūu)V−A|0〉 a2δpu + (−a7 + a9) +〈π+|(ūb)V−A|B 0〉〈K−π0|(s̄u)V−A|0〉 [a1δpu + ap4 + a +〈π0|(d̄b)V −A|B 0〉〈K−π+|(s̄d)V −A|0〉(ap4 − +〈π+|ūb|B0〉〈K−π0|s̄u|0〉(−2ap6 − 2a +〈π0|d̄b|B0〉〈K−π+|s̄d|0〉(−2ap6 + a +〈K−π+π0|(s̄d)V−A|0〉〈0|(d̄b)V−A|B 0〉(ap4 − +〈K−π+π0|s̄(1 + γ5)d|0〉〈0|d̄γ5b|B 0〉(2ap6 − a 8). (A12) 〈K0π−π0|Tp|B−〉 = 〈π0π−|(d̄b)V−A|B−〉〈K 0|(s̄d)V −A|0〉 10 − (a +〈K0π−|(s̄b)V−A|B−〉〈π0|(ūu)V−A|0〉 a2δpu + (−a7 + a9) +〈π0|(ūb)V−A|B−〉〈K π−|(s̄u)V−A|0〉 [a1δpu + ap4 + a +〈π−|(d̄b)V−A|B−〉〈K π0|(s̄d)V−A|0〉 +〈π0|ūb|B−〉〈K0π−|s̄u|0〉(−2ap6 − 2a +〈π−|d̄b|B−〉〈K0π0|s̄d|0〉(−2ap6 + a +〈K0π−π0|(s̄u)V −A|0〉〈0|(ūb)V−A|B−〉(a1δpu + ap4 + a +〈K0π−π0|s̄(1 + γ5)u|0〉〈0|ūγ5b|B−〉(2ap6 + 2a 8). (A13) 〈K0π0π0|Tp|B 0〉 = 〈π0π0|(d̄b)V−A|B 0〉〈K0|(s̄d)V −A|0〉 10 − (a +〈K0π0|(s̄b)V−A|B 0〉〈π0|(ūu)V−A|0〉 a2δpu + (−a7 + a9) +〈π0|(d̄b)V−A|B 0〉〈K0π0|(s̄d)V−A|0〉 +〈π0π0|(ūu) |0〉〈K0|(s̄b) a2δpu + 2a3 + 2a5 + (a7 + a9) +〈K0π0|s̄d|0〉〈π0|d̄b|B0〉(−2ap6 + a +〈π0π0|s̄s|0〉〈K0|s̄b|B0〉(−2ap6 + a +〈K0π0π0|(s̄d)V−A|0〉〈0|(d̄b)V−A|B 0〉(ap4 − +〈K0π0π0|s̄(1 + γ5)d|0〉〈0|d̄γ5b|B 0〉(2ap6 − a 8). (A14) B → πππ 〈π−π+π−|Tp|B−〉 = 〈π+π−|(ūb)V −A|B−〉〈π−|(d̄u)V−A|0〉 a1δpu + a 4 + a 10 − (a 6 + a +〈π−|(d̄b)V−A|B−〉〈π+π−|(ūu)V−A|0〉 a2δpu − ap4 + (a7 + a9) + +〈π−|d̄b|B−〉〈π+π−|d̄d|0〉(−2ap6 + a +〈π−π+π−|(d̄u)V−A|0〉〈0|(ūb)V−A|B−〉(a1δpu + ap4 + a +〈π−π+π−|d̄(1 + γ5)u|0〉〈0|ūγ5b|B−〉(2ap6 + 2a 8). (A15) 〈π0π+π−|Tp|B 0〉 = 〈π+π0|(ūb)V−A|B 0〉〈π−|(d̄u)V−A|0〉 a1δpu + a 4 + a 10 − (a 6 + a +〈π+π−|(d̄b)V −A|B 0〉〈π0|(ūu)V−A|0〉 a2δpu − ap4 + (a (a7 + a9) + +〈π+|(ūb)V−A|B 0〉〈π−π0|(d̄u)V−A|0〉 [a1δpu + ap4 + a +〈π0|(d̄b)V−A|B 0〉〈π+π−|(ūu)V−A|0〉 a2δpu − ap4 + (a7 + a9) + +〈π0|d̄b|B0〉〈π+π−|d̄d|0〉(−2ap6 + a (A16) APPENDIX B: DECAY CONSTANTS, FORM FACTORS AND OTHERS In this appendix we collect the numerical values of the decay constants, form factors, CKM matrix elements and quark masses needed for the calculations. We first discuss the decay constants of the pseudoscalar meson P and the scalar meson S defined by 〈P (p)|q̄2γµγ5q1|0〉 = −ifPpµ, 〈S(p)|q̄2γµq1|0〉 = fSpµ, 〈S|q̄2q1|0〉 = mS f̄S , (B1) and 〈V (p, ε)|Vµ|0〉 = fVmV ε∗µ for the vector meson. For the scalar mesons, the vector decay constant fS and the scale-dependent scalar decay constant f̄S are related by equations of motion µSfS = f̄S, with µS = m2(µ)−m1(µ) , (B2) where m2 and m1 are the running current quark masses. The neutral scalar mesons σ, f0 and a cannot be produced via the vector current owing to charge conjugation invariance or conservation of vector current: fσ = ff0 = fa0 = 0. (B3) However, the decay constant f̄S is non-vanishing. In [48] we have applied the QCD sum rules to estimate this quantity. In this work we folow [48] to use f̄f0(980) = 460MeV, f̄K∗0 (1430) = 550MeV, (B4) at µ = 2.1 GeV. As for the decay constants of vector mesons, we use (in units of MeV). fρ = 216, fK∗ = 218, f̄f0(980) = 460, f̄K∗0 = 550. (B5) Form factors for B → P, S transitions are defined by [41] 〈P (p′)|Vµ|B(p)〉 = (p+ p′)µ − m2B −m2P FBP1 (q m2B −m2P 〈S(p′)|Aµ|B(p)〉 = −i (p + p′)µ − m2B −m2S FBS1 (q m2B −m2S 〈V (p′, ε)|Vµ|B(p)〉 = mB +mV ǫµναβε ∗νpαp′βV (q2), 〈V (p′, ε)|Aµ|B(p)〉 = i (mB +mV )ε ε∗ · p mB +mV (p+ p′)µA ε∗ · p 2)−ABV0 (q2)] , (B6) where q = p− p′, F1(0) = F0(0), A3(0) = A0(0), and mP +mV 2)− mP −mV 2), (B7) where Pµ = (p + p ′)µ, qµ = (p − p′)µ. As shown in [65], a factor of (−i) is needed in B → S transition in order for the B → S form factors to be positive. This also can be checked from heavy quark symmetry [65]. Various form factors for B → S transitions have been evaluated in the relativistic covariant light-front quark model [65]. In this model form factors are first calculated in the spacelike region and their momentum dependence is fitted to a 3-parameter form F (q2) = F (0) 1− a(q2/m2B) + b(q2/m2B)2 . (B8) The parameters a, b and F (0) are first determined in the spacelike region. This parametrization is then analytically continued to the timelike region to determine the physical form factors at q2 ≥ 0. The results relevant for our purposes are summarized in Table XIX. In practical calculations, we shall assign the form factor error to be 0.03. For example, FBK0,1 (0) = 0.35 ± 0.03. The form factor for B to f0(980) is of order 0.25 at q 2 = 0 [48]. In the qq̄ model for the f0(980), 0 = FBf0 sin θ/ For the heavy-flavor independent strong coupling g in HMChPT, we use |g| = 0.59± 0.01± 0.07 as extracted from the CLEO measurement of the D∗+ decay width [39]. The sign is fixed to be negative in the quark model [23]. For the CKM matrix elements, we use the Wolfenstein parameters A = 0.806, λ = 0.22717, ρ̄ = 0.195 and η̄ = 0.326 [52]. The corresponding CKM angles are (sin 2β)CKM = 0.695 +0.018 −0.016 and γ = (59 ± 7)◦ [52]. For the running quark masses we shall use mb(mb) = 4.2GeV, mb(2.1GeV) = 4.95GeV, mb(1GeV) = 6.89GeV, mc(mb) = 1.3GeV, mc(2.1GeV) = 1.51GeV, ms(2.1GeV) = 90MeV, ms(1GeV) = 119MeV, md(1GeV) = 6.3MeV, mu(1GeV) = 3.5MeV. (B9) The uncertainty of the strange quark mass is specified as ms(2.1GeV) = 90± 20 MeV. TABLE XIX: Form factors of B → π,K,K∗0 (1430), ρ transitions obtained in the covariant light- front model [65]. 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Introduction BKKK decays B0KKK decays B-KKK decays BK decays BKK decays B decays Direct CP asymmetries Two-body BVP and BSP decays Conclusions Acknowledgments Decay amplitudes of three-body B decays Decay constants, form factors and others References

0704.1050

Electrical transport properties of polar heterointerface between KTaO3 and SrTiO3

Electrical transport properties of polar heterointerface between KTaO3 and SrTiO3 A. Kalabukhov,1, ∗ R. Gunnarsson,1 T. Claeson,1 and D. Winkler1 1Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology Göteborg, Sweden (Dated: August 10, 2021) Abstract Electrical transport of a polar heterointerface between two insulating perovskites, KTaO3 and SrTiO3, is studied. It is formed between a thin KTaO3 film deposited on a top of TiO2-terminated (100) SrTiO3 substrate. The resulting (KO) 1−(TiO2) 0 heterointerface is expected to be hole- doped according to formal valences of K (1+) and Ti (4+). We observed electrical conductivity and mobility in the KTaO3/SrTiO3 similar to values measured earlier in electron-doped LaAlO3/SrTiO3 heterointerfaces. However, the sign of the charge carriers in KTaO3/SrTiO3 obtained from the Hall measurements is negative. The result is an important clue to the true origin of the doping at perovskite oxide hetero-interfaces. PACS numbers: 73.20.-r,73.21.Ac,73.40.-c http://arxiv.org/abs/0704.1050v1 The mechanism of doping in hetero-interfaces between two insulating perovskite oxides has been intensively discussed since the observation of large electrical conductivity in the hetero-structure between LaAlO3 (LAO) and SrTiO3 (STO). 1,2,3,4,5,6 It was argued that when a thin LAO film is coherently grown on the TiO2-terminated STO substrate, the resulting interface (LaO)+(TiO2) 0 is expected to be polar, provided that the bulk formal valences of Ti and La are maintained at the interface and that the structure of the interface is atomically perfect. The polar structure at the interface results in an infinitely growing electrostatic potential in the (001) direction when the thickness of LAO film is increasing. In order to compensate for the charge discontinuity at the interface, half of an electron per square unit cell may be released leading to conductivity at the interface.7 The possibility of this doping mechanism was supported by theoretical works.8,9 However, there are other possible doping mechanisms in perovskite oxides. It is well known that electrical properties of STO can be changed from insulating to metallic ones by a small reduction of oxygen from its stoichiometric composition.10,11 The possibility that the electrical property at the LAO/STO heterointerface is not due to the oxygen vacancies was presumably ruled out by keeping the STO substrate at deposition conditions which does not result in bulk conductivity due to oxygen vacancies. However, it is still possible that the deposition of the LAO film itself can reduce oxygen from a shallow layer at the STO substrate, as argued by Siemons et al.6 We have previously suggested that oxygen vacancies play an important role in the electrical properties of the LAO/STO heterointerface.4 However, the true microscopic origin of the conductivity at the interface between LAO and STO could not be understood. In this work we treat another polar interface between two insulating perovskite oxides, KTaO3 (KTO) and STO. KTO is a well known material with a cubic structure and lattice parameter of 3.99 Å(compare with 3.905 Åin STO). It is incipient ferroelectric at room temperature with a dielectric constant of about 260.12,13 Tantalum has a formal valence of 5+, and potassium 1+ in KTO. The KTO film should grow as a sequence of layers on a single TiO2-terminated STO substrate in the (001) direction and the resulting interface should have the structure of (KO)−(TiO2) 0. This means that half a hole per square unit cell should be released. This is opposite to the (LaO)+(TiO2) 0 heterointerface, where half an electron per unit cell is transferred to the interface. We have grown thin KTO films on STO substrates and found that the KTO/STO inter- face is indeed conducting with electrical properties very similar to the LAO/STO interface. 0 1 2 3 4 5 6 Position ( m) FIG. 1: (Color online) Atomic force microscope image (top) and cross section (bottom) of the surface of the 6 nm thick KTaO3 film grown on a TiO2-terminated (100) SrTiO3 substrate. Unit cell steps are seen about every 250 nm along the surface. However, the charge of electrical carriers deduced from Hall effect measurements is nega- tive. We discuss possible reasons for this interesting result in view of interface structure and possible doping mechanisms. KTO films were prepared by pulsed laser deposition with in-situ reflection high-energy electron diffraction (RHEED) used to monitor film growth during deposition. The growth conditions were similar to what we used previously to fabricate LAO/STO hetero-interfaces:4 deposition temperature TD = 750 ◦C, oxygen pressure pO2 = 10 −4 mbar, laser energy density J = 1.5 J/cm2. RHEED oscillations could be observed during the initial part of the film growth. However the intensity decreased rapidly and after 3 unit cells it was too low to observe oscillations. The deposition rate estimated from the first RHEED oscillations was 1 unit cell per 10 pulses. The thickness of the KTO films was 13 u.c. layers (approx. 6 nm). Atomic force microscopy (AFM) showed very smooth step-like surface of the KTO film, see -2 -1 0 1 2 6 nm LAO/STO 6 nm KTO/STO T = 300 K H (T) FIG. 2: (Color online) (a) Experimental configuration for determination of Hall coefficient; (b) Hall resistance RXY for KTO/STO and LAO/STO heterostructures measured at room temperature and the same experimental configuration. Fig.1. Electrical measurements were made in a four point van der Pauw configuration14 in the temperature range 2 K – 300 K and in magnetic field up to 5 T. First we proved that the KTO film itself is not conducting by using ”soft” contacts: silver wires glued on the film surface using silver epoxy. In order to reach the interface, we used Ti/Au contact pads fabricated by sputtering through metal mask. The resistance between Ti/Au contacts and contacts glued by silver epoxy was above 10 MΩ, indicating an absence of pinholes in the KTO film. The electrical properties of KTO films may be compared to those of 15 u.c. thick LAO films on TiO2-terminated STO substrates prepared in the same conditions. Both hetero- structures show metallic conductivity with relatively high mobilities and charge carrier con- centrations. Fig.2 shows Hall resistances measured at room temperature under the same experimental configuration (i.e. magnetic field and current direction, see Fig.2a). Both KTO/STO and LAO/STO heterointerfaces had the same sign of Hall coefficient. The sign of the charge carriers is negative according to the magnetic field and bias current directions. The values of the sheet resistance RS, the Hall mobility µH and the charge carrier density nS of the KTO/STO heterointerfaces are very similar to those of LAO/STO, see Fig.3. We 1 10 100 T (K) 6 nm KTO/STO 6 nm LAO/STO 1 10 100 15 (b) T (K) 1 10 100 T (K) FIG. 3: (Color online) Temperature dependence of sheet resistivity RS (a), charge carriers density nS (b) and Hall mobility µH for LAO/STO and KTO/STO heterointerfaces prepared at 10 −4 mbar oxygen pressure. measured three KTO films prepared in similar deposition conditions and they all showed similar transport properties. It is known that potassium deficiency is a significant problem in growth of KTO films due to the high vapor pressure of potassium at high temperatures.13 If this were the case here, the actual heterointerface between KTO/STO may have different reconstruction from the one described above. This possibility needs to be ruled out in a future determination of the microstructure of the hetero-structure by electron microscopy. Independent of the KTO/STO heterointerface microstructure being perfect or not, it is quite remarkable that the electrical properties are very similar to those of LAO/STO heterointerface. This could suggests that there is a common doping mechanism where the type and concentration of charge carriers do not directly depend on the sign of the polar interface deduced from the formal bulk valences. We have previously argued that the high conductivity, mobility, and charge carrier density found in hetero-junctions of LAO/STO prepared at low oxygen pressure mainly are due to oxygen vacancies residing in STO close to the interface. That conclusion is further strengthened by the present findings. The work was supported by the Swedish KAW, SSF, and VR foundations, the EU NANOXIDE, and ESF THIOX programs. ∗ Electronic address: [email protected] 1 A. Ohtomo, and H.Y. Hwang, Nature 427, 423 (2004); 441, 120 (2006). mailto:[email protected] 2 M. Huijben, G. Rijnders,D. H. A. Blank, S. Bals, S. van Aert, J. Verbeeck, G. van Tendeloo, A. Brinkman and H. Hilgenkamp, Nature Materials 5, 556 (2006). 3 S. Thiel, G. Hammerl, A. Schmehl, C. W.Schneider and J. Mannhart, Science 313, 1935 (2006). 4 A. Kalabukhov, R. Gunnarsson, J. Börjesson, E. Olsson, T. Claeson and D. Winkler, Phys. Rev. B 75, 121404(R) (2007). 5 G. Herranz, M. Basletic, M. Bibes, R. Ranchal, A. Hamzic, E. Tafra, K. Bouzehouane, E. Jacquet, J. P. Contour, A. Barthelemy and A. Fert, Phys. Rev. B 73, 064403 (2006). 6 W. Siemons, G. Koster, H. Yamamoto, W. A. Harrison, G. Lukovsky, T. H. Geballe, D. H. A. Blank and M. R. Beasley, cond-mat/0612223 (2006). 7 N. Nakagawa, H. Y. Hwang and D. A. Muller, Nature Mat., 5, 204 (2006). 8 R. Pentcheva and W. E. Pickett, Phys. Rev. B 74, 035112 (2006) 9 M. S. Park, S. H. Rhim and A. J. Freeman, Phys. Rev. B 74, 205416 (2006) 10 O. N. Tufte and P. W. Chapman, Phys. Rev. 155, 796 (1967). 11 C. S. Koonce, M. L. Cohen, J. F. Schooley, W. R. Hosler and E. R. Pfeiffer, Phys. Rev. 163, 380 (1967). 12 D. Z. Bozinis, J. P. Hurrel, Phys. Rev. B 13, 3109 (1976). 13 H.-J. Bae, J. Sigman, S.-J. Park, Y.-H. Heo, L. A. Boatner and D. P. Norton, Solid State Electronics 48, 51 (2004). 14 J. L. van der Pauw Philips Res. Rep. 13, 1 (1958). http://arxiv.org/abs/cond-mat/0612223 References

0704.1051

The Angular Separation of the Components of the Cepheid AW Per

Mon. Not. R. Astron. Soc. 000, 1–?? (2007) Printed 29 October 2018 (MN LATEX style file v2.2) The Angular Separation of the Components of the Cepheid AW Per D. Massa 1⋆ † and N. R. Evans2 ⋆ 1SGT, Inc. NASA’s GSFC, Code 665, Greenbelt, MD 20771, USA 2Center for Astrophysics, 60 Garden St., MS 4, Cambridge, MA 02138 Accepted 2007 December 15. Received 2007 December 14; in original form 2007 August 13 ABSTRACT The 6.4 day classical Cepheid AW Per is a spectroscopic binary with a period of 40 years. Analyzing the centroids of HST/STIS spectra obtained in November 2001, we have determined the angular separation of the binary system. Although we currently have spatially resolved data for a single epoch in the orbit, the success of our approach opens the possibility of determining the inclination, sin i, for the system if the mea- surements are repeated at additional epochs. Since the system is potentially a double lined spectroscopic binary, the combination of spectroscopic orbits for both compo- nents and the visual orbit would give the distance to the system and the masses of its components, thereby providing a direct measurement of a Cepheid mass. Key words: Cepheids – stars: AW Per – binaries: visual – binaries: spectroscopic. 1 INTRODUCTION Cepheids are important stars in many respects, most notably for their roles as fundamental rungs on the cosmic distance ladder and the challenges their structure pose to stellar inte- riors modelling. The use of Cepheids as primary extragalac- tic distance indicators makes a quantitative understanding of their properties extremely valuable. While the Magellanic Clouds are perhaps the best laboratory to study cepheids, the dependence of the period–luminosity relation on metal- icity is still not fully understood (Romaniello et al. 2005). Consequently, accurate distances (absolute magnitudes) to Galactic cepheids are needed to fully understand and quan- tify this dependence and to apply cepheid scale to more metal rich spiral galaxy stars which are more commonly used in extragalactic distance determinations. Cepheids also present important tests for interiors cal- culations since, as evolved stars, their structure is dictated by their evolutionary history. In addition, the models must predict the puslational properties of cepheids, making the modelling especially challenging. This complexity is codified in the term “the Cepheid mass problem”. Forty years ago, when the first hydrodynamic pulsation calculations were made, it was realized that the mass could be derived by either matching the Herzsprung progression of secondary ⋆ E-mail: [email protected]; [email protected] † Based on observations with the NASA/ESA Hubble Space Tele- scope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in As- tronomy, Inc. under NASA contract No. NAS5-26555. maxima or by a parameterization of the pulsation con- stant. These masses were as much as a factor of two smaller than evolutionary calculations. A reconciliation was recently achieved from re-evaluation of the interior opacities (see Si- mon, 1990, for a summary). We see, therefore, that in ad- dition to absolute magnitudes, obtaining accurate Cepheid masses is also important. If we can determine the angular separations of binary systems containing a Cepheid, which are double lined spec- troscopic binaries (SB2s), then the distances and masses of the Cepheids can be derived from basic physics. Because of the central roles of Cepheids in fundamental astrophysics, it is important to have such direct measurements. While several Cepheid distances have been measured directly by the Hipparcos satellite, the quality of these measurements was only sufficient for statistical considerations (e.g., Groe- newegen & Oudmaijer 2000). More recently, a large cam- paign using the Fine Guidance Sensor on HST has begun to yield accurate distances to single Cepheids (Benedict et al. 2002). However, to date the mass of only one cepheid, Polaris, has been directly determined from fundamental ob- servations (Evans, et al. 2007). Although SB2s containing a Cepheid and an A or B star are common (see, Evans 1995), these stars are difficult to re- solve in the optical. This is because of the inevitable, enor- mous magnitude differences of the components in the opti- cal, which result from massive stars evolving toward cooler temperatures at nearly constant luminosity. The top panel Figure 1 shows a typical example of a Cepheid + B star bi- nary, and the contrast between the primary and secondary throughout the optical and IR is obvious. c© 2007 RAS http://arxiv.org/abs/0704.1051v2 2 D. Massa and N. R. Evans Figure 1. Kurucz models for a typical Cepheid (large/red) + hot star (small/blue) binary. The top panel shows how the secondary is roughly 10 times fainter in the optical, making the system ex- tremely difficult to resolve from the ground. On the other hand, the secondary dominates the flux from the system in the UV. The remaining 5 panels demonstrate how the wavelength dependence of the spectrum centroid changes with orientation of the axis of the binary relative to the dispersion for 5 different orientations, shown to the left of each panel. Notice that in the spectral re- gion accessible from the ground, the centroid shifts by less than 10% of the full separation. The “cross-over” point is not reached until λ ∼ 2500Å. A color version of the figure is available in the electronic version of the paper. Thus, while the measurement of a Cepheid mass by di- rectly imaging a double lined spectroscopic binary with a Cepheid primary and an A or B star secondary has been a long-sought goal, ground-based studies have not, as yet, been able to accomplish this (even though they have been able to resolve the stellar disks of some Cepheids, e.g., Kervella et al. 2004, and references therein). As a result, in- direct methods have been developed to determine the masses of Cepheids. The most popular of these uses a combination of UV and optical spectroscopy to obtain radial velocity curves for both components. Then the UV spectral energy distribution (SED) of the hot secondary is used to obtain its temperature. Finally, the mass – temperature relation for main sequence A or B stars is used to infer the mass of the secondary and, thus, (since the system is an SB2) the mass of the primary. This approach has been applied to several systems (SU Cyg, S Mus and V350 Sgr), using IUE or HST spectra to determine the radial velocity curves and SEDs of the secondaries (Evans, et al., 1998). The masses obtained by this approach agree, on average, with the mass- luminosity predictions from evolutionary calculations with moderate convective overshoot (e.g. Schaller, et al. 1992). However, this approach requires an exact understanding of the evolutionary phase of the hot secondary and relies on its spectroscopic parallax to determine the distances to the sys- tems. Clearly, a direct measurement of the masses of both components is more desirable. In this paper, we describe how we used the Space Tele- scope Imaging Spectrograph (STIS) on HST to resolve a the potential SB2 Cepheid binary AW Per using an approach we call “cross-dispersion imaging”. AW Per is a 6.4 day Cepheid which is in a roughly 40 year orbit with its hot secondary (Evans et al. 2000). Evans (1989) studied the system and determined that the secondary is a main sequence B7-8 star and that the color excess of the system is E(B−V ) = 0.52. The Teff of the secondary is expected to be∼ 12000K (Evans 1994). The remainder of the paper is organized as follows: §2 provides an overview the approach used to “resolve” the bi- nary, §3 describes the observations, §4 gives the data anal- ysis, §5 details the analysis of the observations, §6 presents the results, §7 discusses the results and their implications, and §8 summarizes the findings. 2 THE APPROACH(CROSS-DISPERSION IMAGING) 2.1 Basic Principles Massa & Endal (1987) describe how combining imaging and spectroscopy can dramatically increase the effective “resolv- ing power” of an instrument. Specifically, they showed how the wavelength dependence of the centroid of a spectrum can determine the angular separation of an unresolved binary whose components have distinctly different spectra. The ba- sic concept of this approach is quite simple. It is based on an idea put forth by Beckers (1982) and has been indepen- dently discovered by a number of others (see, e.g., Porter et al. 2004, and references therein). Like all cross-dispersion imaging techniques, some sort of a model is required to interpret the observations. This model might be extremely simple, as in the case of a binary where one assumes that the system is composed of exactly two stars, and that one contributes all of the flux at one wavelength and the other contributes all of the flux at an- other wavelength. This crude model would be sufficient to “resolve” the binary from the properties of its spectrum. Consider the image of a highly unresolved binary sys- tem. To first order, the image of the combined light from the system is indistinguishable from a point source. However, the position of an image at any given wavelength will be dis- placed toward the location of the binary component which contributes most of the light that wavelength. In principle, one could obtain images at several different wavelengths and determine how the centers of the images shift from one ex- posure to the next. Analysis of this set of data (along with a model for the flux ratios in each band) would then determine the separation of the two components (Becker 1982). The drawback of this direct approach is that all of the exposures c© 2007 RAS, MNRAS 000, 1–?? AW Per 3 would have to be obtained using different optical elements, making alignment at the sub-pixel level effectively impossi- ble. Instead, Massa & Endal (1987) show that tracking the centroid of the spectrum of the binary has the same effect. Furthermore, because all of the position measurements (the centroid of the spectrum at each wavelength) are obtained at one time, this method is more efficient and the measure- ments are differential in nature, freeing them from several sources of systematic error. To make these notions quantitative, let x and y be the angular coordinates on the detector which are parallel and perpendicular to the wavelength dispersion. Therefore, the wavelengths, λ, are given by λ = λ(x). Now, consider a bi- nary whose components have an angular separation θ and photon fluxes per unit wavelength Np(λ) and Ns(λ) for the primary and secondary, respectively. Further, let φ be the position angle of the binary on the sky (measured c.c. from north toward east of a line from the primary to the sec- ondary) and let α be a similarly measured angle between north and a line in the dispersion direction pointing in the direction of decreasing wavelength. Thus, α can be varied by changing the orientation of the telescope. With these defi- nitions, the wavelength dependence of the centroid of the spectrum of a single observation of a binary is y(λ) = 1 +Np(λ)/Ns(λ) + Const. where (1) ∆y = θ sin(φ− α) (2) (see the appendix). Thus, if Np(λ)/Ns(λ) is known, then measurements at two or more orientations (α’s) enables one to determine θ and φ, the separation and position angle of the binary. Note that if the spectral energy distributions (SEDs) of the components are vastly different, then the po- sition of the centroid shifts from one to the other, depend- ing upon which star dominates the flux at each wavelength. On the other hand, if the binary components have identical SEDs, then no spatial information can be gained from the centroid positions. Figure 1 is a cartoon depicting how the centroid of the spectrum of a binary star, whose components have very dif- ferent effective temperatures, is influenced by the relative energy distributions of the two components and the orien- tation of the binary relative to the dispersion direction of a spectrograph. In this case, the centroid shifts from the cool component at long wavelengths to the hot component at short wavelengths. We define the cross-over wavelength as that wavelength were each binary component contributes equally to the flux. For Cepheid binaries, this wavelength is typically in the near UV (∼ 3000Å for the case shown). In order to infer spatial information from the centroids, it is de- sirable to span as large a wavelength baseline as possible, to maximize the deflections in the centroid positions. The best case would be to cover a large enough wavelength range with a single setting, so that one end of the spectrum is totally dominated by one star and the other end is dominated by the other. If this is not practical, a wavelength band centered on the cross-over wavelength and covering a baseline large enough to experience more than a 50% centroid deflection is adequate. However, in this case, one needs an estimate of the SEDs of the two binary components in order to ex- tract the angular separation. Note that if the absolute flux calibration of the instrument is well-determined, then the Figure 2. Relative error in the angular separation of a binary determined from fitting a cosine curve to measurements obtained at three orientations, {−∆α, 0,+∆α} versus ∆α (abscissa) over the interval ∆α = 1 → 90◦. The different curves are for different values of the orientation of the system on the sky, φ, between φ = 1 → 90◦. flux observations can provide additional information which can be incorporated into the determination of the angular separation (see §5). Finally, to unambiguously determine the separation and position angle of the binary, two or more observations are required in order to solve eq. (2) for θ and φ in terms of the measured quantities ∆y(n) and α(n), for n > 2. The final error associated with the angular separation and the position angle measurements depends upon the band pass of the observation, the signal-to-noise of the data (discussed in the next section), the number of independent orientations obtained and the relation between the these an- gles and φ. We have examined the relative error for sampling three orientations, α(n) = {−∆α, 0,+∆α}, for position an- gles between 1 and 90◦. Figure 2 demonstrates how the rel- ative accuracy of the observations changes as a function of sampling interval, ∆α, and relative orientations, φ, for this case. For most orientations, any sampling with ∆α & 30◦ provides comparable accuracy. The approximations developed in this section are only valid in the sub-Rayleigh regime. Once the sources are re- solved at any wavelength, the entire image must be modeled using a an accurate representation of the point spread func- tion as well as the fluxes of the two objects. 2.2 Exposure Times and Random Errors The counts needed to centroid to an accuracy σ[y(λ)] can be estimated for an instrument whose spread function perpen- dicular to the dispersion is a Gaussian with FWHM = ξ. A single count is equivalent to one estimate of the center of the spectrum drawn from a sample with an RMS disper- sion σ = ξ/ 8 ln 2 = 0.42ξ. Therefore, N samples (counts) determine the centroid to an accuracy of σ[y(λ)] = 0.42ξ . (3) c© 2007 RAS, MNRAS 000, 1–?? 4 D. Massa and N. R. Evans Equation (3) gives the counts needed over a wavelength band to obtain the desired accuracy. The FWHM of the STIS PSF varies from ∼ 0.05 − 0.07′′ (depending on wave- length) and the minimum number of counts obtained in one 10 min observation over a spectral resolution element (2 pix- els) was 4000, and we obtained 3 of these. Therefore, the poorest precision we can expect based upon simple sampling arguments is ∼ 3 × 10−4′′, and this is for a single resolu- tion element. In all, there are 512 independent resolution elements which will be combined to determine a single mea- surement of ∆y through the use of eq. (1). Therefore, ran- dom noise in the angular separation determinations should be . 10−4′′, and not a limiting factor for our observations. However, as is typical for most observations, we shall see that systematic effects will dominate the error budget (see, 3 THE OBSERVATIONS As can be seen from the top panel of Figure 1, a broad wave- length baseline is needed to optimize the extraction process. Furthermore, good spectral resolution is also advantageous, since spectral features provide additional constraints. Con- sequently, we employed the STIS on HST to obtain high spatial resolution, excellent wavelength coverage and good spectral resolution. We used the STIS NUV-MAMA detec- tor together with its G230L grating, since this combination provided good coverage (1600 6 λ 6 3160 Å) of the ex- pected cross-over point (see, Kim Quijano, J., et al. 2003). Spectra were obtained at three distinct roll angles (see, Table 1) which differ by ∼ ±20◦. Although rolls of ±60◦ would be optimal, we were limited to smaller rolls by HST restrictions for objects at the declination of AW Per. Al- though not optimal, Figure 2 shows that this restricted range does not sacrifice very much in theoretical accuracy. After a standard STIS target acquisition, which centers the binary within a 0.1′′ aperture, we obtained the science observations through the 25MAMA aperture, which provides slitless spec- tra of the binary. At each roll, we offset the star by ±0.1′′ and obtained additional science exposures. This procedure allows us to characterize localized distortions in the detec- tor. It is also useful for determining the sensitivity of the observations to their position on the detector, since each spectrum is sampled differently by the pixel lattice. Since the spectrum was repositioned to within 2 pixels (< 0.05′′) after each roll, the dispersion of measurements obtained at the ±0.1′′ offsets should provide a good characterization of the errors that result from all of the changes encountered in the positioning of the spectrum. The reproducibility of these observations also provides a more realistic measurements of the centroiding errors than those based on simple signal-to- noise considerations. As a result of our observing strategy, we obtained 3 observations at each of 3 rolls, for a total of 9 spectra, with exposure times of roughly 10 min each. The orientations mentioned above are measured with respect to the STIS coordinate system, which we define as the x0 − y0 system. In this system, the dispersion direction (from red to blue) makes an angle (measured in the c.c. direction) of −1.4◦ with the x0 axis. 4 DATA REDUCTION 4.1 Centroids The first step in the reduction process was to extract the centroids. This presents a problem, since the STIS detector does not oversample the HST PSF. However, since (as will be explained shortly) only relative centroids will be needed, we can accept some level of bias in the extraction process, as long as it is consistent. This is because the ultimate measure- ments will be differences of the centroids, which will cancel small, uniform biases introduced in the extraction process. We used three separate approaches to extract the cen- troids, y(λ), from the raw images. We chose to analyze the raw images (in their native “highres” 2048×2048 format) be- cause initial experimentation showed that the geometrically corrected images did little to improve the relative positions of the centroids over the a range of 10 pixels or less (which are the scales important to us). Thus it was felt best to avoid the inevitable smoothing which is introduced by the resampling involved in geometric corrections. The first approach we used was a simple cross- correlation technique relative to a set of 0.025′′ FWHM gaus- sians. The second one employed a standard cross-correlation technique using the cross dispersion profiles of a spectrum of a standard star (the wd GD71) which was observed at roughly the same position on the detector with the same grating. We used sinc interpolation in the cross-correlation to compensate for the undersampling of the PSF by the MAMA detector. Finally, we used a non-linear least squares fit to a set of gaussians whose FWHMs, central positions and amplitudes were allowed to vary at each pixel. No sys- tematic differences were found among all three approaches. However, the results from the non-linearly extracted cen- troids produced the results with the lowest pixel-to-pixel scatter, and these were adopted for the following analysis. The 3 sets of centroid measurements at each roll an- gle were rebinned to 512 elements from the 2048 elements available in the raw images, and these were used to con- struct mean centroids at each roll and their standard devi- ations. Because the centroids near the edges of the detec- tor are poorly determined, of the 512 bined pixels (in the wavelength direction) only about 490 are well-behaved. The standard deviations for these 490 pixels determined for each roll angle are over plotted as a function of wavelength in Fig- ure 3. The RMS means for each roll angle are 0.027, 0.024, and 0.027 pixels or (0.67, 0.59, and 0.67 mas). Remember, these are the single observation standard deviations for a single pixel, and there are 9 independent observations with 490 useful pixels. Notice also that this scatter is significantly larger than the one expected from the simple signal-to-noise arguments of the previous section. The reason is that the actual uncertainties are set by random differences between the photometric and geometric centroids of the pixels, and by localized geometric distortions in the detector over the range of a few pixels. Nevertheless, the repeatability of the centroids (to a few percent of a pixel) is considered quite good, and we will use this scatter to characterize the actual measurement errors in the centroid positions. Since the centroids are extracted from the raw images, they contain large scale geometric distortions. Consequently, we will analyze the relative centroids. To construct these, we first combine the centroids determined at each offset for a c© 2007 RAS, MNRAS 000, 1–?? AW Per 5 Table 1. Observation log Obs ID Off Set Roll Obs Date Exp. Time Phasea V (B − V ) arc sec Deg. MJD - 52235 Min. Φ Mag. Mag. o6f104010 +0.0 175.526 0.34765625 10.0 0.906 7.40 1.02 o6f104020 +0.1 175.526 0.35546875 10.0 0.907 7.39 1.01 o6f104020 −0.1 175.526 0.36328125 11.4 0.909 7.38 1.01 o6f105010 +0.0 205.000 0.41406250 10.0 0.916 7.34 1.00 o6f105020 +0.1 205.000 0.42187500 10.0 0.918 7.33 0.99 o6f105030 −0.1 205.000 0.42968750 11.4 0.919 7.32 0.99 o6f106010 +0.0 146.526 0.48046875 10.0 0.927 7.27 0.97 o6f106020 +0.1 146.526 0.48828125 10.0 0.928 7.26 0.97 o6f106030 −0.1 146.526 0.49609375 11.4 0.929 7.26 0.97 a Phase, V and (B − V ) are derived from sources in the literature, as discussed in the text. Figure 3. Standard deviations of the three independent spectra of AW Per obtained at each roll angle. The standard deviations for each roll angle are over plotted. particular roll angle to produce a mean centroid, 〈y〉, at each roll. These measurements contain geometric distortions and any systematic effects introduced by the centroid extrac- tion technique. However, when we analyze the differences between each individual mean and the grand mean of all the observations, these systematic affects are removed. This is because the offsets at each roll are larger than the dis- placements from one roll to another, and the scatter that the former exhibit (Fig. 3) demonstrates that localized ge- ometric distortions are small. Similarly, any systematic af- fects that result from mis-matches between the actual PSF orthogonal to the dispersion and the gaussian used to deter- mine the centroids will cancel out, since the same process is used in each case. Finally, we must account for the fact that y(λ) is not exactly perpendicular to the dispersion. As a result, we must divide the final displacements that we measure by cos(1.4◦). 4.2 Fluxes STIS fluxes were extracted from the images using the CAL- STIS IDL software package developed by Lindler (1998) for Figure 4. Plots of the mean STIS spectrum of AW Per (solid curve) together with the available IUE spectra (dotted), cali- brated to the HST flux system. the STIS Instrument Definition Team. In order to constrain the B star flux contribution, we also incorporate the avail- able IUE spectra (obtained when the Cepheid component was near minimum light), into the analysis given in §5. The IUE fluxes were placed upon the HST/STIS flux system us- ing the transformations described by Massa & Fitzpatrick (2000). Figure 4 compares the IUE and STIS spectra. It is immediately clear that the IUE long wavelength spectra were obtained when the Cepheid was near minimum light (Φ = 0.53, Evans 1989), while the STIS observations were near maximum light (Table 1). The effects of extinction are also clearly apparent, as is the fact that the IUE fluxes are a factor of 1.146 smaller than the STIS fluxes. This discrep- ancy is a constant over the region of overlap, and its origin is unknown. Consequently, we cannot be certain which set of fluxes is correct. In §6 we show that this ambiguity intro- duces a significant uncertainty into our results. The variability of the Cepheid is clearly detectable in the STIS spectra. Figure 5 shows STIS flux ratios for the mean spectra obtained at the second and third roll angles divided by the first. The time lapsed between the mean ob- servations is 1.59 and 3.19 hours, respectively. This plot c© 2007 RAS, MNRAS 000, 1–?? 6 D. Massa and N. R. Evans Figure 5. Plots of the ratios of mean STIS spectra of AW Per obtained at the second and third roll angles divided by the mean flux obtained at the first roll angle. These plots demonstrate how the Cepheid component brightened over the 3.5 hour observing sequence. Notice that the flux at the shortest wavelengths does not change, since it is dominated by the B star secondary. demonstrates two things. First, the Cepheid flux changed significantly throughout the three HST orbits spanned by the observations. Second, the flux ratios decrease with wave- length, becoming unity at the shortest wavelengths. This is contrary to what is normally seen in single Cepheids like δ Cep (Schmidt & Parsons 1982) where the flux changes typically increase with decreasing wavelength. Consequently, this figure shows that the flux at the shortest wavelengths is dominated by the B star, which does not vary. The following analysis also requires the color and mag- nitude of the system the time of the observations. We com- bined the data from Szabados (1980), Moffett & Barnes (1984), Szabados (1991), and Kiss (1998), using the period and HJD for zero phase from Kiss (1998). The combined data were fit with a high order polynomial, and this was used to determine the V and (B − V ) photometry at the times of the STIS observations. The resulting phases and photometry are listed in Table 1. 5 ANALYSIS 5.1 Overview Because our spectra cover a limited band-pass, we require an estimate for the flux ratio of the binary components in order to extract the wavelength dependence of the centroids. This flux ratio is constrained, since it must also satisfy the observed flux of the system, which is the reddened, com- bined flux of the two binary components. Ideally, one would fit the observed flux and centroid positions with a combi- nation of single star spectra obtained with the same instru- ment and which experience the same reddening. However, because there is no such library of single star spectra avail- able, we used an approach which employs a model for the B star SED star and for the UV extinction to construct the combined flux and the centroids. We then used a non-linear least squares fitting procedure1 to fit the centroids and fluxes simultaneously. This method is described in detail in § 5.3. 5.2 Model Components We now describe the components of the model used to fit the observations. In a few instances, refinements might increase the accuracy, but in the interest of expediency, certain ef- fects were ignored for the first attempt. First, we use Kurucz (1991) Atlas 9 models with updated metallicities2 for the B star. We use only models with a micro-turbulent velocity of 2.0 km s−1. The synthetic photometry for the models was calibrated as in Fitzpatrick & Massa (2005). We set log g = 4.0 for the B star atmosphere. The sensitivity of our results to this assumption is tested once a fit is achieved. The model atmosphere fluxes were prepared in the man- ner described by Fitzpatrick & Massa (2005), which is best suited to the IUE fluxes. The dust model is quite general. We use the Fitzpatrick (1999) formulation of the Fitzpatrick & Massa (1990) model since we need a representation of the near-UV extinction, and the original Fitzpatrick & Massa (1990) formulation does not provide one. Although the Fitz- patrick (1999) curve for the near UV is largely untested, it is reasonable and the best currently available. To provide additional flexibility to the Fitzpatrick model, we allow the bump strength (c3), the width of the 2175 Å (γ) and far-UV curvature term (c4) to vary independently. In this way, we can accommodate any observed extinction curve. As a result, the RV parameter (the ratio of visual extinction to color ex- cess) only affects the general slope of the UV extinction and the shape of the near-UV curve, and the wavelength depen- dence of the total extinction to an object can be expressed Aλ ≡ A[RV , E(B − V ), γ, c3, c4;λ] . (4) 5.3 Details of the Fitting Procedures We simultaneously fit the STIS centroids at all three roll angles and the IUE flux from the B star. We constrain the reddened model for the B star by assuming that all of the flux from the system for λ 6 1650 Å is due to the B star. The difference between the observed flux and the reddened B star model provides the Cepheid SED which is used in fitting the centroids. The free parameters of the fit are: The three ∆y(n) (displacements perpendicular to the dispersion at each roll angle), Tseff (the effective temperature of the B star secondary), [m/H] (the abundance parameter for the B star), E(B−V ) (the color excess of the system, consistent with the fluxes), RV (which determines the slope of the UV extinction curve), γ (the width of the 2175 Å bump), c3 (the bump strength), and c4 (the strength of the far UV curvature) – 10 parameters in all. The V magnitude of the B star, Vs, is fixed by the observed flux attributed to the B star at λ = 1650 Å and the extinction at that wavelength relative to V . In addition to the separations, the results also yield an empirical, unreddened UV SED and photometry for the 1 We use the Markwardt non-linear IDL fitting procedure, avail- able at http://astrog.physics.wisc.edu/∼craigm/idl/idl.html. 2 We used the the Kurucz “preferred models” available at http://kurucz.harvard.edu/. c© 2007 RAS, MNRAS 000, 1–?? http://astrog.physics.wisc.edu/~craigm/idl/idl.html http://kurucz.harvard.edu/ AW Per 7 Cepheid. These can then be and compared to models or to actual Cepheids. Since the derived Cepheid flux is identical to the observed flux minus the B star flux for wavelengths longward of 1650 Å, the flux in this region is fit exactly. The equation used to fit the centroids is: − θ2sN(Ts, log gs, vt, [m/H];λ) θ2sN(Ts, log gs, vt, [m/H];λ) and the unreddened flux of the Cepheid is given by p = [N(λ) − θ2sN(Ts, log gs, vt, [m/H];λ)] ×10A[RV ,E(B−V ),γ,c3,c4;λ] (6) where θs is the angular diameter of the B star (fixed by the flux at 1650Å) and n = 1, 2, 3 represents the observations obtained at each roll angle, which are means of the data for the three off-set positions. We cannot use a single mean for the fluxes, since significant changes in V , (B − V ) and the UV SED occur over the course of the observations (see, Table 1, Fig. 5) and must be taken into account. However, the data were averaged at each roll, since the time between off-sets was much smaller than the time between rolls. A major advantage of our approach is that it only relies on a Kurucz Atlas 9 model for the B star, and recent work by Fitzpatrick & Massa (1999, 2005) has demonstrated that these provide excellent representations of low resolution B star SEDs. Further, it avoids using the Atlas 9 models for the Cepheid component, which is desirable since the accuracy of Cepheid model atmospheres has not been fully tested, espe- cially in the UV. This issue is addressed further in §6. The disadvantage of our approach is that we must have extremely well calibrated fluxes, and we have already seen an inconsis- tency between the poorly exposed IUE fluxes and the STIS data. 5.4 Determining the Separations The final step in the analysis is to fit the angular separations derived at each roll angle to a sine curve whose phase and amplitude are related to the position angle and separation of the binary (eq. 2). The amplitude of the curve is the full separation of the system and the phase is the position angle of the system on the sky. The abscissa of the plot is the position angle in the x − y system, which is equal to the values listed in Table 1 minus 1.41◦ (which accounts for the rotation to align the spectra with the y axis). Figure 6 shows the definitions of the different angles used in the analysis, and their relations to one another. 5.5 Weights The non-linear least squares involves fitting an array which consists 3 sets of centroids and the IUE fluxes all at once. To perform the fit, we must provide errors for the different components of this array. The measurement errors affecting the centroids were obtained from the standard deviations of the three independent sets of measurements obtained at each offset position. For the IUE data, we used the error vector which accompanies the MXLO fluxes (see, Nicholes & Linsky 1996). Figure 6. Diagram showing the definitions of the different angles and coordinate systems used in the analysis, and their relations to one another. The position angle on the sky of the binary angle, φ, is defined as the angle measured the c.c. from north to east, with the primary at the origin. The x− y system is the standard STIS coordinate system, with x parallel to the dispersion (increasing in the direction of increasing wavelength) and y perpendicular to it. The angle α (also measured the c.c. from north to east) is defined as the angle between North and x for a given telescope orientation. Thus, φ− α is the angle between the dispersion and a line connecting the binary components and ∆y = θ sin(φ−α) is the displacement of the two spectra of the binary perpendicular to the dispersion. If φ− α = 0 or ±180◦, then ∆y = 0. 6 RESULTS In fitting the data, we assumed a microturbulent velocity of 2.0 km s−1, which is typical for main sequence B stars (e.g., Fitzpatrick & Massa 2005). Because the B star is over- whelmed by the Cepheid in the optical and near-UV, we do not have access to the classical log g diagnostics for B stars, namely, the Balmer jump and Balmer lines. Conse- quently, we fixed the surface gravity at 4.0, again typical for main sequence B stars. We allowed the abundance parame- ter, [m/H] , and the effective temperature of the B star to be optimized by the least squares routine, along with the ∆y’s and the extinction parameters. In addition, we assumed that the IUE fluxes were correct (so the STIS fluxes were divided by 1.146 to make them agree with the IUE data). In apply- ing our model, we also assume that all of the STIS flux in a 30 Å band centered at 1650 Å is due to the B star. We shall examine the effects of our assumptions shortly. Only the IUE fluxes between 1250 and 1700Å are incorporated into the fit of the SED, which constrains the physical prop- erties of the B star. This extends slightly beyond the 1650Å limit used for the STIS data, but recall that the IUE data were obtained when the Cepheid was near minimum light, and nearly a factor of two fainter in the UV (see, Fig. 4). The parameters determined from the fit are given in Ta- ble 2, where parameters that were fixed in the fit are enclosed in parentheses. Figure 7 shows our fits to the centroids. The points are the observed data and the solid curves are the fits obtained simultaneously with the fit to the fluxes. The effects of spectral features on the centroids are clearly seen. c© 2007 RAS, MNRAS 000, 1–?? 8 D. Massa and N. R. Evans Table 2. Parameter Values Parameter Value Parameter Value ∆y1 −0.010 c3 4.13 ∆y2 0.279 c4 0.82 ∆y3 −0.269 γ 0.9686 [6297] Vs (11.084) 15735 (B − V )s0 (−0.156) log gp (4.00) (U −B) 0 (−0.597) log gs [1.60] Vp (7.362) [m/H]p [0.00] (B − V ) 0 (0.494) [m/H]s -0.20 (U −B) 0 (0.359) E(B − V ) 0.53 ∆ logL (0.95) R(V ) 3.11 Values in parenthesis were not involved in the fitting procedure. Values in square brackets were determined from a fit to the Cepheid SED derived from the initial fit. Figure 8 shows the fit to the SED below 1650Å. We do not show the fit to the binary SED longward of 1650Å since it is, by definition, exact. The extinction curve derived from the best fit is also shown in Figure 8, where it is compared to a standard RV = 3.1 curve from Fitzpatrick (1999). We can also estimate the physical parameters of the Cepheid component of the binary by fitting its mean SED inferred from fit. This SED is found by subtracting the red- dened B star model from the observed SED of the system and then correcting this difference for the effects of extinc- tion. The unreddened SED plus its V , (B−V )0 and (U−B)0 (also inferred from the fit) were then fit to an Atlas 9 model. The V , (B − V ) and (U −B) photometry were initially as- signed errors of 0.02, 0.01 and 0.02 mag, respectively. In performing this fit, we fixed the micro-turbulent velocity at 2 km s−1, and allowed T eff (the effective temperature of the primary), log gp (the surface gravity of the primary) and [m/H] (the abundance of the primary), to vary. We had to restrict the surface gravity to be larger than 1.6, other- wise the fitting routine would seek log gp values that were unrealistically small (we expect a log gp ≃ 2.0, e.g., Evans 1994). Furthermore, we had to increase the weight (decrease the error) of the (B − V ) photometry by a factor of 10 in order to obtain reasonable agreement with the photometry. Figure 9 compares the unreddened SED of the Cepheid to the best fit model. The parameters derived from the fit are also listed in Table 2 and are enclosed in square brackets, to distinguish them from the parameters derived from the initial fit to the data. It is also possible to test the reasonableness of the in- ferred UV Cepheid SED by comparing it to IUE observa- tions of the single Cepheid star δ Cep. δ Cep has a period of 5.4 days, compared to 6.5 days for AW Per, and its mean unreddened color is 〈(B − V )〉 = 0.57. To obtain the in- trinsic color of AW Per, we use our derived color excess for the system and the intrinsic colors of the B star sec- ondary from Table 2 and the mean magnitude of the system, 〈V 〉 = 7.49 mag, to correct the observed mean color of the system, 〈(B − V )〉 = 1.06 mag, for both extinction and the presence of the companion. The result is 〈(B−V )p0〉 = 0.57, identical to that of δ Cep (recall that the intrinsic color we derive for AW Per is at Φ ≃ 0.92). Thus, the compar- ison between these two stars is expected to be quite good. Figure 7. Fits to the mean centroids at each roll angle for AW Per. Each mean centroid was fit simultaneously with the corre- sponding fluxes, optical photometry and interstellar extinction. A Kurucz model was used to fit the B star component, and the Cepheid flux was taken to be the difference between the reddened B star model and the observed flux. The bottom plot in Figure 9 compares the unreddened IUE data (points) for δ Cep from several exposures obtained for 0.9 6 Φ(δCep) 6 1.0 to the unreddened Cepheid STIS spectrum (solid curve) of AW Per. Several IUE exposures are required to produce the δ Cep spectrum since the dy- namic range of IUE was so limited and the range of the UV SED of δ Cep is so large. The IUE data had the Massa & Fitzpatrick (2000) corrections applied, were dereddened by an E(B − V ) = 0.09 (Dean et al. 1987) and scaled by 10−0.4(7.37−3.54) , which corresponds to magnitude difference of AW Per at Φ = 0.92 (the mean for the STIS data) and δ Cep at Φ = 0.95 (the mean of the IUE data). Finally, we utilize the ∆y(n) which resulted from the fits to derive the separation of the system and its position angle on the sky. These are found by fitting eq. (2) to the plot of ∆y versus roll angle shown in Figure 10. The error bars at each orientation are the quadratic mean errors for that roll determined from the dispersion in the fits to the three individual sets of observations obtained at each orientation (see, next section). The inverse of the errors squared were c© 2007 RAS, MNRAS 000, 1–?? AW Per 9 Figure 8. Top: Best fit B star (thin curve) compared to the IUE (points) and STIS (thick curve) fluxes. The model includes red- dening. We only show the far-UV region, since the fit is, by defini- tion, exact for wavelengths longward of 1650Å. Bottom: AW Per extinction curve determined by the simultaneous fit of the flux and centroids (solid curve) compared to a standard RV = 3.1 curve (dotted) from Fitzpatrick (1999). used to weight the fit. The final result of the analysis is a separation of θ = 13.74 ± 0.26 mas and a position angle φ = 184.16 ± 1.94 deg, for an accuracy of ∼ 2%. 6.1 Errors in the parameters In this section, we describe the internal, random, errors af- fecting our parameter determinations, and also examine the influence of systematic effects upon the results. The random errors were evaluated in two, independent ways. One is the error estimates calculated by the least squares routine, which are determined by evaluating deriva- tives of the model. These errors are listed in the second col- umn of Table 3. We also obtained error estimates by fitting the sets of observations obtained at the same off-set at each roll angle, independently. These provide 3 sets of indepen- dent observations and we used the parameters determined from each set to obtain standard deviations (S.D.s) of the model parameters. These estimates (divided by 3 applica- ble to the error in the mean) are listed in the third column of Table 3. Notice that the errors in the ∆y(n) determined from the S.D.s are nearly twice as large. To be conservative, Figure 9. Top: Inferred dereddened Cepheid SED (points) com- pared to the best fitting Kurucz model (solid) and the dered- dened flux of the best fit B star (dashed). Bottom: Comparison of the unreddened Cepheid flux (solid curve) and an unreddened IUE spectrum (dots) of δ Cep observations for 0.90 6 Φ 6 0.95. The δ Cep flux is scaled by the difference between V = 3.54 at Φ = 0.925 for δ Cep and V = 7.37, the magnitude of the primary in AW Per at Φ = 0.92 (the mean phase of the STIS observations). As discussed in the text, the δ Cep spectrum is a combination of several IUE spectra. these errors were used as the error shown in Figure 10 and in determining the errors in θ and φ. Beside the random (or measurement) errors, systematic effects will also be present. We characterize these by varying the different assumptions which enter the fitting procedure, and then examining their influence on the result. To begin, we varied the assumed value of log g used to fit the B star by ±0.5, which should encompass all plausible values. The result (the difference divided by 2) is listed in column 4 of Table 3. Next, we tested the affect of assuming that the STIS (and not the IUE) fluxes are correct and allowed for the possibility that the B star accounts for only 95%, instead of 100% of the flux at 1650Å. These results are listed in the last two columns of Table 3 As can be seen from Table 3, the varying the log g can cause a significant change in Teff s, but has little effect on the ∆y(n), which are the object of our analysis. In fact, the only significant change in the ∆y(n) result from our inability to determine whether the STIS or IUE fluxes are correct, c© 2007 RAS, MNRAS 000, 1–?? 10 D. Massa and N. R. Evans Table 3. Errors Param. Prog. S.D. |δ log g| |δ fSTIS fP+fs ∆y(1) 0.004 0.015 1.4× 10−4 6.5× 10−6 5.0× 10−4 ∆y(2) 0.005 0.017 0.0019 1.4× 10−4 0.015 ∆y(3) 0.005 0.014 0.0021 1.3× 10−4 0.015 248 105 1205 9.1 37 [m/H] 0.057 0.025 7.5× 10−5 0.0016 0.0064 E(B − V ) 0.001 0.038 0.018 0.0026 0038 RV 0.031 0.12 0.11 0.0090 2.7× 10 γ 0.015 2.4× 10−4 0.019 8.6× 10−4 0.0025 c3 0.14 0.32 0.049 0.029 0.0055 c4 0.019 0.066 0.014 4.3× 10 −3 0.0068 Figure 10. Determination of the angular separation of AW Per. The observational errors for ∆y were determined from individual fits to the 3 independent offset observations at each roll angle. and even these errors are only of the same order of the errors determined from the repeated observations. As a result, we conclude that the angular separation determined from our analysis is very robust to variations in the assumptions or input parameters. 7 DISCUSSION We have seen that the separation determined from the fit is quite stable. We now discuss the physical parameters de- termined from our fits (Table 2), their reliability and their implications. We first consider the Cepheid SED derived from the fit. It is compared to the best fitting Atlas 9 model in top panel of Figure 9. This “best fitting” model is not a very good fit, since it lies systematically below the observed flux in far-UV flux and over it in the near-UV flux. Furthermore, the agreement with the optical photometry is not very good. The model predicts V = 7.362, (B − V ) = 0.470 and (U − B) = 0.309. The the agreement with the (B − V ) color given in Table 2 is fair, but recall that it was given a very large weight. The agreement with the inferred (U−B) is not very good at all. The poor overall fit probably results from the short comings of Atlas 9 models for Cepheids discussed below. The bottom panel of Figure 9 compares the unreddened SED of the Cepheid component of AW Per to the unred- dened SED of the single Cepheid, δ Cep at approximately the same phase. This figure demonstrates three points. First, the two SEDs agree surprising well. Second, the strong far- UV flux in the derived SED relative to the models is also present (and slightly larger) in δ Cep, so the derived SED is quite reasonable. Third, the flux in δ Cep is extremely small for wavelengths shortward of 1650Å, bolstering our assump- tion that all of the flux in AW Per observed below 1650Å is due to the B star secondary. So, why is the Atlas 9 model fit of the Cepheid so poor? One must remember that Cepheid UV SEDs depend on nu- merous, ill-defined physical processes that are not fully in- corporated into the Atlas 9 models. These include spherical extension, which can enhance the UV flux from an atmo- sphere (see Fig. 4 in Hauschildt et al. 1999), chromospheres (e.g., Sasselov & Lester 1994), the amount of convective en- ergy transport (Castelli, Gratton, & Kurucz 1997) and the details of the line blanketing (Prieto, Hubeny, & Lambert, 2003). In addition, there are inevitably dynamical effects that are not treated by the models. In fact, we initially attempted to fit the data with using an approach that employed models for both the Cepheid and the B star. However, we abandoned it because it produced poor fits and the separations that were ∼ 10% larger than those derived from the adopted technique. The origin of the systematic difference in the centroids can be traced to the gradient in the flux residuals seen in the top of Figure 9. These propagate into the fits of the centroids. Perhaps the use of more detailed Cepheid models could solve this prob- In spite of these difficulties, it is of interest to exam- ine the physical parameters determined from the Cepheid model. To begin, Teff of the best fit model agrees reason- ably well with previous estimates for Cepheid temperatures near maximum light (Evans & Teays 1996, Fry & Carney 1999, Kovtyukh & Gorlova 2000). On the other hand, the fit selects a very low surface gravity and would have set- tled on an even lower value if it had been allowed to do so. It is also interesting that the Cepheid model has a signif- icantly different metallicity than the B star. However, this may not be too strange. Instead, it may simply reflect the fact that the [m/H] parameter in cooler models responds more to spectral features produced by CNO elements, while the same parameter in the B stars responds to the Fe abun- dance (Fitzpatrick & Massa 1999). c© 2007 RAS, MNRAS 000, 1–?? AW Per 11 Next, we consider the parameters determined for the B star. The model fit to the far-UV (Fig. 8) is quite good, and the extinction curve, while distinctly different from the canonical RV = 3.1 curve, is rather unremarkable, with parameters well within normal bounds (e.g., Fitzpatrick & Massa 1990, Valencic et al. 2004). Also, the [m/H] for the B star is well within the expected range for such stars (e.g., Fitzpatrick & Massa 1999, 2005) and the inferred color ex- cess is quite close to previous determinations (Evans 1994). It should not be surprising that these fits are so good, since both the extinction model and the ability of the Atlas 9 mod- els to describe normal B star spectra are well documented. Notice that Teff we derive is considerably hotter than previ- ously estimated by Evans (1994), and lies somewhat closer to the ZAMS (see, Fig. 7 in Evans 1994). However, its prob- able mass, ∼ 5M⊙ (based on its Teff , Andersen, 1991), re- mains significantly less than the lower limit of ∼ 6.6M⊙ determined from the radial velocity orbit of the primary by Evans et al. (2000). Thus, it still appears likely that the B star component of AW Per must also be a binary. 8 SUMMARY We have shown that the signatures of the Cepheid and B star components of AW Per are clearly present in the wave- length dependence of the centroid of its spectrum. This result demonstrates the power of our approach. A simple model was devised to extract the angular separation of the binary from the centroid measurements. The accuracy of the angular separation is ∼ 2%, or ± a few ×10−4′′! We also demonstrated that the results are extremely stable to vari- ations in the expected systematic effects in the data and its analysis. We also showed that one possible source of uncer- tainty in the current data is the absolute level of the far-UV data. Higher quality far-UV observations to secure the B star flux level and secure its parameters would be extremely useful. Our final results are listed in Table 2. In addition to the angular separations and position angle, these include a Cepheid temperature and systemic extinction that agree with previous estimates and a B star secondary temperature that is considerably hotter than previously thought (e.g., Evans, 1994). However, the likely mass of the secondary still appears too small to account for the minimum mass of the secondary inferred by the radial velocity of the pri- mary. Consequently, it is likely that the B star component of AW Per is also be a binary. Finally, the long period of AW Per’s orbit means that it will be a few years before the separation changes enough for the second independent observation needed to determine sin i can be obtained. ACKNOWLEDGMENTS We would like to thank Karla Peterson and Charles Proffit of STScI, who provided valuable guidance in preparing the observations. This work was supported by NASA grants to SGT, Inc. and SAO. REFERENCES Andersen, J. 1991, A&A Rev, 3, 91 Beckers, J.M. 1982, Opt. Acta, 29, 361 Benedict, G.F., McArthur, B.E., Fredrick, L. W., et al. 2002 AJ, 124, 1695 Castelli, F., Gratton, R.G. & Kurucz, R.L. 1997, A&A, 318, 841 Dean, J.F., Warren, P.R. & Cousins, A.W.J. 1987, MN- RAS, 183, 569 Evans, N.R. 1989, AJ, 97, 1737 Evans, N.R. 1994, ApJ, 436, 273 Evans, N.R. 1995, ApJ, 445, 393 Evans, N.R. & Teays, T.J. 1996, AJ, 112, 761 Evans, N.R., Bohm-Vitense, E., Carpenter, K., Beck- Winchatz, B. & Robinson, R. 1998, ApJ, 494, 768 Evans, N.R., Vinko, J. & Wahlgren, G.M. 2000, AJ, 120, Evans, N. 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C.; Dordrect: Kluwer), 441 Lindler, D. 1998, CALSTIS Reference Guide (CALSTIS Version 5.1) (Greenbelt, MD:GSFC) Massa, D. & Endal, A.S. 1987, AJ, 93, 760 Massa, D. & Fitzpatrick, E. L. 2000, ApJS, 126, 517 Moffett, T.J., & Barnes, T.G. 1984, ApJS, 55, 389 Nichols, J.S. & Linsky, J.L. 1996, AJ, 111, 517 Prieto, C.A. , Hubeny, I. & Lambert, D.L. 2003, ApJ, 591, Porter, J.M., Oudmaijer, R.D.& Baines, D. 2004, A&A, 428, 327 Kim Quijano, J., et al. 2003, “STIS Instrument Hand- book”, Version 7.0, (Baltimore: STScI). Romaniello, M., Primas, F., Mottini, M., Groenewegen, M., Bono, G. & Franois, P. 2005, A&A, 429L, 37 Sasselov, D.D. & Lester, J.B. 1994, ApJ, 423, 795 Schaller, G., Schaerer, D., Meynet, G., & Maeder, A. 1992, A&AS, 96, 269 Schmidt, G.S. & Parsons, S.B. 1982, ApJS, 48, 185 Simon, N.R. 1990 in Cacciari, C. & Clementini, G. eds., ASP Conf. Ser., 11, Confrontation between stellar pulsa- tion and evolution, p193 Szabados, L. 1980, Commun. Konkoly Obs. Hung. Acad. Sci., Budapest, No.76 c© 2007 RAS, MNRAS 000, 1–?? 12 D. Massa and N. R. Evans Szabados, L. 1991, Commun. Konkoly. Obs. Hung. Acad. Sci., Budapest, No.96 Valencic, L.A., Clayton, G.C., & Gordon, K.D. 2004, ApJ, 616, 912 APPENDIX A: MATHEMATICAL DETAILS This appendix provides a detailed derivation of how the wavelength dependence of the centroid of the a dispersed image can be used to determine the separation of a binary whose components have different colors. Consider the set of angular coordinates x and y which are parallel and perpendicular to the dispersion, respec- tively, with x increasing in the direction of increasing wave- length (this is the standard STIS coordinate system, Kim et al. 2003). Now, let h(y) be the instrumental profile in the cross-dispersion direction, y. Then the spectrum of a single star located at y = y0 can be expressed as f(λ, y) = N(λ)h(y − y0) (A1) where λ = λ(x), andN(λ) is the photon flux at λ (we assume infinite resolution in the wavelength direction). If the spectra of the primary and secondary components of the binary are centered at yp, and ys, then their spectra are separated by ∆y = ys − yp, and the image of the binary spectrum is given by f(λ, y) = Np(λ)h(y − yp) +Ns(λ)h(y − yp −∆y). (A2) If ∆y is small compared to structure in h(y), this equation can be approximated by f(λ, y) ≃ Np(λ)h(y − yp) +Ns(λ) 4h(y − yp) +∆y dh(y) y=y−yp = [Np(λ) +Ns(λ)]× 4h(y − yp) + Ns(λ) Np(λ) +Ns(λ) dh(y) y=y−yp ≃ [Np(λ) +Ns(λ)]× ∆yNs(λ) Np(λ) +Ns(λ) Therefore, the wavelength dependence of the centroid of the spectrum will vary as y(λ) = yp + ∆yNs(λ) Np(λ) +Ns(λ) = yp + 1 +R(λ) where R(λ) = Np(λ)/Ns(λ) is the flux ratio of the binary components. Now, the separation ∆y depends on both the separation of the binary, θ, and the orientation of the system relative to the dispersion direction. The position angle on the sky of the binary, φ, is defined as the angle measured the c.c. from north to east, with the primary at the origin. The angle α(n) (also measured the c.c. from north to east) is defined as the angle of a line in the dispersion direction pointing in the direction of increasing wavelength for the nth telescope orientation. In this case, φ − α(n) is the angle between the dispersion and a line connecting the binary components and ∆y(n) = θ sin(φ−α(n)) is the displacement of the two spectra of the binary (note that when φ − α = 0, ±180◦, ∆y = 0). Therefore, the observation obtained with the telescope in the nth orientation can be expressed as p + θ sin(φ− α(n))[1 +R(λ)]−1 (A5) where y p is the wavelength independent displacement of the nth exposure in y. To extract both θ and φ from the observed centroids, at least two observations at different α’s are required. There- fore, as long as the relative fluxes of the binary components are known, a linear regression of the wavelength dependence of the centroid against [1+R(λ)]−1 gives ∆y(n) for that ob- servation. The y p are constant terms related to the abso- lute position of the primary star, although in practice the they cannot be reliably disentangled from the large random errors in the absolute position of the binary on the detector at each orientation. Once the ∆y(n) are determined for each orientation, these are plotted against the known quantities, α(n). Since = θ sin(φ− α(n)) (A6) fitting a sine function to the ∆y(n) as a function of the α(n) determines φ and θ, the observables of an astrometric binary at the epoch of the observations. c© 2007 RAS, MNRAS 000, 1–?? INTRODUCTION THE APPROACH(Cross-Dispersion Imaging) Basic Principles Exposure Times and Random Errors THE OBSERVATIONS DATA REDUCTION Centroids Fluxes ANALYSIS Overview Model Components Details of the Fitting Procedures Determining the Separations Weights RESULTS Errors in the parameters DISCUSSION SUMMARY MATHEMATICAL DETAILS

0704.1052

Transverse field effect in graphene ribbons

arXiv:0704.1052v2 [cond-mat.mes-hall] 2 Aug 2007 Transverse field effect in graphene ribbons D. S. Novikov W. I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455 (Dated: July 31, 2007) It is shown that a graphene ribbon, a ballistic strip of carbon monolayer, may serve as a quan- tum wire whose electronic properties can be continuously and reversibly controlled by an externally applied transverse voltage. The electron bands of armchair-edge ribbons undergo dramatic transfor- mations: The Fermi surface fractures, Fermi velocity and effective mass change sign, and excitation gaps are reduced by the transverse field. These effects are manifest in the conductance plateaus, van Hove singularities, thermopower, and activated transport. The control over one-dimensional bands may help enhance effects of electron correlations, and be utilized in device applications. PACS numbers: 73.23.-b 73.63.-b 72.80.Rj Building nanoscale systems with pre-determined prop- erties has long been the focus of basic and applied re- search. Progress in this field is tied to the recent ad- vancements in the synthesis of quantum nanowires [1] and quantum dots [2] via control of the growth process, as well as in the growth and selection of carbon nanotubes [3]. The characteristics of these devices, however, are set by design and are typically difficult to modify. Ideally, one would like to be able to tune the system’s properties reversibly after synthesis. In the present work we suggest that a graphene ribbon (GR), a ballistic strip of recently discovered [4] carbon monolayer, may serve as a quantum wire whose electronic properties can be continuously and reversibly controlled by the external transverse voltage. The setup makes use of the massless relativistic electron dispersion in graphene [4–6], with the valence and conduction bands touching at a conical Dirac point [3, 7]. Electron dispersion in GRs varies depending on their chirality, as the transverse confinement of Dirac fermions is sensitive to the boundary conditions [8–16]. This has prompted proposals for GR applications as field- effect transistiors [17] and valley filters [18]. Further- more, GRs have been suggested as a host of interesting many-body phenomena, including spin polarization on the edges [19, 20], and as a basis for building coupled electron spin qubits [21]. Recently spectral gaps in GRs have been measured [22], scaling approximately inversely with the ribbon width. The basic idea of the present proposal is that the prop- erties at the Dirac point are fragile and can be affected by external fields [23]. Not suprisingly, the proposed strong electric field effect is similar to that considered for carbon nanotubes [24–26]. Unfortunately, small radiusR ∼ 1 nm of single-walled tubes requires very large transverse fields E[MV/cm] ≃ 25/R2[nm]; this has so far hindered observation of band transformation. Remarkably, with GRs, the rib- bon width (that plays the role of the tube circumference) may vary in a broad range, L ∼ 10−200 nm [22], and the effects of strong band transformation become realistic. In the setup shown in Fig. 1, electrons in a GR are con- fined along the x axis, while the longitudinal momentum ky ≡ k is conserved. The effect of the applied transverse voltage V is to induce the potential U(x) = −eE(x− L/2) . (1) The acting field E ∝ Eext ∝ V can be assumed uni- form and proportional to the external field Eext as long as the bands are not strongly mixed (as described be- low); e is the unit charge. We subtracted the average, setting dxU(x) = 0 (the subtracted constant adds to the chemical potential controlled by the gate voltage Vg). The natural units for the transverse voltage and energy u = eEL/∆L , ∆L = h̄v/L ≃ 0.7 eV/L[nm] , (2) where v ≃ 106m/s is graphene’s Fermi velocity. The ballistic limit of transport is implied. We now give an overview of the results. The electron band transformation is shown in Figs. 2 and 3. The lon- gitudinal electron bands change qualitatively when the FIG. 1: (color online). The setup. Left and right electrodes carry the voltage ±V/2, producing the external field Eext. The carrier concentration is controlled by the back gate voltage Vg. http://arxiv.org/abs/0704.1052v2 −10 −5 0 5 10 Wave vector kL along the ribbon 10 20 30 FIG. 2: (color online). Transverse field effect in metallic GRs. Inset: Velocity reversals in metallic GRs occur at the zeros of the function g(u), Eq. (9); first reversal voltage u1 ≈ 9.2. Fine lines show the u ≪ 1 and u ≫ 1 asymptotic behavior of g(u). Top: The voltage u = 15 above the reversal value u1. The Fermi surface acquires a pair of small pockets. Small gaps at k = 0 are due to imperfect boundaries [20]. Bottom: Landauer conductance G (bold integers and colors) in the units of G0 = 2e 2/h, as the number of transverse modes at the Fermi energy Vg. Dashed cut corresponds to the top panel. dimensionless transverse voltage approaches u ∼ 10. A number of effects follow: (i) The Landauer conductance [27] is quantized in the units of G0 = 2e 2/h, similar to that in point contacts in GaAs [28]. The crucial difference is that now the posi- tions and widths of the plateaus can be controlled by the transverse voltage. The sharp steps in Figs. 2 and 3 in real systems will be smoothened out by finite tempera- ture or weak disorder, while the conductance values on the plateaus will remain equal to the quantized values. −4 −3 −2 −1 0 1 2 3 4 Wave vector kL along the ribbon 2 ∆(u) 5 10 15 FIG. 3: (color online). Transverse field effect in semiconduct- ing GRs. For voltages u > ut ≈ 4.5 above threshold the effective mass at k = 0 is negative. Top: Bands transforma- tion, starting from u = 0 (dotted), to just above threshold u = 6 (solid), to above threshold, u = 10 (dashed, only lowest subbands shown), where the gap is minimal at k 6= 0. In- set: Gap suppression occurs above the threshold ut. Bottom: Landauer conductance plateaus in the units of G0 = 2e (ii) The thermopower S ∝ −∂ lnG/∂Vg, being propor- tional to the conductance derivative [29], peaks at the borders between the domains in Figs. 2 and 3 (bottom). (iii) The Fermi velocity in metallic GRs is reduced by the field, v → vg(u) [Fig. 2 inset and Eq. (9) below]. As a consequence, the one-dimensional density of states at the band center ν(0) = 2/{πh̄v|g(u)|} increases [factor 2 is due to spin degeneracy]. This increase magnifies the effects of electron interactions. The latter may manifest themselves via the increase of the Luttinger liquid expo- nent in a sufficiently long ribbon, and through excitonic instabilities (resulting in interaction-induced gaps). (iv) The Fermi velocity changes sign for the field val- ues corresponding to zeroes of g(u), causing strong van Hove singularities in metallic GRs. The Fermi surface fractures, with each sign change adding a pair of small pockets to the Fermi surface (Fig. 2, top). This effect produces extra conductance plateaus (Fig. 2, bottom). (v) There is a threshold voltage ut ≃ 4.5 above which the effective mass of the lowest energy subband in semi- conducting GRs changes sign, so that the longitudinal electron dispersion acquires symmetric minima at small but nonzero k (Fig. 3). The excitation gap is then re- duced by the field (Fig. 3 inset). This effect can be de- tected in the shift of the conductance plateaus (Fig. 3, bottom), and in the activated transport measurements. (vi) The band structure remains electron-hole symmet- ric at any field for both metallic and semiconducting GRs due to the Dirac symmetry of the problem. Thus the conductance plots of Figs. 2 and 3 are independent of the polarity of the gate and transverse voltages. Turning to possible applications, the setup may serve as a field-effect transistor with a tunable working point, in which the “transverse”, V , and the “normal”, Vg, field effects can be utilized separately. Furthermore, one may selectively amplify combinations α(V −V 0)+β(Vg−V 0g ), α = ∂G/∂V |V and β = ∂G/∂Vg|V by choosing an appro- priate working point V = (V 0, V 0g ) on the edge of the conductance plateau. Tuning the parameters to achieve a large gain for say, V − Vg, combined with the device’s large input and low output impedance, is reminiscent of an operational amplifier. By the same token, strong con- ductance nonlinarity in both inputs V and Vg may render this setup into a few-nm size signal multiplier, or even into a logic gate. We now outline the details of the calculation. At the π- band center (ǫ = 0), the electron dispersion is determined by the two inequivalent Dirac points in the Brillouin zone. The low-energy states Ψ(r) = eiKxψ+(r) + e −iKxψ−(r) are represented [3] in terms of the smoothly varying en- velope ψ = {ψ+, ψ−} that consists of the pair of the two- component spinors ψ+ and ψ− with values on the two sublattices of the honeycomb lattice [here K = −4π/3a0, where a0 = 3acc is the graphene lattice constant, and acc = 0.144 nm is the Carbon bond length]. The dy- namics of the envelope is governed by the Dirac equation Hψ = ǫψ, with the effective Hamiltonian , H± = ±ih̄vσ1∂x − h̄vkσ2 + U(x) , where σ1,2 are the Pauli matrices. The boundary condi- tions Ψ(r)|x=0,L = 0 at the armchair edges dictate [14] ψ+(0) + ψ−(0) = 0 , ψ+(L) + e iφnψ−(L) = 0 , (4) where the phase φn = KL = − 2π3 (n + 1 − δ), and L = 1 (n+1− δ)a0 is the effective ribbon width (the dis- tance between the sites on which Ψ vanishes). The phase φn may incorporate corrections coming from imperfect edges, similar to the curvature-induced corrections in nanotubes [30]. (For example, the δt/t ≈ 0.12 change in the hopping amplitude at the edges due to the passi- vated bonds [20] reduces the effective width L and the boundary phase φn by the amount ∝ δ = 3 ≈ 0.20.) The system (3) and (4) is solved numerically (Figs. 2 and 3) via the transfer matrix approach similar to that of Refs. [24, 25, 31]. Eq. (3) is equivalent to ∂xψ± = ±Pψ± , P(x) = kσ3 + iσ1(U − ǫ)/h̄v . (5) The armchair boundary conditions (4) require tr (SS̃) = 2 cosφn for the product of the transfer matrices S = Txe P(x)dx , S̃ = T̃xe P(x)dx , (6) where Tx and T̃x symbolize the “chronological” and “anti- chronological” orderings of the operators P(x) that do not commute for different x. In the absence of the field, the GR spectrum consists of one-dimensional Dirac bands with |ǫk=0| = ∆L× π3 |n+1− 3p− δ|, p = 0,±1,±2, ... . Thus GRs with n = 3p− 1 are metallic (with small gap ∝ δ originating from imperfect boundaries [20]), in which case the lowest energy mode is non-degenerate, and the rest are doubly-degenerate (the latter degeneracy is lifted by the finite δ). The ribbons with n = 3p and n = 3p − 2 are semiconducting, with non-degenerate bands, and excitation gaps |ǫk=0| = ∆L× (1∓ δ) correspondingly. To study the transverse field effect it is convenient to employ the chiral gauge transformation [24, 25] ψ± = e ±iσ1ϕ(x)ψ̃± , ϕ = U(x′)dx′/h̄v , (7) that preserves the boundary conditions (4) and trans- forms the system (3), H± → H̃±, H̃± = h̄v −kσ2e±2iσ1ϕ(x) ± iσ1∂x . (8) The transformation (7) shows that the spectrum at k = 0 is unaffected by the field. For the metallic GRs with ideal edges (eiφn ≡ 1), the two degenerate k = 0, ǫ = 0 eigenstates, each consisting of a pair {ψ̃+, ψ̃−}, are |1〉 = and |2〉 = Projecting the Hamiltonian (8) onto these states, H̃ → h̄vkg(u)σ2, we find the spectrum around k = 0 ǫ = ±h̄v|kg(u)| , g = dξ cos [uξ(1− ξ)] . (9) The function g(u) is plotted in Fig. 2 inset. For |u| ≪ 1, g ≃ 1 − u2/60. Its |u| ≫ 1 form g ≃ π/|u| cos[(|u| − π)/4] determines the successive voltages un ≈ ±(3+4n)π, n = 0, 1, 2, ..., where the k = 0 velocity changes sign. At those voltages the dispersion ǫ ∼ k3 at k = 0, causing the van Hove singularity ν(ǫ) ∼ |ǫ|−2/3 in the density of states at ǫ = 0, and an additional pair of pockets of Fermi surface emerges. In the |u| ≫ 1 limit, such pockets appear at the zeroes of g for both metallic and semiconducting GRs. Electron interactions in graphene result in the RPA screening of the external field Eext. The screening is scale-invariant, E = Eext/κ, for an infinite sheet [32], κ = 1 + 2πe2/4h̄v ≃ 5. The depolarization problem in nanotubes [24–26, 33–36] also yields κ ≃ 5 practi- cally independent of the tube radius and chirality. This linear-screening estimate will remain valid in GRs as long as the subbands are not strongly mixed. In the oppo- site case the field on the ribbon edges, estimated in the Thomas-Fermi fashion, develops an algebraic singular- ity [37], which corresponds to filling the Fermi-surface pockets. As a result, the uniform field model (1) is jus- tified for weak to moderate fields. For the acting field u = 10, the required external field uext ≃ 50 is achieved at EextL ≃ 1V across the ribbon width L = 30 nm. To conclude, the transverse voltage applied across a graphene nanoribbon dramatically affects its longitudinal electronic dispersion. The Fermi surface breaks up into pockets for the metallic ribbons, and the excitation gap closes for the semiconducting ones. 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0704.1053

Domain Switching Kinetics in Disordered Ferroelectric Thin Films

J. Y. Jo et al. Domain Switching Kinetics in Disordered Ferroelectric Thin Films J. Y. Jo,1 H. S. Han,1 J.-G. Yoon,2 T. K. Song,3 S.-H. Kim,4 and T. W. Noh1,∗ ReCOE&FPRD, Department of Physics andAstronomy, Seoul National University, Seoul 151-747, Korea Department of Physics,University of Suwon, Suwon, Gyeonggi-do 445-743, Korea School of Nano Advanced Materials, Changwon National University, Changwon, Gyeongnam 641-773, Korea R&D center, Inostek Inc., Ansan, Gyeonggi-do 426-901, Korea (Dated: October 23, 2018) We investigated domain kinetics by measuring the polarization switching behaviors of polycrys- talline Pb(Zr,Ti)O3 films, which are widely used in ferroelectric memory devices. Their switching behaviors at various electric fields and temperatures could be explained by assuming the Lorentzian distribution of domain switching times. We viewed the switching process under an electric field as a motion of the ferroelectric domain through a random medium, and we showed that the local field variation due to dipole defects at domain pinning sites could explain the intriguing distribution. PACS numbers: 77.80.Fm, 77.80.Dj, 77.84.Dy Domain switching kinetics in ferroelectrics (FEs) un- der an external electric field Eext have been extensively investigated for several decades [1, 2, 3, 4, 5, 6, 7, 8, 9]. The traditional approach to explain the FE switching kinetics, often called the Kolmogorov-Avrami-Ishibashi (KAI) model, is based on the classical statistical theory of nucleation and unrestricted domain growth [10, 11]. For a uniformly polarized FE sample under Eext, the KAI model gives the time (t)-dependent change in polar- ization ∆P (t) as ∆P (t) = 2Ps[1− exp{−(t/t0) n}], (1) where n and t0 are the effective dimension and character- istic switching time for the domain growth, respectively, and Ps is spontaneous polarization. When the nuclei are appearing in time with the same probability, n = 3 for bulk samples and n = 2 for thin films [12]. In addition, t0 is proportional to the average distance between the nuclei, divided by the domain wall speed. Several stud- ies have used the KAI model successfully to explain the ∆P (t) behaviors of FE single crystals and epitaxial thin films [2]. Recently, FE thin films have been intensively investi- gated for FE random access memory (FeRAM) [1]. Most commercial FeRAM use polycrystalline Pb(Zr,Ti)O3 (poly-PZT) films, and their ∆P (t) behaviors determine the reading and writing speeds of the FeRAM. In such non-epitaxial FE films, a domain cannot propagate in- definitely due to pinning caused by numerous defects, so the KAI model cannot be applied. Therefore, it is impor- tant both scientifically and technologically to clarify the domain switching kinetics of polycrystalline FE films. Numerous studies have examined the ∆P (t) behaviors of polycrystalline FE films, and the reported results vary markedly [3, 4, 5, 6, 7]. Lohse et al. measured the polar- ization switching currents of poly-PZT films, and showed that ∆P (t) slowed significantly compared to Eq. (1) [3]. Tagantsev et al. observed similar phenomena for poly- PZT films. To explain these behaviors, they developed the nucleation-limited-switching (NLS) model. They as- sumed that films consist of several areas that have inde- pendent switching kinetics: ∆P (t) = 2Ps [1− exp{−(t/t0) n}]F (log t0)d(log t0), where F (log t0) is the distribution function for log t0 [4]. They assumed a very broad mesa-like function for F (log t0), and could explain their ∆P (t) data. The same 500 ns @ 1.7 V 1.7 V 1.4 V 1.2 V 1.1 V 1.0 V 0.9 V 0.8 V -7 -6 -5 -4 -3 -2 300 K 150 K 80 K 25 K 15 K log t (s) @ 300 K Pole Read P Write A1 A2 Pole Read P 500 ns A1 A2(a) 500 ns FIG. 1: (color online). Schematic diagrams of the pulse trains used to measure (a) non-switching polarization (Pns) and (b) switching polarization (Psw). Time (t)-dependent switched polarization ∆P (t) (c) under various external voltages (Vext) at room temperature and (d) under 1.7 V at various temper- atures. The dotted and solid lines correspond to fitted results using the KAI model and the Lorentzian distribution function in log t0, respectively. http://arxiv.org/abs/0704.1053v1 0 100 200 300 15 K 25 K 80 K (kV/cm) 150 K 300 K -7 -6 -5 -4 -3 log t (s) @ 80 K, 1.7 V Experimental results Lorentzian distribution Gaussian distribution KAI model -6 -4 -2 log t(s) FIG. 2: (color online). (a) Values of n for various T and Eext. (b) ∆P (t) results for (solid symbols) experimental data and fitted results using the Lorentzian (solid line), Gaussian (dashed line), and delta (dotted line) distributions for log t0. The inset shows the distribution functions corresponding to the fitted results. authors also studied La-doped poly-PZT films and found that ∆P (t) at room temperature is limited mainly by nucleation, while at a low temperature (T ), the switching kinetics are governed by domain wall motion, implying the validity of the KAI model [5]. In this Letter, we investigate the polarization switch- ing behaviors of poly-PZT films. We can explain the measured ∆P (t) in terms of the Lorentzian distribution function for F (log t0), irrespective of T . We show that such distribution arises from local field variation in a dis- ordered system with dipole-dipole interactions. Note that (111)-oriented poly-PZT films with a Ti concentration near 0.7 are the most widely used mate- rial in FeRAM applications. We prepared our polycrys- talline PbZr0.3Ti0.7O3 thin film on Pt/Ti/SiO2/Si sub- strates using the sol-gel method. The poly-PZT film had a thickness of 150 nm. X-ray diffraction studies showed that it has the (111)-preferred orientation, and scanning electron microscopy studies indicated that our poly-PZT film consists of grains with a size of about 200 nm. We deposited Pt top electrodes using sputtering with a shadow mask. The areas of the top electrodes were about 7.9×10−9 m2. We obtain the ∆P (t) values of our Pt/PZT/Pt capac- itors using pulse measurements [2, 4, 13, 14]. Figure 1(a) shows the pulse trains used to measure the non- switching polarization change (Pns). Using pulse A1, we poled all the FE domains in one direction. Then, we ap- plied pulse A2 with the same polarity, and measure the current passing a sensing resistor. By integrating the cur- rent, we could obtain the Pns values. Figure 1(b) shows the pulse trains used to measure the switching polariza- tion (Psw). Inserting pulse B with the opposite polarity between pulses A1 and A2, we could reverse some portion of the FE domains, so the difference between the values of Psw and Pns represents the polarization change due to domain switching, namely ∆P (t). We varied t from 200 ns to 1 ms, and Vext from 0.8 to 4 V. The value of Eext can be estimated easily by dividing Vext by the film thickness. At T of 80∼300 K, we used pulses A1 and A2 with a height of 4 V, which was larger than the coercive voltage. Below 80 K, the coercive voltage increases, so we increased the pulse height to 6 V [15]. Figure 1(c) shows the values of ∆P (t)/2Ps at room temperature with numerous values of Vext. Figure 1(d) shows the values of ∆P (t)/2Ps at various T with Vext = 1.7 V. The dotted lines in both figures are the curves best fitting Eq. (1). The KAI model predictions deviated markedly from the experimental ∆P (t) values in the late switching stage, in agreement with Gruverman et al. [6]. In addition, the best fitting results with the KAI model gave unreasonable values of n. As shown in Fig. 2(a), the values of n varied markedly with T and Eext. In ad- dition, in the low Eext region, we obtained n values much smaller than 1, which are not proper as an effective di- mension of domain growth. Therefore, Eq. (1) fails to describe the polarization switching behaviors of our PZT films. To explain the measured ∆P (t), we tried simple func- tions for F (log t0) in Eq. (2). The opposite domain, once nucleated, will propagate inside the film, so we fixed n=2. The solid circles in Fig. 2(b) show the experimen- tal ∆P (t) at 80 K with Vext = 1.7 V. For F (log t0), we tried the delta, Gaussian, and Lorentzian distribu- tion functions, as shown in the inset. The dotted line indicates the fitting results using Eq. (2) with a delta function. Note that this curve corresponds to a fit with the KAI model, and thus the classical theory cannot ex- plain our experimental data. The dashed line shows the Gaussian fitting results. Although this fitting seems rea- sonable, some discrepancies occur. The solid black shows the fitting results with the Lorentzian distribution: F (log t0) = (log t0−log t1)2+w2 , (3) where A is a normalization constant, and w (log t1) is the half-width at half-maximum (a central value) [16]. The Lorentzian fit can account for our observed ∆P (t) behaviors quite well. We applied the Lorentzian fit to all of the other exper- imental ∆P (t) data. As shown by the solid lines in Figs. 1(c) and (d), the Lorentzian fit provides excellent expla- nations. Figure 3(a) presents the Lorentzian distribution -4 -2 0 2 4 0.8 V 1.0 V 1.2 V 1.4 V 1.7 V log t - log t -8 -7 -6 -5 -4 -3 -2 log t 0.8 V 1.0 V 1.2 V 1.4 V 1.7 V @ 300 K @ 300 K FIG. 3: (color online). (a) The Eext-dependent Lorentzian distribution functions at room temperature. (b) Rescaled ∆P (t) using fitting parameters for the Lorentzian distribu- tion function. functions used for the 300 K data. As Vext increases, log t1 and w decrease. We rescaled the experimental ∆P (t)/2Ps data using (log t - log t1)/w. All the data merge into a single line, an arctangent function [16], as shown in Fig. 3(b). Although not indicated in this fig- ure, the experimental data for all other T also merged with this line. This scaling behavior suggests that the Lorentzian distribution function for log t0 is intrinsic. Note that F (log t0) follows not the Gaussian distribu- tion, but the Lorentzian distribution. For a statistically independent random process, it is a basic statistical rule that the resulting distribution should become Gaussian, regardless of the process details [17]. For example, im- purities (or crystal defects) inside a real crystal result in inhomogeneous broadening of the light absorption line, which has a Gaussian line shape. However, some studies have observed that magnetic resonance line broadening of randomly distributed dipole impurities follows the Lorentzian distribution [18]. The first rigorous theoretical result for this problem is that of Anderson, who showed that the distribution of any in- teraction field component in the system of dilute aligned dipoles should be Lorentzian [19, 20]. Polycrystalline FE films should contain many dipole defects that will act as pinning sites for the domain wall motion. To explain our observed Lorentzian distribution of log t0, we assume that a local field E exists at the FE domain pinning sites and that it has a Lorentzian distribution: F (E) = A , (4) where ∆ is the half-width at half-maximum of the E dis- tribution function, related to the concentration of pin- ning sites. In the low Eext region, the domain wall motion should be governed by thermal activation process at the pin- ning sites. Without E effects, thermal activation results in a domain wall speed in the form v ∝ 1/t0 ∝ exp[- (U/kBT )(E0/Eext)], where U is the energy barrier and E0 is the threshold electric field for pinned domains [21]. Since E results in a change in the effective electric field at pinning sites, the associated t0 can be expressed as t0 ∼ exp )( E0 Eext+E . (5) Then, the distribution of E results in a distribu- tion in log t0, using the relation F (log t0) = F (E) · |dE(log t0)/d(log t0)|. With log t1 ≈ w ≈ UE0∆ , (7) we can obtain the desired Lorentzian distribution for F (log t0), i.e., Eq. (3), from Eqs. (4) and (5). Our experimental values for log t1 and w agree with the analytical forms. Figures 4(a) and (b) plot log t1 vs. 1/Eext and w vs. 1/E ext at various T , respectively. Both log t1 and w follow the expected Eext-dependence in the low Eext region. Note that Eq. (6) is consistent with Merz’s law [22], which states that the current coming from FE polarization switching should have a character- istic time of exp(α/Eext), where α is the activation field. Using this empirical law, several studies have measured α values. For example, So et al. reported α ≈1700 kV/cm for 100-nm-thick epitaxial PZT films [2], and Scott et al. reported α ≈ 270 kV/cm for 350-nm-thick poly-PZT films [23]. These values are consistent with our room temperature value of UE0/kBT , i.e., 1400 kV/cm. Our model viewed the FE domain switching kinet- ics as domain wall motion driven by Eext with a ran- dom pinning potential. In the low Eext region, thermal activation at the pinning sites can be important, result- ing in the so-called domain wall creep motion. Applying atomic force microscopy, Tybell et al. [21] and Paruch et al. [24] demonstrated that the domain-switching ki- netics in epitaxial PZT films are governed by the domain wall creep motion. Some theoreticians studied the do- main wall creep motion of an elastic string in a random potential. They found a linear increase in U with an in- crease in T [25]. The insets in Fig. 4(a) show that the 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0 150 300 0 150 300 0.000 0.005 0.010 0.015 0.020 0.025 0.030 15 K 25 K 80 K 150 K 300 K T (K) (cm/kV) 15 K 25 K 80 K 150 K 300 K (cm/kV) FIG. 4: (color online). Eext-dependent (a) log t1 and (b) w at various T . Note that log t1 and w are proportional to 1/Eext and to 1/E2extin the low Eext region, respectively. The insets show UE0/kB and ∆. The solid lines are guidelines for eyes. value of UE0/kB obtained from the linear fits in Fig. 4(a) increase linearly with T , consistent with the theoretical prediction for U [25]. The inset in Fig. 4(b) shows ∆ obtained from the fits to Fig. 4(b). Similar exponential decay behavior was predicted in a magnetic resonance study of randomly distributed dipoles [26]. At this point, we wish to compare our model with the NLS model. Although both models use Eq. (2), the ori- gins and forms for F (log t0) are quite different. In the NLS model, the FE film consists of numerous areas, each with its own and independent t0. Subsequently, it was suggested that the individually switched regions corre- spond to single grains or clusters of grains in which the grain boundaries act as frontiers limiting the propaga- tion of the switched region [8]. Consequently, the NLS model can be applied for polycrystalline films only, and the form of F (log t0) should depend on their microstruc- ture. Conversely, in our model, the interaction between dipole defects inside the FE film induces a distribution in the local field, which results in F (log t0). Therefore, both point defects and the grain boundaries could act as pinning sites. Using the Lorentzian distribution for F (log t0), our model can be used for both epitaxial and polycrystalline FE films [2]. Using Eqs. (2) and (3) with small w values, we could successfully explain the ∆P (t) for FE single crystals or epitaxial thin films [2]. We also found that our model can explain the ∆P (t) data for poly-PZT films with Ti concentrations of 0.48 and 0.65. Note that our model for thermally activated domain switching kinetics can be viewed as the famous prob- lem that treats the propagation of elastic objects driven by an external force in presence of a pinning potential [21, 24, 25]. It can be applied to many FE films, since the domain wall motion with a disordered pinning po- tential should be the dominant mechanism for ∆P (t). Therefore, the ∆P (t) studies can be used to investi- gate numerous intriguing issues concerning nonlinear sys- tems, such as creep motion, avalanche phenomenon, pin- ning/depinning transition, and so on. In summary, we investigated the polarization switch- ing behaviors of (111)-oriented poly-PZT films and found that the characteristic switching time obeyed the Lorentzian distribution. We explained this intriguing phenomenon by introducing the local electric field due to the defect dipole. The authors thank D. Kim for fruitful discussions. This study was financially supported by Creative Re- search Initiatives (Functionally Integrated Oxide Het- erostructure) of MOST/KOSEF. ∗ Electronic address: [email protected] [1] Ferroelectric Memories, edited by J. F. Scott (Springer- Verlag, Berlin, 2000). [2] Y. W. So et al., Appl. Phys. Lett. 86, 92905 (2005) and references therein. [3] O. Lohse et al., J. Appl. Phys. 89, 2332 (2001). [4] A. K. Tagantsev et al., Phys. Rev. B 66, 214109 (2002). [5] I. Stolichnov et al., Appl. Phys. Lett. 83, 3362 (2003). [6] A. Gruverman et al., Appl. Phys. Lett. 87, 082902 (2005). [7] V. Shur et al., J. Appl. Phys. 84, 445 (1998). [8] I. Stolichnov et al., Appl. Phys. Lett. 86, 012902 (2005). [9] B. H. Park et al., Nature 401, 682 (1999). [10] N. Kolmogorov, Izv. Akad. Nauk. Ser. Math. 3, 355 (1937). [11] M. Avrami, J. Chem. Phys. 8, 212 (1940). [12] If all nuclei of opposite polarization arise through whole process, n could be larger than the actual dimension. [13] Y. S. Kim et al., Appl. Phys. Lett. 86, 102907 (2005). [14] J. Y. Jo et al., Phys. Rev. Lett. 97, 247602 (2006). [15] Complications can occur due to charge trapping or do- main pinning, called the imprint effect. Refer to Ref. [1]. To prevent the imprint effect, we applied a pulse with the opposite polarity at the end of each pulse train mea- surement (i.e., after pulse A2). [16] A double exponential function exp[-{10log t/10log t0}n] with n >1 can be approximated as a step function cen- tered at log t0=log t. As a result, Eq. (2) can be approx- imated as 2PsA/π · [arctan{(log t− log t1)/w}+ π/2]. [17] F. Reif, Fundamentals of Statistics and Thermal Physics (McGraw-Hill, Singapore, 1985). [18] J. H. V. Vleck, Phys. Rev. 74, 1168 (1948). [19] P. W. Anderson, Phys. Rev. 82, 342 (1951). [20] J. R. Klauder and P. W. Anderson, Phys. Rev. 125, 912 (1962). [21] T. Tybell et al., Phys. Rev. Lett. 89, 097601 (2002). [22] W. J. Merz, Phys. Rev. 95, 690 (1954). [23] J. F. Scott et al., J. Appl. Phys. 64, 787 (1998). [24] P. Paruch et al., Phys. Rev. Lett. 94, 197601 (2005). [25] A. B. Kolton et al., Phys. Rev. Lett. 94, 047002 (2005). mailto:[email protected] [26] M. W. Klein, Phys. Rev. 173, 552 (1968).

0704.1054

Geometry of Time, Axiom of Choice and Neuro-Biological Quantum Zeno Effect

Geometry of Time, Axiom of Choice and Neuro-Biological Quantum Zeno Effect Moninder Singh Modgil 1 Abstract Role of axiom of choice in quantum measurement is highlighted by suggesting that the conscious observer chooses the outcome from a mixed state. Further, in a periodically repeating universe, these outcomes must be pre-recorded within the non-physical conscious observers, which precludes free will. Free will however exists in a universe with open time, It is suggested that psychol- ogy’s binding problem is connected with Cantor’s original definition of set. Influence of consciousness on material outcome through quantum processes is discussed and interesting constraints derived. For example, it is predicted that quantum mechanical brain states should get frozen if monitored at suffi- ciently small space-time intervals - a neuro-biological version of the so called quantum zeno effect, which has been verified in domain of micro-physics. Existence of a very small micro-mini-black-hole in brain is predicted as a space-time structural interface between consciousness and brain, whose va- porization explains mass-loss reported in weighing experiments, conducting during the moments of death. 1 Department of Physics, Indian Institute of Technology, Kanpur, India. http://arxiv.org/abs/0704.1054v1 1 Introduction Consciousness is probably the most difficult problem attempted by human scientific endeavor, and is developing into an eclectic discipline. In this paper, I introduce certain new set theoretic ideas (among others) in the already inter-disciplinary field of consciousness studies. At the outset, I shall clearly state relevant points of my philosophical stance. A. Nature of Consciousness. Conscious observers are different from brain and bodies, but interact through them. This is identical to the dualistic school of thought, with Ecclles [1] as a representative. This apriori does not rule out emergence of temporary consciousness in matter, as a result of various postulated mecha- nisms [2], such as Bose-Einstein condensations, or self organizing behavior, or phase locked dynamical neural networks, strange attractors etc.. Whereas the material consciousness is temporary (depending upon stability of the physical system), the non-material observer is eternal in time. This does not imply that observer is always conscious, or observer is conscious only in body. The term, ’Soul’, is more accurate, as consciousness and observation are ’temporary phenomena’ accompanying ’Soul’, in certain conditions 1. I shall however continue to use the term, ”conscious observer”, to represent soul. Consciousness may thus be regarded as a common emergent property of both the observer, as well as matter. As can be seen, the stance is flexible, enough to be compatible with almost diverging views on the subject, and may be termed as ’Non-dualistic’ in the sense that it allows simultaneous existence of almost divergent view points. Further, the conscious observers are distinct from each other, and inhabit the same (one) physical universe. B. Nature of Time. It is probably unlikely that complete problem of consciousness can be solved without understanding nature of time. Time has been a subject of many monographs and papers [3, 4, 5] in physics. The unresolved issues here are - 1. Arrow of Time, i.e., its irreversibility, 2. Origin of present moment, alternatively the observed subjective dis- tinction between, past, present and future, with conscious observers’ 1For instancee, during dreamless sleep and coma, both consciousness and observations are absent attention confined to present (light-like hyper-surface), (ignoring out of body and similar psychic experiences [7] , for the time-being). 3. Overall geometry of time, i.e., cyclic or linear etc., [8]. In periodic time, it can be shown that cause and effect are connected by, what in set theory is called an equivalence relation [9]. However, this causal structure could be important, as it has been pointed out that EPR paradox [38], which has been experimentally verified in recent years [11, 12, 13, 14, 15, 16], does suggest that causality is an equivalence relationship [17]. While this circular geometry of time, would appear to lead to the usual causal paradoxes of time loops in physics, such as killing one’s mother before one’s birth (or conception!) - the causal paradoxes are avoided by withholding free- will to the conscious observers or actors or participants (in the time cycle!). Thus, consciousness is only an observer of events pre-recorded within itself, in an over all cyclic time, within this philosophical framework. Any apparent freewill is actually illusory. Further cyclic nature of time blurs the distinction between past and fu- ture, as the two are globally connected. In introduction of his monograph, Zeh [3] quotes Lewis Carroll from the book ”Through the Looking Glass” - White queen to Alice: ”Its a bad memory which works only backwards”. In cyclic time, a good or perfect memory, will be able to remember all events which the possessor (conscious observer) would have observed with in the complete time cycle. I should add that while cyclic time has been a hallmark of ancient cos- mological systems, Poincare , Zermelo, Caratheodoty, and Nietzsche [18, 19, 20, 21] did attempt a mathematical formulation of the idea at the turn of the century, and the idea has recently been evoked by contemporary mathemat- ical physicists such as Segal [23], Guillemin [22] and others [24, 25] , with a view of solving certain problems of observational cosmological physics, and particle physics (macro and micro cosmos). Idea behind introducing cyclic time concept is that if problem of consciousness is going to be solved [or the other way round), it may be so only in cyclic time - or what the mathematical physicists would call the S1 (circle) topology of time. 2 Set Theoretic Connection. Cantor defined set as ”collection into a whole, of objects of our intuition or thought”. The definition is very psychological, and the phrase ”collection into a whole” is of special relevance. The phrase actually implies simulta- neous perception of constituent set elements as a whole. Even a sentence is understood only when perceived as a whole. Role of short term memory in verbal comprehension comes to mind when perceiving or comprehending very large sentences - here 7± 2 chunks of short term memory are probably being used at various levels, and the sentence has to be read many times, before comprehension (collection into a whole) occurs. Interestingly 7±2 has been derived by statistical mechanical considerations, by developing so called Fokker-Plank equation of brain neurons [26]. I equate verbal comprehension with (verbal) perception, as such an equality is the reason for use of phrases such as ”Now I see”, when verbal comprehension occurs. While verbal perception occurring in sentence comprehension involves only a small number of elements, visual perception by contrast requires inte- gration of a very large number of elements. Same can be said of other modes of perception, such as auditory, tactile, kinesthetic, olfactory, propeoceptory. Further, in the conscious experience of an observer, these various percep- tions from different senses, are further integrated into a whole, which even has non-sensory components such as thoughts, memories, feelings, emotions. The complete momentary experience of a conscious observer is thus a set, in context of Cantor’s original definition. Further its a finite set, as it has a finite number of elements, as represented by finite number of brain neurons. 3 Quantum Mechanical Aspects von Neumann [27], Wigner [28] and Pauli [29] have suggested the that wave- packet reduction was occurring when the wave-packet was interacting with the conscious observer. Precise mechanism for this reduction by interac- tion with consciousness has never been worked out. On the other hand cer- tain mechanisms for wave-packet reduction have indeed been worked out by physicists, e.g., wave-packet reduction occurs when a coherent (unreduced) wave-packet approaches a system with infinite degrees of freedom [30, 31]. Hawking [?] has also obtained interesting results concerning wave-packet re- duction near a black-hole when a coherent state approaches it. There exist other interpretations of quantum mechanics which try to do away with concept of wave-packet reduction all together. These include Ev- erett’s many world interpretation [33] , Bohm’s interpretation Bohm 1957, various hidden variable scenarios Bellifante 1973 etc.. However, experimental evidence of quantum zeno effect Itano 1990, and EPR suggests that wave- packet reduction is a good concept to explain the results. I therefore assume that wave-packet reduction does occur, and further it is caused by conscious observer. What are the properties of consciousness which solves these quan- tum measurement problem? Let’s refer to this set of properties of conscious- ness as OM ( O - Observer, M - Measurement). Acronym OM has been selected, because of special significance word ’Om’ has in context of search for one’s true identity in ancient Indian philosophy. von Neumann’s model can be understood in terms of ”Quantum Mea- surement Chain” (QMC). In this when an attempt is made to record state of say Schrodinger’s cat, by a camera, the wave function of camera also passes into a coherent state. Same happens to the wave functions of film, human eyes, retina, brain neurons and so on. In von Neumann’s interpretation, ob- server lies at the end of this quantum measurement chain, and leads to wave packet reduction of all the intermediary links (camera, brain, neurons) in this quantum measurement chain. This chain is originating at Schrodinger’s cat. Its possible for more than one QMC to originate from the same system, e.g., when multiple observers are monitoring state of Schrodinger’s cat. Such quantum measurement chaincs will be called linked. Now, apriori there is no reason, why if consciousness is causing wave- packet reduction (and influencing physical universe), it is not through the two mechanisms already outlined by physicists Hepp, Fakuda, and Hawking (above). In absence of any other description of process of consciousness causing wave-packet reduction, I accept what is not forbidden. Thus, as a conceptual agent responsible for wave-packet reduction, I attribute following properties to consciousness - OM-1. Consciousness is a system of infinite degrees of freedom. OM-2. Embodied Consciousness is associated with a black-hole. Reason for use of letters O and M in OM will become clear in sections 4 and 5. 4 Black-hole in brain! While popular notion of black-holes as massive astro-physical objects still awaits experimental confirmation, theoretical physics has moved ahead with concept of mini-black-holes, which are much less massive, and those which can be light enough to have mass of elementary particles, such as protons [37, 38]. While the astro-physical black-holes are caused by gravitational collapse of stars whose mass exceeds the so called Chandershekhar limit, the concept of micro-mini-black-hole (MMBH) is more like a singularity or hole in fabric of space, i.e., its a place, where physical space ceases to exist, so to say. Its gravitational influence is negligible, being proportional to its mass, and so is its size. Its interesting because of its relativistic, quantum mechanical, and philosophical properties. The reader thus need not be alarmed, that all of his or her gray matter will be sucked down this infernal black-hole in brain. There is an interesting philosophical reason for existence of a black-hole associated with consciousness in brain. While the soul or conscious ob- server, is regarded as a non-physical object, non-localizable in space-time, the present brain studies have almost localized it to be the region within brain. Black-hole provides a escape for the non-material spirit. While the black-hole can be given physical co-ordinates, the area within the event hori- zon, of the black-hole, effectively does not belong to the physical universe - it lies beyond the physical, universe so to say. Thus conscious observer located within such a black-hole strictly speaking does not exist within phys- ical space-time. In the event of physical death, the mind-body connection is severed, (e.g. Moody [39] , and one experts this black-hole to dissolve, or evaporate by a mechanism, analogous to ”Hawking radiation” [40]. Its energy will be carried away by gravitational waves, and therefore will lead to a mass loss equal to mass of MMBH (few grams). Also effect of these grav- itational waves generated at the moment of death, will be similar to high frequency acoustic waves, and would lead to cracking of any glass enclosure containing the physical body. Experiments verifying these phenomena have actually been done at the turn of this century [41]. The kind of tunnel vision reported in near death experiences, i.e., motion through a long dark tunnel, with light at the end of tunnel [39], is actually in accordance with optics near black-holes - an observer escaping through a black-hole would actually experience, similar tunnel vision! Black-holes have many other interesting properties such as existence of local closed time like curves, Morris 1988 which could explain ability of clair- voyants to see past and future events, while existing within the physical body (by motion of point like conscious observer, on one of the local closed time like curves near MMBH). Black holes also provide a handle, or gate- way to other dimension, and non-physical universes, which are of special interest Brahma Kumaris and possibly other workers in Transcendental. In- terestingly, tachyons (particles traveling faster than light) falling through a black-hole, leave the conventional physical universe of three space and one time dimension, and enter a universe with three time and one space dimen- sion Chandola 1986. This latter universe or rather meta-universe, could be of special interest for actual meta-physical experiences. 5 Records within Consciousness Apriori, von Neumann’s interpretation of consciousness causing wave-packet reduction, does not determine, as to which particular outcome is actually selected. Neither do any other mechanisms such as Fakuda’s, Hepp’s or Hawking’s - all they do is reduce a coherent superposition of states into an incoherent mixture - actual outcome then being a psychological process of observation. Thus if embodied consciousness is monitoring quantum states of 109 neurons, and causing their wave-packet reductions, which leads to per- ception, and the complete momentary experience of that observer, then these wave-packet reductions must be recorded within the conscious observer, in a consistent fashion, and cannot be random because - random reductions, will not lead to the perceived order of the universe. Various laws of physics, such as those of continuity, conservation, invariances, etc., are result of per- ceptions, based upon these wave-packet reductions. Hence we can identity another property of consciousness - OM-3. Recorded within conscious observer is outcome of all quantum mea- surements performed by it (as reflected in coherent brain states). These reductions are not random, but have a logical relation to each other, which is the basis for invariances observed in physics, and results of EPR and Bell’s inequality experiments etc. Now this selection of a particular outcome, from the set of all possible out- comes is another psychological process, which was encountered quite early ;in development of set theory. Its called Axiom of Choice [44]. Briefly, it states that, given a set, there exists a choice function, which selects an element of the set. Only problem is, that while it is vital to almost all of mathematics, its use has lead to paradoxical Banach-Tarski theorems, involving duplica- tion of spheres, and show that concept of additive measure is not sound [45]. Details of how axiom of choice relates to quantum measurement process are worked out in [?] There exists another reason for records within consciousness, and evi- dence from it comes from following scenario reminiscent of EPR of quantum mechanics. The argument has a philosophical flavor - if distinct observers A and B separated by a space-like interval, observe state of Schrodinger’s cat in an experiment at a particular instant, it is required that both should observe it either dead or alive. It should not be that observer A, finds cat alive, and observer B finds cat dead. Thus if wave function of Schrodinger’s cat as represented in brain neurons of observer A is collapsing, due to a record- ing within A, this collapse is compatible with a similar collapse occurring in brain of observer B. To ensure this compatibility we require - OM-4. Recording within conscious observers with respect to measurement performed on the same quantum system are mutually compatible. Alterna- tively, outcomes observed by observers lying at ends of distinct but linked quantum measurement chains, are compatible. Support of EPR results of quantum mechanics for records within con- sciousness is as follows. Lets say that observers A and B are separated by a space-like interval, and perform measurement on a correlated photon pair. EPR results indicate that wave functions of both the photons are collapsing only at the moment of measurement, and the collapses are mutually com- patible, which A and B will also notice, when they compare notes latter on. Now, there exists no way for observer A to send a signal to B, regarding his or her outcome, within the frame work of present day physics (light speed limit and all that). The question therefore exists, if the conscious observers A and B are indeed causing these distinct but correlated collapses, how is mutual compatibility being ensured? OM3 is thus related to OM4. This is also where cyclic nature of time may be playing an important role. In cyclic time, the same measurements would have been performed in all the past (infinite) time cycles, and identical outcomes would have been recorded, in all the time cycles. This geometry of time, appears to provide at least a chance for observers to compare note and correlate their outcomes. See [9] for a possible scenario of communication between observers in EPR type experiments using light like signals in cyclic time. I close this section, with an argument, for why the embodied consciousness should exist in a MMBH (micro-mini-black-hole). If outcomes of quantum measurement are pre-recorded in the conscious observers, (OM-3 and OM-4) such a recording constitutes ”hidden variable [35] determining the quantum measurement outcome”. Now Bell’s theorem [47] yielded inequalities, which would distinguish between hidden variable scenario, and actual quantum me- chanics (without hidden variables). Experiments [11]-[16] to test between the two yielded results in accordance with quantum mechanics, i.e., no functions or additional physical hidden variables, which would determine the outcome. Locating the consciousness within a black-hole resolves this problem, because now the recording (hidden variable determining the outcome) is lying beyond the event horizon of the black-hole, and thus effectively outside the universe, and therefore is beyond the purview of present formulation of Bell’s theorem [47]. 6 Neuro-Biological-Quantum-Zeno-Effect Readers would be familiar with ancient Greek Zeno’s paradox [49], which questioned the concept of motion, by arguing that if an arrow in flight was being continuously observed, and occupied a position at every time instant, as to how could the apparent motion observed was actually possible? Though the paradox was resolved in continuum based classical mechanics, it has reap- peared in grab of quantum zeno effect (QZE) - so christened by physicist Sudarshan [49]. Briefly, if a system is in state A, and about to change to state B, before it does so, its wave function (mathematical object describing its physical state), has to go into a superposition of states A and B, i.e., the system exists in a sort of (A+B) state. Now when a quantum measurement is performed on this superposed (A+B) state, the wave function collapses to either A or B. So, if the measurements are being performed sufficiently rapidly, the wave function of the system cannot evolve to first state (A+B) and than state B. As a result it remains frozen in state A. This effect is also called ”watch dog effect”, [50] (thief moves only when watch dog closes its eyes), and the ”boiled kettle phenomena”, (kettle appears to boil over and spill, just when one’s attention is diverted). Thus in terms of quantum me- chanics, paradox of Zeno’s arrow can be formulated and resolved as follows. Where as, wave-function of arrow, which also describes its position is evolv- ing continuously, the actual act of wave-packet reduction, by monitoring or perception of a conscious observer, is a non-continuos phenomena. This is because the process of human perception requires a large number of photons from Zeno’s arrow to reach human eye and retina, where after a time de- lay, a signal is relayed to brain, and a quantum mechanical representation of arrow’s (superposed) wave function is formed by observer’s neurons. This quantum measurement chain of coherent (superposed) state collapses when the conscious observer perceives arrow’s position, and is a non-continuos phe- nomena. Thus, in between these non-continuos perceptions, wave function of arrow can evolve to different positions. QZE has been verified for ensembles of atoms about to make electron transitions from a higher energy state to a lower energy state [36]. Monitoring at progressively smaller intervals, reduces actual number of atoms making the transition, in a given period of time. Neuro-biological-quantum-zeno-effect (NBQZE), as the term suggests, implies that, if brain state of a person is being monitored at sufficiently small space-time scales, (by another person, with data being recorded onto say a computer, all of which is later examined by the experimenter), then neurons of the subject will not be able to evolve to a coherent state and make transition, to another state which would represent transitions from one per- ception to another. Thus person’s subjective experience would be blocked. Even though external sensory stimulus may be applied, the subject would not report perception of the stimulus. The resultant state would be similar to highest states of meditation, which involve complete withdrawal of conscious- ness from the body and senses - effectively the consciousness has ceased to interact with the physical universe, and is no longer performing any quantum measurement - wave-packet reductions, in his or her brain are being caused by the experimenter, and are preventing brain state from evolving along with the wave function of the changing environment or universe. References [1] Ecclles, J. C., and Popper, K. R. : Self and its Brain, Routledge and Kegan Paul (1977). [2] Crick, F. : The Astonishing Hypothesis - The Scientific Search for the Soul, Simon and Schuster, London (1994). [3] Zeh, H.: The Physical Basis of the Direction of Time, Springer-Verlag (1989). [4] Sachs, R.G.: The Physics of Time Reversal, University of Chicago Press (1987). [5] Mackey, M. C. : Time’s Arrow: The Origins of Thermodynamic Behavior, Spriger-Verlag (1992). [6] Davies, P. : The Physics of Time Asymmetry, University of California Press (1977). [7] Blackmore, S.J.: Beyond the Body, Academy Chicago Publishers (1992). [8] Newton-Smith, W.H. : The Structure of Time, Routledge and Kegan Paul (1980). [9] Modgil, M. S. : Time irreversibility implies energy non-conservation or global causality violation, Pre-print (1995). [10] Einstein, A., Podolsky, B., Rosen, N.: Phy. Rev., 47, 777-80 (1935). [11] Freedman, F. J. and Clauser, J.F. : Phys. Rev. Lett., 28, 938-941 (1972). [12] Clauser, J.F. : Phys. Rev. Lett., 36, 1223-6 (1976). [13] Fry, E.S. and Thompson, S.C. : Phys. Rev. Lett., 37, 465-8 (1976). [14] Aspect, A., Grangier, P. and Roger, G. : Phys. Rev. Lett. 47, 460-3 (1981); Phys. Rev. Lett., 49, 91-4 (1982). [15] Aspect, A., Dalibard, J. and Roger, G. : Phys. Rev. Lett., 49, 1804-7 (1982). [16] Aspect, A. : Wave Particle Dualism, pp337-390, ed. Dinner, S., Far- gue, D., Lochak, G. and Selleri F., Reidel, Dordrecht (1984) . [17] de Beauregard, C. : Found. Phys., 15, 871 (1985). [18] Poincare, H. : Acta Mathematica, 13, 1-270 (1890); Revue de Meta- physique et de Morale 1, 534-7 (1893). [19] Zermelo, E. : Annalen der Physik, 57, 485-94 (1896); Annalen der Physik, 59, 793-801 (1896); Caratheodory, C., Sitzber, Preuss. Akad. Wiss., 579 (1919). [20] Brush, S.G. : Kinetic Theory, Vol.2 Irreversible Processes, Pergamon Press 1966. This book also gives English translations of relevant parts of papers of Nitzsche, Poncare, Zermelo and Boltzmann; Kaufman, W. A. : Nietzche: Philosopher, Psychologist, Antichrist, Chapter 11, Princeton University Press (1950) . [21] Nietzsche, C. A.; sa Vie sa Pensee, Gallimard, Paris, Vol. 4, Livre 2, chap. I and 3; Der Wille zur Macht,in his, Gesammelte Werke, Musarion Verlag, Munich (1926), Vol. 19, Book 4, Part 3; English translation by, Manthey-Zorn, O., in Nietzsche, An Anthology of his works, Washington Square Press, New York (1964) p.90. [22] Guillemin, V. : Cosmology in (2+1)-dimensions, cyclic models and deformations of M − 2, 1 , Princeton University Press (1989). [23] Segal, I. E.: Mathematical Cosmology and Extra-galactic Astronomy, Cambridge University Press (1984). [24] Schechter, B.M. : Phy. Rev. D 16, 3015 (1977); Villaroel, J. : I1 Nuo. Cim. A 94, 405 (1986). [25] Castell, L. : Phy. Rev. D 6, 536 (1972). [26] Ingberg, L. : Phys. Rev., A 8, 396 (1983); 29, 346 (1984); 31, 1183 (1985). [27] von Neumann J.: Mathematical foundations of quantum mechanics, Princepton University Press (1955). [28] Wigner, E.W.: A scientist speculates, (1966). [29] W. Pauli quoted in, Mind, Matter, and Quantum Mechanics, (Springer-Verlag, Berlin, 1993), Chap. 7. [30] Hepp, K. : Helv. Phys. Acta., 45, 237 (1972). [31] Fakuda, R. : Phys. Rev., A, 35, 8 (1987); 36, 3023 (1987). [32] Hawking, S. W. : Quantum Coherence down the wormhole, in , Quan- tum Gravity, Proceedings of the Fourth Seminar on Qunatum Grav- ity, Moscow, Markov, M.N., Berezin, V. A., and Frolov, V. P. (eds.), World Scientific, Singapore, p125. [33] Everett E.: Rev. Mod. Phys., 29, 454-462, (1957). [34] Bohm, D. : textbfWholeness and Implicate Order, Routledge, 1980. [35] Bellinfante, F. J.: A Survey of Hidden Variables Theories, Pergamon Press, 1973. [36] Itano, W. M., Heinzen, D.K., Bollinger, J. J. and Wineland, D. J.: Phys. Rev. A 41, 2295 (1990). [37] Recami, E. and Tonin-Zanchin, V.: Unusual Black Holes, in, About some stable (non-evaporating) Extremal Solutions of Einstein Equa- tions, Mukherjee, S., Prasanna, A.R. and Kembhavi, A.K. (eds.), Wiley Eastern Ltd. (1992). [38] Einstein, A. and Rosen, N.: Phys. Rev., 48, 73 (1935). [39] Moody Jr., R. A. : Reflections on Life after Life, Bantam (1977). [40] Hawking, S. W. : Nature, 248, 30 (1974); Commun. Math. Phys. , 43, 199 (1975). [41] MacDougall, D. : Hypothesis concerning Soul Substance Together with Experimental Evidence for the Existence of Such Substance, in, Journal of American Society for Psychical Research, 1, 237-44 (1907). [42] Morris, M.S., Throne, K.S., and Yurtsever, U.: Phys. Rev. Lett., 61, 1446-1449, (1988). [43] Chandola, H. C., Rajput, B.. S., Sagar, R. and Verma, R. C. : Indian J. Pure and Appl. Phys. , 24, 51 (1986). [44] Moore, G. M.: Axiom of Choice, Its Otigin, Development and Influ- ence, Springer-Verlag (1982). [45] Wagon, S.: The Banach-Tarski Paradox, Encylop. of Math. and its applications, Vol. 24, Cambridge University Press (1985). [46] Modgil, M., S. : Axiom of Choice and Quantum Measurement, preprint (1992). [47] Bell, J. S.: Physics, 1, 195 (1964). [48] Grunbaum, A.: Modern Science and Zeno’s Paradoxes, George Allen and Unwin Ltd, London (1967). [49] Mishra, B. and Sudarshan, E. C. G.: J. Math. Phys. , 18, 756 (1977). [50] Joos, E. : Phys. Rev., D 29, 1626 (1984). Introduction Set Theoretic Connection. Quantum Mechanical Aspects Black-hole in brain! Records within Consciousness Neuro-Biological-Quantum-Zeno-Effect

0704.1055

Probing the Structure of Gamma-Ray Burst Jets with Steep Decay Phase of their Early X-ray Afterglows

published in ApJ PROBING THE STRUCTURE OF GAMMA-RAY BURST JETS WITH THE STEEP DECAY PHASE OF THEIR EARLY X-RAY AFTERGLOWS Kentaro Takami1, Ryo Yamazaki1, Takanori Sakamoto2, and Goro Sato2 ABSTRACT We show that the jet structure of gamma-ray bursts(GRBs) can be inves- tigated with the tail emission of the prompt GRB. The tail emission that we consider is identified as a steep decay component of the early X-ray afterglow observed by the X-Ray Telescope on board Swift. Using a Monte Carlo method, we derive for the first time the distribution of the decay index of the GRB tail emission for various jet models. The new definitions of the zero of time and the time interval of a fitting region are proposed. These definitions for fitting the light curve lead us to a unique definition of the decay index, which is useful to investigate the structure of the GRB jet. We find that if the GRB jet has a core-envelope structure, the predicted distribution of the decay index of the tail has a wide scatter and multiple peaks, which cannot be seen for the case of the uniform- and the Gaussian jet. Therefore, the decay index distribution gives us information about the jet structure. Especially if we observe events whose decay index is less than about 2, both the uniform- and the Gaussian jet models will be disfavored, according to our simulation study. Subject headings: gamma-rays: burst — gamma-rays: theory 1. Introduction Gamma-ray burst (GRB) jet structure, that is, the energy distribution E(θ) in the ultra- relativistic collimated outflow, is at present not yet fully understood (Zhang & Mészáros 1 Department of Physics, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan; [email protected], [email protected]. 2 NASA Goddard Space Flight Center, Greenbelt, MD 20771. http://arxiv.org/abs/0704.1055v3 – 2 – 2002b). There are many jet models proposed in addition to the simplest uniform-jet model: the power-law jet model (Rossi et al. 2002; Zhang & Mészáros 2002a), the Gaussian jet model (Zhang et al. 2004), the annular jet model (Eichler & Levinson 2004), the multiple emitting subshell model (Kumar & Piran 2000; Nakamura 2000), the two-component jet model (Berger et al. 2003), and so on. The jet structure may depend on the generation process of the jet and therefore may provide us important information about the central engine of the GRB. For example, in the collapsar model for long GRBs (e.g., Zhang et al. 2003, 2004), the jet penetrates into and breaks out of the progenitor star, resulting in the E(θ) ∝ θ−2 profile (Lazzati & Begelman 2005). For the compact binary merger model for short GRBs, hydrodynamic simulations have shown that the resulting jet tends to have a flat core surrounded by the power-law-like envelope (Aloy et al. 2005). In the pre-Swift era, there were many attempts to constrain the GRB jet structure. Thanks to the HETE-2, statistical properties of long GRBs, X-ray-rich GRBs, and X- ray flashes were obtained (Sakamoto et al. 2005), which were thought to constrain the jet models (Lamb et al. 2004). These observational results constrain various jet models, such as the uniform-jet model (Yamazaki et al. 2004a; Lamb et al. 2005; Donaghy 2006), the multiple subshell model (Toma et al. 2005b), the Gaussian jet model (Dai & Zhang 2005), and so on. For BATSE long GRBs, Yonetoku et al. (2005) derived the distribu- tion of the pseudo-opening angle, inferred from the Ghirlanda (Ghirlanda et al. 2004) and Yonetoku (Yonetoku et al. 2004) relations, as f(θj)dθj ∝ θ j dθj , which is compatible with that predicted by the power-law jet model as discussed in Perna et al. (2003) (however, see Nakar et al. 2004). Afterglow properties are also expected to constrain the jet structure (e.g., Granot & Kumar 2003); however, energy redistribution effects prevent us from reach- ing a definite conclusion. Polarization measurements of optical afterglows bring us important information (Lazzati et al. 2004). In the Swift era, rapid follow-up observation reveals prompt GRBs followed by a steep de- cay phase in the X-ray early afterglow (Tagliaferri et al. 2005; Nousek et al. 2006; O’Brien et al. 2006a). In the most popular interpretations, the steep decay component is the tail emis- sion of the prompt GRB (the so called high-latitude emission), i.e., the internal shock origin (Zhang et al. 2006; Yamazaki et al. 2006; Liang et al. 2006; Dyks et al. 2005), al- though there are some other possibilities (e.g., Kobayashi et al. 2007; Panaitescu et al. 2006; Pe’er et al. 2006; Lazzati & Begelman 2006; Dado et al. 2006). Then, for the uniform-jet case, the predicted decay index is α = 1 − β, where we use the convention Fν ∝ T −αν1+β (Kumar & Panaitescu 2000). For power-law jet case (E(θ) ∝ θ−q), the relation is modified to α = 1−β+(q/2). However, these simple analytical relations cannot be directly compared with observations, because they are for the case in which the observer’s line of sight is along the jet axis and because changing the zero of time, which potentially lies anywhere within – 3 – the epoch where we see the bright pulses, substantially alters the early decay slope. Recently, Yamazaki et al. (2006) (Y06) investigated the tail emission of the prompt GRB, finding that the jet structure can be described and that the global decay slope is not so much affected by the local angular inhomogeneity as it is affected by the global energy distribution. They also argued that the structured jet model is preferable, because steepening GRB tail breaks appeared in some events. In this paper, we calculate for the first time the distribution of the decay index of the prompt tail emission for various jet models and find that the derived distributions can be distinguished from each other, so that the jet structure can be more directly constrained than previous arguments. This paper is organized as follows. We describe our model in § 2. In § 3, we investigate the distribution of the decay index of the prompt GRB emission. Section 4 is devoted to discussions. 2. Tail Part of the Prompt GRB Emission We consider the same model as discussed in the previous works (Y06; Yamazaki et al. 2004b; Toma et al. 2005a,b). The whole GRB jet, whose opening half-angle is ∆θtot, consists of Ntot emitting subshells. We introduce the spherical coordinate system (r, ϑ, ϕ, t) in the central engine frame, where the origin is at the central engine and ϑ = 0 is the axis of the whole jet. Each emitting subshell departs at time t (0 < t < tdur, where j = 1, · · · , Ntot, and tdur is the active time of the central engine) from the central engine in the direction of ~n(j) = (ϑ(j), ϕ(j)), and emits high-energy photons, generating a single pulse as observed. The direction of the observer is denoted by ~nobs = (ϑobs, ϕobs). The observed flux from the jth subshell is calculated when the following parameters are determined: the viewing angle of the subshell θ(j)v = cos −1(~nobs ·~n (j)), the angular radius of the emitting shell ∆θ , the departure time t , the Lorentz factor γ(j) = (1− β2 )−1/2, the emitting radius r 0 , the low- and high- energy photon indices α B and β B , the break frequency in the shell comoving frame ν (Band et al. 1993), the normalization constant of the emissivity A(j), and the source redshift z. The observer time T = 0 is chosen as the time of arrival at the observer of a photon emitted at the origin r = 0 at t = 0. Then, at the observer, the starting and ending times of the jth subshell emission are given by start ∼ t dep + 1 + γ2 , (1) 1 + γ2 , (2) where θ + = θ v +∆θ = max{0, θ(j)v −∆θ }, and we use the formulas β(j) ∼ 1−1/2γ and cos θ ∼ 1 − θ2/2 for γ(j) ≫ 1 and θ ≪ 1, respectively. The whole light curve from the – 4 – GRB jet is produced by the superposition of the subshell emission. Y06 discussed some kinematical properties of prompt GRBs in our model and found that each emitting subshell with θ(j)v ≫ ∆θ produces a single, smooth, long-duration, dim, and soft pulse, and that such pulses overlap with each other and make the tail emission of the prompt GRB. Local inhomogeneities in the model are almost averaged during the tail emission phase, and the decay index of the tail is determined by the global jet structure, that is the mean angular distribution of the emitting subshell because in this paper all subshells are assumed to have the same properties unless otherwise stated. Therefore, we are essentially studying the tail emission from the usual continuous jets at once, i.e., from uniform- or power-law jets with no local inhomogeneity. In the following, we study various energy distributions of the GRB jet through the change of the angular distribution of the emitting subshell. 3. Decay Index of the Prompt Tail Emission In this section, we perform Monte Carlo simulations in order to investigate the jet structure by calculating the statistical properties of the decay index of the tail emission. For a fixed-jet model, we randomly generate 104 observers with their line of sights (LOSs) ~nobs = (ϑobs, ϕobs). For each LOS, the light curve, F (T ) of the prompt GRB tail in the 15–25 keV band is calculated, and the decay index is determined. The adopted observation band is the low-energy end of the Burst Alert Telescope(BAT) detector and near the high- energy end of the X-Ray Telescope(XRT) on Swift. Hence, one can observationally obtain continuous light curves, beginning with the prompt GRB phase to the subsequent early afterglow phase (Sakamoto et al. 2007), so that it is convenient for us to compare theoretical results with observations. However, our actual calculations have shown that our conclusion is not qualitatively altered, even if the observation band is changed, for example, to 0.5–10 keV, as usually considered for other references. For each light curve, the decay index is calculated by fitting F (T ) with a single power- law form, ∝ (T − T∗) −α, as in the following (see Fig. 1). The decay index α depends on the choice of T∗ (Zhang et al. 2006; Yamazaki et al. 2006) 1. Let Ts and Te be the start and end time, respectively, of the prompt GRB, i.e., Ts = min{T start} (3) 1 Recently, Kobayashi & Zhang (2007) have discussed the way to choose the time zero. According to their arguments, the time zero is near the rising epoch of the last bright pulse in the prompt GRB phase. – 5 – Te = max{T } . (4) Then, we take T∗ as the time until 99% of the total fluence, which is defined by Stotal = F (T ′) dT ′, is radiated, that is, F (T ′) dT ′ = 0.99Stotal . (5) Then, the prompt GRB is in the main emission phase for T < T∗, while it is in the tail emis- sion phase for T > T∗. The time interval [Ta, Tb], in which the decay index α is determined assuming the form F (T ) ∝ (T − T∗) −α, is taken to satisfy F (Ta,b) = qa,bF (T∗) , (6) where we adopt qa = 1 × 10 −2 and qb = 1 × 10 −3, unless otherwise stated. We find that in this epoch the assumed fitting form gives a well approximation. At first, we consider the uniform-jet case, in which the number of subshells per unit solid angle is approximately given by dN/dΩ = Ntot/(π∆θ tot) for ϑ < ∆θtot, where ∆θtot = 0.25 rad is adopted. The departure time of each subshell t is assumed to be homogeneously random between t = 0 and t = tdur = 20 sec. The central engine is assumed to produce Ntot = 1000 subshells. In this section, we assume that all subshells have the same values of the following fiducial parameters: ∆θsub = 0.02 rad, γ = 100, r0 = 6.0×10 14 cm, αB = −1.0, βB = −2.3, hν 0 = 5 keV, and A = constant. Our assumption of constant A is justified as follows. Note that the case in which N subshells that have the same brightness A are launched into the same direction, but a different departure time, is equivalent to the case of one subshell emission with the brightness of NA. This is because in the tail emission phase, the second terms in the r.h.s. of Eqs. (1) and (2) dominate the first terms, so that the time difference effect, which arises from the difference of tdep for each subshell, can be obscured. Hence, giving the angular distribution of the emission energy is equivalent to giving the angular distribution of the subshells with constant A. Also, Y06 showed that to obtain a smooth, monotonic tail emission as observed by Swift, the subshell properties, hν ′0 and/or A(j), cannot have wide scatter in the GRB jet. Therefore, we can expect, at least as the zeroth-order approximation, that the subshells have the same properties. The left panel of Fig. 2 shows the decay index α as a function of ϑobs. For ϑobs . ∆θtot (on-axis case), α clusters around ∼ 3. On the other hand, when ϑobs & ∆θtot (off-axis case), α rapidly increases with ϑobs. The reason is as follows. If all subshells are seen sideways (that is, θ(j)v ≫ ∆θ for all j), the bright pulses in the main emission phase followed by the tail emission disappear because of the relativistic beaming effect, resulting in a smaller flux contrast between the main emission phase and the tail emission phase compared with the on- axis case. Then T∗ becomes larger. Furthermore, in the off-axis case, the tail emission decays – 6 – more slowly (|dF/dT | is smaller) than in the on-axis case. Then both Ta−T∗ and Tb−T∗ are larger for the off-axis case than for the on-axis case. As can be seen in Fig. 3 of Zhang et al. (2006), the emission seems to decay rapidly, so that the decay index α becomes large. The left panel of Fig. 3 shows α as a function of the total fluence Stotal which is the sum of the fluxes in the time interval, [Ts, Te]. In Fig. 3, both α and Stotal are determined observationally, so that our theoretical calculation can be directly compared with the observation. A more realistic model is the Gaussian jet model, in which the number of subshells per unit solid angle is approximately given by dN/dΩ = C exp(−ϑ2/2ϑ2c) for 0 ≦ ϑ ≦ ∆θtot, where C = Ntot/2πϑ c [1 − exp(−∆θ tot/2ϑ c)] is the normalization constant. We find only a slight difference between the results for the uniform- and the Gaussian jet models. Therefore, we do not show the results for the Gaussian jet case in this paper. Next, we consider the power-law distribution. In this case, the number of subshells per unit solid angle is approximately given by dN/dΩ = C[1 + (ϑ/ϑc) 2]−1 for 0 ≦ ϑ ≦ ∆θtot, i.e., dN/dΩ ≈ C for 0 ≦ ϑ ≪ ϑc and dN/dΩ ≈ C(ϑ/ϑc) −2 for ϑc ≪ ϑ ≦ ∆θtot, where C = (Ntot/πϑ c)[ln(1 + (∆θtot/ϑc) 2)]−1 is the normalization constant and we adopt ϑc = 0.02 rad and ∆θtot = 0.25 rad. The other parameters are the same as for the uniform-jet case. As can be seen in the right panels of Figs. 2 and 3, both the ϑobs–α and Stotal–α diagrams are complicated compared with the uniform-jet case. When ϑobs . ϑc, the observer’s LOS is near the whole jet axis. Compared with the uniform-jet case, α is larger, because the power- law jet is dimmer in the outer region, i.e., emitting subshells are sparsely distributed near the periphery of the whole jet (see also the solid lines of Figs. 1 and 3 of Y06). If ϑobs ≫ ϑc, the scatter of α is large. Some bursts have an especially small α of around 2. This comes from the fact that the power-law jet has a core region (0 < ϑ . ϑc), where emitting subshells densely distributed compared with the outer region. The core generates the light-curve break in the tail emission phase, as can be seen in Fig. 4 (Y06). In the epoch before the photons emitted by the core arrive at the observer (e.g., T − Ts . 7.5 × 10 2 s for the solid line in Fig. 4), the number of subshells that contribute to the flux at time T , Nsub(T ), increases with T more rapidly than for the uniform-jet case. Then, the light curve shows a gradual decay. If the fitting region [Ta, Tb] lies in this epoch, the decay index α is around 2. In the epoch after the photons arising from the core are observed (e.g., T −Ts & 7.5× 10 2 s for the solid line in Fig. 4), the subshell emission with θ(j)v & ϑobs + ϑc is observed. Then Nsub(T ) rapidly decreases with T , and the observed flux suddenly drops. If the interval [Ta, Tb] lies in this epoch, the decay index becomes larger than 4. To compare the two cases considered above more clearly, we derive the distribution of the decay index α. Here we consider the events whose peak fluxes are larger than 10−4 times of the largest one in all simulated events, because the events with small peak fluxes are not – 7 – observed. Fig. 5 shows the result. For the uniform-jet case (solid line), α clusters around 3, while for the power-law jet case (dotted line), the distribution is broad (1 . α . 7) and has multiple peaks. So far, we have considered the fiducial parameters. In the following, we discuss the dependence on parameters, r0, γ, βB, tdur, and ∆θtot (It is found that the α-distribution hardly depends on the value of αB, ∆θsub, and ν 0 within reasonable parameter ranges). At first, we consider the case in which r0 = 1.0 × 10 14 cm is adopted, with other parameters being fiducial. Fig. 6 shows the result. The shape of the α-distribution is almost the same as that for the fiducial parameters, in both the uniform- and the power-law jet cases. This comes from the fact that in a tail emission phase, the light curve for a given r0 is approximately written as F (T ; r0) ≈ g(cT/r0), where a function g determines the light-curve shape of the tail emission for other given parameters. Then, the light curves in the case of r0 = r0,1 and r0 = r0,2, namely, F (T ; r0,1) and F (T ; r0,2), satisfy the relation F (T ; r0,2) ≈ F ((r0,1/r0,2)T ; r0,1). This can be seen, for example, by comparing the solid line with the dotted one in Fig. 4. Hence, T∗, Ta, and Tb are approximately proportional to r0; in this simple scaling, one can easily find that α remains unchanged for different values of r0. Second, we consider the case of γ = 200 and r0 = 2.4× 10 15 cm, with other parameters being fiducial. In this case, the angular spreading timescale (∝ r0/γ 2) is the same as in the fiducial case, so that the tail emissions still show smooth light curves, although the whole emission ends later, according to the scaling Te ∝ r0 (see the dot-dashed line in Fig. 4). Fig. 7 shows the result. For large γ, the relativistic beaming effect is more significant, so that the events in ϑobs & ∆θtot, which cause large α, are dim compared with the small-γ case. Such events cannot be observed. For the power-law jet case, therefore, the number of large-α events becomes small, although the distribution is still broad (1 . α . 4) and has two peaks. On the other hand, for the uniform-jet case, the distribution of the decay index α is almost the same as for the fiducial parameter set, because the value of the decay index α in the case of ϑobs . ∆θtot is almost the same as that in the case of ϑobs & ∆θtot. Third, we change the value of the high-energy photon index βB from −2.3 to −5.0, with other parameters being fiducial. Fig. 8 shows the result. For the uniform-jet case, the mean value is 〈α〉 ∼ 4, while 〈α〉 ∼ 3 for the fiducial parameters, so that the decay index defined in this paper does not obey the well-known formula α = 1−βB (Kumar & Panaitescu 2000). For the power-law jet case, the whole distribution shifts toward the higher value, and the ratio of the two peaks changes. In the tail emission phase, the spectral peak energy Epeak is below 15 keV (see also Y06), so that the steeper the spectral slope of the high-energy side of the Band function, the more rapidly the emission decays, resulting in the dimmer tail emission (see the dashed line in Fig. 4). Then, the fitting region [Ta, Tb] shifts toward earlier – 8 – epochs, because T∗ becomes small. Therefore, the number of events with small α increases, and the number of events with large α decreases. Furthermore, we comment on the case in which βB is varied for each event in order to more directly compare with the observation. Here we randomly distribute βB according to the Gaussian distribution with a mean of −2.3 and a variance of 0.4. It is found that the results are not qualitatively changed. Next, we change the value of the duration time tdur from 20 sec to 200 sec, with other parameters being fiducial. The epoch of the bright pulses in the main emission phase becomes longer than that in tdur = 20 sec. However, the behavior of the tail emission does not depend on tdur very much (see Fig. 9). Therefore, the distribution of the decay index α is almost the same as that for the fiducial parameters for both the uniform-jet case and the power- law jet case. Even if we consider the case in which tdur is randomly distributed for each event according to the lognormal distribution with an average of log(20 s) and a logarithmic variance of 0.6, the results are not significantly changed. Finally, we discuss the dependence on ∆θtot. Only the uniform-jet case is considered, because the structured jet is usually quasi-universal and because we focus our attention on the behavior of the uniform-jet model. The dotted line in Fig. 10 shows the result for constant ∆θtot = 0.1 rad with other parameters being fiducial. We can see many events with large α. The large α is observed because for small ∆θtot, although the off-axis events (i.e., ∆θtot . ϑobs) are still dim because of the relativistic beaming effect, a fraction of such events survives the flux threshold condition and are observable. Such events have large α & 5 (see the 4th paragraph of this section, which explains the left panel of Fig. 2). This does not occur in the large-∆θtot case. However, we still find in this case that there are no events with α . 2. We consider another case in which ∆θtot is variable. Here we generate events whose ∆θtot distributes as f∆θtotd(∆θtot) ∝ ∆θtot −2d(∆θtot) (0.05 . ∆θtot . 0.4). Then for a given ∆θtot, the quantities ν 0 and A are determined by hν 0 = (∆θtot/0.13) −3.6 keV and A ∝ (∆θtot) −7.3, respectively. Other parameters are fiducial. If the model parameters are chosen in this way, the Amati and Ghirlanda relations (Amati et al. 2002; Ghirlanda et al. 2004) are satisfied, and the event rates of long GRBs, X-ray-rich GRBs and X-ray flashes become similar (Donaghy 2006). The solid line in Fig. 10 shows the result. Again we find that there are no events with α . 2. In summary, when we adopt model parameters within reasonable ranges, the decay index becomes larger than ∼ 2 for the uniform- and the Gaussian jet cases, while a significant fraction of events with α . 2 is expected for the power-law jet case. Therefore, if a non- negligible number of events with α . 2 are observed, both the uniform- and the Gaussian jet models will be disfavored. Furthermore, if we observationally derive the α-distribution, the structure of GRB jets will be more precisely determined. – 9 – 4. Discussion We have calculated the distribution of the decay index, α, for the uniform-, Gaussian, and the power-law jet cases. For the uniform-jet case, α becomes larger than ∼ 2, and its distribution has a single peak. The Gaussian jet model predicts almost the same results as the uniform-jet model. On the other hand, for the power-law jet case, α ranges between ∼ 1 and ∼ 7, and its distribution has multiple peaks. Therefore, we can determine the jet structure of GRBs by analyzing a lot of early X-ray data showing a steep decay component that is identified as a prompt GRB tail emission. However, one of the big challenges in the Swift data for calculating the decay index in our definition is to derive the composite light curve of BAT and XRT. Since the observed energy bands of BAT and XRT do not overlap, we are forced to extrapolate one of the data sets to plot the light curve in a given energy band. To derive the composite light curve unambiguously for a prompt and an early X-ray emission, we need an observation of a prompt emission by current instruments, which overlap the energy range of XRT. The tail behavior with α . 2 does not appear in the uniform- and the Gaussian jet models; hence, it is important to constrain the jet structure. However, in practical observa- tions, such gradually decaying prompt tail emission might be misidentified with the external shock component, as expected in the pre-Swift era. Actually, some events have shown such a gradual decay, without the steep and the shallow decay phases, and their temporal and spectral indices are consistent with a classical afterglow interpretation (O’Brien et al. 2006b). Hence, in order to distinguish the prompt tail emission from the external shock component at a time interval [Ta, Tb], one should study the spectral evolution and/or the continuity and smoothness of the light curve (i.e., whether breaks appear or not) over the entire burst emission. In this paper, we adopt qa = 1×10 −2 and qb = 1×10 −3 when the fitting epoch [Ta, Tb] is determined [see Eq. (6)]. Then, the prompt tail emission in this time interval is so dim that it may often be obscured by the external shock component, causing a subsequent shallow decay phase of the X-ray afterglow. One possible way to resolve this problem is to adopt larger values of qa and qb, e.g., qa = 1/30 and qb = 1 × 10 −2, in which the interval [Ta, Tb] shifts toward earlier epochs, so that the flux then is almost always dominated by the prompt tail emission. We have calculated the decay index distribution for this case (qa = 1/30 and qb = 1 × 10 −2) and have found that the differences between uniform- and power-law jets still arises as can be seen in the case of qa = 1 × 10 −2 and qb = 1 × 10 −3, so that our conclusion remains unchanged. However, the duration of the interval, Tb−Ta, becomes short, which might prevent us from observationally fixing the decay index at high significance. If qa & 1/30, the emission at [Ta, Tb] is dominated by the last brightest pulse. Then, the light- – 10 – curve shape at [Ta, Tb] does not reflect the global jet structure, but reflects the properties of the emitting subshell causing the last brightest pulse. Another way to resolve the problem is to remove the shallow decay component. For this purpose, the origin of the shallow decay phase should be clarified in order to extract the dim prompt tail emission exactly. The other problem is contamination of X-ray flares, whose contribution has to be removed in order to investigate the tail emission component. In any case, if the GRB occurs in an extremely low-density region (a so-called naked GRB), where the external shock emission is expected to be undetectable, our method may be a powerful tool to investigate the GRB jet structure. This work was supported in part by Grants-in-Aid for Scientific Research of the Japanese Ministry of Education, Culture, Sports, Science, and Technology 18740153 (R. Y.). T.S. was supported by an appointment of the NASA Postdoctoral Program at the Goddard Space Flight Center, administered by Oak Ridge Associated Universities through a contract with NASA. REFERENCES Aloy, M. A., Janka, H.-T., & Müller, E. 2005, A&A, 436, 273 Amati, L., et al. 2002, A&A, 390, 81 Band, D. 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E., & Heger, A. 2004, ApJ, 608, 365 This preprint was prepared with the AAS LATEX macros v5.2. – 13 – Ts T Te F(Tb) F(Ta) F(T ) Ta Tb 1/100 1/1000 T [sec]* fitting region Fig. 1.— Example of how the decay index α is determined by the calculated light curve F (T ). The start and end time of the burst are denoted by Ts and Te, respectively. The time T∗ is determined by Eq. (5). The decay index α is determined by fitting F (T ) ∝ (T − T∗) in the time interval [Ta, Tb]. Fig. 2.— Decay index α as a function of ϑobs, the angle between the whole jet axis and the observers’ lines of sight. Red and green points represent events whose peak fluxes are larger and smaller than 10−4 times the largest one in all simulated events, respectively. Left and right panels are for the uniform- and power-law jet cases, respectively. – 14 – Fig. 3.— Decay index α as a function of the total fluence Stotal, the sum of the fluxes in the time interval [Ts, Te]. Red and green points represent events whose peak fluxes are larger and smaller than 10−4 times the largest one in all simulated events, respectively. Left and right panels are for the uniform- and power-law jet cases, respectively. – 15 – 10-12 10-10 102 103 104 T-Ts [sec] fiducial r0=1.0x10 γ=200 , r0=2.4x10 βB=-5.0 Fig. 4.— Examples of light curves of the prompt tail emission in the 15–25 keV band for the power-law jet case and ϑobs > ϑc (ϑobs = 0.27 rad and ϑc = 0.02 rad). The solid line shows the fiducial parameters. A bump caused by the core emission can be seen at T − Ts ∼ 7.5 × 10 2 s. The dotted, dot-dashed, and dashed lines are for r0 = 1.0× 10 14 cm; r0 = 2.4 × 10 15 cm and γ = 200; and βB = −5, respectively, with other parameters being fiducial. Time intervals [Ta, Tb] for each case are denoted by the thick solid lines. The flux is normalized by the peak value. – 16 – 0.05 0.15 0.25 0 1 2 3 4 5 6 7 8 α ( decay index ) Fig. 5.— Distributions of the decay index α for uniform-jet (dN/dΩ = const.;solid line) and power-law jet (dN/dΩ ∝ [1 + (ϑ/ϑc) 2]−1;dotted line) models, respectively. We assume that all subshells have the same values of the following fiducial parameters: ∆θsub = 0.02 rad, γ = 100, r0 = 6.0 × 10 14 cm, αB = −1.0, βB = −2.3, hν 0 = 5 keV, and A = constant. We consider events whose peak fluxes are larger than 10−4 times the largest one in all simulated events (red points in Fig. 2). – 17 – 0.05 0.15 0.25 0 1 2 3 4 5 6 7 8 α ( decay index ) Fig. 6.— Same as Fig. 5, but for r0 = 1.0× 10 14 cm. – 18 – 0.05 0.15 0.25 0.35 0 1 2 3 4 5 6 7 8 α ( decay index ) Fig. 7.— Same as Fig. 5, but for γ = 200 and r0 = 2.4× 10 15 cm. – 19 – 0.05 0.15 0.25 0 1 2 3 4 5 6 7 8 α ( decay index ) Fig. 8.— Same as Fig. 5, but for βB = −5.0. – 20 – 0.05 0.15 0 1 2 3 4 5 6 7 8 α ( decay index ) Fig. 9.— Same as Fig. 5, but for tdur = 200 sec. – 21 – 0.05 0.15 0.25 0.35 0 5 10 15 20 α ( decay index ) Fig. 10.— Distribution of the decay index α for the uniform-jet profile. The dotted line is for ∆θtot = 0.1 rad with other fiducial parameters. The solid line is for the variable-∆θtot case, in which we generate events whose ∆θtot distributes as f∆θtotd(∆θtot) ∝ ∆θtot −2d(∆θtot) (0.05 . ∆θtot . 0.4), and for given ∆θtot, the quantities ν 0 and A are determined by hν ′0 = (∆θtot/0.13) −3.6 keV and A ∝ (∆θtot) −7.3, respectively. Other parameters are fiducial. Introduction Tail Part of the Prompt GRB Emission Decay Index of the Prompt Tail Emission Discussion

0704.1056

Black Hole's Life at colliders

arXiv:0704.1056v1 [hep-ph] 9 Apr 2007 Black hole’s life at colliders Seong Chan Park∗† FRDP, School of astronomy and physics, Seoul National University, Seoul 151-742, Korea E-mail: [email protected] In the series of papers by Ida, Oda and Park, the complete description of Hawking radiation to the brane localized Standard Model fields from mini black holes in the low energy gravity scenarios are obtained. Here we briefly review what we have learned in those papers. From Strings to LHC January 2-10 2007 International Centre Dona Paula, Goa India ∗Speaker. †We thanks to Daisuke Ida and Kin-ya Oda for valuable collaborations. c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ http://arxiv.org/abs/0704.1056v1 mailto:[email protected] Black hole’s Life at colliders Seong Chan Park 1. Introduction We briefly review recent developments in the mini black hole production and evaporation mainly based on the series of works done by Ida, Oda and Park [1, 2, 3, 4]. In the low energy gravity scenarios such as ADD and RS-I, the CERN Large Hadronic collider (LHC) will become a black hole factory [5, 6]. Above the TeV Planck scale, the classical production cross section of the (4+n)-dimensional black hole grows geometrically σ ∼ ŝ1/(n+1), with ŝ being the center of mass energy of the parton scattering. Once produced, black hole loses its energy or mass primarily via Hawking (thermal) radia- tion. The Hawking radiation goes mainly into the standard model quarks and leptons (spinors) and gauge bosons (vectors) localized on the brane, except for a few gravitons and Higgs boson(s). The quanta of Hawking radiation will have characteristic energy spectrum determined by the Hawking temperature and the greybody factor. The process of Hawking radiation in four dimensional rotat- ing black hole has been treated in detail by Teukolsky, Press, Page and others in 1970s’. In higher dimensions, however, it is shown that the process has quite different features. • Hawking temperaure TH ∝ (Mbh)/M∗)1/(n+1) of a (4+ n) dimensional black hole is much higher than 4 dimensional one with the small fundamental scale M∗ ∼ TeV ≪ MPlanck. With this high temperature, the number of available degrees of freedom for Hawking radiation are much bigger in (4+n) dimensions with all the standard model particles localized on the brane [8]. • The near horizon geometry of (4+ n) dimensional black hole is quite complicated. Its ge- ometry is different from that of a four dimensional Kerr black hole. With the highly mod- ified geometry in the vicinity of the event horizon, frequency dependent correction factor of Hawking radiation, i.e., greybody factor, is also largely modified (also see the references [9, 10, 11] as independent studies on the same topic). To understand the physics of those black holes, we have to understand the greybody factor of higher dimensional, rotating black hole [12, 13]. 2. Generalized Teukolsky equation and greybody factor The induced metric on the three-brane in the (4+ n)-dimensional Myers-Perry solution [7] with a single nonzero angular momentum is given by ∆−a2 sin2 ϑ dt2 + 2a(r2 +a2 −∆)sin2 ϑ 2 +a2)2 −∆a2 sin2 ϑ sin2 ϑdϕ2 − Σ dr2 −Σdϑ2, (2.1) where Σ = r2 +a2 cos2 ϑ , ∆ = r2 +a2 −µr1−n. (2.2) Black hole’s Life at colliders Seong Chan Park The parameters µ and a are equivalent to the total mass M and the angular momentum J (2+n)A2+nµ 16πG4+n , J = A2+nµa 8πG4+n (2.3) evaluated at the spatial infinity of the (4+n)-dimensional space-time, respectively, where A2+n = 2π(3+n)/2/Γ((3+ n)/2) is the area of a unit (2+ n)-sphere and G4+n is the (4+ n)-dimensional Newton constant of gravitation. 1.Subtracting outgoing wave contamination at NH and separating ingoing and outgoing wave at FF are described. 2.Here we answer the question :what fraction of energy would be radiated into Hawking radiation in spin-down phase. 2.1 Asymptotic solutions in Kerr-Newman frame We are given a linear, second-order equation, say + τR = 0, (2.4) where η and τ are determined in Kerr-Newman frame as: η = − (s−1)∆′+2iK , (2.5) 2iωr(2s−1)−λ . (2.6) ∆ = r2 +a2 − (r2H +a2) K = (r2 +a2)w−ma. (2.7) The asymptotic solutions are given at NH and FF limits: RNH ∼ Win +Woute2ikr∗∆s, (2.8) RFF ∼ Yinr2s−1 +Youte2ikr∗/r. (2.9) 2.2 BC: Subtracting outgoing contamination at NH The solutions near the horizon r → rH are RNHin = 1+a1(r− rH)+ (r− rH)2 + · · · , RNHout = e 2ikr∗(r− rH)s (1+b1(r− rH)+ · · ·) , (2.10) where the coefficients ai’s and bi’s are straightforward to compute: a1 = − , (2.11) a2 = − (η0 + τ−1)a1 + τ0 1+η−1 , (2.12) · · · , (2.13) Black hole’s Life at colliders Seong Chan Park where η j and τ j are j-th order coefficients of Taylor expansion of η and τ , respectively. For s = 1/2 and 1, τ−1 = 2iωδs,1 −λ , (2.14) τ0 = λ +δs,1 (∆1 −∆2), (2.15) η−1 = δs,1/2 −2i , (2.16) η0 = − (K1∆1 −K0∆2), (2.17) where ∆1 = 2+(n−1)(1+a2), (2.18) ∆2 = 1−n(n−1)(1+a2)/2, (2.19) K0 = (1+a 2)ω −am, (2.20) K1 = 2ω . (2.21) The problem is to integrate Eq.2.4 from purely ingoing initial conditions at r = r0 out to r → ∞. Choosing the positive choice for s makes Yin stable and easily determined by an outward integration. However, for such an integration RNHout is unstable against contaminating the purely ingoing solution. We can avoid the difficulty in mid-integration, relying on a mathematical transformation of the equation to stabilize the solutions in the two asymptotic regimes r → rH and r → ∞ as follows. To counteract the above contamination, let R̃ = R− (1+a1(r− rH)). (2.22) Then f satisfies the equation L R̃ = g, (2.23) where L = d2/dr2+ηd/dr+τ and g=−L (1+a1(r− rH)) =−ηa1−τ (1+a1(r− rH)). Equa- tion 2.23 is now stably integrated through both asymptotic regimes, i.e., from the near horizon to far field regimes. Near the horizon, R̃ becomes R̃(r → rH) = (r− rH)2 + · · · . (2.24) 2.3 Separating solution at FF For s = 1/2, RFFin ∼ 1+ + · · · , RFFout ∼ e2ikr∗ · · ·), (2.25) Black hole’s Life at colliders Seong Chan Park where 1 = −i , (2.26) · · · (2.27) Then, R̃1/2 becomes R̃1/2(r → ∞)≃ (Yin +1−a1)+a1r+ +Yout e2iωr∗ . (2.28) Using this expression, we can easily read out Y ’s without numerical difficulties. Finally, the greybody factor could be written as Γs=1/2 = 1− |c f1 | |Yout|2 |Yin|2 . (2.29) For s = 1, RFFin ∼ r(1+ + · · ·), RFFout ∼ e2ikr∗ + · · ·), (2.30) where cv1 = −i , (2.31) cv2 = − λ 2 −4aω(aω −m) , (2.32) · · · . Then, R̃1 becomes R̃1(r → ∞)≃ (Yin −a1) r+(Yincv1 −1+a1)+ +Yout e2iωr∗ . (2.33) Using this expression, we can easily read out Y ’s without numerical difficulties. Finally, the greybody factor could be written as Γs=1 = 1− |cv2| |Yout|2 |Yin|2 . (2.34) 3. Hawking radiations of mini-black hole Schematically black hole evolution follows five successive steps as is depicted in Fig.1: the production phase, the balding phase, the spin-down phase, the Schwarzschild phase and the Planck phase. When a black hole is produced in high energy collision (production phase), the geometry is highly irregular, and could even be topologically non-trivial. By emitting (bulk) gravitons and other particles, the black hole will be settled down to a rotating black hole which is supposed to be well described by Myers-Perry solution in (4+n) dimensional spacetime (balding phase). Black hole’s Life at colliders Seong Chan Park The decay in spin-down and Schwarzschild phases are calculable in terms of Hawking radi- ation. We are interested in those phases (spin-down phase and Schwarzschild phase) and would answer what fraction of energy will be lost in each of these phases. The rate of energy (and angular momentum) loss by Hawking radiation is given as follows: 2π ∑s,l,m dω〈Ns,l,m〉 , (3.1) where gs is the number of “massless" degrees of freedom at temperature T , namely, the number of degrees of freedom whose masses are smaller than T , with spin s. The expected number of particles of the species of spin s emitted in the mode with spheroidal harmonics l, axial angular momentum 〈Ns,l,m〉= sΓl,m(a,ω) e(ω−mΩ)/T − (−1)2s . (3.2) 1 2 3 4 5 Ω r_h D=10, s=1, a*=0.01 1 2 3 4 5 6 Ω r_h D=10, s=1, a*=0.3 2 4 6 8 Ω r_h D=10, s=1, a*=0.6 2 4 6 8 10 12 Ω r_h D=10, s=1, a*=0.9 2.5 5 7.5 10 12.5 15 Ω r_h D=10, s=1, a*=1.2 5 10 15 20 Ω r_h D=10, s=1, a*=1.5 Figure 1: Hawking radiation from D = 10 black hole. s = 1,a = 0.01,0.3,0.6,0.9,1.2,1.5. Black hole’s Life at colliders Seong Chan Park 4. Time evolution From the ratio of energy and angular momentum in eq.3.1, we can define a scale invariant function γ(as = a/rs) as follows: γ−1(as) ≡ d lnas d lnM (4.1) . (4.2) Now we calculate the ratio of final(M f ) and initial(Mi) energy of black hole by integrating the eq.4.1 with as(ini) for initial angular momentum. = Exp ∫ as(final) as(ini) γ(as) . (4.3) The amount of energy which is radiated in spin-down phase (0 ≈ as(final)6 as 6 as(ini)) is (Mi − M f ) and M f will be also radiated in Schwarzschild phase where the angular momentum of black hole is zero. Next, let us consider the evolution of the black hole. Since time scales as rn+3s in (4+ n) dimensions 1, it is convenient to introduce scale invariant rates for energy and angular momentum as follows. α(as) ≡ −rn+3s d lnM , (4.4) β (as) ≡ −rn+3s d lnJ , (4.5) with these new variables γ(as) can be written as γ−1(as) = β/α(as)− (n+ 2)/(n+ 1). We also introduce dimensionless variables y and z to take angular momentum and mass of the hole: y ≡ − lnas, (4.6) z ≡ − ln , (4.7) then finally we get the time variation of energy and angular momentum in terms of scale-invariant time parameter (τ = r−n−3s (ini)t) with initial mass of the hole: = (β −α n+1 z. (4.8) After finding the solutions z(y) and τ(y) of the coupled differential equations 4.8, one can get y(τ) and z(τ), hence as and M/Mi, as a function of time. From these, one can find how other quantities evolve, such as the area. 1We can easily understand this by simply looking at the formula −dM/dt ∼ AT 4 where the surface area of horizon A ∼ r2s for brane fields and the temperature of the hole T ∼ 1/rs and M ∼ rn+1s . Black hole’s Life at colliders Seong Chan Park 0 0.5 1 1.5 2 2.5 PSfrag replacements 0 0.2 0.4 0.6 0.8 1 PSfrag replacements Figure 2: Evolution of bh in D = 10. Up to now we have used a unit where the size of event horizon is fixed as rh = 1 and angular momentum of the hole is parameterized by (ah = a/rh). For conversion of unit, the following expressions are useful with as = ah/(1+a 1/(n+1). α(as) = −ιn+1n (1+a2h) n+1 r2h , (4.9) β (as) = −κn+1n (1+a2h) n+1 r2h , (4.10) where ιn = rsM− n+1 = (n+2)Ωn+2 , (4.11) κn = ιn( n+1 . (4.12) In Fig.2, black hole evolution in units of the initial mass as a function of rotation parameter as for scalar(s), fermion(f), vector(v), and sum of all the standard model particles(SM) in D = 5(left) and D = 10 (right) are shown. The initial angular momentum parameter is fixed by as = 0.83 and 2.67 in D = 5 and D = 10 that are the maximal rotations allowed by the initial collision, respectively. The mass of the hole goes to zero before the rotation parameter goes to zero when only scalar emission is available. However, when all the standard model fields are turned on, the evolution is essentially determined by the spinor and vector radiation. It is found that a black hole spins down quickly at the first stage with large rotation parameter as and the decrease of rotation parameter slows down as angular momentum of the hole is reduced. When all the standard model fields are turned on (SM), the evolution is essentially determined by the spinor and vector radiation. The figures show that a black hole spins down quickly at the first stage with large rotation parameter and the decrease of rotation parameter slows down as angular momentum of the hole is reduced. 5. Summary and Discussion The complete description of Hawking radiation to the brane localized SM fields and the con- sequent time evolution of mini black hole in the context of low energy gravity scenario has been made. Black hole’s Life at colliders Seong Chan Park We have developed analytic and numerical methods to solve the radial Teukolsky equation which has been generalized to the higher dimension (D = 4+n). Two main points in our numerical methods are as follows. First, we have imposed the proper purely-ingoing boundary condition near the horizon without the growing contamination of the out-going wave by extracting lower order terms explicitly. Second, we have developed the method to fit the in-going and out-going part from the numerically integrated wave solution at far field region by explicitly obtaining the next-to-next order expansion (or next-to-next-to-next order in vector case) of the solution. With these progress in numerical treatment, we can safely integrate the generalized Teukolsky equation up to very large r without out-going wave contamination. Then we have calculated all the possible modes to completely determine the radiation rate of the mass and angular momentum of the hole. Totally 3407 are computed explicitly, other than the modes which are confirmed to be negligible. A black hole tends to lose its angular momentum at the early stage of evolution. However the black hole still have a sizable rotating parameter after radiating half of its mass. More than 70% or 80% of black hole’s mass is lost during the spin down phase. Now that we have completely determined the radiation and evolution of the spin-down and Schwarzschild phases, only remaining hurdle is the evaluation of the balding phase, which is still being disputed due to its non-purturbative nature, to extract the quantum gravitational information at the Planck phase from the experimental data at LHC. References [1] D. Ida, K. y. Oda and S. C. Park, Phys. Rev. D 67, 064025 (2003) [Erratum-ibid. D 69, 049901 (2004)] [arXiv:hep-th/0212108]. [2] D. Ida, K. y. Oda and S. C. Park, arXiv:hep-ph/0501210. [3] D. Ida, K. y. Oda and S. C. Park, Phys. Rev. D 71, 124039 (2005) [arXiv:hep-th/0503052]. [4] D. Ida, K. y. Oda and S. C. Park, Phys. Rev. D 73, 124022 (2006) [arXiv:hep-th/0602188]. [5] S. B. Giddings and S. D. Thomas, Phys. Rev. D 65, 056010 (2002) [arXiv:hep-ph/0106219]. [6] S. Dimopoulos and G. Landsberg, Phys. Rev. Lett. 87, 161602 (2001) [arXiv:hep-ph/0106295]. [7] R. C. Myers and M. J. Perry, Annals Phys. 172, 304 (1986). [8] R. Emparan, G. T. Horowitz and R. C. Myers, Phys. Rev. Lett. 85, 499 (2000) [arXiv:hep-th/0003118]. [9] C. M. Harris and P. Kanti, Phys. Lett. B 633, 106 (2006) [arXiv:hep-th/0503010]. [10] M. Casals, P. Kanti and E. Winstanley, JHEP 0602, 051 (2006) [arXiv:hep-th/0511163]. [11] M. Casals, S. R. Dolan, P. Kanti and E. Winstanley, arXiv:hep-th/0608193. [12] S. C. Park and H. S. Song, J. Korean Phys. Soc. 43, 30 (2003) [arXiv:hep-ph/0111069]. [13] S. C. Park, J. Korean Phys. Soc. 45, 208 (2004).

0704.1057

Working with 2s and 3s

Working with 2s and 3s Diego Dominici ∗ Department of Mathematics State University of New York at New Paltz 1 Hawk Dr. New Paltz, NY 12561-2443 Phone: (845) 257-2607 Fax: (845) 257-3571 October 22, 2018 Abstract We establish an equivalent condition to the validity of the Collatz conjecture, using elementary methods. We derive some conclusions and show several examples of our results. We also offer a variety of exercises, problems and conjectures. 1 Introduction The Collatz conjecture (also known as the 3n + 1 conjecture, Ulam’s con- jecture, the Syracuse problem, Kakutani’s problem, Hasse’s algorithm, etc.) was first proposed by Lothar Collatz in 1937 [2]. In terms of the function T (n), defined by T (n) = , n ≡ 1(mod 2) , n ≡ 0(mod 2) , n ∈ N, (1) ∗e-mail: [email protected] http://arxiv.org/abs/0704.1057v1 the conjecture claims that for all natural numbers n, there exists a natural number k such that T (k) (n) = T ◦ T ◦ · · · ◦ T ︸ ︷︷ ︸ k times (n) = 1. For example, we have T (3) = 5, T (2)(3) = 8, T (3)(3) = 4, T (4)(3) = 2, T (5)(3) = 1, T (7) = 11, T (2)(7) = 17, T (3)(7) = 26, T (4)(7) = 13, T (5)(7) = 20, (2) T (6)(7) = 10, T (7)(7) = 5, T (8)(7) = 8, T (9)(7) = 4, T (10)(7) = 2, T (11)(7) = 1. We define T (∞) (n) = 1. Exercise 1 Prove that if ∀n ∈ N ∃ k ∈ N such that T (k) (n) < n, then the Collatz conjecture is true. The number k is called the stopping time of n. As of February 2007, the Collatz conjecture has been verified for numbers up to 13×258 = 3, 746 , 994, 889, 972, 252, 672 [7]. However, the general case remains open. Introducing the total stopping time function σ∞(n), defined by σ∞(1) = 0 σ∞(n) = inf k ∈ N ∪ {∞} | T (k) (n) = 1 , n ≥ 2, we can reformulate the Collatz conjecture as C = N, (C1) where C = {n ∈ N | σ∞(n) < ∞} . (3) From (2), we have n 2 3 4 5 6 7 8 σ∞(n) 1 5 2 4 6 11 3 Exercise 2 Find σ∞(n) for 9 ≤ n ≤ 100. Hint: The web page http://www.numbertheory.org/php/collatz.html con- tains an implementation which allows the computation of σ∞(n) for large values of n. http://www.numbertheory.org/php/collatz.html One could consider the inverse problem and try to characterize the sets Sk, defined by S0 = {1} and Sk = {n ∈ N | σ∞(n) = k} , k ≥ 1. (4) The first few Sk are S1 = {2} , S2 = {4} , S3 = {8} , S4 = {5, 16} , S5 = {3, 10, 32} , (5) S6 = {6, 20, 21, 64} , S7 = {12, 13, 40, 42, 128} , S8 = {24, 26, 80, 84, 85, 256} . It is clear from (4) that 2k ∈ Sk ∀k ∈ N0, where N0 = N ∪ {0} . In terms of the sets Sk, the Collatz conjecture reads Sk = N. (C2) Exercise 3 Compute Sk for 9 ≤ k ≤ 100. Hint: Consider the inverse map T−1 : N → P (N) , given by T−1(n) = {2n} , n ≡ 0, 1(mod 3) 2n, 1 (2n− 1) , n ≡ 2(mod 3) The sequence of natural numbers n , n ≥ 0 , defined by x 0 = m n+1 = T x(m)n , 0 ≤ n, (6) is called the trajectory or forward orbit of m ∈ N. From (2), we have = {2, 1} , = {3, 5, 8, 4, 2, 1} , = {4, 2, 1} , x(5)n = {5, 8, 4, 2, 1} , x(6)n = {6, 3, 5, 8, 4, 2, 1} , x(7)n = {7, 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1} , x(8)n = {8, 4, 2, 1} . Exercise 4 Find for 9 ≤ m ≤ 100. Using the sequences we can restate Collatz’s conjecture as x(m)n = {2, 1} . (C3) We can also consider higher order recurrences, i.e., instead of (6), use n+i = T , 0 ≤ n, where T (i) (x) = fi,j(x), if x ≡ j mod 2i , 0 ≤ j ≤ 2i − 1. (7) For i = 1, 2, 3, we have f1,0(x) = , f1,1(x) = 3x+ 1 f2,0(x) = , f2,1(x) = 3x+ 1 , f2,2(x) = 3x+ 2 , f2,3(x) = 9x+ 5 f3,0(x) = , f3,1(x) = 9x+ 7 , f3,2(x) = 3x+ 2 , f3,3(x) = 9x+ 5 f3,4(x) = 3x+ 4 , f3,5(x) = 3x+ 1 , f3,6(x) = 9x+ 10 , f3,7(x) = 27x+ 19 Exercise 5 Prove that if the sequence {3k + 4 } ⊂ C, then the Collatz con- jecture is true. In terms of (7), the Collatz conjecture reads ∀n ∈ N ∃ m ∈ N such that n ≡ k (mod 2m) and fm,k(x) < 1. (C4) For example, we have 11 ≡ 11 (mod 25) and f5,10(x) = 3x+ 2 , f5,11(x) = 27x+ 23 Thus, 5 = f5,11(11) = 10, x 10 = f5,10(10) = 1. The literature on the Collatz conjecture is vast and growing rapidly. Rather than attempting to cover it, we refer the reader to the excellent survey papers [5] and [6]. 2 Representation of natural numbers Let the sets Λm be defined by Λm = {2 m} , 0 ≤ m ≤ 3 and n ∈ N | n = , m ≥ 4, for some m, l, b1, . . . , bl ∈ N0, with 0 ≤ l ≤ m− 3 and 0 ≤ b1 < b2 < · · · < bl ≤ m− 4. The first few Λm are Using the (l + 2)−tuple (l, b1, . . . , bl, m) to represent the number we can write Λ4 = {(0, 4) , (1, 0, 4)} , Λ5 = {(0, 5) , (1, 1, 5) , (2, 0, 1, 5)} , Λ6 = {(0, 6) , (1, 0, 6) , (1, 2, 6) , (2, 1, 2, 6)} , (8) Λ7 = {(0, 7) , (1, 1, 7) , (1, 3, 7) , (2, 0, 3, 7), (2, 2, 3, 7)} , Λ8 = {(0, 8) , (1, 0, 8) , (1, 2, 8) , (1, 4, 8) , (2, 1, 4, 8), (2, 3, 4, 8)} . Exercise 6 Compute Λm for 9 ≤ m ≤ 100. Hint: (a) If (v1, v2, . . . , vl+2) ∈ Λm, then (v1, v2 + 1, . . . , vl+2 + 1) ∈ Λm+1. (b) (1, 0, 2m) ∈ Λ2m for all m ≥ 2. Comparing (5) with (8), it seems that Sm = Λm. The next results will show this to be true. Lemma 7 For all m ∈ N0, we have T (Λm+1) ⊂ Λm. (9) Proof. Let n ∈ Λm+1. Then, T (n) = 2bk−1 if b1 > 0 (n even) or T (n) = 2bk+1 2bk+1−1 if b1 = 0 (n odd). In either case, T (n) ∈ Λm. Lemma 8 For all m ∈ N0, we have Λm ⊂ Sm. (10) Proof. We use induction on m. The case of m = 0 is clearly true, since Λ0 = {1} = S0. Assuming (10) to be true for m, let n ∈ Λm+1. From (9) we have T (n) ∈ Λm and therefore σ∞ [T (n)] = m. Thus, σ∞(n) = m+1 and the result follows. Exercise 9 Show that T (n) = . (11) The other inclusion is also true. Theorem 10 For all m ∈ N0, Sm ⊂ Λm. Proof. Clearly, Sm = {2 m} = Λm, 0 ≤ m ≤ 3. Let m ≥ 4 and s ∈ Sm. Using (11) we can write the recurrence (6) as n+1 = x(s)n + 0 = s, (12) where θn = sin x(s)n , i.e., x(s)n ≡ θ n (mod 2) . (13) Assuming {θn} to be a known sequence, the solution of (12) is [1] x(s)n = 2 j + 1 j + 1 or using (13) n = 2 −n3Θ(n−1) 3Θ(k) , (14) Θ (x) = j . (15) Setting n = m and solving for s in (14), we obtain m = 1 3Θ(m−1) 3Θ(k) . (16) Let l = Θ (m− 1) . From (13) and (15), we see that Θ (x) is a step function with unit jumps at x = b1, b2, . . . , bl, m, where 0 ≤ b1 < b2 < · · · < bl < m. Therefore, we can rewrite (16) as Finally, since x m−3 = 8, x m−2 = 4, x m−1 = 2 and x m = 1, the penul- timate jump must occur before or at x = m − 4. Thus, bl ≤ m − 4 and l = Θ (m− 1) ≤ m− 3. Corollary 11 The Collatz conjecture is true if and only if every natural number n can be represented in the form for some m, l, b1, . . . , bl ∈ N0, with 0 ≤ l ≤ m− 3 and 0 ≤ b1 < b2 < · · · < bl ≤ m− 4. Corollary 11 is not a proof of the Collatz conjecture, but it provides a lot of information on the set C and the function σ∞(n). When l = 0, we recover the known fact that 2m ∈ Sm, ∀m ∈ N0. For l = 1, we have the following result. Lemma 12 For all m ∈ N, we have ∈ Sm, 0 ≤ k ≤ , m even, 22k+1 ∈ Sm, 0 ≤ k ≤ , m odd. Proof. Let n ∈ Λm, with l = 1. We have 2m − 2b1 = 2b1 × 2m−b1 − 1 , 0 ≤ b ≤ m− 4. Thus, 2m−b1 ≡ 1 (mod 2) and therefore m− b1 ≡ 0 (mod 2) . Considering the cases m even and m odd, the result follows. When l = 2, the situation is slightly more complicated. To simplify matters, we restrict ourselves to the case of n being odd. Proposition 13 For all m ≥ 5, with m 6= 6, 8, we have 2m−2−6k ∈ Sm, 1 ≤ k ≤ , m ≥ 10 even, 2m−4−6k ∈ Sm, 0 ≤ k ≤ , m ≥ 5 odd. Proof. Let n ∈ Λm, odd, with l = 2. Then, 2m − 3− 2b2 and therefore 2m − 2b2 = 2b2 × 2m−b2 − 1 ≡ 3 (mod 9) . Considering all possible cases, we have 1) 2b2 ≡ 1 (mod 9) and 2m−b2 − 1 ≡ 3 (mod 9), which implies b2 ≡ 0 (mod 6) , m− b2 ≡ 2 (mod 6) . 2) 2b2 ≡ 2 (mod 9) and 2m−b2 − 1 ≡ 6 (mod 9), which implies b2 ≡ 1 (mod 6) , m− b2 ≡ 4 (mod 6) . 3) 2b2 ≡ 4 (mod 9) and 2m−b2 − 1 ≡ 3 (mod 9), which implies b2 ≡ 2 (mod 6) , m− b2 ≡ 2 (mod 6) . 4) 2b2 ≡ 5 (mod 9) and 2m−b2 − 1 ≡ 6 (mod 9), which implies b2 ≡ 5 (mod 6) , m− b2 ≡ 4 (mod 6) . 5) 2b2 ≡ 7 (mod 9) and 2m−b2 − 1 ≡ 3 (mod 9), which implies b2 ≡ 4 (mod 6) , m− b2 ≡ 2 (mod 6) . 6) 2b2 ≡ 8 (mod 9) and 2m−b2 − 1 ≡ 6 (mod 9), which implies b2 ≡ 3 (mod 6) , m− b2 ≡ 4 (mod 6) . Thus, for m even we shall have m − b2 ≡ 2 (mod 6) or b2 ≡ m − 2 (mod 6) and for m odd we need m − b2 ≡ 4 (mod 6) or b2 ≡ m − 4 (mod 6) , with 1 ≤ b2 ≤ m− 4. Writing b2 in terms of m, the result follows. From Corollary 11, we can also get an idea of how the total stopping time σ∞(n) behaves if the Collatz conjecture is true. Solving for m in (C5) we ln (2) 3ln+ 3l In other words, σ∞(n) lies on the family of parametric curves ln (2) 3in+ j , i, j ∈ N0, i ≤ j. For example, we have n 2 3 4 5 6 7 8 σ∞(n) 1 5 2 4 6 11 3 (i, j) (0, 0) (2, 5) (0, 0) (1, 1) (2, 10) (5, 347) (0, 0) Exercise 14 Prove that ln(n) ln(2) ≤ σ∞(n) ∀n ∈ N. 2.0.1 Binary sequences Another approach is to study the sequence , k ≥ 0 , which contains a wealth of information. Definition 15 Let τ : C → N be defined by τ (n) = σ∞(n) 2k. (17) For example, we have n 1 2 3 4 5 6 7 8 τ(n) 1 2 35 4 17 70 2199 8 Clearly, τ (2n) = 2n, ∀n ∈ N0. Exercise 16 Find τ(n) for 9 ≤ n ≤ 100. Let’s study the image of Λm by τ. We have τ (Λ0) = {1} , τ (Λ1) = {2} , τ (Λ2) = {4} , τ (Λ3) = {8} , τ (Λ4) = {16, 17} , τ (Λ5) = {32, 34, 35} , τ (Λ6) = {64, 65, 68, 70} , (18) τ (Λ7) = {128, 130, 136, 137, 140} , τ (Λ8) = {256, 257, 260, 272, 274, 280} . Exercise 17 Let 1 , . . . , λ where Nm = #Λm denotes the number of elements in the set Λm. Prove that ∀m ∈ N0 there exist a sequence 2m > α 1 ≥ · · · ≥ α = 0, (19) such that = 2m + α From (18), we have m 1 2 3 4 5 6 7 8 1 0 0 0 1 3 6 12 24 2 0 0 0 0 2 4 9 18 It follows from (19) that #Λm ≤ α 1 + 1, ∀m ≥ 0. Using (16), we can define an inverse function for τ(n). Definition 18 Let φ : N → Q be defined by φ(n) = 3Φ(m−1) 3Φ(k) where β m−1 . . . β 0 is the binary representation of n, i.e., = n and Φ (x) = For example, we have n 1 2 3 4 5 6 7 8 φ(n) 1 2 1 4 1 2 Exercise 19 Find φ(n) for 9 ≤ n ≤ 100. It follows from Theorem 10 that φ ◦ τ (n) = n, ∀n ∈ N, while (19) implies that Sm = φ 2m, . . . , 2m + α ∩ N, ∀m ≥ 0. In terms of φ, the Collatz conjecture reads N ⊂ φ (N) . (C6) With (C6), we finally reach a statement equivalent to the Collatz con- jecture, which is independent of the original formulation in terms of T (x). Although we have not succeeded in proving (C6), we hope that studying the function φ(n) will shed new light on the Collatz problem. 3 Further problems In the spirit of the Monthly, we offer a series of problems to the curious reader. Those labeled ”Exercise” are relatively easy to prove, ”Problems” denote results strongly supported by numerical evidence and ”Conjectures” are those that we would really wish to prove, but that may turn out to be false. Conjecture 20 Prove that σ∞(n) ≤ δ(n) ln(n) ∀n ≥ 2, (20) where δ(n) is a slowly varying function, which might be eventually constant. Definition 21 The Abby-Normal numbers (AN numbers). Let the scaled total stopping time γ (n) be defined by γ (n) = σ∞(n) ln(n) , n ≥ 2. We say that ak is the k−th AN number if γ (ak) = max {γ (n) | ak−1 ≤ n ≤ ak} , k ≥ 1 with a0 = 2. In other words, {γ (ak)} is an increasing sequence of sharp lower bounds for the function δ(n) defined in (20). Exercise 22 Show that n 1 2 3 4 5 6 7 a(n) 3 7 9 27 230, 631 626, 331 837, 799 γ(n) ≃ 4.55 5.65 5.92 21.24 22.51 23.90 24.12 From the results obtained by Eric Roosendaal [8], it follows that 6, 649, 279, 8, 400, 511, 11, 200, 681, 15, 733, 191, 63, 728, 127, 3, 743, 559, 068, 799, 100, 759, 293, 214, 567, are possible AN numbers. We have γ(100, 759, 293, 214, 567)≃ 35.17. Exercise 23 Find all AN numbers in the interval [1, 000, 000, 100, 759, 293, 214, 567]. Conjecture 24 Prove that there exist infinitely many AN numbers. Problem 25 Let Vm be the m−vector 0 , . . . , x σ∞(m) and L : RN → RN the linear operator defined by L ([v1, v2, . . . vN ]) = [v2, v3, . . . vN , v1] . Let θ (m) be the angle between Vm and L (Vm) . Prove that < cos [θ (m)] < , ∀m ∈ N. < cos [θ (m)] < , ∀m ≡ 1(2). (iii) cos [θ (ak)] = Hint: See [4]. Problem 26 Prove that: (i) ∀k ≥ 6 ∃ mk ∈ Sk such that mk + 1 ∈ Sk. (ii) ∀k ≥ 7 ∃ mk ∈ Sk such that mk + 2 ∈ Sk. (iii) ∀l ∈ N ∃ K ∈ N such that ∀k ≥ K ∃ mk ∈ Sk such that mk + l ∈ Sk . Hint: See [3]. Exercise 27 Prove that #Sk+1 Exercise 28 Let ζ (m) = # {2 ≤ k ≤ m | σ∞(k) = σ∞(k − 1)} . Show that ζ (m) = 0, 2 ≤ m ≤ 12, ζ (m) = 1, 13 ≤ m ≤ 14, ζ (m) = 2, 15 ≤ m ≤ 18, ζ (m) = 3, 19 ≤ m ≤ 20. Conjecture 29 Prove that ζ (m) Problem 30 Prove that 3× 2m−5 , m ≥ 0 (21) 605× 2m−13 , m ≥ 0, where ⌊·⌋ denotes the greatest integer function. Problem 31 Let 1 , . . . , s , Nm = #Sm, 2m = s 1 > s 2 > · · · > s Prove that 2m + 22k k+2, 0 ≤ k ≤ , m ≥ 4, m ≡ 0(mod 2) 2m + 22k+1 k+2, 0 ≤ k ≤ , m ≥ 5, m ≡ 1(mod 2). References [1] R. P. Agarwal. Difference equations and inequalities. Marcel Dekker Inc., New York, 2nd ed., 2000. [2] L. Collatz. On the origin of the (3n+1)−problem. J. Qufu Normal Univ. (Nat. Sci.), 12(3):9–11, 1986. [3] L. E. Garner. On heights in the Collatz 3n+ 1 problem. Discrete Math., 55(1):57–64, 1985. [4] D. Gluck and B. D. Taylor. A new statistic for the 3x+1 problem. Proc. Amer. Math. Soc., 130(5):1293–1301 (electronic), 2002. [5] J. C. Lagarias. The 3x+1 Problem: An Annotated Bibliography (1963– 2000). eprint: arxiv:NT/0309224. [6] J. C. Lagarias. The 3x + 1 Problem: An Annotated Bibliography, II (2001-). eprint: arxiv:math.NT/0608208. [7] T. Oliveira e Silva. Computational verification of the 3x+ 1 conjecture. http://www.ieeta.pt/˜tos/3x+1.html [8] E. Roosendaal. On the 3x + 1 problem. http://www.ericr.nl/wondrous/index.html http://arxiv.org/abs/math/0608208 http://www.ieeta.pt/~tos/3x+1.html http://www.ericr.nl/wondrous/index.html Introduction Representation of natural numbers Binary sequences Further problems

0704.1059

Decartes' Perfect Lens

Descartes’ Perfect Lens Mark B. Villarino Depto. de Matemática, Universidad de Costa Rica, 2060 San José, Costa Rica February 12, 2013 Abstract We give a new, elementary, purely analytical development of Descartes’ theorem that a smooth connected surface is a perfect focusing lens if and only if it is a connected subset of the ovoid obtained by revolving a cartesian oval around its axis of symmetry. Contents 1 Introduction 1 2 The Analytical Problem (in two dimensions) 3 2.1 Both Fixed Points Are Finite . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 The Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 The Solution of the Differential Equation . . . . . . . . . . . . . . . . 6 2.2 One Fixed Point At Infinity; One Fixed Point Finite . . . . . . . . . . . . . 7 2.2.1 The Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 The Solution of the Differential Equation . . . . . . . . . . . . . . . . 8 2.3 Both Fixed Points are at Infinity . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Descartes’ Theorem 9 3.1 Drucker’s Characterization of a Surface of Revolution . . . . . . . . . . . . 9 3.2 Proof of Descartes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1 Introduction Almost two thousand three hundred years ago, the hellenistic mathematician Diokles [4] gave the first proof that a mirror in the shape of a paraboloid of revolution reflects all incident light rays, which are parallel to its axis of symmetry, to a single point, which Kepler [8], in 1604, called the focus. http://arxiv.org/abs/0704.1059v4 After the advent of the calculus it was possible to prove that the only such reflecting surface is generated by revolving a (proper or degenerate) parabola around its axis of sym- metry. This is a very famous and well-known result, and is treated in many easily accessible sources. See, for example, Spiegel [16]. All these proofs are based on Heron’s Law of Reflection, θI = θRf , where θRf is the angle the reflected ray makes with the normal to the reflecting surface at the point of incidence of the incoming ray and θI is the angle the incident ray makes with the normal. One transforms Heron’s equation into the ordinary differential equation (ODE) of the cross-section curve of the mirror. Drucker’s paper [5] would seem to be the final word on the subject. Unfortunately and surprisingly, the corresponding result for a lens, instead of a mirror, is less well-known, at least among mathematicians (although [11] is pleasant attempt to alter that). Yet the case of a lens, too, is quite fascinating and is treatable by elementary means. The purpose of this paper is to remedy the situation and fill this gap. Indeed, it all started in 1637, when Descartes [3] asked for the refractive analogue of the parabolic mirror: Which shape of lens will focus all rays from one radiant point source to one single image point? We will call such a lens a perfect lens . Descartes discovered that the cross-section curve of the perfect lens, assumed to be a surface of revolution, is a fourth degree curve known today as the cartesian oval . It can be defined as the locus of points the ”weighted” sum of whose distances from two fixed points is a constant: d1 + nd2 = c (1.0.1) where d1 and d2 are the distances from any point on the curve to the two fixed points, called the foci, and n is a constant. If one focus is at the origin and the other is at the point (b, 0) where b > 0 , the equation can be written: (1 − n2)(x2 + y2) + 2n2bx + c2 − n2b2 = 4c2(x2 + y2) (1.0.2) If n = ±1, the oval is the conic section : x − b c2−b2 ) = 1 (1.0.3) More information on cartesian ovals can be found in [14] and [15] and [18]. Descartes’ own treatment, which is not altogether easy to read (see [1]), shows that the oval is a solution, but does not show that it is the only solution. The only treatments of Descartes’ result that we have seen in the literature do not appear in books on mathematics (!), but rather on optics (see Hecht [7] and Klein [9]) and use Fermat’s Principle: A light ray traverses the path between two points which takes the least time. A non-trivial computation, based on the calculus of variations, shows that the time the light ray takes to go from the radiant point to the image point is constant for every point of the cross-section curve of the perfect lens (since if the time were different in two points of the curve, it would not be minimal), and therefore its equation is that of the cartesian oval. Moreover such treatments make physical assumptions about the velocity of light in differ- ent media, while, as our treatment will show, the problem is really one in pure mathematics. We have not seen any treatment of the subject which is founded purely on Snell’s law of refraction, which describes the relationship between the angle of incidence and the angle of refraction when light passes the boundary between two isotropic media (media in which the path of a light ray is a straight line). The law states: If θI is the angle the incident ray makes with the normal to the boundary at the point of refraction, and if θR is the angle the refracted ray makes with the normal, then at all points of the boundary the ratio sin θI sin θR where n, called the index of refraction, is constant . Such a treatment of Descartes’ theorem would seem desireable, since it is the immediate generalization of the corresponding treatment of the perfect reflective mirror . In this paper, we will present a new, self-contained, elementary, purely analytical proof, based on Snell’s Law and Drucker’s paper [5], of the following complete form of Descartes’ theorem: Theorem 1. (Descartes’ Theorem) A smooth connected surface is a perfect lens if and only if it is a connected subset of the ovoid obtained by revolving a cartesian oval around its axis of symmetry. 2 The Analytical Problem (in two dimensions) We begin by solving the following two-dimensional purely analytical problem: It is required to find the equation, f(x, y) = 0, of a smooth connected curve, C, for which the straight lines from two fixed points cut the normal in two angles whose sines are in constant ratio. Please note the absence of physical modeling. The problem is purely mathematical, as its its solution. We will find and solve an ordinary differential equation (ODE) for which the equa- tion of the curve is the general solution. The ODE, in fact, will be a restatement of Snell’s Definition 1. We call any curve C that solves the problem a perfect two-dimensional lens with respect to the points F and F ′. 2.1 Both Fixed Points Are Finite 2.1.1 The Differential Equation We assume a cartesian coordinate system in the xy-plane. Let the two fixed points be O(0, 0) and B(b, 0) with b > 0 (Here is we use the assumption that F and F ′ are finite and distinct). Let P(x, y) be a variable point on the curve f(x, y) = 0. We assume P is in the first quadrant and we assume that the curve is concave downwards at P (that is, if y(x) is the function defined implicitly by the equation f(x, y) = 0, then y′′(x0) < 0) . Let l1 be the length of the line segment OP and let l2 be the length of PB. Let MN be the normal to the curve where N is on the concave side of the curve and P is between M and N. Let θ1 := MP̂O, the angle that OP forms with the normal MN , and let θ2 := BP̂N , the angle that BP forms with the normal MN . Let PT be the tangent line to the curve at P where T is the point on the x-axis where the tangent line crosses it. Let φ := P T̂B, the angle, measured counter clockwise, the tangent line forms with the x-axis. We will use the geometry of the figure to obtain formulas for sin θ1 and sin θ2 in terms of x, y, and the derivative, y′. When we substitute these expressions into Snell’s Law, we obtain the desired differential equation for the curve f(x, y) = 0. By the law of cosines applied twice to the △OPB cos PÔB = b2 + l2 cos PB̂O := b2 + l2 cos PÔB = cos(θ1 + φ − 90) cosPB̂O = cos(90 − θ2 − φ) = sin(θ1 + φ) = sin(θ2 + φ) So, we obtain our fundamental formulas: sin(θ1 + φ) = b2 + l2 sin(θ2 + φ) := b2 + l2 (2.1.1) Moreover, it is evident that < θ1 + φ < π 0 6 θ2 + φ < (2.1.2) b2 + l2 b2 + l2 2b2 − 2bx b − x Therefore, equations (2.1.1) become sin(θ1 + φ) = sin(θ2 + φ) = b − x By (2.1.1) and (2.1.2) and the definition of the arcsin function, we obtain π − (θ1 + φ) = arcsin θ2 + φ = arcsin b − x and therefore θ1 = (π − φ) − arcsin θ2 = arcsin b − x whence, sin θ1 = sin(π − φ) cos arcsin − cos(π − φ) sin arcsin = sin φ + cos φ 1 + y′2 1 + y′2 sin θ2 = cos φ sin arcsin b − x − sin φ cos arcsin b − x 1 + y′2 b − x 1 + y′2 b − x 1 + y′2 b − x (1 + y′2) Now, our assumption is that Snell’s Law holds, i.e., that sin θ1 sin θ2 where n is a constant, holds for every point P (x, y) of the curve. Substituting our two formulas for sin θ1 and sin θ2 into this equation gives us the equation: 1 + y′2 1 + y′2 1 + y′2 b − x (1 + y′2) = n (2.1.3) Solving this equation (2.1.3) for y′ we obtain the differential equation of the curve: b − x ) (2.1.4) 2.1.2 The Solution of the Differential Equation We use the “arrow” notation. “P ⇒ Q” means “the proposition P (logically) implies the proposition Q.” (2.1.4) ⇒ b − x yy′ + x nyy′ − n(b − x) yy′ + x x2 + y2 + n · −(b − x) + yy′ (b − x)2 + y2 2yy′ + 2x x2 + y2 + n · −2(b − x) + 2yy′ (b − x)2 + y2 x2 + y2 + n (b − x)2 + y2 x2 + y2 + n (b − x)2 + y2 = c for some (arbitrary) constant c. We have therefore proved: Theorem 2. The general solution for the differential equation (2.1.4) of the perfect two- dimensional lens, C, with respect to the points F = (0, 0) and F ′ = (b, 0), where b > 0, is given by the equation: x2 + y2 + n (b − x)2 + y2 = c. (2.1.5) As we saw, this is the equation (1.0.1) of a cartesian oval with foci at the points (0, 0) and (b, 0). We have assumed that b is finite in this analysis, i.e., that the two foci are a finite distance apart. Now we consider the limiting cases where one or both foci are “at infinity.” We will see that we obtain proper or degenerate conic sections for these cases. 2.2 One Fixed Point At Infinity; One Fixed Point Finite 2.2.1 The Differential Equation We will slightly alter the treatment for the case of two finite foci. To do so, we begin with the following: Definition 2. A point at infinity is specified by means of a line through the origin. The line joining P to a point at infinity is the line through P parallel to the given line. Points at infinity are not considered to be on C. We assume that the fixed point F is at −∞ along the x-axis and that the fixed point F ′ is at the point (b, 0) of the x-axis, where b > 0. Intuitively, this means that a beam of light from −∞, parallel to the x-axis, is brought to a point focus at (b, 0) by a single refracting curve, f(x, y) = 0, of index n. The line joining P to the point at infinity is the line parallel to the x-axis through P . θ1 is the angle the horizontal line through P (x, y) makes with the normal while θ2 is the angle PF makes with the normal. Finally, l be the length of PF . Then, the earlier derivation of the ODE is applicable. We need only observe that θ1 + φ = So, substituting our two new formulas for sin θ1 and sin θ2 into Snell’s Law gives us, instead of (2.1.3), the new equation: 1 + y′2 1 + y′2 b − x (1 + y′2) After some rearrangement, we obtain the differential equation of the curve: (b − x) − yy′ (b − x)2 + y2 · n = 0. (2.2.1) 2.2.2 The Solution of the Differential Equation The ODE (2.2.1) can be solved by the same computations as we did for the ODE (2.1.4) which lead us to the Theorem 3. The general solution for the differential equation (2.2.1) of the perfect two- dimensional lens, C, with the radiant point at −∞ is given by the equation: x + n (b − x)2 + y2 = c. (2.2.2) where c is an arbitrary constant. We observe that the equation (2.2.2) has the following interesting interpretation. The equation (2.2.2) says that the ratio of the distance of the point P from the line x = c − bn 1 − n its distance from the point (b, 0) is the constant ±n, and thererefore, by the focus-directrix definition, is a conic section. This theorem takes a more elegant form if we assume that the curve C passes through the origin. Then, the constant c = nb and, after rationalizing (2.2.2), we obtain ([10], problem B-10, Chapter 20): Theorem 4. The general solution for the differential equation (2.2.1) of the perfect two- dimensional lens, C, is a conic section whose focus is the point where the light is focused and whose excentricity is the reciprocal of the index of refraction. 1. If n2 6= 1, C given by the equation: x − nb ) = 1 (2.2.3) Therefore C is an ellipse if n2 > 1 or an hiperbola if n2 < 1, either one of which is centered at n + 1 2. If n = 1, then C is the segment of the x-axis given by 0 6 x 6 b. 3. If n = −1, then C is the parabola y2 = 4bx (2.2.4) The reader should compare this result with that of the form of the perfect reflecting mirror already cited in [5]. If n < 0, then we get reflection instead of refraction. Maesumi [11] used Fermat’s Principle to treat this case in a very elegant paper, al- though his definition of the index of refraction is the reciprocal of our (standard) one. 2.3 Both Fixed Points are at Infinity Keeping the notation of the case of the radiant point at −∞, we assume that the refracted rays form a parallel beam in the direction such that θ2 + φ = Constant, but, this means that sin(θ2 + φ) = b − x (b − x)2 + y2 where C is some constant. But the condition that C goes through the origin means that C = 1, and rationalizing the resulting equation we obtain: Theorem 5. If both fixed points are at infinity, then the perfect lens C has the equation: x = 0 (2.3.1) That is, it is the vertical y-axis. 3 Descartes’ Theorem 3.1 Drucker’s Characterization of a Surface of Revolution In 1992 [5] Drucker published a very interesting paper in which he treated the problem of finding all perfect mirrors, i.e., mirrors which reflect all rays issuing from one radiant point to one image point. After showing that the two dimensional curve with the perfect reflecting property is a proper or degenerate conic section, he (implicitly) proved the following characterization of a surface of revolution. Drucker, himself, did not state it explicitly. Theorem 6. Let F and F ′ be two fixed points. If, for each point P of the smooth connected surface S the normal ~N at P lies in the subspace spanned by the vectors FP and F ′P , then S is a surface of revolution whose axis of revolution is the line through F and F ′. Proof. We offer a new proof of Drucker’s theorem. It is based on an idea in Salmon [15] which goes back to Monge [12]. Since, by definition, the normal MN is in the subspace spanned by FP and F ′P , it is in the plane of △FPF ′. If MN is always parallel to FF ′, then S is a plane which is perpendicular to FF ′. We exclude this degenerate case for the rest of the argument. (See (2.3.1)). Therefore MN is not always parallel to FF ′. Thus, the infinite line MN intersects FF ′ at some point. This is the characteristic property of the surface S. Let (α, β, γ) be a point on FF ′ and let (l, m, n) be the line’s direction numbers where we assume l · m · n 6= 0. The corollaries deal with the case where one or more coefficients are equal to zero. Then, the equation of the line FF ′ is x − α y − β z − γ = t (3.1.1) where t is the common value of the three fractions. F (x, y, z) = 0 (3.1.2) be the equation of S, where F is a continuously differentiable function of x, y, and z in some open set R, and let (x0, y0, z0) be the point P on S.. Since MN is normal to S at P , its equation is: x − x0 Fx(x0, y0, z0) y − y0 Fy(x0, y0, z0) z − z0 Fz(x0, y0, z0) = T (3.1.3) where T is the common value of the three fractions, and where Fx(x0, y0, z0) ≡ evaluated in (x0, y0, z0), and where the other denominators have a similar interpretation. We assume that all three denominators are different from zero. The corollaries deal with the cases where the denominators are equal to zero. Solving equations (3.1.1) and (3.1.3) for x, y, and z, and then equating the values ob- tained, we get the following homogeneous linear system for the unknowns t, T , and 1: lt − Fx(x0, y0, z0)T + (α − x0) · 1 = 0 mt − Fy(x0, y0, z0)T + (β − y0) · 1 = 0 nt − Fz(x0, y0, z0)T + (γ − z0) · 1 = 0 The analytical condition that this system have a nontrivial solution, which it does by as- sumption, is that the determinant of their coefficients vanish: ∣∣∣∣∣∣ Fx(x0, y0, z0) Fy(x0, y0, z0) Fz(x0, y0, z0) l m n x0 − α y0 − β z0 − γ ∣∣∣∣∣∣ = 0 (3.1.4) The determinant on the left-hand side of (3.1.4) is (one half of ) the Jacobian of the three functions Ω := F (x, y, z), u := lx + my + nz, v := (x − α)2 + (y − β)2 + (z − γ)2, (3.1.5) evaluated at the point P of S. But the point P is totally arbitrary, which means the Jacobian (3.1.4) vanishes in a full neighborhood of P , since F is a continuously differentiable function of x, y, and z in some open set R. According to a classical theorem (see Buck [2], Goursat [6], Osgood [13], and Taylor [17], if the Jacobian of the three functions vanishes identically, then three functions are functionally dependent. That means that there is a function, Ω(u, v), of the two variables u and v, defined and continuously differentiable in a neighborhood of the point (u0, v0), where u0 := lx0 + my0 + nz0, v0 := (x0 − α)2 + (y0 − β)2 + (z0 − γ)2 for which the equation F (x, y, z) = Ω lx + my + nz, (x − α)2 + (y − β)2 + (z − γ)2 (3.1.6) holds identically in a neighborhood of (x0, y0, z0). Now, the equation lx + my + nz = u (3.1.7) represents a plane which cuts the line FF ′ (represented by (3.1.1)) perpendicularly, while the equation (x − α)2 + (y − β)2 + (z − γ)2 = v (3.1.8) represents a sphere of radius v and with center (α, β, γ) on the line FF ′. The points (x, y, z) which are on the plane, (3.1.7), and on the sphere, (3.1.8), simulta- neously, are on their circle of intersection and this circle has its center on the line FF ′. Therefore, the equation (3.1.2) of S, i.e., Ω(u, v) = 0, represents a surface generated by a circle of variable radius whose center moves along the line FF ′ and whose plane is perpendicular to that line. Thus, every planar transverse section of S, perpendicular to FF ′, consists of one or more circles whose centers are on the line FF ′. That is, S is a surface of revolution with axis FF ′. This completes the proof of Drucker’s theorem. Corollary 1. If the z-axis is the axis of revolution, we may take the origin as the point (α, β, γ), and the equation (3.1.2) becomes F (x, y, z) = Ω z, x2 + y2 + z2 (3.1.9) There are similar simplifications in (3.1.6) if we take the other coordinate axes as the axis of revolution. Corollary 2. If Fx(x0, y0, z0) ≡ 0, then the normal is everywhere perpendicular to the x − axis and the equation (3.1.2) becomes the cylinder of revolution: F (x, y, z) = Ω my + nz, (y − β)2 + (z − γ)2 (3.1.10) There are similar simplifications if the other components of the normal are zero. Remark 1. In order to apply the theorem on functional dependence which we used in the above proof, we have to make sure that we comply with all the hypotheses. The only one, which we did not explicitly state in the body of the proof is, using the notations of (3.1.7) and (3.1.8), is that at least one of the three jacobians ∂(u, v) ∂(x, y) ∂(u, v) ∂(y, z) ∂(u, v) ∂(x, z) , (3.1.11) is different from zero at (x0, y0, z0). We claim that even more is true in our case. We will prove that at least two of the jacobians (3.1.11) are different from zero. Suppose, to the contrary, that at least two of them are equal to zero, say ∂(u, v) ∂(x, y) ∂(u, v) ∂(y, z) = 0 (3.1.12) This leads to x − α y − β y − β z − γ , (3.1.13) respectively. By (3.1.13) x − α y − β z − γ (3.1.14) which is the equation of the axis FF ′. But, this means that S is just the straight line axis, which is excluded by the hypothesis that S is a smooth surface. Therefore, at least two of the jacobians (3.1.11) are different from zero and the theorem on functional dependence is applicable. Remark 2. The proof shows that the characteristic property of a surface S of revolution is that the normal to any point of S intersects the axis of revolution. 3.2 Proof of Descartes’ Theorem We adapt Drucker’s definition Definition 3. Let S be a smooth connected surface and let F and F ′ be points not in S. We say that S is a perfect lens relative to F and F ′ if, for each point P in S: 1. the normal ~N at P lies in the subspace spanned by the vectors FP and F ′P , and 2. the sines of the angles which FP and F ′P form with that normal are in constant ratio for every point P in S. By condition 2 of the definition, the cross-section of S sliced out by the xy-plane is a plane curve C which is a perfect two-dimensional lens relative to F and F ′. That means that C is either (part of) a cartesian oval, or (part of) a conic section, or a degenerate case of either one. Therefore, by condition 1 and Drucker’s Theorem, a three dimensional perfect lens S is (part of) a surface of revolution with axis FF ′ obtained by rotating a two-dimensional perfect lens S around it. This completes the proof of Descartes’ Theorem. Acknowledgment Support from the Vicerrectoŕıa de Investigación of the University of Costa Rica is acknowl- edged. References [1] Barbin, E; Guitart, R., La pulsation entre les conceptions optiques, algbriques, articules, et projetive des ovales cartsiennes, Actes de la septime universit d’t interdis- ciplinaire sur l’historie des mathmatiques, (Nantes, juillet 1997) [2] Buck, R.C., Advanced Calculus, McGraw-Hill, Inc., New York, 1978 [3] R. Descartes, The Geometry, Part II, Great Books, Volume 31, Encyclopedia Bri- tannica, New York, 1952 [4] Diokles, On Burning Mirrors, G J Toomer, Sources in the History of Mathematics and the Physical Sciences 1, Springer, (New York, 1976). [5] Drucker, D., Reflection Properties of Curves and Surfaces, Mathematics Magazine, 65 (1992), 147-157 [6] Goursat, E., A Course in Mathematical Analysis, Vol. 1, Dover Publications New York, 1964 [7] Hecht, E., Optics, 4th edition, Barnes and Noble New York, 2001 [8] Kepler, Astronomia, pars optica iv.4, 1604. [9] Klein, M.V., Optics, John Wiley & Sons New York, 1970 [10] Leighton, R.B. ; Vogt, R.E., Exercises in Introductory Physics, Addison-Wesley, Phillipines, 1969 [11] Maesumi, M., Parabolic Mirrors, Elliptic and Hyperbolic Lenses, AMM, 99 (1992), 558-560 [12] Monge, G., R Taton (ed.), L’Oeuvre scientifique de Monge Paris, 1951., [13] Osgood, W.F., Functions of Real Variables, G.E. Stechert & Co., New York, 1938 [14] Ferrol, R., Mandonnet, J., Ovale de Descartes From http://www.mathcurve.com/courbes2d/descartes/descartes.shtml(2007) [15] Salmon, G., A Treatise on the Analytic Geometry of Three Dimensions, Hodges, Fig- gis, & Co, Dublin, 1882 [16] Spiegel, M., Applied Differential Equations, Prentice-Hall, New York, 1958 [17] Taylor, A.E.; Mann, W.R., Advanced Calculus, 3rd edition, John Wiley & Sons New York, 1986 [18] Weisstein, Eric W. ”Cartesian Ovals.” From MathWorld–A Wolfram Web Re- source. http://mathworld.wolfram.com/CartesianOvals.html http://www.mathcurve.com/courbes2d/descartes/descartes.shtml(2007 http://mathworld.wolfram.com/CartesianOvals.html Introduction The Analytical Problem (in two dimensions) Both Fixed Points Are Finite The Differential Equation The Solution of the Differential Equation One Fixed Point At Infinity; One Fixed Point Finite The Differential Equation The Solution of the Differential Equation Both Fixed Points are at Infinity Descartes' Theorem Drucker's Characterization of a Surface of Revolution Proof of Descartes' Theorem

0704.1060

Deformation of Dijkgraaf-Vafa Relation via Spontaneously Broken N=2 Supersymmetry

April, 2007 OCU-PHYS 262 Deformation of Dijkgraaf-Vafa Relation via Spontaneously Broken N = 2 Supersymmetry H. Itoyamaa,b∗ and K. Maruyoshia† a Department of Mathematics and Physics, Graduate School of Science Osaka City University b Osaka City University Advanced Mathematical Institute (OCAMI) 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan Abstract It is known that the fermionic shift symmetry of the N = 1, U(N) gauge model with a superpotential of an adjoint chiral superfield is replaced by the second (spontaneously broken) supersymmetry in the N = 2, U(N) gauge model with a prepotential and Fayet-Iliopoulos parameters. Based on a diagrammatic analysis, we demonstrate how the well-known form of the effective superpotential in the former model is modified in the latter. A set of two equations on the one-point functions stating the Konishi anomaly is modified accordingly. ∗e-mail: [email protected] †e-mail: [email protected] http://arxiv.org/abs/0704.1060v3 I. Introduction For more than two decades, effective superpotential has been a central object in the nonperturbative study of N = 1 supersymmetric theories. This object is protected from perturbative corrections in the conventional sense [1], and yet receives important nonpertur- bative corrections (see for example [2, 3]). In recent years, analyses from superstring theory have revealed an interesting perturbative window into nonperturbative physics with the use of the gluino condensate superfield variable [4, 5, 6, 7]. In [8], field theoretic discussion based on the model with U(N) gauge group and rigid N = 1 supersymmetry (see eq. (2.2) for its action SN=1) is given and this is in accord with the string theory based developments. Superstring theory, on the other hand, insists upon maximally extended supersymmetry with no adjustable parameter. A scenario that one may draw is that this extended super- symmetry becomes spontaneously broken to N = 1. Along this vein, a field theory model with U(N) gauge group and rigid N = 2 supersymmetry spontaneously broken to N = 1 has been introduced in [9, 10, 11] (see eq. (2.1) for its action SN=2), generalizing the abelian counterpart of [12]. (See also [13] for N = 2 supergravity and [14] for related discussions.) Several properties of this model have been derived. In this letter, we make a first analysis on the interplay between the effective superpotential and partially as well as spontaneously broken N = 2 supersymmetry, shedding a light upon the comparison of the two models mentioned above. A key aspect of this comparison is that the fermionic shift symmetry of SN=1 gets replaced by the second (spontaneously broken) supersymmetry of SN=2. In fact, this is one of the original motivations/results of [9]. The fermionic shift symmetry of SN=1 supplies the well-known formula [7, 8] constraining the form of the effective superpotential, which is originally proposed from flux compactifi- cation of string theory [15, 16]. Based on a diagrammatic analysis [17] (for a review see [18]), we are able to state how this form undergoes modifications in the model SN=2. After giving a few accounts of the model in the next section, we present a diagrammatic analysis of Weff in section III. Our final understanding is summarized in eq. (3.10). This is followed by a computation of the two-loop contribution to Weff in section IV. In the final section, we derive a set of two equations on the two generating functions R(z) and T (z) of the one-point functions, generalizing the argument based on the chiral ring and the Konishi anomaly in [8]. We observe a modification from that given in [8] here as well. II. The U(N) gauged model with spontaneously broken N = 2 supersymmetry Let us briefly recall a few ingredients of the model, which are needed in what follows. The action [9] given in the Wess-Zumino gauge can be written as SN=2 = d4xd4θ Φ̄eadV ∂F(Φ) − h.c. + ξV 0 d4xd2θ ∂2F(Φ) ∂Φa∂Φb WaWb + eΦ0 +m ∂F(Φ) + h.c. , (2.1) where V = V ata and W α are the vector superfield and the gauge superfield strength re- spectively and Φ = Φata (a = 0, 1, . . . , N 2 − 1) is the chiral superfield ∗. There are three Fayet-Iliopoulos parameters (e,m, ξ) which are all real. For simplicity, we choose the prepo- tential as a single trace function of degree n + 2: F(Φ) = k=1 gkTrΦ k+1/(k + 1)!. While this action is shown to be invariant under the N = 2 supersymmetry transformations [9, 10], the vacuum breaks half of the N = 2 supersymmetries. Extremizing the scalar potential, we obtain the condition 〈 ∂ ∂Φ0∂Φ0 〉 = −(e± iξ)/m, which is a polynomial of order n and this determines the expectation value of the scalar field. The action SN=2 in (2.1) is to be compared with that of the N = 1, U(N) gauge model with a single trace tree level superpotential W (Φ): SN=1 = d4xd4θTr Φ̄eadV Φ+ d4xd2θTr (iτWW +W (Φ)) + h.c. , (2.2) where τ is a complex gauge coupling τ = θ/2π + 4πi/g2. In [9], it is checked that the second supersymmetry reduces to the fermionic shift symme- try in the limit m → ∞. The action SN=2 in fact reduces to SN=1 in the limit m, e, ξ → ∞ with mgk (k ≥ 2) fixed [19]. We show that our result reduces to that of [7, 17] in this limit. III. Diagrammatic analysis of the effective superpotential In this letter, we consider the matter-induced part of the effective superpotential by integrating out the massive degrees of freedom Φ: d4x(d2θWeff+h.c.+d 4θ(nonchiral terms)) = DΦDΦ̄eiSN=2. (3.1) ∗a = 0 corresponds to the overall U(1) part. Let us take Wα (or V ) as the background field †. We consider the case of unbroken U(N) gauge group. For simplicity, we choose 〈Φ〉 = 0 by setting g1 = −(e± iξ)/m. We are interested in the holomorphic superpotential which does not contain the anti- holomorphic couplings ḡk. We can take ḡk = 0 for k ≥ 3 without loss of generality. Collecting the Φ̄ dependent terms, we obtain SΦ̄ = d4xd4θ Φ̄eadV ∂F(Φ) − (ḡ1Φ̄ + Φ̄2)eadV Φ d4xd2θ̄ TrΦ̄2 d4xd4θTr Φ̃ḡ2 ḡ1Φ− . (3.2) In the last expression, we have introduced a covariantly anti-chiral superfield Φ̃ = Φ̄eadV , which satisfies ∇αΦ̃ = 0 (∇α = e −adV Dαe adV ). Eq. (3.2) is quadratic in Φ̃ and can be integrated straightforwardly. As a result, we obtain the following terms, 16ḡ2 ḡ1Φ− ḡ1Φ− (Img1) 8mḡ2 Φ∇2Φ+ . . . , (3.3) where . . . denotes the higher order interaction terms, which we will not consider here. Indeed, these interaction vertices are higher order in m−1 compared to the vertices which we consider below. These contribute to our main result (3.10) as higher order corrections in m−1 and do not spoil our conclusion that the effective superpotential is modified from the case of SN=1 (2.2). Replacing d2θ̄ integration by −∇̄2/4 and collecting the terms which are not in SΦ̄, we obtain an action after the Φ̄ integration ‡: d4xd2θTr (Img1) 32mḡ2 Φ∇̄2∇2Φ +m (WΦsWΦk−1−s) . (3.4) The first two terms are already present in the integrations with regard to the action SN=1 (2.2). The last term is new and originates from the gauge kinetic term in eq. (2.1). As we will see below, this last term does contribute to the effective superpotential and becomes responsible for the violation of the well-known relation [7, 8] between the effective superpo- tential of the gauge theory and the planar free energy of the matrix model having the tree level (bare) superpotential as its potential. †The simplest background is that consisting of a vanishing gauge field Aµ and a constant gaugino λ which satisfies {λα, λβ} = 0 [18]. This configuration implies that traces of more than two W vanish. ‡In eq. (3.4), it is understood that the generating functional has a renormalized perturbation expansion in which a nonvanishing tadpole is always canceled by a nonvanishing value of the source coupled to Φ. This implies that the tadpole can in practice be ignored. After rescaling Φ → aΦ with a2 = mḡ2/(Img1) 2, the quadratic part of the action (3.4) reduces to −�+m′ + adWαDα (2WWΦ2 +WΦWΦ), where we have used the relation ∇̄2∇2Φ = 16(�Φ − adWαDαΦ/2) and introduced m a2mg2 and g 3 = a 2g3/12. The propagator in the momentum space is ∆(p, π) = dse−s(p 2+m′+ 1 adWαπα−ig The Grassmann momentum πα is Fourier transformation of superspace coordinate θα and the matrix M is Mabcd = (WW)daδbc + (WW)bcδda +WdaWbc, (3.5) where we have exhibited the gauge index dependence explicitly. This matrix is not present in the propagator of [17]. Using eq. (3.5), we are able to insert W without involving the momentum πα. The interaction terms in eq. (3.4) are divided into the following two types: type I. m Tr Φk, k = 3, . . . , n+ 1. type II. − Tr(WΦsWΦk−1−s), k = 4, . . . , n+ 1. Type I vertices are already present in [17]. Type II vertices are not present in [17]. They insert two W in specific ways. Before going on to consider loop diagrams, let us first demonstrate that we have only to consider planar diagrams in our case as well [17, 18]. For a given diagram, we denote by V the number of vertices, by P the number of propagators and by h the number of holes (or index loops). There are V sets of chiral superspace integrations from V vertices. One of them becomes the chiral superspace integration over the effective superpotential, and the number of remaining πα momentum integrations is P − V + 1. These Grassmann integrations must be saturated by 1 adWαπα terms in the propagators. Furthermore, we can freely insert W both from the M terms in the propagators and from the type II vertices. If we denote the number of these additional insertions by 2α, the total number of W insertions is 2(P − V + 1 + α). On the other hand, one index loop can accommodate at most two W. Thus we have h ≥ P − V + 1+ α. This implies that only the planar diagrams contribute to the effective superpotential as the Euler number of the diagram is χ = V − P + h. A planar diagram with h index loops has (h − 1) loop momenta. Let us consider the (h− 1)-loop planar diagrams (contributing to the (h− 1)-loop vacuum amplitude) in which all vertices are type I. Let us, for a moment, ignore the M term of (3.5). The calculation is then the same as that of [17] which we briefly describe. Each diagram is a product of the bosonic part obtained by integrating over the momentum p and the fermionic one coming from the πα integrations. As we have seen in the last paragraph, we have exactly 2(h − 1) W insertions in the fermion part. There are two possibilities for these W insertions. The one is to keep one of the index loops empty, filling the remaining index loops with two W. This yields NSh−1 term, where S = − 1 TrU(N)W αWα. The other is to fill each of two index loops chosen with single W, which yields Sh−2wαwα terms where w α = 1 TrWα. After calculating the both parts, we perform the Schwinger parameter integrals. Clearly this procedure is universal to every (h− 1)-loop planar diagram up to the multiplications by the symmetric factor and by the coupling constants. Therefore every such diagram is a product of these factors with the following expression {NhSh−1 + hC22S h−2wαwα} ≡ (h−1) 0 ,(3.6) where we have introduced A (h−1) 0 . The factor h of the first term comes from the choice of the empty index loop, and hC2 of the second term is the combination of inserting two W into different index loops. The most important fact is that the dependence on Schwinger parameters of the bosonic part is cancelled by that of the fermionic part. This explains that the calculation of the effective superpotential of the gauge theory reduces to that of the matrix model [17]. There are two types of corrections to A (h−1) 0 . The one is due to the presence of the M terms in the propagators, which we denote by A (h−1) 1 . The other is due to the type II vertices, which is obtained by replacing one of the type I vertices in A (h−1) 0 by the corresponding type II vertex and by summing over all possibilities. We denote this by A (h−1) 2 . We consider them in order. Let us see the effects of the M term, namely, eq. (3.5). It plays a role of inserting two W further. Thus we will obtain terms which are proportional to Sh. Note that we cannot insert more than two W because, in such case, at least one of the index loops has more than two insertions of W. For the parts contributing to NSh−1, which have an empty index loop, we can further insert WαWα from the first two terms in (3.5). In the case in which they are inserted in the a-th index loop, we obtain TrWW, where ia labels the propagators which form the a-th index loop. The absence of factor N is explained by the absence of an empty index loop. The factor h is not present as we have so far restricted ourselves to the a-th index loop. Summing over all index loops, we obtain the first contribution to A (h−1) TrWW = 2ig′3 TrWW, where we have used that when all index loops are summed, they pass through each double line propagator exactly twice. Let us note that the parts contributing to the second term of eq. (3.6) can receive further insertions of W as well. They have two index loops with a single W insertion, for which we can exploit the last term of M . An insertion of this term requires that two index loops share a propagator. Let us define the index A = 1, . . . , hC2 as labeling the combinations of such two index loops and the index à labeling the cases which have a common propagator in the two index loops chosen. Let us further introduce the index ià labeling the common propagator in case Ã. With these notations, we obtain the second contribution to A (h−1) 2Sh−2 WαabWαcdW Wβdc = ig TrWαWα. Putting all these together, we obtain the contributions from the vertices of type I, (h−1) (h−1) 1 (si)) )h−1( 16π2iPg3S wαwα. (3.7) It is important that the above new term has Schwinger parameter dependence aside from the exponential factor. In [17], it was pointed out that the cancellation of this dependence represents the reduction of the system to the matrix model. The appearance of this new term with Schwinger parameter dependence may spoil this reduction. Note also that this new term does not have an overall factor N , indicating the violation of the well-known relation due to Dijkgraaf-Vafa [7]. We now turn to the vertices of type II which contain two W insertions. The ℓ-th order vertex in Φ is Tr(2WWΦℓ +WΦWΦℓ−1 + . . .+WΦℓ−1WΦ). (3.8) where we have omitted the overall factors. The first term inserts two W into an index loop while the remainder insert them into two different index loops. Having done 2(h − 1) πα Figure 1: two-loop planar diagrams integrations, we obtain 2(h − 1) W insertions. We can therefore use vertex (3.8) only once in a diagram. When this is done, insertion of the M term from the propagator is disallowed. Let us consider A (h−1) 2 and suppose that one of the type I vertices, TrΦ ℓ, is replaced by the above vertex (3.8). The first term connects ℓ index loops and we can insert W2 into ℓ different ways. Thus we obtain 2ℓTrWW as a contribution to A (h−1) 2 . For the other terms of eq. (3.8), there are in total ℓ(ℓ − 1) ways of inserting two W into different index loops. These give 2Sh−2 ℓ(ℓ− 1) WαabWαcdW baWβdc = ℓ(ℓ− 1)TrWW. Summing the above two contributions, we obtain ℓ(ℓ + 1)TrWW. Thus, in any (h − 1)-loop diagram, changing a vertex from type I to type II is equivalent to considering only NSh−1 terms in eq. (3.6) and changing the coupling constant by mgℓ → 16π2igℓ+1S , for ℓ ≥ 3. (3.9) Therefore, we obtain a formula for the contribution from the (h− 1)-loop diagrams with P propagators to Weff in (2.1), (h−1) ∂F (h−1) ∂2F (h−1) wαwα − 16π2iPmg3 ∂F (h−1) (h−1) 2 , (3.10) where W (h−1) 2 is defined by replacing, in the first term, one coupling constant according to eq. (3.9) and summing over all possibilities. We have denoted by F (h−1) the (h − 1)-loop contribution to the planar free energy of the matrix model. IV. Example As a sample computation, let us take the two-loop contribution to the effective superpo- tential. There are two two-loop planar diagrams depicted in Fig.1. Collecting all possible insertions of W, we obtain (mg3) 32(mg2)3 NS2 − (mg3) 16(mg2)3 Swαwα + π2i(mg3) 2(mg2)4 π2i(mg3)(mg4) 3(mg2)3 . (4.1) The first two terms are the ones which are present in the computation based on [7, 8] with SN=1. The third one comes from the M term in the propagator and the last one from the type II vertices. Note that, in the limit m → ∞ with mgk (k ≥ 2) fixed, we reproduce the result of [17]. In an arbitrary loop amplitude, the situation is the same: new terms are of order m−1 in this limit. The overall U(1) part does not decouple from the SU(N) part. This can be easily seen by translating S into the glueball superfield Ŝ = − 1 TrSU(N) W αWα and extracting the factor in front of wαwα. By the existence of the last two terms in eq. (3.10), it is nonvanishing. For example, in the two-loop example, this part in (4.1) reads πi(mg3)[2(mg2)(mg4)− 3(mg3) 2(mg2)4 wαwα 6= 0, V. The chiral ring and the generalized Konishi anomaly An alternative approach to the effective superpotential is to exploit and extend the prop- erties of the N = 1 chiral ring and the generalized Konishi anomaly equations based on reference [20, 8]. The anomalous Ward identity of our model for the general transformation δΦ = f(Φ,W) is = 〈TrfW ′(Φ)〉Φ − Tr(fF ′′′(Φ)WαWα) , (5.1) where W ′′(Φ) = mF ′′′(Φ). In terms of the two generating functions of chiral one-point functions R(z) = − TrWαWα z − Φ T (z) = z − Φ the anomalous Ward identities (5.1) are R(z)2 = W ′(z)R(z) + f(z), 2R(z)T (z) = W ′(z)T (z) + c(z) + 16π2iF ′′′(z)R(z) + c̃(z), where f(z) and c(z) are polynomials of degree n− 1 in z and c̃(z) is a polynomial of degree n− 2: f(z) = − (W ′(Φ)−W ′(z))WαWα z − Φ c(z) = 4 W ′(Φ)−W ′(z) z − Φ c̃(z) = −i (F ′′′(Φ)− F ′′′(z))WαWα z − Φ The last term of eq. (5.1) does not contribute to the equation for R(z) because of the chiral ring relation TrWαWαW βWβ = 0. The equation for R(z) is the same as that of [8], which is the loop equation of the matrix model. On the other hand, the equation for T (z) alters from that of [8]. The final step of this approach is to express the effective superpotential in terms of R(z) and T (z). Taking a variational derivative of (3.1) with respect to the coupling gk, we obtain ∂Weff dzzkT (z) + 16π2i (k − 1)! dzzk−1R(z). Hence we can determine the effective superpotential up to gk independent terms. Acknowledgements We thank Kazuhito Fujiwara, Yosuke Imamura, Hiroaki Kanno, Hironobu Kihara, Ya- sunari Kurita, Kazutoshi Ohta and Makoto Sakaguchi for useful discussions. We are grateful to Hiraku Yonemura for his collaboration at an early stage. This work is supported in part by the Grant-in-Aid for Scientific Research (18540285) from the Ministry of Education, Science and Culture, Japan. Support from the 21 century COE program “Constitution of wide-angle mathematical basis focused on knots” is gratefully appreciated. The preliminary version of this work was presented in YITP workshop “Fundamental Problems and Applications of Quantum Field Theory”, YITP-W-06-16 in Yukawa Institute for Theoretical Physics, Kyoto University (December 14-16 2006). 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0704.1061

Comment on "Chiral Suppression of Scalar Glueball Decay"

Comment on “Chiral Suppression of Scalar Glueball Decay” Kuang-Ta Chao1, Xiao-Gang He2, and Jian-Ping Ma3,1 Department of Physics, Peking University, Beijing Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei Institute Of Theoretical Physics, Academia Sinica, Beijing PACS numbers: PACS numbers: 12.39.Mk, 12.38.Bx In a recent letter, based on an effective Lagrangian, Chanowitz[1] showed that in the limit that the mass mq of a light quark q goes to zero, the decay amplitude for a scalar glueball Gs decaying into qq̄ goes to zero, and conjectured further that this chiral suppression also oc- curs at the hadron level for Gs decays into ππ,KK with the ratio of these two branching ratios to be of the or- der O(m2u,d/m s) for finite quark masses. Here we show that the decay Gs → qq̄ is forbidden in the chiral limit in QCD without assumptions. More essentially, we show that this chiral suppression may be spoiled and may not materialize itself at the hadron level. A glueball here is assumed to be a pure gluonic state. It decays into a qq̄ pair through a multi-gluon annihila- tion process. The decay amplitude for Gs → q(p1)q̄(p2) can be written as a product of a spinor pair ū(p1) and v(p2) with a product of any number of γ matrices sand- wiched between the spinors. Because vector-like coupling in QCD, for mq = 0 the number of the γ-matrices is an odd number which can always be reduced to one γ- matrix. Therefore the amplitude can be written as: Tqq̄ = ū(p1)γµA µv(p2). Lorentz covariance of the amplitude then dictates Aµ(p1, p2) to be of the form a1p + a2p . Therefore in the chiral limit mq = 0, Tqq̄ = 0. The result also applies to a pseudoscalar glueball decays into a qq̄ pair. To study whether there is a chiral suppression in Gs → ππ,KK or not, we work with an effective La- grangian, Ls = fgG a,µνGaµνGs, as in [1], and employ QCD factorization[2] to calculate the amplitude Tππ for Gs → π +π−. To the leading twist-2 order, there are two diagrams with the two gluons splitting into two quarks and two anti-quarks, and then form two pions. The two gluons are off-shell by the scale at order of MGs . A direct calculation gives: Tππ = −αsfg du1du2φπ+(u1)φπ−(u2) (1 − u1)(1 − u2) [1 +O(αs, λ/MGs)] , where φπ is normalized as duφπ(u) = 1. ui(i = 1, 2) is the momentum fraction carried by the anti-quark in the meson. In the above, λ can be any soft scale, such as quark mass, ΛQCD and mπ. Clearly, Tππ is not zero in the chiral limit mq = 0. The amplitude for Gs → K +K− decay can be ob- tained by replacing quantities related to π by those re- lated to K correspondingly. We would obtain, R = B(Gs → ππ)/B(Gs → KK) ≈ f K = 0.48, which is substantially different from 1. This suppression is much milder compared with the one at the quark level. This is due to the fact that in perturbative QCD (pQCD) calcu- lation the decay of Gs → ππ,KK is related to the cou- pling of Gs to two pairs of qq̄ compared with conjectured by Chanowitz in [1], where it is assumed that Gs just cou- ples to one qq̄ pair. We should point out that whether the chiral suppression at quark level can be realized still waits for better non-perturbative calculation for the di- rect two quark hadronization into ππ and KK. If the pQCD contribution dominates, the result of R ≈ f4π/f can be obtained without the assumption of the effective Lagrangian. Because glueball is a pure gluon state, the amplitude of the decay Gs → π +π− can always be writ- ten with QCD factorization as Tππ = f πHg ⊗φπ+ ⊗φπ− , where the higher-twist effects related to π’s are neglected andHg consists of some perturbative coefficient functions and some quantities related to the structure of Gs. Al- though Hg is unknown, one can easily find the result of R ≈ f4π/f The f0(1710) is a candidate for scalar glueball. Early measurement obtained R ≤ 0.11[3], and a larger one by BES[4] R = 0.41+0.11 −0.17 recently. It is interesting to notice that the later is consistent with our result and may fa- vor that the f0(1710) is a gluebal. However one should remember that the prediction R ≈ f4π/f K can have sub- stantial non-perturbative corrections and there may be further complication by mixing effects of a glueball with qq̄ states. A more detailed study can be found in [5]. Acknowledgments: This work was supported in part by grants from NSC and NNSFC (No 10421503). [1] M.S. Chanowitz, Phys. Rev. Lett. 95, 172001(2005) . [2] S.J. Brodsky and G.P. Lepage, Phys. Rev. D24, 2848(1981), G.P. Lepage and S.J. Broadsky, Phys. ReV. D22, 2157(1980). [3] W.-.M Yao et al., (Particle Data Group), J. Phys. G33, 1(2006). http://arxiv.org/abs/0704.1061v1 [4] M. Ablikim et al. (BES Collaboration), Phys. Lett. B642, 441(2006). [5] K.T. Chao, Xiao-Gang He and J.P. Ma, hep-ph/0512327. http://arxiv.org/abs/hep-ph/0512327

0704.1062

Scaling p_T distributions for p and \bar{p} produced in Au+Au collisions at RHIC

Scaling pT distributions for p and p̄ produced in Au+Au collisions at RHIC W.C. Zhang, Y. Zeng, W.X. Nie, L.L. Zhu and C.B. Yang Institute of Particle Physics, Hua-Zhong Normal University, Wuhan 430079, P.R. China With the experimental data from STAR and PHENIX on the centrality dependence of the pT spectra of protons and anti-protons produced at mid-rapidity in Au+Au collisions at 200 GeV, we show that for protons and anti-protons there exists a scaling distribution independent of the colliding centrality. The scaling functions can also describe data from BRAHMS for both proton and anti-proton spectra at y = 2.2 and 3.2. The scaling behaviors are shown to be incompatible with the usual string fragmentation scenario for particle production. PACS numbers: 25.75.Dw,13.85.Ni I. INTRODUCTION One of the most important quantities in investigating properties of the medium produced in high energy col- lisions is the particle distribution for different species of final state particles. RHIC experiments have found a lot of novel phenomena from the particle spectra, such as the unexpectedly large p/π ratio at pT ∼ 3 GeV/c [1], the constituent quark number scaling of the elliptic flows [2], and strong nuclear suppression of the pion spectrum in central Au+Au collisions [3], etc. From the spectrum one can learn a lot on the dynamics for particle production. In many studies, searching for a scaling behavior of some quantities vs suitable variables is useful for unveil- ing potential universal dynamics. A typical example is the proposal of the parton model from the x-scaling of the structure functions in deep-inelastic scatterings [4]. Quite recently, a scaling behavior [5] of the pion spectrum at mid-rapidity in Au+Au collisions at RHIC was found, which related spectra with different collision centralities. In [6] the scaling behavior was extended to non-central region, up to η = 3.2 for both Au+Au and d+Au colli- sions. The same scaling function can be used to describe pion spectra for pT up to a few GeV/c from different col- liding systems at different rapidities and centralities. The shape of pion spectrum in those collisions is determined by only one parameter 〈pT 〉, the mean transverse momen- tum of the particle. It is very interesting to ask whether similar scaling behaviors can be found for spectra of other particles produced in Au+Au collisions at RHIC. In this paper, the scaling property of the spectra for protons and anti-protons is investigated and compared with that for pions. The organization of this paper is as follows. In Sec. II we will address the procedures for searching the scaling behaviors. Then in Sec. III the scaling properties of the spectra for protons and anti-protons produced in Au+Au collisions at RHIC at sNN = 200 GeV will be studied. We discuss mainly the centrality scaling of the spectra at mid-rapidity and extend the discussion to very forward region with rapidity y = 2.2 and 3.2 briefly. Sec. IV is for discussions on the relation between the scaling behaviors and the string fragmentation scenario. II. METHOD FOR SEARCHING THE SCALING BEHAVIOR OF THE SPECTRUM As done in [5, 6], the scaling behavior of a set of spectra at different centralities can be searched in a few steps. First, we define a scaled variable z = pT /K , (1) and the scaled spectrum Φ(z) = A 2πpTdpTdy pT=Kz , (2) with K and A free parameters. As a convention, we choose K = A = 1 for the most central collisions. With this choice Φ(z) is nothing but the pT distribution for the most central collisions. For the spectra with other centralities, we try to coalesce all data points to one curve by choosing proper parameters A and K. If this can be achieved, a scaling behavior is found. The detailed expression of the scaling function depends, of course, on the choice of A and K for the most central collisions. This arbitrary can be overcome by introducing another scaling variable u = z/〈z〉 = pT /〈pT 〉 , (3) and the normalized scaling function Ψ(u) = 〈z〉2Φ(〈z〉u)/ Φ(z)zdz . (4) Here 〈z〉 is defined as 〈z〉 ≡ zΦ(z)zdz/ Φ(z)zdz . (5) By definition, Ψ(u)udu = uΨ(u)udu = 1. This scaled transverse momentum distribution is in essence similar to the KNO-scaling [7] on multiplicity distribu- tion. III. SCALING BEHAVIORS OF PROTON AND ANTI-PROTON DISTRIBUTIONS Now we focus on the spectra of protons and anti- protons produced at mid-rapidity in Au+Au collisions at http://arxiv.org/abs/0704.1062v2 sNN = 200 GeV. STAR and PHENIX Collaborations at RHIC published spectra for protons and anti-protons at mid-rapidity for a set of colliding centralities [8, 9]. STAR data have a pT coverage larger than PHENIX ones. As shown in Fig. 1, all data points for proton spectra at different centralities can be put to the same curve with suitably chosen A and K, by the procedure explained in last section. The parameters are shown in Table I. Except a few points for very peripheral collisions (cen- tralities 60-92% for PHENIX data and 60-80% for STAR data), all points agree well with the curve in about six orders of magnitude. The larger deviation of data at centralities 60-92% for PHENIX and 60-80% for STAR from the scaling curve may be due to the larger central- ity coverage, because the size of colliding system changes dramatically in those centrality bins. For simplicity we define v = ln(1+ z), and the curve can be parameterized Φp(z) = 0.052 exp(14.9v − 16.2v2 + 3.3v3) . (6) 2 4 6 8 10 12 PHENIX 0−10% 20−30% 40−50% 60−92% 0−12% 10−20% 20−40% 40−60% 60−80% p in Au+Au FIG. 1: Scaling behavior of the spectrum for protons pro- duced at mid-rapidity in Au+Au collisions at RHIC. The data are taken from [8, 9]. Feed-down corrections are considered in the data. The solid curve is from Eq. (6). Similarly, one can put all data points for anti-proton spectra at different centralities to a curve with other sets of parameters A and K which are given also in TABLE I. The agreement is good, as can be seen from Fig. 2, with only a few points in small pT region for peripheral colli- sions departing a little from the curve. For anti-proton the scaling function is Φp̄(z) = 0.16 exp(13v − 14.9v2 + 2.9v3) , (7) with v defined above. To see how good is the agreement between the fitted curves in Figs. 1 and 2 and the experimental data, one can calculate a ratio B = experimental data/fitted results , 2 4 6 8 10 12 STARPHENIX 0−12%0−10% 10−20%20−30% 20−40%40−50% 40−60%60−92% 60−80% 40−80% pbar in Au+Au FIG. 2: Scaling behavior of the spectrum for anti-protons produced at mid-rapidity in Au+Au collisions at RHIC. The data are taken from [8, 9]. Feed-down effects are not corrected in the STAR data for p̄. The solid curve is from Eq. (7). STAR p p̄ centrality K A K A 0-12% 1 1 1 1 10-20% 0.997 1.203 1.005 1.417 20-40% 0.986 2.009 0.991 2.305 40-60% 0.973 4.432 0.993 5.414 60-80% 0.941 13.591 0.959 16.686 40-80% 0.986 8.126 PHENIX p p̄ centrality K A K A 0-10% 1.042 1.226 1.068 2.404 20-30% 1.026 2.532 1.045 4.901 40-50% 1.031 6.253 1.013 11.754 60-92% 0.934 39.056 0.935 69.31 BRAHMS p p̄ centrality K A K A y = 2.2 0.930 0.921 y = 3.2 1.079 0.754 1.153 6.985 TABLE I: Parameters for coalescing all data points to the same curves in Figs. 1 and 2. and show B as a function of pT in linear scale for all the data sets, as shown in Fig. 3 for the case of proton. From the figure one can see that almost all the points have values of B within 0.7 to 1.3, which means that the scaling is true within an accuracy of 30%. This is quite a good fit, considering the fact that the data cover about 6 orders of magnitude. For anti-protons, the agreement is better than for protons. Now one can see that the transverse momentum dis- tributions for protons and anti-protons satisfy a scaling law. For large pT (thus large z) the scaling functions in Eqs. (6) and (7) behave as powers of pT , though the 2 4 6 8 10 12 FIG. 3: Ratio between experimental data and the fitted re- sults shown in Fig. 1. STAR and PHENIX data are taken from [8, 9]. Symbols are the same as in Fig. 1. 2 4 6 8 10 12 PHENIX 0−10% 20−30% 40−50% 60−92% BRAHMS y=3.2 p in Au+Au 0−12% 10−20% 20−40% 40−60% 60−80% FIG. 4: Normalized scaling distribution for protons produced at mid-rapidity and very forward direction in Au+Au col- lisions at RHIC with the scaling variable u. STAR and PHENIX data are taken from [8, 9] and BRAHMS data from [10]. expressions are not in powers of z or pT . The scaling functions in Eqs. (6) and (7) depend on the choices of A and K for the case with centrality 0-12% for STAR data. With the variable u defined in Eq. (3) this dependence can be circumvented. 〈z〉’s for protons and anti-protons are 1.14 and 1.08, respectively, with integration over z in the range from 0 to 12, roughly corresponding to the pT range measured by STAR. The normalized scaling func- tions Ψ(u) for protons and anti-protons can be obtained easily from Eqs. (6) and (7) and are shown in Figs. 4 and 5, respectively together with scaled data points as in Figs. 1 and 2. A simple parameterization for the two normalized scaling functions in Figs. 4 and 5 can be given as follows Ψp(u) = 0.064 exp(13.6v − 16.67v2 + 3.6v3) , Ψp̄(u) = 0.086 exp(12.41v − 15.31v2 + 3.16v3) , with v = ln(1 + u). 2 4 6 8 10 12 PHENIX 0−10% 20−30% 40−50% 60−92% BRAHMS y=3.2 y=2.2 pbar in Au+Au 0−12% 10−20% 20−40% 40−60% 60−80% 40−80% FIG. 5: Normalized scaling distribution for anti-protons pro- duced at mid-rapidity and very forward direction in Au+Au collisions at RHIC with the scaling variable u. STAR and PHENIX data are taken from [8, 9] and BRAHMS data from [10]. As in the case for pion distributions, one can also in- vestigate the pT distributions of protons and anti-protons in non-central rapidity regions in Au+Au collisions. The only data set we can find is from BRAHMS [10] at rapid- ity y = 2.2 and 3.2 with centrality 0-10%. It is found that the BRAHMS data can also be put to the same scaling curves, as shown in Figs. 4 and 5. The values of corre- sponding parameters A and K are also given in TABLE I. Thus the scaling distributions found in this paper may be valid in both central and very forward regions for pro- tons and anti-protons produced in Au+Au collisions at RHIC at sNN=200 GeV. Now one can ask for the difference between the scaling functions for protons and anti-protons. After normaliza- tion to 1 the difference between the scaling distributions Ψ(u) for protons and anti-protons is shown in Fig. 6. In log scale the difference between the two scaling functions is invisible at low u. To show the difference clearly a ra- tio r = Ψp(u)/Ψp̄(u) is plotted in the inset of Fig. 6 as a function of u. The increase of r with u is in agreement qualitatively with data shown in [9] where it is shown that p̄/p decreases with pT monotonically. The difference in the two scaling functions can be understood physically. In Au+Au collisions there are much more quarks u, d than ū and d̄ in the initial state. In the central region in the state just before hadronization, more u and d quarks can be found because of the nuclear stopping effect in the interactions. As a consequence, more protons can be formed from the almost thermalized quark medium than anti-protons in the small pT regime. Experimen- tal data show that in low pT region the yield of anti- proton is about 80% that of protons in central Au+Au collisions at RHIC. This difference contributes to the net baryon density in the central region in Au+Au collisions at RHIC. On the other hand, in the large pT region, pro- tons and anti-protons are formed mainly from fragmen- tation of hard partons produced in the QCD interactions with large momentum transfer. As shown in [11], the gluon yield from hard processes is about five times that of u and d quarks. The fragmentation from a gluon to p and p̄ is the same. The amount of u, d quarks from hard processes is about 10 times that of ū, d̄ when the hard parton’s transverse momentum is high enough. It is well-known that the fragmentation function for a gluon to p or p̄ is much smaller than that for a u or d (ū or d̄) to p (p̄) because of the dominant valence quark contribu- tion to the latter process. As a result, the ratio of yields of proton over anti-proton at large pT is even more than that at small pT . After normalizing the distributions to the scaling functions the yield ratio of proton over anti- proton increases approximately linearly with u when u is large. It should be mentioned that no such difference for π+, π− and π0, because they all are composed of a quark and an antiquark. 2 4 6 8 10 12 2 4 6 8 10 12 FIG. 6: Comparison between the scaling functions for pro- tons and anti-protons produced at mid-rapidity in Au+Au collisions at RHIC with the scaling variable u. The inset if for the ratio Ψp(u)/Ψp̄(u). The scaling behaviors of the pT distribution functions for protons and anti-protons can be tested experimen- tally from studying the ratio of moments of the mo- mentum distribution, 〈pnT 〉/〈pT 〉n = unΨ(u)udu for n = 2, 3, 4, · · ·. From the determined normalized distri- butions, the ratio can be calculated by integrating over u in the range from 0 to 12, as mentioned above, and the results are tabulated in TABLE II. The values of the ratio are independent of the parameters A and K in the fitting process but only on the functional form of the scaling distributions. If the scaling behaviors of particle distributions are true, such ratios should be constants in- n p p̄ π 2 1.194 1.215 1.65 3 1.717 1.775 4.08 4 2.978 3.064 14.4 5 6.415 6.417 64.73 6 19.045 17.253 373.82 TABLE II: Ratio of moments 〈pnT 〉/〈pT 〉 n for protons, anti- protons and pions produced in Au+Au collisions at RHIC. dependent of the colliding centralities and rapidities. For comparison, the corresponding values of the ratio for pi- ons produced in the same interactions, calculated in [6], are also given in TABLE II. Because of very small dif- ference in the scaling distributions for protons and anti- protons at small u, the ratio for protons increases with n at about the same rate as for anti-proton for small n. For large n, the ratio for p becomes larger than that for p̄ because of the big difference in the scaling functions for p and p̄ at large u. Because of the very strong suppression of high transverse momentum proton production relative to that of pions, the ratio for pions increases with n much more rapidly than for p and p̄. Another important question is about the difference be- tween the scaling functions for protons in this paper and for pions in [5, 6]. Experiments at RHIC have shown that the ratio of proton yield over that of pion increases with pT up to 1 in the region pT ≤ 3 GeV/c and saturates in large pT region. This behavior should be seen from the scaling functions for these two species of particles. For the purpose of comparing the scaling distributions we define a ratio R = Ψp(u)/Ψπ(u) , (8) and plot the ratio R as a function of u in Fig. 7. The ratio increases with u, when u is small, reaches a maximum at u about 1 and then decreases. Finally it decreases slowly to about 0.1 for very large u. The highest value of R is about 1.6, while the experimentally observed p over π ratio is about 1 at pT ∼ 3 GeV/c. The reason for this difference is two-fold. One is the normalization difference in defining R and the experimental ratio. Another lies in the different mean transverse momenta 〈pT 〉’s for pions and protons with which the scaling variable u is defined and used in getting the ratio R. The existence of difference in the scaling distributions for different species of particles produced in high energy collisions is not surprising, because the distributions re- flect the particle production dynamics which may be dif- ferent for different particles. In the quark recombina- tion models [12, 13, 14] pions are formed by combining a quark and an anti-quark while protons by three quarks. Because different numbers of (anti)quarks participate in forming the particles, their scaling distributions must be different. In this sense, our investigation results urge more studies on particle production mechanisms. 0 1 2 3 4 5 FIG. 7: Ratio Ψp(u)/Ψπ(u) between the scaling functions for protons and pions produced in Au+Au collisions at RHIC as a function of the scaling variable u. The pion scaling distri- bution is from [5, 6]. IV. DISCUSSIONS From above investigation we have found scaling distri- butions for protons and anti-protons produced in Au+Au collisions at RHIC in both mid-rapidity and forward re- gion. The difference between those two scaling distri- butions is quite small, but they differ a lot from that for pions and the ratio Ψp/Ψπ exhibits a nontrivial behavior. Investigations in [5, 6] and in this paper have shown that particle distributions can be put to the same curve by linear transformation on pT . Though we have not yet a uniform picture for the particle productions in high energy nuclear collisions, the scaling behaviors can, in some sense, be compared to that from the string frag- mentation picture [15]. In that picture if there are n strings, they may overlap in an area of Sn and the average area for a string is then Sn/n. It is shown that the momentum distributions can be related to the case in pp collisions also by a linear variable change pT → pT ((Sn/n)AuAu/(Sn/n)pp)1/4. Viewed from that picture, our fitted K gives the degree of string overlap. The average area for a string in most central Au+Au collisions is about 70 percent of that in peripheral ones from the values of K obtained from fitting the spectra of proton. If string fragmentation is really the production mechanism for all species of particles in the collisions, one can expect that the overlap degree obtained is the same from the changes of spectrum of any particle. In the lan- guage in this work, values of K are expected the same for pions, protons and other particles in the string frag- mentation picture for particle production. Our results show the opposite. Comparing the values of K from [5] and this work, one can see that for pion spectrum K is larger for more peripheral collisions but smaller for pro- ton and anti-proton spectra. Our results indicate that other particle production mechanisms may also provide ways to the scaling distributions. Obviously more de- tailed studies, both theoretically and experimentally, are needed. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 10635020 and 10475032, by the Ministry of Education of China under Grant No. 306022 and project IRT0624. [1] S.S. Adler et al., PHENIX Collaboration, Phys. Rev. C 69, 034909 (2004). [2] D. Molnár and S.A. Voloshin, Phys. Rev. Lett. 91, 092301 (2003); P.Sorensen,STAR Collaboration, J.Phys. G30, S217 (2004). [3] See, for example, S.S.Adler,PHENIX Collaboration, Phys. Rev. Lett. 91, 072301 (2003). [4] J.D. Bjorken and E.A. Paschos, Phys. Rev. 185, 1975 (1969). [5] R.C. Hwa and C.B. Yang, Phys. Rev. Lett. 90, 212301 (2003). [6] L.L. Zhu and C.B. Yang, Phys. ReV. C 75, 044904 (2007). [7] Z. Koba, H.B. Nielsen and P. Olesen, Nucl. Phys. B 40, 317 (1972). [8] S.S. Adler et al., PHENIX Collaboration, Phys. Rev. Lett. 91, 172301 (2003). [9] B.I. Abelev et al., STAR Collaboration, Phys. Rev. Lett. 97, 152301 (2006). [10] R. Karabowicz for the BRAHMS Collaboration, talk at Quark Matter 2005, Budapest, Hungary, Nucl. Phys. A 774, 447 (2006); I. Arsene et al., BRAHMS collaboration, nucl-ex/0610021. [11] D.K. Srivastava, C. Gale and R.J. Fries, Phys. Rev. C 67, 034903 (2003). [12] R.C. Hwa, and C.B. Yang, Phys. Rev. C 67, 034902 (2003). [13] V. Greco, C.M. Ko, and P. Lévai, Phys. Rev. Lett. 90, 202302 (2003); Phys. Rev. C 68, 034904 (2003). [14] R.J. Fries, B. Müller, C. Nonaka and S.A. Bass, Phys. Rev. Lett. 90, 202303 (2003); Phys. Rev. C 68, 044902 (2003). [15] M.A. Braun, F. del Moral, and C. Pajares, Nucl. Phys. A 715, 791 (2003). http://arxiv.org/abs/nucl-ex/0610021

0704.1063

SiO maser observations of a wide dust-temperature range sample

Astrophysical Masers and Their Environments Proceedings IAU Symposium No. 242, 2007 J. Chapman & W. Baan, eds. c© 2007 International Astronomical Union DOI: 00.0000/X000000000000000X SiO maser observations of a wide dust-temperature range sample Jun-ichi Nakashima1 and Shuji Deguchi2 1Academia Sinica Institute of Astronomy and Astrophysics, P.O. Box 23-141, Taipei 10617, Taiwan email: [email protected] 2Nobeyama Radio Observatory, National Astronomical Observatory, Minamimaki, Minamisaku, Nagano 384-1305, Japan Abstract. We present the results of SiO line observations of a sample of known SiO maser sources covering a wide dust-temperature range. The aim of the present research is to investigate the causes of the correlation between infrared colors and SiO maser intensity ratios among different transition lines. We observed in total 75 SiO maser sources with the Nobeyama 45m telescope quasi-simultaneously in the SiO J = 1−0 v = 0, 1, 2, 3, 4 and J = 2−1 v = 1, 2 lines. We also observed the sample in the 29SiO J = 1−0 v = 0 and J = 2−1 v = 0, and 30SiO J = 1−0 v = 0 lines, and the H2O 61,6−52,3 line. As reported in previous papers, we confirmed that the intensity ratios of the SiO J = 1−0 v = 2 to v = 1 lines clearly correlate with infrared colors. In addition, we found possible correlation between infrared colors and the intensity ratios of the SiO J = 1−0 v = 3 to v = 1&2 lines. Keywords. masers — stars: AGB and post-AGB — stars: late-type — stars: mass loss — stars: statistics 1. Introduction An important problem in the studies on the SiO maser was that SiO maser sources ever known were considerably biased. Specifically, the dust (effective) temperature of known SiO maser sources, which was calculated from mid-infrared flux densities (such as the IRAS andMSX flux densities), was limited roughly in a range of 250 K. Tdust . 2000 K. This is because the previous SiO maser surveys have been limited to relatively warm dust-temperature ranges. Consequently a non-negligible number of potential SiO maser sources (especially with a low dust-temperature) have been slipped from the previous SiO maser surveys. Nyman et al.(1993) first realized the importance of SiO maser sources exhibiting a low dust-temperature. They investigated how SiO maser emission behaves in a low dust-temperature range by observing OH/IR stars in the SiO J = 1−0 v = 1&2 and J =2−1 v = 1 lines. The OH/IR stars often exhibit a low dust-temperature less than Tdust = 250 K. In their observation cold objects clearly show a larger intensity ratio of the SiO J = 1−0 v = 2 to v = 1 lines. Both collisional and radiative schemes cannot fully ex- plain this observational properties of the SiO masers (Bujarrabal 1994; Deol et al. 1995). Nyman et al.(1993) suggested that an infrared H2O line (116,6 ν2 = 1→127,5 ν2 = 0) overlapping with the SiO J = 0 v = 1→J = 1 v = 2 transition might play an important role. However, in early 1990s the number of cold SiO maser sources (like OH/IR stars) was quite limited, and it was difficult to statistically investigate the relation between infrared colors and intensity ratios of SiO maser lines. Nakashima & Deguchi(2003b) recently extended the Nyman’s study by surveying the SiO maser emission in cold, dusty IRAS sources exhibiting low dust temperature less http://arxiv.org/abs/0704.1063v1 2 J. Nakashima & S. Deguchi -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 corr: 0.59 A log(F25/F12) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 corr: 0.40 log(F25/F12) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 log(F25/F12) corr: 0.35 Figure 1. Infrared colors versus intensity ratio of SiO maser lines. The horizontal axes represent infrared colors. F25 and F12 denote the IRAS flux densities at λ = 25 and 12 µm, respectively. The filled dots (•), upward triangles (△) and downward triangles (▽) respectively represent the intensity ratios of the SiO maser lines, lower limits of the ratio and upper limits of the ratio. Correlation coefficients are given in the upper-left corners of each panel. The dashed lines are the results of least-square fitting of a first order polynomial. than 250 K. They found roughly 40 new SiO maser sources in the cold dusty objects, and in conjunction with the results of another SiO maser survey of relatively warm IRAS objects (Nakashima & Deguchi 2003a) they clearly demonstrated that the intensity ratio of the SiO J = 1−0 v = 2 to v = 1 lines increases in inversely proportional to the dust temperature. Nakashima & Deguchi(2003b) again suggested that the overlap line of H2O might explain this correlation if the overlap line becomes stronger with decrease of the dust temperature. To consider further this problems, we need to confirm whether properties of the SiO lines other than J = 1−0 v = 1 and 2 lines are consistent with the existence of the H2O overlap line. In this contributed paper we present the result of quasi-simultaneous observations in the multiple different SiO rotational lines with the Nobeyama 45m telescope. The main aim of the observation is to check the behavior of SiO maser intensity ratios including lines other than the J = 1−0 v = 1 and 2 lines. 2. Observations and Results The observing targets were selected from Nakashima & Deguchi(2003a, b) and the Nobeyama SiO maser source catalog (Gorny et al. in preparation) in terms of the IRAS colors and flux densities. The targets are distributed roughly in the right ascension range between 18h and 22h, because the cold SiO maser sources found by Nakashima & Deguchi(2003b) are distributed roughly in this range. We selected the observing targets basically in order of the brightness at λ = 12 µm, but we also paid attention to the source distribution in the IRAS two-color diagram so that the observing targets continuously cover the entire color range. SiO line observations with the Nobeyama 45m telescope were made in two separated periods: May 11–19, 2004 and February 15–19, 2006. In the first period we observed, in total, 38 objects. The observed SiO transitions in the first period were J = 1−0 v =1, 2, 3 and J = 2−1 v =1, 2. We also observed in the 29SiO J = 1−0 v =0 and J = 2−1 v =0 lines. In addition, we observed 27 objects in the H2O maser line at 22 GHz (61,6−52,3) as a backup observation under rainy/heavy cloudy condition. In the second period we observed, in total, 53 objects. The observed transitions in the second period were SiO J = 1−0 v =0, 1, 2, 3, 4, 29SiO J = 1−0 v =0 and 30SiO J = 1−0 v =0. The technical details of the observations will be presented in our future paper (Nakashima & Deguchi, in preparation). In this paper, we focus on the properties of the SiO J = 1−0 v = 1, 2 and 3 lines, in SiO maser intensity ratios 3 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 log(F25/F12) 8 micron flux (MSX) SiO J=1-0 v=1 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 SiO J=1-0 v=2 log(F SiO J=1-0 v=3 C1.71+0.26 2.47+0.25 2.12+0.20 Figure 2. [Panels A, B and C]: Relation between infrared colors and absolute intensity of SiO maser lines. The notation of the infrared colors is same with that used in Figure 1. The intensity of SiO maser lines is standardized at the distance of 1 kpc. The thick dashed lines represent the results of least-square-fitting of a first order polynomial. The inclinations of the fitted lines (thick dashed lines) are given at the lower-right corners of each panel with statistical uncertainty. In the panel A, only the data points below log(F25/F12) = 0.5 were fitted by the polynomial. The data points above log(F25/F12) = 0.5 are independently fitted by a first order polynomial, and the results of the fitting is given as the chain line. [Panel D]: Relation between 8µm absolute flux density and infrared color. The 8µm flux is standardized at the distance of 1 kpc. which we detected an enough number of objects for statistical analysis. Figure 1 shows the relations between infrared colors and intensity ratios among the SiO maser lines. The line intensities used to calculate the intensity ratios are velocity-integrated intensities. In the panel A of Figure 1 we can clearly confirm the positive correlation between the log(F25/F12) color and the intensity ratio of the SiO J = 1−0 v = 2 to v = 1 lines as reported by Nakashima & Deguchi(2003b). Interestingly, the intensity ratios of the J = 1−0 v = 3 to v = 2 lines and of the J = 1−0 v = 3 to v = 1 lines seem to also correlate with the log(F25/F12) color (see, panels B and C) even though the correlation coefficients are slightly smaller than that of panel A. Figure 2 shows the relations between infrared colors and absolute intensities of the SiO maser lines. The intensity of the SiO maser lines is standardized at the distance of 1 kpc using the luminosity distances. The panels A, B and C of Figure 2 show the relations between the log(F25/F12) color and the absolute intensity of the SiO J = 1−0 v = 1, 2 and 3 lines. A notable feature seen in these panels is that the SiO maser absolute intensities undoubtedly correlate with the log(F25/F12) color. Another clear feature is that the higher the vibrational transitions, the steeper the inclination of the dashed lines representing the results of least-square-fitting of a first order polynomial. This tendency is consistent with the correlation seen in Figure 1. In the panel A in Figure 2, the values of the absolute intensity of SiO maser emission seem to maximize at log(F25/F12) ∼ 0.5, and the values tend to decrease with increase of the color in the red region above log(F25/F12) = 0.5. The log(F25/F12) color of 0.5 corresponds the boundary between distributions of AGB and post-AGB stars in the log(F25/F12) color. In fact, the panel A 4 J. Nakashima & S. Deguchi in Figure 1 (and Figure 8 in Nakashima & Deguchi 2003b) shows a sudden change of the feature at log(F25/F12) ∼ 0.5. No such change is seen in the panels B and C in Figure 2, simply because the SiO J = 1−0 v = 2 and 3 lines have not been detected above log(F25/F12) = 0.5 in the present observations. A possible reason for the correlation seen in panels A, B and C in Figure 2 is that the energy input to the SiO maser region increases with the infrared colors. To confirm this possibility, in panel D in Figure 2 we plotted the 8µm flux densities as a function of the infrared colors. The values of the 8µm flux densities were taken from the MSX point source catalog. If we rely on the radiative scheme the 8µm flux should well represent the energy input to the SiO maser region, because the λ = 8µm corresponds to the ∆v = 1 SiO transition (e.g., Deguchi & Iguchi 1976). In panel D in Figure 2 the 8µm flux densities are standardized at the distance of 1kpc using the luminosity distances. The distribution of the data points seen in panel D is, in fact, strikingly similar with those seen in panels A, B and C, supporting that the 8µm flux tightly correlates with the SiO maser intensity as suggested by Bujarrabal et al.(1987). 3. Discussion In this section we discuss the possible explanation for the correlation between infrared colors and SiO maser intensity ratios among the v = 1, 2 and 3 lines at 43 GHz. One possible explanation is to introduce the overlap line of H2O (116,6 ν2 = 1 → 127,5 ν2 = 0), which has been first suggested by Olofsson et al.(1981) to explain the anomalous, weak intensity of the SiO J = 2−1 v = 2 line in oxygen-rich (O-rich) stars. This H2O line overlaps with the SiO J = 0 v = 1→J = 1 v = 2 transition with a velocity difference of 1 km s−1. With this line overlap, the J = 1 v = 2 level is overpopulated, and the weakness of the SiO J = 2−1 v = 2 line is explained by this overpopulation. The overpopulation at the J = 1 v = 2 level is also consistent with the strong intensity of the J = 1−0 v = 2 line. Thus, the correlation between the infrared colors and the intensity ratio of the SiO J = 1−0 v = 2 to v = 1 lines may be explained if this overlap line of H2O becomes stronger with increase of the infrared colors. One problem in this interpretation is that the intensity ratios of the SiO J = 1−0 v = 3 to v = 1&2 lines cannot be explained only by the H2O 116,6 ν2 = 1 → 127,5 ν2 = 0 line. However Cho et al.(2007) recently reported an interesting detection of the SiO J = 2−1 v = 3 line toward an S-type star, χ Cyg. They also confirmed that the SiO J = 2−1 v = 3 line is weak in O-rich stars. The S- type stars have almost same amount of oxygen and carbon atoms in their envelopes, and consequently they have few H2O molecules in the envelopes. These results potentially suggest that another overlap line of H2O affects on the population distribution of SiO in O-rich stars, and Cho et al.(2007) have suggested that the H2O 50,5 ν2 = 2→63,4 ν2 = 1 line overlapping with the SiO J = 0 v = 2→J = 1 v = 3 line (with a velocity difference of about 1.5 km s−1) acts on the population distribution of SiO. Thus, if both H2O 116,6 ν2 = 1→127,5 ν2 = 0 and 50,5 ν2 = 2→63,4 ν2 = 1 lines becomes stronger with increase of infrared colors, all correlations between infrared colors and the SiO maser intensity ratios among the J = 1−0 v = 1, 2 and 3 lines might be explained. The line intensity of the H2O 50,5 ν2 = 2→63,4 ν2 = 1 line is usually weaker than that of the 116,6 ν2 = 1→127,5 ν2 = 0line. This fact also seems to be consistent with the relatively weak intensity of the SiO J = 1−0 v = 3 line. However, there are some other problems on the explanation with the overlap line of H2O. First, we have to explain how the H2O infrared lines overlapping with the SiO lines become stronger with increase of infrared colors. The relative abundance of H2O molecules possibly increases with infrared colors, but this is not conclusive. Second, the SiO maser intensity ratios 5 correlation between infrared colors and the intensity ratios of the SiO J = 1−0 v = 2 to v = 1 lines might be explained without the overlap line of H2O. In the envelopes of very cold objects, strong 8µm emission comes from every direction to the SiO masing region, causing ineffective pumping through the SiO ∆v = 1 transition. On the other hand, 4µm emission corresponding to the SiO ∆v = 2 transition is more effectively pump the SiO population instead of the 8µm. These processes might explain the correlation seen in Figure 3 (e.g., Doel et al. 1995). Third, a recent theoretical calculation predicted that if we introduce the overlap line of H2O the spatial distribution of the maser spots cannot be theoretically reproduced (Soria-Ruiz et al. 2004). Thus, this problem will be remain controversial for some more time. 4. Summary In this research we observed 75 known SiO maser sources quasi-simultaneously in the SiO J = 1−0, v = 0, 1, 2, 3 and 4 lines, SiO J = 2−1 v = 1 and 2, 29SiO J = 1−0 v = 0 and J = 2−1 v = 0, and 30SiO J = 1−0 v = 0 lines. We also observed the targets in the H2O 61,6−52,3 line under rainy/heavy cloudy condition. The sample continuously covers a very wide dust-temperature range from 150 K to 2000 K. The correlation between infrared colors and the intensity ratio of the SiO J = 1−0 v = 2 to v = 1 lines is confirmed as reported by Nakashima & Deguchi(2003b). The intensity rations of SiO J = 1−0 v = 3 to v = 1&2 lines possibly correlate with infrared colors. The overlap lines of H2O might explain the correlations between the infrared colors and the SiO maser intensity ratios among the J = 1−0 v = 1, 2 and 3 lines, although there are alternative ways to interpret the phenomena. Acknowledgements The present research has been supported by the Academia Sinica Institute of Astron- omy & Astrophysics in Taiwan. References Bujarrabal, V. 1994, A&A 285, 953 Bujarrabal, V., Planesas, P., & del Romero, A. 1987, A&A 175, 164 Cho, S.-H., Lee, C. W., & Park, Y.-S. 2007, ApJ 657, 482 Deguchi, S., & Iguchi, T. 1976, PASJ 28, 307 Doel, R. C., et al. 1995, A&A 302, 797 Nakashima, J., & Deguchi, S. 2003a, PASJ 55, 203 Nakashima, J., & Deguchi, S. 2003b, PASJ 55, 229 Nyman, L.-Å., Hall, P. J., & Le Bertre, T. 1993, A&A 280, 551 Olofsson, H., Rydbeck, O. E. H., Lane, A. P., & Predmore, C. R. 1981, ApJ 247, L81 Olofsson, H., Rydbeck, O. E. H., & Nyman, L.-A. A&A 150, 169 Soria-Ruiz, R., et al. 2004, A&A 426, 131 Discussion Elitzur: In our recent work, infrared intensity and colors are uniquely determined by the optical depth of the dust shell. SiO maser intensity is also correlated to the optical depth. Therefore, what you are finding is somehow related to the effect of the activity of the photosphere such as the variation of mass loss rates. Nakashima: Thanks for useful comments. (We took account of Elitzur’s comments in the text.) Introduction Observations and Results Discussion Summary

0704.1064

Excitation Spectrum Gap and Spin-Wave Stiffness of XXZ Heisenberg Chains: Global Renormalization-Group Calculation

Excitation Spectrum Gap and Spin-Wave Velocity of XXZ Heisenberg Chains: Global Renormalization-Group Calculation Ozan S. Sarıyer1, A. Nihat Berker2−4, and Michael Hinczewski4 1Department of Physics, Istanbul Technical University, Maslak 34469, Istanbul, Turkey, 2College of Sciences and Arts, Koç University, Sarıyer 34450, Istanbul, Turkey, 3Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A., and 4Feza Gürsey Research Institute, TÜBİTAK - Bosphorus University, Çengelköy 34684, Istanbul, Turkey The anisotropic XXZ spin- 1 Heisenberg chain is studied using renormalization-group theory. The specific heats and nearest-neighbor spin-spin correlations are calculated thoughout the entire tem- perature and anisotropy ranges in both ferromagnetic and antiferromagnetic regions, obtaining a global description and quantitative results. We obtain, for all anisotropies, the antiferromagnetic spin-liquid spin-wave velocity and the Isinglike ferromagnetic excitation spectrum gap, exhibiting the spin-wave to spinon crossover. A number of characteristics of purely quantum nature are found: The in-plane interaction sxi s j + s induces an antiferromagnetic correlation in the out-of-plane szi component, at higher temperatures in the antiferromagnetic XXZ chain, dominantly at low temper- atures in the ferromagnetic XXZ chain, and, in-between, at all temperatures in the XY chain. We find that the converse effect also occurs in the antiferromagnetic XXZ chain: an antiferromagnetic szi s j interaction induces a correlation in the s component. As another purely quantum effect, (i) in the antiferromagnet, the value of the specific heat peak is insensitive to anisotropy and the temperature of the specific heat peak decreases from the isotropic (Heisenberg) with introduction of either type (Ising or XY) anisotropy; (ii) in complete contrast, in the ferromagnet, the value and temperature of the specific heat peak increase with either type of anisotropy. PACS numbers: 67.40.Db, 75.10.Pq, 64.60.Cn, 05.10.Cc I. INTRODUCTION The quantum Heisenberg chain, including the possi- bility of spin-space anisotropy, is the simplest nontrivial quantum spin system and has thus been widely studied since the very beginning of the spin concept in quan- tum mechanics [1, 2, 3]. Interest in this model continued [4, 5, 6, 7, 8, 9, 10] and redoubled with the exposition of its richly varied low-temperature behavior [11, 12, 13] and of its relevance to high-temperature superconductiv- ity [14, 15, 16, 17, 18]. It has become clear that antifer- romagnetism and superconductivity are firmly related to each other, adjoining and overlapping each other. A large variety of theoretical tools have been employed in the study of the various isotropic and anisotropic regimes of the quantum Heisenberg chain, including finite-systems extrapolation [6, 19], linked-cluster [7] and dimer-cluster [20] expansions, quantum decimation [21], decoupled Green’s functions [22], quantum transfer ma- trix [23, 24], high-temperature series expansion [25], and numerical evaluation of multiple integrals [26]. The XXZ Heisenberg chain retains high current interest as a theo- retical model [27, 28] with direct experimental relevance [29]. In the present paper, a position-space renormalization- group method introduced by Suzuki and Takano [30, 31] for d = 2 dimensions and already applied to a number of d > 1 systems [30, 31, 32, 33, 34, 35, 36] is used to compute the spin-spin correlations and the specific heat of the d = 1 anisotropic quantum XXZ Heisenberg model, easily resulting in a global description and de- tailed quantitative information for the entire temperature and anisotropy ranges including the ferromagnetic and antiferromagnetic, the spin-liquid and Isinglike regions. We obtain, for all anisotropies, the antiferromagnetic spin-liquid spin-wave velocity and the Isinglike ferromag- netic excitation spectrum gap, exhibiting the spin-wave to spinon crossover. A number of other characteristics of purely quantum nature are found: The in-plane interac- tion sxi s j induces an antiferromagnetic correlation in the out-of-plane szi component, at higher temperatures in the antiferromagnetic XXZ chain, dominantly at low temperatures in the ferromagnetic XXZ chain, and, in- between, at all temperatures in the XY chain. We find that the converse effect also occurs in the antiferromag- netic XXZ chain: an antiferromagnetic szi s j interaction induces a correlation in the s i component. As another purely quantum effect, (i) in the antiferromagnet, the value of the specific heat peak is insensitive to anisotropy and the temperature of the specific heat peak decreases from the isotropic (Heisenberg) with introduction of ei- ther type (Ising or XY) anisotropy; (ii) in complete con- trast, in the ferromagnet, the value of the specific heat peak is strongly dependent on anisotropy and the tem- perature of the specific heat peak increases with either type of anisotropy. This purely quantum effect is a pre- cursor to different phase transition temperatures in three dimensions [32, 36, 37, 38]. Our calculational method is relatively simple, readily yields global results, and is overall quantitatively successful. http://arxiv.org/abs/0704.1064v2 II. THE ANISOTROPIC QUANTUM HEISENBERG MODEL AND THE RENORMALIZATION-GROUP METHOD A. The Anisotropic Quantum Heisenberg Model The spin- 1 anisotropic Heisenberg model (XXZ model) is defined by the dimensionless Hamiltonian −βH = sxi s j + s + Jzs , (1) where β = 1/kBT and 〈ij〉 denotes summation over nearest-neighbor pairs of sites. Here the sui are the quan- tum mechanical Pauli spin operators at site i. The ad- ditive constant G is generated by the renormalization- group transformation and is used in the calculation of thermodynamic functions. The anisotropy coefficient is R = Jz/Jxy. The model reduces to the isotropic Heisen- berg model (XXX model) for |R| = 1, to the XY model for R = 0, and to the Ising model for |R| → ∞. B. Renormalization-Group Recursion Relations The Hamiltonian in Eq.(1) can be rewritten as − βH = {−βH(i, i+ 1)} . (2) where βH(i, i+ 1) is a Hamiltonian involving sites i and i + 1 only. The renormalization-group procedure, which eliminates half of the degrees of freedom and keeps the partition function unchanged, is done approximately [30, Trodde −βH =Trodde {−βH(i,i+1)} (3) =Trodde {−βH(i−1,i)−βH(i,i+1)} {−βH(i−1,i)−βH(i,i+1)} ′H′(i−1,i+1) ≃ e i {−β′H′(i−1,i+1)} = e−β ′H′ . Here and throughout this paper, the primes are used for the renormalized system. Thus, at each successive length scale, we ignore the non-commutativity of the operators beyond three consecutive sites, in the two steps indicated by ≃ in the above equation. Since the approximations are applied in opposite directions, one can expect some mutual compensation. Earlier studies [30, 31, 33, 34, 35] have been successful in obtaining finite-temperature behavior on a variety of quantum systems. The transformation above is summarized by p s ms Two-site basis eigenstates + 1 1 |φ1〉 = | ↑↑〉 0 |φ2〉 = {| ↑↓〉+ | ↓↑〉} − 0 0 |φ4〉 = {| ↑↓〉 − | ↓↑〉} TABLE I: The two-site basis eigenstates that appear in Eq.(8). These are the well-known singlet and triplet states. The state |φ3〉 is obtained by spin reversal from |φ1〉, with the same eigenvalue. ′H′(i,k) = Trj e {−βH(i,j)−βH(j,k)}, (4) where i, j, k are three successive sites. The operator −β′H′(i, k) acts on two-site states, while the operator −βH(i, j)−βH(j, k) acts on three-site states, so that we can rewrite Eq.(4) in the matrix form, 〈uivk|e−β ′H′(i,k)|ūiv̄k〉 = 〈uiwj vk|e−βH(i,j)−βH(j,k)|ūiwj v̄k〉 , (5) where state variables u, v, w, ū, v̄ can take spin-up or spin- down values at each site. The unrenormalized 8× 8 ma- trix on the right-hand side is contracted into the renor- malized 4× 4 matrix on the left-hand side of Eq.(5). We use two-site basis states vectors {|φp〉} and three-site ba- sis states vectors {|ψq〉} to diagonalize the matrices in Eq.(5). The states {|φp〉}, given in Table I, are eigen- states of parity, total spin magnitude, and total spin z- component. These {|φp〉} diagonalize the renormalized matrix, with eigenvalues J ′z +G ′, Λ2 = + J ′xy − J ′z +G Λ4 = − J ′zxy − J ′z +G ′. (6) The states {|ψq〉}, given in Table II, are eigenstates of parity and total spin z-component. The {|ψp〉} diagonal- ize the unrenormalized matrix, with eigenvalues Jz + 2G, λ4 = 2G, (7) λ2 = − 8J2xy + J + 2G, λ3 = − 8J2xy + J + 2G. With these eigenstates, Eq.(5) is rewritten as γp ≡ 〈φp|e−β ′H′(i,k)|φp〉 = u,v,ū, v̄,w,q 〈φp|uivk〉〈uiwjvk|ψq〉· 〈ψq|e−βH(i,j)−βH(j,k)|ψq〉〈ψq |ūiwj v̄k〉〈ūiv̄k|φp〉 . (8) p ms Three-site basis eigenstates + 3/2 |ψ1〉 = | ↑↑↑〉 1/2 |ψ2〉 = µ{| ↑↑↓〉 + σ| ↑↓↑〉 + | ↓↑↑〉} |ψ3〉 = ν{−| ↑↑↓〉 + τ | ↑↓↑〉 − | ↓↑↑〉} − 1/2 |ψ4〉 = {| ↑↑↓〉 − | ↓↑↑〉} TABLE II: The three-site basis eigenstates that appear in Eq.(8) with coefficients σ = (−Jz + 8J2xy + J z )/2Jxy , τ = (Jz + 8J2xy + J z )/2Jxy and normalization factors µ, ν. The states |ψ5−8〉 are obtained by spin reversal from |ψ1−4〉, with the same respective eigenvalues. Thus, there are three independent γp that determine the renormalized Hamiltonian and, therefore, three renor- malized interactions in the Hamiltonian closed under renormalization-group transformation, Eq.(1). These γp γ1 = e J′z+G = e2G− Jz + cosh 8J2xy + J Jz sinh 8J2xy + J 8J2xy + J γ2 = e J′xy− J′z+G =2e2G− 8J2xy + J Jz sinh 8J2xy + J 8J2xy + J γ4 = e +G′ = 2e2G , (9) which yield the recursion relations J ′xy = ln , J ′z = ln , G′ = γ21γ2γ4 As expected, J ′xy and J z are independent of the additive constant G and the derivative ∂GG ′ = bd = 2, where b = 2 is the rescaling factor and d = 1 is the dimensionality of the lattice. For Jxy = Jz, the recursion relations reduce to the spin- isotropic Heisenberg (XXX) model recursion relations, while for Jxy = 0 they reduce to spin- Ising model re- cursion relations. The Jz = 0 subspace (XY model) is not (and need not be) closed under these recursion rela- tions [30, 31]: The renormalization-group transformation induces a positive Jz value, but the spin-space easy-plane aspect is maintained. In addition, there exists a mirror symmetry along the Jz-axis, so that J xy (−Jxy, Jz) = J ′xy (Jxy, Jz) and J ′z (−Jxy, Jz) = J ′z (Jxy, Jz). The thermodynamics of the system remains unchanged under flipping the in- teractions of the x and y spin components, since the renormalization-group trajectories do not change. In fact, this is part of a more general symmetry of the XYZ model, where flipping the signs of any two interactions leaves the spectrum unchanged [8]. Therefore, with no loss of generality, we take Jxy > 0. Independent of the sign of Jxy, Jz > 0 gives the ferromagnetic model and Jz < 0 gives the antiferromagnetic model. C. Calculation of Densities and Response Functions by the Recursion-Matrix Method Just as the interaction constants of two consecutive points along the renormalization-group trajectory are re- lated by the recursion relations, the densities are con- nected by a recursion matrix T̂ , which is composed of derivatives of the recursion relations. For our Hamilto- nian, the recursion matrix and density vector ~M are ∂J′xy szi s . (11) These are densities Mα associated with each interaction ∂ lnZ , (12) where Nα is the number of α-type interactions and Z is the partition function for the system, which can be expressed both via the unrenormalized interaction con- stants as Z( ~K) or via the renormalized interaction con- stants as Z( ~K ′). By using these two equivalent forms, one can formulate the density recursion relation [39] Mα = b M ′βTβα , Tβα ≡ ∂K ′β . (13) Since the interaction constants, under renormalization- group transformation, stay the same at fixed points such as critical fixed points or sinks, the above Eq.(13) takes the form of a solvable eigenvalue equation, bd ~M∗ = ~M∗ · T̂ , (14) at fixed points, where ~M = ~M ′ = ~M∗. The fixed point densities are the components of the left eigenvector of the recursion matrix with left eigenvalue bd [39]. At ordinary points, Eq.(13) is iterated until a sink point is reached under successive renormalization-group transformations. In algebraic form, this means ~M (0) = b−nd ~M (n)T̂ (n)T̂ (n−1) · · · T̂ (1) , (15) where the upper indices indicate the number of iteration (transformation), with ~M (n) ≃ ~M∗. This method is applied on our model Hamiltonian. The sink of the system is at infinite temperature J∗xy = J∗z = 0 for all initial conditions (Jxy, Jz). Response functions are calculated by differentiation of densities. For example, the internal energy is U = szi s , employing T = 1/Jxy, and U = szi s , employing T = 1/|Jz|. The spe- cific heat C = ∂TU follows from the chain rule, C = J2xy szi s , for T = 1/Jxy, C = J2z szi s ∂|Jz| , for T = 1/|Jz|. III. CORRELATIONS SCANNED WITH RESPECT TO ANISOTROPY The ground-state and excitation properties of the XXZ model offer a variety of behaviors [11, 12, 40, 41]: The antiferromagnetic model with R < −1 is Isinglike and the ground state has Néel long-range order along the z spin component with a gap in the excitation spectrum. For −1 ≤ R ≤ 1, the system is a ”spin liquid”, with a gapless spectrum and power-law decay of correlations at zero temperature. The ferromagnetic model with R > 1 is also Isinglike, the ground state is ferromagnetic along the z spin component, with an excitation gap. Our calculated szi s sxi s nearest-neighbor spin-spin correlations for the whole range of the anisotropy coefficient R are shown in Fig.1, for various temperatures. The xy correlation is always non-negative. Recall that we use Jxy > 0 with no loss of generality. In the Isinglike antiferromagnetic (R < −1) region, the z correlation is expectedly anti- ferromagnetic. As the szi s correlation saturates for large |R|, the transverse correlation is some- what depleted. In the Isinglike ferromagnetic (R > 1) region, the szi s correlation is ferromagnetic, saturates quickly as the correlation quickly goes to zero. In the spin-liquid (|R| < 1) region, the szi s correlation monotonically passes through zero in the feromagnetic side, while the correlation is maximal. The re- markable quantum behavior of szi s around R = 0 is discussed in Sec.V below. It is seen in the figure that these changeovers are increasingly sharp as temperature is decreased and, at zero temperature, become discon- tinuous at R = 1. As seen in Fig.1(b), at zero temper- ature, our calculated szi s correlations FIG. 1: (a) Calculated nearest-neighbor spin-spin correlations szi s (thick curves from lower left) and (thin curves from upper left) as a function of anisotropy coefficient R for temperatures 1/Jxy = 0, 0.1, 0.2, 0.4, 0.8. (b) Calculated zero- temperature nearest-neighbor spin-spin correlations (thin and thick curves, as in the upper panel) compared with the exact points of Ref.[4, 40, 42, 43, 44] shown with filled and open symbols for szi s respectively. At R = 1, the calculated szi s discontinuously goes from antiferromagnetic to the exact result of 0.25 [40] of saturated ferromagnetism and the calculated discontinuously goes from ferro- magnetic to the exact result of constant zero [40]. show very good agreement with the known exact points [4, 42, 43, 44]. Also, our results for R > 1 fully overlap the exact results of szi s = 0.25 and = 0 [40]. We also note that zero-temperature is the limit in which our approximation is at its worst. FIG. 2: Calculated nearest-neighbor spin-spin corre- lations (upper panels) and szi s (lower panels) for the antiferromagnetic XXZ chain, as a function of temperature, for anisotropy coefficients R = 0,−0.25,−0.50,−0.75,−1,−2,−4,−8,−∞ spanning the spin-liquid (left panels) and Isinglike (right panels) regions. Note that, in every one of the panels, the correlation curves cross each other. This remarkable quantum phenomenon is discussed in the text. IV. ANTIFERROMAGNETIC XXZ CHAIN For the antiferromagnetic XXZ chain, our calculated szi s nearest-neighbor spin-spin correla- tions as a function of temperature are shown in Fig.2 for various anisotropy coefficients R. We find that when Jxy is the dominant interaction (spin liquid), the corre- lations are weakly dependent on anisotropy R. When Jz is the dominant interaction (Isinglike), the correlations are weakly dependent on anisotropy R only at the higher temperatures. Our results are compared with multiple- integral results [26] in Fig.3. In every one of the panels of Fig.2, the correlation curves cross each other, revealing a remarkable quantum phenomenon. In a classical system, the correlation be- tween a given spin component (e.g., ) is expected to decrease when the coupling of another spin component (e.g., |Jz |) is increased. It is found from the antiferromag- netic XXZ chain in Fig.2 that the opposite may occur in a quantum system: In this figure, an increase in Jxy causes an increase in | szi s | for 1/|Jz| > 0.9 and 0.4 in the spin-liquid and Isinglike regions respectively. Con- versely, an increase in |Jz| causes an increase in for 1/Jxy > 0.4 and 2.1 in the spin-liquid and Isinglike regions respectively. This quantum effect can be called cross-component spin correlation. FIG. 3: Comparison of our results (thick lines) for the cor- relation functions of the antiferromagnetic XXZ chain, with the multiple-integral results of Ref.[26] (thin lines), for var- ious anisotropy coefficients R spanning the spin-liquid and Isinglike regions. The antiferromagnetic specific heats calculated with Eq.(16) are shown in Fig.4 for various anisotropy coeffi- cients and compared, in Figs.5, 6, with finite-lattice ex- pansion [6, 19], quantum decimation [21], transfer matrix [24], high-temperature series expansion [25] results and, for the R = 0 case, namely the XY model, with the exact result [5] C = (1/4πT ) cosω/ cosh dω. The C(T ) peak temperature is highest for the isotropic case (Heisenberg) and decreases with anisotropy increasing in either direction (towards Ising or XY). The peak value of C(T ) is only weakly dependent on anisotropy, espe- cially for the Isinglike systems. A strong contrast to this behavior will be seen, as another quantum mechanical phenomenon, in the ferromagnetic XXZ chain. The linearity, at low temperatures, of the spin liq- uid (|R| ≤ 1) specific heat with respect to temperature is expected on the basis of spin-wave calculations for the antiferromagnetic XXZ model [45, 46]. This linear form of C(T ) reflects the linear energy-momentum dis- persion of the low-lying excitations, the magnons. The low-temperature magnon dispersion relation is ~ω = ckn, where c is the spin-wave velocity and n = 1 for the an- tiferromagnetic XXZ model in d = 1 [40]. The internal energy, given by U = (1/2π) dk~ω(k)/(eβ~ω(k)−1), is dominated by the magnons at low temperatures, yielding U ∼ T 2 and C ∼ T for n = 1 in the dispersion relation. From this relation, our calculated spin-wave velocity c as a function of anisotropy R is given in Fig.7 and compares well with the also shown exact result [47]. A simultaneous fit to the dispersion relation exponent n, expected to be 1, yields 1.00± 0.02. However, for the Isinglike −R > 1, the unexpected linearity instead of an exponential form FIG. 4: Calculated specific heats C of the antiferromagnetic XXZ chain, as a function of temperature for anisotropy coeffi- cients R = 0,−0.25,−0.50,−0.75,−1,−2,−4,−8,−∞ span- ning the spin-liquid (upper panel) and Isinglike (lower panel) regions. caused by a gap in the excitation spectrum, points to the approximate nature of our renormalization-group calcu- lation. The correct exponential form is obtained in the large −R limit, where the renormalization-group calcu- lation becomes exact. Rojas et al. [25] have obtained the high-temperature expansion of the free energy of the XXZ chain to order β3, where β is the inverse temperature. The specific heat from this expansion is 2 +R2 J2xy − J3xy + 6− 8R2 −R4 J4xy. (17) This high-temperature specific heat result is also com- pared with our results, in Fig.6, and very good agreement is seen. In fact, when in the high-temperature region of 0 < β < 0.1, we fit our numerical results for C(β) to the fourth degree polynomial C = Σ4i=0Aiβ i, and we do find (1) the vanishing A0 < 10 −5 and A1 < 10 −7 for all R and (2) the comparison in Fig.8 between our results for A2 and A3 and those of Eq.(17) from Ref.[25], thus obtaining excellent agreement for all regions of the model. V. FERROMAGNETIC XXZ CHAIN For the ferromagnetic (i.e., R > 0) systems in Fig.1, szi s expectation value becomes rapidly negative at lower temperatures for R < 1, even though for R ≥ 0 all couplings in the Hamiltonian are ferromag- FIG. 5: Comparison of our antiferromagnetic specific heat re- sults (thick lines) with the results of Refs.[5] (open circles), [6] (dotted), [19] (thin lines), [21] (dash-dotted), and [23, 24] (dashed), for anisotropy coefficients R = 0,−0.5,−1,−2 span- ning the spin-liquid and Isinglike regions. FIG. 6: Comparison of our antiferromagnetic specific heat results (thick lines) with the high-temperature J → 0 behav- iors (thin lines) obtained from series expansion in Ref.[25], for anisotropy coefficients R = 0,−0.50,−0.75,−1,−2,−∞ spanning the spin-liquid and Isinglike regions. netic. This is actually a real physical effect, not a nu- merical anomaly. In fact, we know the spin-spin corre- lations for the ground state of the one-dimensional XY model (the R = 0 case of our Hamiltonian), and we can compare our low-temperature results with these exact values. The ground-state properties of the spin- 1 model are studied by making a Jordan-Wigner trans- FIG. 7: Our calculated antiferromagnetic spin-wave veloc- ity c versus the anisotropy coefficient R. The dashed line, 2π sin(γ)/γ where γ ≡ cos−1(−R), is the exact result [47]. FIG. 8: Comparison of our results with the high-temperature expansion of Ref. [25] for all regions: antiferromagnetic (outer panels) and ferromagnetic (inner panels), spin-liquid (left panels) and Isinglike (right panels). Triangles and circles de- note our results, while solid and dashed lines denote the re- sults of Ref.[25] for A2 and A3, respectively. The error bars, due to the statistical fitting procedure of the coefficients A2 and A3, have half-heights of 1.7×10 −4 and 2.6×10−3 respec- tively. formation, yielding a theory of non-interacting spinless fermions. Analysis of this theory yields the exact zero- temperature nearest-neighbor spin-spin correlations [4] shown in Table III. Our renormalization-group results in the zero-temperature limit, also shown in this table, compare quite well with the exact results, as with the other exact points in Fig.1(b), although in the worst re- gion for our approximation. Finally, by continuity, it is reasonable that for a range of R positive but less than one, the z component correlation function is as we find, intriguingly but correctly negative at low temperatures. Thus, the interaction sxi s j (irrespective of its sign, due to the symmetry mentioned at the end of Sec.IIB) induces an antiferromagnetic correlation in the szi com- ponent, competing with the szi s j interaction when the FIG. 9: Calculated nearest-neighbor spin-spin correlations (upper panels) and szi s (lower panels) for the ferromagnetic XXZ chain, as a function of temperature, for anisotropy coefficients R = 0, 0.25, 0.50, 0.75, 1, 2, 4, 8,∞ spanning the spin-liquid (left panels) and Isinglike (right pan- els) regions. FIG. 10: Left panel: Calculated nearest-neighbor spin- spin correlations szi s for the ferromagnetic XXZ chain, as a function of temperature 1/Jxy in the spin liquid, for anisotropy coefficients R = 0, 0.25, 0.50, 0.75, 1. Right panel: The sign-reversal temperature T0 of the nearest-neighbor cor- relation 〈szi s j 〉: our results (full curve) and the analytical re- sult from the quantum transfer matrix method (dashed) [23]. latter is ferromagnetic. For finite temperatures, our calculated nearest- neighbor spin-spin correlations are shown in Figs.9, 10, for different values of R. These results are compared with Green’s function calculations [22] in Fig.11. As expected from the discussion at the beginning of this section, in the spin-liquid region, the correlation 〈szi szj 〉 is negative at low temperatures. Thus, a competition occurs in the FIG. 11: Comparison of our ferromagnetic R = 1, 5 results with Green’s function calculations [22] . FIG. 12: Calculated specific heats C of the ferromagnetic XXZ chain, as a function of temperature for anisotropy coef- ficients R = 0, 0.25, 0.50, 0.75, 1, 2, 4, 8,∞ spanning the spin- liquid (upper panel) and Isinglike (lower panel) regions. correlation 〈szi szj 〉 between the XY-induced antiferromag- netism and the ferromagnetism due to the direct cou- pling between the sz spin components. In fact, the rein- forcement of antiferromagnetic correlations of 〈szi szj 〉 by increasing Jxy (and also its converse) was seen in the antiferromagnetic XXZ chain discussed in the previous section. Thus, we see that whereas this cross-component effect is dominant at low temperatures in the ferromag- netic XXZ chain, it is seen at higher temperatures in the antiferromagnetic XXZ chain and, in-between, through- out the temperature range in the XY chain. In the ferromagnetic XXZ chain, as a consequence of the competition mentioned above, a sign reversal in 〈szi szj 〉 occurs from negative to positive correlation, at temperatures J−1xy = T0(R).[48] At this temperature, by cancelation of the competing effects, the nearest-neighbor FIG. 13: Comparison of our ferromagnetic specific heat re- sults (thick lines) with the high-temperature J → 0 behaviors (thin lines) obtained from series expansion [25], for anisotropy coefficients R = 0.25, 0.50, 0.75, 1, 2,∞ spanning the spin- liquid and Isinglike regions. Zero-temperature correlations of the spin- 1 XY chain Exact values from Ref. [4] results 〉 0.15915 0.17678 〈szi s j 〉 −0.10132 −0.12500 TABLE III: Zero-temperature nearest-neighbor correlations of the spin- 1 XY chain. correlation 〈szi szj 〉 is zero. Our calculated T0(R) curve is shown in Fig.10, and has very good agreement with the exact result T0 = ( 3 sin γ/4γ) tan[π(π − γ)/2γ] where γ ≡ cos−1(−R) [23]. The calculated ferromagnetic specific heats are shown in Fig.12 for various anisotropy coefficients and com- pared, in Figs.13, 14, with finite-lattice expansion [6], quantum decimation [21], decoupled Green’s functions [22], transfer matrix [23, 24], high-temperature se- ries expansion [25] results and, for the R = 0 case, namely the XY model, with the exact result [5] C = (1/4πT ) cosω/ cosh dω. In sharp contrast to the antiferromagnetic case in Sec.IV, the peak C(T ) temperature is highest for the most anisotropic cases (XY or Ising) and decreases with anisotropy decreas- ing from either direction (towards Heisenberg). In the same contrast, the peak value of C(T ) is dependent on anisotropy, decreasing, eventually to a flat curve, as anisotropy is decreased. This contrast between the fer- romagnetic and antiferromagnetic systems is a purely quantum phenomenon. Specifically, the marked contrast between the specific heats of the isotropic antiferromag- netic and ferromagnetic systems, seen in the full curves of Figs.4 and 12 respectively, translates into the different critical temperatures of the respective three-dimensional systems.[32, 36, 37, 38] Classical ferromagnetic and anti- FIG. 14: Comparison of our ferromagnetic specific heat re- sults (thick lines) with the results of Refs.[5] (dash-double- dotted), [6] (dotted), [21] (dash-dotted), [22] (open cir- cles), and [23, 24] (dashed), for anisotropy coefficients R = 0, 0.5, 1, 5 , 2, 5 spanning the spin-liquid and Isinglike re- gions. ferromagnetic systems are, on the other hand, identically mapped onto each other. The low-temperature specifics heats are discussed in detail and compared to other results in Sec.VI. VI. LOW-TEMPERATURE SPECIFIC HEATS Properties of the low-temperature specific heat of the ferromagnetic XXZ chain have been derived from the thermodynamic Bethe-ansatz equations [40]. For anisotropy coefficient |R| ≤ 1, the model is gapless [11, 12] and, except at R = 1, the specific heat is lin- ear in T = J−1xy in the zero-temperature limit, C/T = 2γ/(3 sinγ) where again γ ≡ cos−1(−R). Note that this result contradicts the spin-wave theory prediction of C ∼ T 1/2 for the ferromagnetic chain (n = 2 for the fer- romagnetic magnon dispersion relation of the kind given above in Sec.IV). The spin-wave result is valid only for R = 1, the isotropic Heisenberg case. From the expres- sion given above, we see that C/T diverges as R → 1−, and at exactly R = 1 it has been shown that C ∼ T 1/2 [40]. In the Isinglike region R > 1, the system exhibits a FIG. 15: The calculated excitation spectrum gap ∆ versus anisotropy. FIG. 16: Calculated specific heat coefficient C/T as a function of anisotropy R, for T = 0.10, 0.05, 10−10 . gap in its excitation spectrum and the specific heat be- haves as C ∼ T−a exp(−∆/T ), with ∆ being the excita- tion spectrum gap [11, 12, 40]. There exist two gaps for the energy, called the spinon gap and the spin-wave gap, given by ∆spinon = 1−R−2 and ∆spinwave = 1−R−1. These are the minimal energies of elementary excita- tions [10, 40]. A crossover between them occurs atR = 5 below this value, the spinon gap is lower, while above this value the spin-wave gap is lower. We have double- fitted our calculated specific heats with respect to the gap ∆ and the leading exponent a, for the entire range of anisotropy R between 0 < R−1 < 1 (Fig.15). Our calculated gap ∆ behaves linearly in R−1 for R−1 close to 1, and crosses over to 1/2 at R−1 = 0, as expected. We also obtain the exponent a = 1.99± 0.02 in the Ising FIG. 17: Calculated specific heat coefficient C/T as a func- tion of temperature for anisotropy coefficient R = −5 (thin grey), R = −2 (thick grey), -1 (dotted), -0.5 (dash-dotted), 0.5 (dashed), and 2 (thin black). limit R−1 ≤ 0.2 and a = 1.52 ± 0.10 in the Heisenberg limit R−1 ≥ 0.9. These exponent values are respectively expected to be 2 and 1.5 [9, 10]. We now turn to the discussion of our specific heat re- sults for the entire ferromagnetic and antiferromagnetic ranges. Our calculated C/T curves are plotted as a func- tion of anisotropy and temperature in Figs.16 and 17 respectively. We discuss each region of the anisotropy R separately: (i) R > 1 : The specific heat coefficient C/T vanishes in the T → 0 limit and has the expected exponential form as discussed above in this section. The spin-wave to spinon excitation gap crossover is obtained. (ii) R ≈ 1 : The double-peak structure of C/T in Fig.16 is centered at R = 1. As temperature goes to zero, the peaks narrow and diverge. (iii) −1 ≤ R < 1 : The specific heat coefficient is C/T = 2γ/(3 sinγ) in this region [11, 40], and our calcu- lated specific heat is indeed linear at low temperatures. The C/T curves for R = −1,−0.5, 0.5 in Fig.17 all ex- trapolate to nonzero limits at T = 0. The spin-wave dispersion relation exponent and velocity, for the antifer- romagnetic system, is correctly obtained for the isotropic case and for all anisotropies, as seen in Fig.7. Fig.18 directly compares C/T = 2γ/(3 sinγ) with our results: The curves have the same basic form, gradually rising from R = −1, with a sharp divergence as R nears 1. At R = 1+, we expect C/T = 0. Our T = 10−10 curve diverges at R = 1 and indeed returns to zero at R = 1.0000001. (iv) R < −1 : We expect a vanishing C/T , which we do find as seen in Fig.16 and in the insets of Fig.17. The exponential behavior of the specific heat is clearly seen in the Ising limit. VII. CONCLUSION A detailed global renormalization-group solution of the XXZ Heisenberg chain, for all temperatures and anisotropies, for both ferromagnetic and antiferromag- netic couplings, has been obtained. In the spin-liquid region, the linear low-temperature specific heat and, for the antiferromagnetic chain, the spin-wave dispersion re- lation exponent n and velocity c have been obtained. In the Isinglike region, the spin-wave to spinon crossover of the excitation spectrum gap of the ferromagnetic chain has been obtained from the exponential specific heat, as well as the correct leading algebraic behaviors in the Heisenberg and Ising limits. Purely quantum mechani- cal effects have been seen: We find that the xy corre- lations and the antiferromagnetic z correlations mutu- FIG. 18: Calculated specific heat coefficient C/T as a func- tion of anisotropy coefficient R in the spin-liquid region, −1 ≤ R ≤ 1, at constant temperature T = 10−10 . Our renormalization-group result (grey curve) is compared to the zero-temperature Bethe-Ansatz result (black curve). Inset: our calculation (grey curve) at constant T = 10−2 is again compared to the zero-temperature Bethe-Ansatz result (black curve). ally reinforce each other, for different ranges of temper- atures and anisotropies, in ferromagnetic, antiferromag- netic, and XY systems. The behaviors, with respect to anisotropy, of the specific heat peak values and locations are opposite in the ferromagnetic and antiferromagnetic systems. The sharp contrast found in the specific heats of the isotropic ferromagnetic and antiferromagnetic sys- tems is a harbinger of the different critical temperatures in the respective three-dimensional systems. When com- pared with existing calculations in the various regions of the global model, good quantitative agreement is seen. 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0704.1065

Precise Control of Band Filling in NaxCoO2

Microsoft Word - preprint.doc April 9, 2007, submitted to J. Phys. Soc. Jpn. Layered sodium cobaltate NaxCoO2 1) provides us with a fascinating playground to study the physics of strongly correlated electrons on the frustrated triangular lattice in a wide range of band filling. Recent two important discoveries have accelerated the research of NaxCoO2: one is an unusually large thermoelectric power for x ~ 2/32) and the other is superconductivity below 4.5 K in a hydrated compound with x ~ 1/3.3) Particularly, a lot of studies have focused on the hydrated superconductor to clarify the mechanism of the superconductivity and to understand underlying electronic states realized in the triangular lattice. In spite of extensive research, however, there still remain some controversial issues possibly resulting from difficulty in the chemistry of the compounds that put obstacles in the way of obtaining high-quality samples.4, 5) The electronic phase diagram of NaxCoO2 as a function of x has been proposed by several groups. Foo et al. gave a typical one where a charge-ordered magnetic insulator exists at x = 0.5 with a "paramagnetic (PM) metal" on the left (x < 0.5) and a "Curie-Weiss (CW) metal" on the right (x > 0.5).6) In contrast, Yokoi et al. found that the boundary between the two metals is located approximately at x = 0.6.7) The reason for this discrepancy is not known. Moreover, the origin of the change in metallic character has not yet been interpreted in a clear manner. On the charge-ordered insulator at x = 0.5, it is known that the ordering of Na ions, which is already present at high temperature, triggers a magnetic as well as a metal-insulator transitions at low temperatures.6-8) The electronic states of NaxCoO2 near the Fermi level come from the Co d derived t2g state that splits into an a1g singlet and an e'g doublet. These bands are filled progressively with x, because one Na+ ion donates one electron to the CoO2 layer. According to band structure calculations,9-14) the a1g band always crosses the Fermi level, irrespective of band filling, which gives a large circular (cylindrical) Fermi surface (FS) around the Γ point in the Brillouin zone. In addition, two important features on the FS are found in the local-density approximation calculations: one is a set of small hole pockets originating from the e'g band near the K points, which are expected to appear at low fillings such as x = 0.3. The other is a small concentric electron FS around Γ coming from a dip in the band energy of the a1g state, which may appear at high fillings such as x = 0.7. However, the absence of the latter was suggested for any doping levels by more sophisticated band structure calculations incorporating spin polarizations or the on-site Coulomb interaction U.11) Experimentally, angle-resolved photoemission spectroscopy (ARPES) study on Na0.7CoO2 observed only a large circular FS around Γ and failed to detect the other FSs.15) Recently, Mochizuki and Ogata pointed out in their tight binding model that band dispersions and FS topology change sensitively with the thickness of the CoO2 layer: 16, 17) the hole pockets near K would appear for a thinner CoO2 layer, while the small electron pocket around Γ would be present for a thicker CoO2 layer. Thus, the current status is miles away from a complete understanding of the basic electronic structures of NaxCoO2. In this letter, we study systematically the transport and thermodynamic properties of NaxCoO2 using a series of polycrystalline samples. The key point of the present study is a novel method used to prepare samples: most of samples studied so far were obtained by a soft-chemistry method at ambient temperature,3) which might cause an inhomogeneous distribution of Na ions or otherwise lead to Precise Control of Band Filling in NaxCoO2 Daisuke YOSHIZUMI, Yuji MURAOKA**, Yoshihiko OKAMOTO, Yoko KIUCHI, Jun-Ichi YAMAURA, Masahito MOCHIZUKI1, ***, Masao OGATA2 and Zenji HIROI* Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581 1RIKEN, Hirosawa, Wako, Saitama 351-0198 2Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033 Electronic properties of the sodium cobaltate NaxCoO2 are systematically studied through a precise control of band filling. Resistivity, magnetic susceptibility and specific heat measurements are carried out on a series of high-quality polycrystalline samples prepared at 200°C with Na content in a wide range of 0.35 ≤ x ≤ 0.70. It is found that dramatic changes in electronic properties take place at a critical Na concentration x* that lies between 0.58 and 0.59, which separates a Pauli paramagnetic and a Curie- Weiss metals. It is suggested that at x* the Fermi level touches the bottom of the a1g band at the Γ point, leading to a crucial change in the density of states across x* and the emergence of a small electron pocket around the Γ point for x > x*. KEYWORDS: sodium cobaltate, resistivity, magnetic susceptibility, specific heat, Sommerfeld coefficient, Fermi surface, band filling *E-mail: [email protected] **Present address: The Graduate School of Natural Science and Technology, Okayama University, 1-1, Naka-3, Tsushima, Okayama 700-8530 ***Present address: Tokura Multiferroics Project, ERATO, Japan Science and Technology Agency (JST) a strong tendency for Na ordering at certain fractional compositions such as x = 0.5. Particularly, the Na ordering strongly influences the electronic state of the (CoO2) x- layer and may mask the intrinsic features. We adopted alternatively a solid-state reaction at higher temperatures starting from two end members of Na0.35CoO2 and Na0.70CoO2 and succeeded in controlling x on a scale of 0.01. Moreover, since the high-temperature reaction favoured disordering of Na ions, more intrinsic properties of the (CoO2) x- layer have been clarified. We prepared a series of polycrystalline samples of NaxCoO2 by a solid-state reaction of Na0.35CoO2 and Na0.70CoO2. First, powders of Na0.70CoO2 were synthesized by a reaction of stoichiometric amounts of Na2CO3 and Co3O4 in air at 860°C. Sodium deintercalation was then carried out in a 1.0 M Br2 solution in acetnitrile to obtain powders of Na0.35CoO2. The Na content was determined by the inductively coupled plasma-atomic emission spectroscopy (ICP) method. A dozen of samples with intermediate compositions were prepared by reacting the two powders of Na0.35CoO2 and Na0.70CoO2 in an appropriate ratio in a sealed quartz tube at 200°C for 24 hours, followed by slow cooling to room temperature. The Na content of the products was also examined by the ICP analysis. The phase purity was confirmed by means of powder x-ray diffraction. More detail on the sample preparation and characterization will be reported elsewhere.18) Resistivity and specific heat were measured in a Quantum Design PPMS system, and magnetic susceptibility measurements were performed in a Quantum Design MPMS system. Figure 1 shows the systematic variation of resistivity ρ with x. For clarity, each set of data is divided by the value at 250 K. The ρ of x = 0.50 exhibits a smooth, semiconductor-like increase at low temperature with no anomalies for a metal-insulator transition, in contrast to the previous report where a soft-chemically prepared sample of x = 0.50 exhibits a sharp rise in ρ below 53 K.6, 7) We examined our sample by electron diffraction and found that a superstructure coming from Na ordering was almost absent.18) We think that this semiconducting behavior is due to a partial ordering of Na ions, because even at room temperature the Na ions tend to align.18) As x increases from 0.50, such a semiconducting increase in ρ is progressively suppressed, and finally a metallic T dependence appears above 0.55. Nevertheless, a small upturn is observed below 20 K, as shown in the inset to Fig. 1, which may be ascribed to weak localization due to a random electrostatic potential from disordered Na distributions. Compared with the T dependence in ρ for x ≤ 0.58, which is always concave-downward, those of x ≥ 0.59 are apparently dissimilar; almost T linear for x = 0.59 and concave-upward for x = 0.66 and 0.70. The last one shows a steep decrease below 30 K, which is similar as reported previously for x = 0.75.6) Therefore, the T dependence of ρ changes significantly across x = 0.58 ~ 0.59. Another related change is found in the evolution of magnetic susceptibility χ, as shown in Fig. 2. The χ for x ≤ 0.58 shown in the left panel is relatively small in magnitude and weakly T dependent, while, in distinct contrast, the χ for x ≥ 0.59 is large in magnitude and shows a characteristic CW divergence that is progressively enhanced with increasing x. Hence, a substantial change in magnetism must occur across a critical Na concentration x* between 0.58 and 0.59, accurately corresponding to the above results from resistivity. Looking in more detail, the χ of x = 0.35 decreases gradually with decreasing T from 300 K, exhibits a minimum at 120 K and then shows a CW divergence at low Fig. 1. Evolution of the temperature dependence of resistivity ρ normalized to the value at 250 K for 0.50 ≤ x ≤ 0.58 (left) and 0.59 ≤ x ≤ 0.70 (right). Fig. 2. Magnetic susceptibility χ measured on heating in a magnetic field of 10 kOe for 0.35 ≤ x ≤ 0.58 (left) and 0.59 ≤ x ≤ 0.70 (right). Note the difference in the vertical scale. temperature. This low-temperature CW component may be due to a minor portion of spins that come from defects or impurities and are noninteracting with majority spins. The positive slope above 120 K suggests the existence of a peak in χ at a higher temperature. In fact, the 0.50 and 0.52 samples seem to show a broad maximum near 300 K. Moreover, a broad peak is clearly observed at around 150 K for x = 0.56. The peak is possibly further shifted to a lower T for x = 0.58 and becomes obscure. This broad peak indicates the development of antiferromagnetic (AF) short- range order on cooling. Then, its systematic shift to lower temperatures implies that the characteristic energy of AF spin fluctuations or effective superexchange Jeff decreases with increasing x. Since the density of spins (1-x) decreases as x increases, we expect that Jeff between nearest-neighbour spins also decreases, consistent with the observed behavior in χ. For x ≥ 0.59, on the other hand, the Weiss temperature deduced from fitting the data to the CW law χ = C/(T - Θ) is always negative in sign (AF) and gradually increases from - 156 K (x = 0.59) to -99 K (x = 0.70). This fact indicates that AF fluctuations are suppressed or additional ferromagnetic interactions are enhanced with increasing x. In order to demonstrate more explicitly the dramatic changes across x*, the ρ and χ data of x = 0.58 and 0.59 are compared in Fig. 3. The ρ of 0.58 is proportional to T2 in a wide T range below 70 K, as shown in the inset, provided that the low-temperature upturn is ignored. Such T2 behavior is what one expects generally for a strongly correlated electron system. In contrast, the T dependence in ρ of 0.59 is rather unusual, showing an almost linear variation in a similarly wide T range below 80 K. Hence, there must be a substantial difference in the scattering mechanism of carriers between the two samples. On the other hand, it is apparent from Fig. 3 that χ is enhanced enormously from 0.58 to 0.59. Moreover, the T dependence above 50 K changes from linear to CW-like. This T linear behavior for 0.58 may be accidental as a few contributions of different origins coexist. The unusual T- linear dependence in ρ for 0.59 must be related to the appearance of the CW component in χ: additional magnetic scattering can be the source. To evaluate the change in the T dependence of χ, the slope at 100 K is plotted in Fig. 4, following the previous analysis done by Yokoi et al.7) The slope is close to zero or takes small negative values for x ≤ 0.58, while it decreases almost discontinuously to a relatively large negative value at 0.59, followed by a further decrease with increasing x. Another experimental evidence to support the existence of a critical x value has been obtained from specific heat measurements. The x dependence of the Sommerfeld coefficient γ determined from the intercept of the C/T versus T2 plot is shown in Fig. 4. Obviously, there is a change in γ at x*: γ increases slightly with x for x ≤ 0.58 and suddenly rises at x* by 17 %, from 16.8 mJ K-2 mol-1 for 0.58 to 19.7 mJ K-2 mol-1 for 0.59. Then, it increases rapidly to be saturated at 31.5 mJ K-2 mol-1 for x = 0.70. Note that the γ of 0.50 is finite because of the metallic nature of our sample in the absence of charge order. Since γ is proportional to the density of states (DOS), we conclude that the DOS at the Fermi level suddenly increases above x*. All the above results indicate that the electronic structure of NaxCoO2 does not change smoothly with x, but there is a well-defined boundary at x*. It is found that the critical concentration lies between 0.58 and 0.59, not at 0.5 as reported by Foo et al.,6) but close to 0.60 as reported by Fig. 4. x dependences of the slope in χ at 100 K (top) and the Sommerfeld coefficient γ (bottom). Crosses in the bottom panel represent the data reported by Yokoi et al.7) Dotted lines serve as guides to the eye. Fig. 3. Comparisons of resistivity (left) and magnetic susceptibility (right) between x = 0.58 and 0.59. Inset in the left panel shows a ρ - ρ0 versus T 2 plot for 0.58, where ρ0 is the residual resistivity and is 1.83 mΩ cm. Inset in the right panel shows the T dependence of inverse χ for 0.59. Yokoi et al.7) The sharp boundary breaks the phase diagram into two regions: a Pauli paramagnetic metal with a relatively small DOS for x < x*and a CW metal with an enhanced DOS for x > x* (Fig. 5). A main reason why we observed such sharp changes in properties at x* in the present study may be ascribed to the high quality of our samples prepared at high temperature as well as the reduced influence of the Na ordering. On the basis of these experimental lines of evidence, we are now ready to consider what is going on with x in NaxCoO2 in terms of band structures. As x decides filling in the t2g band, it is reasonable to assume that the band structure changes at a certain filling in the rigid band picture. According to band structure calculations,9-13) there is a dip around the Γ point in the a1g band, though it has not yet been detected experimentally.15) Then, it is likely that at x* the Fermi level with energy E* touches the bottom of the a1g band exactly at the Γ point, as schematically depicted in Fig. 5. An approximate profile of DOS expected from the band structure is also illustrated in Fig. 5, which is constant below E* due to the two dimensionality and exhibits a discontinuous jump at E*, followed by a further increase above E*. This profile of DOS is qualitatively in good agreement with the observed x dependence of γ shown in Fig. 4. A minor difference between them may come from the actual three dimensionality of the band structure that splits the a1g band into two. 9, 12) On the other hand, the top of the e'g band must be below the Fermi level even at the lowest filling, because no enhancement in γ is observed for 0.35 ≤ x < x* in Fig. 4. Furthermore, one expects a distinct change in the topology of the FS at x*: an additional small electron pocket should appear for x > x*. Because of this small electron pocket, strong ferromagnetic correlations are expected for x > x*, which has been predicted theoretically10, 13, 19) and evidenced by neutron diffraction20, 21) and other experiments.22-24) The observed change from a PM metal to a CW metal is obviously attributed to the emergence of this small electron FS. It seems difficult to estimate the accurate value of x* from band structure calculations, because the band structure is highly sensitivity to the crystal structure or the effect of electron correlations.9-14) However, Korshunov et al. estimated x* (xm in their report) to be 0.56 based on a tight- binding fit to an LAPW calculation, which is close to our value.13, 14) They pointed out that taking account of electron correlations would push it up to 0.68,13, 14) which suggests that electron correlations may not be so important in NaxCoO2. Concerning superconductivity found in the hydrated compound, Sakurai et al. found that superconductivity appears at two specific Co valences of +3.48 and +3.40.4, 5) Interestingly, the latter corresponds to x = 0.60 in NaxCoO2, just above our x*. Kuroki and his coworkers pointed out theoretically that the presence of the two concentric FSs around the Γ point leads to an enhanced spin fluctuation and thus gives rise to an extended s-wave superconductivity in the hydrated compound.25) We think that superconductivity would show up at just above x* even in nonhydrated NaxCoO2, if the influence of disordered Na ions is appropriately taken away. The other Co valence of +3.48 corresponds to x = 0.52 in NaxCoO2, which is a simple PM metal. However, a neutron diffraction experiment found that hydration squashes the CoO2 layer. 26) It is theoretically predicted that this structural change with hydration pushes the e'g band above the Fermi level, leading to an enhancement in DOS and thus the occurrence of a spin- triplet superconductivity.16, 17) Therefore, the two superconducting states may be associated with the two corresponding FSs with enhanced DOS. In summary, we have studied the electronic properties of NaxCoO2 with varying x, using a series of high quality samples prepared at high temperature. Dramatic changes in various quantities and thus in the electronic structure are found at a critical Na concentration x* between 0.58 and 0.59. This provides strong evidence of the presence of an electron pocket around the Γ point for high band fillings of x > x*. The intrinsic phase diagram of the (CoO2) x- layer is rather simple, as depicted in Fig. 5, in the absence of an electrostatic potential superimposed from the Na layers above and below. Fig. 5. Schematic representation of the band structure of NaxCoO2. Band dispersions along the Γ-K line (left), an expected profile of density of states (middle) and an x-T phase diagram (right) are depicted. The critical Na content x* in the phase diagram corresponds to a band filling with the Fermi energy equal to E*, as shown by a broken line, where the Fermi level touches the bottom of the a1g band at the Γ point. Hexagons represent the Brillouin zone with the Γ point at the center and the K points at the corners. A small electron Fermi surface appears for the Fermi energy above E* in addition to a large Fermi surface around Γ. In the phase diagram, a Curie-Weiss metal exists above x*, while a Pauli paramagnetic metal below x*. A charge-ordered insulator reported at x = 0.5 is excluded in this phase diagram, because it is extrinsic, coming from the Na ordering. Acknowledgment We thank M. Ichihara for his help in electron microscopy observations. 1) C. Fouassier et al.: J. Solid State Chem. 6 (1973) 532. 2) I. Terasaki, Y. Sasago and K. Uchinokura: Phys. Rev. B 56 (1997) R12685. 3) K. Takada et al.: Nature 422 (2003) 53. 4) H. Sakurai et al.: J. Phys. Soc. Jpn. 74 (2005) 2909. 5) H. Sakurai et al.: Phys. Rev. B 74 (2006) 092502. 6) M. L. Foo et al.: Phys. Rev. Lett. 92 (2004) 247001. 7) M. Yokoi et al.: J. Phys. Soc. Jpn. 74 (2005) 3046. 8) G. Gaparovi et al.: Phys. Rev. Lett. 96 (2006) 046403. 9) D. J. Singh: Phys. Rev. B 61 (2000) 13397. 10) D. J. Singh: Phys. Rev. B 68 (2003) 020503(R). 11) P. Zhang et al.: Phys. Rev. Lett. 93 (2004) 236402. 12) M. D. Johannes et al.: Europhys. Lett. 68 (2004) 433. 13) M. M. Korshunov et al.: JETP Letters 84 (2007) 650. 14) M. M. Korshunov et al.: Phys. Rev. B 75 (2007) 94511. 15) M. Z. Hasan et al.: Phys. Rev. Lett. 92 (2004) 246402. 16) M. Mochizuki and M. Ogata: J. Phys. Soc. Jpn. 75 (2006) 113703. 17) M. Mochizuki and M. Ogata: J. Phys. Soc. Jpn. 76 (2007) 013704. 18) D. Yoshizumi et al.: in preparation. 19) K. Kuroki et al.: Phys. Rev. Lett. 98 (2007) 136401. 20) S. P. Bayrakci et al.: Phys. Rev. B 69 (2004) 100410(R). 21) A. T. Boothroyd et al.: Phys. Rev. Lett. 92 (2004) 197201. 22) K. Ishida et al.: J. Phys. Soc. Jpn. 72 (2003) 3041. 23) I. R. Mukhamedshin et al.: Phys. Rev. Lett. 94 (2005) 247602. 24) Y. Ihara et al.: J. Phys. Soc. Jpn. 75 (2006) 124714. 25) K. Kuroki et al.: Phys. Rev. B 73 (2006) 184503. 26) J. W. Lynn et al.: Phys. Rev. B 68 (2003) 214516.

0704.1066

Observations on degenerate saddle point problems

7 Observations on degenerate saddle point problems Andrew V. Knyazev Department of Mathematical Sciences University of Colorado at Denver and Health Sciences Center P.O. Box 173364, Campus Box 170, Denver, CO 80217-3364 Abstract We investigate degenerate saddle point problems, which can be viewed as limit cases of standard mixed formulations of symmetric problems with large jumps in coefficients. We prove that they are well-posed in a standard norm despite the degeneracy. By wellposedness we mean a stable dependence of the solution on the right-hand side. A known approach of splitting the saddle point problem into separate equations for the primary unknown and for the Lagrange multiplier is used. We revisit the traditional Ladygenskaya–Babuška–Brezzi (LBB) or inf–sup condition as well as the standard coercivity condition, and analyze how they are affected by the degeneracy of the corresponding bilinear forms. We suggest and discuss generalized conditions that cover the degenerate case. The LBB or inf–sup condition is necessary and sufficient for wellposedness of the problem with respect to the Lagrange multiplier under some assumptions. The generalized coercivity condition is necessary and sufficient for wellposedness of the problem with respect to the primary unknown under some other assumptions. We connect the generalized coercivity condition to the positiveness of the minimum gap of relevant subspaces, and propose several equivalent expressions for the minimum gap. Our results provide a foundation for research on uniform wellposedness of mixed formulations of symmetric problems with large jumps in coefficients in a standard norm, independent of the jumps. Such problems appear, e.g., in numerical simulations of composite materials made of components with contrasting properties. Key words: Wellposedness, mixed, symmetric, saddle point, Lagrange multiplier, Ladygenskaya–Babuška–Brezzi (LBB) condition, inf–sup condition, coercivity, minimum gap between subspaces. PACS: 46.15.Cc, 02.30.Sa, 02.60.Lj 1. Introduction Degenerate saddle point problems, e.g., can be viewed as limit cases of mixed formulations of symmetric problems with large jumps in coefficients, corresponding to an infi- nite jump.We prove that the degeneracy does not affect the wellposedness in a standard norm under some natural as- sumptions, using ideas that are initiated by [3, 4, 5, 6, 7, 14, 15]. By wellposedness, contrary to illposedness, we mean a stable dependence of the solution on the right-hand side. Results of this paper provide a foundation for research on uniform wellposedness of mixed formulations of symmet- ric problems with large jumps in coefficients in a standard norm, independent of the jumps. Email address: andrew.knyazev[AT]cudenver.edu (Andrew V. Knyazev). URL: http://math.cudenver.edu/˜ aknyazev/ (Andrew V. Knyazev). 1 Partially supported by the National Science Foundation award DMS-0612751. The necessary and sufficient condition, e.g., [9, 10], of the standard wellposedness of an operator equation with an arbitrary right–hand side is the existence of a bounded inverse of the operator. We argue that in some practical cases the equation is degenerate, i.e. the inverse operator does not exist. Assuming that the right–hand side is in the operator range, a solution exists, but is not unique. Tomake the solution unique we factor out the operator null–space. This leads to a natural generalization, where boundedness of the pseudoinverse of the operator is used as the necessary and sufficient condition of wellposedness of a degenerate operator equation, by analogy with [13, 15]. With this idea in mind, we revisit necessary and sufficient conditions of wellposedness of an abstract mixed problem. In the symmetric case we consider here, the mixed problem can be interpreted as a variational saddle point problem. For generalized saddle point problems we refer the reader, e.g., to [11]. We start in Section 2 with a standard abstract symmetric mixed problem as in [9, 10]. By analogy with [14, 17], we Preprint submitted to Comp. Meth. Applied Mech. Engineering, accepted and published as doi:10.1016/j.cma.2006.10.01926 October 2018 http://arxiv.org/abs/0704.1066v1 split the saddle point problem into two equations, for the primary unknown and for the Lagrange multiplier. This split is somewhat implicit in [9, 10]. The equation for the primary unknown is self-consistent, since here we eliminate the Lagrange multiplier from the mixed system using an orthogonal projector. Following, e.g., [10], we discuss the traditional neces- sary and sufficient conditions of wellposedness, namely, the Ladygenskaya–Babuška–Brezzi (LBB) or inf-sup condition and the coercivity condition. The LBB or inf-sup condition, considered in Section 3, is necessary and sufficient for a sta- ble dependence of the Lagrange multiplier on an arbitrary right-hand side. We review the traditional point of view that the coerciv- ity condition is a necessary and sufficient condition of well- posedness of the problem. In Section 4, an operator form of the dual variational problem without assuming the co- ercivity condition is considered. We examine the unique- ness of the solution and describe all possible multiple solu- tions for a given right-hand side. All admissible right-hand sides are determined. We formulate several equivalent nec- essary and sufficient conditions of wellposedness in terms of closedness of relevant subspaces. We also derive a geo- metrical condition—a positiveness of a minimum gap [12] between relevant closed subspaces. A possible application of our theory is the Hellinger– Reissner formulation, e.g., [1], of nonhomogeneous Lamé equations for media with (almost) rigid inclusions, where the Lagrange multiplier is the displacement, and we get an operator equation for the stress on the closed subspace of divergence free (in a weak sense) stresses. Infinitely large Lamé coefficients λ and µ, in a subdomain, result in a null-space of the operator in the equation for the stress, so the inverse operator does not exist and the problem is not wellposed in a traditional sense. Our abstract geomet- rical condition of generalized wellposedness in this example is equivalent to a possibility of extension of displacements preserving the energy norm of the Lamé operator. It has been proved in [4, 7] that such an extension is possible un- der some assumptions. We expect that in the limit case of infinitely large Lamé coefficients λ and µ in a subdomain the pseudoinverse of the operator is bounded, which makes the problem wellposed for the stresses in the L2 sense, i.e. the L2 norm of the stress is stable even if the Lamé coef- ficients are large in a subdomain. We plan to address this application in the future. 2. Abstract symmetric saddle point problems In this section we essentially follow well known argu- ments, e.g., [10], with some simplifications due to the sym- metry of the saddle point problem and our unwillingness to introduce dual spaces. Straightforward manipulations, using a pair of complementary closed subspaces, allow us, as in [14, 17], to formulate separate equations for the pri- mary unknown and for the Lagrange multiplier of the sad- dle point problem; see, e.g., survey [8, Sec. 6] for similar matrix null-space methods. We start by formulating and investigating the problem using bilinear forms, and then repeat the arguments for operator-based formulations that are used in the last section of the paper. 2.1. Formulations using bilinear forms LetH andV be two real Hilbert spaces with scalar prod- ucts and norms denoted by (·, ·)H, ‖ · ‖H and (·, ·)V, ‖ · ‖V correspondingly. Let a(·, ·) : H × H → R and b(·, ·) : H ×V → R be two continuous bilinear forms with a(·, ·) symmetric and nonnegative definite. We consider the fol- lowing problem: for a given g ∈ H and f ∈ H find σ ∈ H, called the “primary unknown,” and u ∈ V, called the “La- grange multiplier,” such that a(σ, ǫ) + b(ǫ, u) = (g, ǫ)H, ∀ǫ ∈ H, b(σ − f, v) = 0, ∀v ∈ V. We place the right-hand side f “inside” of the form b as it allows us to take f ∈ H, not to introduce the dual space V′, and makes several statements somewhat simpler. We call (1) a saddle point problem, since equations (1) are the optimality conditions and their solution is a saddle point for the Lagrangian, e.g., [10], defined by a(σ, σ) + 2b(σ − f, u)− 2(g, σ)H. We call a linear manifold, not necessarily closed, a “sub- space” and a closed linear manifold a “closed subspace.” Let us introduce a special notation N ⊆ H for the closed subspace, which is the null-space of the bilinear form b(·, ·) with respect to its first argument, i.e. N = {ǫ ∈ H : b(ǫ, v) = 0, ∀v ∈ V}. Let us denote by P ≡ N⊥ ⊆ H the closed subspace which is H-orthogonal (complementary) to N. Closed subspaces N and P play important roles in this paper, so let us introduce anH-orthogonal projector P on H such that N(P ) = N and R(P ) = P and the com- plementary projector P⊥ = I − P with R(P⊥) = N and N(P⊥) = P, where by R(P ) we denote the range of opera- tor P and, with a slight abuse of the notation, by N(P ) we denote the null-space of operator P . We assume through- out the paper, unless stated otherwise, that a bounded op- erator is defined everywhere on a corresponding space. As an orthogonal projector, operator P : H → H is bounded H-selfadjoint, P = P ∗, and satisfies P = P 2. In the first equation of system (1), let us split it into two equations, by plugging separately ǫ = Pǫ ∈ P and ǫ = P⊥ǫ ∈ N and using the fact that b(P⊥ǫ, u) = 0, ∀ǫ ∈ H. The second equation in system (1) has a simple equivalent geometric interpretation: σ − f ∈ N, or (σ − f, ǫ)H = 0, ∀ǫ ∈ P. We then rewrite system (1) in the following equivalent form: a(σ, ǫ) + b(ǫ, u) = (g, ǫ)H, ∀ǫ ∈ P, a(σ, ǫ) = (g, ǫ)H, ∀ǫ ∈ N, (σ − f, ǫ)H = 0, ∀ǫ ∈ P. Nowwemake an important observation that we can treat the first line in system (2) as an equation for the Lagrange multiplier u, given the primary unknown σ, i.e. b(ǫ, u) = (g, ǫ)H − a(σ, ǫ), ∀ǫ ∈ P. (3) The last two lines in system (2) involve neither the Lagrange multiplier u, nor the bilinear form b, and can be used to determine the primary unknown σ: a(σ, ǫ) = (g, ǫ)H, ∀ǫ ∈ N, (σ − f, ǫ)H = 0, ∀ǫ ∈ P. System (4) describes, e.g., [10], the optimality conditions of the constrained minimization problem inf {a(σ, σ) − 2(g, σ)H}, σ ∈ H : (σ − f, ǫ)H = 0, ∀ǫ ∈ P. 2.2. Operator-based formulations In addition to the formulations above involving bilin- ear forms, it is convenient to consider equivalent operator- based formulations.We associatewith the forms a and b two linear continuous operators A : H → H and B : H → V defined by (Aσ, ǫ)H = a(σ, ǫ), (Bσ, v)V = b(σ, v), ∀ǫ, σ ∈ H, v ∈ V. In this definition of A and B we follow a slightly simplified, e.g., [11, 17], rather than standard [10], approach, namely, we do not need dual spaces H′ and V′. Now, we reformulate the main statements of subsection 2.1 using the just defined operators A and B. The following operator formulation Aσ + B∗u = g in H, B(σ − f) = 0 in V is equivalent to the original problem (1) with the bilinear forms, where the adjoint operator B∗ : V → H is defined, as usual, by (σ,B∗v)H = (Bσ, v)V, ∀σ ∈ H, v ∈ V. The operator A is selfadjoint and nonnegative definite, A = A∗ ≥ 0 on H since it is defined by the symmetric and nonnegative definite form a. We notice that the second equation in system (5) has the same geometric interpretation as in the case of bilinear forms-based system (1): σ − f ∈ N(B). The null-space N(B) ⊆ H and its H-orthogonal complement R(B∗) ⊆ H have already been denoted by N and P, correspondingly, and introduced together with the H-orthogonal projector P on H such that N = N(P ) = N(B) and P = R(P ) = R(B∗) in the previous subsection. We split the first equation in system (5) in two orthogonal parts corresponding to N and P, using that PB∗u = B∗u and P⊥B∗u = 0, since R(B∗) ⊆ P. We replace B with P , since they share the same null-space, in the second equation in system (5) to get the following equivalent form of system PAσ +B∗u = Pg in H, P⊥Aσ = P⊥g in H, P (σ − f) = 0 in H. We notice that the first line in system (6) is an equation for the Lagrange multiplier u, given the primary unknown σ, as in (3), i.e. B∗u = P (g −Aσ). We next discuss the necessary and sufficient conditions from [10] of wellposedness of the problem and make it clear why one can find weaker necessary and sufficient conditions. To simplify our arguments, we take advantage in the rest of the paper of the split of the original system into separate equations for the Lagrange multiplier u and the primary unknown σ that we have described in this section. It is important to realize, however, that we have not made any substitutions, neither in the solutions u and σ, nor in the right-hand sides f and g. So whatever statements we next prove concerning the dependence of the solutions u and σ on the right-hand sides f and g, these statements are equally applicable to both the separate equations and to the original system in either bilinear form- or operator-based context. 3. Inf-sup or LBB condition In this section, we discuss a traditional assumption, be- ing recently referred to as Ladygenskaya–Babuška–Brezzi (LBB) condition, see Babuška and Aziz [2], Brezzi and Fortin [10], Ladyzhenskaya [16], that the range of operator B : H → V, denoted by R(B), is closed. The closedness of a range of a closed operator is ultimately connected to the boundedness of the operator (pseudo-)inverse, e.g., [12]. In our specific situation, operator B is bounded with the closed domain H and, thus, is closed, so its (pseudo- )inverse B−1 : R(B) → H/N(B) is also closed. It is neces- sary to use a factor-space here to define the inverse, since the standard operator inverse B−1 : R(B) → H does not exist if N(B) is nontrivial. We note that N(B) is closed and that the factor-space H/N(B) is a Hilbert space, as is H. In a Hilbert space, a convenient set of representants for the classes in the factor-space is simply the correspond- ing orthogonal complement, e.g., H/N(B) is isometrically isomorphic to P = (N(B))⊥ ⊆ H, so we set ‖σ‖H/N(B) = ‖Pσ‖H. The subspaceR(B) is the domain of the closed op- erator B−1 : R(B) → H/N(B) therefore, R(B) is closed if and only if B−1 : R(B) → H/N(B) is bounded. Closed- ness of R(B) is equivalent to closedness of R(B∗), so all the arguments above can be equivalently reformulated for the adjoint operator B∗ and its (pseudo-)inverse. When written in terms of inequalities involving the bi- linear form b : b(σ, v) ‖σ‖H/N(B)‖v‖V = inf ‖Bσ‖V ‖σ‖H/N(B) ‖B−1‖R(B)→H/N(B) or, equivalently, b(σ, v) ‖σ‖H‖v‖V/N(B∗) = inf ‖B∗v‖H ‖v‖V/N(B∗) ‖B−∗‖R(B∗)→V/N(B∗) the LBB condition is also known as the inf-sup condition, see Babuška and Aziz [2], Brezzi and Fortin [10], where V/N(B∗) means the factor-space of V with respect to the closed subspace N(B∗). We implicitly assume that the ar- guments in the inf-sup formulas above and throughout the paper do not make both the numerator and the denomi- nator vanish. In Ladyzhenskaya [16], the inf-sup condition does not appear to be explicitly formulated, instead, closed- ness of a range of the gradient operator is investigated in connection with wellposedness of the diffusion equation. We note that the induced norms of an operator and its adjoint are equal, so both inf-sup expressions above are equal to the same constant that we call cb. If at least one of the spaces H or V is finite dimensional then the value cb is positive automatically, so it becomes important how cb depends on some parameters, e.g., on the dimension. Let us mention that in many practical applications the space V can be naturally defined such that N(B∗) = {0}, so the latter inf-sup expression of the LBB condition takes the form b(σ, v) ‖σ‖H‖v‖V = cb > 0, which can be most often seen in publications on the subject. We now contribute our own equivalent formulations of the LBB condition. Lemma 3.1 Subspaces R(B) ⊆ V and R(BB∗) ⊆ V are closed simultaneously. Moreover, if either of them is closed we have R(BB∗) = R(B). Proof. If BB∗v = 0 then (B∗v,B∗v)H = 0, i.e. B ∗v = 0, which proves that N(BB∗) = N(B∗). Taking an orthogo- nal complement to both parts givesR(BB∗) = R(B) as the operator BB∗ is selfadjoint. Trivially, R(BB∗) ⊆ R(B). If the range R(BB∗) is closed then R(B) = R(BB∗) = R(BB∗) ⊆ R(B), but clearlyR(B) ⊆ R(B), which proves closedness of R(B) = R(BB∗). To prove the inverse statement, assuming that R(B) is closed, we invoke the orthogonal decomposition argu- ment 2 : H = R(B∗) ⊕ (R(B∗))⊥ = R(B∗) ⊕ N(B) since R(B) and thus R(B∗) are closed. Multiplying this equal- ity by B gives R(B) = BH = B(R(B∗) ⊕ N(B)) = BR(B∗) = R(BB∗). ✷ 2 This proof is suggested by an anonymous referee We use the previous lemma to introduce (BB∗)−1 : R(BB∗) → V/N(B∗) in the next Lemma 3.2. It is nec- essary to use the factor-space V/N(B∗) here, since the standard inverse (BB∗)−1 : R(BB∗) → V does not exist if N(B∗) is nontrivial. Lemma 3.2 Closedness of R(B) ⊆ V is equivalent to boundedness of the operator (BB∗)−1 : R(BB∗) → V/N(B∗). Proof. By Lemma 3.1, closedness of R(B) ⊆ V is equiv- alent to closedness of R(BB∗) ⊆ V. We use several well- known statements on closed operators, e.g., [12], applied to the operator BB∗, that we have already reviewed in the second paragraph of this section for the operator B. The operator BB∗ is bounded and has the closed domain V, so the operator is closed and its (pseudo-)inverse (BB∗)−1 : R(BB∗) → V/N(B∗) with the domain R(BB∗) ⊆ V is also closed. The domain R(BB∗) ⊆ V of the closed opera- tor B−1 : R(BB∗) → H/N(B) is closed if and only if the operator is bounded. ✷ If R(B) is closed then, using Lemmas 3.1 and 3.2, R(B) = R(BB∗) and we can derive the following useful formula P = B∗(BB∗)−1B : H → H. (7) Indeed, we first note that R((BB∗)−1) ⊆ V/N(B∗) is multiplied by B∗ in (7), so the product is independent of the choice of a representant from the equivalence class V/N(B∗) and, thus, is correctly defined. Second, righ-hand side of (7) is a linear and bounded operator as a product of linear and bounded operators. Moreover, it is an orthogonal projector on H since it is selfadjoint and idempotent, and has the null-space the same as the orthoprojector P has. If the LBB condition is not satisfied, i.e. R(B) is not closed, then the domain of definition of the operator B∗(BB∗)−1B is the subspace R(B∗) ⊕ N(B), which is not closed, and formula (7), where P is the orthogonal projector on H with N(P ) = N(B), clearly does not hold. Let us note that in the case of finite dimensional spaces H and V the range R(B) is evidently closed, the opera- tor (B∗)+ = (BB∗)+B is the well-known Moore–Penrose pseudo inverse of B∗, and P = B∗(B∗)+ is the well known formula for the orthogonal projector onto the range of B∗. If σ is an exact solution of system (5), then u in (5) can be found from the equationB∗u = −Aσ+g ∈ R(B∗). If σ is an approximate solution of system (5) such that the condition Aσ − g ∈ R(B∗), which is necessary and sufficient for the existence of u, does not hold, then u can be computed from the projected equation B∗u = P (−Aσ + g) ∈ P. Both the original and the projected equations for u are wellposed by the LBB assumption, i.e. R(B∗) = P and ‖u‖V/N(B∗) ≤ ‖σ‖H + ‖g‖H. Whether the LBB assumption is necessary for wellposed- ness of the equation for u depends on if the set of all possible right-hand sides g − Aσ gives the whole subspace R(B∗), see [10]. For example, in a practically important case g = 0 we have B∗u = −Aσ = −PAσ ∈ R(PA) ⊆ R(P ). If the latter inclusion is strict, it opens up an opportunity for a weaker, compared to the original LBB, assumption of well- posedness of the above equation for u. In the present paper, however, we are concerned with finding σ, not u. The LBB condition for the bilinear form b appears to be of no importance for our results in the next section where we analyze wellposedness of system (5) with respect to the σ unknown only, assuming that the u unknown is of no interest, or can be found for a given σ using some postprocessing. 4. Coercivity conditions 4.1. The standard coercivity condition We finally get to the main topic of the paper: an assump- tion on A which is a condition of wellposedness of (5) with respect to σ. For the reader’s convenience, we briefly repeat the necessary notation and the system of equations for σ to make this section self-consistent. LetH be a real Hilbert space and P be an orthoprojector in H with a null-space N(P ) = N and a rangeR(P ) ≡ P—we emphasize that the range of any orthoprojector in a Hilbert space is closed. Let A be a linear and bounded operator such that 0 ≤ A∗ = A on H. The last two lines in system (6) represent an oper- ator form of system (4); they do not involve the Lagrange multiplier u or the operator B and determine the primary unknown σ ∈ H: P⊥(Aσ − g) = 0 in H, P (σ − f) = 0 in H, where g ∈ H and f ∈ H are given and P⊥ ≡ I−P.We can also replace system (8) with the following equivalent single equation: P⊥A |N ψ = P⊥g − P⊥APf ∈ N, σ = ψ + Pf, (9) where in (9) we take a restriction of the operator P⊥A on its invariant closed subspace N, and we are looking for a solutionψ ∈ N. Then the necessary and sufficient condition of wellposedness of problem (9) for an arbitrary g ∈ H is, clearly, that the range of P⊥A |N is N. This leads to the traditional assumption, see [10], a(σ, σ) ≥ ca > 0, ∀σ ∈ N, ‖σ‖H = 1 or, in an operator form, A ≥ caI on N ⊆ H, since A is selfadjoint nonnegative. Thus, this assumption is also necessary and sufficient [9, 10] for wellposedness of system (5) with respect to σ for an arbitrary g ∈ H. In the rest of the section, we analyze the scenario, where A is selfadjoint nonnegative on H, but may be degenerate on N, so we impose necessary restrictions on g ∈ H, and determine a generalized coercivity condition that covers the case of the degeneracy. 4.2. Existence, uniqueness, and wellposedness Before we investigate the existence and uniqueness of the solution σ, we prove the following technical, but important, lemma. Lemma 4.1 Let P be an orthoprojector in H with a null- space N(P ) = N and a range R(P ) ≡ P = N⊥, and A be a linear and bounded operator such that 0 ≤ A∗ = A on H. N(P⊥A) ∩N = N(A) ∩N, (10) {N(P⊥A) ∩N}⊥ = R(A) +P, (11) N(P⊥A) ∩N ⊕ P⊥R(A). (12) Proof.We first verify (10). It follows fromN(P⊥A) ⊇ N(A) that the right-hand side of (10) is included in the left-hand side. To prove the reverse inclusion, let ϕ ∈ N and P⊥Aϕ = 0, then 0 = (P⊥Aϕ,ϕ) = (Aϕ,ϕ) = ‖A1/2ϕ‖2 (recall that A ≥ 0). Then A1/2ϕ = 0 and Aϕ = 0. Therefore, equality (10) holds. Equality (11) follows from (10), by substituting N(A) = F⊥ and R(A) = F in the well-known simple identity F⊥ ∩ P⊥ = (F + P)⊥ and noting that (R(A) + P)⊥⊥ = R(A) +P = R(A) +P by properties of the closure. Finally, to obtain the second term in the orthogonal de- composition (12) of N we see that by (11) {N(P⊥A) ∩ N}⊥ ∩N = R(A) +P ∩N; at the same time R(A) +P ∩N= P⊥R(A) +P ∩N P⊥R(A)⊕P ∩N = P⊥R(A), which completes the proof of the lemma. ✷ We start with the solution uniqueness. Lemma 4.2 Suppose that for some fixed g ∈ H and f ∈ H there exists a solution σ of (8). Then it is unique provided that N(A) ∩N = {0};otherwise, all possible solutions yield the hyperplane σ+ {N(A)∩N} and there exists the unique normal (with minimal norm in H) solution of (8) that can be also defined as a common element of the above hyperplane and the closed subspace R(A) +P, which is the set of all normal solutions for all possible f and g. Proof. All solutions of (8) with g = f = 0 constitute the closed subspace N(P⊥A) ∩N(may be 0-dimensional), which by (10) is the same as N(A) ∩ N. Hence, all solu- tions of (8) with the given g and f, provided that there exists at least one solution σ, constitute the hyperplane σ + N(A) ∩ N. It is known that each closed hyperplane in a Hilbert space has a unique element with the minimal norm, i.e. the element that is orthogonal to the directing closed subspace N(A)∩N of the hyperplane. The orthogo- nal complement to the directing closed subspace is already given by (11). ✷ In the rest of the subsection we use the following equation equivalent to (9): (P⊥A+ P )σ = P⊥g + Pf. (13) The assumptions on the right-hand side of the system (8) which ensure the existence of a solution are rather standard and follow from (13) easily. Lemma 4.3 For any f ∈ H there exists a solution of (8) if and only if g ∈ R(A)+P, i.e. P⊥g+Pf ∈ P⊥R(A)+P = R(A) +P. Proof. The subspace (not necessarily closed) P⊥R(A) +P is simply the range of the operator P⊥A + P of equation (13). ✷ The subspaceR(A)+P that appears in Lemmas 4.2 and 4.3 plays the central role in the following necessary and sufficient conditions of wellposedness. Theorem 4.1 The following statements are equivalent: (i) The subspace R(A) +P is closed. (ii) The subspace AN+P is closed. (iii) The subspace P⊥R(A) is closed. (iv) The subspace P⊥AN is closed. (v) Problem (13) with f ∈ H and g ∈ R(A) +P is well- posed in the factor-space, ‖σ‖H/{N(A)∩N} ≤ c(‖g‖ + ‖f‖), or, equivalently, ‖σ‖ ≤ c(‖g‖ + ‖f‖) for the normal solution σ ∈ R(A) +P. Proof. (1)⇔(3) We have R (A)+ P= P⊥R(A)⊕P. (1)⇔(2) The subspace P⊥R(A) ⊕ P = R(A) + P is the range of the operator P⊥A + P. The range of a bounded operator is closed if and only if the range of the conjugate operator is closed. (2)⇔(4) Using the same arguments as above, AN + P = P⊥AN⊕P. (1)⇔(5) The operator P⊥A + P is bounded and defined everywhere on a Hilbert space, thus it is closed. Therefore, the (pseudo)inverse operator (P⊥A+ P )−1 : R(P⊥A+ P ) → H/{N(A) ∩N} is closed. It is bounded if and only if its domain of definition R(P⊥A + P ) is closed. A normal solution is a convenient representant of a factor-class in a Hilbert space. ✷ 4.3. Generalized coercivity conditions Statements (1)–(4) in Theorem 4.1 may not be so easily verifiable in practice, so we want to find a somewhat eas- ier assumption that generalizes the standard coercivity as- sumption A ≥ caI on N ⊆ H, which itself does not hold if the operatorA vanishes on a nontrivial subspace ofN ⊆ H. Let us return back to equation (9). We remind the reader that the first equation in (8) is equivalent to the orthog- onal expansion σ = ψ + Pf, where ψ = P⊥σ ∈ N. This and the second equation in (8) lead to (9) that we present here, introducing a special notation K = P⊥A |N, in the equivalent form Kψ = φ, ψ ∈ P⊥R(A), φ = P⊥g − P⊥APf ∈ P⊥R(A).(14) under the assumption that g ∈ R(A) +P. The operatorK is bounded, selfadjoint, and nonnegative definite on N, where N ⊆ H inherits the scalar product and the norm of H, so there exists a bounded, selfadjoint, and nonnegative definite square root K on N. Applying the inf-sup condition to the operator K on N, by direct analogy with Lemmas 3.1 and 3.2 and their proofs, we have that N = N(K) and Theorem 4.2 The following statements are equivalent: (i) The subspace R ⊆ N is closed. (ii) The subspace R(K) ⊆ N is closed. (iii) The inf-sup condition for the operator K on N Kǫ, σ ‖ǫ‖N/N(K)‖σ‖N = inf ‖ǫ‖N/N(K) > 0 (15) holds. (iv) The norm of the operator K−1 : R(K) → N/N(K) is equal to ρ <∞. Moreover, under either of the assumptions we have = R(K). Noticing that R(K) = P⊥AN, we immediately see that statements (4) in Theorem 4.1 and (2) in Theorem 4.2 are the same, so all statements of Theorems 4.1 and 4.2 are equivalent. Our last goals in this subsection are to present statement (3) of Theorem 4.2 in original terms, so that it resembles the coercivity condition, and to bound the norm of the solution in terms of the norms of the right-hand sides, using statement (4) of Theorem 4.2. Theorem 4.3 For any g ∈ R(A) + P the following as- sumption A ≥ 1 I on the subspace P⊥R(A) (16) with a (finite) constant ρ > 0 is necessary and sufficient for the normal solution σ with P⊥σ ∈ P⊥R(A) to exist and to be unique and continuous in f ∈ H and g ∈ R(A) + P. Moreover, assumption (16) implies ‖σ‖2 ≤ ‖f‖2 + ρ2‖g −APf‖2. (17) Proof. First, we note that inequality (16) on the subspace P⊥R(A) is equivalent to the same inequality on its clo- sure P⊥R(A) because of the continuity of A and the scalar product. Second, as (ǫ,Kǫ) = (ǫ, P⊥Aǫ) = (ǫ, Aǫ) for all ǫ ∈ P⊥R(A) ⊆ N, inequality (16) is also equivalent to I on the closed subspace P⊥R(A). (18) Now we show that (18) is equivalent to (15), which is condition (3) of Theorem 4.2. For the numerator in (15), we have = (ǫ,Kǫ). To handle the denominator in (15), we remind the reader the orthogonal decomposition N(P⊥A) ∩N ⊕P⊥R(A) stated as (12) and proved in Lemma 4.1. Splitting ǫ ∈ N according to this orthogo- nal decomposition, we see that its first component—from N(P⊥A) ∩ N = N(K)—vanishes both in the numerator, since it is in the null-space of K, and in the denominator of (15), by the definition of the factor-norm, which gives (18), where only the second component—from P⊥R(A)— survives. We conclude that (16) is equivalent to (15), which is condition (3) of Theorem 4.2, and thus, to all statements of Theorems 4.1 and 4.2. Finally, if (16) holds then the subspace R(P⊥A) is closed, the operator K : P⊥R(A) 7→ P⊥R(A) is an isomorphism and problem (14) is wellposed for f ∈ H and g ∈ R(A) +P, i.e. ‖ψ‖ ≤ ρ‖P⊥g − P⊥APf‖ ≤ ρ‖g −APf‖ (19) by Theorem 4.2. Estimate (17) follows from σ = ψ + Pf and (19) due to the statement of Lemma 4.2 that the nor- mal solution σ ∈ R(A) +P, that is, ψ ∈ P⊥R(A) = P⊥R(A) +P = P⊥ ∩ (P⊥ ∩N(A))⊥ is the corresponding part of the orthogonal expansion σ = ψ + Pf for the nor- mal solution. ✷ 4.4. Minimum gap between subspaces The rest of the section concerns the case where the range of A is closed, so assumption (16) can be equivalently re- formulated using the minimum gap between some relevant subspaces. We first find a simple way to check if the range of A is closed. Lemma 4.4 Condition A ≥ 1 I on the subspaceR(A) ≡ D (20) with a (finite) constant ρD > 0 is equivalent to closedness of D. Proof. The operator A is a linear, bounded, and every- where defined. Thus, it is closed and its inverse A−1 : D → H/N(A) is also closed. Boundedness of the inverse is equiv- alent, on the one hand, to condition (20) and, on the other hand, to closedness of D. ✷ Now we are ready to present a simplified version of the necessary and sufficient condition of wellposedness (16), assuming that the range of A is closed. Theorem 4.4 Let the range R(A) ≡ D be closed, the or- thoprojector onD be denoted byD, and the constant ρD > 0 be defined by (20). Then inequality (16) is equivalent to the inequality κ ≡ inf ψ∈P⊥D > 0. (21) In particular, (20) and (21) lead to (16) with ρ = ρD/κ Proof. We have R(P⊥A) = P⊥R(A) = P⊥D, i.e. the subspaces indicated in (16) and (21) coincide. Now the main assertion of the Lemma is a consequence of relations (Dψ,Dψ) ≤ (Aψ,ψ) = (ADψ,Dψ) ≤ ‖A‖(Dψ,Dψ) which hold for an arbitrary ψ ∈ H. ✷ The next two lemmas provide alternative assumptions, equivalent to (21), which are necessary and sufficient for wellposedness, assuming that the range of A is closed. It is important to have a choice of a criterion that may be easier to check in a practical application. For aesthetic reasons we denote N ≡ P⊥. Lemma 4.5 Let D and P be orthogonal projectors onto closed subspaces D and P, and let D⊥ = I −D and P⊥ = I − P be orthogonal projectors onto the orthogonal comple- ments D⊥ and P⊥, respectively. The following statements are equivalent: (i) The subspace P⊥D is closed. (ii) The subspace D+P is closed. (iii) The subspace D⊥ +P⊥ is closed. (iv) The subspace PD⊥ is closed. Proof. (1)⇔(2) The subspace P⊥D is closed iff the sub- space P⊥D⊕P = D+P is closed as the terms are orthog- onal in the first expression. (2)⇔(3) By Theorem IV-4.8 of [12], a sum of closed sub- spaces in a Hilbert space is closed if and only if the sum of their orthogonal complements is closed. (3)⇔(4) Using the same arguments as above, P⊥ +D⊥ = P⊥ ⊕ PD⊥. ✷ Lemma 4.6 Using the notation of Lemma 4.5, the follow- ing equalities hold: ψ∈P, ψ 6∈D dist{ψ;D} dist{ψ;D ∩P} = inf ψ∈D, ψ 6∈P dist{ψ;P} dist{ψ;P ∩D} = inf ψ∈D⊥, ψ 6∈P⊥ dist{ψ;P⊥} dist{ψ;P⊥ ∩D⊥} = inf ψ∈P⊥, ψ 6∈D⊥ dist{ψ;D⊥} dist{ψ;D⊥ ∩P⊥} = inf ψ∈P⊥D = inf ψ∈D⊥P = inf ψ∈PD⊥ ‖D⊥ψ‖ = inf ψ∈DP⊥ ‖P⊥ψ‖ Moreover, each statement in the previous Lemma is equiv- alent to the positiveness κ > 0 in (21). Proof. The first three equalities are derived in Section IV-4 of [12] on the minimum gap between subspaces, along with a statement that positiveness of the minimum gap between two given subspaces is a necessary and sufficient condition of the sum of the subspaces, in our case,D+P, to be closed. We now prove that ψ∈P⊥, ψ 6∈D⊥ dist{ψ;D⊥} dist{ψ;D⊥ ∩P⊥} = inf ψ∈P⊥D\{0} All other equalities can be then trivially derived from the previous ones just by interchanging P and D. We first notice that in the right-hand side we can apply the inf to the closure P⊥D \ {0} as well, because a norm is a continuous function, ψ∈P⊥D\{0} ‖ψ‖ = infψ∈P⊥D\{0} ‖ψ‖ . We have, P⊥D = P⊥ ∩ (P⊥ ∩D⊥)⊥ as N(DP⊥) = P ⊕ (P⊥ ∩D⊥). The latter can be checked directly. We always have dist{ψ;D⊥} = ‖Dψ‖. If ψ ∈ P⊥D = P⊥∩(P⊥∩D⊥)⊥ ⊆ (P⊥∩D⊥)⊥,we also have dist{ψ;D⊥∩ P⊥} = ‖ψ‖. Thus, dist{ψ;D⊥} dist{ψ;D⊥ ∩P⊥} , ψ ∈ P⊥D \ {0}. Finally, using the orthogonal representation P⊥ = (P⊥ ∩ D⊥)⊕P⊥D, everyϕ ∈ P⊥ can be written as the orthogonal sum ϕ = (ϕ−ψ)⊕ψ, where ϕ−ψ ∈ P⊥ ∩D⊥, ψ ∈ P⊥D. Then dist{ψ;D⊥} = dist{ϕ;D⊥} and also dist{ψ;D⊥ ∩ P⊥} = dist{ϕ;D⊥ ∩P⊥}; so the value of the ratio dist{ψ;D⊥} dist{ψ;D⊥ ∩P⊥} dist{ϕ;D⊥} dist{ϕ;D⊥ ∩P⊥} does not depend on ϕ−ψ and its two infimum values, taken with respect to ψ ∈ P⊥D \ {0} and ϕ ∈ P⊥, ϕ 6∈ D⊥, coincide. ✷ Finally, we notice that g = 0 if we apply a saddle point approach to diffusion or linear elasticity equations. Indeed, in the Hellinger–Reissner formulation of nonhomogeneous Lamé equations, our σ represents the stress tensor, the La- grange multiplier u is the displacement, and if we also in- troduce the stain ǫ by the stain-displacement relation ǫ = −B∗u, then the first line in system (5) becomes Aσ − ǫ = g, which is the constitutive equation (3-D Hooke’s law), where of course g = 0. The second line in (5) is the equi- librium equation, where all body and traction forces are represented by f 6= 0. The assumption g = 0 allows us to look for even weaker conditions of wellposedness that we plan to investigate in the future. Acknowledgments The author thanks Ivo Babuška and Franco Brezzi for discussions. This work has been stimu- lated by collaboration with Nikolai S. Bakhvalov. The au- thor thanks an anonymous referee, who has made numer- ous useful suggestions to improve the original version of the paper, and CU-Denver students Donald McCuan and Christopher Harder for proofreading the paper. References [1] D. N. Arnold and R. Winther. Mixed finite elements for elasticity. Numer. Math., 92(3):401–419, 2002. [2] I. Babuška and A. K. Aziz. Survey lectures on the mathematical foundations of the finite element method. In The mathematical foundations of the fi- nite element method with applications to partial differ- ential equations, pages 1–359. Academic Press, New York, 1972. With the collaboration of G. Fix and R. B. Kellogg. [3] N. S. Bakhvalov and A. V. Knyazev. A new iterative algorithm for solving problems of the fictitious flow method for elliptic equations. Soviet Math. Doklady, 41(3):481–485, 1990. [4] N. S. Bakhvalov and A. V. Knyazev. Fictitious domain methods and computation of homogenized properties of composites with a periodic structure of essentially different components. In Gury I. Marchuk, editor, Numerical Methods and Applications, pages 221–276. CRC Press, Boca Raton, 1994. [5] N. S. Bakhvalov and A. V. Knyazev. Preconditioned iterative methods in a subspace for linear algebraic equations with large jumps in the coefficients. In D. Keyes and J. Xu, editors, Domain Decomposition Methods in Science and Engineering, volume 180 of Contemporary Mathematics, pages 157–162.American Mathematical Society, Providence, 1994. [6] N. S. Bakhvalov, A. V. Knyazev, and G. M. Kobel’kov. Iterative methods for solving equations with highly varying coefficients. In Roland Glowinski, Yuri A. Kuznetsov, Gérard A. Meurant, Jacques Périaux, and Olof Widlund, editors, Fourth International Sympo- sium on Domain Decomposition Methods for Partial Differential Equations, pages 197–205, Philadelphia, PA, 1991. SIAM. [7] N. S. Bakhvalov, A. V. Knyazev, and R. R. Parashkevov. Extension theorems for Stokes and Lame equations for nearly incompressible media and their applications to numerical solution of problems with highly discontinuous coefficients. Numerical Linear Algebra with Applications, 9(2):115–139, 2002. [8] M. Benzi, G. H. Golub, and J. Liesen. Numerical solu- tion of saddle point problems. Acta Numer., 14:1–137, 2005. [9] F. Brezzi. On the existence, uniqueness and approx- imation of saddle point problems arising from La- grangian multipliers. RAIRO Anal. Numer., 2:129– 151, 1974. [10] F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer–Verlag, New York, 1991. [11] P. Ciarlet, Jr., J. Huang, and J. Zou. Some observa- tions on generalized saddle-point problems. SIAM J. Matrix Anal. Appl., 25(1):224–236, 2003. [12] T. Kato. Perturbation Theory for Linear Operators. Springer–Verlag, New–York, 1976. [13] A. V. Knyazev. Analysis of transmission problems on Lipschitz boundaries in stronger norms. J. Numer. Math., 11(3):225–234, 2003. [14] A. V. Knyazev. Iterative solution of PDEwith strongly varying coefficients: algebraic version. In R. Beauwens and P. de Groen, editors, Iterative Methods in Linear Algebra, pages 85–89, Amsterdam, 1992. Elsevier. [15] A. V. Knyazev and O. Widlund. Lavrentiev regular- ization + Ritz approximation= uniform finite element error estimates for differential equations with rough coefficients. Mathematics of Computation, 72:17–40, 2003. [16] O. A. Ladyzhenskaya. The boundary value problems of mathematical physics, volume 49 of Applied Mathe- matical Sciences. Springer-Verlag, New York, 1985. [17] J. Xu and L. Zikatanov. Some observations onBabuška and Brezzi theories. Numer. Math., 94(1):195–202, 2003. Introduction Abstract symmetric saddle point problems Formulations using bilinear forms Operator-based formulations Inf-sup or LBB condition Coercivity conditions The standard coercivity condition Existence, uniqueness, and wellposedness Generalized coercivity conditions Minimum gap between subspaces

0704.1067

Expected Planets in Globular Clusters

EXPECTED PLANETS IN GLOBULAR CLUSTERS Noam Soker1 & Alon Hershenhorn 1 ABSTRACT We argue that all transient searches for planets in globular clusters have a very low detection probability. Planets of low metallicity stars typically do not reside at small orbital separations. The dependance of planetary system properties on metallicity is clearly seen when the quantity Ie ≡ Mp[a(1 − e)] is considered; Mp, a, e, are the planet mass, semi-major axis, and eccentricity, respectively. In high metallicity systems there is a concentration of systems at high and low values of Ie, with a low-populated gap near Ie ∼ 0.3MJ AU 2, where MJ is Jupiter’s mass. In low metallicity systems the concentration is only at the higher range of Ie, with a tail to low values of Ie. Therefore, it is still possible that planets exist around main sequence stars in globular clusters, although at small numbers because of the low metallicity, and at orbital periods of & 10 days. We discuss the implications of our conclusions on the role that companions can play in the evolution of their parent stars in globular clusters, e.g., influencing the distribution of horizontal branch stars on the Hertzsprung-Russell diagram of some globular clusters, and in forming low mass white dwarfs. Subject headings: stars: horizontal-branch globular clusters: general - stars: rotation - planets 1. INTRODUCTION Evolved sun-like stars that burn helium in their cores occupy the horizontal branch (HB) in the Hertzsprung-Russel (HR) diagram. HB stars that have low metallicity and/or low envelope mass are blue, and are termed blue HB (BHB) stars in globular clusters (GCs), and sdB or sdO (sdOB together; termed also extreme HB stars or EHB) in the field (not in GCs). There are strong indications that many of the sdOB stars in the field are in close binary systems (e.g., Maxted et al. 2001; Maxted 2004), and there is a strong support to 1Department of Physics, Technion−Israel Institute of Technology, Haifa 32000 Israel; [email protected] http://arxiv.org/abs/0704.1067v2 – 2 – the idea that the sdOB phenomena is caused by binary companions (e.g., Han et al. 2003; Maxted 2004). The large fraction of stars in the field that are likely to have planets around them (Lineweaver & Grether 2003) hint that planets can also lead to the formation of sdOB stars (Soker 1998). We note the recent tentative claim for a planet orbiting an sdB star at an orbital separation of a = 1.7 AU (Silvotti et al. 2007), that might hint that closer planets existed in the system before the progenitor turned into a red giant star. The distributions of stars on the HB (the HB morphology; also referred to as the color- magnitude diagram [CMD]) differ substantially from one GC to another. It has long been known that metallicity is the main, but not sole, factor which determines the HB morphology (for an historical review see, e.g., Fusi Pecci & Bellazzini 1997). The other factor (or factors) which determines the HB morphology is termed the ‘second parameter’. There is a debate on what are the main processes influencing the second parameter (Catelan 2007 and references therein), with a binary interaction being one of these processes. In the low mass binary second parameter model the companion is very light (a very low mass main sequence [MS] star, a brown dwarf, or a massive planet), and in most cases will be destroyed as it falls deeper into the envelope (Soker 1998; Soker & Harpaz 2007). Therefore, the non-detection of companions to HB stars in GCs (Moni Bidin et al. 2006) is not in contradiction with the model. Although close brown dwarf companions exist (e.g., Zucker & Mazeh 2000), and are included in the low mass binary second parameter models, they are rare (e.g., Mazeh et al. 2003; Grether & Lineweaver 2006), and so we concentrate on planets and low mass MS stars. More problematic to the binary second parameter model might seem to be the null detection of planets to MS stars in GCs (Weldrake et al. 2007b). Weldrake et al. (2007b) looked for transits in the GCs ω Cen and 47 Tuc, and did not find any close planets, but did find stellar companions (Weldrake et al. 2007a). The null detection of planets in 47 Tuc (Gilliland et al. 2000) does not contradict the binary model (Soker & Hadar 2001). The reason is that this GC has only few stars on its blue HB. If there were many planets in this GC, then they would be swallowed by the red giant branch (RGB) star progenitor of the HB star, causing high mass loss rate (Livio & Soker 2002) and the formation of many BHB stars. The null detection of planets in the GCs ω Cen (Weldrake et al. 2007b) can be com- patible with the planet binary model for the second parameter if the planets don’t reside in close orbits. Namely, orbital periods & 10 days. For that, Soker & Harpaz (2007) predicted-conjectured that planet companions might exist in GCs, but at orbital separations of 0.3 AU . a . 3 AU. Planetary systems in the upper orbital separation range will be destroyed in GCs, but those in the lower range can survive (Fregeau et al. 2006). – 3 – Soker & Harpaz (2007) only based their prediction on the requirement of their model, but did not bring any observations to support this conjecture. Our main goal is to further explore the binary model for the second parameter, and in particular to examine the possible role of planets. For that, we turn to examine what can be learned from the wealth of data acquired in the field of exoplants. We use recent results from the field of exoplants to support the claim that if planets exist in GCs, they are at larger orbital separations than planets around stars close to the sun. Unlike planets, low mass MS companions can reside close to the parent star in GCs (Weldrake et al. 2007a), and be much more significant in influencing the evolution of their parent star. Namely, the low mass binary second parameter model in low metallicity GCs must be based mainly on low mass main sequence stellar companions, but with contribution of massive planets as well. 2. PLANET COMPANIONS To support the conjecture that if planets exist in GCs they don’t reside at small orbital separations we present several correlations between properties of known extra solar planets. We start by presenting the well known distribution of planets by their orbital period P (Figure 1), but motivated by the recent results of Grether & Lineweaver (2007) we do so separately for three ranges of metallicity of the parent stars: [Fe/H] < −0.1 (black), −0.1 ≤ [Fe/H] ≤ 0.1 (gray), and 0.1 < [Fe/H] (white). Here and in the rest of the diagrams in the paper, each bin shows the number of planets with a period (or other relevant quantities) greater than the number to the left of the bin and smaller than the number to the right of the bin. The leftmost bin shows the number of planets with a period smaller than the number to the right of the bin. The rightmost bin shows the number of planets with a period greater than the number to the left of the bin. All planets data used here are from the Extrasolar Planets Encyclopaedia maintained by Schneider (2007, and references therein), as of June 1, 2007. We skip the comparison of the planet hosting star metallicity distribution with that of other field stars (see, e.g., Santos et al. 2001). There is a population gap in this histogram, i.e., the planets are concentrated in two regions. The gap exists only for the high and medium metallicity systems, as marked by the two thick horizontal arrows in Figure 1. The long period region is limited from above by selection effects. This gap is well known, e.g., Udry et al. (2003) noted the shortage of planets in the range 0.06 AU ≤ a ≤ 0.6 AU. We seek a better quantity to distinguish between close and wide planets, and between the metallicity ranges. For that, and motivated by known correlations between planets’ mass and period (e.g., Mazeh et al. 2005), we plot in Figure 2 planets in the Mp − a plane, where – 4 – Log[ Period/Day ] Period/Day 1 2 4 7 10 20 40 70 10 [Fe/H] < -0.1 -0.1 [Fe/H] 0.1 0.1 < [Fe/H] Period Fig. 1.— Histogram of the number of planets as a function of the orbital period P in days, for three ranges of metallicity as indicated. Each bin shows the number of planets with a period greater than the number to the left of the bin and smaller than the number to the right of the bin. The leftmost bin shows the number of planets with a period smaller than the number to the right of the bin. The rightmost bin shows the the number of planets with a period greater than the number to the left of the bin. The horizontal thick arrows mark the gaps in the respective two high metallicity ranges. Data from the Extrasolar Planets Encyclopaedia maintained by Schneider (2007, June 1, and references therein). – 5 – Mp is the minimum planet mass, used here in units of Jupiter mass MJ , and a is the orbital semi-major axis, used here in units of AU. Filled and empty circles are for systems where the host star metallicity is below and above solar metallicity, respectively. We note that there is a morphological structure along a few lines of α = constant. (1) Two lines are marked by their α value on Figure 2. These lines bound a low-populated stripe between them, and emphasize two populated clumps: one consists of planets having high masses and large orbital separations, and the other consists of planets having low masses and small orbital separations. Based on this, we take α = 2 as our standard value to further analyze the correlations. In Figures 3-9 we show the distribution of the entire sample as a function of the following quantities: Mpa 2, Mpa Ie ≡ Mp[a(1− e)] 2, (2) [a(1 − e)]2, Mp[a(1 − e)] 3, [a(1 − e)]2/Mp, and [a(1 + e)] 2/Mp, respectively. Planets with unknown eccentricity were calculated with e = 0. The quantity Ie has the dimension of moment of inertia, and might therefore indicate the importance of some kind of interaction between the planet and the parent star, as will be discussed in section 4.3. As the strongest interaction occurs near periastron, the relevant distance is a(1− e) rather than a alone. From Figures 1-9 we learn the following. 1. As is well known (e.g., Udry et al. 2003; Marcy et al. 2005) there is a concentration of planets at very short periods of days. Then there is a low-populated range, the gap, and a rise to a second grouping of planets at hundreds to thousands of days. The gap exists only at the two higher metallicity ranges, as marked on Figure 1 by the horizontal thick arrows. 2. At lower metallicities planets tend to have longer orbital periods. The ratio of planets with P > 100 day to planets with P < 100 day is 28/16 = 1.75 and 31/17 = 1.8 for [Fe/H] < −0.1 and −0.1 ≤ [Fe/H] ≤ 0.1, respectively, while it is only 64/53 = 1.2 for 0.1 < [Fe/H]. This trend was found also by Marchi (2007) in the C2 and C3 sub samples defined there. This trend will have to be checked with much larger samples in the future. 3. We find that the double-peak distribution of planets at high metallicity is more pro- nounced when instead of the period other quantities are used, e.g., Mpa 2 or [a(1 − e)]2/Mp, or [a(1 + e)] 2/Mp, and even more so when the quantity Ie = Mp[a(1 − e)] used. – 6 – Log[ a/AU ] -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1 2 4 7 10 [Fe/H] < 0 [Fe/H] > 0 1.7α = 2.36α = Fig. 2.— The distribution of all known planets (from The Extrasolar Planets Encyclopaedia) in the minimum mass (in MJ)−semi-major axis (in AU) plane. Filled and empty circles are for host stars metallicity below and above solar, respectively. Two lines are drawn to show morphological features in the distribution, with the value of α marked (eq. 1). The morphological feature we refer to is a low-populated stripe between the two lines, and two populated clumps, one above the upper line and one below the lower line. – 7 – Log[ (M p/MJ)(a/AU) .5 0. [Fe/H] < -0.1 -0.1 [Fe/H] 0.1 0.1 < [Fe/H] Fig. 3.— Histogram of the number of planets as a function of Mpa 2 (in MJ AU 2) for the three metallicity ranges. – 8 – Log[ (M p/MJ)(a/AU) 1/2 ] [Fe/H] < -0.1 -0.1 [Fe/H] 0.1 0.1 < [Fe/H] Fig. 4.— Histogram of the number of planets as a function of Mpa 1/2 (in MJ AU 1/2) for the three metallicity ranges. – 9 – Log[ (Mp/MJ)([a/AU][1-e]) .5 0. [Fe/H] < -0.1 -0.1 [Fe/H] 0.1 0.1 < [Fe/H] ( )[ ] M a e− Fig. 5.— Histogram of the number of planets as a function of Ie = Mp[a(1−e)] 2 (inMJ AU for the three metallicity ranges. – 10 – Log[ ([a/AU][1-e]) .5 0. [Fe/H] < -0.1 -0.1 [Fe/H] 0.1 0.1 < [Fe/H] ( )[ ] 1a e− Fig. 6.— Histogram of the number of planets as a function of [a(1 − e)]2 (in AU2) for the three metallicity ranges. – 11 – Log[ (Mp/MJ)([a/AU][1-e]) .5 0. [Fe/H] < -0.1 -0.1 [Fe/H] 0.1 0.1 < [Fe/H] ( )[ ] M a e− Fig. 7.— Histogram of the number of planets as a function of Mp[a(1 − e)] 3 (in MJ AU for the three metallicity ranges. – 12 – Log[ ([a/AU][1-e])2/(Mp/MJ) ] [Fe/H] < -0.1 -0.1 [Fe/H] 0.1 0.1 < [Fe/H] ( )[ ] a e M− Fig. 8.— Histogram of the number of planets as a function of [a(1− e)]2/Mp (in AU 2 M−1 for the three metallicity ranges. – 13 – Log[ ([a/AU][1+e])2/(Mp/MJ) ] [Fe/H] < -0.1 -0.1 [Fe/H] 0.1 0.1 < [Fe/H] ≤ ≤( )[ ] a e M+ Fig. 9.— Histogram of the number of planets as a function of [a(1 + e)]2/Mp (in AU 2 M−1 for the three metallicity ranges. – 14 – 4. From the different parameters we have tried, Ie = Mp[a(1 − e)] 2 shows most clearly the two groups of planets at high metallicity, and the differences between planets with different metallicity of their parent star. The criteria we use to prefer the parameter Ie are (a) a smooth variation in the peaks of the histogram, i.e., small fluctuations in peaks (e.g., the peaks of the graph of [a(1 − e)]2/Mp in Fig. 8 have larger fluctuations than the peaks of Ie); (b) a sharp jump between the group of planets with large separations and the deep gap for the two high metallicity ranges (e.g., for Mpa 2, presented in Fig. 3, the difference between the peaks and gap is not as sharp as for Ie presented in Fig. 5); (c) A wide gap (e.g., for Mpa 1/2 presented in Fig. 4 the gap is very narrow); and (d) A clear different behavior of the low metallicity range and the two high metallicity ranges. In particular, when using Ie such a jump does not exist for the low metallicity range. For the highest metallicity range used here there is a large jump at Ie ≃ 0.3MJ AU 2 (log Ie = −0.5), which clearly separates two Ie-populations of planets. While in high metallicity systems there are two well defined populations of planets, in low metallicity systems there is only one peak: Planets of low metallicity stars have typically larger orbital separations. Because Mp is the minimum mass, in the histogram showing [a(1 − e)]2/Mp (Fig. 8) of the real distribution, systems will move to the left smearing the peak of planets in the lowest metallicity range. In the histogram using Ie (Fig. 5), on the other hand, using the real mass will move systems to the right. This might make the peak on the right for the lowest metallicity range more pronounced. 5. We have tried to use the same quantities listed above with 1 + e instead of 1− e. For most cases the separation between close and wide planets is worse than when 1 − e is used. This implies that the periastron is physically more influential than the apastron for defining close and wide planets. However, for the quantity [a(1 + e)]2/Mp shown in figure 9 a partition to two groups is evident. Still, we regard Ie to be the best indicator of the two planet populations. Burkert & Ida (2007) find that a gap in the semimajor axis distribution does not exist if only a sample of systems with hosting stellar mass of M < 1.2M⊙ is used. We find that when using only hosting stars with masses of M < 1.2M⊙ a gap does exist in the Ie distribution for the high metallicity range. There is a gap in the medium metallicity range as well. We also find that ∼ 70% of the hosting stars with masses of M > 1.2M⊙ are in our high metallicity range. The planets in lower metallicity systems with stellar mass of M > 1.2M⊙ tend to have larger values of Ie than planets in systems with stellar mass of M < 1.2M⊙. We therefore propose that both stellar mass (Burkert & Ida 2007) and metallicity are fundamental quantities influencing the distribution of planets around stars. – 15 – Burkert & Ida (2007) also find that the gap in the high hosting stellar mass sample exists for planets with Mp sin i > 0.8M⊙, but not for M sin i ≤ 0.8M⊙. The dependance on the planet mass is included together with the semimajor axis and eccentricity in Ie. We find that the Ie distribution does not have a gap for systems with both Mp > 0.8MJ and M > 1.2M⊙, unlike the semimajor axis distribution found by Burkert & Ida (2007). We do see the gap in the Ie distribution of systems with Mp > 0.8MJ and in the Ie distribution of systems with M < 1.2M⊙. It is not clear if stars in GCs have planets at all, and in particular close-in planets that could be found with the transit technique, as the fraction of detected planetary systems decreases sharply with decreasing metallicity (e.g., Santos et al. 2001; Fischer & Valenti 2005; Grether & Lineweaver 2007). According to Grether & Lineweaver (2007) the most probable value of this fraction for the metallicity range appropriate for GCs is . 0.1%, although the uncertainty is large, and values of ∼ 1% are still possible. Contrary to the general trend is the low metallicity of M-dwarfs (low mass MS stars) hosting planets (Bean et al. 2006), which still leaves hope for planetary systems in GCs. If some planets do exist in GCs, the implications of the results presented here are clear: the planets will not be on short orbital periods. Therefore, we conclude, all transient searches for planets in GCs have a very low detection probability. (At least in the statistical sense, as rare close planets might exist.) However, planets might be detected in metal-rich clusters. The open cluster NGC 6791 has [Fe/H]∼ 0.4, and has a large population of EHB stars and low mass WDs (Kalirai et al. 2007), both of which were formed by stars having increased mass loss on the RGB. Stellar companions are not likely to cause this increased mass loss (Kalirai et la. 2007). We propose that the increased mass loss on the RGB in this cluster is partially caused by planets swallowed by the RGB progenitors of EHB stars and low mass WDs. We therefore predict that many transient events can be detected in this cluster. 3. LOW MASS MAIN SEQUENCE COMPANIONS A similar analysis was conducted as in the previous section but for stellar companions based on the same sample of 135 systems analyzed by Grether & Lineweaver (2006). We present two histograms of the period of the secondary. In figure 10 the sample was divided into three ranges of metallicity as in the previous section. In figure 11 we follow the division of Grether & Lineweaver (2006) and divide the sample into two ranges of color of the parent star as marked on the figure. – 16 – Log[Period/Day] Period/Day 1 2 4 7 10 20 40 70 10 [Fe/H] < -0.1 -0.1 [Fe/H] 0.1 0.1 < [Fe/H] ≤≤Period Fig. 10.— Histogram of the number of stellar companions as a function of the orbital period P in days, for three ranges of metallicity as indicated. Data from Grether & Lineweaver (2006, 2007). – 17 – Log[Period/Day] Period/Day 1 2 4 7 10 20 40 70 10 B-V < 0.75 0.75 < B-V Period Fig. 11.— Histogram of the number of Stellar companions as a function of the orbital period P in days, for two ranges of color as indicated. – 18 – From figures 10 and 11 we can deduce the following: 1. As with planets, there are two concentrations of stellar companions: at short and long orbital periods. 2. Contrary to the behavior of planets, this double-distribution is more pronounced in the low and medium metallicity ranges, with longer orbital periods at higher metallicities. The ratio of stellar companions with P > 100 day to P < 100 day is 27/49 = 0.55, 15/27 = 0.56 and 10/7 = 1.43 for [Fe/H] < −0.1, −0.1 ≤ [Fe/H] ≤ 0.1 and 0.1 < [Fe/H], respectively. 3. Redder systems tend to have a slightly shorter orbital periods. The ratio of stellar companions with P > 100 to P < 100 day is 34/50 = 0.68 and 18/33 = 0.55 for B− V < 0.75 and B− V > 0.75, respectively. 4. The width of the gap between long and short orbital periods is ∼ 70 (from ∼ 30 to ∼ 100 day) and ∼ 460 day (from ∼ 100 to ∼ 550 day) for [Fe/H] < −0.1 and −0.1 ≤ [Fe/H] ≤ 0.1, respectively. The center of the gap is located at P ≈ 56 and P ≈ 247 day for [Fe/H] < −0.1 and −0.1 ≤ [Fe/H] ≤ 0.1, respectively. We have tried to use a finer classification of metallicity and color ranges, namely [Fe/H] < −0.3, −0.3 < [Fe/H] < −0.1, −0.1 ≤ [Fe/H] ≤ 0.1, and 0.1 < [Fe/H], for metal- licity, and B− V < 0.6, 0.6 < B− V < 0.7, 0.7 < B− V < 0.8, and 0.8 < B−V for the color (data not shown). These did not add new information. We find that the ratio between the number of systems with P < 100 days to systems with P > 100 days is 49/27 = 1.8, 33/15 = 2.2, and 28/10 = 2.8, for [Fe/H] < −0.1, [Fe/H] < −0.3, and [Fe/H] < −0.4, respectively. This shows that the tendency of low metallicity systems to harbor short period companions is robust. Although we find that low metallicity stars tend to be slightly closer (shorter orbital periods) than higher metallicity systems, this does not automatically imply the same trend to low mass systems. Maxted & Jeffries (2005) find that a large fraction of very low mass stars seem to be in binary systems, but not very close ones. In addition, a large fraction of stellar companions to low mass stars can have very low mass M2 < 0.3M⊙ (Mazeh et al. 2003), and low mass stellar companions tend to be at large orbital separation (Grether & Lineweaver 2007). – 19 – 4. DISCUSSION AND SUMMARY 4.1. Main Results As is well known and can be seen in figures 1-9, there are two regions highly-populated with planets, with a low-populated gap between them. What we have found here (sections 2 and 3) is the following. 1. In planets the partition to two groups is more significant for high metallicity systems. 2. From the different quantities we have tried, the quantity Ie ≡ Mp[a(1 − e)] 2 both distinguishes between high and low metallicity systems, and shows best this partition to two planet populations in the high metallicity range. 3. In high metallicity systems planets tend to reside on an average closer orbital periods. We note that these two properties depend also on the hosting stellar mass (Burkert & Ida 2007). 4. This trend for metallicity dependance is opposite for stellar companions (section 3). 5. This trend for stellar companions is mainly due to metallicity and not to the parent star’s mass. There is only a small difference in the ratio of stellar companions with P > 100 to P < 100 day for the two color ranges used. We note that there are other properties for which stars with planetary systems and with stellar companions show opposite behavior. The most important is the finding that as metallicity decreases the star is much more likely to have a stellar companion than to harbor a planetary system (Grether & Lineweaver 2007). Our results have implications for two areas. 4.2. Implications for Globular Clusters Since in galactic GCs the metallicity is very low, close-in planets are not expected there. However, if planets do exist around some stars in GCs, they will most likely have large orbital separations. Therefore, the transit search for planets in GCs has a very low detection probability, and the non-detection of close-in planets should not be considered as evidence against the presence of planets in GCs. We should stay open to the possibility that wide (large orbital periods) planetary systems exist in GCs. – 20 – The large fraction of low metallicity stars that have stellar companions (Grether & Lineweaver 2007) implies that a large fraction of stars in GCs might have formed with stellar companions around them. It seems that binary systems are indeed common in GCs (Leigh et al. 2007). Not only metallicity, but other properties at the formation epoch of globular clusters (Soker & Hadar 2001) might determine the presence of companions and planets. For example, density in star forming regions can determine stellar rotation (Wolff et al. 2007). Put together, our results support the low mass binary second parameter model, but the companions in low metallicity star clusters are more likely to be very low mass stellar companions, as observed by, e.g., Weldrake et al. (2007a), rather than massive planets. In that model a low mass companion (a very low mass main sequence star, a brown dwarf, or a massive planet) influences the post-main sequence evolution of stars, in particular the properties of the parent stars on the horizontal branch. (Soker 1998; Soker & Harpaz 2007). In high metallicity clusters, such as NGC 6791 (Kalirai et al. 2007), planets might be more important than stellar companions in forming EHB stars and low mass (undermassive) white dwarfs. We predict that many transient events can be detected in this cluster. 4.3. Implications for Planetary Systems This topic is beyond our scope. However, our findings suggest that the migration of planets from large to small orbital separation depends on a combination of parameters ex- pressed by Ie (eq. 2). This quantity has the dimension of moment of inertia, and may imply that a process reminiscent of the Darwin instability (e.g., Eggleton 2006) is in operation. In particular, the strong dependance on eccentricity, in that 1− e is a much better factor than 1 + e in showing the two planet populations, suggests that some sort of tidal interaction is operating. Although the situation cannot be simple, let us try the following. The left limit of the right populated area in figure 5 is Ie ≃ 0.3. Let us substitute this in the condition for the Darwin instability to occur (Eggleton 2006, sec. 4.2) > λcrit, (3) where M1 is the stellar mass, R1 is the stellar radius, kR1 the radius of gyration with k2 ≃ 0.05 − 0.1 for main sequence stars, Ω is the stellar angular velocity, and ω = 2π/P is the mean orbital angular velocity. The critical value λcrit rapidly decreases with increasing eccentricity, with λcrit = 1/3, 0.171, 0.102, and 0.052, for e = 0, 0.3, 0.4, and 0.5, respectively. – 21 – As most planets have e . 0.4, in this range we see that λcrit ≃ (1− e)2 for e . 0.5. (4) Substituting approximation (4) in equation (3), and taking M1 = 1M⊙, k 2 = 0.075, and Ω ≃ ω, the condition for the Darwin instability to occur becomes R1 & 8 0.3MJ AU R⊙ (5) This explanation requires that the planets interact with an inflated pre-main sequence star. This possibility will be studied in a future paper. We thank Charles Lineweaver and Daniel Grether for giving us their data on stellar companions. We thank David Weldrake for useful comments, and an anonymous referee for comments that improved the presentation of our results. This research was supported by a grant from the Asher Space Research Institute at the Technion. REFERENCES Bean, J. L., Benedict, G. 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E., Dror, D. & Venn, K. 2007, AJ, 133, 1092 Zucker, S., & Mazeh, T. 2000, ApJ, 531, L67 This preprint was prepared with the AAS LATEX macros v5.2. http://arxiv.org/abs/astro-ph/0702349 http://arxiv.org/abs/0705.0910 http://exoplanet.eu/ http://arxiv.org/abs/astro-ph/0703753 http://arxiv.org/abs/astro-ph/0612215 INTRODUCTION PLANET COMPANIONS LOW MASS MAIN SEQUENCE COMPANIONS DISCUSSION AND SUMMARY Main Results Implications for Globular Clusters Implications for Planetary Systems

0704.1068

Fast paths in large-scale dynamic road networks

Fast paths in large-scale dynamic road networks Giacomo Nannicini1,2, Philippe Baptiste1, Gilles Barbier2, Daniel Krob1, Leo Liberti1 LIX, École Polytechnique, F-91128 Palaiseau, France Email:{giacomon,baptiste,dk,liberti}@lix.polytechnique.fr Mediamobile, 10 rue d’Oradour sur Glane, Paris, France Email:{giacomo.nannicini,gilles.barbier,contact}@v-trafic.com October 22, 2018 Abstract Efficiently computing fast paths in large-scale dynamic road networks (where dynamic traffic information is known over a part of the network) is a practical problem faced by several traffic information service providers who wish to offer a realistic fast path computation to GPS terminal enabled vehicles. The heuristic solution method we propose is based on a highway hierarchy-based shortest path algorithm for static large-scale networks; we maintain a static highway hierarchy and perform each query on the dynamically evaluated network. 1 Introduction Several cars are now fitted with a Global Positioning System (GPS) terminal which gives the exact geographic location of the vehicle on the surface of the earth. All of these GPS terminals are now endowed with detailed road network databases which allow them to compute the shortest path (in terms of distance) between the current vehicle location (source) and another location given by the driver (destination). Naturally, drivers are more interested in the source-destination fastest path (i.e. shortest in terms of travelling time). The greatest difficulty to overcome is that the travelling time depends heavily on the amount of traffic on the chosen road. Currently, some state agencies as well as commercial enterprises are charged with monitoring the traffic situation in certain pre-determined strategic places. Furthermore, traffic reports are collected from police cars as well as some taxi services. The dynamic traffic information, however, is as yet limited to a small proportion of the whole road network. The problem faced by traffic information providers is currently that of offering GPS terminal enabled drivers a source-destination path subject to the following constraints: (a) the path should be fast in terms of travelling time subject to dynamic traffic information being available on part of the road network; (b) traffic information data are updated approximately each minute; (c) answers to path queries should be computed in real time. Given the data communication time and other overheads, constraint (c) practically asks for a shortest path computation time of no more than 1 second. Constraint (b) poses a serious problem, because it implies that the fastest source-destination path may change each minute, giving an on-line dimension to the problem. A source-destination query spanning several hundred kilometers, which would take several hours to travel, would need a system recomputing the fastest path each minute; this in turn would mean keeping track of each query for potentially several hours. As the estimated computational cost of this requirement is superior to the resources usually devoted to the task, a system based on dynamic traffic information will not, in practice, ever compute the on-line fastest path. As a typical national road network for a large European country usually counts several million junctions and road segments, constraint (c) implies that a straight Dijkstra’s algorithm is not a viable option. In view of constraint (a), in our solution method fast paths can be efficiently computed by means of a point-to-point hierarchy-based shortest path algorithm for static large-scale networks, where the hierarchy is built using static information and each query is answered on the dynamically evaluated network. http://arxiv.org/abs/0704.1068v2 {giacomon,baptiste,dk,liberti}@lix.polytechnique.fr {giacomo.nannicini,gilles.barbier,contact}@v-trafic.com 1 INTRODUCTION 2 This paper makes two original scientific contributions (i) We extend a known hierarchy-based shortest path algorithm for static large-scale undirected graphs (the Highway Hierarchies algorithm [SS05]) to the directed case. The method has been developed and tested on real road network data taken from the TeleAtlas France database [NV05]. We note that the original authors of [SS05] have extended the algorithm to work on directed graphs in a slightly different way than ours (see [SS06]). (ii) We propose a method for efficiently finding fast paths on a large-scale dynamic road network where arc travelling times are updated in quasi real-time (meaning very often but not continuously). In the rest of this section, we discuss the state of the art as regards shortest path algorithms in dynamic and large-scale networks, and we describe the proposed solution. The rest of the paper is organized as follows. In Section 2 we briefly review the highway hierarchy-based shortest path algorithm for static large- scale networks, which is one of the important building blocks of our method, and discuss the extension of the existing shortest-path algorithm to the directed case. Section 3 discusses the computational results, and Section 4 concludes the paper. 1.1 Shortest path algorithms in road networks The problem of computing fastest paths in graphs whose arc weights change over time is termed the Dynamic Shortest Path Problem (DSPP) [BRTed]. The work that laid the foundations for solving the DSPP is [CH66] (a good review of this paper can be found in [Dre69], p. 407): Dijkstra’s algorithm is extended to the dynamic case through a recursion formula based on the assumption that the network G = (V,A) has the FIFO property: for each pair of time instants t, t′ with t < t′: ∀ (u, v) ∈ A τuv(t) + t ≤ τuv(t ′) + t′, where τuv(t) is the travelling time on the arc (u, v) starting from u at time t. The FIFO property is also called the non-overtaking property, because it basically says that if A leaves u at time t and B at time t′ > t, B cannot arrive at v before A using the arc (u, v). The shortest path problem in dynamic FIFO networks is therefore polynomially solvable [Cha98], even in the presence of traffic lights [AOPS03]. Dijkstra’s algorithm applied to dynamic FIFO networks has been optimized in various ways [BRTed, Cha98]; the A∗ one-to-one shortest path algorithm has also been extended to dynamic networks [CS02]. The DSPP is NP-hard in non-FIFO networks [Dea04]. Although in this paper we do not assume any knowledge about the statistical distribution of the arc weights in time, it is worth mentioning that a considerable amount of work has been carried out for computing shortest paths in stochastic networks. A good review is [FHK+05]. The computation of exact shortest paths in large-scale static networks has received a good deal of attention [CZ01]. The established practice is to delegate a considerable amount of computation to a preprocessing phase (which may be very slow) and then perform fast source-destination shortest path queries on the pre-processed data. Recently, the concept of highway hierarchy was proposed in [SS05, Sch05, SS06]. A highway hierarchy of L levels of a graph G = (V,A) is a sequence of graphs G = G0, . . . , GL with vertex sets V 0 = V, V 1 ⊇ . . . ⊇ V L and arc sets A0 = A,A1 ⊇ . . . ⊇ AL; each arc has maximum hierarchy level (the maximum i such that it belongs to Ai) such that for all pairs of vertices there exists between them a shortest path (a1, . . . , ak), where ai are the consecutive path arcs, whose search level first increases and then decreases, and each arc’s search level is not greater than its maximum hierarchy level. A more precise description is given in Section 2. The A∗ algorithm has also been extended to use a concept, reach, which has turned out to be closely related to highway hierarchies (see [GKW05]). 1.2 Description of the solution method The solution method we propose in this paper efficiently finds fast paths by deploying Dijkstra-like queries on a highway hierarchy built using the static arc weights found in the road network database, 2 HIGHWAY HIERARCHIES ALGORITHM ON DYNAMIC DIRECTED GRAPHS 3 but used with the dynamic arc weights reflecting quasi real-time traffic observations. This implies using two main building blocks: highway hierarchy construction (the Highway Hierarchies1 algorithm extended to directed graphs), and the query algorithm. Consequently, the implementation is a complex piece of software whose architecture has been detailed in the appendix. • Highway hierarchy. Apply the directed graph extension of the HH algorithm (see Section 2) to construct a highway hierarchy using the static road network information. In particular, arc travelling times are average estimations found in the database. This is a preprocessing step that has to be performed only when the topology of the road network changes. The CPU time taken for this step is not an issue. • Efficient path queries. Efficiently address source-destination fast path requests by employing a multi-level bidirectional Dijkstra’s algorithm on the dynamically evaluated graph using the highway hierarchy structure constructed during preprocessing. This algorithm is carried out each time a path request is issued; its running time must be as fast as possible, in any case not over 1 second. 2 Highway Hierarchies algorithm on dynamic directed graphs The Highway Hierarchies algorithm [SS05, Sch05] is a fast, hierarchy-based shortest paths algorithm which works on static undirected graphs. HH algorithm is specially suited to efficiently finding shortest paths in large-scale networks. Since the HH algorithm is one of our main building blocks, we briefly review the necessary concepts. The Highway Hierarchies algorithm is heavily based on Dijkstra’s algorithm [Dij59], which finds the tree of all shortest paths from a root vertex r to all other vertices v ∈ V of a weighted digraph G = (V,A) by maintaining a heap H of reached vertices u with their associated (current) shortest path length c(u) (elements of the heap are denoted by [u, c(u)]. Vertices which have not yet entered the heap (i.e. which are still unvisited) are unreached, and vertices which have already exited the heap (i.e. for which a shortest path has already been found) are settled. Dijkstra’s algorithm is as follows. 1. Let H = {[r, 0]}. 2. If H = ∅, terminate. 3. Let u be the vertex in H with minimum associated path length c(u). 4. Let H = H r {u}. 5. For all v ∈ δ+(u), if c(u) + τuv < c(v) then let H = (H r {[v, c(v)]}) ∪ {[v, c(u) + τuv]}. 6. Go to 2. A bidirectional Dijkstra algorithm works by keeping track of two Dijkstra search scopes: one from the source, and one from the destination working on the reverse graph. When the two search scopes meet it can be shown that the shortest path passes through a vertex that has been reached from both nodes ([Sch05], p. 30). A set of shortest paths is canonical2 if, for any shortest path p = 〈u1, . . . , ui, . . . , uj . . . , uk〉 in the set, the canonical shortest path between ui and uj is a subpath of p. The HH algorithm works in two stages: a time-consuming pre-processing stage to be carried out only once, and a fast query stage to be executed at each shortest path request. Let G0 = G. During the first stage, a highway hierarchy is constructed, where each hierarchy level Gl, for l ≤ L, is a modified subgraph 1From now on, simply HH 2Dijkstra’s algorithm can easily be modified to output a canonical shortest paths tree (see [Sch05], Appendix A.1 — can be downloaded from http://algo2.iti.uka.de/schultes/hwy/). http://algo2.iti.uka.de/schultes/hwy/ 2 HIGHWAY HIERARCHIES ALGORITHM ON DYNAMIC DIRECTED GRAPHS 4 of the previous level graph Gl−1 such that no canonical shortest path in Gl−1 lies entirely outside the current level for all sufficiently distant path endpoints: this ensures that all queries between far endpoints on level l−1 are mostly carried out on level l, which is smaller, thus speeding up the search. Each shortest path query is executed by a multi-level bidirectional Dijkstra algorithm: two searches are started from the source and from the destination, and the query is completed shortly after the search scopes have met; at no time do the search scopes decrease hierarchical level. Intuitively, path optimality is due to the fact that by hierarchy construction there exist no canonical shortest path of the form 〈a1, . . . , ai, . . . , aj . . . , ak . . .〉, where ai, aj , ak ∈ A and the search level of aj is lower than the level of both ai, ak; besides, each arc’s search level is always lower or equal to that arc’s maximum level, which is computed during the hierarchy construction phase and is equal to the maximum level l such that the arc belongs to Gl. The speed of the query is due to the fact that the search scopes occur mostly on a high hierarchy level, with fewer arcs and nodes than in the original graph. 2.1 Highway hierarchy As the construction of the highway hierarchy is the most complicated part of HH algorithm, we endeavour to explain its main traits in more detail. Given a local extensionality parameter H (which measures the degree at which shortest path queries are satisfied without stepping up hierarchical levels) and the maximum number of hierarchy levels L, the iterative method to build the next highway level l+1 starting from a given level graph Gl is as follows: 1. For each v ∈ V , build the neighbourhoodN lH(v) of all vertices reached from v with a simple Dijkstra search in the l-th level graph up to and including the H-st settled vertex. This defines the local extensionality of each vertex, i.e. the extent to which the query “stays on level l”. 2. For each v ∈ V : (a) Build a partial shortest path tree B(v) from v, assigning a status to each vertex. The initial status for v is “active”. The vertex status is inherited from the parent vertex whenever a vertex is reached or settled. A vertex w which is settled on the shortest path 〈v, u, . . . , w〉 (where v 6= u 6= w) becomes “passive” if |N lH(u) ∩N H(w)| ≤ 1. (1) The partial shortest path tree is complete when there are no more active reached but unsettled vertices left. (b) From each leaf t of B(v), iterate backwards along the branch from t to v: all arcs (u,w) such that u 6∈ N lH(t) and w 6∈ N H(v), as well as their adjacent vertices u,w, are raised to the next hierarchy level l + 1. 3. Select a set of bypassable nodes on level l + 1; intuitively, these nodes have low degree, so that the benefit of skipping them during a search outweights the drawbacks (i.e., the fact that we have to add shortcuts to preserve the algorithm’s correctness). Specifically, for a given set Bl+1 ⊂ Vl+1 of bypassable nodes, we define the set Sl+1 of shortcut edges that bypass the nodes in Bl+1: for each path p = (s, b1, b2, . . . , bk, t) with s, t ∈ Vl+1 r Bl and bi ∈ Bl+1, 1 ≤ i ≤ k, the set Sl+1 contains an edge (s, t) with c(s, t) = c(p). The core G′l+1 = (V l+1, E l+1) of level l + 1 is defined as:V ′ = Vl+1 rBl+1, E = (El+1 ∩ (V × V ′ )) ∪ Sl+1. The result of the contraction is the contracted highway network G′l+1, which can be used as input for the following iteration of the construction procedure. It is worth noting that higher level graphs may be disconnected even though the original graph is connected. 2.1 Example Take the directed graph G = (V,A) given in Fig. 1 (above). We are going to construct a road hierarchy with H = 3 and L = 1 on G. First we compute N0 (v) for all v ∈ V = {v0, . . . , v6}. 2 HIGHWAY HIERARCHIES ALGORITHM ON DYNAMIC DIRECTED GRAPHS 5 v0, v1, v2 {0, 1, 2} v3, v4, v5 {3, 4, 5} v6 {3, 5, 6} Next, we compute B(v) for all v ∈ V and raise the hierarchy level of the relevant arcs from the leaves to B(v) to v. We only discuss the computation of B(v0) in detail as the others are similar. 1. Vertex v0 is initialized as an active vertex. 2. Dijkstra’s algorithm is started. (a) v0 is settled (cost 0) on the empty path, so the passivity condition (1) does not apply; (b) v1 and v2 are reached from v0 with costs resp. 1 and 2, and inherit its active status; (c) v1 is settled (cost 1) on the path 〈v0, v1〉 and condition (1) does not apply; (d) v6 is reached from v1 with cost 1 + 4 = 5 and set to active; (e) v2 (cost 2) is settled on 〈v0, v2〉; (f) v4 is reached from 2 with cost 2 + 6 = 8 and set to active; (g) v6 (cost 5) is settled on the path 〈v0, v1, v6〉: since N (v1) ∩ N (v6) = ∅, condition (1) is verified, and v6 is labeled passive; (h) v3 is reached from v6 with cost 1 + 4 + 4 = 9 and set to passive. (i) v4 (cost 8) is settled on the path 〈v0, v2, v4〉: since N 3 (v2) ∩ N 3 (v4) = ∅, condition (1) is verified, and v4 is labeled passive; (j) v5 is reached from v4 with cost 2 + 6 + 2 = 10 and set to passive; (k) the only unsettled vertices are v3 and v5. Since both are reached and passive, the search terminates. 3. The leaf vertices of B(v0) are v4 and v6. (a) From t = v4, we iterate backwards along the arcs on the path 〈v0, v2, v4〉: the arc (v2, v4) has the property that v2 6∈ N (v4) and v4 6∈ N (v2), so its hierarchy level is raised to l + 1 = 1 (the other arc on the path, (v0, v2), stays at level l = 0); (b) from t = v6, we iterate backwards along the arcs on the path 〈v0, v1, v6〉: the arc (v1, v6) has the property that v1 6∈ N 3 (v6) and v6 6∈ N 3 (v1), so its hierarchy level is raised to 1 (the other arc on the path stays at level 0). Fig. 1 shows the hierarchy at level 1. 2.2 Extension to directed graphs The original description of the HH algorithm [SS05] applies to undirected graphs only; in this section we provide an extension to the directed case. It should be noted that the HH algorithm was extended to the directed case by the authors (see [SS06]) in a way which is very similar to that described here. However, we believe our slightly different exposition helps to clarify these ideas considerably. The algorithm for hierarchy construction, as explained in Section 2.1, works with both undirected and directed graphs. However, storing all neighbourhoods N lH(v) for each v and l has prohibitive memory requirements. Thus, the original HH implementation for checking whether a vertex v is in N lH(u) is based on comparing the distance d(u, v) with the “distance-to-border” (also called slack) from u to the border of its neighbourhood N lH(u). The “distance-to-border” d H(u) is a measure of a neighbourhood’s 2 HIGHWAY HIERARCHIES ALGORITHM ON DYNAMIC DIRECTED GRAPHS 6 Figure 1: The graph of Example 2.1 (left) and its highway hierarchy for H = 3, L = 1 (right): the dashed lines indicate arcs at level 0, the solid lines indicate arcs at level 1. radius, and is defined as the distance d(u, v) where v is the farthest node in N lH(u), i.e. the cost of the shortest path from u to the H-th settled node when applying Dijkstra’s algorithm on node u at level l. This is the basis of the slack-based method in [Sch05], p. 19 (from which we draw our notation). In the partial shortest paths tree B(s0) computed in Step 2a of the algorithm in Section 2.1, the slack ∆(u) is recursively computed for all u ∈ B(s0) starting from the leaves t0 of B(s0), as follows. 1. Initialise a FIFO queue Q to contain all nodes u of B(s0), ordering them from the farthest one to the nearest one with respect to s0. 2. Set ∆(u) = dlH(u) for u a leaf of B(s0) and +∞ otherwise. 3. If Q is empty, terminate. 4. Remove u from Q, and let p be its predecessor in B(s0). 5. If ∆(p) = +∞ and p 6∈ N lH(s0), p is added to Q. 6. Let ∆(p) = min(∆(p),∆(u)− d(p, u)). 7. If ∆(p) < 0, the edge (p, u) is lifted to the higher hierarchical level. 8. Return to Step 3. The algorithm works because Thm. 2 in [Sch05] proves that condition ∆(p) < 0 is equivalent to the condition of Step 2b of the algorithm in Section 2.1. The cited theorem is based on the following assumption: ∀u ∈ V (u 6∈ N lH(t0) → d H(t0)− d(u, t0) < 0). (2) This condition may fail to hold for directed graphs, since d(u, t0) 6= d(t0, u). To make Assumption 2 hold, we have to consider a neighbourhood radius computed on the reverse graph, that is the graph G = (V,A) such that (u, v) ∈ A ⇔ (v, u) ∈ A. Thus, we modified the original implementation to compute, for each node, a reverse neighbourhood N H(v) (see Figure 2), so that we are able to store the corresponding reverse neighbourhood radius d H(u)∀u ∈ V . We replace Step 2 in the algorithm above by: 2a. Set ∆(u) = d H(u) for u a leaf of B(s0) and +∞ otherwise. We are now going to prove our key lemma. 3 COMPUTATIONAL RESULTS 7 2.2 Lemma Let u, s ∈ V and t a leaf in B(s). If u 6∈ N H(t) then d H(t)− d(u, t) < 0. Proof. Suppose d(u, t) ≤ d H(t). By definition, this means that there is a shortest path in N H(t) which connects u to t. Therefore, u ∈ N H(t) against the hypothesis. ✷ It is now straightforward to amend Thm. 2 in [Sch05] to hold in the directed case; all other theorems in [Sch05] need similar modifications, replacing N lH(t) with N H(t) and d H(t) with d H(t) whenever t is target node or is “on the right side” of a path - it will always be clear from the context. The query algorithm must me modified to cope with these differences, using d H(t) instead of d H(t) whenever we are searching in the backwards direction. Figure 2: An example which shows neighbourhoods and reverse neighbourhoods with H = 3; only solid arcs are lifted to a higher level in the hierarchy. Note that arcs (p, t) and (p′, t) are not lifted even if p, p′ /∈ N lH(t); this is because p, p ′ ∈ N H(t), and for target node we consider the reverse neighbourhood. Interestingly, the problem with the slack-based method was first detected when our original imple- mentation of the HH algorithm failed to construct a correct hierarchy for the Paris urban area. This shows that the extension of the algorithm to the directed case actually arises from real needs. 2.3 Heuristic application to dynamic networks The original Highway Hierarchies algorithm, as described above, finds shortest paths in networks whose arc weights do not change in time. By forsaking the optimality guarantee, we adapt the algorithm to the case of networks whose arc weights are updated in quasi real-time. Whereas the highway hierarchy is constructed using the static arc travelling times from the road network database, each point-to-point path query is deployed on a dynamically evaluated version of the highway hierarchy where the arcs are weighted using the quasi real-time traffic information. In particular, in all tests that involved a comparison with neighbourhood radius we use the static arc travelling times, while for all evaluations of path lengths or of node distances we use the real-time (dynamic) travelling times. This means that the static travelling times are used to determine neighbourhood’s crossings, and thus to determine when to switch to a higher level in the hierarchy, while the key for the priority queue for HH algorithm is computed using only dynamic travelling times. 3 Computational results In this section we discuss the computational results obtained by our implementation. As there seems to be no other readily available software with equivalent functionality, the computational results are not comparative. However, we establish the quality of the heuristic solutions by comparing them against 3 COMPUTATIONAL RESULTS 8 the fastest paths found by a plain Dijkstra’s algorithm. We mention here two different approaches: dynamic highway-node routing ([SS07]), which uses a selection of nodes operated by the HH algorithm to build an overlay graph (see [HSW06]), and dynamic ALT ([DW07]), which is a dynamic landmark-based implementation of A∗. Both approaches, however, although very performing with respect to query times, require a computationally heavy update phase (which takes time in the order of minutes), and thus are not suitable for our scenario, where, supposedly, each arc can have its cost changed every 2 minutes (roughly). We performed the tests on the entire road network of France, using a highway hierarchy with H = 65 and L = 9. The original network has 7778913 junctions and 17154042 road segments; the number of nodes and arcs in each level is as follows. level 0 1 2 3 4 5 6 7 8 9 nodes 7778913 1517291 433286 182474 91888 53376 34116 23338 16445 11790 arcs 17154042 3461385 1283000 583380 308249 183659 119524 81170 57235 41092 We show the results for queries on the full graph without dynamic travelling times in Table 1; in this case, all paths computed with the HH algorithm are fastest paths. In Table 2, instead, we record our results on a graph with dynamic travelling times; we also report the relative distance of the solution found with our heuristic version of the HH algorithm and the fastest path computed with Dijkstra, and, for comparative reasons, the results of the naive approach which consists in computing the traffic-free optimal solution with the HH algorithm (i.e., on the static graph) and then applying dynamic times on the so-found solution. Dynamic travelling times were taken choosing, for each query, one out of five sets of values recorded in different times of the day for each of the 29384 arcs with dynamic information. Although this number is small with respect to the total number of arcs in the graph, it should be noted that most of these arcs correspond to very important road segments (highways and national roads). All arcs (i, j) that did not have a dynamic travelling time were assigned a different weight at each query, chosen at random with a uniform distribution over [τij , 15τij ], where τij is the reference time for arc (i, j). This choice has been made in order to recreate a difficult scenario for the query algorithm: even if the number of arcs with real traffic information is still small, it is going to increase rapidly as the means for obtaining dynamic information increase (e.g. number of road cameras, etc.), and thus, to simulate an instance where most arcs have their travelling time changed several times per hour, we generated each arc’s cost at random. The interval [τij , 15τij ] is simply a rough estimation of a likely cost interval, based on the analysis of historical data. All tables report average values over 5000 queries. All computational results in Table 1 and 2 have been obtained on a multiprocessor Intel Xeon 2.6 GHz with 8GB RAM running Microsoft Windows Server 2003, compiling with Miscrosoft Visual Studio 2005 and optimization level 2. Computational results show that, although with no guarantee of optimality, our heuristic version of the algorithm works well in practice, with 0.55% average deviation from the optimal solution and a recorded maximum deviation of 17.59%; query times do not seem to be influenced by our changes with respect to the original version of the algorithm. The naive approach of computing the shortest path on the static graph, and then applying dynamic times, records an average error of 2%, but it has a much higher variance, and a maximum error of 27.95%; although the average error is not high, it’s still almost 4 times the average error of the more sofisticated approach, and the high variance suggests lack of stability in the solution’s quality. The low value recorded for the average error with the naive approach (in absolute terms) can be explained as a consequence of the following two facts: travelling times generated at random on arcs without real-time traffic information cannot simulate real traffic situation, because they lack spatial coherence (i.e. they do not simulate congested nearby zones) and traffic behaviour information (i.e. the fact that during peak hours important road segments are likely to be congested, while less important roads are not), thus making the task of finding a fast path easier; besides, the average query on such a large graph corresponds to a very long path (296 minutes on the traffic-free graph, 2356 minutes on the dynamic graph), and on long paths it is usually necessary to use highways 4 CONCLUSION 9 or national roads regardless of their congestion status - which is exactly what the HH algorithm does. This last sentence is supported by the fact that, if we consider only the 500 shortest queries in terms of path length, the average error of the naive approach increases to 3.60%, while the average error of the heuristic version increases to 0.97%; this is in accord with the fact that on short paths the influence of traffic is greater, because alternative routes that do not use highways are more appealing, while on long paths using highways is often a necessary step. However, in relative terms, the heuristic version of the HH algorithm performs significantly better than the naive approach proposed for comparison, and we expect the difference to increase (in favour of the heuristic algorithm) if applied to a graph fully covered with real traffic information. Figure 3 shows how the optimal and the heuristic path may differ; since the hierarchy built on the static graph emphasizes important roads, the heuristic algorithm applied on the dynamically weighted graph still tends to use highways and national roads even when they are congested (up to a certain degree), thus sometimes losing optimality. Dijkstra’s algorithm HH algorithm # settled nodes 2275563 18966 # explored nodes 2587112 36200 query time [sec] 11.830 0.099 Table 1: Computational results on the static graph: average values Dijkstra’s algorithm HH algorithm HH algorithm naive approach heuristic version # settled nodes 2280872 19174 19099 # explored nodes 2594361 36581 36492 query time [sec] 11.917 0.100 0.099 distance from optimum (variance) 0% 2.00% (5.00) 0.55% (0.45) Table 2: Computational results on the graph with dynamic times: average values 4 Conclusion We present a heuristic algorithm for efficiently finding fast paths in large-scale partially dynamically weighted road networks, and benchmark its application on real-world data. The proposed solution is based on fast multi-level bidirectional Dijkstra queries on a highway hierarchy built on the statically weighted version of the network using the Highway Hierarchies algorithm, and deployed using the dynamic arc weights. Computational results show that, although with no guarantee of optimality, the proposed solution works well in practice, computing near-optimal fast paths quickly enough for our purposes. Acknowledgements We are grateful to Ms. Annabel Chevaux, Mr. Benjamin Simon and Mr. Benjamin Becquet for invaluable practical help with Oracle and the real data, and to the rest of the Mediamobile’s energetic and youthful staff for being always friendly and helpful. REFERENCES 10 Figure 3: Fast paths calculated with different algorithms; each number identifies a path, paths are par- tially overlapping. 1: Dijkstra’s algorithm (optimal solution) with real-time arc costs; dynamic travelling time: 24 minutes and 6 seconds. 2: HH algorithm (heuristic solution) with real-time arc costs; dynamic travelling time: 25 minutes and 5 seconds. 3: HH algorithm without real-time arc costs (traffic-free optimal solution); dynamic travelling time: 37 minutes and 5 seconds. References [AOPS03] R.K. Ahuja, J.B. Orlin, S. Pallottino, and M.G. Scutellà. Dynamic shortest paths minimizing travel times and costs. Networks, 41(4):197–205, 2003. [BRTed] L.S. Buriol, M.G.C. Resende, and M. Thorup. Speeding up dynamic shortest path algorithms. INFORMS Journal on Computing, submitted. [CH66] K.L. Cooke and E. Halsey. The shortest route through a network with time-dependent intern- odal transit times. Journal of Mathematical Analysis and Applications, 14:493–498, 1966. [Cha98] I. Chabini. Discrete dynamic shortest path problems in transportation applications: complex- ity and algorithms with optimal run time. Transportation Research Records, 1645:170–175, 1998. [CS02] I. Chabini and L. Shan. Adaptations of the A∗ algorithm for the computation of fastest paths in deterministic discrete-time dynamic networks. IEEE Transactions on Intelligent Transportation Systems, 3(1):60–74, 2002. REFERENCES 11 Figure 4: Highway hierarchy near the Champs Elysées, Paris; colour intensity and line width increase with hierarchy level, in that order (a wide line with a lighter colour has a higher hierarchy level than a thin line with dark colour). [CZ01] E.P.F. Chan and N. Zhang. Finding shortest paths in large network systems. In GIS ’01: Proceedings of the 9th ACM international symposium on Advances in geographic information systems, pages 160–166, New York, NY, USA, 2001. ACM Press. [Dea04] B.C. Dean. Shortest paths in fifo time-dependent networks: theory and algorithms. Technical report, MIT, Cambridge MA, 2004. [Dij59] E.W. Dijkstra. A note on two problems in connexion with graphs. Numerische Mathematik, 1:269–271, 1959. [Dre69] S.E. Dreyfus. An appraisal of some shortest-path algorithms. Operations Research, 17(3):395– 412, 1969. [DW07] D. Delling and D. Wagner. Landmark-based routing in dynamic graphs. InWEA 2007, volume 4525 of Lecture Notes in Computer Science. Springer, 2007. [FHK+05] T. Flatberg, G. Hasle, O. Kloster, E.J. Nilssen, and A. Riise. Dynamic and stochastic aspects in vehicle routing – a literature survey. Technical Report STF90A05413, SINTEF, Oslo, Norway, 2005. [GKW05] A.V. Goldberg, H. Kaplan, and R.F. Werneck. Reach for A∗: Efficient point-to-point shortest path algorithms. Technical Report MSR-TR-2005-132, Microsoft Research, 2005. [HSW06] M. Holzer, F. Schulz, and D. Wagner. Engineering multi-level overlay graphs for shortest-path queries. In SIAM, volume 129 of Lecture Notes in Computer Science, pages 156–170. Springer, 2006. [Ker04] B. S. Kerner. The Physics of Traffic. Springer, Berlin, 2004. [NV05] TeleAtlas NV. Tele Atlas Multinet ShapeFile 4.3.1 Format Specifications. TeleAtlas NV, May 2005. REFERENCES 12 [Sch05] D. Schultes. Fast and exact shortest path queries using highway hierarchies. Master Thesis, Informatik, Universität des Saarlandes, June 2005. [SS05] P. Sanders and D. Schultes. Highway hierarchies hasten exact shortest path queries. In G. Stølting Brodal and S. Leonardi, editors, ESA, volume 3669 of Lecture Notes in Computer Science, pages 568–579. Springer, 2005. [SS06] P. Sanders and D. Schultes. Engineering highway hierarchies. In ESA 2006, volume 4168 of Lecture Notes in Computer Science, pages 804–816. Springer, 2006. [SS07] P. Sanders and D. Schultes. Dynamic highway-node routing. In WEA 2007, volume 4525 of Lecture Notes in Computer Science, pages 66–79. Springer, 2007. Introduction Shortest path algorithms in road networks Description of the solution method Highway Hierarchies algorithm on dynamic directed graphs Highway hierarchy Extension to directed graphs Heuristic application to dynamic networks Computational results Conclusion

0704.1069

Optical Zeno Gate: Bounds for Fault Tolerant Operation

Optical Zeno Gate: Bounds for Fault Tolerant Operation Patrick M. Leung∗ and Timothy C. Ralph Centre for Quantum Computer Technology, Department of Physics, University of Queensland, Brisbane 4072, Australia (Dated: September 8, 2021) In principle the Zeno effect controlled-sign gate of Franson et al’s (PRA 70, 062302, 2004) is a deterministic two-qubit optical gate. However, when realistic values of photon loss are considered its fidelity is significantly reduced. Here we consider the use of measurement based quantum processing techniques to enhance the operation of the Zeno gate. With the help of quantum teleportation, we show that it is possible to achieve a Zeno CNOT gate (GC-Zeno gate) that gives (near) unit fidelity and moderate probability of success of 0.76 with a one-photon to two-photon transmission ratio κ = 104. We include some mode-mismatch effects and estimate the bounds on the mode overlap and κ for which fault tolerant operation would be possible. PACS numbers: 03.67.Lx, 42.50.-p I. INTRODUCTION Photons are a natural choice for making qubits be- cause the quantum information encoded can have a long decoherence time and is easy to manipulate and mea- sure. Also, photonic qubits are the only type of qubits that are feasible for long distance quantum communica- tion. Quantum information processing requires universal two-qubit entangling gates. Knill et al [1] showed that it is theoretically possible to do scalable quantum com- puting with linear optics by using measurement induced interactions to perform the two-qubit gates. However, de- spite a continuous effort in reducing the resource require- ment [2, 3, 4, 5], the resource overhead is still high for linear optical quantum computing. Franson et al[6] pro- posed the use of two-photon absorption non-linearity and exploiting the quantum Zeno effect to implement a con- trol sign (CZ) gate that requires much less resources than linear optics schemes. However, the problem with the Zeno gate is that photon loss affects the performance of the Zeno gate significantly and the single photon to two- photon loss ratio requirement is very stringent. One solu- tion couldbe to combine measurement induced quantum processing with the Zeno gate. Previously we have shown that when using the Zeno gate for qubit fusion [8], state distillation [9] with post-selection can boost the gate fi- delity to unity and that for less stringent absorption ra- tios the gate has an advantage in success probability over linear optics gates for fusing clusters of qubits. These clusters of qubits can then be used for cluster state quan- tum computing [10]. In addition, Myers and Gilchrist [7] have shown that the performance of the Zeno gate may be enhanced by using error correction codes such as re- dundancy and parity encoding. Here we design a high fidelity Zeno CNOT gate suitable for circuit-based quantum computing. Although with the current estimate of the photon loss ratio, only a poor fidelity Zeno gate is directly achievable, here we show that it is possible to use two of these Zeno gates to do Bell measurements and implement a Gottesman- Chuang[11] teleportation type of CNOT gate (GC-Zeno gate) that, like the fusion gate, gives high fidelity via state distillation and moderate success probability via partially off-line state preparation. We include the effect of mode-mismatch and detector efficiency on the scheme and estimate lower bounds on the parameter which in principle allow fault tolerant operation. The paper is arranged in the following way. The in- troduction continues with a subsection on the Zeno CZ gate, which describes the scheme and modelling of the gate and give descriptions on the modelling parameters that are also used for modelling the GC-Zeno CZ gate. In section II, we discuss the GC-Zeno gate and the effect of mode-mismatch and detector efficiency on the gate. In section III, we give estimates of the lower bounds on the photon loss ratio and mode-matching, followed by a sub- section on the advantage in using state distillation. We conclude and summarize in section IV. Ia. Zeno CZ Gate Franson et al’s control sign gate scheme consists of a pair of optical fibers weakly evanescently coupled and doped with two-photon absorbing atoms. The purpose of the two-photon absorbers is to suppress the occurrence of two photon state components in the two fibre modes via the Quantum Zeno effect. This allows the state to remain in the computational basis. After a length of fibre corresponding to a complete swap of the two fibre modes, a π phase difference is produced between the |11〉 term and the other three basis terms. After swapping the fibre modes by simply crossing them, a CZ gate is achieved. The gate becomes near deterministic and performs a near unitary operation when the Quantum Zeno effect is strong and photon loss is insignificant. However, with current technology, the strength of the http://arxiv.org/abs/0704.1069v1 Quantum Zeno effect is a few orders of magnitude below what is required, and thus the Zeno gate has significant photon loss. In [8], the gate is modelled as a succession of n weak beamsplitters followed by two-photon absorbers as shown in Fig. 1. As n → ∞ the model tends to the continu- ous coupling limit envisaged for the physical realization. The gate operates on the single-rail encoding for which |0〉L = |0〉 and |1〉L = |1〉 with the kets representing pho- ton Fock states. Fig. 2 shows how the single rail CZ can be converted into a dual rail [13] CZ with logical encoding |0〉L = |H〉 and |1〉L = |V 〉. Let L be the total length of n absorbers. Also, let γ1 = exp( ) and γ2 = exp( the probability of single photon and two-photon trans- mission respectively for one absorber. Here the param- eter λ = χL, where L is the length of the absorber and χ is the corresponding proportionality constant related to the absorption cross section. Furthermore, κ specifies the relative strength of the two transmissions and relates them by γ2 = γ 1 . This CZ gate does the following oper- ation: |00〉 → |00〉 |01〉 → γn/2 |10〉 → γn/2 |11〉 → −γn τ |11〉+ f(|02〉, |20〉) (1) where the new expression for τ is given by: τn,λ = (gn,λ + dn,λ√ 2dn,λ − hn,λ) +(gn,λ − dn,λ√ 2dn,λ + hn,λ) dn,λ = (1 + cos )(1 + γ2) + 2 γ2(cos( )− 3) gn,λ = (cos γ2 + 1) hn,λ = 2(cos γ2 − 1) (2) The explicit form of the |02〉, |20〉 state components are suppressed, as they lie outside the computational basis and so do not explicitly contribute to the fidelity. FIG. 1: Construction of our CZ gate. FIG. 2: CZ gate in dual rail implementation. From equation 1, it is clear that the amplitude of the four computational states are unequal and this lowers the gate fidelity. With the current best estimate of κ = 104, the unherald fidelity is only 0.94. If the gate is used in a measurement based strategy then state distillation can be used and the fidelity of the gate can be improved by trading off some success probability. Figure 3 shows the distillated Zeno CZ gate circuit. The τ gate is simply an interferometer consisting of two 50-50 beam splitters with a two-photon absorber in each arm, which gives op- eration: |00〉 → |00〉, |01〉 → |01〉, |10〉 → |10〉, and |11〉 → γ′ τ |11〉. Here γ′ = τ1/κ is the single photon transmission coefficient of the absorber. The distillation beam splitters labelled 1 to 3 have transmission coeffi- cient τ respectively. With these dis- tillations in place, the operation of the distillated Zeno CZ gate is: |00〉 → γn τ |00〉 |01〉 → γn1 γ 1τ |01〉 |10〉 → γn τ |10〉 |11〉 → −γn τ |11〉+ f(|02〉, |20〉) (3) After measuring the output (detectors at output are not shown in fig.3) and treating the photon bunching terms (|02〉, |20〉) and the terms with photon loss as fail- ures (excluded from Eq. 3 for clarity), renormalising the states gives unit heralded fidelity independent of λ and probability of success Ps = γ τ2 = e−2λ/κτ2+2/κ. FIG. 3: Schematic of distillated Zeno CZ gate. II. ZENO GATE USING GOTTESMAN-CHUANG SCHEME As argued above, state distillation can improve the fidelity of the Zeno gate to unity by trading off success probability. However, the output of the distillated Zeno FIG. 4: Schematic of GC-Zeno gate. The state |χ〉 is ((|00〉 + |11〉)|00〉 + (|01〉 + |10〉)|11〉)/2 gate contains terms outside the computational basis due to photon loss and photon bunching. Hence if we want the gate to have unit fidelity, it is necessary to measure the output and exclude these failure terms by post-selection. However, such post-selection means that the gate can no longer be used directly as a CNOT gate for circuit-based quantum computing. Gottesman and Chuang [11] showed the viabil- ity in using state teleportation and single qubit operations to implement a CNOT gate. The scheme requires the four qubit entangled state |χ〉 = ((|00〉+ |11〉)|00〉+ (|01〉+ |10〉)|11〉)/2. Preparing the entangled state requires a CZ operation, which can be done off-line with linear optics with high fidelity. Bell measurements are made between the input qubits and the first and last qubits of |χ〉. The measurement results are fed forward for some single qubit corrections such that the circuit gives a proper CNOT operation. Here we propose using such scheme, as shown in figure 4, to implement a GC-Zeno CNOT gate with high fidelity. Since this gate includes state distillation, post-selection and off-line state preparation, the gate has unit fidelity (under perfect mode-matching) and moderate success probability. Figure 5 plots the probability of success against the one-photon to two-photon transmission ratio κ. It shows that with κ = 104 (current best estimate), the probability of success is about 0.76, which is better than the break even point of 0.25 for the linear optics version of this gate [1]. IIa. Effect of Mode-Mismatch From source preparation to gate operation to detec- tion, mode-mismatch is an unavoidable issue in optical quantum computing that causes unlocated errors which lowers the fidelity of the device[15]. Fortunately, with the help of quantum error correction, a certain amount of unlocated error rate, including but not limited to mode- mismatch errors, can be tolerated. A reliable quantum FIG. 5: Plot of probability of success versus log(κ) (in base 10) for GC-Zeno gate. Note that the success probability is not one at κ = 108, but that the curve asymptotically approaches one for very large κ. Result is per two input qubits. Detector inefficiency is taken into account in accord with Dawson et al’s bound. gate must therefore have unlocated error rates below this threshold. The dominant source of mode-mismatch error in the GC-Zeno gate is from the CZ gate and τ gate, where two-photon interaction occurs. Here we follow Rohde et al’s [14] analysis to examine the effect of such error. We take the simplest model in which the mode-mismatch is present between the photons entering the gate but re- main constant through the gate. In this case, the mode- mismatch in two-photon interaction can be analysed as having two-photons fail to interact with some probabil- ity. This allows us to write the operations for the CZ gate as follow, where 0 < Γ < 1 quantifies the over- lap of the two wavepackets. Γ2 is the probability that the two photons successfully interacted and Γ = 0 for completely mode-mismatched and Γ = 1 for completely mode-matched. The bar in the |1̄1〉 term indicates mode- mismatched component of the state. |00〉 → |00〉 |01〉 → |10〉 → |11〉 → −Γγn1 τ |11〉+ 1− Γ2γn1 |1̄1〉 +f(|02〉, |20〉) (4) And similarly for the operations of τ gate: |00〉 → |00〉 |01〉 → |10〉 → |11〉 → Γγ τ |11〉+ 1− Γ2γ |1̄1〉 +f(|02〉, |20〉) (5) With the equations for the CZ and τ gate[16], and given a normalized input state (α|00〉 + β|01〉 + δ|10〉 + ǫ|11〉), we can obtain analytical expression for the fidelity F (per qubit) and success probability Ps (per qubit) of the GC-Zeno gate as follow. Equation 6 and 7 show that both the fidelity and success probability are state dependent due to mode-mismatch. The worst case of fidelity occurs when the input state is the equal super- position state (|00〉+ |01〉+ |10〉+ |11〉)/2 (i.e. α = δ = β = ǫ = 1/2) and the worst case of success probability occurs when the input state is the pure state |11〉 (i.e. α = β = δ = 0 and ǫ = 1). α∗A1 + β ∗A2 + δ ∗A3 + ǫ |A1|2 + |A2|2 + |A3|2 + |A4|2 e−2λ/κτ2/κ 2(1 + e−λ/κτ2+1/κ) |A1|2 + |A2|2 + |A3|2 + |A4|2 (7) where a1 = (τ+τΓ+ 1− Γ2), a2 = (τ−τΓ+ 1− Γ2), a3 = (τ − τΓ− 1− Γ2), a4 = (τ + τΓ− 1− Γ2), and A1 = αa 1 + βa1a2 + δa1a2 + ǫa 2, A2 = αa1a3 + βa2a3 + δa1a4 + ǫa2a4, A3 = αa1a3 + βa1a4 + δa2a3 + ǫa2a4, A4 = αa 3 + βa3a4 + δa3a4 + ǫa IIb. Effect of Detector Efficiency In practice, even for the most advanced photon detec- tor, detector inefficiency is always present. The effect of this noise is to reduce the probability of success of the gate but not the fidelity because the errors are locatable. III. ESTIMATE OF BOUNDS FOR FAULT TOLERANCE We now wish to estimate lower bounds on the mode- matching, Γ, and photon loss ratio, κ, that will still allow fault-tolerant operation. We allow a small amount of detector inefficiency but assume all other parameters are ideal. To make this estimate we directly use the bounds obtained by Dawson et al [12] for a deterministic error correction protocol. For this protocol, they numerically derived one bound using the 7-qubit Steane code and another bound using the 23-qubit Golay code. In order to use the Dawson et al’s bounds we need to identify the unlocated and located error rates for our gate. In general, the unlocated error rate is less than 1 − F but here we take it to be 1 − F because in our analysis, γ is almost 1, which means the other terms in- volved are very small. The located error rate is simply 1 − Ps (both F and Ps are per qubit). Using these re- lationships, we convert each of the bounds into a fidelity versus success probability bound. For a gate built with two-photon absorbers that have a certain single-photon to two-photon transmission ratio κ, we can find an op- timal λ (i.e. choosing an optimal absorber length) that FIG. 6: Lower bounds of amount of mode-matching Γ required for a fault tolerant GC-Zeno gate versus single- photon to two-photon transmission ratio κ. The bounds are derived from Dawson et al’s [12] results on deterministic error correction protocol. The top and bottom curves are for the 7-qubit Steane code and the 23-qubit Golay code respectively. Above the curves are the regions where the amount of mode-mismatch is tolerable. Here we have used the worst case input. gives a maximum success probability. Hence, by match- ing the success probability with the bound, we can deter- mine the corresponding fidelity threshold and therefore find the least amount of mode-matching required for fault tolerant gate operation. We note that the error model used by Dawson et al is specific to their optical cluster state architecture and will differ in detail from the appro- priate error model for the GC-Zeno gate. Nonetheless we assume that a comparison based on the total error rates will give a good estimate of the bounds. Figure 6 shows the lower bounds on the mode- matching parameter Γ for a gate with a certain κ. Since the fidelity and success probability are state dependent due to mode-mismatch, in that figure, we have plotted for the case of worst fidelity input state (i.e. the equal superposition state). The top and bottom curves are best fit curves for using the 7-qubit Steane code and the 23-qubit Golay code respectively. The curves show that highly mode-matched photons are essential for robust gate operation. With the worst fidelity input state, (|00〉 + |01〉 + |10〉 + |11〉)/2, for the Steane code, the lowest Γ required for fault tolerant operation is about 0.998, where κ = 106, and for the Golay code, the lowest Γ required is about 0.996, where κ = 5 × 105. With the worst success probability input state, |11〉, for the Steane code, the lowest Γ required for fault tolerant operation is about 0.995, where κ = 106, and for the Golay code, the lowest Γ required is about 0.989, where κ = 5 × 105. Figure 6 also shows that under (near) perfect mode-matching, the required κ can be as low as approximately 6000 for the Steane code and 2000 for the Golay code. Two-photon absorbers with such κ values may be achievable with the best of current technology. IIIa. Advantage of Using State Distillation State distillation allows us to trade off some success probability against fidelity for the GC-Zeno gate, or in other words, reducing the unlocated error rate by hav- ing a larger located error rate. Since the determinis- tic error correction protocol can tolerate both unlocated and located errors, therefore we should ask whether state distillation is truly advantageous? We can answer this question by comparing two GC-Zeno gates in the case of perfect mode matching, where one has complete distil- lation and the other has no distillation. For the case of complete distillation, the deterministic error correction protocol with the 7-qubit Steane code can tolerate errors of a GC-Zeno gate with κ = 6100, and with the 23-qubit Golay code, it can tolerate errors of a GC-Zeno gate with κ = 2100. For state distillation to be advantageous un- der the same protocol, these values of κ must be smaller than the values of κ for the case of no distillation[17]. For the case of no distillation, the fidelity and suc- cess probability of the gate becomes state dependent. In the parameters space of interest, the input that gives the worst fidelity is (|00〉 + |01〉 + |10〉 − |11〉)/2. With this input state, we find that for the protocol using the 7- qubit Steane code and no distillation, the critical κ is 12000. Similarly, for the protocol using the 23-qubit Go- lay code and no distillation, the critical κ is 4300. With an arbitrary amount of distillation, the value of κ lies between the limit of no distillation and full distillation cases. Hence it is evident that state distillation is advan- tageous. Also, it should be noted it is better to have only located error, which is the case when there is full distilla- tion, than have both located and unlocated errors, which is the case when no or some distillation is utilized. IV. CONCLUSION In this paper, we have shown that it is possible to build a high fidelity Zeno CNOT gate with two distil- lated Zeno gates implemented in the Gottesman-Chuang teleportation CNOT scheme. For one-photon to two- photon transmission ratio κ = 104 (current best esti- mate), the gate has a success probability of 0.76 under perfect mode-matching. When including measurement noise that equals one-tenth of the gate’s noise, and the effect of mode-mismatch in the CZ and τ gate, we find that with the deterministic error correction protocol us- ing the 7-qubit Steane code, the lowest Γ required for fault tolerant gate operation is 0.998, where κ = 106. For using the 23-qubit Golay code, the lowest Γ required is 0.996, where κ = 5 × 105. Hence, the requirement on mode-matching is stringent for a fault tolerant GC-Zeno gate. ∗ Electronic address: [email protected] [1] E. Knill, R. Laflamme, and G.J. Milburn, Nature 409, 46-52 (2001) [2] N.Yoran and B.Reznik, Phys. Rev. Lett. 91, 037903 (2003) [3] M.A. Nielsen, Phys. Rev. Lett. 93, 040503 (2004) [4] A.J.F.Hayes, A.Gilchrist, C.R.Myers and T.C.Ralph, J.Opt.B 6, 533 (2004) [5] D.E.Browne and T.Rudolph, Phys. Rev. Lett. 95, 010501 (2005) [6] J.D. Franson, B.C. Jacobs, and T.B. Pittman, Phys. Rev. A 70, 062302 (2004) [7] C.R. Myers and A. Gilchrist, quant-ph 0612176 (2006) [8] P.M. Leung and T.C. Ralph, Phys. Rev. A 74, 062325 (2006) [9] R.T. Thew and W.J. Munro, Phys. Rev. A 63, 030302(R) (2001) [10] R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. 86 5188 (2001) [11] D. Gottesman, I.L. Chuang, Nature 402, 6760 (1999) [12] C.M. Dawson, H.L. Haselgrove, and M.A. Nielsen, Phys. Rev. Lett. 96, 020501 (2006) [13] A.P. Lund, T.C. Ralph, Phys. Rev. A, 66, 032307 (2002) [14] P.P. Rohde and T.C. Ralph, Phys. Rev. A 73, 062312 (2006) [15] Note that mode mismatch in CZ and τ gate causes unlo- cated error that lowers the fidelity, in which we find that it cannot be improved with state distillation. [16] Due to mode-mismatch, the τ gate is less effective in two- photon distillation. It is true that we can increase the two-photon absorption strength in the τ gate to make up for the inefficiency. However, here we assume that we do not know the mode-matching parameter Γ of the in- put wavepackets, and therefore this adjustment cannot be made. In addition, increasing the two-photon distilla- tion will increase single-photon loss as well, which lowers the probability of success. [17] An ideal Zeno gate requires strong quantum Zeno effect and strong quantum Zeno effect corresponds to a large κ value. However such large non-linearity is difficult to engineer. Hence it is desirable to have a gate that works with modest κ. mailto:[email protected]

0704.1070

Differential Diversity Reception of MDPSK over Independent Rayleigh Channels with Nonidentical Branch Statistics and Asymmetric Fading Spectrum

Differential Diversity Reception of MDPSK over Independent Rayleigh Channels with Nonidentical Branch Statistics and Asymmetric Fading Spectrum Hua Fu and Pooi Yuen Kam ECE Department, National University of Singapore Singapore 117576, Email: {elefh, elekampy}@nus.edu.sg Abstract— This paper is concerned with optimum diversity re- ceiver structure and its performance analysis of differential phase shift keying (DPSK) with differential detection over nonselective, independent, nonidentically distributed, Rayleigh fading chan- nels. The fading process in each branch is assumed to have an arbitrary Doppler spectrum with arbitrary Doppler bandwidth, but to have distinct, asymmetric fading power spectral density characteristic. Using 8-DPSK as an example, the average bit error probability (BEP) of the optimum diversity receiver is obtained by calculating the BEP for each of the three individual bits. The BEP results derived are given in exact, explicit, closed-form expressions which show clearly the behavior of the performance as a function of various system parameters. I. INTRODUCTION The receiver structure and bit error probability (BEP) per- formance of differential phase shift keying (DPSK) with differ- ential detection over nonselective, independent and identically distributed (i.i.d.), Rayleigh fading channels with combining diversity reception have been well known in the literature [1]−[4]. However, reaserch shows that in some practical systems, the independent, non-identically distributed (i.n.i.d.) channel model is more accurate [5], [6]. In i.n.i.d. channel, the fading processes and possibly the additive, white Gaussian noise (AWGN) on the diversity branches have non-uniform power profiles which are distinct from one another. The effect of the nonidentical diversity branch statistics on the receiver structure is studied in [7]. Recently, based on the maximum a posteriori probability (MAP) criterion, an explicit structure of the optimum combining differential receiver and a complete set of closed-form BEP expressions and their Chernoff upper bounds, for 2-, 4- and 8-DPSK, both with optimum combining reception and suboptimum combining reception, are derived in [8]−[10]. The purpose of this paper is to provide a further extension. The results derived in this paper, together with those in [8]−[10], form a benchmark counterpart to the classic ones for the i.i.d. channel given in [1]−[4]. In a Rayleigh channel, the fading gain is usually modeled as a zero-mean, stationary, complex, Gaussian random pro- cess. The most widely accepted model [1]−[10] is that the spectrum of the fading process over each diversity branch is symmetric around the carrier so that the quadrature processes are independent of each other. This assumption is valid for various fading spectra. For example, see [11] and its refer- ences. However, in some fading environments such as the land mobile channel with Jakes model [12], the Doppler spectrum becomes asymmetric when the multipath signals are absorbed by obstacles or the propagation environment is characterised by directional non-isotropic scattering [13]−[15]. Thus, it is of great practical importance to take account of the effect of the asymmetric fading spectrum on the receiver structure and the performance analysis of differentially detected DPSK over i.n.i.d. channels, the topic of this paper. The paper is orgainzed as follows. In Section II, the signal model is introduced and different optimum diversity receivers are derived for different Rayleigh fading scenarios (see eqs. (17)−(20) below). In Section III, we use 8-DPSK as an example to study the BEP performance. Here, the average BEP of the optimum diversity receiver is obtained by calculating the BEP for each of the three individual bits. The results are given in exact, explicit, closed-form expressions which show clearly the behavior of the performance as a function of signal-to- noise ratio (SNR), fading correlation coefficient, and diversity order. Section IV presents numerical examples. Throughout this paper, overhead ∼ denotes a complex quantity, superscript ∗ will denote its conjugate, E is the ensemble average operator, δ represents the Kronecker delta, and [·]T denotes transposition of the vector and matrix. II. SIGNAL MODEL AND RECEIVER STRUCTURE With space diversity reception over L frequency nonse- lective, i.n.i.d., Rayleigh fading branches with AWGN, the received signal over the ith branch, i = 1, 2, · · · , L, during the kth symbol interval kT ≤ t < (k + 1)T is given, after matched filtering and sampling at time t = (k + 1)T , by the statistic r̃i(k), where r̃i(k) = E jφ(k)c̃i(k) + ñi(k). (1) Here, Es is the energy per symbol, and for DPSK, φ(k) is the data-modulated phase with Gray encoding of bits onto the phase transition ∆φ(k) = φ(k) − φ(k − 1). The kth data symbol is conveyed in ∆φ(k). We assume here that all symbol points are equally likely. In (1), a rectangular data pulse shape g(t), where g(t) = 1/ T for 0 ≤ t < T and zero elsewhere, is assumed so that each matched filter has a rectangular low- pass-equivalent impulse response hi(t) = g(T − t) for all i. Thus, the filtered noise ñi(k) is given by ñi(k) = ∫ (k+1)T ñi(t)√ dt. (2) Here, {ñi(t)}Li=1 is a set of i.n.i.d., lowpass, complex AWGN processes with E [ñi(t)] = 0 and E[ñi(t)ñ i (t− τ)] = Niδ(τ) so that {ñi(k)}k is a sequence of zero-mean, complex Gaus- sian variables with covariance function for each branch i E[ñi(k)ñ i (j)] = Ni δkj (3) The multiplicative distortion c̃i(k) in (1) is given by c̃i(k) = ∫ (k+1)T c̃i(t) dt. (4) http://arxiv.org/abs/0704.1070v1 Here, c̃i(t) = ai(t) + jbi(t) is a set of i.n.i.d., lowpass, zero-mean, stationary, complex, Gaussian random processes. Each c̃i(t) represents the complex gain due to frequency nonselective Rayleigh fading of the ith branch. For asymmetric spectrum in each i, the inphase fading process ai(·) and the quadrature phase fading process bi(·) are generally correlated. At any time instant t, however, ai(t) and bi(t) are always uncorrelated. With reference to Fig. 1, it is shown in [16] that the covariance function of ai(·) and bi(·) can be obtained as E[ai(t)bi(t)] = 0 (5a) E[ai(t− τ)ai(t)] = E[bi(t− τ)bi(t)] = Ri(τ) (5b) E[ai(t)bi(t− τ)] = −E[bi(t)ai(t− τ)] = Qi(τ) (5c) Note that if the spectrum of each c̃i(t) is symmetric, the processes ai(·) and bi(·) will be independent (i.e., we have Qi(τ) = 0) with the same covariance function Ri(τ). Letting c̃i(k) = ai(k) + jbi(k), it follows from (4) and (5) that both {ai(k)}k and {bi(k)}k are sequences of zero-mean, real-valued, Gaussian random variables with E[ai(k)bi(k)] = 0 (6a) E[ai(k − l)ai(k)] = E[bi(k − l)bi(k)] = Ci(l) (6b) ∫ (k+1)T ∫ (k+1−l)T (k−l)T Ri(u − v) du dv E[ai(k)bi(k − l)] = −E[bi(k)ai(k − l)] = Di(l) (6c) ∫ (k+1)T ∫ (k+1−l)T (k−l)T Qi(u− v) du dv Thus, the covariance matrix can be obtained as Γi = E ai(k) ai(k − l) bi(k) bi(k − l) ai(k) ai(k − l) bi(k) bi(k − l) Ci(0) Ci(l) 0 Di(l) Ci(l) Ci(0) −Di(l) 0 0 −Di(l) Ci(0) Ci(l) Di(l) 0 Ci(l) Ci(0)  (7) For each i , c̃i(k) and ñi(k) are mutually independent. For i 6= j, {c̃i(k), ñi(k)} are independent of {c̃j(k), ñj(k)}. The diversity branches are nonidentical since the covariance functions Ri(τ), Qi(τ) and Niδ(τ) depend on the branch index i. For convenience of later application, the following parameters are defined. The fading correlation coefficient at the matched filter output over a symbol interval of T for the ith diversity branch is defined as ρ̃i = E[c̃i(k)c̃ i (k − 1)]√ |c̃i(k)|2 |c̃i(k − 1)|2 Ci(1)− jDi(1) Ci(0) From (8), we note that ρ̃i is a complex quantity. It is a measure of the fluctuation rate of the channel fading process. The mean received SNR per symbol over the ith branch is defined as |E1/2s ejφ(k)c̃i(k)|2 2EsCi(0) We consider 2-, 4- and 8-DPSK with Gray encoding of bits onto ∆φ(k) as shown in [4, Fig.1] for 4- and 8-DPSK, the mean received SNR per bit γbi is given by γ i = γi for 2-DPSK, γbi = γi/2 for 4-DPSK, and γ i = γi/3 for 8-DPSK. Using the MAP criterion, the aim of the receiver is to determine from the received signals {r̃i(k), r̃i(k − 1)}Li=1 which one of the possible values 2πm/M , m = 0, 1, · · · ,M− 1, of the phase difference ∆φ(k) has maximum probability of occurrence. Following [9], it can be shown that MAP detection is equivalent to maximum log-likelihood detection. Specifically, based on {r̃i(k), r̃i(k − 1)}Li=1, we decide that ∆φ(k) = 2πn/M whenever the log-likelihood function logΨm = r̃i(k) ∣∣∣r̃i(k − 1),∆φ(k) = is maximized for m = n. To proceed with evaluating (10), we need to verify that c̃i(k) = ai(k)+jbi(k) and c̃i(k−1) = ai(k−1)+jbi(k−1) are jointly complex Gaussian. By being jointly complex Gaussian, it means that if x̃ = xR + jxI and ỹ = yR + jyI are two column complex random vector, then [xR has a real multivatiate Gaussian probability density function (PDF), and furthermore, if u = [xR T ]T and v = T ]T , then the real covariance matrix of [uT vT ]T has a special form given in [18, Theorem 15.1] that satis- fies Goodman’s theorem [19]. After careful examination, it follows from (7) that c̃i(k) and c̃i(k − 1) are indeed jointly complex Gaussian1. Thus, conditioned on c̃i(k − 1), c̃i(k) is conditionally complex Gaussian with mean [18] E [c̃i(k)|c̃i(k − 1)] = ρ̃i c̃i(k − 1) (11) and variance {∣∣c̃i(k)− E[c̃i(k)|c̃i(k − 1)] ∣∣∣c̃i(k − 1) = 2Ci(0)− 2 C2i (1) +D i (1) Ci(0) Moreover, conditioned on the vector [ai(k − 1) bi(k − 1)]T , the vector [ai(k) bi(k)] T is conditionally Gaussian with covariance matrix given by Ci(0)− C (1)+D2 Ci(0) 0 Ci(0)− C (1)+D2 Ci(0) which is a diagonal matrix. This shows that Re[c̃i(k)|c̃i(k−1)] and Im[c̃i(k)|c̃i(k − 1)] are independent. Applying (11), (12) and (13) to (10), we obtain logΨm = ζ + (14) 2Es [Ci(1) + jDi(1)] e −j 2πm M r̃i(k)r̃ i (k − 1) [2EsCi(0) +Ni]2 − 4E2s [C2i (1) +D2i (1)] or, equivalently logΨm = ζ + (15) |ρ̃i| γi e−j∠ρ̃i (1 + γi)2 − (|ρ̃i|γi)2 r̃i(k) r̃ i (k − 1)e−j where ζ represents the constant term which does not affect the decision. In (15), the quantities |ρ̃i| = (1)+D2 ∠ρ̃i = − tan−1 Di(1) Ci(1) represent the magnitude and phase of the correlation coefficient ρ̃i given in (8), respectively. 1We also call them the proper complex Gaussian random variables [17]. Defining the real-valued weighting factors |ρ̃i|γi (1 + γi)2 − (|ρ̃i|γi)2 , (16) it follows from (15) that the optimum combining differential receiver will now compute, for the kth symbol, the decision statistics {Λm(k)}M−1m=0 , and declares that ∆φ(k) = Λn(k) = maxm {Λm(k)}, where Λm(k) = Re wi r̃i(k) r̃ i (k − 1) e−j∠ρ̃i If the spectrum of the channel complex gain is symmetric, ρ̃i is a real-valued quantity. Then, the optimum combining differential receiver (17) will become [9] Λ′m(k) = Re wi r̃i(k) r̃ i (k − 1) If the diversity branches are i.i.d., but the fading gains have asymmetric spectrum, the optimum receiver will become Λ′′m(k) = Re M e−j∠ρ̃ r̃i(k) r̃ i (k − 1) where ρ̃ = ρ̃i for i = 1, 2, · · · , L. For i.i.d. branches with fading gains having symmetric spectrum, the optimum receiver is the well-known product detector, given by [4] Λ′′′m(k) = Re r̃i(k) r̃ i (k − 1) Comparing (20) with (17), we see that in the case of i.n.i.d. channels with asymmetric power spectrum, the receiver first rotates the product phasor r̃i(k)r̃ i (k − 1) between the two received signal samples at each diversity branch by the angle −∠ρ̃i, then scales each resulting phasor by the weight wi, and finally sums all L rotated and scaled phasors to form a decision variable. Clearly, in order to form the optimum detector (17), besides the received signal samples r̃i(k) and r̃i(k − 1), the receiver requires the a priori knowledge of the channel statistics, including the power spectral densities of AWGN Ni, both the magnitude and phase of the fading correlation coefficient ρ̃i, and the mean received SNR γi. These quantities can be pre-computed according to our knowledge of the channel statistics at the receiver. III. PERFORMANCE ANALYSIS In this section, we will derive exact, explicit and closed- form BEP expressions for differentially detected DPSK for the optimum receiver (17). Due to space limitation, we only consider 8-DPSK in this paper. The signal constellation, bit mapping and the decision region Rm for 8-DPSK is shown in Fig. 2. In [4] and [9], the average BEP is computed using the binary reflected Gray code (BRGC) approach through Hamming weight spectrum [20]. It is shown in [21] that the BRGC approach with Hamming weight is less accurate for M ≥ 16. In this paper, we adopt a new approach, namely, the average BEP is obtained by calculating the BEP for each of the three individual bits in 8-DPSK. This approach has the advantage of showing explicitly the BEP performance differently for the three different transmitted information bits. Therefore, using the bit which has lower BEP to convey more important information can improve communication reliability. From Fig. 2, we see that each signal point is represented by a 3-bit symbol (j1, j2, j3). We use Pj1 , Pj2 and Pj3 to denote the corresponding individual BEP. Since the three bits are equally likely, the average BEP is given by (Pj1 + Pj2 + Pj3) (21) We begin with computing Pj1 . Without loss of generality, it is assumed that j1 = 0. The case where j1 = 1 gives an identical result. From Fig. 2, we see that the bit j1 = 0 is associated with the symbols 000 (∆φ(k) = 0), 001 (∆φ(k) = π/4), 011 (∆φ(k) = π/2), and 010 (∆φ(k) = 3π/4). Thus, conditioning on j1 = 0, the BEP Pj1 will be given by Pj1 = Pj1 (e|∆φ(k) = 0) + Pj1(e|∆φ(k) = π/4) (22) +Pj1(e|∆φ(k) = π/2) + Pj1 (e|∆φ(k) = 3π/4) Here, Pj1(e|∆φ(k) = mπ/4),m = 0, 1, 2, 3, is the probabil- ity that conditioning on ∆φ(k) = mπ/4, the decision j1 = 1 is made. With reference to Fig. 2, this is equivalent to the probability that conditioning on ∆φ(k) = mπ/4, the phasor∑L i=1 wi r̃i(k) r̃ i (k − 1) e−j∠ρ̃i lies outside the half-plane region R0+R1+R2+R3 (i.e., in the region R4+R5+R6+R7). The BEP Pj1(e|∆φ(k) = mπ/4) is thus obtained as e|∆φ(k) = mπ/4 8 (23) wir̃i(k)r̃ i (k − 1)e−j∠ρ̃i ∣∣∣∣∣ ∆φ(k) = To evaluate (23), first, it follows from (11) and (12) that con- ditioning on ∆φ(k) = mπ/4 and on r̃i(k− 1) ej∠ρ̃i ej3π/8 = α̃i, for i = 1, 2, · · · , L, the quantity r̃i(k) is condition- ally Gaussian with mean α̃i ρ̃iγi e−j∠ρ̃ie−j3π/8ejmπ/4 = |ρ̃i|γi e−j3π/8ejmπ/4, where ρ̃i = |ρ̃i|ej∠ρ̃i has been used, and variance (1+γi) 2−(|ρ̃i|γi)2 Ni. Then, in (23) the quantity Re[e−j i=1 wir̃i(k)r̃ i (k − 1)e−j∠ρ̃i)] is conditionally Gaussian with mean cos mπ/4− 3π/8 i=1 wi |ρ̃i|γi |α̃i|2 and variance 1 i=1 w (1+γi) 2−(|ρ̃i|γi)2 Ni |α̃i|2. Finally, following the derivation procedure detailed in [9], the BEP in (23) can be obtained as e|∆φ(k) = mπ = (24) + 1/λi where the quantity Gi is given by j=1,j 6=i λi − λj , and λi = (|ρ̃i|γi)2 (1 + γi)2 − (|ρ̃i|γi)2 Putting (24) into (22) leads to the BEP Pj1. An interesting observation from (24) is that the BEP does not depend on the phase, ∠ρ̃i, of the fading correlation coefficient ρ̃i. Intuitively, this is because the optimum receiver (17) can provide “phase compensation” for each diversity branch before combining using the channel statistic knowledge e−j∠ρ̃i . As such, we expect that the receivers (18) and (20) are suboptimum over the channel with asymmetric fading spectrum. Next, we compute Pj2 in (21). The procedure for obtaining the conditional BEP for j2 = 0 is parallel to that followed in the case for j1 = 0. From Fig. 2, the bit j2 = 0 is associated with the symbols 001 (∆φ(k) = π/4), 000 (∆φ(k) = 0), 100 (∆φ(k) = 7π/4), and 101 (∆φ(k) = 3π/2). Hence, conditioning on j2 = 0, the BEP Pj2 is given by Pj2 = Pj2(e|∆φ(k) = π/4) + Pj2(e|∆φ(k) = 0) (26) +Pj2(e|∆φ(k) = 7π/4) + Pj2 (e|∆φ(k) = 3π/2) where Pj2(e|∆φ(k) = nπ/4), n = 0, 1, 6, 7, is the conditional probability that the phasor i=1 wir̃i(k)r̃ i (k− 1)e−j∠ρ̃i lies in the half-plane region R2 +R3 +R4 +R5, i.e., e|∆φ(k) = nπ/4 8 (27) wir̃i(k)r̃ i (k − 1)e−j∠ρ̃i ∣∣∣∣∣ ∆φ(k) = which has solution Pj2 (e|∆φ(k) = nπ/4)= (28) + 1/λi Putting (28) into (26) leads to the BEP Pj2. Finally, we compute Pj3 in (21). From Fig. 2, the bit j3 = 0 is associated with the symbols 100 (∆φ(k) = 7π/4), 000 (∆φ(k) = 0), 010 (∆φ(k) = 3π/4), and 110 (∆φ(k) = π). Thus, conditioning on j3 = 0, the BEP Pj3 is given by Pj3 = Pj3 (e|∆φ(k) = 7π/4) + Pj3(e|∆φ(k) = 0) (29) +Pj3(e|∆φ(k) = 3π/4) + Pj3 (e|∆φ(k) = π) where Pj3(e|∆φ(k) = lπ/4), l = 0, 3, 4, 7, is the conditional probability that the phasor i=1 wir̃i(k)r̃ i (k− 1)e−j∠ρ̃i lies in the region R1 + R2 + R5 + R6. This is equivalent to the conditional probability that after rotating by −π/8, the product of the inphase and quadrature-phase components of the phasor∑L i=1 wir̃i(k)r̃ i (k − 1)e−j∠ρ̃i is greater than zero, i.e., e|∆φ(k) = l π/4 = (30) wir̃i(k)r̃ i (k − 1)e−j∠ρ̃i wir̃i(k)r̃ i (k − 1)e−j∠ρ̃i ∣∣∣∣∣ ∆φ(k) = From the argument for deriving (24), we note that condi- tioning on ∆φ(k) = lπ/4 and on r̃i(k − 1) ej∠ρ̃i ejπ/8 = β̃i, for i = 1, 2, · · · , L, the inphase component in (30), Re[e−j i=1 wir̃i(k)r̃ i (k − 1)e−j∠ρ̃i)] is conditionally Gaussian with mean cos lπ/4− π/8 i=1 wi |ρ̃i|γi |β̃i|2 and variance 1 i=1 w (1+γi) 2−(|ρ̃i|γi)2 Ni |β̃i|2. Similarly, the component Im[e−j i=1 wir̃i(k)r̃ i (k − 1)e−j∠ρ̃i)] in (30) is also a conditionally Gaussian random variable, with mean sin lπ/4− π/8 i=1 wi |ρ̃i|γi |β̃i|2 and vari- ance 1 i=1 w (1+γi) 2−(|ρ̃i|γi)2 Ni |β̃i|2. Moreover, it fol- lows from (13) and the properties of the complex Gaussian ran- dom variables [18] that the conditional inphase and quadrature- phase components Re[e−j i=1 wir̃i(k)r̃ i (k − 1)e−j∠ρ̃i)] and Im[e−j i=1 wir̃i(k)r̃ i (k−1)e−j∠ρ̃i)] in (30) are also independent. Therefore, conditioning on ∆φ(k) = lπ/4 and on r̃i(k − 1) ej∠ρ̃i ejπ/8 = β̃i, and denoting the inphase and quadrature-phase components as X ∼ N lπ/4− π/8 u, η2 Y ∼ N lπ/4− π/8 u, η2 where u and η2 are given, respectively, by |ρ̃i|γi 1 + γi |β̃i|2 (1 + γi) 2 − (|ρ̃i|γi)2 1 + γi Ni |β̃i|2 (32) the conditional BEP Pj3 ∣∣∆φ(k) = lπ , β̃i is given by ∣∣∆φ(k) = , β̃i X Y > 0 ∣∣∆φ(k) = , β̃i This is probability that the product of two independent real- valued Gaussian random variables with non-zero, nonidentical means and identical variances is greater than zero. This is a special case of the results given in [2, Appendix B] concerning the probability that a general quadratic form in complex-valued Gaussian random variables is less than zero. Using [2, (B-21) of Appendix B], (33) can be evaluated as ∣∣∆φ(k) = , β̃i = 1− (34) g[1− sin (lπ/2− π/4)], g[1 + sin (lπ/2− π/4)] I0 [g| cos (lπ/2− π/4)|] exp(−g) where, Q1(a, b) is first-order Marcum’s Q-function and Ik(x) is the kth-order modified Bessel function of the first kind. In (34), the quantity g = i=1 w i |β̃i|2 has PDF given by [9] p(g) = w′i Ni (1 + γi) w′iNi (1 + γi) where w′i = (|ρ̃i|γi)2 (1+γi)[(1+γi)2−(|ρ̃i|γi)2] . Averaging the condi- tional probability (34) over g using the PDF (35) gives the BEP Pj3 ∣∣∆φ(k) = lπ/4 in (30), i.e., ∣∣∆φ(k) = ∣∣∆φ(k) = , β̃i p(g)dg Substituting (34) and (35) into (36), we obtain, after manipu- lation and simplification, ∣∣∆φ(k) = A2i − cos2 ( lπ2 − 1− | cos ( −cos2 ( lπ − 1/2√ A2i − cos2 ( lπ2 − where Ai is given by 1 + γi |ρ̃i|γi Putting (37) into (29) leads to the BEP Pj3. Substituting (22), (26) and (29) in (21), we obtain the average BEP P . IV. NUMERICAL EXAMPLE Fig. 3 plots the BEP performance for the three individual bits in (22), (26) and (29) and the average BEP in (21) of 8-DPSK against the total average received SNR per bit. The order of diversity is set to L = 2. The abscissa represents the total mean SNR per bit which is given by γb = i=1 γ i=1 γi. The average received bit energy distribution among the two branches is set to γb1 : γ 2 = 30% : 70%. It is assumed that the fading correlation coefficient (the normalized covariance function) model follows [14, eq.(10)], given by E[c̃i(t)c̃ i (t− τ)] |c̃i(t)|2 κ2 − 4π2f2dτ2 + j4πκfdτ I0(κ) where fd is the Doppler frequency, and κ is a parameter that controls the width of the angle of arrival of scatter components [14, eq.(1)]. Note that if κ = 0, (38) results in the correlation coefficient for the Jakes two-dimensional isotropic scattering model, i.e., E[c̃i(t)c̃ i (t− τ)]/E |c̃i(t)|2 = I0(j2πfmτ) = J0(2πfmτ), where J0(·) is the zeroth-order Bessel function. We assume that the normalized Doppler spread fdT = 0.03 and 0.05 for diversity branches 1 and 2, respectively, and the parameter κ is set to 3. Thus, we have ρ̃1 = 0.9871+ j0.1519 and ρ̃2 = 0.9642 + j0.2511. It is seen from Fig. 3 that the third bit j3 has the lowest BEP, whereas, the BEP Pj1 for the first bit j1 is equal to the BEP Pj2 for the second bit j2. REFERENCES [1] M. Schwartz, W.R. Bennett, and S. Stein, Communication Systems and Techniques, New York: McGraw-Hill, 1966. [2] J. G. Proakis, Digital Communications, 4th edition, New York: McGraw- Hill, 2001. [3] M. K. Simon and M. S. Alouini,, Digital communication over fading channels, 2nd Edition, New York: John Wiley & Sons, 2005. [4] P. Y. Kam, “Bit error probabilities of MDPSK over the nonselective Rayleigh fading channel with diversity reception,” IEEE Trans. Com- mun., vol.39, pp.220-224, February 1991. [5] M.Z. Win and J.H. Winters, “Analysis of hybrid selection/maximal-ratio combining of diversity channels with unequal SNR in Rayleigh fading,” Proc. 49th IEEE VTC, pp.215-220, May 16-20, 1999. [6] P. Polydorou and P. Ho, “Error performance of MPSK with diversity combining in non-uniform Rayleigh fading and non-ideal channel esti- mation,” Proc. 51st IEEE VTC, pp.627-631, May 15-18, 2000. [7] F. Adachi, “Postdetection optimal diversity combiner for DPSK differen- tial detection,” IEEE Trans. Veh. Technology, vol.42, pp.326-337, August 1993. [8] H. Fu and P. Y. Kam, “Performance of Optimum and Suboptimum Com- bining Diversity Reception for Binary DPSK over Independent, Non- identical Rayleigh Fading Channels,” Proc. 40th IEEE ICC, pp.2367- 2371, May 16-20, 2005. [9] H. Fu and P. Y. Kam, “MDPSK diversity receiver over Rayleigh fading channels with differential detection and nonidentical branch statistics,” Proc. 63rd IEEE VTC, pp.1660-1664, May 7-10, 2006. [10] H. Fu and P. Y. Kam, “Performance of Optimum and Suboptimum Combining Diversity Reception for Binary and Quadrature DPSK over Independent, Nonidentical Rayleigh Fading Channels,” to appear in the IEEE Trans. Commun., May 2007. [11] L. J. Mason “Error probability evaluation for systems employing differ- ential detection in a Rician fast fading environment and Gaussian noise,” IEEE Trans. Commun., vol.35, pp.39-46, January 1987. [12] W.C. Jakes, Microwave Mobile Communications, NJ: IEEE Press, 1974. [13] M. Patzold, Y.C. Li and F. Laue “A study of a land mobile satel- lite channel model with asymmetrical Doppler power spectrum and lognormally distributed line-of-sight component,” IEEE Trans. Veh. Technology, vol.47, pp.297-310, February 1998. [14] A. Abdi, J.A. Barger and M. Kaveh “A parametric model for the distribution of the angle of arrival and the associated correlation function and power spectrum at the mobile station,” IEEE Trans. Veh. Technology, vol.51, pp.425-434, May 2002. [15] K. Anim-Appiah “Complex envelope correlations for nonisotropic scat- tering,” Electron. Lett., vol.34, pp.918-919, April 1998. [16] W. B. Davenport and W. L. Root, An introduction to the theory of random signals and noise, New York: McGraw-Hill, 1958 [17] F. D. Neeser and J.L. Massey “Proper complex random processes with applications to information theory,” IEEE Trans. Inform. Theory, vol.39, pp.1293-1302, July 1993. [18] S.M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, New Jersey: Prentice-Hall, 1998. [19] A. Papoulis and S.U. Pillai, Probability, Random Variables and Stochas- tic Processes, 4th Ed., MA: McGraw-Hill, 2002. [20] P. J. Lee, “Computation of the bit error rate of coherent M-ary PSK with Gray code bit mapping,” IEEE Trans. Commun., vol.34, pp.488- 491, May 1986. [21] J. Lassing, E.G. Strom, E. Agrell and L. Ottosson, “Computation of the exact bit-error rate of coherent M-ary PSK with Gray code bit mapping,” IEEE Trans. Commun., vol.51, pp.1758-1760, November 2003. Fig. 1. Illustration of complex channel fading process. Fig. 2. 8-DPSK constellation and decision region. 10 15 20 25 30 35 40 SNR (γ ), dB Fig. 3. BEP comparison of the three individual bits and the average of all bits for 8-DPSK. INTRODUCTION SIGNAL MODEL AND RECEIVER STRUCTURE PERFORMANCE ANALYSIS NUMERICAL EXAMPLE References

0704.1071

Bulk viscosity of superfluid neutron stars

Bulk viscosity of superfluid neutron stars Mikhail E. Gusakov Ioffe Physical Technical Institute, Politekhnicheskaya 26, 194021 Saint-Petersburg, Russia (Dated:) The hydrodynamics, describing dynamical effects in superfluid neutron stars, essentially differs from the standard one-fluid hydrodynamics. In particular, we have four bulk viscosity coefficients in the theory instead of one. In this paper we calculate these coefficients, for the first time, assuming they are due to non-equilibrium beta-processes (such as modified or direct Urca process). The results of our analysis are used to estimate characteristic damping times of sound waves in superfluid neutron stars. It is demonstrated that all four bulk viscosity coefficients lead to comparable dissipation of sound waves and should be considered on the same footing. PACS numbers: 97.60.Jd, 26.60.+c, 47.37.+q, 47.75.+f I. INTRODUCTION The matter in pulsating neutron stars is not (even lo- cally) in chemical equilibrium. Particles of different kinds turn into one another so that the system evolves to equi- librium. If a deviation from the equilibrium is small then the processes of mutual transformations of particles can be described in terms of an effective bulk viscosity (see, e.g., Ref. [1]). This viscosity influences the ‘instability windows’, that are the regions of physical parameters (e.g., the rotation period and temperature of a star) at which the neutron star becomes unstable against the ra- diation of gravitational waves [2, 3, 4, 5]. The bulk vis- cosity, generated by non-equilibrium processes of particle transformations, was calculated in a series of papers for neutron-star matter of various composition (for example, for matter composed of neutrons, protons, and electrons with an admixture of muons; for hyperon or quark mat- ter). A short review and references to these papers can be found in Ref. [6]. It is generally agreed that the stellar matter becomes superfluid at a certain stage of neutron star thermal evo- lution [7, 8, 9]. A lot of attention has been paid to the question how superfluidity affects the bulk viscosity (see, e.g., Refs. [10, 11, 12, 13, 14, 15]). In analogy with the or- dinary hydrodynamics of non-superfluid liquid, the only one ‘standard’ bulk viscosity coefficient has been calcu- lated and analyzed in all these papers. Meanwhile, it is well known [1, 16, 17] that a superfluid liquid, composed of identical particles, is generally described by the four bulk viscosity coefficients. So, what can be expected from neutron stars, which contain a mixture of many super- fluid species? In this paper we show that non-equilibrium processes of particle transformations lead to the appearance of at least four bulk viscosity coefficients. Each of them is im- portant for analyzing dissipative processes in superfluid neutron stars. To be specific, we consider the simplest model of stellar matter composed of neutrons, protons, and electrons (npe-matter). In this case the bulk vis- cosity is associated with the non-equilibrium direct or modified Urca process. The paper is organized as follows. In Sec. II we phe- nomenologically obtain the general form of the dissipa- tive corrections to the equations of relativistic hydrody- namics [18, 19], describing a superfluid liquid composed of identical particles. In Sec. III this dissipative hydro- dynamics is generalized to describe superfluid mixtures and applied to npe-matter. In Sec. IV we calculate and analyse all four bulk viscosity coefficients provided by non-equilibrium beta-processes. For illustration of these results, in Sec. V we calculate the characteristic damping times of sound waves in superfluid npe-matter. Sec. VI presents summary. II. THE DISSIPATIVE RELATIVISTIC HYDRODYNAMICS OF ONE-COMPONENT SUPERFLUID LIQUID In this section we obtain the general form of dissipa- tive terms entering the equations of relativistic superfluid hydrodynamics of electrically neutral liquid composed of identical particles. For that purpose, we need to choose a version of non-dissipative hydrodynamics. There is a number of equivalent formulations of non-dissipative rel- ativistic superfluid hydrodynamics [18, 20, 21, 22, 23, 24]. The most elegant and general amongst them seems to be the formulation of Carter [20, 21, 22] in which the hydro- dynamic equations follow from a convective variational principle. Most of the relativistic calculations (see Refs. [25, 26, 27]) modelling pulsations of superfluid neutron stars have been made within this approach (see, however, [19]). In this paper, we will not use the Carter’s hydrody- namics because it is an essentially phenomenological the- ory and it does not allow easy interpretation in terms of quantities calculated from microscopic theory. Since our main goal is the calculation of bulk viscosity coefficients, we will employ the hydrodynamics of Son [18]. It was initially proposed in the context of heavy-ion collisions and is derived directly from microscopic theory. There- fore, it is straightforward to relate various parameters of this hydrodynamics to microphysics. Using the notations of Ref. [19] it can be rewritten in a particularly simple form, which is a natural relativistic generalization of the http://arxiv.org/abs/0704.1071v2 standard non-relativistic superfluid hydrodynamics pio- neered by Tisza [28], Landau [29, 30], and Khalatnikov [31]. Although the Son’s description is ideal for comparing with microphysics, it has some serious disadvantages. In contrast to the Carter’s hydrodynamics, in which the ba- sic fluid variables are the particle number density current and the entropy density current, the Son’s hydrodynam- ics is a hybrid in a sense that its fluid variables are the rescaled entropy density current and the rescaled momen- tum of a particle (or a Cooper pair) from the condensate (in the literature it is traditionally and somewhat confus- edly referred to as ‘the superfluid velocity’). As a con- sequence, the Landau-type hydrodynamics of Son has a lower symmetry than that of Carter. However, it can be simply translated into the hydrodynamics of Carter as was demonstrated, for example, in the review paper by Andersson and Comer [32] (see their section 16.2). Below we will use the hydrodynamics of Son [18], em- ploying the notations of Ref. [19], convenient for our problem. Unless otherwise stated, the speed of light is set equal to c = 1. The hydrodynamic equations take the standard form µ = 0, (1) µν = 0, (2) where ∂µ ≡ ∂/∂xµ (space-time indices are denoted by Greek letters). Neglecting dissipation, the particle cur- rent density jµ and the energy-momentum tensor T µν can be presented in the form [19] jµ = nuµ + Y wµ, (3) T µν = (P + ε)uµuν + Pηµν +Y (wµwν + µwµuν + µwνuµ) . (4) Here n, P , ε, and µ (not to be confused with the space- time index µ!) are the number density, pressure, energy density, and relativistic chemical potential of particles, respectively; ηµν = diag(−1,+1,+1,+1) is the special- relativistic metric; Y is the relativistic analogue of su- perfluid density ρs. In the non-relativistic limit, we have Y = ρs/m 2, where m is the mass of a free particle. Fur- ther, uµ is the four-velocity of normal (non-superfluid) liquid component, normalized so that µ = −1; (5) wµ is the four-velocity, which describes the motion of superfluid liquid component. It can be expressed through some scalar function φ, wµ = ∂µφ− µuµ. (6) It is easy to verify that φ is related to the wave function phase Φ of the Cooper-pair condensate by the equality ▽▽▽φ = (~/2) ▽▽▽Φ (see Ref. [19]). In the non-relativistic limit spatial components of the four-vectors uµ and wµ are equal to uuu = VVV q, www = m(VVV s −VVV q), (7) where VVV s = ▽▽▽φ/m and VVV q are, respectively, the su- perfluid and normal velocities of the well known non- relativistic theory of superfluid liquids (see, e.g., Ref. [17]). The four-velocity wµ must satisfy one additional con- straint. This constraint fixes a comoving frame, that is the frame where we measure (and define) all the thermo- dynamic quantities. Choosing the constraint in the form µ = 0, (8) from Eqs. (3) and (4) one obtains that in this particu- lar case comoving is the frame where four-velocity equals uµ = (1, 0, 0, 0). It is straightforward to show that in this frame the basic thermodynamic quantities n, ▽▽▽φ, and ε are defined by j0 = n, (9) jjj = Y ▽▽▽φ, (10) T 00 = ε. (11) Using some equation of state and taking into account the second law of thermodynamics (T is the temperature, S is the entropy density) dε = T dS + µ dn+ d (wµwµ) , (12) as well as the definition of the pressure P ≡ −ε+ µn+ TS, (13) we can express all other thermodynamic quantities as functions of ε, n, and▽▽▽φ. Eqs. (1)–(13) fully describe the non-dissipative relativistic hydrodynamics of uncharged superfluid liquid. As a consequence of these formulae, one can easily derive the continuity equation for the entropy, ∂µ(Su µ) = 0. (14) Now let us include dissipation in the hydrodynam- ics described above. As in the non-dissipative case, we assume that in the comoving frame [in which uµ = (1, 0, 0, 0)], the basic thermodynamic quantities ε, n, and ▽▽▽φ are still defined by Eqs. (9)–(11). Being written in a relativistically invariant form, the conditions (9) and (10) imply that the particle current density jµ is still given by Eq. (3), where some four-velocity wµ sat- isfies the constraint (8) [as in the non-dissipative case]. In view of Eqs. (9) and (10), the most general expression for the four-velocity wµ is wµ ≡ ∂µφ− (µ+ κ)uµ. (15) Here a scalar κ is a small dissipative correction to be determined below. If we neglect dissipation, then κ = 0 and Eq. (15) coincides naturally with (6). Without any loss of generality, the expression for the energy-momentum tensor T µν can be presented in the form: T µν = (P + ε)uµuν + Pηµν +Y (wµwν + µwµuν + µwνuµ) + τµν . (16) Here τµν is an unknown dissipative tensor. In view of Eqs. (8) and (11), τµν satisfies the constraint uµuντ µν = 0. (17) Let us determine the dissipative corrections τµν and κ assuming they are linear in small gradients of hydrody- namic variables. For this aim we need to derive an en- tropy generation equation. It can be easily obtained from Eqs. (1)–(2) if we make use of Eqs. (3), (8), (12), (13), (15), and (16), µ = − ∂µ (Y w µ)− τµν ∂µ +Y wµ ∂µT + u ν Y wµ ∂νuµ. (18) Here the entropy current density Sµ is given by Sµ = Suµ − τµν − Y wµ, (19) and satisfies the natural constraint uµS µ = −S. Let us analyze the last two terms in Eq. (18). In ad- dition to a quadratic dependence on small gradients of hydrodynamic variables, they also depend on the four- velocity wµ. As follows from Eq. (7), the spatial part of the four-vector wµ is proportional to the difference be- tween the superfluid and normal velocities. In a large variety of problems this difference is small (as a conse- quence, the time component w0 is also small because of the constraint 8). In particular, it cannot exceed some (not very large) critical value ∆VVV cr at which superfluidity breaks down (see Sec. IV). For instance, if we study small perturbations of matter which is initially at rest (in ther- modynamic equilibrium with uµ = 0 and wµ = 0), then the last two terms in Eq. (18) are much smaller than the first two terms (a typical example of such situation is pro- vided by sound waves in superfluid npe-matter, see Sec. V). Below, we will neglect the last two terms in Eq. (18) when obtaining the dissipative corrections τµν and κ. Moreover, in the dissipative corrections we will also neglect small dissipative terms, explicitly depending on wµ. For example, we will neglect the terms of the form wµ∂µT and u µwν∂γu γ in the expressions for κ and τµν , respectively. An inclusion of these small terms would result in 13 kinetic coefficients describing dissipation in superfluid liquid. In the non-relativistic case the same approximation is used, for instance, in the textbook by Landau and Lifshitz [1] (see their §140) and in the mono- graph by Khalatnikov [17]. Since the entropy does not decrease, the right-hand side of Eq. (18) must be positive. This requirement puts certain restrictions on a general form of τµν and κ (linear in gradients). The standard consideration (see, e.g., Ref. [33]) shows that τµν = −κ (Hµγ uν +Hνγ uµ) ∂γT + Tu δ ∂δuγ − η Hµγ Hνδ ∂δuγ + ∂γuδ − ηγδ ∂εu − ξ1 H µν ∂γ (Y w γ)− ξ2 H µν ∂γu γ , (20) κ = −ξ3 ∂µ (Y w µ)− ξ4 ∂µu µ. (21) In these equations Hµν ≡ ηµν + uµuν ; κ and η are, re- spectively, the thermal conductivity and shear viscosity coefficients; ξ1,. . .,ξ4 are the bulk viscosity coefficients. From the Onsager symmetry principle it follows that ξ1 = ξ4. (22) For the positive definiteness of the quadratic form in the right-hand side of Eq. (18) it is necessary to have the kinetic coefficients κ, η, ξ2, and ξ3 positive and the coef- ficient ξ1 satisfying the inequality ξ21 ≤ ξ2ξ3. (23) In the non-relativistic limit the dissipative hydrody- namics proposed here coincides with the well known the- ory of Khalatnikov (see, e.g., Refs. [1, 17, 34]). For illus- tration, let us indicate how the bulk viscosity coefficients ξKh1,. . .,ξKh4 of Khalatnikov are related to those intro- duced in this paper. It is easy to demonstrate that ξKh1 = , ξKh2 = ξ2, (24) ξKh3 = , ξKh4 = . (25) Thus, in this section we have constructed the relativistic dissipative hydrodynamics of a superfluid liquid, com- posed of identical particles. It should be noted, that the dissipation was first included into the hydrodynamics [18] by Pujol and Davesne [35]. However, it is difficult to use their dissipative hydrodynamics in applications. The point is that the authors do not specify the comoving frame, where they define thermodynamic quantities. It is easy to verify that the frame which is defined as co- moving in our paper, cannot serve as comoving in Ref. [35]. III. VISCOSITY IN SUPERFLUID MIXTURES Let us apply the general formulae obtained in Sec. II to neutron star matter. As already mentioned in Sec. I, we consider the simplest model of neutron star cores composed of neutrons (n), protons (p), and electrons (e). All results of this and following sections can be easily generalized to the case of matter with more complicated composition (e.g., npe-matter with admixture of muons or hyperon matter). It is generally agreed that as a neutron star cools down the neutrons and protons become superfluid in its core. In such a system we have three velocity fields (instead of two, as in the previous section). They are superfluid velocity of neutrons, superfluid velocity of protons, and normal velocity uµ of neutron and proton Bogoliubov quasiparticles and electrons. In this section we do not consider the dissipative effects related to the diffusion of particles. Neglecting the diffusion, nucleon Bogoliubov excitations and electrons move with the same velocity uµ. In analogy with Sec. II, it is convenient to introduce four- velocities w and w instead of superfluid velocities of neutrons and protons, respectively. The Son’s version of non-dissipative hydrodynamics was extended to the case of npe-mixture in Ref. [19] (for earlier formulations see, e.g., Refs. [25, 36, 37, 38, 39, 40]). The main goal of this section is to include the viscous dissipative terms into the hydrodynamics [19]. Below the subscripts i and k refer to nucleons, i, k = n, p. Unless otherwise stated, the summation is assumed over repeated nucleon indices i and k. The full set of hydrodynamic equations describing su- perfluid mixtures consists of i) energy-momentum con- servation law (2) with the energy-momentum tensor T µν given by T µν = (P + ε)uµuν + Pηµν wν(k) + µiw uν + µk w + τµν ; (26) ii) particle conservation laws written for neutrons, pro- tons, and electrons (l = n, p, e), = 0, j = niu µ + Yikw = neu µ; (27) iii) constraints on the four-velocities w = 0, (28) and iv) the second law of thermodynamics, dε = T dS + µi dni + µe dne + w(k)µ . (29) To take into account potentiality of superfluid motion, four-velocities w should be expressed through some scalar functions φi and presented in the form = ∂µφi − qiA µ − (µi + κi)u µ. (30) Note that one can avoid introduction of these new func- tions φi in the hydrodynamics of superfluid mixtures if one formulates the potentiality condition (30) in the equivalent way + qiA µ + (µi + κi)u wν(i) + qiA ν + (µi + κi)u . (31) Below we will use the latter formulation because it is more suitable for our purpose. In this approach four- velocities w should be treated as independent hydro- dynamic variables. In Eqs. (26)–(31) µl and nl are, respectively, the rel- ativistic chemical potential and the number density of particle species l = n, p, e; Aµ is the four-potential of the electromagnetic field; qi is the electric charge of nu- cleon species i. Furthermore, Yik = Yki is a 2 × 2 sym- metric matrix which naturally appears in the theory as a generalization of the superfluid density to the case of superfluid mixtures. In the non-relativistic limit this ma- trix is related to the entrainment matrix ρik (see Refs. [19, 41, 42, 43]) by the equality Yik = ρik/(mimk), where mi is the mass of nucleon species i. The pressure P is defined in the same way as for non-superfluid npe-matter (compare with Eq. 13), P ≡ −ε+ µini + µene + TS. (32) The dissipative hydrodynamics formulated above differs from the hydrodynamics [19], describing superfluid mix- tures, only by the dissipative terms τµν , κn, and κp. The general form of these terms can be found from the entropy generation equation which is analogous to Eq. (18) [see Sec. II], µ = − − τµν ∂µ +Yikw ∂µT + u ν Yikw ∂νuµ, (33) where the entropy density current Sµ is Sµ = Suµ − τµν − . (34) Using the requirement that the entropy does not de- crease, one can easily obtain the dissipative terms τµν , κn, and κp from Eq. (33), τµν = −κ (Hµγ uν +Hνγ uµ) ∂γT + Tu δ ∂δuγ − η Hµγ Hνδ ∂δuγ + ∂γuδ − ηγδ ∂εu − ξ1i H µν ∂γ − ξ2 H µν ∂γu γ , (35) κn = −ξ3i ∂µ − ξ4n ∂µu µ, (36) κp = −ξ5i ∂µ − ξ4p ∂µu µ. (37) Here, as in Sec. II, we omit small dissipative terms, ex- plicitly depending on w and, in addition, we neglect particle diffusion. An inclusion of these terms would result in 19 kinetic coefficients as it has been recently demonstrated by Andersson and Comer [40] for the case of npe-matter (they obtained their result using a non- relativistic version of the Carter’s hydrodynamics). In Eqs. (35)–(37) ξ1i, ξ2, ξ3i, ξ4i, and ξ5i are the bulk viscosity coefficients (i = n, p). Some of them are related by the Onsager symmetry principle, ξ1i = ξ4i, ξ3p = ξ5n. (38) In addition, for positive definiteness of the quadratic form in the right-hand side of Eq. (33) one needs the following inequalities κ ≥ 0, η ≥ 0, ξ3n ≥ 0, ξ5p ≥ 0, ξ2 ≥ 0, ξ5pξ2 ≥ ξ 1p, ξ3nξ2 ≥ ξ 1n, ξ3nξ5p ≥ ξ 2ξ1nξ1pξ3p + ξ2ξ3nξ5p − ξ 1pξ3n − ξ 3pξ2 − ξ 1nξ5p ≥ 0. (39) Equations (33)–(39) are derived under the assump- tion that electrons and protons can move independently. However, this is not the case since they are charged. Any macroscopic motion of electrons is accompanied by that of protons to ensure quasineutrality condition (see, e.g., [19]), ne = np. (40) One can obtain then from the continuity equations (27) for protons and electrons (neglecting small ‘diffusive’ terms), = 0. (41) In principle (if we are not interested in the distribution of the electromagnetic field, which couples together elec- trons and protons), we can use this equation instead of the constraints (28) and (31) for protons. Equations (33)–(39) should be modified to take into account the conditions (40) and (41). As follows from Eq. (41), the only bulk viscosity coefficients which contribute to τµν and κn (see Eqs. 35 and 36) are ξ1n, ξ2, ξ3n, and ξ4n. Since we neglect the last two terms in the right- hand side of the entropy generation equation (33), only these four coefficients are responsible for dissipation of mechanical energy of macroscopic motion in superfluid npe-matter. IV. NON-EQUILIBRIUM BETA-PROCESSES AND THE CALCULATION OF BULK VISCOSITY In Sec. III we phenomenologically considered npe- matter and found the viscous terms in the relativistic hydrodynamic equations for superfluid mixtures. When doing this, we have ignored the fact that because of non- equilibrium beta-processes, the number of particles in the system is not conserved. In this section we demonstrate that the effect of non-equilibrium beta-processes is equiv- alent to the appearance of four effective bulk viscosity coefficients ξ1n, ξ2, ξ3n, and ξ4n in the hydrodynamic equations. In the npe-matter we have two types of beta-processes responsible for beta-equilibration. They are the direct Urca process and the modified Urca process. The power- ful direct Urca process is open only for some equations of state with large symmetry energy (and at large enough densities, when pFn ≤ pFp + pFe, pFn, pFp, and pFe being the Fermi momenta of neutrons, protons, and electrons, respectively). The direct and inverse reactions of this process have the form (see, e.g., Refs. [7, 9, 44]) n → p + e + ν̄, p + e → n + ν. (42) Here ν and ν̄ stand for neutrino and antineutrino, re- spectively. If the direct Urca process is forbidden by momentum conservation, then the main mechanism of beta-equilibration is the modified Urca process [7, 9, 44] N +n → N +p + e+ ν̄, N +p+ e → N +n+ ν. (43) An additional nucleon N = n, p here is needed to take an excess of momentum away and open the process. The full thermodynamic equilibrium includes beta- equilibrium and we have [44] δµ ≡ µn − µp − µe = 0. (44) In this case the number of direct reactions in a matter ele- ment per unit time is equal to the number of inverse reac- tions. Thus, the total number of particles of any species remains constant. In particular, the electron generation rate ∆Γ (that is the net number of electrons generated in beta-reactions in a unit volume per unit time) is zero, ∆Γ = 0. If we perturb the system, the condition (44) will not necessarily hold (δµ 6= 0) and the direct and inverse reac- tions will not precisely compensate each other (∆Γ 6= 0). In this paper we assume that the deviation from the equi- librium is small, δµ ≪ kBT . Then ∆Γ can be presented in the form [10, 11, 12, 45, 46] ∆Γ = λ δµ. (45) Here λ is a function of various thermodynamic quanti- ties defined in the equilibrium state (e.g., of the temper- ature and the particle number densities). For superfluid npe-matter this function was calculated by Haensel, Lev- enfish, and Yakovlev [10, 11]. Note, that these authors neglected the dependence of ∆Γ on the scalars wα w(k)α though (in principle) they can be non-zero in thermody- namic equilibrium. However, it seems that the results of Refs. [10, 11] are accurate as long as (as an example, we take the case of i = k = n) wµ(n)w ∼ mn (VVV sn −VVV q) ≪ kBT, (46) where VVV sn is the superfluid velocity of neutrons. A nu- merical estimate of Eq. (46) gives |∆VVV n| ≡ |VVV sn −VVV q| ≪ 0.01 c . (47) For clarity, we introduce the velocity of light c in this condition. On the other hand, as follows from the Landau criterion (see, e.g., Ref. [47]), superfluidity of neutrons breaks down if |∆VVV n| > |∆VVV cr|, where |∆VVV cr| ∼ kBTcn ≈ 0.0003 c Here ∆n is the energy gap in the neutron dispersion re- lation; Tcn is the critical temperature of neutron super- fluidity onset; n0 = 0.16 fm −3 is the number density of nucleons in saturated nuclear matter. It is worth not- ing that the criterion (48) is actually an upper limit on |∆VVV cr|. In reality, a superfluid state can be destroyed at much lower |∆VVV cr| due to the formation of vortices in superfluid matter [16, 48]. Comparing Eqs. (47) and (48) one can see that if the neutrons are superfluid then at not very low temperatures the condition (47) is always justified. To calculate the bulk viscosity coefficients we will use the non-dissipative hydrodynamics of superfluid mixtures [19] (see also Sec. III). The energy-momentum tensor T µν for such a hydrodynamics is given by Eq. (26), while the four-velocity w satisfies the conditions (28) and (31). Notice, that the dissipative components τµν and κi in Eqs. (26) and (31) should be taken zero, τ µν = 0 and κi = 0. Further, we assume that the quasineutrality condition (40) is fulfilled in both equilibrated and non- equilibrated matter. Using Eq. (40), the second law of thermodynamics for mixtures (29) can be rewritten in the form: dε = T dS + µndnb − δµ dne + w(k)µ . (49) Finally, let us assume that the four-velocities w = 0 in equilibrium. To take the non-equilibrium beta-processes into con- sideration it is necessary to add corresponding sources in the right-hand sides of the continuity equations for electrons, protons, and neutrons, ∂µ (neu µ) = ∆Γ, (50) µ + Ypkw = ∆Γ, (51) µ + Ynkw = −∆Γ. (52) When writing Eqs. (50)–(52) we bear in mind that every neutron decay is accompanied by the appearance of an electron and a proton (see the reactions 42 and 43). Taking into account the quasineutrality condition (40), one gets from Eqs. (50) and (51) the equality (41). Thus, Eq. (41) remains the same as in the absence of beta- processes. It is more convenient to use the continuity equation for baryons instead of Eqs. (51) and (52), be- cause non-equilibrium beta-processes do not influence the total number of baryons per unit volume, nb = nn + np. Summing together Eqs. (51) and (52) and using Eq. (41), one obtains µ + Ynkw = 0. (53) Let us assume that npe-matter is slightly perturbed out of thermodynamic equilibrium so that deviations from the equilibrium are small and one can linearize the hydrodynamic equations. Below we will work in the comoving (at one particular moment) frame asso- ciated with some element of npe-matter. In such a frame uµ = (1, 0, 0, 0). We can assume further that perturbations in the comoving frame depend on time t as exp(iωct), where ωc is the frequency of perturbation (measured in this frame). From the normalization condition (5) and Eq. (28) it follows that w0(i) = 0, (54) 0 = 0, ∂tw (i) = 0. (55) Using these equalities, one gets from Eqs. (50) and (53) ∂tne + div (neuuu) = ∆Γ, (56) ∂tnb + div nbuuu+ Ynkwww(k) = 0. (57) Here uuu and www(k) are the spatial components of four- vectors uµ and w , respectively. The number densi- ties of electrons ne and baryons nb can be presented as ne = ne0 + δne, nb = nb0 + δnb, where ne0 and nb0 are the equilibrium number densities, while δne and δnb are small non-equilibrium terms depending on time as exp(iωct). Here and hereafter the thermodynamic quan- tities related to the equilibrium state will be denoted by the subscript ‘0’. Using these notations as well as formula (45) and linearizing Eqs. (56)–(57), we get δne = [λ δµ− ne0 div(uuu)] , (58) δnb = − nb0 div(uuu) + div Ynkwww(k) . (59) Notice, that the chemical potential disbalance δµ in Eq. (58) depends on δne and δnb. Actually, δµ can gener- ally be presented as a function of nb, ne, T , and the scalars wµ(i)w (the proton number density is equal to the electron one, see the quasineutrality condition 40). One can neglect the temperature dependence of δµ in a strongly degenerate npe-matter (see, e.g., Refs. [49, 50]). Moreover, since the scalars wµ(i)w are of the second order smallness, their contribution to δµ is also negligi- ble (we recall that w = 0 in equilibrium). Expanding δµ(nb, ne) in Taylor series in the vicinity of its equilib- rium value (which is zero, δµ(nb0, ne0) = 0, see Eq. 44), one obtains in the first approximation δµ(nb, ne) = ∂δµ(nb0, ne0) δnb + ∂δµ(nb0, ne0) Analogous formulae can be written for perturbations of pressure δP ≡ P (nb, ne) − P0, neutron chemical poten- tial δµn ≡ µn(nb, ne) − µn0, and energy density δε ≡ ε(nb, ne)− ε0, ∂P (nb0, ne0) δnb + ∂P (nb0, ne0) δne, (61) δµn = ∂µn(nb0, ne0) δnb + ∂µn(nb0, ne0) δne, (62) ∂ε(nb0, ne0) δnb. (63) In the last equation we have neglected the term of the form [∂ε(nb0, ne0)/∂ne0] δne. From the second law of thermodynamics (49) we have ∂ε(nb0, ne0)/∂ne0 = −δµ. Therefore, this term is quadratically small and can be omitted. Using Eqs. (58)–(60) one finds δne = i ne0 ωc div(uuu) + ∂δµ(nb0, ne0) nb0 λ div(uuu) ∂δµ(nb0, ne0) λ div Ynkwww(k) , (64) where F ≡ ω2c + i ωc λ∂δµ(nb0, ne0)/∂ne0. For non- equilibrium Urca-processes and typical (for neutron stars) pulsation frequencies ωc ∼ 10 3 − 104 s−1, we have (see, e.g., Refs. [10, 11]) ωc ≫ λ |∂δµ(nb0, ne0)/∂ne0|. In this case Eq. (64) can be simplified by keeping only terms linear in λ. The result can be written as δne = δne1 + δne2, (65) where the first term equals δne1 = div(uuu) (66) and describes compression and decompression of the pul- sating matter. This term remains the same even in the absence of non-equilibrium beta-processes. The second term δne2 is due to non-equilibrium beta-processes δne2 = ∂δµ(nb0, xe0) div(uuu) ∂δµ(nb0, ne0) Ynkwww(k) . (67) Notice, that in this formula the partial derivative ∂δµ(nb0, xe0)/∂nb0 is taken at constant value of xe0 ≡ ne0/nb0. When obtaining Eq. (67) we used the identity ∂Ψ(nb0, xe0) = nb0 ∂Ψ(nb0, ne0) ∂Ψ(nb0, ne0) where Ψ is an arbitrary function of nb0 and ne0. Our further strategy is as follows. We take the energy- momentum tensor T µν for superfluid mixtures (with τµν=0) from Eq. (26) and expand all the thermodynamic quantities (e.g., the pressure P and the energy density ε), which determine this tensor, around their equilib- rium values. Restricting ourselves to linear perturbation terms, we obtain for the tensor T µν (in the comoving frame), T 00 = ε0 + δε, T 0j = T j0 = µi0Yik w T jm = (P0 + δP ) δjm. (69) Here, the spatial indices j and m are equal to 1, 2, 3; the relativistic entrainment matrix Yik is taken in equilib- rium; δP and δε are given by Eqs. (61) and (63), respec- tively. Let us assume for a while that there are no non- equilibrium beta-processes in the matter. In this case δne2 = 0 (see Eq. 67) and the matter is reversibly pulsat- ing around the equilibrium. Then the mechanical energy is not dissipating, and the entropy is conserved. Thus it is obvious that dissipative are only those terms in the tensor T µν which are directly related to δne2. Writing out these terms in the form of a separate tensor τ bulk, we τ00bulk = 0, bulk = τ bulk = 0, bulk = ∂P (nb0, ne0) δne2 δjm. (70) This tensor can be easily rewritten in an arbitrary frame if we take into account Eq. (67), bulk = ∂P (nb0, ne0) ∂δµ(nb0, ne0) ∂δµ(nb0, xe0) . (71) Comparing the tensor τ bulk with the phenomenological dissipative tensor τµν (see Eq. 35), we find the expres- sions for the effective bulk viscosity coefficients ξ1n and ξ2, generated by non-equilibrium beta-processes ξ1n = − ∂P (nb0, ne0) ∂δµ(nb0, ne0) , (72) ξ2 = − ∂P (nb0, ne0) ∂δµ(nb0, xe0) . (73) Let us do the same with the potentiality condition (31) on the four-velocity of neutrons w . As a result, we obtain in the comoving frame the dissipative component κn, appearing because of non-equilibrium beta-processes, ∂µn(nb0, ne0) δne2. (74) In a fully covariant form, this component is given by Eq. (36) where the effective bulk viscosity coefficients ξ3n and ξ4n are ξ3n = − ∂µn(nb0, ne0) ∂δµ(nb0, ne0) , (75) ξ4n = − ∂µn(nb0, ne0) ∂δµ(nb0, xe0) , (76) and, in addition, the condition (41) is taken into account. Thus, we have calculated the four effective bulk viscos- ity coefficients ξ1n, ξ2, ξ3n, and ξ4n. As will be shown in Sec. V, each of them makes a comparable contribution to characteristic damping times of mechanical energy. No- tice, that only the coefficient ξ2 is usually analyzed in the literature devoted to non-equilibrium beta-processes in superfluid matter. The expression (73) for ξ2 coincides with earlier results (see, e.g., Refs. [10, 11]). Not all of the coefficients (72)–(73) and (75)–(76) are independent. The coefficients ξ1n and ξ4n are equal be- cause of the Onsager principle (38). This can be shown if one applies the following relation for npe-matter (see, e.g., Ref. [10]), ∂P (nb0, xe0) = −n2b0 ∂δµ(nb0, xe0) . (77) Furthermore, it is easy to verify, that instead of one of the inequalities (39) relating the coefficients ξ1n, ξ2, and ξ3n, we have the strict equality ξ21n = ξ2ξ3n. (78) It is fulfilled only for those non-equilibrium processes, for which the expansion (65) is valid. Therefore, we have only two independent bulk viscosity coefficients. To prove Eq. (78) it is instructive to consider the en- tropy generation equation. Neglecting all the dissipative processes except for the non-equilibrium beta-processes (e.g., neglecting thermal conductivity, diffusion, shear viscosity), one can obtain from the hydrodynamics dis- cussed in this section T ∂µS µ = δµ∆Γ = λ δµ2. (79) Here we made use of Eq. (45). Since we are interested only in terms linear in λ, we can substitute δne1 for δne into Eq. (60) which determines δµ. On the other hand, the entropy generation equation in terms of the effective bulk viscosities takes the form (see Eqs. 33, (35)–(37), and 41), T ∂µS µ = ξ3n +(ξ1n + ξ4n) ∂µ µ + ξ2 (∂µu . (80) Let us compare the right-hand sides of Eqs. (79) and (80). It follows from Eqs. (60) and (79) that for any given uµ it is always possible to choose four-velocities w such a way, that δµ = 0 and the entropy generation rate vanishes (at some point and at some particular moment). In terms of the bulk viscosity formalism this means that one can vanish the quadratic form in the right-hand side of Eq. (80) by an appropriate choice of these velocities. This is possible only if the equality (78) is satisfied. V. DAMPING OF SOUND WAVES IN SUPERFLUID NPE-MATTER Let us illustrate the results of previous sections by calculating characteristic damping times of sound waves propagating in a homogeneous superfluid npe-matter. For simplicity, we consider only the damping due to the effective bulk viscosity. Neglecting dissipation, the sound modes of superfluid npe-matter have been thoroughly in- vestigated starting from the pioneering paper by Epstein [51] in which he argued that there would be two types of sound modes in neutron stars (see, e.g., Refs. [19, 38, 39]). In particular, Gusakov and Andersson [19] were the first who considered in full relativity sound modes in npe- matter at finite temperatures. Here we closely follow their analysis. The pulsation equations (81) and (82) of Ref. [19] can be used to describe sound waves tak- ing into account dissipation. Thus, there is no need to derive these equations from the hydrodynamics of super- fluid mixtures (Secs. III and IV) once again. Instead, we will rewrite them using the notations adopted in our paper. The result is (P0 + ε0)uuu+ µn0Ynkwww(k) = −▽▽▽δP, (81) µn0uuu+www(n) = −▽▽▽δµn. (82) The first equation is a consequence of the relativistic Eu- ler equation, which can be derived from Eq. (2) in a stan- dard way (see, e.g., Ref. [1]). The second equation follows from the condition (31) written for neutrons. To fully define the system, Eqs. (81) and (82) should be supple- mented by the condition (41). Using Eqs. (54) and (55), this condition can be presented in the form Ypkwww(k) = 0. (83) Now assuming that all the perturbations are plane waves proportional to exp(iωt− ikkkrrr), one obtains the following compatibility condition for Eqs. (81)–(83) [s = ω/k is the velocity of sound in units of c] y s4 + C1s 2 + C2 + δA = 0, (84) where Ypp nb0 µn0 (YnnYpp − YnpYpn) − 1, (85) µn0nb0 (β1 − γ1 − γ1y) + γ2 − β2 , (86) µn0nb0 (β2γ1 − β1γ2) , (87) ∂P (nb0, xe0) , γ2 = ∂µn(nb0, xe0) , (88) ∂P (nb0, ne0) , β2 = ∂µn(nb0, ne0) . (89) A small complex term δA appears in the compatibility condition (84) because of the bulk viscosity. It is given δA = − µ2n0 nb0 A1 + s , (90) A1 = µn0nb0 γ2 ξ1n − µn0 β2 ξ2 −P0nb0 γ1 ξ3n + P0 β1 ξ4n, (91) A2 = µn0 y ξ2 + ξ2 + n b0 ξ3n − nb0 ξ1n − nb0 ξ4n .(92) We remind that the bulk viscosity coefficients (and the quantities A1 and A2) depend on the frequency ω, A1,2 ∼ ω−2. The biquadratic equation (84) has two non-trivial solutions for two possible sound velocities. Neglecting dissipation, these modes have been analyzed in details in Ref. [19]. In particular, the sound velocities s 1 and s have been calculated there for the first and second modes. The dissipation leads to the appearance of small complex corrections δs1,2 to the velocities s 1,2 and consequently to decrements of sound waves. Since δA is small in com- parison with other terms in Eq. (84), one can use the perturbation theory in deriving the characteristic damp- ing times τ1,2. The parameters τ1 and τ2 are e-folding times of the pulsation amplitude for the first and second sound modes, respectively, τ1,2 ≈ k δs1,2 2 i s 2 y s 1,2 + C1 . (93) As follows from Eqs. (90)–(92), they are independent of ω. At T → Tcn we have Ynn, Ynp, Ypn → 0 and y ≈ nb0/(µn0Ynn) → ∞ (see Ref. [19] for a more detailed dis- cussion). In this limit the characteristic damping times 2P0 γ1 , (94) τ2 ≈ − 2µn0P0 γ1 D ω2 (γ1 A1 + µn0 D ξ2) . (95) Here we introduce the parameter D ≡ β2γ1 − β1γ2. At T > Tcn the neutrons are non-superfluid. In this case the second mode does not exist [formally, s 2 = 0], while the first mode is the usual sound wave. The characteristic FIG. 1: Characteristic damping times τ1,2 of sound waves versus temperature T for the first mode (two upper curves) and for the second mode (two lower curves). The solid curves demonstrate the damping of sound taking into account all four bulk viscosity coefficients. The dashed curves are calcu- lated assuming that only ξ2 is non-zero. The neutron critical temperature is indicated by the vertical dot-dashed line. The baryon number density is nb = 3n0 = 0.48 fm damping time for an ordinary sound wave is given by Eq. (94). As expected, its damping is governed by the only one bulk viscosity coefficient ξ2. For illustration, in Fig. 1 we present the character- istic damping times τ1,2 of sound waves (in years) as a function of temperature T for two sound modes. The figure is plotted for npe-matter with the baryon number density nb0 = 3n0. The critical temperature of neutrons is taken to be Tcn = 10 9 K. The protons are assumed to be non-superfluid. When calculating thermodynamic quantities and their derivatives we employed the equa- tion of state from Ref. [52]. It opens the direct Urca process at baryon number density of 5.84n0. Therefore, the process is forbidden for nb0 = 3n0. In this case, the main mechanism of energy dissipation is the non- equilibrium modified Urca process. To calculate the func- tion λ, which enters Eqs. (72)–(73) and (75)–(76) for the bulk viscosity coefficients, we have used the results of Ref. [11]. For calculating the relativistic entrainment matrix Yik we have employed the BJ v6 nucleon-nucleon poten- tial [19, 43, 53]. Actually, the microphysics input we use here to plot the figure is taken from Ref. [19] (see this reference for more details). The two upper curves correspond to the first sound mode, while the two lower curves – to the second mode. The solid curves are plotted taking into account all four bulk viscosity coefficients. The dashed curves are ob- tained under the assumption that all coefficients but ξ2 are equal to zero, ξ1n = ξ3n = ξ4n ≡ 0. As seen from the figure, the characteristic damping times of sound waves increase as the temperature de- creases. This is natural, because when the neutrons are superfluid, the Urca processes (and hence the bulk vis- cosity) are exponentially suppressed at T ≪ Tcn. Let us emphasize that at temperatures T <∼ 5 × 10 8 K, the shear viscosity of electrons can exceed the bulk viscosity generated by the non-equilibrium modified Urca process. As a result, the damping of sound waves will be mainly due to the shear viscosity. The first mode turns into the ordinary sound at T > Tcn. As follows from the figure, in the vicinity of neutron critical temperature the dissipation is primarily deter- mined by the bulk viscosity coefficient ξ2 (in accordance with Eq. 94). Consequently, near the transition point the solid and the dashed curves for the first mode coincide. On the contrary, the difference between the solid and the dashed curves for the second mode remains significant at any T < Tcn. The characteristic damping times for these two curves differ approximately by a factor of 3. It is worth noting that we would come to the similar conclusions if we considered sound waves in denser mat- ter, where the direct Urca process is open. In that case the characteristic damping times would be 6–7 orders of magnitude smaller, but the relative difference between the solid and the dashed curves will be approximately the same. Therefore, the main result of the present section is that the bulk viscosity coefficients ξ1n, ξ3n, and ξ4n es- sentially influence the dissipative properties of superfluid npe-matter and cannot be ignored. All four bulk viscos- ity coefficients should be considered on the same footing. VI. SUMMARY We performed a self-consistent analysis of the influ- ence of non-equilibrium beta-processes on dissipation of mechanical energy in superfluid matter of neutron stars. We start with the Son’s version of non-dissipative one- fluid relativistic hydrodynamics to describe superfluid mixtures (see Refs. [18, 19]). We determined the gen- eral form of dissipative terms entering the equations of this hydrodynamics. For simplicity, the effects of parti- cle diffusion were ignored. The equations of dissipative hydrodynamics were applied to the matter composed of neutrons, protons, and electrons (npe-matter). In this case the hydrodynamic equations contain four bulk vis- cosity coefficients rather than one, as in non-superfluid matter. It was demonstrated, that non-equilibrium beta- processes generate all four bulk viscosity coefficients, and only two of them are independent. The other two coef- ficients can be expressed through the first two by Eqs. (38) and (78). It is worth to emphasize that only the bulk viscosity coefficient ξ2 has been considered in the astrophysical literature so far. The expression (73) for ξ2 coincides with similar expressions of previous works (see, e.g., Refs. [10, 11, 45, 46]). To illustrate the results obtained in the present paper we considered a problem of damping of sound waves via the bulk viscosity due to non-equilibrium beta-processes in superfluid homogeneous npe-matter. It was shown that all four bulk viscosity coefficients make compara- ble contributions to the characteristic damping times of sound waves. Our results can be important for the analysis of vari- ous gravitational-driven instabilities in neutron stars, in particular, the r-mode instability (see, e.g., Ref. [4]). The additional bulk viscosity coefficients lead to a more effec- tive damping of these instabilities. Moreover, the results can be applied to the problems of rotochemical and gravi- tochemical heating of millisecond pulsars with superfluid cores. In the absence of superfluidity these problems were carefully analyzed in Refs. [49, 54, 55, 56]. The first at- tempt to discuss qualitatively the effects of superfluidity has been made in Ref. [57]. In conclusion let us note that the method of the bulk viscosity calculation, used here in the simple case of npe- matter, can be extended to matter with more compli- cated composition (npe-matter with admixture of muons, hyperon or quark matter). Such a generalization is be- yond the scope of the present study and will be consid- ered elsewhere. Acknowledgments The author is grateful to D.P. Barsukov and E.M. Kan- tor for discussions, to A.I. Chugunov for technical assis- tance, to D.G. Yakovlev for reading the manuscript and critical comments, and to A. Reisenegger and anonymous referee for very useful remarks. This research was sup- ported by RFBR (grants 05-02-16245 and 05-02-22003) and by the Federal Agency for Science and Innovations (grant NSh 9879.2006.2). [1] L. D. Landau and E. M. Lifshitz, Fluid mechanics, Course of theoretical physics, (Pergamon Press, Oxford, 1987). [2] L. Lindblom, Astrophys. J. 438, 265 (1995). [3] N. Andersson and K. D. Kokkotas, Int. J. Mod. Phys. D10, 381 (2001). [4] N. Andersson, Class. Quantum Grav. 20, R105 (2003). [5] N. Andersson, Proceedings of the conference Isolated Neutron Stars: from the Interior to the Surface (April 2006, London), eds. D. Page, R. Turolla, and S. Zane, As- trophysics and Space Science, in press, astro-ph/0610192. [6] H. Dong, N. Su, and Q. Wang (2007), astro-ph/0702181. http://arxiv.org/abs/astro-ph/0610192 http://arxiv.org/abs/astro-ph/0702181 [7] D. G. Yakovlev, K. P. Levenfish, and Yu. A. Shibanov, Physics-Uspekhi 42, 737 (1999). [8] U. Lombardo and H.-J. Schulze, in Physics of Neu- tron Star Interiors, Springer Lecture Notes in Physics (Springer, Berlin), Eds. D. Blaschke, N. K. Glen- denning, and A. Sedrakian, v. 578, p. 30 (2001), astro-ph/0012209. [9] D. G. Yakovlev and C. J. Pethick, Ann. Rev. Astron. Astrophys. 42, 169 (2004). [10] P. Haensel, K. P. Levenfish, and D. G. Yakovlev, Astron. Astrophys. 357, 1157 (2000). [11] P. Haensel, K. P. Levenfish, and D. G. Yakovlev, Astron. 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Carter, in: A Random Walk in Relativity and Cosmol- ogy, Proceedings of the Vadya-Raychaudhuri Festschrift, IAGRG, 1983, edited by Dadhich N., Krishna Rao J., Narlikar J.V., and Vishveshwara C.V. (Wiley Eastern, Bombay, 1985), pp. 48–62 [23] I.M. Khalatnikov and V.V. Lebedev, Phys. Lett. A91, 70 (1982). [24] V.V. Lebedev and I.M. Khalatnikov, Zh. Eksp. Teor. Fiz. 83, 1623 (1982) [Sov. Phys. JETP 56, 923 (1982)]. [25] G. L. Comer, D. Langlois, and L. M. Lin, Phys. Rev. D60, 104025 (1999). [26] N. Andersson, G. L. Comer, and D. Langlois, Phys. Rev. D66, 104002 (2002). [27] S. Yoshida and U. Lee, Phys. Rev. D67, 124019 (2003). [28] L. Tisza, Nature 141, 913 (1938). [29] L.D. Landau, Zh. Eksp. Teor. Fiz. 11, 592 (1941). [30] L.D. Landau, J. Physics 11, 91 (1947). [31] I.M. Khalatnikov, Zh. Eksp. Teor. Fiz. 23, 169 (1952). [32] N. Andersson and G. L. Comer, Living Reviews in Rela- tivity 10, 1 (2007). [33] S. Weinberg, Astrophys. J. 168, 175 (1971). [34] I. M. Khalatnikov, Zh. Eksp. Teor. Fiz. 23, 169 (1952). [35] C. Pujol and D. Davesne, Phys. Rev.C67, 014901 (2003). [36] G. Mendell, Astrophys. J. 380, 515 (1991a). [37] G. Mendell, Astrophys. J. 380, 530 (1991b). [38] L. Lindblom and G. Mendell, Astrophys. J. 421, 689 (1994). [39] N. Andersson and G. L. Comer, Mon. Not. R. Astron. Soc. 328, 1129 (2001). [40] N. Andersson and G. L. Comer, Class. Quant. Grav. 23, 5505 (2006). [41] A. F. Andreev and E. P. Bashkin, Zh. Eksp. Teor. Fiz. 69, 319 (1975) [Sov. Phys. JETP 42, 164 (1976)]. [42] M. Borumand, R. Joynt, and W. Kluźniak, Phys. Rev. C54, 2745 (1996). [43] M. E. Gusakov and P. Haensel, Nucl. Phys. A761, 333 (2005). [44] D. G. Yakovlev, A. D. Kaminker, O. Y. Gnedin, and P. Haensel, Phys. Rep. 354, 1 (2001). [45] R. F. Sawyer, Phys. Rev. D39, 3804 (1989). [46] P. Haensel, R. Schaeffer, Phys. Rev. D45, 4708 (1992). [47] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2, (Pergamon Press, Oxford, 1980). [48] R. P. Feynman, Statistical Mechanics, (Benjamin, Mas- sachusetts, 1972). [49] A. Reisenegger, Astrophys. J. 442, 749 (1995). [50] M. E. Gusakov, D. G. Yakovlev, and O. Y. Gnedin, Mon. Not. Astron. Soc. 361, 1415 (2005). [51] R. I. Epstein, Astrophys. J. 333, 880 (1988). [52] H. Heiselberg and M. Hjorth-Jensen, Astrophys. J. 525, L45 (1999). [53] A. D. Jackson, E. Krotscheck, D. E. Meltzer, and R. A. Smith, Nucl. Phys. A386, 125 (1982). [54] R. Fernández and A. Reisenegger, Astrophys. J. 625, 291 (2005). [55] A. Reisenegger, P. Jofre, R. Fernandez, and E. Kantor, Astrophys. J. 653, 568 (2006). [56] P. Jofré, A. Reisenegger, and R. Fernández, Phys. Rev. Lett. 97, 131102 (2006). [57] A. Reisenegger, Astrophys. J. 485, 313 (1997). http://arxiv.org/abs/astro-ph/0012209 http://arxiv.org/abs/astro-ph/0607643

0704.1072

A Faddeev Calculation for Pentaquark $\Theta^+$ in Diquark Picture with Nambu-Jona-Lasinio Type Interaction

A Faddeev Calculation for Pentaquark Θ+ in Diquark Picture with Nambu-Jona-Lasinio Type Interaction H. Mineo1,2,∗), J.A. Tjon1,3,∗∗), K. Tsushima4,5,∗∗∗), and Shin Nan Yang1,†) 1 Department of Physics, National Taiwan University, Taipei 10617, Taiwan 2 Institute of Atomic and Molecular Sciences, Academia Sinica, P.O.Box 23-166, Taipei 10617, Taiwan 3KVI, University of Groningen, The Netherlands 4National Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan 5Grupo de F́ısica Nuclear and IUFFyM, Universidad de Salamanca, E-37008 Salamanca, Spain A Bethe-Salpeter-Faddeev (BSF) calculation is performed for the pentaquark Θ+ in the diquark picture of Jaffe and Wilczek in which Θ+ is a diquark-diquark-s̄ three-body system. Nambu-Jona-Lasinio (NJL) model is used to calculate the lowest order diagrams in the two- body scatterings of s̄D and DD. With the use of coupling constants determined from the meson sector, we find that s̄D interaction is attractive while DD interaction is repulsive. However, we can not find a bound 1 pentaquark state. Instead, a bound pentaquark Θ+ channel is obtained with unphysically strong vector mesonic coupling constants. §1. Introduction Experimental situation on the existence of Θ+ is currently filled with conflicting positive and negative evidence and the question is still not yet fully settled.1) Even if it turns out that Θ+ does not exist, it would still be theoretically interest- ing to understand the absence of such an exotic. Many theoretical approaches have been used, including quark models, QCD sum rules, and lattice QCD, in addition to the chiral soliton model, to understand the properties and structure of Θ+ since its first sighting.2) One of the most intriguing theoretical ideas suggested for Θ+ is the diquark picture of Jaffe and Wilczek (JW)3) in which Θ+ is considered as a three- body system consisted of two scalar, isoscalar, color 3̄ diquarks (D’s) and a strange antiquark (s̄). It is based, in part, on group theoretical consideration. It would hence be desirable to examine such a scheme from a more dynamical perspective. It is known that diquark arises naturally from Nambu-Jona-Lasinio (NJL) model, an effective quark theory in the low energy region.4) NJL model conveniently incor- porates chiral symmetry and its spontaneously breaking which dictates the hadronic physics at low energy. Models based on NJL type of Lagrangians have been very successful in describing the low energy meson physics. Based on relativistic Fad- ∗) e-mail address: [email protected] ∗∗) e-mail address: [email protected] ∗∗∗) e-mail address: [email protected] †) e-mail address: [email protected] typeset using PTPTEX.cls 〈Ver.0.9〉 http://arxiv.org/abs/0704.1072v2 2 H. Mineo, J.A. Tjon, K. Tsushima, and S.N. Yang deev equation, the NJL model has also been applied to the baryon systems.5), 6) It has been shown that, using the quark-diquark approximation, one can explain the nucleon static properties and the qualitative features of the empirical valence quark distribution reasonably well.6) Consequently, we will employ NJL model to describe the dynamics of a (s̄DD) three-particle system in Faddeev formalism. We use relativistic equations to describe both the three-particle and its two-particle sub- systems, namely, the Bethe-Salpeter-Faddeev (BSF) equation7) and Bethe-Salpeter (BS) equations. In practice, Blankenbecler-Sugar reduction scheme is used to reduce the four-dimensional integral equation into three-dimensional ones. §2. SU(3)f NJL model and the diquark The SU(3)f NJL model is a chirally symmetric four-fermi contact interaction Lagrangian. With the use of Fierz transformations, the original NJL interaction Lagrangian LI can be rewritten, for the qq̄ channel, as LI,qq̄ = G1 (ψ̄λafψ) 2 − (ψ̄γ5λafψ) (ψ̄γµλafψ) 2 + (ψ̄γµγ5λafψ) (ψ̄γµλ0fψ) 2 + (ψ̄γµγ5λ0fψ) (ψ̄γµλ0fψ) 2 − (ψ̄γµγ5λ0fψ) + · · · , (2.1) where a = 0 ∼ 8, and λ0 I. For later use, we define G5 = G2 + Gv , with Gv ≡ G3 + G4. For the scalar, isoscalar diquark channel, interaction Lagrangian is given by LI,s = Gs ψ̄(γ5C)λ2fβ ψT (C−1γ5)λ2fβ , (2.2) where βAc = λA(A = 2, 5, 7) corresponds to one of the color 3̄c states. C = iγ is the charge conjugation operator, and λ′s are the Gell-Mann matrices. The constituent quark and diquark masses can be obtained from the gap equation and t-matrix of the diquark. Since we are only interested in a qualitatively study of the interactions inside Θ+, we will use the empirical values of the constituent quark masses Mu,d = 400 MeV, Ms = 600 MeV, and the diquark mass MD = 600 MeV as obtained in Ref.8) §3. Two-body interactions for s̄D and DD channels In the JW model for Θ+,3) symmetry consideration requires that the the spatial wave function of the two scalar-isoscalar, color 3̄ diquarks must be antisymmetric and the lowest possible state is p-state. Since Θ+ is of JP = 1 , s̄ would be in relative s-wave to the DD pair. Accordingly, we will consider only the configuration where s̄D and DD are in relative s- and p-waves, respectively. Fig. 1 shows the lowest order diagram, i.e., first order in LI,qq̄ in s̄D scat- tering. Trace properties in Dirac and flavor space limit the vertex Γ to only the vector-isoscalar term, (ψ̄γµλ0 ψ)2. For the DD interaction, the quark rearrangement Faddeev Calculation for Pentaquark Θ+ 3 diagram gives no contribution because of its color structure. The lowest order non- vanishing diagram of the first order in LI,qq̄ is given in Fig. 2 where only the direct term is shown. The corresponding exchange diagram vanishes again because of the color structure. Γ=λ f aγµ (a=0,8) Fig. 1. s̄D potential of the lowest order in LI,qq̄ . Fig. 2. Lowest order diagrams in DD scattering. With the use of the interaction Lagrangians of Eqs. (2.1-2.2), we obtain the following driving terms of Fig. 1 and 2, in the BS equations for s̄D and DD two- particle systems, < s̄fDf |V |s̄iDi > = (−v̄(ps̄i))(−iVs̄D)(pDi, pDf )v(ps̄f ), Vs̄D = GvFv(q 2)Ṽs̄D(pDi, pDf ), Ṽs̄D(pDi, pDf ) = (6pDi + 6pDf )/2 (3.1) − iVDD(~pDi, ~pDf ) = 128i 2)−G5(pD1i + pD1f ) · (pD2i + pD2f )F ,(3.2) where ps̄ and pD denote the four-momentum of the s̄-quark and diquark etc. The Fv and Fs are the vector and scalar form factors of the scalar diquark. For simplicity, we will assume that both take the dipole form, (1−q2/Λ2)−2, with Λ = 0.84 GeV. In the NJL model calculation with the Pauli-Villars cutoff,8) the coupling constants are related to the mesonic coupling constants by G1 = Gπ/2, G2 = Gρ/2 and G5 = Gω/2 which give Gv = −0.78 GeV −2. We remark that the sign of Gv is definitely negative since omega meson is heavier than the rho meson. The potential matrix elements of Eqs. (3.1-3.2) can then be used in the scattering equations obtained with the use of Blankenbecler-Sugar three-dimensional reduction scheme7) for the BS equation for both the s̄D and DD systems. The resultant scattering equations are solved to obtain the two-body t−matrix elements and the phase shifts. §4. Results for Θ+ and discussion Our results for the phase shifts are shown in Fig. 3(a). We see that the s-wave phase shifts for s̄D is positive which indicates the interaction is attractive, while the p-wave DD interaction is repulsive since their phase shift is negative. In Fig. 3(b) 4 H. Mineo, J.A. Tjon, K. Tsushima, and S.N. Yang we show the Gv dependence of the two-body s̄D binding energy. We see that with the type of interaction constructed in Sec. 3, a s̄D bound state begins to appear only when Gv becomes less than −5 ∼ −6 Gev −2, far too negative as compared to the physical value of -0.78 Gev−2 determined from the ρ− ω mass difference. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 pE [GeV](a) 6 8 10 12 14 -Gv [GeV](b) Fig. 3. (a) Phase shifts δl for the s̄D in s-wave with Gv = −0.78 GeV −2 (solid line) and DD in p-wave (dashed line). (b) Gv dependence of the s̄D binding energy. The three-body BSF equation7) takes the same form as the nonrelativistic one, Ti(s) = ti(s) + ti(s)G0(s) [Tj(s) + Tk(s)] , (4.1) where G0 is the free three-particle Green’s function and ti(s) is the two-particle t−matrix of particles j and k with (i, j, k) being a cyclic permutation of (1, 2, 3). If one uses the Blankenbecler-Sugar approximation for G0 and the two-body t−matrix elements obtained in Sec. 3, then the homogeneous equation of Eq. (4.1) can be solved9) to look for a possible three-body s̄DD bound state. We could not find a bound pentaquark in JP = 1 channel. However, we do see a pentaquark in JP = 1 channel when Gv becomes less that ∼ −8.0 GeV −2. Pentaquark binding energy EB(5q) grows from 77 to 505 MeV as Gv decreases from -8.0 to -14.0 GeV The effect of consierdable weaker attraction in the JP = 1/2+ channel is caused by the spectator particle being in a p-wave state. Acknowledgements One of the authors (S.N.Y.) thanks the Yukawa Institute for Theoretical Physics at Kyoto University, for warm hospitality extended to him during the YKIS2006 on ”New Frontiers on QCD”. References 1) T. Nakano, talk at YKIS2006 (Kyoto, Japan, November 20 - December 8, 2006). 2) T. Nakano et al. [LEPS Collaboration], Phys. Rev. Lett. 91, (2003) 012002. 3) R.L. Jaffe and F. Wilczek, Phys. Rev. Lett. 91, (2003) 232003. 4) Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1960); 124, 246 (1961). 5) S. Huang and J. Tjon, Phys. Rev. C49, 1702 (1994). 6) H. Mineo et al., Phys. Rev. C60, 065201 (1999); Nucl. Phys. A703, 785 (2002). 7) G. Rupp and J.A. Tjon, Phys. Rev. C37, (1988) 1729. 8) H. Mineo et al., Phys. Rev. C72, (2005) 025202. 9) A. Ahmadzadeh and J. Tjon, Phys. Rev. 147, (1966) 1111. Introduction SU(3)f NJL model and the diquark Two-body interactions for D and DD channels Results for + and discussion

0704.1075

New Terms for the Compact Form of Electroweak Chiral Lagrangian

arXiv:0704.1075v2 [hep-ph] 3 May 2007 Preprint typeset in JHEP style - HYPER VERSION TUHEP-TH-07156 New Terms for the Compact Form of Electroweak Chiral Lagrangian Hong-Hao Zhang, Wen-Bin Yan, and J. K. Parry Center for High Energy Physics & Department of Physics, Tsinghua University, Beijing 100084, China E-mail: [email protected], [email protected], [email protected] Xue-Song Li Science College, Hunan Agricultural University, Changsha 410128, China E-mail: [email protected] Abstract: The compact form of the electroweak chiral Lagrangian is a reformulation of its original form and is expressed in terms of chiral rotated electroweak gauge fields, which is crucial for relating the information of underlying theories to the coefficients of the low- energy effective Lagrangian. However the compact form obtained in previous works is not complete. In this letter we add several new chiral invariant terms to it and discuss the contributions of these terms to the original electroweak chiral Lagrangian. Keywords: Electroweak Chiral Lagrangian, Beyond Standard Model. http://arxiv.org/abs/0704.1075v2 mailto:[email protected] mailto:[email protected] mailto:[email protected] mailto:[email protected] http://jhep.sissa.it/stdsearch So far the postulated Higgs particle of the standard model has not been observed in ex- periments. We do not know what the electroweak symmetry breaking mechanism in nature is. The electroweak chiral Lagrangian is a general low-energy description for electroweak symmetry breaking patterns [1, 2], especially for those strong dynamical electroweak sym- metry breaking mechanisms [3]. All the coefficients of the electroweak chiral Lagrangian, in principle, can be fixed by experiments [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Due to the non- perturbative property of the possible strong dynamics, it is difficult to relate the measured coefficients of the electroweak chiral Lagrangian to underlying theories. Recently, the series of work of Ref. [14, 15, 16, 17, 18, 19, 20, 21] successfully produced the predictions for the coefficients of the chiral Lagrangian from the underlying theory of QCD, which lights up the hope of building up the relationship between the coefficients of the electroweak chiral Lagrangian and underlying strong dynamical models. As seen in Ref. [14], a crucial first step of the derivation of the chiral Lagrangian from QCD is to reformulate the original chiral Lagrangian in terms of chiral rotated external source fields. For the case of the elec- troweak chiral Lagrangian, we need to similarly reformulate it in terms of chiral rotated electroweak gauge fields in order to deduce the information of underlying theories. An attempt of this line of thought is the work of Ref. [22], which tries to give a so-called com- pact form of the electroweak chiral Lagrangian in terms of chiral rotated electroweak gauge fields. However, there are still several relevant terms not included in the reformulation of Ref. [22]. In this letter, we shall present 3 new terms for this reformulation and give their contributions to the original electroweak chiral Lagrangian. We refer the interested reader to Ref. [22] for all the details on the compact form of the electroweak chiral Lagrangian and the relations between the compact form and the original form. Besides the inner-product terms c2g b′µV b ′ν , c8g ν − dνV a′µAν , and c9g (dµV a ν already included in the compact form of Ref. [22], there should be 3 new cross-product terms which are given by, ∆L = c′ 2 + c′ a′b′(dµV ν − dνV ν + c′ a′b′(dµV a where a′, b′ run from 1 to 2, and ǫ12 = −ǫ21 = 1. Here and henceforth V a ′µ and Aµ are short for the chiral rotated gauge fields V and A in Ref. [22] respectively. There are also 3 cross-product terms corresponding to the c15,17,25-terms in that paper, but they vanish. Let us consider Eq. (1) term by term. From the definitions of Ref. [22], it is straightforward to obtain the relation between the first term of Eq. (1) and the ordinary terms of the electroweak chiral Lagrangian as follows, 2 = −4c′ [tr(XµXν)] − [tr(XµX µ)]2 − tr(XµXν)tr(τ µ)tr(τ3Xν) +tr(XµX µ)[tr(τ3Xν)] , (2) where Xµ ≡ U †(DµU). And the last two terms of Eq. (1) are respectively given by, ′b′(dµV ν − dνV b′µAν – 1 – tr(τ3Xν) tr(τ3Xµ)tr(XµXν)− tr(τ 3Xν)tr(XµX + itr(W µνX tr(τ3W µν)tr(τ , (3) with Wµν ≡ U W aµνU , and ′b′(dµV a (1− 4β1)c tr(τ3Xµ)tr(τ3Xν)tr(XµXν) −[tr(τ3Xµ)tr(τ . (4) From Eqs. (2), (3) and (4), we obtain the contributions of these 3 new terms to the original electroweak chiral Lagrangian as follows, ∆α3 = , ∆α4 = −4c , ∆α5 = 4c ∆α6 = 4c (1− 4β1)c , ∆α7 = −4c ∆α9 = − , ∆α10 = (1− 4β1)c . (5) The coefficients c′ 2,8,9 and ci (i = 1, 2, . . . , 25) in this compact form of the electroweak chiral Lagrangian are determined by the underlying ultraviolet theories. For example, if we take the one-doublet technicolor model as the underlying theory, it can be shown that these 3 new coefficients c′ 2,8,9 are all non-vanishing, and full details will be presented in forthcoming publications [23]. In summary, we have provided 3 new terms to the compact form of the electroweak chiral Lagrangian introduced in Ref. [22]. These additional terms were not considered in the previous work. In this letter we have related these new terms to the original electroweak chiral Lagrangian, which will be crucial in forthcoming studies of strong dynamical models. Acknowledgments We are indebted to Qing Wang for all our knowledge about the chiral Lagrangian and his helps and supports for this work. This work is supported in part by the National Natural Science Foundation of China. References [1] T. 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0704.1076

Seeing maximum entropy from the principle of virtual work

Microsoft Word - Virtualwork_maxent_revision.doc Seeing maximum entropy from the principle of virtual work Qiuping A. Wang Institut Supérieur des Matériaux et Mécaniques Avancées du Mans, 44 Av. Bartholdi, 72000 Le Mans, France Abstract We propose an extension of the principle of virtual work of mechanics to random dynamics of mechanical systems. The total virtual work of the interacting forces and inertial forces on every particle of the system is calculated by considering the motion of each particle. Then according to the principle of Lagrange-d’Alembert for dynamical equilibrium, the vanishing ensemble average of the virtual work gives rise to the thermodynamic equilibrium state with maximization of thermodynamic entropy. This approach establishes a close relationship between the maximum entropy approach for statistical mechanics and a fundamental principle of mechanics, and constitutes an attempt to give the maximum entropy approach, considered by many as only an inference principle based on the subjectivity of probability and entropy, the status of fundamental physics law. PACS numbers : 05.20.-y, 05.70.-a, 02.30.Xx, 1) Introduction The principle of maximum entropy (maxent) is widely used in the statistical sciences and engineering as a powerful tool and fundamental rule. The maxent approach in statistical mechanics can be traced back to the works of Boltzmann and Gibbs[3] and finally be given the status of principle thanks to the work of Jaynes[4] who used it with Boltzmann-Gibbs- Shannon (BGS) entropy (see below) to derive the canonical probability distribution for statistical mechanics in a simple manner. However, in spite of its success and popularity, maxent has always been at the center of scientific and philosophical discussions and has raised many questions and controversies[4][5][6]. A central question is why a thermodynamic system chooses the equilibrium microstates such that the BGS entropy gets to maximum. As a basic assumption of scientific theory, maxent is not directly or indirectly related to observation and undoubted facts. In the literature, maxent is postulated as such or justified either a priori by the second laws with additional hypothesis such as the entropy functional (Boltzmann or Shannon entropy)[6], or a posteriori by the correctness of the probability distributions derived from it[4]. In statistical inference theory, it was often justified by intuitive arguments based on the subjectivity of probability[4] or by relating it to other principles such as the consistency requirement and the principle of insufficient reason of Laplace, which have been the object of considerable criticisms[5]. Another important question about maxent is whether or not the BGS entropy is unique as the measure of uncertainty or disorder that can be maximized in order to determine probability distributions. This was already an question raised 40 years ago by the scientists who tried to generalize the Shannon entropy by mathematical considerations [9][10]. In the present work, we try to contribute to the debate around maxent by an attempt to derive maxent from a well known fundamental principle of classical mechanics, the virtual work principle or Lagrange-d’Alembert principle (LAP) [1][2] without additional hypotheses to LAP and about entropy property. LAP is widely used in physical sciences as well as in mechanical engineering. It is a basic principle capable of yielding all the basic laws of statics and of dynamics of mechanical systems. It is in addition a simple, clearly defined, easily understandable and palpable law of physics. It is hoped that this derivation is scientifically and pedagogically beneficial for the understanding of maxent and of the relevant questions and controversies around it. In this work, the term entropy, denoted by S, is used in the sense of the second law of thermodynamics for equilibrium system. 2) Principle of virtual work The variational calculus in mechanics has a long history which may be traced back to Galilei and other physicists of his time who studied the equilibrium problem of statics with LAP (or virtual displacement1). LAP gets unified and concise mathematical forms thanks to Lagrange[1] and d’Alembert[2] and is considered as a most basic principle of mechanics from which all the fundamental laws of statics and dynamics can be understood thoroughly. LAP says that the total work done by all forces acting on a system in static equilibrium is zero on all possible virtual displacements which are consistent with the constraints of the system. Let us suppose a simple case of a system of N points of mass in equilibrium under the action of N forces Fi (i=1,2,…N) with Fi on the point i, and imagine virtual displacement of each point ir δ for the point i. According to LAP, the virtual work Wδ of all the forces Fi on all ir δ cancels itself for static equilibrium, i.e. i rFW (1) This principle for statics was extended to dynamical equilibrium by d’Alembert[2] in the LAP by adding the initial force iiam − on each point: =⋅−∑= i ramFW (2) where mi is the mass of the poin i and ia its acceleration. From this principle, we can not only derive Newtonian equation of dynamics, but also other fundamental principles such as least action principle. 3) Why maximum thermodynamic entropy ? We suppose that the mechanics laws are usable not only for mechanical system containing small number of particles in regular motion, but also for large number of particles in random and stochastic motion for which one has to use statistical approach introducing probability distribution of mechanical states. Let us first consider an ensemble of equilibrium systems, 1 In mechanics, the virtual displacement of a system is a kind of imaginary infinitesimal displacement with no time passage and no influence on the forces. It should be perpendicular to the constraint forces. each composed of N particles in random motion with vi v the velocity of the particle i. It will be shown that the result for canonical ensemble can be easily extended to microcanonical ensemble and grand-canonical ensemble. Without loss of generality, let us look at a system without macroscopic motion, i.e., 0 v . We imagine that the system in thermodynamic equilibrium leaves the equilibrium state by a reversible infinitesimal virtual process. Let Fi be the force on a particle i of the system at that moment. Fi includes all the interacting forces particles-particles and particles-walls of the container. During the virtual process, each particle with acceleration ir&& has a virtual displacement r i vδ . The total virtual work on this displacement is given by i rrmFW δδ ⋅−∑= (3) Although the sum of the accelerations of all the particles vanishes, i.e., 0 irm && v , the acceleration ir&& on each particle can be nonzero. So in general 0 i rrm v&&v δ . As a matter of fact, we have kiiiiii ermrrmrrm δδδδ ==⋅=⋅ )2 v&v&vv&&v where eki is the kinetic energy of the particle. On the other hand, we suppose these are no dissipative forces in the system or on the particles. It means that the energy of the system will not change if the system is completely closed and isolated. Let epi be the potential energy of a particle i subjet to the force Fi , we should have pii eF −∇= and i ererF ∑−=⋅∑ ∇−=⋅∑ === 111 vvv (4) So finally it follows that ∑−=∑ +−= kipi eeeW )( δδδδ (5) where eiδ is a virtual variation of the total energy eee kipii += of the particle i and ∑= the total energy of the N particles. At this stage, no statistics has been done. The particles are treated as if they had regular dynamics. As a matter of fact, when the dynamics is random such as in a thermodynamic system, a microscopic process can leads the N particles from a given microstate to different microstates j with different probability pj (j=1,2 … w). If we looks at the system in the phase space, the considered process with given virtual displacements can take different directions or paths each leading to a given microstate with some likelihood. Hence the virtual work given by Eq.(5) is not a correct and complete expression for the random dynamics. It is in fact the virtual work of a possible process leading to a microstate j. It should be written as ( )∑−= jij eW δδ instead of Eq.(5). This "partial" virtual work cannot be used in the LAP since it is only a possible part of the total virtual work whose correct expression needs the introduction of the probability distribution of microstates pj. Logically, the total virtual work should be an average of the work given by Eq.(5) over all the possible microstates, i.e., WpW j jδδ ∑= (6) This expression is essential in the application of LAP, an approach originally for regular dynamics, to irregular and random dynamics. Eq.(6) makes it possible to introduce the dynamic uncertainty (entropy) into the variational approach as shown below. In terms of thermodynamic ensemble, Eq.(6) is the ensemble average of the virtual works of all the members of an ensemble of systems distributed over the microstates. It is this average which is measurable and has a physical sense in the case of random dynamics just as the usual average energy in thermodynamics. It is not conceivable to let the partial virtual work of Eq.(5) vanish because this would signifies that the random motion in each direction in phase space of the virtual process is regular according to LAP and there would be only one direction or phase path of the virtual process leading to only one microstate, which is contradictory with the hypothesis of the random dynamics. This reasoning is the essential difference of the present approach from the simple search for mechanics principle and a direct use of the latter to each possible state or trajectory in phase space. The use of mechanics principle in regular way in general yields regular mechanical laws irrelevant to thermodynamics. The dynamical randomness is the fact that not all the possible states or trajectories follow the regular mechanical laws due to noises, certain chaos, or to quantum mechanics in which the Newtonian laws are obeyed only statistically. The statistical Newtonian second law given in [11] is an example. Eq.(6) can be accounted for in an explicit way as follows. A microstate j is some distribution of the N particles over the one particle states k with energy εk where k varies from, say, 1 to g (g can be very large). We imagine Nj identical particles distributed over the g states at a microstate j which is here a combination of g numbers nk of particles over the g states, i.e., j={n1, n2, … ng, }. We naturally have ( )∑= jkj nN and ( ) ε k jkj nE ∑= . During the process of virtual work, only the energy of the particle can change (the fact that virtual work does not affect nk can be understood from quantum point of view since kε is discrete but virtual work is infinitesimal and continuous). For a given j with probability pj, the virtual work given in Eq.(5) is now ∑+−=∑ ∑+−∑ =−= kjkkjk kjkj nEnnnW εδδεδεδδεδ )()()()( . (7) The first term of the right hand side is the total energy variation due to the one particle energy variation kδε caused by the virtual work as well as to the variation in particle number jNδ of the system. The second term is just the energy variation caused by the particle number variation ∑= jkj nN )(δδ . Hence Eq.(6) reads NEnEnpEpW j δμδδεδδεδδ +−=∑+−=∑∑+∑−= )( . (8) where we put an expression for the chemical potential Nn kk δδεμ /∑= with ( ) ∑=∑ ∑=∑= j nnpNpN δδδδ and j j EpE δδ ∑= . Since ∑−= jj pEEE δδδ and pNNN δδδ with j jEpE ∑= and j jNpN ∑= , we get ∑ −++−∑ =−∑ ++−= jj pNENEpNNpEEW δμμδδδμμδδδδ )( (9) Now using the first law NWQE μδδδδ +−= for Grand-canonical ensemble, we identify the heat transfer ∑ −= jjj pNEQ δμδ )( . For a reversible virtual process, we can write ∑ −== jjj pNEQS δμββδδ )( and get NEW ++−= . (10) where S is the thermodynamic entropy of the second law. The following variational calculus for different ensemble is straightforward. According to LAP 0=Wδ , we have 0)( =+− NES βμβδ (11) which is the usual algorithm of maxent for grand-canonical ensemble. The only difference is that here the "constraints" associated with energy and particle number appear in the variational calculus as a simple consequence of LAP, in contrast to the introduction of these constraints in the inference theory or inferential statistical mechanics[4] by the argument that an averaged value of an observable quantity represents a factual information to be put into the maximization of information in order to derive unbiased probability distribution[5]. In order to see further details about this maxent, let us suppose the entropy is a function of the probability distribution pj of the considered moment, i.e., ...)...( ,2,1 jpppfS = . We can write j S δδ ∑ = due to the variations of the virtual process. On the other hand, we have jjj pNES δμβδ )( which implies 0)( =+−∑ δβμβ . (12) By virtue of the normalization condition 0=∑ jpδ , one can prove [12] that αβμβ =+− (13) with a constant α. Eq.(13) can be used for deriving the probability distribution of the nonequilibrium component of the dynamics if the functional f is given. Inversely, if the probability distribution is known, one can derive the functional of S. For canonical ensemble, we have 0=Nδ and 0)( =− ES βδ (14) or, by the same argument as above, αβ =− (15) For microcanonical ensemble, the system is completely closed and isolated with constant energy 0=Eδ and constant particle number 0=Nδ . When the virtual displacements occur, the total virtual work would be transformed into virtual heat such that Eq.(10) becomes 0=− QW δδ . LAP 0=Wδ leads to 0=Sδ or α= (16) which necessarily yields uniform probability distribution over the different microstates j, i.e., pj =1/w whatever is the form of the entropy S. Note that here the uniform distribution over the microstates is not an a priori assumption, but a consequence of LAP. This equiprobability can be proven as follows only by supposing that ...)...( ,2,1 jpppfS = is a strictly increasing or decreasing function of all pj throughout the interval 10 ≤≤ jp , i.e., its derivatives 0or 0 <> and are zero only at some finite number of points on the interval. However, Eq.(16) tells us that α= is a constant independent of pj, implying that ...)...( ,2,1 jpppfS = is either a linear function of all pj, or all pj are identical. It is evident that entropy cannot be linear function of pj. The equal probability of all microstates follows. The conclusion of this section is that, at thermodynamic equilibrium, the maxent under the constraints of energy is a consequence of the equilibrium condition LAP extended to random motion. From Eq.(8), one notices that maxent can be written in the following concise form for any ensemble with n random variables Xi (i=1,2 …n): ii Xδχ where iχ is some constant corresponding to iχ . For grand-canonical ensemble, (17) this is 0=− NE δμδ and for canonical ensemble, it is 0=Eδ . We stress that in the above derivation, the only essential assumptions or fundamental physical hypotheses used before the LAP are the first and second laws of thermodynamics for equilibrium system and reversible process. Hence the three algorithms of maxent for the three statistical ensembles are in principle valid for all systems for which the first and second laws are valid. We would like to mention here that this derivation of maxent is not associated with any given form of entropy like in the original version of Jaynes principle. 4) Concluding remarks This work shows that the maximum entropy principle has a close connection with the fundamental principle of classical mechanics, the principle of virtual work, i.e., for a mechanical system to be in thermodynamics equilibrium with maximum entropy, the total virtual work of all the forces on all the elements (particles) of the system should vanish. Indeed, if one admits that thermodynamic entropy is a measure of dynamical disorder and randomness, it is natural to say that this disorder must get to maximum in order that all the random forces act on each degree of freedom of the motion in such a way that over any possible (virtual) displacement, the work of all the forces is zero. In other words, this vanishing work can be obtained if and only if the randomness of the forces is at maximum over all degree of freedom allowed by the constraints to get stable equilibrium state. To our opinion, the present result is helpful not only for the understanding of maxent derived from a more basic and well understood mechanical principle, it also shows that entropy in physics is not necessarily a subjective quantity reaching maximum for correct inference, and that maximum entropy is a law of physics but not merely an inference principle. After finishing this paper, the author became aware of a work of Plastino and Curado[12] on the equivalence between the particular thermodynamic relation ES βδδ = and maxent in the derivation of probability distribution. They consider the particular thermodynamic process affecting only the microstate population in order to find a different way from maxent to derive probability. The work part is not considered in their work. Their analysis is pertinent and consequential. The present work provides a substantial support of their reasoning from a basic principle of mechanics. References [1] J.L. Lagrange, Mécanique analytique, Blanchard, reprint , Paris (1965) (Also: Oeuvres, Vol. 11.) [2] J. D’Alembert, Traité de dynamique, Editions Jacques Cabay , Sceaux (1990) [3] J. Willard Gibbs, Principes élémentaires de mécanique statistique (Paris, Hermann, 1998) [4] E.T. Jaynes, The evolution of Carnot's principle, The opening talk at the EMBO Workshop on Maximum Entropy Methods in x-ray crystallographic and biological macromolecule structure determination, Orsay, France, April 24-28, 1984; Gibbs vs Boltzmann entropies, American Journal of Physics, 33,391(1965) ; Where do we go from here? in Maximum entropy and Bayesian methods in inverse problems, pp.21-58, eddited by C. Ray Smith and W.T. Grandy Jr., D. Reidel, Publishing Company (1985) [5] Jos Uffink, Can the maximum entropy principle be explained as a consistency requirement, Studies in History and Philosophy of Modern Physics, 26B (1995): 223-261 [6] L.M. Martyushev and V.D. Seleznev, Maximum entropy production principle in physics, chemistry and biology, Physics Reports, 426, 1-45 (2006) [7] Y.P. Terletskii, Statistical physics, North-Holland Publishing Company, Amsterdam, 1971 [8] Q.A. Wang, Some invariant probability and entropy as a maximizable measure of uncertainty, to appear in J. Phys. A (2008); cond-mat/0612076 [9] A. Rényi, Calcul de probabilité, Paris, Dunod, 1966, P522 A. Wehrl, Rev. Mod. Phys., 50(1978)221 [10] M.D. Esteban, Kybernetika, 31(1995)337 [11] R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals, McGraw-Hill Publishing Company, New York, 1965 [12] A. Plastino and E.F.M. Curado, Phys. Rev. E, 72(2005)047103; E.F.M. Curado and A. Plastino, arXiv:cond-mat/0601076; also cond-mat/0509070

0704.1077

Microlocal Asymptotic Analysis in Algebras of Generalized Functions

Microlocal Asymptotic Analysis in Algebras of Generalized Functions Antoine Delcroix Equipe Analyse Algébrique Non Linéaire – Laboratoire A O C Faculté des sciences - Université des Antilles et de la Guyane, BP 250, 97157 Pointe à Pitre Cedex, Guadeloupe (France) [email protected] Jean-André Marti Equipe Analyse Algébrique Non Linéaire – Laboratoire G T S I Faculté des sciences – Université des Antilles et de la Guyane, BP 250, 97157 Pointe à Pitre Cedex, Guadeloupe (France) [email protected] Michael Oberguggenberger∗ Institute of Basic Sciences in Engineering – Unit of Engineering Mathematics Faculty of Civil Engineering – University of Innsbruck, Technikerstraße 13 – 6020 Innsbruck, Austria [email protected] November 6, 2018 Abstract We introduce a new type of local and microlocal asymptotic analysis in algebras of gen- eralized functions, based on the presheaf properties of those algebras and on the properties of their elements with respect to a regularizing parameter. Contrary to the more classical frequential analysis based on the Fourier transform, we can describe a singular asymptotic spectrum which has good properties with respect to nonlinear operations. In this spirit we give several examples of propagation of singularities through nonlinear operators. Keywords: microlocal analysis, generalized functions, nonlinear operators, presheaf, propaga- tion of singularities, singular spectrum. Mathematics Subject Classification (2000): 35A18, 35A27, 46E10, 46F30, 46T30 1 Introduction Various nonlinear theories of generalized functions have been developed over the past twenty years, with contributions by many authors. These theories have in common that the space of distributions is enlarged or embedded into algebras so that nonlinear operations on distribu- tions become possible. These methods have been especially efficient in formulating and solving nonlinear differential problems with irregular data. Most of the algebras of generalized functions possess the structure of sheaves or presheaves, which may contain some sub(pre)sheaves with particular properties. For example, the sheaf ∗Supported by FWF (Austria), grant Y237. http://arxiv.org/abs/0704.1077v1 G of the special Colombeau algebras [2, 7, 15] contains the subsheaf G∞ of so-called regular sections of G such that the embedding: G∞ → G is the natural extension of the classical one: C∞ → D′. This notion of regularity leads to G∞-local or microlocal analysis of generalized functions, extending the classical results on the C∞-microlocal analysis of distributions due to Hörmander [8]. This concept has been slightly extended in [4] to less restrictive kinds of measuring regularity. In [14], microlocal regularity theory in analytic and Gevrey classes has been generalized to algebras of generalized functions. Many results on propagation of singularities and pseudodifferential techniques have been obtained during the last years (see [5, 6, 9, 10, 11]). Nevertheless, these results are still mainly limited to linear cases, since they use frequential methods based on the Fourier transform. In this paper, we develop a new type of asymptotic local and microlocal analysis of gener- alized functions in the framework of (C, E ,P)-algebras [12, 13], following first steps undertaken in [12]. An example of the construction is given by taking G as a special case of a (C, E ,P)- structure (see Subsection 2.2 for details). Let F be a subsheaf of vector spaces (or algebras) of G and (uε)ε a representative of u ∈ G (Ω) for some open set Ω ⊂ R n. We first define OFG (u) as the set of all x ∈ Ω such that uε tends to a section of F above some neighborhood of x. The F-singular support of u is Ω\OFG (u). For fixed x and u, Nx(u) is the set of all r ∈ R+ such that εruε tends to a section of F above some neighborhood of x. The F-singular spectrum of u is the set of all (x, r) ∈ Ω × R+ such that r ∈ R+\Nx(u). It gives a spectral decomposition of the F-singular support of u. This asymptotic analysis is extended to (C, E ,P)-algebras. This gives the general asymptotic framework, in which the net (εr)ε is replaced by any net a satisfying some technical conditions, leading to the concept of the (a,F)-singular asymptotic spectrum. The main advantage is that this asymptotic analysis is compatible with the algebraic structure of the (C, E ,P)-algebras. Thus, the (a,F)-singular asymptotic spectrum inherits good properties with respect to nonlin- ear operations (Theorem 15 and Corollary 16). The paper is organized as follows. In Section 2, we introduce the sheaves of (C, E ,P)-algebras and develop the local asymptotic analysis. Section 3 is devoted to the (a,F)-microlocal analysis and specially to the nonlinear properties of the (a,F)-singular asymptotic spectrum. In Section 4 various examples of the propagation of singularities through non linear differential operators are given. 2 Preliminary definitions and local parametric analysis 2.1 The presheaves of (C, E ,P)-algebras: the algebraic structure We begin by recalling the notions from [12, 13] that form the basis for our study. (a) Let: (1) Λ be a set of indices; (2) A be a solid subring of the ring KΛ (K = R or C); this means that whenever (|sλ|)λ ≤ (|rλ|)λ for some ((sλ)λ, (rλ)λ) ∈ K Λ ×A, that is, |sλ| ≤ |rλ| for all λ, it follows that (sλ)λ ∈ A ; (3) IA be a solid ideal of A ; (4) E be a sheaf of K-topological algebras over a topological space X . Moreover, suppose that (5) for any open set Ω in X, the algebra E(Ω) is endowed with a family P(Ω) = (pi)i∈I(Ω) of semi-norms such that if Ω1, Ω2 are two open subsets of X with Ω1 ⊂ Ω2, it follows that I(Ω1) ⊂ I(Ω2) and if ρ 1 is the restriction operator E(Ω2) → E(Ω1), then, for each pi ∈ P(Ω1) the semi-norm p̃i = pi ◦ ρ 1 extends pi to P(Ω2) . (6) Let Θ = (Ωh)h∈H be any family of open sets in X with Ω = ∪h∈HΩh. Then, for each pi ∈ P(Ω), i ∈ I(Ω), there exist a finite subfamily of Θ: Ω1, . . . , Ωn(i) and corresponding semi-norms p1 ∈ P(Ω1), . . . , pn(i) ∈ P(Ωn(i)), such that, for any u ∈ E(Ω) pi (u) ≤ p1 (u |Ω1 ) + . . .+ pn(i)(u |Ωn(i)). (b) Define |B| = {(|rλ|)λ , (rλ)λ ∈ B}, B = A or IA, and set H(A,E,P)(Ω) = (uλ)λ ∈ [E(Ω)] | ∀i ∈ I(Ω), ((pi(uλ))λ ∈ |A| J(IA,E,P)(Ω) = (uλ)λ ∈ [E(Ω)] | ∀i ∈ I(Ω), (pi(uλ))λ ∈ |IA| C = A/IA, Note that, from (2), |A| is a subset of A and that A+ = {(bλ)λ ∈ A, ∀λ ∈ Λ, bλ ≥ 0} = |A|. The same holds for IA. Furthermore, (2) implies also that A is a K-algebra. Indeed, it suffices to show that A is stable under multiplication by elements of K. Let c be in K and (aλ)λ ∈ A. Then (caλ)λ satisfies (|caλ|)λ ≤ (|naλ|)λ for some n ∈ N. We have (naλ)λ ∈ A since A is stable under addition. Thus, using (2), we get that (caλ)λ ∈ A. For later reference, we recall the following notions entering in the definition of a sheaf A on X. Let (Ωh)h∈H be a family of open sets in X with Ω = ∪h∈HΩh. (F1) (Localization principle) Let u, v ∈ A(Ω). If all restrictions u|Ωh and u|Ωh , h ∈ H, coincide, then u = v in A(Ω). (F2) (Gluing principle) Let (uh)h∈H be a coherent family of elements of A(Ωh), that is, the restrictions to the non-void intersections of the Ωh coincide. Then there is an element u ∈ A(Ω) such that u|Ωh = uh for all h ∈ H. Proposition 1 (i) H(A,E,P) is a sheaf of K-subalgebras of the sheaf E (ii) J(IA,E,P) is a sheaf of ideals of H(A,E,P). Proof. The proof can be found in [12, 13], so we just recall the main steps. We start from the statement that E and EΛ are already sheaves of algebras. From (5), we infer that H(A,E,P) and J(IA,E,P) are a presheaves (the restriction property holds) and that the localization property (F1) is valid. To obtain the gluing property (F2) we need property (6), which generalizes the situation from C∞ to E . Theorem 2 The factor H(A,E,P)/J(IA,E,P) is a presheaf satisfying the localization principle (F1). Proof. From the previous proposition, we know that A = H(A,E,P)/J(IA,E,P) is a presheaf. For Ω1 ⊂ Ω2, the restriction is defined by A (Ω2) −→ A (Ω1) u 7−→ u |Ω1 := [uλ |Ω1 ] where (uλ)λ is any representative of u ∈ A(Ω2) and [uλ |Ω1 ] denotes the class of (uλ |Ω1)λ. The definition is consistent and independent of the representative because for each (uλ)λ∈Λ ∈ H(A,E,P)(Ω2) and (ηλ)λ∈Λ ∈ J(IA,E,P)(Ω2), we have (uλ)λ |Ω1 := (uλ |Ω1 )λ ∈ H(A,E,P)(Ω1) , (ηλ)λ |Ω1 := (ηλ |Ω1 )λ ∈ J(IA,E,P)(Ω1) The localization principle is also obviously fulfilled because J(IA,E,P) is itself a sheaf. Proposition 3 Under the hypothesis (2), the constant sheaf H(A,K,|.|)/J(IA,K,|.|) is exactly the ring C = A/IA. Proof. We clearly have H(A,K,|.|) = A and J(IA,K,|.|) = IA. Definition 1 The factor presheaf of algebras over the ring C = A/IA: A = H(A,E,P)/J(IA,E,P) is called a presheaf of (C, E ,P)-algebras. Notation 1 We denote by [uλ] the class in A(Ω) defined by (uλ)λ∈Λ ∈ H(A,E,P)(Ω). For u ∈ A, the notation (uλ)λ∈Λ ∈ u means that (uλ)λ∈Λ is a representative of u. Remark 1 The problem of rendering A a sheaf (and even a fine sheaf) is not studied here. It is well known that the Colombeau algebra G, which is a special case of a (C, E ,P)-algebra (see Subsection 2.2), forms a fine sheaf [1, 7]. The sheaf property can be inferred from the existence of a C∞-partition of unity associated to any open covering of an open set Ω of Rd. This existence is fulfilled because X = Rd is a locally compact Hausdorff space. On the other hand, C∞ is a fine sheaf because multiplication by a smooth function defines a sheaf homomorphism in a natural way. Hence the usual topology and C∞-partition of unity defines the required sheaf partition of unity. Observing that G is a sheaf of C∞-modules and using the well known result that a sheaf of modules on a fine sheaf is itself a fine sheaf, we obtain the corresponding assertion about G. In the general case, turning A into a sheaf requires additional hypotheses, which are not necessary for the results in this paper. Indeed, the presheaf structure of A and the (F1)-principle are sufficient to develop our local and microlocal asymptotic analysis. Remark 2 The map ι : K → A defined by ι (r) = (r)λ is an embedding of algebras and induces a ring morphism from K → C if, and only if, A is unitary (Lemma 14, [13]). Indeed, if A is unitary, (r)λ = r (1λ)λ is an element of A since A is a K-algebra, and ι is clearly an injective ring morphism. The converse is obvious. Moreover, if Λ is a directed set with partial order relation ≺ and if (7) IA ⊂ (aλ)λ ∈ A | lim aλ = 0 then the morphism ι is injective. Indeed, if [ι (r)] = 0, relation (7) implies that the limit of the constant sequence (r)λ is null, thus r = 0. 2.2 Relationship with distribution theory and Colombeau algebras One main feature of this construction is that we can choose the triple (C, E ,P) such that the sheaves C∞ and D′ are embedded in the corresponding sheaf A. In particular, we can multiply (the images of) distributions in A. We consider the sheaf E = C∞ over Rd, whereP is the usual family of topologies (PΩ)Ω∈O(Rd). Here O denotes the set of all open sets of Rd; this notation will be used in the sequel. Let us recall that PΩ is defined by the family of semi-norms (pK,l)K⋐Ω,l∈N with ∀f ∈ C∞ (Ω) , pK,l (f) = sup x∈K,|α|≤l |∂αf (x)| . From Lemma 14 in [13], it follows that the canonical maps, defined for any Ω ∈ O σΩ : C ∞ (Ω) → H(A,E,P)(Ω) f 7→ (f)λ , are injective morphism of algebras if, and only if, A is unitary. Under this assumption, these maps give rise to a canonical sheaf embedding of C∞ into H(A,E,P) and (using a partition of unity in C∞ inducing a sheaf structure on A) to a canonical sheaf morphism of algebras from C∞ into A. This sheaf morphism turns out to be a sheaf morphism of embeddings if Λ is a directed set with respect to a partial order ≺ and if relation (7) holds. We shall address the question of the embedding of D′ for the simple case of Λ = (0, 1]. For a net (ϕε)ε of mollifiers given by ϕε (x) = , x ∈ Rd where ϕ ∈ D(Rd) and ϕ (x) dx = 1, and T ∈ D′ , the net (T ∗ ϕε)ε is a net of smooth functions in C , moderately increasing in . This means that (8) ∀K ⋐ Rd,∀l ∈ N, ∃m ∈ N : pK,l (T ∗ ϕε) = o(ε −m), as ε→ 0. This justifies to choose (rε)ε ∈ R (0,1] | ∃m ∈ N : |uε| = o(ε −m), as ε→ 0 (rε)ε ∈ R (0,1] | ∀q ∈ N : |uε| = o(ε q), as ε→ 0 In this case (with E = C∞), the sheaf of algebras A = H(A,E,P)/J(IA,E,P) is exactly the so-called special Colombeau algebra G [2, 7, 16]. Then, for all Ω ∈ O , C∞ (Ω) is embedded in A (Ω) σΩ : C ∞(Ω) → A(Ω) f 7→ [fε] with fε = f for all ε in (0, 1] , because the constant net (f)ε belongs to H(A,E,P) and (f)ε ∈ J(IA,E,P) implies f = 0 in C∞(Ω). Furthermore, D′ is embedded in A by the mapping ι : T 7→ (T ∗ ϕε)ε Indeed, relation (8) implies that (T ∗ ϕε)ε belongs to H(A,E,P) and (T ∗ ϕε)ε ∈ J(IA,E,P) implies that T ∗ϕε → 0 in D , as ε→ 0 and T = 0. Thus, ι is a well defined injective map. With the help of cutoff functions, we can define analogously, for each open set Ω in Rd, an embedding ιΩ of D ′ (Ω) into A (Ω), and finally a sheaf embedding D′ → A. This embedding depends on the choice of the net of mollifiers (ϕε)ε. We refer the reader to [3, 15] for more complete discussions about embeddings in Colombeau’s case and to [13] for the case of (C, E ,P)- algebras. 2.3 An association process We return to the general case with the assumption that A is unitary and Λ is a directed set with partial order relation ≺ . Let us denote by: • Ω an open subset of X, • F a given sheaf (or presheaf) of topological K-vector spaces (resp. K-algebras) over X containing E as a subsheaf of topological algebras, • a a map from R+ to A+ such that a(0) = 1 (for r ∈ R+, we denote a (r) by (aλ (r))λ). In the Colombeau case, a typical example would be aε(r) = ε r, ε ∈ (0, 1]. For (vλ)λ ∈ H(A,E,P) (Ω), we shall denote the limit of (vλ)λ for the F-topology by lim F(Ω) vλ when it exists. We recall that lim F(V ) uλ |V = f ∈ F(V ) iff, for each F-neighborhood W of f , there exists λ0 ∈ Λ such that λ ≺ λ0 =⇒ uλ|V ∈W. We suppose also that we have, for each open subset V ⊂ Ω, (9) J(IA,E,P)(V ) ⊂ (vλ)λ ∈ H(A,E,P)(V ) : lim F(V ) vλ = 0 Definition 2 Consider u = [uλ] ∈ A(Ω), r ∈ R+, V an open subset of Ω and f ∈ F(V ). We say that u is a (r)-associated with f in V : F(V ) if lim F(V ) (aλ (r) uλ |V ) = f. In particular, if r = 0, u and f are called associated in V . To ensure the independence of the definition with respect to the representative of u, we must have, for any (ηλ)λ ∈ J(IA,E,P)(Ω), that lim F(V ) aλ (r) ηλ |V = 0. As J(IA,E,P)(V ) is a module over A, (aλ (r) ηλ |V )λ is in J(IA,E,P)(V ). Thus, our claim follows from hypothesis (9). Example 1 Take X = Rd, F = D′, Λ =]0, 1], A = G, V = Ω, r = 0. The usual association between u = [uε] ∈ G (Ω) and T ∈ D ′ (Ω) is defined by u ∼ T ⇐⇒ u D′(Ω) T ⇐⇒ lim D′(Ω) uε = T. 2.4 The F-singular support of a generalized function We use the notations of Subsection 2.3. According to the hypothesis (9), we have, for any open set Ω in X, J(IA,E,P)(Ω) ⊂ (uλ)λ ∈ H(A,E,P)(Ω) : lim F(V ) uλ = 0 FA(Ω) = u ∈ A(Ω) | ∃ (uλ)λ ∈ u, ∃f ∈ F(Ω) : lim F(V ) uλ = f FA(Ω) is well defined because if (ηλ)λ belongs to J(IA,E,P)(Ω), we have lim F(V ) ηλ = 0. Moreover, FA is a sub-presheaf of vector spaces (resp. algebras) of A. Roughly speaking, it is the presheaf whose sections above some open set Ω are the generalized functions of A (Ω) associated with an element of F (Ω). Thus, for u ∈ A (Ω), we can consider the set OFA (u) of all x ∈ Ω having an open neighbor- hood V on which u is associated with f ∈ F (V ), that is: OFA (u) = {x ∈ Ω | ∃V ∈ Vx : u |V ∈ FA(V )} , Vx being the set of all the open neighborhoods of x. This leads to the following definition: Definition 3 The F-singular support of u ∈ A(Ω) is denoted SFA (u) and defined as SFA (u) = Ω\O A (u) . Remark 3 (i) The validity of the gluing principle (F2) is not necessary to get the notion of support (and of F-singular support) of a section u ∈ A(Ω). More precisely, the localization principle (F1) is sufficient to prove the following: The set A (u) = {x ∈ Ω | ∃V ∈ Vx, u |V = 0} is exactly the the union ΩA (u) of the open subsets of Ω on which u vanishes. Indeed, (F1) allows to show that u vanishes on an open subset O of Ω if, and only if, it vanishes on an open neighborhood of every point of O. This leads immediately to the required assertion. Moreover, ΩA (u) = O A (u) is the largest open set on which u vanishes, S A (u) = Ω\O A (u) is exactly the support of u in its classical definition, and the F-singular support of u is a closed subset of its support. (ii) In contrast to the situation described above for the support, we need the gluing principle (F2) if we want to prove that the restriction of u to O A (u) belongs to FA(O A (u)). We make this precise in the following lemma. Lemma 4 Take u ∈ A(Ω) and set ΩFA (u) = ∪i∈IΩi, (Ωi)i∈I denoting the collection of the open subsets of Ω such that u |Ωi ∈ FA (Ωi). Then, if FA is a sheaf (even if A is only a prehesaf), (i) ΩFA (u) is the largest open subset O of Ω such that u |O belongs to FA (O); (ii) ΩFA (u) = O A(u) and S A (u) = Ω \ Ω A (u). Proof. (i) For i ∈ I, set u |Ωi = fi ∈ FA (Ωi). The family (fi)i∈I is coherent by assumption: From (F2), there exists f ∈ FA(Ω A (u)) such that f |Ωi = fi. But from (F1), we have f = u on ∪i∈IΩi = Ω A (u). Thus u |ΩF (u) ∈ FA(Ω A (u)), and Ω A (u) is clearly the largest open subset of Ω having this property. (ii) First, OFA (u) is clearly an open subset of Ω. For x ∈ O A (u), set u |Vx = fx ∈ FA (Vx) for some suitable neighborhood Vx. The open set O A (u) can be covered by the family (Vx)x∈OF As the family (fx) is coherent, we get from (F2) that there exists f ∈ FA ∪x∈OF (u)Vx such that f |Vx = fx. From (F1), we have u = f on ∪x∈OF (u)Vx and, therefore, u |OF (u) ∈ FA(O A (u)). Thus OFA (u) is contained in Ω A (u). Conversely, if x ∈ Ω A (u), there exists an open neigh- borhood Vx of x such that u |Vx ∈ FA (Vx). Thus x ∈ O A (u) and the assertion (ii) holds. Proposition 5 For any u, v ∈ A(Ω), if F is a presheaf of topological vector spaces, (resp. algebras), we have: SFA (u+ v) ⊂ S A (u) ∪ S A(v). Moreover, in the resp. case, we have SFA (uv) ⊂ S A(u) ∪ S A(v). Proof. If x ∈ Ω belongs to OFA(u) ∩ O A(v), there exist V and W in Vx such that u |V ∈ FA(V ) and v |W ∈ FA(W ). Thus (u + v)|V ∩W ∈ FA(V ∩W ) (resp. (uv)|V ∩W ∈ FA(V ∩W )), which implies OFA(u) ∩ O A(v) ⊂ O A(u+ v) (resp. O A(u) ∩O A(v) ⊂ O A (uv) ). The result follows by taking the complementary sets in Ω. This proposition leads easily to the following: Corollary 6 Let (uj)1≤j≤p be any finite family of elements in A(Ω). If F is a presheaf of topological vector spaces, (resp. algebras), we have SFA ( 1≤j≤p uj) ⊂ 1≤j≤p SFA(uj). Moreover, in the resp. case, we have SFA ( 1≤j≤p uj) ⊂ 1≤j≤p SFA(uj). In particular, if uj = u for 1 ≤ j ≤ p, we have S p) ⊂ SFA(u). Example 2 Taking E = C∞; F = D′; A = G leads to the D′-singular support of an element of the Colombeau algebra. This notion is complementary to the usual concept of local association in the Colombeau sense. We refer the reader to [12, 13] for more details. Example 3 In the following examples we consider X = Rd, E = C∞ and A = G. (i) Take u ∈ σΩ (C ∞ (Ω)), where σΩ : C ∞ (Ω) → G (Ω) is the canonical embedding defined in Subsection 2.2. Then SC G (u) = ∅, for all p ∈ N. (ii) Take ϕ ∈ D (R), with ϕ (x) dx = 1, and set ϕε (x) = ε −1ϕ (x/ε). As ϕε D′(R) δ, we have: G ([ϕε]) = {0}. We note also that S G ([ϕε]) = {0}. Indeed, for any K ⋐ R ∗ = R\ {0} and ε small enough, ϕε is null on K and, therefore, ϕε C∞(R∗) (iii) Take u = [uε] with uε(x) = ε sin(x/ε). We have that lim pK,0(uε) = 0, for all K ⋐ R, whereas lim pK,1(uε) does not exist for l ≥ 1. Therefore G (u) = ∅ , S G (u) = R. Remark 4 For any (p, q) ∈ N with p ≤ q, and u ∈ G, it holds that SC G (u) ⊂ S G (u). 3 The concept of (a,F)-microlocal analysis Let Ω be an open set in X. Fix u = [uλ] ∈ A(Ω) and x ∈ Ω. The idea of the (a,F)-microlocal analysis is the following: (uλ)λ may not tend to a section of F above a neighborhood of x, that is, there exists no V ∈ Vx and no f ∈ F (V ) such that lim F(V ) uλ = f . Nevertheless, in this case, there may exist V ∈ Vx, r ≥ 0 and f ∈ F (V ) such that lim F(V ) aλ(r)uλ = f , that is [aλ(r)uλ |V ] belongs to the subspace (resp. subalgebra) FA(V ) of A(V ) introduced in Subsection 2.4. These preliminary remarks lead to the following concept. 3.1 The (a,F)-singular parametric spectrum We recall that a is a map from R+ to A+ such that a(0) = 1 and F is a presheaf of topological vector spaces (or topological algebras). For any open subset Ω of X, u = [uλ] ∈ A(Ω) and x ∈ Ω, set N(a,F),x (u) = r ∈ R+ | ∃V ∈ Vx, ∃f ∈ F(V ) : lim F(V ) (aλ(r)uλ |V ) = f r ∈ R+ | ∃V ∈ Vx : [aλ (r)uλ |V ] ∈ FA(V ) It is easy to check that N(a,F),x (u) does not depend on the representative of u. If no confusion may arise, we shall simply write N(a,F),x (u) = Nx(u). Theorem 7 Suppose that: (a) For all λ ∈ Λ ∀ (r, s) ∈ R+, aλ(r + s) ≤ aλ(r)aλ(s), and, for all r ∈ R+\ {0}, the net (aλ (r))λ converges to 0 in K. (b) F is a presheaf of separated locally convex topological vector spaces. Then we have, for u ∈ A(Ω): (i) If r ∈ Nx(u), then [r,+∞) is included in Nx(u). Moreover, for all s > r, there exists V ∈ Vx such that: lim F(V ) (aλ(s)uλ |V ) = 0. Consequently, Nx(u) is either empty, or a sub-interval of R+. (ii) More precisely, suppose that for x ∈ Ω, there exist r ∈ R+, V ∈ Vx and f ∈ F(V ), nonzero on each neighborhood of x included in V , such that lim F(V ) (aλ(r)uλ |V ) = f . Then Nx(u) = [r,+∞) . (iii) In the situation of (i) and (ii), we have that 0 ∈ Nx(u) iff Nx(u) = R+. Moreover, if one of these assertions holds, the limits lim F(V ) (aλ (s) uλ |V ) can be non null only for s = 0. Proof. (i) If r ∈ Nx(u), there exist V ∈ Vx and f ∈ F(V ) such that lim F(V ) (aλ(r)uλ|V ) = f. As F(V ) is locally convex, its topology may be described by a family QV = (qj)j∈J(V ) of semi-norms. For all s > r, we have, for any j ∈ J (V ), qj(aλ(s) (uλ |V )) = aλ(s) qj(uλ |V ) ≤ aλ(s− r) aλ(r) qj(uλ |V ) ≤ aλ(s − r) qj(aλ(r) uλ |V ). From lim qj (aλ(r) (uλ |V − f)) = 0, we have qj(aλ(r) uλ |V ) < +∞ and lim qj (aλ(s)(uλ |V )) = 0, since aλ(s− r) → 0. Thus lim F(V ) (aλ(s)uλ |V ) = 0. (ii) From (i), we have [r,+∞) ⊂ Nx(u). Suppose that there exists t < r in Nx(u). Then we get W ∈ Vx, which can be chosen included in V , and g ∈ F(W ) such that lim F(W ) (aλ(t)uλ |W ) = g. With the notations of the proof of (i), we have qj(aλ(r) (uλ |W )) ≤ aλ(r − t)qj(aλ(t)uλ |W ). As qj(aλ(t)uλ |V ) is bounded, it follows that lim qj(aλ(r) (uλ |W )) = 0, which is in contradiction with lim F(V ) (aλ(r) (uλ |V ) = f 6≡ 0 on W. (iii) The first assertion follows directly from (i) and the second from (ii). From now on, we suppose that the hypotheses (a) and (b) of Theorem 7 are fulfilled. We Σ(a,F),x(u) = Σx(u) = R+\Nx(u), R(a,F),x (u) = Rx(u) = inf Nx(u). According to the previous remarks and comments, Σ(a,F),x(u) is an interval of R+ of the form[ 0, R(a,F),x (u) 0, R(a,F),x (u) , the empty set, or R+. Definition 4 The (a,F)-singular spectrum of u ∈ A(Ω) is the set (a,F) A (u) = {(x, r) ∈ Ω× R+ | r ∈ Σx(u)} . Example 4 Take X = Rd, E = C∞, F = Cp (p ∈ N = N∪{+∞}), f ∈ C∞ (Ω). Set u =[( and v = ε−1 |ln ε| f in A (Ω) = G (Ω). Then, for all x ∈ R, N(a,Cp),x (u) = [1,+∞) , N(a,Cp),x (v) = (1,+∞) , R(a,Cp),x (u) = R(a,Cp),x (v) = 1. Remark 5 We have: Σ(a,F),x(u) = ∅ iff N(a,F),x(u) = R+ and, according to Theorem 7, iff 0 ∈ N(a,F),x(u), that is, there exist (V, f) ∈ Vx×F(V ) such that lim F(V ) (aλ(0)uλ |V ) = f . As aλ(0) ≡ 1, this last assertion is equivalent to x ∈ O A (u). Thus Σ(a,F),x(u) = ∅ iff x /∈ S A (u). This remark implies directly the: Proposition 8 The projection of the (a,F)-singular spectrum of u on Ω is the F-singular support of u. 3.2 Example: The Colombeau case In this subsection we investigate the relationship between the (a,F)-singular spectrum and the sharp topology for X = Rd, E = C∞, F = Cp (p ∈ N), A = G, aε (r) = ε r. First, let us remark that, for u = [uε] ∈ G (Ω), x ∈ Ω (Ω ∈ O ), N(a,Cp),x (u) is never empty. Indeed, consider V ∈ Vx with V ⋐ Ω. There exists m > 0 such that pp,V (uε) = o (ε −m) as ε → 0. Thus, p (uε) = o (ε −m) for all k ≤ p and lim Cp(V ) (ε muε |V ) = 0. Thus [m,+∞) ⊂ N(a,Cp),x (u) . Let us now recall the construction of the sharp topology on G (Ω) . For u = [(uε)ε] ∈ G (Ω), K ⋐ Ω, l ∈ N, set vK,l(u) = inf r ∈ R ∣∣ pK,l (uε) = o(ε−r) as ε→ 0 The real number vK,l(u) is well defined, i.e. does not depend on the representative of u, and is called the (K, l)-valuation of u. It has the usual properties: (i) ∀λ ∈ C\{0}, ∀u ∈ G (Ω) , vK,l(λu) = vK,l(u) ; (ii) ∀u, v ∈ G (Ω) , vK,l(u+ v) ≤ sup(vK,l(u), vK,l(v)). The family (vK,l) permits to define the (K, l)-pseudodistances dK,l on G (Ω) by ∀ (u, v) ∈ G (Ω) , dK,l (u, v) = exp (vK,l(u− v)) , which turns out to be ultrametric: ∀ (u, v, w) ∈ G (Ω) , dK,l (u, v) ≤ sup(dK,l(u,w), dK,l(w, v)). The topology defined by the family (dK,l)K,l is called the sharp topology on G (Ω). As we are interested here in valuations greater or equal to 0, we set, for u ∈ G (Ω), νK,l(u) = sup (vK,l(u), 0) . We can define, for x ∈ Ω, the l-valuation of u at x by νx,l(u) = inf (u) |V ∈ V (x) , V relatively compact and set, for any p ∈ N, νpx(u) = sup 0≤l≤p νx,l(u). Proposition 9 For all p ∈ N, [uε] ∈ G (Ω) and x ∈ Ω, we have νpx(u) = R(a,Cp),x (u) = infN(a,Cp),x (u) . Proof. Take r > ν x(u). Then, for any l with 0 ≤ l ≤ p, one has r > νx,l(u) and there exists V ∈ V (x) , V relatively compact, such that v (u) < r. Thus, p (uε) = o(ε −r), as ε → 0, and lim Cp(V ) (ε ruε |V ) = 0, which implies that r > R(a,Cp),x (u) and ν x(u) ≥ R(a,Cp),x (u). Conversely, if r > R(a,Cp),x (u), there exists V ∈ V (x) such that lim Cp(V ) (ε ruε |V ) = 0. For any relatively compact neighborhood W of x included in V , we get p (uε) = o(ε −r) and r > v (u) > νx,l(u). Thus, r ≥ ν x(u) and ν x(u) ≤ R(a,Cp),x (u). 3.3 Some properties of the (a,F)-singular parametric spectrum Notation 2 For u = [uλ] ∈ A (Ω), lim F(V ) (aλ(r)uλ |V ) ∈ F (V ) means that there exists f ∈ F (V ) such that lim F(V ) (aλ(r)uλ |V ) = f . 3.3.1 Linear properties Proposition 10 For any u, v ∈ A(Ω), we have (a,F) A (u+ v) ⊂ S (a,F) A (u) ∪ S (a,F) A (v) . Proof. Let r be in Nx(u) ∩Nx(v). Then there exist V ∈ Vx and W ∈ Vx such that F(V ) (aλ(r)uλ |V ) ∈ F (V ) and lim F(W ) (aλ(r) vλ |W ) ∈ F (W ) . Thus lim F(V ∩W ) (aλ(r) (uλ + vλ) |V ∩W ) ∈ F (V ∩W ) and r ∈ Nx(u+ v). Consequently, Nx(u) ∩Nx(v) ⊂ Nx(u+ v). We obtain the result by taking the complementary sets in R+. Corollary 11 For any u, u0, u1 in A(Ω) with (i) u = u0 + u1 (ii) S (a,F) A (u0) = ∅, we have (a,F) A (u) = S (a,F) A (u1) . Proof. Proposition 10 and condition (ii) give S (a,F) A (u) ⊂ S (a,F) A (u1). As (i) implies u0 = u− u1, we obtain the converse inclusion, and thus the equality. 3.3.2 Differential properties We suppose that F is a sheaf of topological differential vector spaces (resp. algebras), with continuous differentiation, admitting E as a subsheaf of topological differential algebras. Then the sheaf A is also a sheaf of differential algebras with, for any α ∈ Nd and u ∈ A (Ω), ∂αu = [∂αuλ] , where (uλ)λ is any representative of u. The independence of ∂αu on the choice of representative follows directly from the definition of J(IA,E,P).) Proposition 12 Let u be in A(Ω). For all ∂α, α ∈ Nd, we have (a,F) αu) ⊂ S (a,F) A (u) . Proof. Take u ∈ A(Ω), α ∈ Nd, x ∈ Ω, r ∈ Nx(u). There exists V ∈ Vx, f ∈ F (V ) such F(V ) (aλ(r)uλ |V ) = f. The continuity of ∂α implies that F(V ) (aλ(r)∂ α uλ |V ) = ∂ Thus Nx(u) ⊂ Nx(∂ αu). The result is proved. In the following two results we require that F is a sheaf of topological modules over E , in addition. The proofs are straightforward. Proposition 13 Let g be in E(Ω) and u in A(Ω). We have (a,F) A (gu) ⊂ S (a,F) A (u) . Propositions 10, 12 and 13 finally imply: Corollary 14 Let P (∂) = |α|≤m α be a differential polynomial with coefficients in E(Ω). For any u ∈ A(Ω), we have (a,F) A (P (∂)u) ⊂ S (a,F) A (u) . 3.3.3 Nonlinear properties Theorem 15 For given u and v ∈ A(Ω), let Di (i = 1, 2, 3) be the following disjoint sets: D1 = S A (u)�(S A (u) ∩ S A (v)) ; D2 = S A (v)�(S A (u) ∩ S A (v)) ; D3 = S A (u) ∩ S A (v). Then the (a,F)-singular asymptotic spectrum of uv verifies (a,F) A (uv) ⊂ {(x,Σx(u)), x ∈ D1} ∪ {(x,Σx(v)), x ∈ D2} ∪ {(x,Ex(u, v)), x ∈ D3} where (for any x ∈ D3) Ex(u, v) = [0, supΣx(u) + supΣx(v)] if Σx(u) 6= R+ and Σx(v) 6= R+ R+ if Σx(u) = R+ or Σx(v) = R+ Proof. Suppose that x belongs to D1. Then x is not in S A (v) and we have Σx(v) = ∅, Nx(v) = R+. If Nx(u) is not empty, let r be in Nx(u). As Nx(v) = R+, we have r ∈ Nx(v). Thus there exists V ∈ Vx (resp. W ∈ Vx) such that [aλ(r)uλ |V ] ∈ FA(V ) (resp. [aλ(r)vλ |W ] ∈ FA(W )). As F is a sheaf of topological algebras we have [aλ(r) (uλvλ) |V ∩W ] ∈ FA(V ∩W ). Thus, r belongs to Nx(uv). Therefore, we have proved that Σx(uv) ⊂ Σx(u). If Nx(u) is empty, we have Σx(u) = R+ and the above inclusion is obviously fulfilled. For x in D2, the same proof gives Σx(uv) ⊂ Σx(v). Consider x in D3. Then, Σx(u) and Σx(v) are not empty. We suppose first that both of them are not equal to R+. Set R = supΣx(u) and S = supΣx(v). If r > R, there exists r ′ ∈ Nx(u) such that R < r′ < r and then, from the part (i) of Theorem 7, there exists V ∈ Vx such that F(V ) (aλ(r)uλ |V ) = 0. Similarly, if s > S, there exists W ∈ Vx such that F(W ) (aλ(s) vλ |W ) = 0. Then lim F(V ∩W ) (aλ(r)aλ(s) (uλvλ) |V ∩W ) = 0. By expressing this limit in terms of semi- norms, as in the proof of Theorem 7 and by using the inequality aλ(r+ s) ≤ aλ(r)aλ(s), we get that lim F(V ∩W ) (aλ(r + s) (uλvλ) |V ∩W ) = 0. Thus [r + s,∞[ ⊂ Nx(uv) or [0, r + s[ ⊃ Σx(uv) for any r > R and s > S. Thus Σx(uv) ⊂ [0, R+ S] = [0, supΣx(u) + supΣx(v)]. If Σx(u) or Σx(v) is equal to R+, the obvious inclusion Σx(uv) ⊂ R+ gives the last result. Corollary 16 For given u ∈ A(Ω) and p ∈ N∗, we have (a,F) (x,Hp,x(u)), x ∈ S A (u) where Hp,x(u) = [0, p supΣx(u)] if Σx(u) 6= R+ R+ if Σx(u) = R+ Proof. When Σx(u) = R+, the result is obvious. Suppose now Σx(u) 6= R+. We shall prove the result by induction. If p = 1, the result is a simple consequence of the definitions. Suppose that the result holds for some p ≥ 1. Set v = up in the previous theorem. We have D1 = S A (u)�S p) ; D2 = ∅ ; D3 = S (a,F) (x,Σx(u)), x ∈ S A (u)�S (x, [0, (p + 1) supΣx(u)]), x ∈ S by using the induction hypothesis. It follows a fortiori that (a,F) (x, [0, (p + 1) supΣx(u)]), x ∈ S A (u) 4 Applications to partial differential equations In this section we shall compute various (a,F)-singular spectra of solutions to linear and nonlin- ear partial differential equations. Throughout we shall suppose that Λ =]0, 1], X = Rd, E = C∞, F = Cp (1 ≤ p ≤ ∞) or F = D′, aε(r) = ε r. The results will hold for any (C, E ,P)-algebra A = H(A,E,P)/J(IA,E,P) such that (aε(r))ε ∈ A+ for all r ∈ R+ and property (9) holds. Example 5 The (a,Cp)-singular spectrum of powers of the delta function. Given a mollifier of the form ϕε (x) = , x ∈ Rd where ϕ ∈ D(Rd), ϕ ≥ 0 and ϕ (x) dx = 1, its class in A(Rd) defines the delta function δ(x) as an element of A(Rd). Its powers are given by (m ∈ N) Clearly, the C0-singular spectrum is given by (a,C0) 0, [0,md] Differentiating ϕm(x) and observing that for each derivative there is a point x at which it does not vanish we see that (a,Ck) 0, [0,md + k] Example 6 The (a,D′)-singular spectrum of powers of the delta function. Given a test function ψ ∈ D(Rd), we have ∫ ϕmε (x)ψ(x) dx = εmd−d ϕm(x)ψ(εx) dx, (a,D′) m) = ∅ for m = 1, S (a,D′) 0, [0,md − d[ for m > 1. 4.1 The singular spectrum of solutions to linear hyperbolic equations Consider the Cauchy problem for the d-dimensional linear wave equation ∂2t uε(x, t)−∆uε(x, t) = 0, x ∈ R d, t ∈ R uε(x, 0) = u0ε(x), ∂tuε(x, 0) = u1ε(x), x ∈ R where u0ε, u1ε ∈ C ∞(Rd) represent elements u0, u1 of an algebra A(R d) as outlined at the beginning of this section. Under suitable assumptions on the ring A, the corresponding net of classical smooth solutions represents a unique solution u in the algebra A(Rd+1); for example, this holds in the Colombeau case [16]. Let t → E(t) ∈ C∞(R : E ′(Rd)) be the fundamental solution of the Cauchy problem. Then uε(·, t) = E(t) ∗ εru0ε + E(t) ∗ ε ru1ε. If for some r ≥ 0 and u0 ∈ D ′(Rd), εru0ε(x)ψ(x) dx → 〈u0, ψ〉 for all ψ ∈ D(Rd), then εruε(x)ψ(x) dx = 〈E(t) ∗ ε ru0ε, ψ〉 = 〈ε ru0ε, Ě(t) ∗ ψ〉 → 〈u0, Ě(t) ∗ ψ〉 for all ψ ∈ D(Rd) and t ∈ R as well. We arrive at the following assertion. Proposition 17 Assume that S (a,D′) A (u0) and S (a,D′) A (u1) are contained in R d×I, where I = ∅, I = [0, r[ or I = [0, r] for some r, 0 ≤ r ≤ ∞. Let u ∈ A(Rd+1) be the solution to the linear wave equation (10). Then S (a,D′) A (u(·, t)) ⊂ R d × I for all t ∈ R. This upper bound may or may not be reached, depending on the effects of finite propagation speed or the Huyghens principle in odd space dimension d ≥ 3. We just illustrate some of the possible effects for the one-dimensional wave equation with powers of delta functions as initial data. Thus we consider the problem ∂2t uε(x, t)− ∂ xuε(x, t) = 0, x ∈ R, t ∈ R uε(x, 0) = c0ϕ ε (x), ∂tuε(x, 0) = c1ϕ ε (x), x ∈ R, where ϕ is a mollifier as in Example 5 and c0, c1 ∈ R. The solution to (11) is given by uε(x, t) = ϕmε (x− t) + ϕ ε (x+ t) ∫ x+t ϕnε (y) dy. We observe that uε(x, t) = 0 for sufficiently small ε when |x| > |t|, that is, outside the light cone, and uε(x, t) = sign(t) εn−1‖ϕn‖L1(R) for sufficiently small ε when |x| < |t|. Example 7 If in equation (11) c0 6= 0, c1 = 0 then (a,D′) A (u) = {(x, t, r) : |x| = |t|, 0 ≤ r < m− 1} with the provision that S (a,D′) A (u) = ∅ when m = 1. If in equation (11) c0 = 0, c1 6= 0 then (a,D′) A (u) = {(x, t, r) : |x| ≤ |t|, 0 ≤ r < n− 1}. When both c0 and c1 are nonzero the singular spectrum is obtained as the union of the two spectra. For the C0-singular spectrum the following results hold: If in equation (11) c0 6= 0, c1 = 0 then (a,C0) A (u) = {(x, t, r) : |x| = |t|, 0 ≤ r ≤ m}. If c0 = 0, c1 6= 0 then (a,C0)) A (u) = {(x, t, r) : |x| < |t|, 0 ≤ r < n− 1} ∪ {(x, t, r) : |x| = |t|, 0 ≤ r ≤ n− 1}. 4.2 The singular spectrum of solutions to semilinear hyperbolic equations In this subsection we study the paradigmatic case of a semilinear transport equation ∂tuε(x, t) + λ(x, t)∂xuε(x, t) = F (uε(x, t)), x ∈ R, t ∈ R uε(x, 0) = u0ε(x), x ∈ R where λ and F are smooth functions of their arguments. In this situation, the singular spectrum of the initial data may be decreased or increased, depending on the function F . We observe that by a change of coordinates we may assume without loss of generality that λ ≡ 0. In fact, denote by s → γ(x, t, s) the characteristic curve of (12) passing through the point x at time s = t, that is the solution to γ(x, t, s) = λ(γ(x, t, s), s), γ(x, t, t) = x. The function v(y, s) = u(γ(y, 0, s), s) is a solution of the initial value problem ∂sv(y, s) = F (v(y, s), v(y, 0) = u0(y), at least as long as the characteristic curves exist. Example 8 (The dissipative case) The equation ∂tuε(x, t) = −u ε(x, t), x ∈ R, t > 0 uε(x, 0) = u0ε(x), x ∈ R has the solution uε(x, t) = u0ε(x)√ 2tu20ε(x) + 1 2t+ 1/u20ε(x) When the initial data are given by a power of the delta function, u0ε(x) = ϕ ε (x), the solution formula shows that uε(x, t) is a bounded function (uniformly in ε) and supported on the line {x = 0}. Thus uε(x, t) converges to zero in D ′(R×]0,∞[), and so (a,D′) A (u0) = 0, [0,m − 1[ (a,D′) A (u) = ∅. Example 9 The equation ∂tuε(x, t) = 1 + u2ε(x, t), x ∈ R, t > 0 uε(x, 0) = u0ε(x), x ∈ R has the solution uε(x, t) = u0ε(x) cosh t+ 1 + u20ε(x) sinh t. We first take a delta function as initial value, that is, u0ε(x) = ϕε(x). Then uε(x, t)ψ(x, t) dxdt = ϕ(x) cosh t+ ε2 + ϕ2(x) sinh t ψ(εx, t) dxdt ϕ(x) cosh t+ |ϕ(x)| sinh t ψ(0, t) dxdt for ψ ∈ D(R2). Thus in this case (a,D′) A (u0) = S (a,D′) A (u) = ∅. On the other hand, taking the derivative of a delta function as initial value, u0ε(x) = ϕ ε(x), a similar calculation shows that uε(x, t)ψ(x, t) dxdt = ϕ(x) cosh t+ ε4 + (ϕ′)2(x) sinh t ψ(εx, t) dxdt and so (a,D′) A (u0) = ∅, S (a,D′) A (u) = {(0, t, r) : t > 0, 0 ≤ r < 1}. The next example shows that it is quite possible for the singular spectrum to increase with time. Example 10 The equation ∂tuε(x, t) = uε(x, t) + 1) uε(x, t) + 1 , x ∈ R, t > 0 uε(x, 0) = u0ε(x), x ∈ R has the solution uε(x, t) = u0ε(x) + 1 provided u0ε > −1 in which case the function on the right hand side of the differential equation is smooth in the relevant region. To demonstrate the effect, we take a power of the delta function as initial value, that is u0ε(x) = ϕ ε (x). Then (a,D′) A (u0) = {(0, r) : 0 ≤ r < m− 1}, S (a,D′) A (u) = {(0, t, r) : t > 0, 0 ≤ r < me t − 1}. In situations where blow-up in finite time occurs, microlocal asymptotic methods allow to extract information beyond the point of blow-up. This can be done by regularizing the initial data and truncating the nonlinear term. We demonstrate this in a simple situation. Example 11 Formally, we wish to treat the initial value problem ∂tu(x, t) = u 2(x, t), x ∈ R, t > 0 u(x, 0) = H(x), x ∈ R where H denotes the Heaviside function. Clearly, the local solution u(x, t) = H(x)/(1− t) blows up at time t = 1 when x > 0. Choose χε ∈ C ∞ (R) with 0 ≤ χε(z) ≤ 1 ; χε(z) = 1 if |z| ≤ ε −s , χε(z) = 0 if |z| ≥ 1 + ε −s , s > 0. Further, let Hε(x) = H ∗ ϕε(x) where ϕε is a mollifier as in Example 5. We consider the regularized problem ∂tuε(x, t) = χε uε(x, t) u2ε(x, t), x ∈ R, t > 0 uε(x, 0) = Hε(x), x ∈ R. When x < 0 and ε is sufficiently small, uε(x, t) = 0 for all t ≥ 0. For x > 0, uε(x, t) = 1/(1− t) as long as t ≤ 1− εs. The cut-off function is chosen in such a way that |χε(z)z 2| ≤ (1 + ε−s)2 for all z ∈ R. Therefore, ∂tuε ≤ (1 + ε −s)2 always and ∂tuε = 0 when |uε| ≥ 1 + ε Continuing the regularized solution beyond time t = 1 − εs, we infer by combining the two inequalities that ε−s ≤ uε(x, t) ≤ 1 + ε −s for t ≥ 1 − εs when x > 0 and ε is sufficiently small. Finally, as long as t < 1, the regularized solution remains bounded with respect to ε near (0, t) for ε small enough; after t = 1, the asymptotic growth of order ε−s spills over into any neighborhood of every point (x, t) for x ≥ 0. Collecting all previous estimates, we obtain the following C0-singular support and singular spectrum (for aε(r) = ε r) of u = [uε]: A (u) = S1(u) ∪ S2(u) with S1(u) = {(0, t) : 0 ≤ t < 1} ; S2(u) = {(x, t) : x ≥ 0, t ≥ 1}, (a,C0) A (u) = (S1(u)× {0}) ∪ (S2(u)× [0, s]) . The C0-singularities (resp. -singularities) of u are described by means of two sets: S1(u) and S2(u) (resp. S1(u) × {0} and S2(u) × [0, s]). The set S1(u) (resp. S1(u) × {0}) is related to the data C0 (resp. )-singularity. The set S2(u) (resp. S2(u) × [0, s]) is related to the singularity due to the nonlinearity of the equation giving the blow-up at t = 1. The blow-up locus is the edge {x ≥ 0, t = 1} of S2(u) and the strength of the blow-up is measured by the length s of the fiber [0, s] above each point of the blow-up locus. This length is closely related to the diameter of the support of the regularizing function χε and depends essentially on the nature of the blow-up: Changing simultaneously the scales of the regularization and of the cut-off (i.e. replacing ε by some function h(ε) → 0 in the definition of ϕε and χε) does not change the fiber and characterizes a sort of moderateness of the strength of the blow-up. 4.3 The strength of a singularity and the sum law When studying the propagation and interaction of singularities in semilinear hyperbolic systems, Rauch and Reed [18] defined the strength of a singularity of a piecewise smooth function. We recall this notion in the one-dimensional case. Assume that the function f : R → R is smooth on ] − ∞, x0] and on [x0,∞[ for some point x0 ∈ R. The strength of the singularity of f at x0 is the order of the highest derivative which is still continuous across x0. For example, if f is continuous with a jump in the first derivative at x0, the order is 0; if f has a jump at x0, the order is −1. Travers [21] later generalized this notion to include delta functions. Slightly deviating from her definition, but in line with the one of [18], we define the strength of singularity of the k-th derivative of a delta function at x0, ∂ xδ(x− x0), by −k − 2. The significance of these definitions is seen in the description of what Rauch and Reed termed anomalous singularities in semilinear hyperbolic systems. We demonstrate the effect in a paradigmatic example, also due to [18], the (3× 3)-system (∂t + ∂x)u(x, t) = 0, u(x, 0) = u0(x) (∂t − ∂x)v(x, t) = 0, v(x, 0) = v0(x) ∂tw(x, t) = u(x, t)v(x, t), w(x, 0) = 0 Assume that u0 has a singularity of strength n1 ≥ −1 at x1 = −1 and v0 has a singularity of strength n2 ≥ −1 at x2 = +1. The characteristic curves emanating from x1 and x2 are straight lines intersecting at the point x = 0, t = 1. Rauch and Reed showed that, in general, the third component w will have a singularity of strength n3 = n1 + n2 + 2 along the half-ray {(0, t) : t ≥ 1}. This half-ray does not connect backwards to a singularity in the initial data for w, hence the term anomalous singularity. The formula n3 = n1 + n2 + 2 is called the sum law. Travers extended this result to the case where u0 and v0 were given as derivatives of delta functions at x1 and x2. We are going to further generalize this result to powers of delta functions, after establishing the relation between the strength of a singularity of a function f at x0 and the singular spectrum of f ∗ ϕε. We consider a function f : R → R which is smooth on ]−∞, x0] and on [x0,∞[ for some point x0 ∈ R; actually only the local behavior near x0 is relevant. We fix a mollifier ϕε(x) = as in Example 5 and denote the corresponding embedding of D′(R) into the (C, E ,P)-algebra A(R) by ι. In particular, ι(f) = [f ∗ ϕε]. If f is continuous at x0, then limε→0 f ∗ ϕε = f in C 0. If f has a jump x0, this limit does not exist in C0, but limε→0 ε rf ∗ ϕε = 0 in C 0 for every r > 0. We have the following result. Proposition 18 Let x0 ∈ R. If f : R → R is a smooth function on ]−∞, x0] and on [x0,∞[ or f(x) = ∂kxδ(x − x0) for some k ∈ N, then the strength of the singularity of f at x0 is −n if and only if Σ(a,C1),x0 = [0, n]. Here n ∈ N and aε(r) = ε Proof. When n = 0, the function f is continuous and its derivative has a jump at x0. From what was said before Proposition 18 it follows that Σ(a,C1),x0 = {0}. When n = 1, the function f has a jump itself at x0 and its distributional derivative contains a delta function part. Thus limε→0 ε rf ∗ϕε = 0 in C 0 for every r > 0 and limε→0 ε r∂xf ∗ϕε = 0 in C 0 for every r > 1, and neither of the two limits exists for smaller r. Therefore, Σ(a,C1),x0 = [0, 1]. When n ≥ 2, f(x) = ∂n−2x δ(x− x0) and the assertion is straightforward. We shall now return to the model equation (13) and demonstrate that the sum law remains valid when the initial data are powers of delta functions. We work in suitable (C, E ,P)-algebras A(R) and A(R2) in which the initial value problem (13) can be uniquely solved (see the discus- sion at the beginning of Subsection 4.1). We still consider the scale aε(r) = ε Proposition 19 Let u0(x) = δ m(x+1), v0(x) = δ n(x−1) for some m,n ∈ N∗. Let w ∈ A(R2) be the third component of the solution to problem (13). Then w(x, t) vanishes at all points (x, t) with x 6= 0 as well as (0, t) with t < 1, and Σ(a,C1),(0,t) ⊂ [0,m+ n] for t ≥ 1. Proof. A representative of w is given by wε(x, t) = ϕmε (x+ 1− s)ϕ ε (x− 1 + s) ds. The fact that the mollifier ϕ has compact support entails that wε(x, t) vanishes for sufficiently small ε whenever x 6= 0 or t < 1. We have wε(x, t) = (x+ 1− s (x− 1 + s ∂twε(x, t) = (x+ 1− t (x− 1 + t ∂xwε(x, t) = εm+n+1 (x+ 1− s (x+ 1− s (x− 1 + s εm+n+1 (x+ 1− s (x− 1 + s (x− 1− s If the support of ϕ is contained in an interval [−κ, κ], say, then the t-integrations extend at most from x+ 1− κε to x+ 1 + κε at fixed x. Therefore, all terms converge to zero uniformly on R2 when multiplied by εr with r > m+ n. This proves the assertion. Using the correspondence between the singular spectrum and the strength of a singularity formulated in Proposition 18, as well as Example 5, we may say that the strength of the singularity of δm(x + 1) at x0 = −1 is n1 = −m − 1, while the strength of the singularity of δn(x−1) at x0 = +1 is n2 = −n−1. The strength of the singularity of the solution w at points (0, t) with t ≥ 1 is −m− n = n1 + n2 + 2 and is seen to satisfy the sum law. 4.4 Regular Colombeau generalized functions The subsheaf G∞ of regular Colombeau functions of the sheaf G is defined as follows [16]: Given an open subset Ω of Rd, the algebra G∞(Ω) comprises those elements u of G(Ω) whose representatives (uε)ε satisfy the condition (14) ∀K ⋐ Ω ∃m ∈ N ∀l ∈ N : pK,l(uε) = o(ε −m) as ε→ 0. The decisive property is that the bound of order ε−m is uniform with respect to the order of derivation on compact sets. The algebra G∞(Ω) satisfies G∞(Ω) ∩ D′(Ω) = C∞(Ω) and forms the basis for the investigation of hypoellipticity of linear partial differential operators in the Colombeau framework. We are going to characterize the G∞-property in terms of the C∞-singular spectrum. The scale a is still given by aε(r) = ε Proposition 20 Let u ∈ G(Ω). Then u belongs to G∞(Ω) if and only if Σ(a,C∞),x 6= R+ for all x ∈ Ω. Proof. If u ∈ G∞(Ω), x ∈ Ω and Vx is a relatively compact open neighborhood of x, property (14) says that there is m ∈ N such that limε→0 ε muε = 0 in C ∞(Vx). Thus Σ(a,C∞),x 6= R+. Conversely, if Σ(a,C∞),x 6= R+ we can find an open neighborhood Vx of x and m(x) ∈ N such that limε→0 ε ruε = 0 in C ∞(Vx) for all r ≥ m. Any compact set K can be covered by finitely many such neighborhoods. Letting m be the maximum of the numbers m(x) involved, we obtain property (14). In relation with regularity theory of solutions to nonlinear partial differential equations, a further subalgebra of G(Ω) has been introduced in [17] – the algebra of Colombeau functions of total slow scale type. It consists of those elements u of G(Ω) whose representatives (uε)ε satisfy the condition (15) ∀K ⋐ Ω ∀r > 0 ∀l ∈ N : pK,l(uε) = o(ε −r) as ε→ 0. The term slow scale refers to the fact that the growth is slower than any negative power of ε as ε→ 0. This property can again be characterized by means of the singular spectrum. Proposition 21 An element u ∈ G(Ω) is of total slow scale type if and only if Σ(a,C∞),x ⊂ {0} for all x ∈ Ω. Proof. If u is of total slow scale type, x ∈ Ω and Vx is a relatively compact open neigh- borhood of x, property (15) implies that limε→0 ε suε = 0 in C ∞(Vx) for every s > 0. Thus Σ(a,C∞),x ⊂ {0}. To prove the converse, we take a compact subset K and r > 0 and cover K by finitely many neighborhoods Vx of points x ∈ K such that limε→0 ε ruε = 0 in C ∞(Vx). Then property (15) follows. References [1] Aragona, J., Biagioni, H.: Intrinsic definition of the Colombeau algebra of generalized functions. Anal. Math. 17, 75–132 (1991) [2] Colombeau, J.F.: Multiplication of Distributions: a tool in mathematics, numerical engi- neering and theoretical physics. Lecture Notes in Mathematics, vol. 1532. Springer-Verlag, Berlin (1992) [3] Delcroix, A: Remarks on the embedding of spaces of distributions into spaces of Colombeau generalized functions. Novi Sad J. Math. 35(2), 27–40 (2005) [4] Delcroix, A.: Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity. J. Math. Anal. Appl. 327, 564–584 (2007) [5] Garetto, C., Gramchev, T., Oberguggenberger, M.: Pseudo-Differential operators and reg- ularity theory. Electron J. Diff. Eqns. 2005(116), 1–43 (2005) [6] Garetto, C., Hörmann, G.: Microlocal analysis of generalized functions: pseudodifferential techniques and propagation of singularities. Proc. Edinb. Math. Soc. 48, 603–629 (2005) [7] Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.: Geometric Theory of Generalized Functions with Applications to General Relativity. Kluwer Academic Publ., Dordrecht (2001) [8] Hörmander, L.: The Analysis of Linear Partial Differential Operators, I: Distribution Theory and Fourier Analysis. Grundlehren der mathematischen Wissenschaften, vol. 256, 2nd edition. Springer-Verlag, Berlin (1990) [9] Hörmann, G., De Hoop, M.V.: Microlocal analysis and global solutions for some hyperbolic equations with discontinuous coefficients. Acta Appl. Math. 67, 173–224 (2001) [10] Hörmann, G., Kunzinger, M.: Microlocal properties of basic operations in Colombeau algebras. J. Math. Anal. Appl. 261, 254–270 (2001) [11] Hörmann, G., Oberguggenberger, M., Pilipović, S.: Microlocal hypoellipticity of linear differential operators with generalized functions as coefficients. Trans. Amer. Math. Soc. 358, 3363–3383 (2006) [12] Marti, J.A.: (C, E ,P)-Sheaf structure and applications. In: Grosser, M., Hörmann, G., Kunzinger, M., Oberguggenberger, M. (Eds.) Nonlinear Theory of Generalized Functions. Chapman & Hall/CRC Research Notes in Mathematics, vol. 401, pp.175–186. Boca Raton (1999) [13] Marti, J.A.: Nonlinear algebraic analysis of delta shock wave to Burgers’ equation. Pacific J. Math. 210(1), 165–187 (2003) [14] Marti, J.A.: GL-microanalysis of generalized functions. Integral Transf. Spec. Funct. 2–3, 119–125 (2006) [15] Nedeljkov, M., Pilipović, S., Scarpalézos, D.: The linear theory of Colombeau general- ized functions. Pitman Research Notes in Mathematics, vol. 385. Longman Scientific & Technical, Harlow (1998) [16] Oberguggenberger, M.: Multiplication of Distributions and Applications to Partial Differ- ential Equations. Pitman Research Notes in Mathematics, vol. 259. Longman Scientific & Technical, Harlow (1992) [17] Oberguggenberger, M.: Generalized solutions to nonlinear wave equations. Matemática Contemporânea 27, 169–187 (2004) [18] Rauch, J., Reed, M.: Jump discontinuities of semilinear, strictly hyperbolic equations in two variables: Creation and propagation. Comm. Math. Phys. 81, 203-227 (1981) [19] Scarpalézos, D.: Colombeau’s generalized functions: topological structures; microlocal properties. A simplified point of view. Bull. Cl. Sci. Math. Nat. Sci. Math. 25, 89–114 (2000) [20] Schwartz, L.: Théorie des Distributions. Hermann, Paris (1966) [21] Travers, K.: Semilinear hyperbolic systems in one space dimension with strongly singular initial data. Electron. J. Diff. Eqns. 1997(14), 1–11 (1997) Introduction Preliminary definitions and local parametric analysis The presheaves of (C,E,P)-algebras: the algebraic structure Relationship with distribution theory and Colombeau algebras An association process The F-singular support of a generalized function The concept of (a,F)-microlocal analysis The (a,F)-singular parametric spectrum Example: The Colombeau case Some properties of the (a,F)-singular parametric spectrum Linear properties Differential properties Nonlinear properties Applications to partial differential equations The singular spectrum of solutions to linear hyperbolic equations The singular spectrum of solutions to semilinear hyperbolic equations The strength of a singularity and the sum law Regular Colombeau generalized functions

0704.1078

1H-NMR Study on the Magnetic Order in the Mixture of Two Spin Gap Systems (CH3)2CHNH3CuCl3 and (CH3)2CHNH3CuBr3

untitled 1H-NMR Study on the Magnetic Order in the Mixture of Two Spin Gap Systems (CH3)2CHNH3CuCl3 and (CH3)2CHNH3CuBr3 Keishi Kanada1, Takehiro Saito1, Akira Oosawa1, Takayuki Goto1 and Takao Suzuki1,2 1Department of Physics, Faculty of Science and Technology, Sophia University 7-1 Kioicho, Chiyodaku Tokyo 102-8554 2Advanced Meson Science Laboratory, RIKEN 2-1 Hirosawa, Wako, Saitama 351-0198 (Received ) The antiferromagnetic ordering in the solid-solution of the two spin-gap systems (CH3)2CHNH3CuCl3 and (CH3)2CHNH3CuBr3 has been investigated by 1H-NMR. The sample with the Cl-content ratio x=0.85 showed a clear splitting in spectra below TN=13.5 K, where the spin-lattice relaxation rate T1−1 showed a diverging behavior. The critical exponent of the temperature dependence of the hyperfine field is found to be 0.33. KEYWORDS: disordered spin-gap systems, NMR, magnetic order 1. Introduction In classical spin systems, disorder destroys magnetic order by cutting off the correlation between neighboring spins. However, in quantum spin systems, where the magnetic order is instabilized by the quantum spin fluctuation, disorder often brings the non-trivial effect to their ground state. A low number of holes drastically destruct the antiferromagnetic order in parent phase of high-TC cuprates through frustration effect.1) In spin liquid systems represented by CuGeO3, disorder induces an unpaired spin among singlet dimer and brings a static magnetic order to the disordered state.2) The title compounds of (CH3)2CHNH3CuCl3 and (CH3)2CHNH3CuBr3 abbreviated as IPA-CuCl3 and IPA-CuBr3 are isostructural spin gap systems with a non-magnetic ground state. Both the mechanism and the magnitude of the gap are quite different in the two systems. In the Cl-system, which was discovered to have a ladder-like magnetic structure based on an inelastic neutron experiment5), the neighboring two S=1/2 spins on the rung interact ferromagnetically to form effective S=1 spins at low temperatures. The interaction among these integer spins is weak and antiferromagnetic, so that the ground state is gapped one as Haldane conjectured6). In a Br-system, the interaction between the two neighboring spins is antiferromagnetic, so that they form a singlet dimer state at low temperatures. The spin excitation gaps of these two systems have been reported to be 18 and 98 K based on magnetization experiments7-9). The former was re-examined in the neutron experiment and reported to be 13.9 K5). The aim of this paper is a microscopic investigation on the ground state of the solid solution of the two spin gap systems. Manaka et al. have been working intensively on the macroscopic measurements in IPA-Cu(ClxBr1-x)3, and they report that the system becomes gapless only within the limited region x=0.44-0.87, where a magnetic order occurs at low temperatures9,10). However, a microscopic investigation on the spin state has not been conducted except for our preliminary result by µSR11). In this paper, we report the existence of an antiferromagnetic phase transition in IPA-Cu(ClxBr1−x)3 (x=0.85) through the use of 1H-NMR spectra and the spin-lattice relaxation rate. 2. Experimental Evaporation method was utilized to grow single crystals of IPA-Cu(ClxBr1−x)3 (x=1 and 0.85). An isopropanol solution of isopropylamine hydrochloride, copper(II) chloride dihydrate, isopropylamine hydrobromide, copper (II) bromide was placed in a bowl, which was maintained at 30(±0.1)°C, and in an atmosphere of flowing nitrogen gas during an entire period of crystal growth, approximately two months. A typical size of the obtained crystals was around 3×5×10mm with a rectangular shape as was reported in a previous paper 9). The content of Cl, x=0.85(5) was determined using the ICP method for three tiny fragments chipped off from different points of the crystal. Macroscopic quantities of obtained crystals, the specific heat and the susceptibility were measured by PPMS and MPMS manufactured by Quantum Design Co Ltd. The uniform susceptibility of x=0.85 showed a kink at around 12K, below which it steeply decreased for H⊥C and slightly increases for H⊥A. The specific heat also showed a small but distinct cusp at the same temperature. The sample x=1 showed no anomalies in neither of the two quantities. These results are consistent with reports by Manaka 9,10). 1H-NMR measurements were performed using a 4K cryogen-free refrigerator set in a 6T cryogen-free-superconducting magnet. The spectra were measured by recording a spin-echo amplitude simultaneously ramping up or down the magnetic field. The spin-lattice relaxation rate T1−1 was measured by the saturation-recovery method with a pulse train. Relaxation curves were traced until the difference of the nuclear magnetization from its saturation value becomes one percent. 3. Results and Discussion Figure 1 shows the field-swept spectra of the two samples x=0.85 and 1.0 at various temperatures under the field around 3T. At high temperatures both the samples show a sharp paramagnetic resonance line. The width around 18K is approximately 200 and 80 Oe for x=0.85 and 1.0 respectively. As the temperature decreases, the x=1.0 sample maintains a sharp width, and only the x=0.85 sample shows a drastic splitting in the peak below TN=13.5 K. This temperature coincides with that for an anomaly in 0χ and the specific heat. Five distinct peaks overlap each other at the lowest temperature of 4 K. These multiple peaks in spectra correspond to the inequivalent proton sites exposed to a hyperfine field produced by the antiferromagnetically-ordered spins. Ten inequivalent proton sites are in the unit cell, and these proton sites have different distances from the nearest Cu site, varying from 3.3 to 6.8 Å. They should feel different hyperfine fields from the ordered moment and should contribute to each split peak. Actually, observed peak separations around 300 Oe are comparable with the estimation by the classical dipole-dipole interaction between electronic and nuclear spins. Though we are still in the process of conducting a detailed site assignment, which requires measurements of spectra involving various directions of the field, we conclude that the spin structure is simple antiferromagnetic rather than incommensurate or glass-like. Recently, Nakamura numerically investigated the disordered bond-alternating spin chain to report that an antiferromagnetic state is more stable than a glass-like state expected in a classical point of view4). He pointed out that the emergence of Néel state is totally to due the quantum effect. In our previous µSR report11), we have pointed out that in the sample with x=0.95 which belongs to the gapped region, the antiferromagnetic spin fluctuation is anomalously enhanced at low temperatures. This fact suggests that the magnetic instability is inherently present in the gapped sample. Figure 2 shows the dependence on the temperature of a peak separation H∆ , which is defined as the distance between the positions at 20% of maxima of the left most and right most peaks, and is considered to be a good measure for the magnitude of the staggered moment and hence of the order parameter. The dependence of H∆ on temperature is aptly described by the scaling function ( )βN1 TT− , where β is a critical exponent estimated from data fitting to be 0.33, the value of which is close to the value of 0.327 which is expected 3D-Ising model16), and is consistent with our report by µSR11). The reason for the appearance of dimensionality of the universality class is 3D, simply because the phase transition in 1D-spin systems must be set off by the weak interaction paths that runs three dimensionally in the system. In previous paper11), we have explained the appearance of the Ising-type universality class in terms of D*, an effective single-ion anisotropy that appears at low temperatures owing to the formation of effective S=1 spins in ferromagnetic dimers. However, an existence of effective S=1 spins12-14) are proved only in the pure system of x=1.0 but the disordered system x=0.85 containing both ferromagnetic and antiferromagnetic dimers. It is, therefore, not self-evident whether or not the disordered system has a universality class identical to the pure system. According to Harris criterion15), the disorder does not affect the universality class only for those systems with the negative critical exponent of the specific heat α, which does not hold in the present case. Therefore, the universality class of x=0.85 sample should be determined carefully through experiments. Generally, in the presence of finite magnetic anisotropy D, the universality class of the spin system with Heisenberg interaction J shows a crossover from the isotropic Heisenberg class to Ising-like one as temperature getting closer to TN, at the temperature defined as 1~φ−t , where t is the reduced temperature, and 1<φ , the crossover index16). The critical exponent is expected to show simultaneously the gradual change from 0.367 of 3D-Heisenberg to 0.327 of 3D-Ising. However, as is clear in Fig. 2, the value of β, the gradient of the fitting line, shows a further decrease from 0.33 as the fitting temperature region is expanded. We conclude that our previous interpretation of the observed β to be the Ising model arisen from D* is possibly be misleading. The reason of appearance of β=0.33 experimentally confirmed by both µSR and NMR is still open question at this stage, and must be resolved in the future both theoretically and experimentally. Figure 3 shows the dependence of the spin-lattice relaxation rate 11 −T on temperature. In a ordered state, 11 −T was measured on a left most peak. There was no significant difference in 11 −T on any peaks except for the center peak which bears rather a lower relaxation rate. While 11 −T of the gapped sample (x=1.0) decreases exponentially as the temperature decreases, reflecting the gapped ground state, the gapless sample (x=0.85) shows diverging behavior around NT . This clearly demonstrates the fact that the observed phase transition is a second order one. In the vicinity of the second order phase transition, the fluctuation in the magnetic field shows a critical slow down and enhances NMR- 11 −T , a measure for the Fourier component of the Larmor frequency in the fluctuation. The dominant q-component of the fluctuation is considered to be located at far apart from zero, that is, possibly, around π=q . The reason for this is the uniform susceptibility that probes the fluctuation around q=0 shows no diverging behavior around NT . The scaling plot of 11 −T in the temperature region both above and below NT is shown in Fig. 4, where the dynamical critical exponent was obtained by fitting ( )nTT N1− to the observed temperature dependence as n=1.0(4) for NTT < and 0.5(4) for NTT > . The latter value above TN coincides with that for the classical 3D localized spin system. The universality class belongs to the three dimensional one, for which the same argument as that on β stated above holds. In conclusion, we investigated 1H-NMR on a bond-disordered spin-gap system IPA-Cu(Cl0.85Br0.15)3. The existence of an antiferromagnetic long-range order was clearly demonstrated by peak splitting in the spectra and by the divergence of 11 −T . The critical exponent was obtained from the temperature dependence of hyperfine field to be β=0.33, which is close to the value expected for the 3D-Ising model. Acknowledgment We thank Dr. H. Manaka and Prof. T. Ohtsuki for their valuable advice, and Dr. K. Noda at Kuwahara Lab., Sophia University for his assistance with specific heat measurements using PPMS. We also thank Prof. T. Nojima at the Center of Low Temperature Science of Tohoku Univ. for his assistance with magnetization measurements using MPMS. This work was supported by the Kurata Memorial Hitachi Science and Technology Foundation, Saneyoshi Scholarship Foundation and by a Grant-in-Aid for Scientific Research on priority Areas “High Field Spin Science in 100T” (No.451) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT). 1) Y. Kitaoka, S. Hiramatsu, K. Ishida, T. Kohara, K. Asayama: J. Phys. Soc. Jpn. 56, 3024 (1987) . 2) M. Hase, I. Terasaki, Y. Sasago, K. Uchinokura and H. Obara: Phys. Rev. Lett. 71, 4059 (1993) . 3) M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher: Phys. Rev. 40, 546 (1989). 4) T. Nakamura: Phys. Rev. B71, 144401 (2005). -10- 5) T. Masuda, A. Zheludev, H. Manaka, L.-P. Regnault, J.-H. Chung and Y. Qiu: Phys. Rev. Lett. 96, 047210 (2006). 6) F. D. Haldane: Phys. Rev. Lett. 50, 1153 (1983). 7) H. Manaka, I. Yamada, and K. Yamaguchi: J. Phys. Soc. Jpn. 66, 564 (1997). 8) H. Manaka and I. Yamada: J. Phys. Soc. Jpn. 66, 1908 (1997). 9) H. Manaka, I. Yamada and H. Aruga Katori: Phys. Rev. B63, 104408 (2001); In early papers of Manaka, three surfaces of the crystal were referred as A, B and C-plane in ascending order of area. Recent neutron experiment by Masuda has revealed that b*-axis is perpendicular to C-plane, and c-axis, A-plane. 10) H. Manaka, I. Yamada, H. Mitamura and T. Goto: Phys. Rev. B66, 064402 (2002). 11) T. Saito, A. Oosawa, T. Goto, T. Suzuki and I. Watanabe: Phys. Rev. B74, 134423. 12) H. Manaka, I. Yamada, M. Hagiwara and M. Tokunaga: Phys. Rev. B63, 144428 (2001). 13) H. Manaka,I. Yamada: Physica B284-288, 1607 (2000). 14) H. Manaka,I. Yamada: Phys. Rev. B62, 14279 (2000). 15) A. B. Harris: J. Phys. C 7, 1671 (1974). 16] J. Cardy: “Scaling and Renomalization in Statistical Physics” (Cambridge U. P. 1996). -11- −0.05 0 0.05 H−1γ/ν0 (T) 18.0K x=0.85(gapless) x=1.0 (gapped) 15K Fig. 1 1H-NMR spectra of x=0.85 and 1.0 at various temperatures. The gyromagnetic ratio of 1H, 42.5774MHz/T is denoted as 1γ. 0 5 10 15 Temperature (K) TN=13.5K 0.01 0.05 0.1 0.5 1 H⊥C x=0.85 (gapless) 1−T(K)/TN β=0.33 ∆H~ (1−|T |/TN) -12- Fig. 2 Dependence of the splitting width ∆H on the temperature, definition of which is shown in Fig. 1. The curve shows the scaling function ( )βN1 TT− , where =NT 13.5 K and 33.0=β are determined by data fitting. 0 10 20 300 Temperature (K) x=0.85 (gapless) x=1 (gapped) 0 1.0 2.0 0 10 20 Delay (msec) ↑ 3.7K ↓ 14K Fig. 3 Dependence of the spin-lattice relaxation rate 11 −T on temperature of gapped (x=1.0) and gapless (x=0.85) samples with the field direction H⊥C and H⊥B respectively. The inset shows typical relaxation curves for the x=0.85 sample. -13- 0.05 0.1 0.5 1 1−|T |/TN x=0.85 (gapless) T <TN 1.0 T >TN 0.5 −1~ (1−|T |/TN) Fig. 4 Scaling plot of 11 −T against the reduced temperature N1 TT− . The dynamical critical exponent was obtained from data fitting (dashed lines).

0704.1079

Instanton Induced Neutrino Majorana Masses in CFT Orientifolds with MSSM-like spectra

IFT-UAM/CSIC-07-12 CERN-PH-TH/2007-061 hep-th/yymmnnn Instanton Induced Neutrino Majorana Masses in CFT Orientifolds with MSSM-like spectra L.E. Ibáñez1 A.N. Schellekens2 and A. M. Uranga3 1 Departamento de F́ısica Teórica C-XI and Instituto de F́ısica Teórica C-XVI, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain 2 NIKHEF, Kruislaan 409, 1009DB Amsterdam, The Netherlands; IMAPP, Radboud Universiteit, Nijmegen, The Netherlands; Instituto de Matemáticas y F́ısica Fundamental, CSIC, Serrano 123, Madrid 28006, Spain. 3 PH-TH Division, CERN CH-1211 Geneva 23, Switzerland (On leave from IFT, Madrid, Spain) Abstract Recently it has been shown that string instanton effects may give rise to neutrino Majorana masses in certain classes of semi-realistic string compactifications. In this paper we make a systematic search for supersymmetric MSSM-like Type II Gepner orientifold constructions admitting boundary states associated with instantons giving rise to neutrino Majorana masses and other L- and/or B-violating operators. We analyze the zero mode structure of D-brane instantons on general type II orientifold compactifications, and show that only instantons with O(1) symmetry can have just the two zero modes required to contribute to the 4d superpotential. We however discuss how the addition of fluxes and/or possible non-perturbative extensions of the orientifold compactifications would allow also instantons with Sp(2) and U(1) symmetries to generate such superpotentials. In the context of Gepner orientifolds with MSSM-like spectra, we find no models with O(1) instantons with just the required zero modes to generate a neutrino mass superpotential. On the other hand we find a number of models in one particular orientifold of the Gepner model (2, 4, 22, 22) with Sp(2) instantons with a few extra uncharged non-chiral zero modes which could be easily lifted by the mentioned effects. A few more orientifold examples are also found under less stringent constraints on the zero modes. This class of Sp(2) instantons have the interesting property that R-parity conservation is automatic and the flavour structure of the neutrino Majorana mass matrices has a simple factorized form. http://arxiv.org/abs/0704.1079v2 1 Introduction String Theory, as the leading candidate for a unified theory of Particle Physics and Gravity, should be able to describe all observed particle phenomena. One the most valuable experimental pieces of information obtained in the last decade concerns neu- trino masses. Indeed the evidence from solar, atmospheric, reactor and accelerator experiments indicates that neutrinos are massive. The simplest explanation of the smallness of neutrino masses is the see-saw mechanism [1]. The SM gauge symmetry allows for two types of operators bilinear on the neutrinos (with dimension ≤ 4) : Lν = MabνaRνbR + habνaRH̄Lb (1.1) where νR is the right-handed neutrino, L is the left-handed lepton doublet and H̄ is the Higgs field. In supersymmetric theories, this term arise from a superpotential with the above structure, upon replacing fields by chiral superfields. If Mab is large, the lightest neutrino eigenvalues have masses Mν = < H̄ > 2 hTM−1h (1.2) For M ∼ 1010 − 1013 GeV and Dirac neutrino masses of order charged lepton masses, the eigenvalues are consistent with experimental results. What is the structure of neutrinos and their masses in string theory? In specific compactifications giving rise to the MSSM spectra singlet fields corresponding to right- handed neutrinos νR generically appear. Dirac neutrino masses are also generically present but the required Majorana νR masses are absent. This is because most MSSM- like models constructed to date have extra U(1) symmetries, under which the right- handed neutrinos are charged, which hence forbid such masses. In many models, such symmetries are associated to a U(1)B−L gauge boson beyond the SM. In order to argue for the existence of νR masses, string model builders have searched for non- renormalizable couplings of the type (νRνRN̄RN̄R) with extra singlets NR. Once the latter fields get a vev, U(1)B−L is broken and a Majorana mass appears for the νR. Although indeed such couplings (or similar ones with higher dimensions) exist in some semi-realistic compactifications, such a solution to the neutrino mass problem in string theory has two problems: 1) The typical νR masses so generated tend to be too small due to the higher dimension of the involved operators and 2) The vevs for the NR fields breaks spontaneously R-parity so that dimension 4 operators potentially giving rise to fast proton decay are generated. This is in a nutshell the neutrino problem in string compactifications (see [2] for a recent discussion in heterotic setups). In [3] (see also [4]) two of the present authors pointed out that there is a built- in mechanism in string theory which may naturally give rise to Majorana masses for right-handed neutrinos. It was pointed out that string theory instantons may generate such masses through operators of the general form Mstring e −U νRνR . (1.3) Here U is a linear combination of closed string moduli whose imaginary part gets shifted under a U(1)B−L gauge transformation in such a way that the operator is fully gauge invariant. The exponential factor comes from the semi-classical contribution of a certain class of string instantons. This a pure stringy effect distinct from the familiar gauge instanton effects which give rise to couplings violating anomalous global symmetries like (B+L) in the SM. Here also (B−L) (which is anomaly-free) is violated. This operator is generated due to existence of instanton fermionic zero modes which are charged under (B − L) and couple to the νR chiral superfield. Although the effect can take place in different constructions, the most intuitive description may be obtained for the case of Type IIA CY orientifold compactifications with background D6-branes wrapping 3-cycles in the CY. In the simplest configurations one has four SM stacks of D6-branes labeled a,b, c,d which correspond to U(3), SU(2) (or U(2)), U(1)R and U(1)L gauge interactions respectively, which contain the SM group. One can construct compactifications with the MSSM particle spectrum in which quarks and leptons lie at the intersections of those SM D6-branes. Then the relevant instantons correspond to euclidean D2-branes wrapping 3-cycles in the CY (satisfying specific properties so as to lead to the appropriate superspace interaction). The D2-D6 intersections lie the additional fermionic zero modes which are charged under (B−L). For instantons with the appropriate number of intersections with the appropriate D6-branes, and with open string disk couplings among the zero modes and the νR chiral multiplet (see fig.(3.1)), the operator in (1.3) is generated. The fact that the complex modulus U transforms under U(1)B−L gauge transfor- mations indicates that the U(1)B−L gauge boson gets a mass from a Stückelberg term. So a crucial ingredient in the mechanism to generate non-perturbative masses for the νR’s is that there should be massless U(1)B−L gauge boson which become massive by a Stückelberg term. It turns out that not many semi-realistic models with U(1)B−L mass from Stückelberg couplings have been constructed up to date. In the literature there are two main classes of RR tadpole free models with massive B-L. The first class are non-susy type IIa toroidal orientifold models first constructed in [5]. The second class are the type II Gepner orientifold models constructed by one of the present authors and collaborators [6, 7]. The former were already considered in [3]. In the present paper we will concentrate on the RCFT Gepner model constructions, which lead to a large class of MSSM like models, more representative of the general Calabi-Yau compactifications (for a recent discussion of instanton-induced neutrino masses in a model with no RR tadpole cancellation, see [8]). The class of constructions in [6, 7] start with any of the 168 Type II compacti- fications obtained by tensoring N = 2 SCFT minimal models. In addition one can choose a number of modular invariant partition functions (MIPF), leading to a total of 5403. Then different consistent orientifold projections are performed on the different models. This yields a total of 49304 Type II orientifolds. The open string sector of the theory is defined in terms of the boundary states of the theory. Intuitively, they play the same role as D-branes wrapping cycles in the geometrical settings. Thus one associates boundary states a,b, c,d to the gauge groups giving rise to the SM. Differ- ent choices for the SM boundary states lead to different spectra. In the present paper we will make use of the data in [6] which contains 211634 different MSSM-like spectra (including also different hidden sectors). Although this number is huge, most of these models are really extensions of the MSSM, since they have either an extra U(1)B−L or SU(2)R × U(1)B−L group factor beyond the SM group. As we said, we are actually only interested in models in which the U(1)B−L gets a Stückelberg mass. Then we find that the number of MSSM-like models with these characteristics is dramatically reduced: only 0.18 percent of the models (391) have a massive U(1)B−L. As we said, in the geometrical setting of IIA orientifolds with intersecting D6-branes [9, 10] (see [11] for reviews and [12, 13] for the IIB counterparts), instantons are associ- ated to D2-branes wrapping 3-cycles, like the background D6-branes do. Analogously, in the RCFT setting the same class of boundary states appearing in the SM construc- tions are the ones corresponding to instantons. The zero modes on the instanton is computable from the overlaps of instanton brane boundary states (zero modes un- charged under the 4d gauge group) or of instanton and 4d spacefilling brane boundary states (zero modes charged under the corresponding gauge factor). We find that the criteria for a non-perturbative superpotential to be generated [14] are only fulfilled if the Chan-Paton (CP) symmetry of the instantons is O(1). For instantons with CP symmetry 1 Sp(2) or U(1) we find that there are a few extra uncharged fermionic zero modes which would preclude the formation of the searched superpotentials. On the other hand we argue that the addition of fluxes and/or possible non-perturbative 1We adopt the convention that the fundamental representation of Sp(m) is m-dimensional. extensions of the orientifold compactifications would allow also instantons with Sp(2) and U(1) symmetries to generate such superpotentials. We thus include all O(1), Sp(2) and U(1) instantons 2 in our systematic search. The computation of charged and un- charged fermion zero modes may be easily implemented as a routine in a systematic computer search for instanton zero modes in Gepner MSSM-like orientifolds. Results of such a systematic computer search are presented in this article. We find that out of the 391 models with massive U(1)B−L, there are very few ad- mitting instantons with the required minimal O(1) CP symmetry, and in fact none of them without additional vector-like zero modes. On the other hand we do find 32 models admitting Sp(2) symmetric instantons with just the required charged zero mode content (and the minimal set of non-chiral fermion zero modes). In fact they are all variations of the same orientifold Gepner model based on the tensor product (k1, k2, k3, k4) = (2, 4, 22, 22). These models all in fact correspond to the same MIPF and orientifold projection, they only differ on which particular boundary states corre- sponding to the four a,b, c,d SM ‘stacks’. All models have the same chiral content, exactly that of the MSSM , with extra vectorlike chiral fields which may in principle become massive in different points of the CY moduli space. They have no hidden sector, i.e., the gauge group is just that of the SM. For each of the models there are 8 instantons with Sp(2) CP symmetry with just the correct charged zero mode structure allowing for the superpotential coupling (1.3) giving rise to νR Majorana masses. As we said, they have extra uncharged fermion zero modes beyond the two required to generate a superpotentials. However one would expect that these unwanted zero modes might be lifted in more generic situations in which e.g. NS/RR fluxes are added. We thus see that, starting with a ’large’ landscape of 211634 MSSM-like models, and searching for instantons inducing neutrino masses, we find there are none admitting the O(1) instantons with exactly the required zero mode structure, and only few (32) examples with Sp(2) instantons with next-to-minimal uncharged zero mode structure (and exactly the correct charged zero modes). Let us emphasize though that it is the existence of massive U(1)B−L models which is rare. Starting with the subset of models with a massive U(1)B−L, finding models with correct instanton charged zero modes within that class is relatively frequent, 10 percent of the cases. Furthermore, we will see that there are further models beyond those 32 which contain extra non-chiral instanton zero modes and which may also be viable if these modes get massive by some 2We refer to the different kinds of instanton by their Chan-Paton symmetry on their volume. Since we are not interested in gauge theory instantons, this notation should not be confusing. effect (like e.g. the presence of RR/NS fluxes). Instantons may generate some other interesting superpotential couplings in addition to νR masses, some possibly beneficial and others potentially dangerous. In particular we find that in the models which contain Sp(2) instantons which might induce νR masses, there are also other instantons which would give rise directly to the Weinberg operator [15] (LHLH) (1.4) Once the Higgs field gets a vev, this gives rise directly to left-handed neutrino masses. Thus we find that in that class of models both the see-saw mechanism (which also gives rise to a contribution to the Weinberg operator) and an explicit Weinberg operator might contribute to the physical masses of neutrinos. Which effect dominates will depend on the relative size of the corresponding instanton actions as well as on the size of the string scale. Among potentially dangerous operators which might be generated stand the R-parity violating operators of dimension < 5, which might give rise e.g. to fast proton decay. We make an study of the possible generation of those, and find that in all models in which νR masses might be generated R-parity is exactly conserved. This is a very encouraging result. A natural question to ask is whether one can say something about the structure of masses and mixings for neutrinos. As argued in [3] generically large mixing angles are expected, however to be more quantitative we also need to know the structure of Yukawa couplings for leptons. In principle those may be computed in CFT but in practice this type of computation has not yet been developed for CFT orientifolds. Nevertheless we show that, in the case of instantons with Sp(2) CP symmetry, a certain factorization of the flavor structure takes place, which could naturally give rise to a hierarchical structure of eigenvalues for neutrino masses. The structure of this article is as follows. In the next section we present a discussion of instanton induced superpotentials in Type II orientifolds. This discussion will apply both to Type IIA and Type IIB CY orientifolds as well as to more abstract CFT orientifolds. We discuss the structure of both uncharged and charged instanton zero modes. In particular we show that only instantons with O(1) CP symmetry have the appropriate uncharged zero mode content to induce a superpotential contribution. We also discuss how Sp(2) and U(1) might still generate superpotential contributions if extra ingredients are added to the general setting. In section 3 we apply that discussion to the specific case of the generation of νR Majorana masses, showing what is the required zero mode structure in this case. We show how the flavor structure of the Majorana mass term factorizes in the case of instantons with Sp(2) CP symmetry, leading potentially to a hierarchical structure of eigenvalues. We further discuss the generation of other B/L-violating operators including the generation of the Weinberg operator as well as R-parity violating couplings. In section 4 we review the RCFT Type II orientifold constructions in [6, 7]. A general discussion of zero fermion modes for instantons in RCFT orientifolds is presented in section 5. The results of our general search for instantons generating νR masses are presented in section 6. We provide a list of all Gepner orientifolds which admit instanton con- figurations potentially giving rise to νR Majorana masses. We describe the structure of the models with Sp(2) instantons having the required charged zero modes for that superpotential to be generated. We also describe the boundary states of the corre- sponding instantons and provide the massless spectrum of the relevant MSSM-like models. Other orientifolds with zero mode structure close to the minimal one are also briefly discussed. A brief discussion about the possible generation of R-parity violating superpotentials is included. We leave section 7 for some final comments. Some nota- tion on the CFT orientifold constructions, and a discussion of the CFT symmetries in the Sp(2) examples are provided in two appendices. As this paper was ready for submission, we noticed [16, 17], whose discussion of instanton zero modes partially overlaps with our analysis in Section 2.2. 2 Instanton induced superpotentials in Type II ori- entifolds In this Section we review the generation of superpotentials involving 4d charged fields via D-brane instantons in type II compactifications. The discussion applies both to type IIA and IIB models, and to geometrical compactification as well as to more abstract internal CFT’s. For recent discussions on D-brane instantons, see [4, 3, 25, 8] . Before starting, a notational remark in in order. Our notation is adapted to working in the covering theory, namely the type II compactification, and orientifolding in a further step. Thus we describe the brane configurations as a system of branes (described by boundary states for abstract CFT’s), labeled k, and their orientifold images labeled k′. Similarly, we denote M the brane / boundary state corresponding to the instanton brane, andM ′ its orientifold image. If a brane is mapped to itself under the orientifold action, we call it a ‘real’ brane / boundary, and ‘complex’ otherwise. 2.1 D-brane instantons, gauge invariance and effective opera- A basic feature of type II orientifold compactifications with D-branes is the generic presence of Stückelberg couplings between the U(1) gauge fields on the D-branes, and certain 4d RR closed string 2-forms. These couplings are required by the Green- Schwarz mechanism when the U(1)’s have non-zero triangle contributions to mixed anomalies [18, 19], but can also exist for certain non-anomalous U(1)’s [5, 20]. These couplings make the U(1) gauge bosons massive, but the symmetry remains as a global symmetry exact in perturbation theory. Since the closed string moduli involved are scalars (0-forms) in the RR sector, the natural candidate non-perturbative effects to violate these U(1) symmetries are instantons arising from euclidean D-branes coupling to these fields. In computing the spacetime effective interaction mediated by the instanton, one needs to integrate over the instanton zero modes. In the generic case (and in particular for the case of our interest) there are no bosonic zero modes beyond the universal ones (namely, the four translational bosonic zero modes associated to the position of the instanton). On the other hand, the structure of fermion zero modes will be crucial. Since we are interested in models with non-trivial 4d gauge group, arising from a set of background 4d spacetime filling branes, we consider separately fermion zero modes which are uncharged under the 4d gauge group and those which are charged. In this paper we restrict our discussion to 4d N = 1 supersymmetric models, and this will simplify the analysis of zero modes. Fermion zero modes which are uncharged under the 4d gauge group determine the kind of 4d superspace interaction which is generated by the instanton. We are interested in generating superpotential interactions, which, as is familiar, requires the instanton to have two fermion zero modes to saturate the d2θ superspace integration. For this, a necessary (but not sufficient!) condition is that the D-branes are half-BPS, so these fermion zero modes are the Goldstinos of the two broken supersymmetries. In the string description, uncharged zero modes arise from open strings in the MM sector (in our notation, the one leading to adjoint representations), which in particular contain these Goldstinos, and the MM ′ sector (in our notation, the one leading to two-index symmetric or antisymmetric tensors). Note that both are the same for ‘real’ branes. Hence we are primarily interested in D-branes whose MM sector contains just two fermion zero modes, and whose MM ′ sector (for ‘complex’ branes) does not contain additional fermion zero modes. In analogy with the familiar case of gauge theory instantons [21], charged fermion zero modes determine the violation of perturbative global symmetries by the instanton- induced interaction. Namely, in order to saturate the integration over the charged fermions zero modes, the spacetime interaction must contain insertions of fields charged under the 4d gauge symmetry, and in particular under the global U(1) factors, which are thus violated by the D-brane instanton. Notice that this holds irrespectively of the number of uncharged fermion zero modes, namely of the kind of superspace interaction induced by the instanton. Restricting to superpotential interactions, the resulting operator in the 4d effective action has roughly the form Winst = e −U Φ1 . . .Φn (2.1) Here the fields Φ1, . . . ,Φn are 4d N = 1 chiral multiplets charged under the 4d gauge group, and in particular also under the U(1) symmetries. Note also that the instanton amplitude contains a prefactor (which in general depends on closed and open string moduli) arising from the Gaussian path integral over (massive) fluctuations of the instanton (hence described by an open string annulus partition function, see [22, 23] for related work), which we can ignore for our purposes in this paper. For D-brane instantons, U is the closed string modulus to which the euclidean D- brane couples. In the D-brane picture, instanton fermion zero modes charged under the gauge factor carried by the kth stack of 4d space-filling branes (and its image k′) arise from open strings in the Mk and Mk′ sectors, transforming as usual in the ( M , k) and ( M , k) representations, respectively (with both related in the case of ‘real’ branes). The (net) number of instanton fermion zero modes with such charges is given by certain multiplicities 3 IMk, IMk′. A D-brane instanton, irrespectively of the superspace structure of the 4d interac- tions it may generate, violates U(1)k charge conservation by an amount IMk − IMk′ for ‘complex’ branes and IMk for ‘real’ branes. In particular, this is the total charge of the field theory operator Φ1 . . .Φn in (2.1). From the Stückelberg couplings, it is possible to derive [3] (see [23, 24, 25] for related work, also [26]) 3In geometric type IIA compactifications with 4d spacetime-filling branes and instanton branes given by D6- and D2-branes wrapped on Special Lagrangian 3-cycles, IMk corresponds to the inter- section number between the 3-cycles corresponding to the kthD6- and the D2-brane M (and similarly for IMk′ ). In geometric type IIB compactifications, it corresponds to the index of a suitable Dirac operator. In general (even for abstract CFT’s) it can be defined as the Witten index for the 2d theory on the open string with the boundary conditions corresponding to the two relevant branes. We will often abuse language and refer to this quantity as intersection number, even in Section 6 where we work in the non-geometric regime of type IIB compactifications. that for ‘complex’ instantons, gauge transformations of the U(1)k vector multiplets Vk → Vk + Λk, transform U as U → U + Nk(IMk − IMk′)Λk (2.2) For ‘real’ brane instantons, which were not considered in [3], the shift is given by 4 U → U + NkIMkΛk (2.3) (this new possibility will be an important point in our instanton scan in Section 6). The complete interaction (2.1) is invariant under the U(1) gauge symmetries. How- ever, from the viewpoint of the 4d low-energy effective field theory viewpoint, it leads to non-perturbative violations of the perturbative U(1) global symmetries, by the amount mentioned above. In the string theory construction there is a simple microscopic explanation for the appearance of the insertions of the 4d charged fields (related to the mechanism in [27]). The instanton brane action in general contains cubic terms αΦ γ, involving two instanton fermions zero modes α in the ( M , k) and γ in the ( p, M) coupling to the 4d spacetime field Φ in the ( k, p) of the 4d gauge group 5. Performing the Gaussian path integral over the instanton fermion zero modes leads to an insertion of Φ in the effective spacetime interaction. In general, and for a ‘complex’ instanton, there are several fermion zero modes αi, γi in the fundamental (resp. antifundamental) of the instanton gauge group, coupling to a 4d spacetime chiral operators Oij (which could possibly be elementary charged fields, or composite chiral operators). Gaussian integration over the fermion zero modes leads to an insertion of the form detO (for ‘real’ brane instantons, detO should be interpreted as a Pfaffian). It is straightforward to derive our above statement on the net charge violation from this microscopic mechanism. Note that the above discussions show that instantons in different topological sec- tors (namely with different RR charges, and different intersection numbers with the 4d spacefilling branes) contribute to different 4d spacetime operators. In particular, multiwrapped instantons, if they exist as BPS objects, contribute to operators different 4Equivalently, one may use (2.2), but must include an additional factor of 1/2 from the reduction of the volume for a real brane (which is invariant under the orientifold action). 5 Although there is no chirality in 0+0 dimension, the fermion zero modes α and γ are distinguished by their chirality with respect to the SO(4) global symmetry of the system (which corresponds to rotations in the 4d spacetime dimensions, longitudinal to the space-filling branes, and transverse to the instanton brane). Supersymmetry of the instantons constrains the couplings on the instanton action (such as the cubic couplings above) to have a holomorphic structure. from the singly wrapped instanton. This implies that the instanton expansion for a fixed operator is very convergent, and could even be finite. Another important implication of the above discussion is that, in order to generate a specific operator via an instanton process, a necessary condition is that the instanton has an appropriate number and structure of charged zero modes. However, this is not sufficient. Insertions of 4d fields appear only if the fields couple to the instanton fermion zero modes via terms at most quadratic in the zero modes. In equivalent terms, only zero modes appearing in the Gaussian part of the instanton action can be saturated by insertions of 4d fields (those to which they couple). The requirement that the zero modes have appropriate couplings to the 4d fields may be non-trivial to verify in certain constructions. This is the case for the Gepner model orientifolds in coming sections, whose couplings are computable in principle, but unknown in practice. In such cases we will assume that any coupling which is not obviously forbidden by symmetries will be non-vanishing. Unfortunately there are no arguments to estimate the actual values of such non-vanishing couplings, hence we can argue about the existence of certain instanton induced operators, but not about the coefficients of such terms. 2.2 Zero mode structure for D-brane instantons In this section we describe more concretely different kinds of instantons and the struc- ture of interesting and unwanted zero modes. Our discussion will be valid for general D-brane models in perturbative type II orientifolds without closed string fluxes, al- though we also make some comments on more general F-theory vacua and the effects of fluxes. A more specific discussion is presented in Section 5. 2.2.1 Uncharged zero modes We start discussing zero modes uncharged under the 4d gauge group. These are crucial in determining the kind of superspace interaction induced by the instanton on the 4d theory. In particular, we are interested in instantons contributing to the 4d superpo- tential, namely those which contain just two fermion zero modes in this sector. We are also interested in instantons which may contain additional fermion zero modes, and the possible mechanisms that can be used to lift them. Let us discuss ‘real’ and ‘complex’ brane instantons in turn. • Real brane instantons Real brane instantons correspond to branes which are mapped to themselves by the orientifold action, hence M = M ′. Uncharged zero modes arise from the MM open string sector. As discussed in Section 5, there is a universal sector of zero modes, in the sense that it is present in any BPS D-brane instanton, which we now describe. Before the orientifold projection, we have a gauge group U(n) on the volume of n coincident instantons. Notice that, although there are no gauge bosons in 0 + 0 dimensions, the gauge group is still well-defined, since it acts on charged states (open string ending on the instanton brane). There are four real bosonic zero modes and four fermion zero modes in the adjoint representation. For the minimal U(1) case, the four bosons are the translational Goldstones. The four fermions arise as follows. This sector is insensitive to the extra 4d spacefilling branes, and feels an accidental 4d N = 2 supersymmetry. The BPS D-brane instanton breaks half of this, and leads to four Goldstinos, which are the described fermions 6. The orientifold projection acts on this universal sector as follows (see Section 5 for further discussion). The gauge group is projected down to orthogonal or symplectic. Hence instanton branes with symplectic gauge group must have even multiplicity (a related argument, in terms of the orientifold action on Chan-Paton indices, is given in Section 5). For instantons with O(n) gauge symmetry, the orientifold projects the four bosonic zero modes and two fermion zero modes (related by the two supercharges of 4d N = 1 supersymmetry broken by the instanton) to the two-index symmetric represen- tation, and the other two fermion zero modes (related by the other two supercharges of the accidental 4d N = 2 in this sector) to the antisymmetric representation. Hence for O(1) instantons (namely instantons with O(1) gauge group on their volume), we have just two fermion zero modes, which are the Goldstinos of 4d N = 1 supersymmetry, and the instanton can in principle contribute to the superpotential (if no additional zero modes arise from other non-universal sectors). For instantons with Sp(n) gauge symmetry, the orientifold projects the four bosonic zero modes and two fermion zero modes to the two-index antisymmetric representation, and the other two fermion zero modes to the symmetric representation. Hence, even for the minimal case of Sp(2) instantons, we have just two fermion zero modes in the triplet representation, in ad- dition to the two 4d N = 1 Goldstinos. Hence Sp(2) instantons cannot contribute to the superpotential in the absence of additional effects which lift these zero modes (see later) 7. 6We thank F. Marchesano for discussions on this point. 7 For D-brane instantons corresponding to gauge instantons, the additional fermion zero modes in the universal sector couple to the boson and fermion zero modes from open strings stretched between the instanton and the 4d spacefilling brane. They act as Lagrange multipliers which impose In addition to this universal sector, there exist in general additional modes, whose presence and number depends on the detailed structure of the branes. Namely, on the geometry of the brane in the 6d compact space in geometric compactifications, or on the boundary state of the internal CFT in more abstract setups. They lead to a number of boson and fermion zero modes in the symmetric or antisymmetric represen- tation. The computation of these multiplicities in terms of the precise properties of the instanton branes is postponed to Section 5. In order to generate a superpotential, one must require these modes to be absent, except for the case of antisymmetrics of O(1) instantons, which are actually trivial. An important point is that extra fermion zero modes (including the extra triplet fermion zero modes in the universal sector of Sp(2) instantons, and any two-index tensor fermion zero mode in the non-universal sectors) are in principle not protected against acquiring non-zero masses once the model is slightly modified. In other words, such fermions are non-chiral, in terms of the SO(4) chirality in footnote 5. One such modification is motion in the closed string moduli space, which can lift the non- universal modes if there are non-trivial couplings between them and closed string mod- uli (unfortunately, such correlators are difficult to compute, even in cases where the CFT is exactly solvable, like the Gepner models). Note that extra zero modes in the universal sector of Sp(2) instantons cannot be lifted by this effect, since it does not break the accidental 4d N = 2 in this sector. A second possible modification which in general can lift extra zero modes is the addition of fluxes, generalizing the results for D3-brane instantons in geometric compactifications [29] (for non-geometric CFT compactifications, we also expect a similar effect, once fluxes are introduced following [30]). Note that fluxes can lift extra zero modes in the universal sector as well, since they can break the accidental 4d N = 2 susy in this sector. A last mechanism arising in more general F-theory compactifications and discussed below for complex instantons, is valid for real instanton branes as well. The bottom line is that in the absence of such extra effects, only O(1) instantons can contribute to superpotential terms. However, in modifications of the model such extra effects can easily lift the extra fermion zero modes. Hence, this kind of extra vector-like zero modes will not be considered catastrophic, and real instantons (including the O(1) and Sp(2) cases) with such zero modes are considered in our scan in Section 6. the fermionic constraints in the ADHM construction [28], and may not spoil the generation of a superpotential. • Complex brane instantons Zero modes uncharged under the 4d gauge group can arise from the MM and MM ′ open string sectors. Notice that the orientifold action maps the MM sector to the M ′M ′, hence we simply discuss the former and impose no projection. The discussion of the MM sector is as for real brane instantons before the orientifold projection. The universal sector leads to a U(n) gauge symmetry, and four bosonic and four fermionic zero modes in the adjoint representation. The bosons are translational Goldstones, while the fermions are Goldstinos of the accidental 4d N = 2 present in this sector. Hence, even in the minimal case of U(1) brane instantons there are two extra fermion zero modes, beyond the two fermion zero modes corresponding to the 4d N = 1 Goldstinos. Hence U(1) instanton (except for those corresponding to gauge instantons, see footnote 7) cannot contribute to superpotential terms in the absence of additional effects, like closed string fluxes . However, keeping in mind the possibility of additional effects lifting them in modifications of the model, we include them in the discussion. Also, in what follows we will use the U(n) notation for the different fields to keep track of the Chan-Paton index structure. The above statement would seem in contradiction8 with computations of non- perturbative superpotentials [14] induced by M5-branes instantons in M-theory com- pactifications on Calabi-Yau fourfolds, which are dual to D3-brane instantons (with world-volume U(1) gauge group) on type IIB compactifications. The resolution is that the M5-branes that contribute are intersected by different (p, q) degenerations of the elliptic fiber. This implies that U(1) D3-brane instanton only contribute if they are intersected by mutually non-local (p, q) 7-branes. The two extra fermion zero modes exist locally on the D3-brane volume, but cannot be defined globally due to the 7-brane monodromies. Hence such effect can take place only on non-perturbative type IIB com- pactifications including (p, q) 7-branes. Note that in perturbative compactifications, namely IIB orientifolds, the (p, q) 7-branes are hidden inside orientifold planes [31] with their monodromy encoding the orientifold projection; hence the only branes that can contribute to the superpotential are real branes, for which the projection/monodromy removes the extra fermion zero modes, as discussed above. In addition to this universal sector, theMM sector may contain a non-universal set of fermions and bosons, in the adjoint representation (hence uncharged under U(1)). They depend on the specific properties of the brane instanton, and will be discussed in Section 5. These additional zero modes should be absent in order for the instanton 8We thank S. Kachru for discussions on the ideas in this paragraph. to induce a non-trivial superpotential. Notice however that these zero modes are un- charged under any gauge symmetry, and hence vector-like. Thus, there could be lifted in modifications of the model, as discussed for real instantons. The MM ′ sector is mapped to itself under the orientifold action. Hence it leads to a number of bosons and fermions in the two index symmetric or antisymmetric repre- sentations. Notice that the two-index antisymmetric representation is trivial for U(1), so these modes are actually not present. On the other hand, fermion zero modes in the two-index symmetric representation are chiral and charged under the brane instanton gauge symmetry. Hence they cannot be lifted by any of the familiar mechanisms, and thus spoil the appearance of a non-perturbative superpotential, even if the model is modified. Such fermion zero modes are considered catastrophic and we will look for models avoiding them in our scan in Section 6. 2.2.2 Charged fermion zero modes • Real brane instantons Instanton zero modes charged under the 4d gauge group arise fromMk open string sectors (and their image Mk′). In the generic case, there are no scalar zero modes in these sectors. This is because in mixed Mk open string sectors the 4d spacetime part leads to DN boundary conditions, which already saturate the vacuum energy in the NS sector. Only in the special case where the internal structure of the spacetime filling brane k and the instanton brane are the same, there may be NS ground states of the internal CFT which do not contribute extra vacuum energy, hence leading to massless scalars. However, this corresponds to brane instantons which can be interpreted as instantons of the 4d gauge theory on the 4d space-filling branes. The instantons we are interested in for the generation of neutrino Majorana mass terms are not of this kind [3] (see e.g. [32, 33, 28, 25, 34] for discussions on gauge theory instantons from D-brane instantons). Hence we focus on charged fermion zero modes, which are generically present in any mixed Mk sector. Let us define LMk, LMk′ the (positive by definition) number of left-handed chiral fermion zero modes in the representations ( M , k), ( M , k), respectively. The net number of chiral fermion zero modes in the ( M , k) is given by IMk = LMk′ − LMk. This controls the violation of the U(1)a global charge by the instanton. Namely, such fermion zero modes in the Mk, Mp sectors lead (if suitable couplings are present) to the insertion of 4d charged fields Φkp and/or Φkp′. In addition, there are PMk = min(LMk′ , LMk) vector-like pairs of fermion zero modes. Since they are vector-like, they may be lifted by slight modifications of the model, like moving in the closed string moduli space, or by introducing additional ingredients, like fluxes. In addition, they may be lifted by moving in the open string moduli space of the 4d spacefilling branes, as follows. The zero modes may lead to insertions of 4d fields Φkk, if the kk sector contains such 4d chiral multiplets (or to insertions of composite 4d operators in the adjoint of the kth 4d gauge factor), and if they couple to the zero modes. Although this may not be generically not the case, many of our models in coming section contain such adjoint fields. Hence, a non- trivial vev for the latter can lift these extra vector-like zero modes, hence leading to instanton generating the superpotential of interest. Given these diverse mechanisms to lift these zero modes, their presence of such zero modes is thus unwanted, but again not necessarily catastrophic. One last comment, related to the concrete kind of instanton search we will be interested in. Namely, we will be searching for instantons leading to a specific operator, carrying non-trivial charges under a specific set of 4d gauge factors. Postponing the detailed discussion to Sections 3.1, 4.2 , let us denote a, b, c, d the set of branes leading to a field theory sector, denoted ‘observable’ (and which reproduces the SM in our examples). We will require the instanton to have a prescribed number of chiral fermion zero modes charged under these branes, namely we require the intersection numbers of the instanton with these branes IMa, . . . , IMd to have specific values (as mentioned above, in the most restrictive scan we forbid vector-like pairs of zero modes under these branes). In addition, the model in general contains an additional sector of branes, denoted ‘hidden’ (since there is zero net number of chiral multiplets charged under both sectors) and labeled hi, required to fulfill the RR tadpole cancellation conditions. In general there may be instanton fermion zero modes from e.g. the Mh1, h2M sectors, which would contribute to insertions of the 4d fields in the h1h2 sector if there are suitable cubic couplings. These extra insertions could be avoided if such 4d fields in the hidden sector acquire vevs (note that vevs for the (vector-like) fields charged under the visible and hidden sectors would typically break hypercharge, and should be avoided), and hence lift the zero modes. Equivalently, from the 4d perspective, the unwanted extra h1h2 field insertion is replaced by its vev. However, this renders the discussion very model dependent. Moreover, the possibility of hidden brane recombination was not included in the search for SM-like models in [6, 7] (namely, the possibility of allowing for chiral fields charged under the observable and hidden gauge groups, which may become non-chiral and possibly massive upon hidden brane recombination). Hence we will consider these chiral fermion zero modes as unwanted (as usual, non-chiral modes are unwanted but not catastrophic, hence they are allowed for in a more relaxed search). • Complex brane instantons The discussion of ‘complex’ brane instantons is somewhat analogous to the previous one, with the only complication that the braneM and its imageM ′ lead to independent modes, leading to a more involved pattern of fermion zero modes. Instanton zero modes charged under the 4d gauge group arise from the Mk,Mk′ and related sectors. As for ‘real’ brane instantons, there are generically no scalars in these sectors (and certainly not in our case of interest). Hence we focus on charged fermion zero modes, which are generically present in any mixed sector. In contrast with ‘real’ brane instantons, a net combination of fermion zero modes in the ( M , k) + ( M , k) is not vector-like, but chiral under the instanton gauge symmetry. Such a pair cannot therefore be lifted even by modifications of the theory. In general, if the instanton has a mismatch in the total numbers nα, nγ of fermion zero modes αi in the M and γj in the M , the instanton amplitude automatically vanishes. Namely, the matrix of operators Oij coupling to the zero modes necessarily has rank at most min(nα, nγ). That is , if nα > nγ there are linear combinations of the αi which do not couple, and cannot lead to insertions. Moreover, they are not liftable by the familiar mechanisms 9, thus in our instanton search in Section 6 such excess zero modes are forbidden even in relaxed scans. Let us thus discuss a sector of fermion zero modes with equal number nα = nγ . Considering a given 4d space-filling brane k, let us denote LMk, LM ′k′, LMk′, LM ′k the (positive by definition) number of left-handed chiral fermion zero modes in the repre- sentations ( M , k), ( M , k), ( M , k), and ( M , k) respectively. The net number of chiral fermion zero modes in the ( M , k) and ( M , k) is given by IMk = LMk′ −LM ′k′ and IMk′ = LMk−LM ′k, respectively. This net number of fermions zero modes controls the violation of the U(1)a global charge by the instanton. Namely, such fermion zero modes in the Mk, Mp, Mk′, Mp′ sectors lead (if suitable couplings are present) to the insertion of 4d charged fields Φkp and/or Φkp′. 9Note that such a mismatch is always correlated with the existence of extra chiral zero modes in the MM ′ sectors, discussed above. Denoting ~Qa, ~Qorient the vector of RR charges of the 4d space- filling branes and orientifold planes, they satisfy the RR tadpole conditions Na ~Qa + Na ~Qa′ + ~Qorient. = 0. By taking the ‘intersection’ bilinear with the RR charges ~QM of the brane instanton, we have IMa+IMa′+IM,orient = 0. This implies that the number of fundamentals minus anti-fundamentals of the instanton gauge group is related to the number of two-index tensors. The remaining fields in this sector are vector-like pairs, in the ( M , k) + ( M , k) or the ( M , k) + ( M , k), which in principle lead to the vanishing of the instanton amplitude, but which can be lifted by additional effects (motion in closed or open string moduli space, or addition of fluxes), in a way consistent with the gauge symmetries in 4d spacetime and on the instanton. Just like for ‘real’ brane instantons, we conclude by commenting on our concrete instanton search in models with a set of visible branes a, b, c, d and a set of hidden branes hi. The requirement that the instanton leads to an operator with specific charges under the visible branes fixes the values of the quantities IMa − IMa′, etc. As we described for real branes, one may still have fermion zero modes charged under the hidden sector branes, but they lead to additional insertions, hence we rather focus on instantons with IMhi − IMh′j = 0. The two kinds of conditions, on intersection numbers with visible and hidden branes, still leave the possibility of combinations of fermion zero modes of the kind ( M , k) + ( M , k), which do not contribute to IMk, or of the kind ( M , k) + ( M , k), which does not contribute to IMk′. Such combinations are automatically vector-like, and thus may be lifted in modifications of the theory. But the condition also allow combinations like ( M , k)+( M , k), which exploit a cancellation between IMk and IMk′ (as also does ( M , k) + ( M , k)). Such combinations are chiral by themselves, and in general imply a mismatch of modes in the M and the M . The total mismatch can be arranged to vanish using combinations of the kind ( M , k) + ( M , k) and ( M , p) + ( M , p) for different branes. However, the only way to lift these pairs is by breaking the gauge symmetry on the 4d space-filling branes k and p. This can be done without damage to the visible sector if these are hidden branes, but this corresponds to the recombination of hidden branes that, as mentioned already, we are not going to consider. Hence only vector-like pairs with respect to each brane are considered to be liftable in simple modifications of the theory. In our instanton search, these are the only additional fermion zero modes which we allow in relaxed scans (but they are clearly not allowed for in restricted scans) 3 Instanton induced Majorana neutrino masses In this Section we discuss the possible physical effects of D-brane instantons in string models with SM-like spectrum. In particular we describe the conditions to generate right-handed neutrino Majorana masses. We also comment on other possible B and/or L violating operators that can be generated by instantons. In this section we will again Sp(2),U(2) U , D gluon d− Leptonic a− Baryonic b− Left c− Right Figure 1: Quarks and leptons at intersecting branes use the geometrical language of IIA intersecting D-branes but it should be clear that our discussion equally applies to general CFT orientifolds like the ones presented in the next section. 3.1 The MSSM on the branes Let us now specify the discussion in the previous section to the case of the generation of a right-handed neutrino mass term. In order to do that we need some realistic orientifold construction with the gauge group and fermion spectrum of the Standard Model (SM). In the context of Type II orientifolds perhaps the most economical brane configuration leading to a SM spectrum is the one first considered in [5]. This consists of four stacks, labelled a,b, c,d. The gauge factor on branes a is U(3), and contains QCD and baryon number. The d factor is U(1)d, and corresponds to lepton number. Stack b contains SU(2)Weak either embedded in U(2) or Sp(2). Finally brane c can either provide a U(1) or an O(2) factor. In the brane intersection language, the chiral fermions of the SM live at the intersections of these branes, as depicted in Fig. 3.1. The U(1)Y factor of the standard model is embedded in the Chan-Paton factors of branes a,c and d as (QB−L −QR) (3.1) where Qx denotes the generator of the U(1) of brane stack x (in case the Chan-Paton factor of brane c is O(2) one should use the properly normalized O(2) generator). Note that in this convention the Qd generator appears with sign opposite to other conventions in the literature, e.g. in [3]. In addition to Y these models have two additional U(1) gauge symmetries: Qanom = 3Qa +Qd = 9QB +QL Y ′ = Qa +Qc −Qd = QB−L +QR (3.2) The first is anomalous whereas the second, which we will call B − L (with a slight abuse of language, since it is in fact a linear combination of B − L and hypercharge), is anomaly free. In models in which the electroweak gauge group is embedded in U(2), rather than in Sp(2), there is a second anomalous U(1)b. The charges of the SM particles under these U(1) symmetries are given in table 1. Intersection D = 4 fields/ zero modes Qa Qc Qd Y QM (ab),(ab’) QL 3(3, 2) 1 0 0 1/6 0 (ca) UR 3(3̄, 1) -1 1 0 -2/3 0 (c’a) DR 3(3̄, 1) -1 -1 0 1/3 0 (db),(db’) L 3(1, 2) 0 0 1 -1/2 0 (c’d) ER 3(1, 1) 0 -1 -1 1 0 (cd) νR 3(1, 1) 0 1 -1 0 0 (Mc) αi 2(0, 0) 0 -1 0 1/2 1 (dM) γi 2(0, 0) 0 0 1 -1/2 -1 Table 1: Standard model spectrum and U(1) charges of particles and zero modes. QM stands for the world-volume gauge symmetry in the case of U(1) complex instantons. The U(1)k gauge symmetries have couplings with the RR 2-forms Br of the model, as follows SBF = Nk(pkr − pk′r) Br ∧ Fk (3.3) where pkr, pk′,r are given by the RR charges of the D-branes. These imply that under a U(1)k gauge transformation Ak → Ak + dΛk the scalar ar dual to Br transforms as ar → ar + Nk (pkr − pk′r) Λk (3.4) This has two effects: 1) The linear combination of axion fields r(pkr − pk′,r)ar is eaten up by the U(1)k massless gauge boson, making it massive. 2) For anomalous U(1)k, the anomalies cancel through a 4d version of the Green-Schwarz mechanism. This works due to the existence of appropriate ar F ∧F couplings, involving the gauge fields in the theory. It is obvious that all anomalous U(1)’s become massive by this mechanism. However it is important to realize [5] that gauge bosons of anomaly-free symmetries like U(1)B−L may also become massive by combining with a linear combination of axions. This is interesting since it provides a mechanism to reduce the gauge symmetry of the model without needing explicit extra Higgsing. In the models in which U(1)B−L becomes massive in this way, the gauge group left over is purely that of the SM. Moreover, we will see that having (B-L) massive by this Stückelberg mechanism is crucial to allow the generation of instanton-induced Majorana neutrino masses. Note that the B∧F couplings may also be potentially dangerous, since in principle they could also exist for hypercharge, removing U(1)Y from the low-energy spectrum. As we will see in our RCFT examples later on, having massless U(1)Y but massive U(1)B−L turns out to be a strong constraint in model building. 3.2 Majorana mass term generation As discussed in the previous section, string instantons can give rise to non-perturbative superpotentials breaking explicitly the perturbative global U(1) symmetries left-over from U(1) gauge bosons made massive through the Stückelberg mechanism. The kind of operator we are interested in has the form W ≃ e−Sins νRνR (3.5) where νR is the right-handed neutrino superfield 10. Here Sins transforms under both U(1)B−L and U(1)R in such a way that the overall operator is gauge invariant. This operator may be created if the mixed open string sectors lead to fermionic zero modes αi, γi , i = 1, 2, appropriately charged under the 4d gauge factors. As we discussed in the previous section, to generate a superpotential one needs instanton with O(1) Chan- Paton symmetry, in order to lead to two uncharged fermion zero modes to saturate the d2θ 4d superspace integration. On the other hand, as we argued, instantons with Sp(2) or U(1) CP symmetries may also induce the required superpotentials if there is some additional dynamics getting rid of the extra uncharged zero modes which in principle appear in instantons with these symmetries. We thus consider all O(1), Sp(2) and U(1) instantons in our discussion. In order to to get a νR bilinear, the intersection numbers of instanton M and d, c branes are as follows Sp(2) case : IMc = 1 ; IMd = −1 (3.6) (since there is an extra multiplicity from the two branes required to produce Sp(2)) O(1) case : IMc = 2 ; IMd = −2 (3.7) 10 Actually we denote by νR the left-handed ν L field following the usual (a bit confusing) convention. ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� ��������������� Figure 2: Disk amplitude coupling two charged zero modes to νR in the geometrical Type IIA intersecting brane approach. U(1) case : IMc = 2 ; IMd = −2 or IMd′ = 2 ; IMc′ = −2 (3.8) Furthermore there must be cubic couplings involving the right-handed neutrino super- field νa in the ath family and the fermionic zero modes αi, γj Lcubic ∝ dija (αi νaγj) , a = 1, 2, 3 (3.9) In type IIA geometric compactifications, this coupling arises from open string disk instantons, see Fig. 3.1. In general type IIA models (resp. IIB models), the coefficients dija depend on the Kähler (resp. complex structure) moduli, and possibly on open string moduli. In simple CFT models (like e.g. in toroidal cases) these quantities may be in principle explicitly computed. These trilinear couplings appear in the instanton action and after integration of the fermionic zero modes αi, γi one gets a superpotential coupling proportional to d2α d2γ e−d a (αiν aγj) = d2θ νaνb ( ǫijǫkld b ) (3.10) yielding a right-handed neutrino mass term. This term is multiplied by the exponential of the instanton euclidean action so that the final result for the right-handed neutrino mass (up to a 1-loop prefactor) has the form MRab = Ms( ǫijǫkld b ) exp(− qM,rar ) (3.11) For geometric compactifications VΠM is roughly related to the wrapped volume. We keep the same notation to emphasize that the effect is non-perturbative in gs. In supersymmetric models the term in the exponential is the linear combination U of complex structure moduli to which the instanton D-brane couples, as described in the previous section. As explained, the gauge U(1)c, U(1)d transformation of the bilinear piece and the e−SD2 factor nicely cancel. Note that from the viewpoint of the 4d SM effective field theory, the instanton has generated a Majorana neutrino mass violating B−L. Notice also that since this symmetry is non-anomalous, its violation cannot be associated to a gauge instanton, hence this is a pure string theory instanton effect. 3.3 Flavor and the special case of Sp(2) instantons In order to extract more specific results for the flavor structure of the obtained Majo- rana mass operator, one needs to know more details about the quantities dija coming from the disk correlators. However in the particular case of Sp(2) instantons, the la- bels i, j are Sp(2) doublet indices, and the symmetry requires dija = daǫ ij. The mass matrix for the three neutrinos is given by MRab =2Msdadb exp (−U), so that the flavour dependence on a, b = 1, 2, 3 factorizes. More generally, as we will see in our RCFT search in Section 6, there are typically several different instantons contributing to the amplitude, so that we actually have a result for the mass MRab = 2Ms d(r)a d −Ur (3.12) where the sum goes over the different contributing instantons. One thus has a structure of the form e−Urdiag (d 1 , d 2 , d 3 ) · 1 1 1 1 1 1 1 1 1 · diag (d(r)1 , d 2 , d 3 ) . (3.13) This structure is very interesting. Indeed, each instanton makes one particular (instanton- dependent) linear combination of the neutrinos massive, leaving two linear combina- tions massless. Hence, for three or more instantons, one generically has a matrix with three non-zero eigenvalues. It is easy to imagine a hierarchical structure among the three eigenvalues if e.g. the exponential suppression factors exp(−ReUr) are different for each instanton. 3.4 Other B− and L−violating operators Our main focus in this paper is on the generation of right-handed neutrino Majorana masses. However instantons may induce other L- and B-violating operators which we briefly summarize in this subsection. 3.4.1 The Weinberg operator A right-handed neutrino Majorana mass term is not the only possible operator violating lepton number. Instanton effects may also give rise to dimension 5 operators not involving νR. Specifically, the Weinberg operator (LHLH) . (3.14) might be generated. Once Higgs fields get a vev v this operator gives rise directly to Majorana masses for the left-handed neutrinos of order ≃ v2/M . Indeed, it is easy to check that in this case the required instanton M must verify Sp(2) case : IMc = −1 ; IMd = 1 (3.15) O(1) case : IMc = −2 ; IMd = 2 (3.16) U(1) case : IMc = −2 ; IMd = 2 or IMc′ = 2 ; IMd′ = −2 (3.17) (here we are assuming SU(2)weak to be embedded in an Sp(2)). Note that these intersection numbers are different to those giving rise to νR mass terms. In particular they lead to a transformation under B − L opposite to that of νR mass operators 11. In the present case there are altogether four fermionic zero modes αi,γi corresponding to the intersections of the instanton M with the branes c, d. These zero modes can have couplings involving the left-handed leptons L and the u-type Higgs multiplet H Ldisk ∝ cija (αi(LaH)γj) . (3.18) Upon integration over the fermionic zero modes one recovers the Weinberg operator. In the present case the scale M of the Weinberg operator will be the string scale Ms and the coupling λ ≃ exp(−Sins). Again, in the particular case of Sp(2) instantons the situation simplifies (cija = c aǫij) and one gets left-handed neutrino Majorana masses MLab = < H >2 2c(r)a c −Sr (3.19) where r runs over the different contributing instantons and Sr is their corresponding action. The flavour structure of this left-handed neutrino mass matrix is the same as in eq.(3.13) and again may potentially lead to a hierarchical structure of left-handed neutrino masses, as is experimentally observed. 11Instantons with these intersection numbers will be denoted with a plus sign in the instanton search later on In a given model both this kind of instanton and the one giving rise to right- handed neutrino masses (which is different) may be present. This contribution to the left-handed neutrino Majorana mass is in principle sub-leading compared to the see-saw contribution MLab(see-saw) = < H >2 d(r)a d −Sr)−1 hD (3.20) where is the ordinary Yukawa coupling constant habD (ν b). In principle the former is doubly suppressed both by 1/Ms and the exponential factor. On the other hand if the exponential suppression is not too large this mechanism involving directly the Weinberg operator may be the most relevant source of neutrino masses. This is because the see-saw contribution coming from νR exchange is proportional to the square of the ordinary Yukawa couplings habD which could be small. One could even think of having just the Weinberg operator as the unique source of the observed left-handed neutrino masses. Note however that in string vacua like this, in which the νR’s are present and massless at the perturbative level, having just the Weinberg operator would not be phenomenologically correct, and instantons of the first class are still needed so that the νR’s get a sufficiently large mass. 3.4.2 R-parity violating operators In the case ofN = 1 SUSYmodels like the MSSM there might be operators of dimension 3 and 4 violating lepton and/or baryon number. These are the superpotential couplings WRp = µ aH + λabcQ aDbLc + λ′abcU aDbDc + λ′′abcL aLbEc (3.21) in standard notation. Unlike the neutrino operators mentioned above, these operators violate B −L in one unit (rather than 2). It is well known that the standard R-parity of the MSSM may be identified with a Z2 subgroup of U(1)B−L, so these terms are odd under R-parity. The simultaneous presence of all these couplings is phenomenologically unacceptable. Indeed, the third coupling violates baryon number, and the other three violate lepton number. Together they lead to proton decay at an unacceptably large rate. On the other hand couplings violating either B or L are phenomenologically allowed. It is an interesting question whether any of these operators may be induced by string instanton effects. A first point to note is that instantons with Sp(2) Chan-Paton symmetry can never generate operators of this type. The reason is that all charged zero modes will necessarily come in Sp(2) doublets and hence the charged operators induced will always involve an even number of charged D = 4 fields and R-parity is automatically preserved. On the other hand O(1) and U(1) instantons may generate R-parity violating operators. In particular, the LH bilinear is essentially the square root of the Weinberg operator, and may be induced if a U(1) or O(1) instanton M exists with IMc = −1 ; IMd = 1 or IMc′ = 1 ; IMd′ = −1 . (3.22) (in the O(1) case the second option is not independent from the first). Again, if the appropriate disk couplings are non-vanishing a term with µaL ∼ Ms exp(−Sins) is generated. The rest of the operators in WRp may also be generated. Possible instanton zero modes which may induce them are shown in table 2. For example, the QDL operator may be induced if a U(1) instanton M with intersection numbers IMb = −1 ; IMc′ = 1 ; IMd = 1 (3.23) is present and in addition couplings Ldisk ∝ cab (α(UaQbj)γj) + c′a(βLajγj) (3.24) exist. Here α, β, γ are zero modes corresponding to (Mc′), (Md) and (bM) intersections and a, b(j) are flavor(SU(2)L) indices. Analogous trilinear or quartic disk amplitudes involving two charged zero modes should exist to generate the rest of the R-parity violating amplitudes in table 2. D = 4 Operator IMa IMa′ IMb IMc IMc′ IMd IMd′ νRνR 0 0 0 2 0 -2 0 LH̄LH̄ 0 0 0 -2 0 2 0 LH̄ 0 0 0 -1 0 1 0 QDL 0 0 -1 0 1 1 0 UDD -1 0 0 1 2 0 0 LLE 0 0 -1 0 1 1 0 QQQL 1 0 -2 0 0 1 0 UUDE -1 0 0 2 2 -1 0 Table 2: Zero modes required to generate Lepton/Baryon-number violating superpoten- tial operators. Sp(2) instantons cannot give rise to R-parity violating operators whereas O(1),U(1) instantons may in principle contribute to all of them. In the case of U(1) instantons there are additional zero mode possibilities which are obtained by exchanging IMx ↔ −IMx′ . 3.4.3 Dimension 5 proton decay operators There are also superpotential dimension-5 operators violating B and L which may be constructed from the MSSM matter superfields. Indeed the dimension 5 operators )QQQL ; ( )UUDE (3.25) are in fact the leading source of proton decay in SUSY GUT models with R-parity. Unlike the other operators considered here these ones preserve B−L (hence R-parity) but not B +L. These operators do not contribute directly to a proton decay but need to be ’dressed’ by a one loop exchange of some fermionic SUSY particle. This makes that, even although they are suppressed only by one power of the relevant fundamental scale, the loop factor and the corresponding couplings make the overall rate in SUSY- GUTS (barely) consistent with present experimental bounds for M of order the GUT scale or larger. These dimension 5 operators may also be induced in D-brane models of the class here considered by the presence of instantons with appropriate intersection numbers. For instance, the first operator may be induced through O(1) or U(1) instantons M IMb = IMb′ = −2 ; IMa = 1 ; IMd = 1 (3.26) Again Sp(2) instantons cannot induced this operator, since the the Ma intersection would yield 6 (rather than 3) colored fermionic zero modes. The proton decay rate obtained from these operators depend on the ratio exp(−Sins) × 1/Ms. For Ms of order 1016 GeV, the rate is consistent with present bounds if exp(−Sins) provides a suppression of a few orders of magnitude. On the other hand, models with a low string scale may be in danger unless the exponential suppression is sufficiently large (or such particular instantons are absent). As a general conclusion, these phenomenological aspects of instanton induced oper- ators very much depend on the action of the instanton, e.g. the volume of the wrapped D2-instanton in the intersecting D-brane constructions. In any event it is clear that the instantons here considered may indeed induce proton decay at a model-dependent rate. However in certain models R-parity will be preserved and prevent too rapid pro- ton decay. Indeed, this is what we find in our instanton search in Gepner orientifolds. As we said Sp(2) instantons automatically preserve R-parity. More generally, models that violate R-parity are rare, and the corresponding instantons actually generate very high dimensional operators, so R-parity breaking effects seems quite suppressed. In fact in our search within MSSM-like models in Gepner model orientifolds we do not find instantons with just the correct charged zero modes to generate the low dimen- sional couplings discussed above. So, at least within our class of RCFT constructions, R-parity preservation is quite a common feature. 4 CFT orientifolds In this section we describe the 4d string models we consider, namely orientifolds of type IIB Gepner model compactifications. This is a very large class, on which one can carry out large scans for certain desired properties. And moreover at present the only known class of (SUSY) models with massive B − L. 4.1 Construction of the models In general, RCFT orientifolds are orientifold projections of closed string theories con- structed using rational conformal field theory. Although this includes in principle rational tori and orbifolds, the real interest lies in cases where the two-dimensional CFT is interacting, because such theories are hard to access by other methods. A disadvantage of the use of RCFT is that this method is algebraic, and not geometric in nature, so that one cannot easily explore small deformations of a certain string theory. It is best thought of as a rational scan of moduli spaces. The most easily accessible examples are the orientifolds of tensor products of mini- mal N = 2 conformal field theories (“Gepner models”) forming a type IIB closed string theory. During the last decade, examples in this class have been studied by many au- thors (see [35][36][37][38][39][40][41][42]), and searched systematically in [6] and [7]. Although the Gepner models form only a small subset of RCFT’s, they already offer a large number of possibilities. The total number of tensor products with the required central charge c = 9 is 168. On top of this, one can choose a large number of distinct modular invariant partition functions on the torus. The orientifold formalism is not available for all of them, but it has been completely worked out [43] for all simple cur- rent invariants (based on the charge conjugation invariant). This yields a total of 5403 distinct MIPFs. On top of this, we may choose various orientifold projections. Here the only known possibilities are a class of simple-current based choices [44][45][46][47]. This then yields a total of 49304 orientifolds. For each orientifold choice, the full open string partition function is a,b,i NaNbA abχi( aχ̂i(  (4.1) Here Aiab are the annulus coefficients, M a the Moebius coefficients, Na the Chan-Paton multiplicities and χ(τ) are the closed string characters, and χ̂i(τ) = T −1/2χi(τ). The set of integers i is simply the set of primary fields of the closed string CFT, and depends only on the tensor product. The integers a, b are the boundary labels; this set depends on the MIPF. Our notation and labelling conventions for these CFT quantities are explained in Appendix A. The integers Aiab and M a depend in addition also on the orientifold choice; in the case of Aiab the latter dependence is very simple: all distinct annuli can be written as A cb, where Ω is the orientifold choice (which we usually do not specify explicitly) and CΩcb is the boundary conjugation matrix, which acts as an involution on the set of boundaries. Suppressing some details (which can be found in [43]) we may write these integers m,J,K SimRa,(m,J)g JK Rb,(m,K) (4.2) MΩ,ia = m,J,K P imRa,(m,J)g (m,K) (4.3) Here m is the label of an Ishibashi-state (the set of states that propagates in the transverse (or closed string) channel of the the annulus or Moebius diagrams). It is a subset of the set of closed string labels i, but in general there are degeneracies, so that more than one distinct Ishibashi state belongs to a given closed string label. These degeneracies are distinguished by the labels J,K (see Appendix A). The complex numbers R and U are respectively the boundary and crosscap coefficients. Note that the latter depend on the orientifold choice, but the former do not. The only dependence of the annulus coefficients on the orientifold choice is through the Ishibashi metric gΩJK , which is a matrix on each Ishibashi degeneracy space, and which can be a sign if there are no degeneracies. Finally, the matrix P is given by P = TST 2S T , where S and T are the generators of the modular group of the torus. Similar expressions exist for the Klein bottle multiplicities defining the unoriented closed sector, but they will not be needed in this paper. The boundary labels a, b, . . . refer to all boundaries that respect the bulk symmetries of the CFT. This includes the individual N = 2 chiral algebras of the factors in the ten- sor product, the alignment currents12 that ensure the proper definition of world-sheet supersymmetry and the space-time supersymmetry generator that imposes a general- ized GSO-projection on the spectrum. The latter implies that all characters χi respect (at least) N = 1 space-time supersymmetry. By construction, the boundary states are then supersymmetric as well. Both conditions (boundary and bulk space-time su- persymmetry) can in principle be relaxed within the formalism, but this leads to a much larger set of bulk and boundary states. The precise labelling of the boundaries is explained in Appendix A and involves a subset of the closed string labels i and a degeneracy label, distinct from the one used for the Ishibashi states. The set of bound- ary labels is complete in the sense of [45]. This means that no additional boundary states exist that respect all the aforementioned symmetries. It also means that the matrices R are square matrices (although their rows and columns are defined in terms of different index sets). It is in principle possible to write down additional boundary states that break some of the world-sheet symmetries. This is an important possibility to keep in mind, but we will not consider it here. The massless spectrum is obtained by restricting the characters χi to massless states. Since the characters are supersymmetric those massless states are either vector multiplets or chiral multiplets. The latter can be restricted to one chirality (e.g left- handed); the other choice merely produces the CPT conjugates. Boundaries are called real if a = a′, where the conjugate boundary a′ is defined by CΩa,a′ = 1, and complex otherwise. The Chan-Paton multiplicities Na give rise to gauge groups U(Na) for complex boundaries and SO(Na) or Sp(Na) for real ones. In the latter case Na must be even. To count bi-fundamentals we define Lab ≡ Aiabχi( )massless,L . (4.4) Note that because of the factor 1 in (4.1) and the fact that Lab is symmetric, the value of Lab is indeed precisely the number of bi-fundamentals in the representation (Na, Nb). 12These are spin-3 currents consisting of products of the world-sheet supercurrents of the factors in the tensor product, including the NSR space-time factor. It is convenient to introduce the intersection matrix13 Iab ≡ Lab′ − La′b , (4.5) which is manifestly antisymmetric in a and b. Note that for a pair of complex bound- aries a, b with conjugates a′, b′ one can define four quantities that are relevant for the massless spectrum, two of which are chiral, namely Iab and Iab′ . It is often convenient to associate a geometric picture to these integers. Thus we will often refer to the boundary labels and their multiplicities as “stacks of branes”, and view the integers Iab as brane intersection numbers. This is only done for convenience and does not imply a concrete brane realization; indeed, it does not make sense to say that a given boundary label corresponds to a Dp-brane for some give p. Such an interpretation might be valid in a large radius limit, assuming such a limit exists. In general, for a choice of Chan-Paton multiplicities Na there will be tadpoles in the one-point closed string amplitudes on the disk and the crosscap. These have to be cancelled in order to make the theory consistent (since we work with supersymmetric strings we do not have the option of cancelling RR and NS-NS tadpoles separately). This leads to a condition on the Chan-Paton multiplicities: NaRa,(m,J) = 4ηmUm,J (4.6) where η0 = 1 and all other η’s are −1; there is such a condition for any Ishibashi label (m, J) that leads to a massless scalar in the transverse channel. The one for m = 0 (which is non-degenerate) is the dilaton tadpole condition. It has the special feature that all coefficients Ra0 are real and positive. The crosscap coefficient U0 is also real and can be chosen positive (in the CFT both signs are acceptable). If U0 6= 0 (4.6) limits the Chan-Paton multiplicities; if U0 = 0 the only solution is Na = 0 for all a, which rules out any realization of the Standard Model. This reduces the number of usable orientifolds to 33012. Tadpole cancellation condition implies cancellation of RR-charges coupling to long- range fields, and absence of local anomalies. There is a second condition that has 13Note that Lab is a symmetric matrix giving the number of chiral multiplets in the ( a, b) bi- fundamental. This is a natural quantity in unoriented CFT’s, where a symmetric definition for the annulus amplitude exists. In oriented CFT the annulus is, in general, not symmetric, but on the other hand it is possible to choose the branes in such a way that only ( , ) bi-fundamentals appear. This has become the customary way of counting states in the intersecting brane literature, even for orientifold models. The quantity Iab is defined in such a way that it is anti-symmetric in a and b. This is why boundary conjugations appear in the right hand side. This has the additional advantage of making I a more familiar quantity for readers used to the standard intersection brane conventions. to be taken into account, which has to do with Z2 charges that do not couple to long-range fields, usually referred to as “K-theory charges” in geometric constructions. Uncancelled K-theory charges may lead to global anomalies in symplectic factors of the gauge group. But even if this symptom is absent, the disease may still exist. A much more general way to probe for uncancelled K-theory charges is to require the absence of global anomalies not only in the Chan-Paton gauge group but also on all symplectic brane-anti-brane pairs that can be added to it as “probe-branes” [48]. Presently this is the most general constraint that be imposed in these models, but it is not known if additional ones are required. This probe brane constraint leads to a large number of mod-2 constraint and is potentially very restrictive, but almost harmless in practice [49]. It is satisfied by all models we consider in the present paper. 4.2 Search for SM-like models The complete set of solutions to these conditions is finite but huge, but the vast major- ity is of no phenomenological interest. In the last few years systematic searches have been carried out for models that contain the Standard Model. The models that were considered have the property that the set of Chan-Paton labels can be split into two subsets, the observable and the hidden sector. The former has been limited, for prac- tical reasons, to at most four complex brane stacks, required to contain the Standard Model gauge group and the right intersections to yield three families of quarks and leptons, plus (in general) some non-chiral (vector-like) additional matter. The hidden sector is only constrained by the requirement that there be no net number of chiral multiplets charged under both the observable and hidden sector, and by practical com- putational limitations. The main purpose of the hidden sector in these models is to provide variables that can be used to satisfy the tadpole and global anomaly condi- tions, since the multiplicities in the observable sector are already fixed. In some cases the observable sector already satisfies the constraints by itself, and there is no hidden sector. The observable sector can be realized in many different ways if one only imposes the constraint that the standard model should be contained in it. These possibilities were recently explored in [7]. We will focus on the realization described in Section 3.1, first considered in [5]. There are four stacks, namely a (containing QCD and baryon number as U(3)), b (containing electroweak SU(2) embedded as U(2) or Sp(2)), c (providing a U(1) or an O(2) factor14, and d (providing another U(1) factor). 14In [6] also Sp(2) was considered, but this requires an additional Higgs mechanism. The standard model hypercharge generator is , defined in (3.1): Qd (4.7) where Qx denotes the generator of the U(1) of brane stack x; in case the Chan-Paton factor of brane c is O(2) one should use the properly normalized O(2) generator. In addition to Y these models have two or three additional U(1) gauge symmetries (the latter case if electroweak SU(2) arises from U(2)). These (except the combination B − L) are anomalous, with anomaly cancelled by the Green-Schwarz mechanism, implying the existence of a B ∧ F coupling making them massive. In fact, as already mentioned, such Stückelberg couplings may be present for non-anomalous U(1)’s as well. We are interested in models where the hypercharge gauge boson does not have such couplings (otherwise the model would be phenomenologically unacceptable), but where the B−L gauge boson is massive by such couplings (both in order that the gauge group reduces to the SM one, and that neutrino Majorana masses may be induced by string instantons, as discussed in previous sections). The combined requirements of having a massive B − L and a massless Y turn out to be difficult to satisfy. In fact, if the group on brane c is O(2) they are impossible to satisfy simultaneously, because the O(2) component of the vector boson does not couple to any axions, and hence the B−L and Y bosons have the same mass. But even in models with a U(1) group on brane c it happens rather rarely that both constraints are satisfied simultaneously, at least in the searches that have been done so far. We will make use here of the data presented in [6, 7], which are available in slightly improved form on the website www.nikhef.nl/∼t58/filtersols.php. This database con- sist of 211634 distinct spectra. Here “distinct” means that they are physically different for a given MIPF15 if the hidden sector is ignored. Hence the differences can be the number of vector-like states of various kinds or the dilaton couplings of branes a, b, c, d. Geometrically, these spectra may originate from the same moduli space, but then in any case from different points on this moduli space. The improvements in comparison with the data presented in [6] consist of taking into account the full global anomaly conditions from probe branes. In some cases this required nothing more than checking these conditions for an existing solution of the tadpole conditions, but in other cases a new solution had to be found. As a result, a few models disappeared from the original database, but due to improved algorithms a few new ones could be added. The net re- 15Rare cases of identical spectra and couplings originating from different MIPFs are treated as distinct. http://www.nikhef.nl/~t58/filtersols.php sult is some small but inconsequential changes in the total number of models of various kinds. The numbers we will mention below are based on the improved database. The total number of models in that database with a Chan-Paton group U(3) × Sp(2)× U(1)× U(1) is 10587. Of these, 391 (about 4%) have a massive B − L vector boson. For U(3)×U(2)×U(1)×U(1) these numbers are, respectively, 51 and 0. Hence no examples of the latter type were found, although they were found with 1,2 and 4 families (in a limited search), in a few percent of the total number of models. It seems therefore reasonable to expect that U(3) × U(2) × U(1) × U(1) with massive B − L do exist, and that their absence is just a matter of statistics. Just for comparison, the total number of U(3)× Sp(2)× O(2)× U(1) models is 56627. 5 Fermion zero modes for instantons on RCFT’s In this section we discuss D-brane instantons for general compactifications, including abstract CFT ones. We also provide the spectrum of zero modes on an instanton brane, using the information about their internal structure i.e. in the compactified dimension in geometric models, or of the internal CFT in more abstract setups like in previous section. We will be interested in the latter case. A first question that should be addressed is what this internal structure is. For instance, in type IIA geometric compactifications, it corresponds to a supersymmetric (i.e. special lagrangian) 3-cycle. Notice that these are the same kind of 3-cycles already used to wrap the D6-branes that give rise to the 4d gauge symmetry of such models. For general CFT’s, D-branes are described as boundary states. To describe instantons, one can simply use the same boundary state of the internal CFT to describe the 4d space-filling branes present in the model and the instanton branes. The only difference is that boundaries satisfy Neumann conditions in the 4d space-filling case, and Dirichlet in the instanton case. This exploits the fact that whenever a boundary state of the internal CFT, and with Neumann boundary conditions in the 4d space is an acceptable state of the full CFT, the same boundary state of the internal CFT, combined with Dirichlet boundary conditions in the 4d space also gives an acceptable state of the full CFT. For geometric compactifications this is related to Bott periodicity of the K-theory classes associated to the D-brane charges, but it is possible to show it in general. Since instanton D-branes can thus be naturally associated to the boundary states of 4d space-filling branes, it is convenient to express the spectrum of zero modes of the former in terms of the massless states of the latter. This is particularly useful, since the computation of the spectra on 4d space-filling branes for Gepner model orientifolds has already been described (although the arguments below are valid also for geometric compactifications). Hence, let us denote by M a 4d space-filling brane associated with the same boundary state of the internal CFT as the instanton brane M of interest. Note that the 4d space-filling brane M is an auxiliary tool, and need not be (and, for our instantons of interest, will not be) one of the 4d space-filling branes present in the model. ‘Real’ brane instantons Let us first consider the case of ‘real’ brane instantons. Consider a set ofm 4d space- filling branes M, and focus first on the massless spectrum in the MM sector. Before the orientifold projection, it leads to a universal 4d N = 1 U(m) vector multiplet, and a number LMM of adjoint chiral multiplets. The orientifold operation maps this sector to itself, acting on the Chan-Paton with a matrix γΩ,M. This matrix satisfies γTΩ,Mγ Ω,M = ±1m (5.1) The two possibilities can be chosen to correspond to γΩ,M = 1m or γΩ,M = ǫm, with −1r 0 , and m = 2r hence necessarily even in the latter case. They corre- spond to the SO and Sp projections, respectively. The orientifold projection on the N = 1 vector multiplet Chan-Paton matrices is given by λ = −γΩ,M λT γ−1Ω,M (5.2) and leads to SO(m) or Sp(m) vector multiplets for the SO or Sp projection (hence the name). Concerning the N = 1 chiral multiplets, they fall in two classes of p−, p+ (with p− + p+ = LMM) which suffer the projections λ = ±γΩ,M λT γ−1Ω,M (5.3) For the SO projection, this leads to p+, p− chiral multiplets in the , representation. For the Sp projection, there are p+, p− chiral multiplets in the , representation. The sectors Ma (where a is a 4d space-filling branes present in the model) are mapped to sectors Ma′, so it is enough to focus on the former. After the orientifold projection one gets LMa, LMa′ chiral multiplets in the ( M, a), ( M, a). Let us now obtain the zero modes for a set of m instanton branes M in terms of the above spectrum. The MM sector is closely related to the MM sector, by changing the NN boundary conditions in 4d spacetime to DD boundary conditions (which can be done in a covariant formalism, but not in the light-cone gauge). Before the orientifold projection, one obtains the same set of states (since moddings for NN and DD boundary conditions are identical, both in the NS and R sector), but with different world-volume interpretation. Also, the change in boundary conditions implies that some polarization states which are unphysical for the 4d spacefilling brane are physical in the instanton brane. Hence, the U(m) gauge bosons on the 4d space-filling brane M correspond to four adjoint real scalars in the instanton brane M . Similarly, the 4d spinors in M, correspond to four fermion zero modes on M , transforming as two spinors of opposite chiralities θα, θ̃α̇ of the SO(4) rotation group in transverse space. The orientifold projection maps the MM sector to itself, acting on Chan-Paton indices with a matrix γΩ,M . In close analogy with the argument in [50] for the familiar D5-D9-brane system in type I (see [51, 52] for related derivations), one can show that the condition (5.1) flips sign upon changing four NN boundary conditions to DD, hence γTΩ,Mγ Ω,M = ∓1m (5.4) Namely, the instanton brane has Sp(m) gauge group when the 4d space-filling brane (with same internal boundary state) has gauge group O(m), and vice-versa. We still refer to these projections as SO and Sp, hoping no confusion arises. Note that, as mentioned in Section 2.2, although there are no gauge bosons in 0+ 0 dimensions, the gauge group is present on the instantons in that it acts on open string endpoints. Let us consider the effect of the orientifold projection on the MM states, as com- pared with the effect onMM states. Again, following arguments familiar in the D5-D9 brane system in type I, one can show that the signs in conditions like (5.2), (5.3) re- main unchanged upon changing four NN dimensions to DD, except for bosonic modes polarized along the directions longitudinal to these four dimensions (and for fermions related to them by the unbroken susy of the total system). To be concrete, consider- ing the four MM adjoint bosons, and two MM adjoint fermions θα associated to the universal MM vector multiplets, they suffer the projection λ = +γΩ,M λ T γ−1M (5.5) Hence they transform in the of Sp(m) for the SO projection, and in the of SO(m) for the Sp projection. On the other hand, for the two fermion zero modes θ̃α̇, the projection is λ = −γΩ,M λT γ−1M (5.6) and leads to two fermion zero modes in the of Sp(m) for the SO projection, and in the of SO(m) for the Sp projection. This implies that in order to obtain two fermion zero modes from this univer- sal multiplet, in order to generate a superpotential, one should consider instantons with orthogonal gauge group and multiplicity one (O(1) instantons). For instantons with symplectic gauge group and multiplicity two (Sp(2) instantons), there are two additional fermion zero modes in the triplet representation. As mentioned, we will continue to consider such instantons in our relaxed scan. Multiple instantons, i.e. boundary states with higher multiplicity, lead to a larger amount of additional fermion zero modes (due to the larger gauge representations for the fermions), and do not contribute to superpotentials; we will not consider such cases even in relaxed scans, since they also very often lead to too many charged fermion zero modes and cannot contribute to the operators of interest (except possibly for O(2) and U(2) instantons with low intersections, which are kept in our scan as a curiosity). Similarly, for the p± sets of MM scalars and fermions associated to the MM 4d chiral multiplets, the projection is λ = ±γΩ,M λT γ−1Ω,M (5.7) with the same sign choice as in (5.3). The different structure of γΩ implies that, for the SO projection we get p+, p− sets of scalars and fermions in the , , while for the Sp projection there are p+, p− sets of scalars and fermions in the , . This concludes the discussion of the MM sector. Let us not consider the Ma sec- tors, from the information from the Ma sectors. Notice that this implies changing four NN boundary conditions to DN, which have different moddings. Hence the states are different in both situations, but the information on the multiplicities is preserved. Specifically, in the NS sector the DN boundary condition introduce an additional vac- uum energy which generically makes all states massive. Hence there are no massless scalar zero modes in generic Ma sectors. In the R sector, the change in the mod- dings reduces the dimension of the massless ground state, leading to a single (chiral) fermionic degree of freedom. Since the orientifold action maps the Ma sector to Ma′ sectors, there are no subtleties in the orientifold projection. The end result is LMa, LMa′ fermion zero modes in the ( M, a), ( M, a). The net number of chiral fermion zero modes in the ( M, a) is given by IMa = LMa′ − LMa, i.e. the net number of chiral multiplets in the related Ma sector. The results for orientifold projections for real branes are shown in table 3. Proj. Multiplet in M M (before orient.) M (after orient.) M (after orient.) SO N = 1 vect. mult. U(m) O(m) Sp(m) 2 f + 2 f + 4 b N = 1 ch. mult. (p+ + p−) Ad p+ + p− 2p+ ( f + b ) + 2p− ( f + b ) Sp N = 1 vect. mult. U(m) Sp(m) O(m) 2 f + 2 f + 4 b N = 1 ch. mult. (p+ + p−) Ad p+ + p− 2p+ ( f + b ) + 2p− ( f + b ) Any N = 1 ch. mult. LMa′( M, a)+ LMa′( M, a)+ LMa′( M , a) f LMa( M, a) LMa( M, a) LMa( M , a) f net IMa( M, a) net IMa( M , a) f Table 3: Orientifold projection for real branes: Massless modes of the 4d space-filling branes M (before and after the orientifold projection) and zero modes on the instanton branes M (denoted with sub-indices b, f for bosonic and fermionic modes) Complex brane instantons We now consider the case of complex brane instantons. The arguments are very similar, hence the discussion is more sketchy. Consider m 4d spacefilling branes M, associated to the internal boundary state of the instanton brane M of interest. The MM leads to a 4d N = 1 U(m) vector multiplet and a number LMM′ of adjoint chiral multiplets. The orientifold action maps it to the M′M′ sector, hence we may keep just the former and impose no projection. The MM′ sector is mapped to itself under the orientifold projection. Denoting by γΩ,M the action on Chan-Paton indices, the MM′ modes split into sets L±MM , L M ′M ′, which suffer a projection λ = ±γΩ,M λT γ−1Ω,M (5.8) leading, for γΩ,M = 1m, to L chiral multiplets in the , , and L+ chiral multiplets in the , . The net number of chiral multiplets in the , is I+ M,M − L+M′M′, I−MM′ = L−M,M − L−M′M′ . And oppositely for γΩ,M = ǫm. Finally, the Ma, Ma′ and related sectors lead, after the orientifold projection, to LMa′ , LMa, LM′a′ , LM′a chiral multiplets in the ( M, a), ( M, a), ( M, a), ( M, a). In order to simplify notation, we replaceM →M in these expressions in our discussions of instanton zero modes. Let us now consider m brane instantons M and compute their zero mode spectrum in terms of the above. In the MM (and its image M ′M ′) sector there are four scalar modes and four fermions in the adjoint of the U(m) gauge symmetry group; these are related to the 4d vector multiplet in the MM sector. In addition, there are LMM ′ sets of scalars and fermions in the adjoint, related to the LMM′ non-universal chiral multiplets in the MM sector. The MM ′ sector is mapped to itself, and one has to impose the orientifold projection (recalling that the matrix γΩ,M differs from γΩ,M). For γΩ,M = 1, hence γΩ,M = ǫ, we obtain L MM , L MM chiral multiplets in the , , and L+M ′M ′ , L M ′M ′ chiral multiplets in the , . The net number of chiral multiplets in the , is I+MM ′ = L MM − L+M ′M ′, I−MM ′ = L−MM − L−M ′M ′ . And oppositely for γΩ,M = ǫ hence γΩ,M = 1. In theMa,Ma′ and related sectors, there are generically no bosonic zero modes, and there are LMa, LM′a′ , LMa′ , LM′a chiral fermion zero modes in the ( M , a), ( M , a), ( M , a), and ( M , a) respectively. The net number of chiral fermion zero modes in the ( M , a) and ( M , a) is given by IMa = LMa′ −LM′a′ and IMa′ = LMa−LM′a. In order to simplify notation, we replace M →M in these expressions in our discussions of instanton zero modes. The results for orientifold projections for real branes are shown in table 4. 6 Search for M instantons In this section we perform a search of models which admit an instanton inducing a right-handed neutrino Majorana mass operator. Namely, for each model with the chiral content of the SM in the classification described in Section 4.2, we first scan over boundary states, searching for all instantons with the required uncharged and charged fermion zero mode structure to yield neutrino masses. We then relax our criteria a bit and allow for instantons with correct charged zero mode structure but having extra non-chiral zero modes (both charged and uncharged). The idea is that these non-chiral zero modes could be lifted by diverse effects, as discussed. It is important to recall that the cubic couplings between instanton zero modes and 4d chiral multiplets are difficult to compute in Gepner model orientifolds. Hence, we will simply assume that such couplings are non-zero if there is no symmetry forbidding them. Proj. Multiplet in M M (before orient.) M (after orient.) M (after orient.) Any N = 1 vect. mult. U(m)× U(m)′ U(m) U(m) 4Ad f + 4Ad b N = 1 ch. mult. padj Ad + padjAd ′ padjAd 2padj ( Ad f + Ad b ) SO N = 1 ch.mult. LMM( M, M′) L 2L+MM b,f + 2L MM b,f LM′M′( M, M′) L M′M′ M M′M′ M 2L+M ′M ′ b,f + 2L M ′M ′ b,f Sp N = 1 ch.mult. LMM( M, M′) L 2L+MM b,f + 2L MM b,f LM′M′( M, M′) L M′M′ M M′M′ M L+M ′M ′ b,f + L M ′M ′ b,f Any N = 1 ch. mult. LMa′( M, a)+ LMa′( M, a) LMa′( M , a) f . . . LMa( M, a) LMa( M , a) f . . . LM′a′( M, a) LM ′a′( M , a) f . . . LM′a( M, a) LM ′a( M , a) f net IMa( M, a) net IMa( M , a) f net IMa′( M, a) net IMa′( M , a) f Table 4: Orientifold projection for complex branes: Massless modes of the 4d space-filling branesM (before and after the orientifold projection) and zero modes on the instanton branes M (denoted with sub-indices b, f for bosonic and fermionic modes) 6.1 The instanton scan Our detailed strategy will become clear along the description of the results. Given a set of a,b,c,d standard model branes, we must look for additional boundary states M that satisfy the requirements of a (B−L)-violating instanton. From the internal CFT point of view this is just another boundary state, differing from 4d spacefilling branes only in the fully localized 4d spacetime structure. The minimal requirement for such a boundary state is B − L violation, which means explicitly IMa − IMa′ − IMd + IMd′ 6= 0 (6.1) It is easy to see that the existence of such an instanton implies (and hence requires) the existence of a Stückelberg coupling making B−L massive. To see this, consider adding to the Standard Model configuration a 4d spacefilling brane M (in fact used in Section 5) associated to the boundary stateM (RR tadpoles can be avoided by simultaneously including M antibranes, which will not change the argument). The new sector in the chiral spectrum charged under the branes M can be obtained by reversing the argument in Section 5, and is controlled by the intersection numbers of M . From the above condition it follows that the complete system has mixed U(1)B−L × (GM)2 anomalies, where GM is the Chan-Paton-factor of brane M. These anomalies are cancelled by a Green-Schwarz mechanism involving a (B −L)-axion bilinear coupling, which ends up giving a mass to B−L via the Stückelberg mechanism. This coupling is in fact not sensitive to the presence of the brane M, hence it must have been present already in the initial model (without M). Hence the existence of a boundary labelM that satisfies (6.1) implies that B−L is massive. Unfortunately the converse is not true: even if B−L has a Stückelberg mass, this still does not imply the existence of suitable instantons satisfying (6.1)16 Indeed, in several models we found not a single boundary state satisfying (6.1). Note that, since hypercharge must be massless, one can use the reverse argument and obtain that IMa − IMa′ − IMc + IMc′ − IMd + IMd′ = 0 (6.2) in all models. We verified this for all models we considered as a check on the compu- tations. As already discussed in Section 4.2, in the search for SM constructions in Gepner orientifold, there are 391 models with massless hypercharge and massive B−L. In these models we found a total of 29680 instantons with B−L violation, i.e. with intersection numbers satisfying (6.1). Of course, in order to serve our purpose of generating a Majorana mass superpotential, the instantons have to satisfy some more conditions. Let us consider them in order of importance, and start with the conditions on the net number of chiral fermion zero modes charged under the 4d observable sector. Clearly we need IMa = IMa′ and IMb = IMb′ . The latter condition is automatically satisfied in this case, because the b-brane is real in all 391 models. The chiral conditions on the zero modes charged under the branes c and d are as in [3]17 and are given in equations (3.6), (3.7) (3.8) of the present paper. These are the instantons of most interest, and on which we mainly focus. However, as discussed in Section 3.4, other important B- and/or L- violating operators (such as the Weinberg operator or the LH operator) can 16From intuition in geometric compactifications, one expects that there may always exist a D-brane with the appropriate topological pairings, but there is no guarantee that there is a supersymmetric representative in that topological sector, and even less that it would have no additional fermion zero modes. Note also that even if such D-brane instantons exists, there is no guarantee that it will fall in the scan over RCFT boundary states. 17Note that there is a sign change in the contribution of the U(1)d generator to Y in comparison to [3] be generated by instantons with similar intersection numbers, up to a factor of 2 and a sign, see table 2. For this reason we also allow at this stage any instanton which has the correct number of charged zero modes to generate them. Imposing these conditions reduces the number of candidate instantons potentially contributing to neutrino masses in any of the models to 1315. All instantons satisfying these requirements are summarized in the table 5. In columns 1,2 and 3 we list the tensor combination, MIPF and orientifold choice for which the model occurred. The latter two numbers codify simple current data that describe respectively a MIPF and an orientifold. MIPFs are in general defined by means of a subgroup H of the simple current group G, plus a certain matrix X of rational numbers [55]. Orientifolds are defined by a simple current and a set of signs [43]. In previous work [6] we have enumerated these quantities (up to permutation symmetries) and assigned integer labels to them for future reference. We only refer to these numbers here, but further details are available upon request. Usually for each MIPF and orientifold which contains the standard model there are several choices a,b,c,d for which it is obtained. For a given choice of tensor combination, MIPF and orientifold and SM branes there may be several instantons. For clarity we put all such instantons together in the information in table 5. In column 4 we indicate which type of instanton branes were found. Five types are distinguished: O1, O2, S2, U1 and U2, corresponding to O(1), O(2), Sp(2), U(1) and U(2) Chan-Paton symmetry on the instanton volume. The number indicates the instanton brane multiplicity that gives the correct number of instanton charged zero modes from the a, b, c, d branes, to lead to right-handed neutrino Majorana masses. The number of zero modes is in general the product of the instanton brane multiplicity and ‘intersection number’ with the corresponding 4d spacefilling brane. As discussed in Section 5, for symplectic branes the smallest possible brane multiplicity is 2. As we discussed there, only O1 instantons may have the required universal minimal set of two zero modes in the uncharged sector. Still we look for all O(1), Sp(2) and U(1) instantons which may yield a superpotential if the extra uncharged fermion zero modes. In this vein we also include a search for O2 and U2 instantons. Note also that such O2 or U2 instantons imply the existence of other instantons involving the same boundary state, but with multiplicity 1, which may lead to the R-parity violating operator LH . We will discuss the generation of R-parity violating operators at the end of this section. The third character (+ or −) in the instanton in table 5 is the sign of IMc′ − IMc. For the instantons giving rise to right-handed neutrino Majorana masses this sign should be negative, whereas it should be positive for instantons giving rise to the Weinberg operator (or the LH operator), see table 2. The 1315 instantons are divided in the following way over the different types: 3 of types O1+ and O1−, 46 of type U1+, 24 of type U1−, 550 S2+, 627 S2−, 27 of types U2+ and U2− and four of types O2+ and O2−. Notice that the vast majority (97.5%) of the instanton solutions are of type S2+ and S2−. This is encouraging given the nice properties of such instantons, concerning e.g. R-parity conservation. Note also that in almost all cases both S− and S+ are simultaneously present,18 so both sources of physical neutrino Majorana masses (from the see-saw mechanism or the Weinberg operator) are present. The other instanton classes possibly generating right-handed neutrino masses are O1− and U1−, which are much less abundant. There is just one orientifold with O1− instantons, for which one can obtain cancellation of RR tadpoles, see below. On the other hand we have found no orientifold with U1− instantons and cancellation of tadpoles, see below. Table 5: Summary of instanton branes. Tensor MIPF Orientifold Instanton Solution (1,16,16,16) 12 0 S2+, S2− Yes (2,4,12,82) 19 0 S2−! ? (2,4,12,82) 19 0 U2+!, U2−! No (2,4,12,82) 19 0 U1+, U1− No (2,4,14,46) 10 0 (2,4,14,46) 16 0 (2,4,16,34) 15 0 (2,4,16,34) 15 1 (2,4,16,34) 16 0 S2+, S2− Yes (2,4,16,34) 16 1 (2,4,16,34) 18 0 S2− Yes (2,4,16,34) 18 0 U1+, U1−, U2+, U2− No (2,4,16,34) 49 0 U2+, S2−!, U1+ Yes Continued on next page 18In some models contributing many instantons there is an exact symmetry between S− and S+. This explains the approximate symmetry in the full set. In some cases this symmetry can be understood in terms of flipping the degeneracy labels of boundary states. We regard it as accidental, since it is not found in all models. Table 5 – continued from previous page Tensor MIPF Orientifold Instanton Solution (2,4,16,34) 49 0 U1− No (2,4,18,28) 17 0 (2,4,22,22) 13 3 S2+!, S2−! Yes! (2,4,22,22) 13 2 S2+!, S2−! Yes (2,4,22,22) 13 1 S2+, S2− No (2,4,22,22) 13 0 S2+, S2− Yes (2,4,22,22) 31 1 U1+, U1− No (2,4,22,22) 20 0 (2,4,22,22) 46 0 (2,4,22,22) 49 1 O2+, O2−, O1+, O1− Yes (2,6,14,14) 1 1 U1+ No (2,6,14,14) 22 2 (2,6,14,14) 60 2 (2,6,14,14) 64 0 (2,6,14,14) 65 0 (2,6,10,22) 22 2 (2,6,8,38) 16 0 (2,8,8,18) 14 2 S2+!, S2−! Yes (2,8,8,18) 14 0 S2+!, S2−! No (2,10,10,10) 52 0 U1+, U1− No (4,6,6,10) 41 0 (4,4,6,22) 43 0 (6,6,6,6) 18 0 Most models have a hidden sector containing extra boundary states beyond the SM ones. In the same spirit of imposing chiral conditions first, we should require that IMh = IMh′, where h is a hidden sector brane. This is to guarantee that the generated superpotential does not violate some hidden sector gauge symmetry which would require the presence of hidden sector fields along with the νR bilinear. The latter condition is not imposed on the previously known hidden sector (i.e. the one in [6, 7]), but instead a new search for tadpole solutions was performed, for each M , restricting the candidate hidden sector branes to those satisfying IMh = IMh′ (as discussed in Section 5). This is because in general the known hidden sector in [6, 7] is just a sample out of a huge number of possibilities. In column 5 we indicate for which instantons it was possible to satisfy the tadpole conditions with this additional constraint. With regard to observable-hidden matter we use the same condition as in [6], namely that it is allowed only if it is vector-like. Such a solution could be found for 879 of the 1315 instantons, with ten cases inconclusive (i.e it was computationally too difficult to decide if a solution does or does not exist). The latter are indicated with a question mark in column 5 (for most of the undecidable cases there is a tadpole solution for a different instanton with the same characteristics; for that reason just one question mark appears). Independently of the RR tadpole condition (since there may be alternative sources for its cancellation, or hidden sectors which fall beyond the reach of RCFT), we can also consider the further constraint that the number of charged fermion zero modes is exactly right, not just in the chiral sense. This means IMa = IMa′ = IMb = IMb′ = 0, IMc = 2, IMc′ = 0 and IMd = −2, IMd′ = 0 or vice-versa. Furthermore we require that there are no adjoint or rank-2 tensor zero-modes (note that the latter could be chiral if the instanton brane is complex, and indeed they are in some of the 1315 cases). This reduces the 1315 instantons to 263. In column 4 we indicate those cases with an exclamation mark. It is noteworthy that the success rate for solving the tadpole conditions is highest for these instantons: 254 of the 263 allow a solution (with 3 undecided). If an exclamation mark appears in column 4, this only indicates that some of the instantons are free of the aforementioned zero modes, not that all of them are. But in all cases, if there are tadpole solutions, they exist in particular for the configurations with an exclamation mark. Finally we may impose the condition that IMh and IMh′ are separately zero. This is indicated with and exclamation mark in column 5. This turns out to be very restrictive. The only cases where this happens have no hidden sector at all. It is worth remarking that the only instantons having exactly the correct set of charged zero modes and cancelling tadpoles are of S2± type. Also those instantons are the only cases marked with an exclamation mark in column 4 and 5. These examples, which will be discussed below in some detail, also have just the minimal set of fermion zero modes, except for the universal sector (which for Sp(2) instantons contains two extra triplets). The main conclusion about this scan is that we did not find any instantons with exactly the zero mode fermions to generate the neutrino mass superpotential. However we have found a number of examples which come very close to that, with exactly the required charged zero modes and a very reduced set of extra uncharged zero modes from the universal sector. These extra zero modes are non-chiral and hence one expects that e.g. RR/NS fluxes or other effects may easily lift them, as we discussed in section 2. Concerning O(1) instantons, which have just the two required fermion zero modes in the universal sector, we have found one example, with the appropriate net structure of charged zero modes. However, it has plenty of other extra zero modes. We discuss examples of O(1) and Sp(2) instantons in the following subsections. 6.2 An O1 example Let us first discuss the case of O(1) instantons. In principle they would be the more attractive since they have no undesirable universal zero modes at all. Unfortunately this type of instanton is rare within the set we scanned, and we found just one example with a solution to the tadpole equations without any unwanted chiral zero-modes. The instanton however has a very large number of uncharged and charged vector-like zero modes. The standard model brane configuration occurs for tensor product (2, 4, 22, 22), MIPF 49, orientifold 1, boundaries (a,b,c,d) = (487, 1365, 576, 486). As usual we only provide this information in order to locate this model in the database. Further details are available on request. The bi-fundamental fermion spectrum of this model in the (a,b,c,d) sector is fairly close to the MSSM: there is an extra up-quark mirror pair, two mirror pairs of lepto- quarks with down quark charges and one with up-quark charges, plus two extra right- handed neutrinos (i.e. a total of five right-handed neutrinos). There are three MSSM Higgs pairs. The tensor spectrum is far less appealing, in particular for brane c: this has 25 adjoints and 7 vector-like pairs of anti-symmetric tensors. As we said, there is just one instanton brane of type O1−. It has exactly the right number of zero-modes with brane d, but five superfluous pairs of vector-like zero- modes with brane c, plus one vector-like pair with brane a. In addition there are four symmetric tensor zero-modes on the instanton brane (which of course are vector-like, since it is a real brane): the parameter p+ in table 3 is equal to 2. The tadpole solution that is (chirally speaking) compatible with this instanton has a large hidden sector: O(1)×O(2)4×O(3)×U(1)2×Sp(2)2×U(3) (there are other pos- sibilities, but no simple ones). This hidden sector introduces more undesirable features: vector-like observable/hidden matter, vector-like instanton/hidden sector modes, plus chiral and non-chiral matter within the hidden sector. Finally the coupling ratios are as follows: α3/α2 = .54, sin 2θw = .094, and the instanton coupling is 3.4 times weaker than the QCD coupling (α3/αInstanton = 3.4). Despite these unappealing features this model does demonstrate the existence of this kind of solution. 6.3 The S2 models As we have mentioned, these are the examples which come closer to the minimal set of fermion zero modes. As we see in Table 5, all such instantons satisfying the criteria on the zero mode structure (except for the extra universal zero modes) appear for models based on the same CFT orientifold. It is the one obtained from the (2, 4, 22, 22) Gepner model with MIPF 13 and orientifold 3 in the table. The model is obtained as follows. 6.3.1 The closed string sector We start with the tensor product (2,4,22,22). This yields a CFT with 12060 primary fields, 48 of which are simple currents, forming a discrete group G = Z12×Z2×Z2. After taking into account the permutation symmetry of the last two factors, we find that this tensor product has 54 symmetric MIPFs, and we choose one of them to build the model of interest. For convenience we specify all quantities in terms of a standard minimal model notation, but also in terms of the labelling of the computer program “kac” that generates the spectrum. This particular MIPF is nr. 13. To build it we choose a subgroup of G, which is isomorphic to H = Z12 × Z2. The generator of the Z12 factor is primary field nr. 1, (0, 0, 0, {24,−24, 0}, {24, 20, 0}), and the Z2 factor is generated by primary field nr. 24, (0, 0, 0, 0, {24, 20, 2}). The representations are specified on a basis (NSR, k = 2, k = 4, k = 22, k = 22), i.e. the boundary conditions of the NSR- fermions and the four minimal models in the tensor product. Here 0 indicates the CFT vacuum, and for all other states we use the familiar (l, q, s) notation for the N = 2 minimal models. The first generator has conformal weight h = 11 and has ground state dimension 1. The second has weight h = 11 and has ground state dimension 2: the ground state contains both (0, 0, 0, 0, {24, 20, 2}) and (0, 0, 0, {24, 20, 2}, 0). The matrix X defining the MIPF according to the prescription given in [53][54][55] is (6.3) This simple current modification is applied to the charge conjugation invariant of the tensor product. This defines a MIPF that corresponds to an automorphism of the fusion rules, and that pairs all the primaries in the CFT off-diagonally. The number of Ishibashi states, and hence the number of boundary states is 1080. The MIPF is invariant under exchange of the two k = 22 factors: this maps current 24 to itself, and current 1 to current 11, which is also in H. Hence this symmetry of the tensor product maps H into itself, and it also preserves the matrix X . To define an orientifold, we must specify a “Klein bottle current” plus two signs defined on the basis of the simple current group. For the current K we use the generator of the second Z2 in G, primary field nr. 12. This is the representation (0, 0, {4,−4, 0}, {(24, 16, 2)}, {(24,−12, 2)}) which is degenerate with nine other states, all of dimension 1 and conformal weight 7. The crosscap signs are chosen, on the afore- mentioned basis of H as (+,−). This results in a crosscap coefficient of 0.0464731, and it is orientifold nr. 3 of a total of 8. The orientifold is also invariant under permutation of the identical factors. The closed string spectrum contains 14 vector multiplets and 60 chiral multiplets. 6.3.2 The standard model branes To build a standard model configuration we have to specify the boundary state labels. It turns out that we have four choices for label a and b, one for c and two for d. This leads to a total of 32 possibilities. Among these 32 there are 22 have distinct spectra (distinguished by the number of vector-like states), but for all 32 choices one obtains the same set of dilaton couplings. It seems plausible that these choices simply correspond to putting the a, b and d branes in slightly different positions, so that we move the configuration in brane moduli space. The choices are as follows (these are boundary labels assigned by the computer program, and can be decomposed in terms of minimal model representations; this will be explained in table 6 below) a : 10, 22, 130, 142 b : 210, 282, 290, 291 c : 629 d : 712, 797 There are additional possibilities, but they do not give rise to additional distinct spec- Table 6: Branes appearing in standard model configurations Label Orbit/Deg. Reps Weight Dimension 10 240 (0, 0, 0, 0, {10, 0, 0}) 5/4 1 130 2760 (0, 0, 0, {10, 0, 0}, 0) 5/4 1 22 [528,0] (0, 0, 0, {1,−1, 0}, {11, 1, 0}) 3/2 1 (0, 0, 0, {1, 1, 0}, {11,−1, 0}) 3/2 1 142 [3048,0] (0, 0, 0, {11,−1, 0}, {1, 1, 0}) 3/2 1 (0, 0, 0, {11, 1, 0}, {1,−1, 0}) 3/2 1 210 4248 (0, 0, {3, 3, 0}, {3,−3, 0}, {9,−9, 0}) 1/2 1 282 5760 (0, 0, {3, 3, 0}{9,−9, 0}{3,−3, 0}) 1/2 1 290 [5952,0] (0, 0, {1, 1, 0}{9, 7, 0}{11,−11, 0}) 5/6 1 291 [5952,24] (0, 0, {1, 1, 0}{9, 7, 0}{11,−11, 0}) 5/6 1 629 [9348,30] (0, (1,−1, 0), 0, {9, 9, 0}{5,−3, 0} 7/12 1 712 [9852,0] (0, {1, 1, 0}{3,−3, 0}{1, 1, 0}{5, 5, 0}) 1/2 2 (0, {1, 1, 0}{1,−1, 0}{1, 1, 0}{5,−3, 0}) 1/2 2 797 [10356,30] (0, {1, 1, 0}{3,−3, 0}{5, 5, 0}{1, 1, 0}) 1/2 2 (0, {1, 1, 0}{1,−1, 0}{5,−3, 0}{1, 1, 0}) 1/2 2 The second column gives the boundary labels in terms of a primary field label and a degeneracy label (boundaries not indicated by square brackets are not degenerate). The labels appearing in columns 1 and 2 are assigned by the computer program, and are listed here only for the purpose of reproducing the results using that program. In column 2, the boundary labels are expressed in terms of primary field labels, as in formula (A.4). If a single number appears, this is a representative of an H-orbit corresponding to the boundary. If square brackets are used, this means that the H- orbit has fixed points, and that it corresponds to more than one boundary label. The second entry in the square brackets is the degeneracy label, and refers to a character of the “Central Stabilizer” defined in [43]; the details of the definition and the labelling will not be important here. In this case the first entry within the square brackets refers to an orbit representative. These orbit representatives can also be expressed in a standard form for minimal model tensor products. This is done in column 3. This is basically the same expansion shown in (A.4), except that the degeneracy label ΨI turns out to be trivial in all cases, both for the standard model and for the instanton branes shown below (although the theory does contain primaries with non-trivial Ψ’s). In columns 4 and 5 we specify the weight and ground state dimension of the corresponding highest weight representation. These data are not directly relevant for the boundary state, but helps in identifying it. Since boundaries are specified by orbit representatives, it is not straightforward to compare them, since the standard choice (the one listed in column 2) is arbitrary. For this reason we have used another representative in columns 3, 4 and 5, selected by an objective criterion: we choose the one of minimal dimension and minimal conformal weight (in that order). If there is more than one representative satisfying these criteria we list all. 6.3.3 The open string spectrum In Table 7 we summarize the spectra of the 32 models. The first four columns list the a,b,c,d brane labels. The last eight columns specify the total number of multiplets of types Q (quark doublet), U (up quark singlet), D (down quark singlet), L (lepton dou- blet), E (charged lepton singlet), N (neutrino singlet), Y (lepto-quark) and H (Higgs). The numbers given are for the total number of lefthanded fermions in the represen- tation, plus their complex conjugates. So for example a 7 in column “Q” means that there are 5 quark doublets in the usual representation (3, 2, 1 ), plus two in the complex conjugate representation (3∗, 2,−1 This yields the required three families of quark doublets, plus two mirror pairs. Hence the smallest number that can occur in the six columns QUDLEN is three, if there are no mirrors (note that cubic anomaly cancellation requires three right-handed neutrinos in this class of models). The lepto-quarks Y are all in the same representation as the down-quarks (D), or the conjugate thereof, and they occur only as vector-like mirror pairs. They differ from D-type mirror quarks because they carry lepton number, because they come from open strings ending on the d-brane instead of the c-brane. In general, there can also exist U-type lepto-quarks, but in these models they do not occur. Finally the numbers 10, 18 and 26 in column ’H’ mean that there are 5, 9 or 13 MSSM Higgs pairs H + H̄. It is worth noticing that right-handed quarks U,D and neutrinos N = νR do not have vectorlike copies. On the other hand right-handed leptons E always have one and the left-handed fields Q,L may have up to 3 vector-like copies. Table 7: Spectrum all 32 configurations. U(3) Sp(2) U(1) U(1) Q U D L E N Y H 10 210 629 712 7 3 3 9 5 3 6 10 22 210 629 712 7 3 3 9 5 3 6 10 130 210 629 712 3 3 3 9 5 3 2 10 142 210 629 712 3 3 3 9 5 3 2 10 10 282 629 712 3 3 3 5 5 3 6 26 22 282 629 712 3 3 3 5 5 3 6 26 130 282 629 712 7 3 3 5 5 3 2 26 142 282 629 712 7 3 3 5 5 3 2 26 10 290 629 712 3 3 3 3 5 3 6 18 22 290 629 712 3 3 3 3 5 3 6 18 130 290 629 712 3 3 3 3 5 3 2 18 142 290 629 712 3 3 3 3 5 3 2 18 10 291 629 712 3 3 3 3 5 3 6 18 22 291 629 712 5 3 3 3 5 3 6 18 130 291 629 712 3 3 3 3 5 3 2 18 142 291 629 712 3 3 3 3 5 3 2 18 10 210 629 797 7 3 3 5 5 3 2 10 22 210 629 797 7 3 3 5 5 3 2 10 130 210 629 797 3 3 3 5 5 3 6 10 142 210 629 797 3 3 3 5 5 3 6 10 10 282 629 797 3 3 3 9 5 3 2 26 22 282 629 797 3 3 3 9 5 3 2 26 130 282 629 797 7 3 3 9 5 3 6 26 142 282 629 797 7 3 3 9 5 3 6 26 10 290 629 797 3 3 3 3 5 3 2 18 22 290 629 797 3 3 3 3 5 3 2 18 130 290 629 797 3 3 3 3 5 3 6 18 142 290 629 797 3 3 3 3 5 3 6 18 10 291 629 797 3 3 3 3 5 3 2 18 22 291 629 797 5 3 3 3 5 3 2 18 130 291 629 797 3 3 3 3 5 3 6 18 Continued on next page Table 7 – continued from previous page U(3) Sp(2) U(1) U(1) Q U D L E N Y H 142 291 629 797 3 3 3 3 5 3 6 18 In the following table we list the multiplicities Laa and Laa′ of the branes that occur in these models, leading to vector-like sets of adjoints and rank-2 tensors. Since brane b is symplectic, the number of adjoints is equal to the number of symmetric tensors. Table 8: 4d matter from the aa and aa′ sectors. Boundary Adjoints Anti-symm. Symm. a(10) 2 2 6 a(22) 2 2 2 a(130) 2 2 6 a(142) 2 2 2 b(210) - 14 10 b(282) - 14 10 b(290) - 14 6 b(291) - 14 6 c(629) 9 - 14 d(712) 3 - 6 d(797) 3 - 6 It should be emphasized that CFT constructions generically correspond to par- ticular points in moduli space of CY orientifolds. Due to this, they usually have an ‘enhanced’ massless particle content with extra vector-like matter and closed string gauge interactions. Thus one would expect that many of the massless vector-like chiral fields present in this class of models could gain masses while moving to a nearby point in moduli space. 6.3.4 The instantons Each of these 32 Standard Model compactifications admits 8 instantons. The instanton labels are identical for all the 32 models. They are listed in Table 9. The first five columns use the same notation as for the standard model boundary labels. In column 6 we list the numerical value of the dilaton coupling to the instanton brane. This quantity is proportional to 1 . It is instructive to compare these couplings to the gauge couplings, in order to gain intuition on the suppression factor for our instantons. In these models the U(3) dilaton couplings are 0.00622, so that the instantons are more strongly coupled than QCD19 On the other hand in this particular model the ratio α3/α2 at the string scale is 3.23 (the value of sin 2θw at the string scale is 0.527). All of these couplings are subject to renormalization group running, and there are plenty of vector-like states to contribute to this, if one assumes that they acquire masses at a sufficiently low scale. One should perform a detailed renormalization group analysis to check whether one may obtain consistency with the gauge couplings measured at low-energies. Let us emphasize however that one expects that moving in moduli space many of these vector-like states will gain masses and also the values of the different gauge couplings will also generically vary. Since the value of the Type II dilaton is a free parameter at this level, one can get the appropriate (intermediate) mass scale for the right-handed neutrino Majorana masses by choosing an appropriate value for the dilaton. In this context, it is satisfactory to verify that the instanton couplings are unrelated to the gauge couplings, as expected since they do not correspond to gauge instantons [3], and are in fact less suppressed than the latter. Note that the 8 instantons fall into two distinct classes (evidently not related by any discrete symmetry, since the conformal weight on the boundary orbit is distinct, and the coupling is different as well). Within each class, the orbits of the four instan- ton boundaries appear to be related by the Z2 symmetries of interchange of the last two tensor factors, and simultaneous inversion of the charge q of the minimal model. However, one has to be very careful in reading off symmetries directly from the labels in columns 3 of Tables (6) and (9) for a number of reasons. First of all the entries in column 3 are representatives of boundary orbits, and these representatives themselves are merely representatives of extension orbits. Secondly the action of any discrete 19 Note that the Type II dilaton in this compactifications is an arbitrary parameter which can always be chosen so that we consistently work at weak coupling. It is the relative value of gauge couplings which we are comparing here. Table 9: Instantons for all 32 configurations Lbl. Orbit/Deg. Reps Weight Dim. coupling 414 [8064,0] (0, {1, 1, 0}, 0, {22,−22, 0}, {20, 16, 0}) 5/2 1 0.0016993 417 [8076,30] (0, {1,−1, 0}, 0, {22, 22, 0}, {20,−16, 0}) 5/2 1 0.0016993 456 [8316,0] (0, {1, 1, 0}, 0, {20, 16, 0}, {22,−22, 0}) 5/2 1 0.0016993 459 [8328,30] (0, {1,−1, 0}, 0, {20,−16, 0}, {22, 22, 0}) 5/2 1 0.0016993 418 [8088,0] (0, {1, 1, 0}, 0, {22,−22, 0}, {18, 16, 0}) 5/3 1 0.0027033 420 [8100,0] (0, {1,−1, 0}, 0, {22, 22, 0}, {18,−16, 0}) 5/3 1 0.0027033 502 [8592,0] (0, {1, 1, 0}, 0, {18, 16, 0}, {22,−22, 0}) 5/3 1 0.0027033 505 [8604,30] (0, {1,−1, 0}, 0, {18,−16, 0}, {22, 22, 0}) 5/3 1 0.0027033 symmetry on the degeneracy labels can be non-trivial. In appendix B we discuss these symmetries in more detail. 6.4 Other examples The Sp(2) instanton examples just discussed are the ones which get closer to the required minimal set of fermion zero modes. Under slightly weaker conditions, we find many more solutions. In all these cases some additional mechanism beyond exact RCFT will be needed to lift some undesirable zero modes. The simplest such case is the following. The tensor product is (2, 8, 8, 18), MIPF nr. 14, orientifold 2 (the precise spectra may be found using this information in the database www.nikhef.nl/∼t58/filtersols.php). There are three distinct brane configu- rations for which almost perfect instantons exist, namely (a,b, c,d) = (64, 562, 389, 67) and (64, 577, 389, 67) and (65, 560, 189, 66). Each has six instantons, three of type S2+ and three of type S2−. As in the foregoing example, the six instantons are identical for the three standard model configurations. In this example, they have three differ- ent dilaton coupling strengths: .00254, .00665 and .0108 (each value occurs once for S2+ and once for S2−). By comparison, the U(3)-brane dilaton coupling strength is 0.0119338, so that the instanton brane coupling is quite a bit stronger than the QCD coupling. This is again an interesting point if we want that νR masses are not too much http://www.nikhef.nl/~t58/filtersols.php suppressed. Furthermore in this example there are three distinct instanton couplings, so that one may expect three non-zero eigenvalues (with a hierarchy) in the mass ma- trix. As in the previous examples there is not gauge coupling unification, one rather has α3/α2 = .4813 and sin 2(θw) = .183 at the string scale. Again a full renormalization group analysis should be performed in order to check consistency with the measured low-energy gauge coupling values. These models all have a hidden sector consisting of a single Sp(2) factor. They have respectively 3, 1 and 3 susy Higgs pairs, and a spectrum of bi-fundamentals that is closer to that of the standard model than the previously discussed Sp(2) examples: quarks and leptons do not have vector-like copies (there are only some vector-like leptoquarks), and even one of the three models have the minimal set of Higgs fields of the MSSM. The rest of the spectrum is purely vector-like, and contains a number of rank-2 tensors, including eight or six adjoints of U(3). Furthermore there is vector- like observable-hidden matter. The only undesirable instanton zero-mode is a single bi-fundamental between the hidden sector Sp(2) brane and the instanton brane. Still, these SM brane configurations without the hidden sector, provide interesting and very simple local models of D-brane sectors admitting instantons generating neutrino masses (with the additional ingredients required to eliminate the extra universal triplets of fermion zero modes). 6.5 R-parity violation We now turn to the generation of other possible superpotentials violating B − L. An instanton violates R-parity if the amount of B − L violation, IMa − IMa′ − IMd + IMd′ (6.4) is odd. Examples of instantons with that property were found in the following ten- sor product/MIPF/orientifold combinations: [(1, 16, 16, 16), 12, 0], [(2, 4, 16, 34), 49, 0], [(2, 4, 12, 82), 19, 0] [(2, 4, 22, 22), 49, 0] and [(2, 4, 16, 34), 18, 0]. Note that all cases for which O2 or U2 instantons were found necessarily have R-parity violating instantons as well: the corresponding O1 and U1 instantons have IMd or IMd′ equal to ±1, whereas the intersection with the a is non-chiral. In principle, there are many more ways to obtain R-parity violating instantons (either due to non-vanishing contributions to IMa − IMa′ or higher values of IMd − IMd′), and indeed, many such instantons turn out to exist. But the number of tensor product/MIPF/orientifold combinations where they occur hardly increases: only in the case [(1, 16, 16, 16), 12, 0] we found R-parity violating instantons, but no U1 or O1 instantons. This suggests that in the other cases R-parity is a true symmetry of the model. Unfortunately we have no way of rigorously ruling out any other non-perturbative effects, but at least the set we can examine re- spects R-parity. This includes in particular the models without hidden sector (found for [(2, 4, 22, 22), 13, 3] ) discussed above. The following table list the total number of instantons with the chiral intersections listed in table 2. The total number of instantons (boundaries violating the sum rule, as defined in (6.1)) is 29680, for all standard model configurations combined. The last four columns indicate how many unitary instantons satisfy the sum rule exactly as listed in table (6.1), how many satisfy it with IMx ↔ −IMx′ (the column U’), and how many O-type and S-type instantons there are. Here ‘S’ refers to boundaries with a symplectic Chan-Paton group if the boundary is used as an instanton brane. All intersection numbers for type S have been multiplied by 2 before comparing with table 2. For real branes, the relevant quantities used in the comparison are IMa − IMa′, IMc−IMc′ and IMd−IMd′ , while IMb = 0. There are fewer unitary instantons possibly generating Majorana masses then the numbers mentioned above because the conditions we use here are stricter: we require here that IMx and IMx′ match exactly, not just their difference. Note however that this still allows additional vector-like zero-modes. If we only wish to consider cases without any spurious zero-modes, we may limit ourselves to the O-type instantons in the last column. There are very few to inspect, and all of them turn out to have a few non-universal zero modes. D = 4 Operator U U’ S O νRνR 1 2 627 3 LH̄LH̄ 0 5 550 3 LH̄ 3 3 0 4 QDL 8 4 0 4 UDD 0 0 0 4 LLE 8 4 0 4 QQQL 0 4 0 3892 UUDE 4 0 0 3880 Table 10: Number of instantons in our search which may induce neutrino masses (first 2 rows), R-parity violation (next 4 rows) or proton decay operators (last 2 rows). The last two cases are B − L preserving dimension five operators, and obviously do not come from the set of 29680 B −L violating instantons. They were searched for separately, but the search was limited to the same 391 models we used in the rest of the paper. Obviously, one could equally well look for such instantons in the full database, since their existence does not require a massive B − L. It is interesting to note that in the classes of MSSM-like models discussed earlier in this section with the closest to minimal zero mode structure, there are no instantons al all generating either R-parity violating or the B − L dim=5 operators in the table. This makes them particularly attractive. Note that all numbers in table 10 refer to the occurrence of instantons in the set of 391 tadpole-free models with massive B-L, but without checking the presence of zero-modes between the hidden sector and the instanton. It makes little sense to use the hidden sector in the database for such a check, since this is just one sample from a (usually) large number of possibilities. A meaningful question would be: can one find a hidden sector that has no zero-modes with the instanton. We have done such a search for the B − L violating instantons (see the exclamation marks in the last column of table (5)), but not for the B − L preserving instantons. 7 Conclusions and outlook In this paper we have presented a systematic search for MSSM-like Type II Gepner orientifold models allowing for boundary states associated to instantons giving rise to neutrino Majorana masses. This search is very well motivated since neutrino masses are not easily accommodated in the semi-realistic compactifications constructed up to now. String instanton induced Majorana masses provides a novel and promising way to understand the origin of neutrino masses in the string theory context. The string instantons under discussion are not gauge instantons. Thus, for example, they not only break B + L symmetry (like ’t Hooft instantons do) but also B − L, allowing for Majorana neutrino mass generation. The obtained mass terms are of order Ms exp(−V/g2) but this suppression is unrelated to the exponential suppression of e.g. electroweak instantons and may be mild. In fact we find in our most interesting examples that the instanton action is typically substantially smaller than that of QCD or electroweak instantons, and hence these effects are much less suppressed than those coming from gauge theory instantons. To perform our instanton search we have analyzed the structure of the zero modes that these instantons must have in order to induce the required superpotential. This analysis goes beyond the particular context of Gepner orientifolds and has general validity for Type II CY orientifolds. We have found that instantons with O(1) CP symmetry have the required universal sector of just two fermionic zero modes for the superpotential to be generated. Instantons with Sp(2) and U(1) CP symmetries have extra unwanted universal fermionic zero modes, which however may be lifted in a va- riety of ways in more general setups, as we discuss in the text. In fact we find in our search that around 98 % of the instantons with the correct structure of charged zero modes have Sp(2) CP symmetry. Indeed, from a number of viewpoints the Sp(2) instantons are specially interesting. The instantons we find with the simplest structure of fermionic zero modes are Sp(2) instantons which are also the ones which are present more frequently in the MSSM-like class of Gepner constructions considered. They have also some interesting features from the phenomenological point of view. Indeed, due to the non-Abelian structure of the CP symmetry, the structure in flavor space of the neutrino Majorana masses factorizes. This makes that, irrespective of what particu- lar compactification is considered, Sp(2) instantons may easily lead to a hierarchical structure of neutrino masses. It would be important to further study the possible phenomenological applications of the present neutrino mass generating mechanism. String instanton effects can also give rise to other B- or L-violating operators. Of particular interest is the dimension 5 Weinberg operator giving direct Majorana masses to the left-handed neutrinos. We find that in the most interesting cases, different instantons giving rise to the Weinberg operator and to νR Majorana masses are both simultaneously present. Which effect is the dominant one in the generation of the physical light neutrino masses depends on the values of the instanton actions and amplitudes as well as on the value of the string scale. Instantons may also generate dim< 5 operators violating R-parity. We find however that instantons inducing such operators are extremely rare, and in fact are completely absent in the Gepner models with the simplest Sp(2) instantons inducing neutrino masses. There are many avenues yet to be explored. It would be important to understand better the possible sources (moving in moduli space, addition of RR/NS backgrounds etc.) of uplifting for the extra uncharged fermionic zero modes in the most favoured Sp(2) instantons. A second important question is that we have concentrated on check- ing the existence of instanton zero modes appropriate to generate neutrino masses; one should further check that the required couplings among the fermionic zero modes and the relevant 4d superfields (i.e. νR or LH̄) are indeed present in each particular case. This is in principle possible in models with a known CFT description but could be difficult in practice for the Gepner models here described. Instantons can also generate other superpotentials with interesting physical appli- cations. One important example is the generation of a Higgs bilinear (i.e. a µ-term) in MSSM-like models [4, 3]. Thus, e.g., one could perform a systematic search for instantons (boundary states) generating a µ-term in the class of CFT Gepner orien- tifolds considered in the present article. Other possible application is the search for instantons inducing superpotential couplings involving only closed string moduli. The latter may be useful for the moduli-fixing problem, or for non-perturbative corrections to perturbatively allowed couplings [56]. Finally, it would be important to search for analogous instanton effects inducing neutrino masses in other string constructions (heterotic, M-theory etc.). A necessary condition is that the anomaly free U(1)B−L gauge boson should become massive due to a Stückelberg term. The importance of neutrino masses in physics beyond the Standard Model is un- questionable. We have shown that string theory instantons provide an elegant and simple mechanism to implement them in semi-realistic MSSM-like string vacua, and a powerful constraint in model building. In our opinion, the conditions of the existence of appropriate instantons to generate neutrino masses should be an important guide in a search for a string description of the Standard Model. Acknowledgements We thank M. Bertolini, R. Blumenhagen, S. Franco, M. Frau, S. Kachru, E. Kiritsis, A. Lerda, D. Lüst, F. Marchesano, T. Weigand for useful discussions. A.M.U. thanks M. González for encouragement and support. The research of A.N. Schellekens was funded in part by program FP 57 of the Foundation for Fundamental Research of Matter (FOM), and Research Project FPA2005-05046 of de Ministerio de Educacion y Ciencia, Spain. The research by L.E. Ibáñez and A.M. Uranga has been supported by the European Commission under RTN European Programs MRTN-CT-2004-503369, MRTN-CT-2004-005105, by the CICYT (Spain), and the Comunidad de Madrid under project HEPHACOS P-ESP-00346. Appendix A CFT Notation Here we summarize the labelling conventions for various CFT quantities. Further details and explanations can be found in [43]. It is important to keep in mind that there are four steps in the construction, each involving choices of some quantities. The steps are • A CFT tensor product • An extension of the chiral algebra of this tensor product • The choice of a MIPF • The choice of an orientifold The second and third step are easily confused. A MIPF can itself be of extension type (although it can also be of automorphism or mixed type), meaning that it implies an extension of the chiral algebra. The crucial difference between step two and three in that case is that in step 2 all fields that are non-local with respect to the extension are projected out, and the symmetry of the extension is imposed on all states of the CFT, i.e in particular on all boundary states. The extension in step three acts as a bulk invariant, but the boundary states are not required to respect the symmetry implied by the extension. Primary fields of N = 2 minimal models are labeled in the usual way by three integers (l, q, s). In addition to these minimal models, one building block of our CFT’s is of course a set of NSR fermions in four dimensions. They can be represented by the four conjugacy classes (0), (v), (s), (c) analogous to those of a root lattice of type D. Primary fields in a tensor product of M factors are therefore labelled as I = ((x), (l1, q1, s1), . . . , (lM , qM , sM)) (A.1) where x = 0, v, s or c. This tensor product is extended by the alignment currents and the spin-1 field corresponding to the space-time supersymmetry generator. This organizes the tensor product fields into orbits, which can be labelled by one of the elements of the orbit. We always choose the field of minimal conformal weight (or one of them, in case there are more) as the orbit representative labelling the orbit. The supersymmetry generator may have fixed points, leading to orbits appearing more than once as primary fields of the extended theory. In those cases we need an additional degeneracy label to distinguish them. It is convenient to choose for this label a character of the discrete group that is causing the degeneracy, the “untwisted stabilizer”, which depends on I. Denoting this character as ΨI we get then the following set of labels for the primaries of the extended CFT i = [I,ΨI ] (A.2) where I has the form (A.1). If there are no degeneracies we will leave out the square brackets and the ΨI . In boundary CFT’s two new labels appear: the labels of Ishibashi-states that propa- gate in the transverse channel of the annulus, and the boundary labels. In the simplest, “Cardy” case both sets of labels are in one-to-one correspondence with the extended CFT labels i. But if we consider non-trivial MIPFs Zij both sets of labels are different. The Ishibashi states are in one-to-one correspondence with the fields i with Ziic 6= 0. Degeneracies can occur here if Ziic 6> 0. This requires the introduction of a degeneracy label. Such degeneracies may occur if the stabilizer of i (the set of simple currents that fix i) is non-trivial. It is convenient to use elements J of the stabilizer as degeneracy labels, so that the Ishibashi labels get the following form m = (i, J) , (A.3) where i is an extended CFT label, as defined above (to be precise, in some cases a non-trivial degeneracy label is introduced even if Ziic = 1. The details will not matter here). Boundary states correspond to orbits of the simple current group H that defines a MIPF. To label such orbits we choose a representative. There is no obvious canonical representative (one could use one of minimal conformal weight, but the conformal weight of orbit members of a boundary state does not play any rôle in the formalism, unlike the conformal weight of a primary). So in this case we just make an arbitrary choice. Once again there can be degeneracies. In this case they are due to a subgroup of the stabilizer called the “Central Stabilizer”. It is convenient to label the boundary states by an orbit representative i and a character ψi of the central stabilizer. If we expand the boundary state label in all of its components we get a = [i, ψi] = [[I,ΨI ], ψi] = [((x), (l1, q1, s1), . . . , (lM , qM , sM)),ΨI ], ψi] . (A.4) Note that i is just a representative of a boundary orbit, and that I is just a represen- tative of an orbit of the extension of the CFT. B Instanton boundary symmetries In the hidden-sector free example discussed in some detail in section 6 we have en- countered Sp(2)-type instantons, the most common kind in our scan. This particular model is the one that comes closest to the required zero mode count, although the only superfluous zero modes are rather awkward. Let us assume that the effect of these su- perfluous universal zero-modes instantons can be avoided. Then there is still another problem we have to face, namely that the two zero modes αi and γi are related by an Sp(2) transformation of the label i. Then we we need at least three independent instantons (with unrelated couplings) to generate three non-zero neutrino masses, as discussed in Section 3.3. Since the technology to compute the couplings is not yet available, we cannot be completely sure that the relevant couplings are distinct, or indeed that they are non-vanishing, but at least we can inspect if there are obvious symmetries relating them. The unextended tensor product (2, 4, 22, 22) has 64 discrete symmetries: five sep- arate charge conjugations of the factors (including the NSR space-time factor) and the interchange of the two identical k = 22 minimal models. To get space-time su- persymmetry this tensor product is extended with the product of the simple current Ramond ground states of each factor. These Ramond ground states are not invariant under charge conjugation. Therefore this extension breaks the discrete symmetries to Z2 × Z2, the combined charge conjugation of all five factors and the permutation of the two identical factors. The combined charge conjugation also acts non-trivially on the Ramond ground states in each factor, but the result is the charge conjugate of the space-time supersymmetry generator, which is in the chiral algebra. The combined conjugation is in fact the charge conjugation symmetry of the extended CFT. It turns out that only a Z2 subgroup of Z2 × Z2 acts non-trivially on the simple currents of the extended CFT: the permutation of the k = 22 factors acts in the same way as charge conjugation. The action of these symmetries on the complete set of primary fields is more complicated. It is easy to see that the permutation acts differently than charge conjugation. In general the primary fields of the extended CFT are labelled as i = [((x), (i1), . . . , (i4)),Ψ] (see Appendix A). The action of the permutation is to interchange i3 and i4, but in cases with a non-trivial degeneracy it is not a priori clear which of the degenerate states is the image of the map. This can be resolved by examining the fusion rules, which should be invariant under the permutation: Nijk = Nπ(i)π(j)π(k) , (B.1) where N is a fusion coefficient and π a permutation or other automorphism. In general there may be more than one way to resolve these ambiguities, resulting in additional automorphism of the CFT. The standard example of this situation is the extension of the affine algebra A1 level 4 by the simple current. The resulting CFT has an outer automorphism, non-trivial charge conjugation, that has no counterpart in A1 level 4. As mentioned above, the Z2 permutation symmetry is respected by the MIPF and the orientifold, and since charge conjugation acts in the same way on the simple currents as the permutation, charge conjugation is respected as well. In this way we end up with (at least) a surviving Z2×Z2 discrete symmetry acting on the boundary labels, or a larger discrete symmetry if that symmetry is extended by the action on the degeneracy labels of the extension. The foregoing story repeats itself for the action on the boundary labels. The boundary labels are given in terms of the CFT labels plus a second degeneracy label, the one indicated by the second entry in the square brackets in column 2 of tables (6) and (9). Once again one has to determine not only how a symmetry acts on the first entry (this is just the action of the symmetry in the extended CFT, respecting its fusion rules), but also how it acts on the degeneracy labels. In this case the precise action can be determined from the invariance of the annulus coefficients Aiab = A π̂(a)π̂(b) , (B.2) where π is the action on the primaries of the extended CFT (as determined above) and π̂ is the action on the boundary labels induced by π. Since the orientifold choice is non-trivial, boundary charge conjugation does not coincide with CFT charge conjugation. Indeed, the eight instanton boundary states are invariant under boundary charge conjugation (which they must be in order to produce a “real” Sp(2)-type instanton). However, just as permutations, CFT charge conjugation may induce a non-trivial discrete symmetry on the boundary states. In addition to these “outer automorphisms” there is the notion of boundary simple currents, introduced in the appendix of [6]. These may be thought of as remnants of the original simple currents, and imply relations between annulus amplitudes of the Aiab′ = A Ja(Jb)′ (B.3) All of the aforementioned symmetries might relate instanton couplings, and hence threaten their numerical independence. However, in order to do that they have to be symmetries of the full standard model/instanton configuration, not just relate some of the eight instantons to each other. It is easy to see that the permutation of the k = 22 factors changes the standard model brane configuration. Consider brane c: it turns out that under permutation boundary state 629 it is mapped to boundary state 544 or 545 (depending on the action on the degeneracy label), which in any case is distinct. Hence even if the instanton boundaries 414 and 456 resp. 418 and 502 are mapped to each other by boundary permutation, at the same time the standard model configuration is mapped to a distinct one. This means that we may expect at least four distinct couplings, which should be sufficient. It is of course possible to work out the discrete symmetries exactly, but in view of this argument this would not yield any additional insight. We do know the exact boundary orbits. The orbit of instanton label 414 is (414, 415, 416, 417), so that instantons 414 and 417 are related. But the orbit of brane c under the same action is (629, 628, 626, 627). Hence the action that relates 414 and 417 maps 629 to 627. In fact all four standard model boundaries a,b,c,d are mapped to different ones. This implies that instantons 414 and 417 may produce different couplings as well, so that all eight instantons may contribute in a different way. These distinctions concern the disk correlators d(r)a in (3.12). 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D54 (1996) 1667, arXiv:hep-th/9601038. [51] G. Pradisi and A. Sagnotti, Phys. Lett. B 216 (1989) 59. [52] E. Witten, Nucl. Phys. B 460 (1996) 541 [arXiv:hep-th/9511030]. [53] B. Gato-Rivera and A. N. Schellekens, Commun. Math. Phys. 145, 85 (1992). [54] B. Gato-Rivera and A. N. Schellekens, Nucl. Phys. B 353, 519 (1991). [55] M. Kreuzer and A. N. Schellekens, Nucl. Phys. B 411, 97 (1994) [arXiv:hep-th/9306145]. [56] S.A. Abel, M.D. Goodsell, arXiv:hep-th/0612110. http://arxiv.org/abs/hep-th/0403262 http://arxiv.org/abs/hep-th/0603217 http://arxiv.org/abs/hep-th/0007174 http://arxiv.org/abs/hep-th/9909114 http://arxiv.org/abs/hep-th/0004100 http://arxiv.org/abs/hep-th/0011048 http://arxiv.org/abs/hep-th/0510074 http://arxiv.org/abs/hep-th/9601038 http://arxiv.org/abs/hep-th/9511030 http://arxiv.org/abs/hep-th/9306145 http://arxiv.org/abs/hep-th/0612110 Introduction Instanton induced superpotentials in Type II orientifolds D-brane instantons, gauge invariance and effective operators Zero mode structure for D-brane instantons Uncharged zero modes Charged fermion zero modes Instanton induced Majorana neutrino masses The MSSM on the branes Majorana mass term generation Flavor and the special case of Sp(2) instantons Other B- and L-violating operators The Weinberg operator R-parity violating operators Dimension 5 proton decay operators CFT orientifolds Construction of the models Search for SM-like models Fermion zero modes for instantons on RCFT's Search for M instantons The instanton scan An O1 example The S2 models The closed string sector The standard model branes The open string spectrum The instantons Other examples R-parity violation Conclusions and outlook Appendix CFT Notation Instanton boundary symmetries

0704.1081

Asymmetry of in-medium rho-mesons as a signature of Cherenkov effects

Asymmetry of in-medium ρ-mesons as a signature of Cherenkov effects I.M. Dremin∗, V.A. Nechitailo† P.N.Lebedev Physics Institute RAS, 119991 Moscow, Russia December 3, 2018 Abstract Cherenkov gluons may be responsible for the asymmetry of dilep- ton mass spectra near ρ-meson observed in experiment. They can be produced only in the low-mass wing of the resonance. Therefore the dilepton mass spectra are flattened there and their peak is slightly shifted to lower masses compared with the in-vacuum ρ-meson mass. This feature must be common for all resonances. PACS: 12.38Bx, 13.87.-a There exist numerous experimental data [1, 2, 3, 4, 5, 6, 7, 8, 9] about the in-medium modification of widths and positions of prominent vector-meson resonances. They are mostly obtained from the shapes of dilepton mass and transverse momentum spectra in nucleus-nucleus collisions. Such in-medium effects were tied theoretically to chiral symmetry restoration a long time ago [10]. The dilepton mass spectra decrease approximately exponentially with in- crease of masses albeit with substantial declines from the average approx- imation of the general trend by the exponent in the low-mass region. A significant excess of low-mass dilepton pairs yield over expectations from hadronic decays is observed in experiment. The shape of the excess mass spectra shown in [1, 2, 3] is dominated by ρ-mesons. Their ratio to other vector meson resonances can be estimated as ρ : ω : φ=10:1:2. ∗email: [email protected] †email: [email protected] http://arxiv.org/abs/0704.1081v2 Several approaches have been advocated for explanation of the excess. Strong dependence of the parameters of the effective Lagrangian on the tem- perature and the chemical potential was assumed in [11, 12]. The hydro- dynamical evolution was incorporated in [13] to describe the spectra. The QCD sum rules and dispersion relations have been used [14, 15] to show that condensates decrease in the medium leads to both broadening and slight downward mass shift of resonances. The similar conclusions have been ob- tained from more traditional attempts using either the empirical scattering amplitudes with parton-hadron duality [16, 17] or the hadronic many-body theory [18, 19, 20]. In the latest approach, which pretends on the best description of experi- mental plots, the in-medium V-meson spectral functions are evaluated. The excess of dilepton pairs below ρ-mass is ascribed to anti-/baryonic effects. This conclusion is the alternative to more common ideas about the chiral restoration at high energies. It asks for some empirical constraints to fit the observed excess. In this paper we propose another possible source of low-mass lepton pairs. Namely, the emission of Cherenkov gluons may provide a substantial contri- bution to the low mass region. Considered first for processes at very high energies [21], the idea about Cherenkov gluons was extended to resonance production [22, 23]. For Che- renkov effects to be pronounced in ordinary or nuclear matter, the (either electromagnetic or nuclear) index of refraction of the medium n should be larger than 1. Qualitatively, the observed low mass excess of lepton pairs is easy to ascribe to the gluonic Cherenkov effect if one reminds that the index of refraction of any medium exceeds 1 within the lower wing of any resonance (the ρ-meson, in particular). This feature is well known in electrodynamics (see, e.g., Fig. 31-5 in [24]) where the atoms behaving as oscillators emit as Breit-Wigner resonances when get excited. This results in the indices of refraction larger than 1 within their low-energy wings. In QCD, one can imagine that the nuclear index of refraction for gluons in the hadronic medium behaves in a similar way in the resonance regions. This statement is more general and can be valid also at other energies if the relation (see, e.g., [25]) between the index of refraction and the forward scattering amplitude F (E, 0o) is fulfilled not only for photons but for gluons as well: ∆n = Ren− 1 ∝ ReF (E, 0o)/E. (1) Here E is the photon (gluon) energy. In classical electrodynamics, it is the dipole excitation of atoms in the medium by light which results in the Breit- Wigner shape of the amplitude F (E, 0o). In hadronic medium, there should be some modes (quarks, gluons or their preconfined bound states, conden- sates, blobs of hot matter...?) which can get excited by the impinging parton, radiate coherently if n > 1 and hadronize at the final stage as hadronic res- onances [22, 23]. The hadronic Cherenkov effect can provide insight into the substructure of the medium formed in nucleus-nucleus collisions. The resonance amplitudes are chosen for F (E, 0o) at comparatively low energies. The scenario, we have in mind, is as follows. The initial parton, belonging to a colliding nucleus, emits a gluon which traverses the nuclear medium. On its way, it collides with some internal modes. Therefore it affects the medium as an ”effective” wave which accounts also for the waves emitted by other scattering centers (see, e.g., [25]). Beside incoherent scattering, there are processes which can be described as the refraction of the initial wave along the path of the coherent wave. The Cherenkov effect is the induced coherent radiation by a set of scattering centers placed on the way of propagation of a gluon. That is why the forward scattering amplitude plays such a crucial role in formation of the index of refraction. At low energies its excess over 1 is related to the resonance peaks as dictated by the Breit-Wigner shapes of the amplitudes. In experiment, usual resonances are formed during the color neutralization process. However, only those gluons whose energies are within the left-wing resonance region of n > 1 give rise also to collective Cherenkov effect proportional to ∆n. Thus, apart from the ordinary Breit-Wigner shape of the cross section for resonance production, the dilepton mass spectrum would acquire the additional term proportional to ∆n at masses below the resonance peak. Therefore its excess near the ρ-meson can be described by the following formula1 (m2ρ −M 2)2 +M2Γ2 1 + w m2ρ −M θ(mρ −M) HereM is the total c.m.s. energy of two colliding objects (the dilepton mass), mρ=775 MeV is the in-vacuum ρ-meson mass. The first term corresponds to the Breit-Wigner cross section. According to the optical theorem it is proportional to the imaginary part of the forward scattering amplitude. The second term is proportional to ∆n where it is taken into account that the ratio of real to imaginary parts of Breit-Wigner amplitudes is ReF (M, 0o) ImF (M, 0o) m2ρ −M . (3) 1We consider only ρ-mesons here. To include other mesons, one should evaluate the corresponding sum of similar expressions. This term vanishes for M > mρ because only positive ∆n lead to the Che- renkov effect. Namely it describes the distribution of masses of Cherenkov states. In these formulas, one should take into account the in-medium mod- ification of the height of the peak and its width. In principle, one could consider mρ as a free in-medium parameter as well. We rely on experimental findings that its shift in the medium is small. All this asks for some dynamics to be known. In our approach, it is not determined. Therefore, first of all, we just fit the parameters A and Γ by describing the shape of the mass spec- trum at 0.75 < M < 0.9 GeV measured in [3] and shown in Fig. 1. In this way we avoid any strong influence of the φ-meson. Let us note that w is not used in this procedure. The values A=104 GeV3 and Γ = 0.354 GeV were obtained. The width of the in-medium peak is larger than the in-vacuum ρ-meson width equal to 150 MeV. Thus the low mass spectrum at M < mρ depends only on a single pa- rameter w which is determined by the relative role of Cherenkov effects and ordinary mechanism of resonance production. It is clearly seen from Eq. (2) that the role of the second term in the brackets increases for smaller masses M . The excess spectrum in the mass region from 0.4 GeV to 0.75 GeV has been fitted by w = 0.19. The slight downward shift about 40 MeV of the peak of the distribution compared with mρ may be estimated from Eq. (2) at these values of the parameters. This agrees with the above statement about small shift compared to mρ. The total mass spectrum (the dashed line) and its widened Breit-Wigner component (the solid line) according to Eq. (2) with the chosen parameters are shown in Fig. 1. The overall description of experimental points seems quite satisfactory. The contribution of Cherenkov gluons (the excess of the dashed line over the solid one) constitutes the no- ticeable part at low masses. The formula (2) must be valid in the vicinity of the resonance peak. Thus we use it for masses larger than 0.4 GeV only. The experimental data plotted in Fig. 1 have not been corrected for the acceptance of the experiment, which strongly depends on mass and transverse momentum of the muon pairs. However, due to an approximate cancellation between the variations of the thermal radiation mediated by the rho and those of the acceptance, the data as shown can roughly be interpreted as spectral function of the rho, averaged over momenta and the complete space- time evolution of nuclear collision [3]. To use these data without further corrections is therefore justified as long as the pT spectra of the radiation and those of the Cherenkov process are not dramatically different. From general principles one would expect slightly lower pT for low-mass dilepton pairs from coherent Cherenkov processes than for incoherent scattering. Qualitatively, this conclusion is supported by experiment [3]. The Cherenkov dominance region of masses from 400 MeV to 600 MeV below the ρ-resonance has softer Figure 1: Excess dilepton mass spectrum in semi-central In(158 AGeV)-In of NA60 (dots) compared to the in-medium ρ-meson peak with additional Cherenkov effect (dashed line). pT -distribution compared to the resonance region from 600 MeV to 900 MeV filled in by usual incoherent scattering. More accurate statements can be obtained after the microscopic theory of Cherenkov gluons developed. We should mention that the expression (2) may be applied for ∆n ≪ 1. The RHIC experiments revealed rather large ∆n ≈ 2. If the same values are typical at lower energies of SPS then the more general formulas (see [23]) should be used. The qualitative conclusions stay valid. Whether the in-medium Cherenkov gluonic effect is as strong as shown in Fig. 1 can be verified by measuring the angular distribution of the lepton pairs with different masses. The trigger-jet experiments similar to that at RHIC are necessary to check this prediction. One should measure the angles between the companion jet axis and the total momentum of the lepton pair. The Cherenkov pairs with masses between 0.4 GeV and 0.7 GeV should tend to fill in the rings around the jet axis. The angular radius θ of the ring is determined by the usual condition cos θ = as discussed in more detail in [22]. Another way to demonstrate it is to measure the average mass of lepton pairs as a function of their polar emission angle (pseudorapidity) with the companion jet direction chosen as z-axis. Some excess of low-mass pairs may be observed at the angle (4). Baryon-antibaryon effects can not possess signatures similar to these ones. In practice, these procedures can be quite complicated at comparatively low energies if the momenta of decay products are comparable to the trans- verse momentum of the resonance. It can be a hard task to pair leptons in reliable combinations. The Monte Carlo models could be of some help. In non-trigger experiments like that of NA60 there is another obstacle. Everything is averaged over directions of initial partons. Different partons are moving in different directions. The angle θ, measured from the direction of their initial momenta, is the same but the total angles are different, corre- spondingly. The averaging procedure would shift the maxima and give rise to more smooth distribution. Nevertheless, some indications on the substruc- ture with maxima at definite angles have been found at the same energies by CERES collaboration [27]. It is not clear yet if it can be ascribed to Che- renkov gluons. To recover a definite maximum, it would be better to detect a single parton jet, i.e. to have a trigger. The prediction of asymmetrical in-medium widening of any resonance at its low-mass side due to Cherenkov gluons is universal. This universality is definitely supported by experiment. Very clear signals of the excess on the low-mass sides of ρ, ω and φ mesons have been seen in [5, 6]. This effect for ω-meson is also studied in [8]. Slight asymmetry of φ-meson near 0.9 - 1 GeV is noticeable in the Fig. 1 shown above but the error bars are large there. We did not try to fit it just to deal with as small number of parameters as possible. There are some indications at RHIC (see Fig. 6 in [7]) on this effect for J/ψ-meson. At much higher energies one can expect better alignement of the mo- menta of initial partons. This would favour the direct observation of emitted by them rings in non-trigger experiments. The first cosmic ray event [26] with ring structure gives some hope that at LHC energies the initial partons are really more aligned and this effect can be found. The possible addi- tional signature at high energies could be the enlarged transverse momenta of particles within the ring. To conclude, the new mechanism is proposed for explanation of the low- mass excess of dilepton pairs observed in experiment. It is the Cherenkov gluon radiation which adds to the ordinary processes at the left wing of any resonance. Acknowledgments We thank S. Damjanovic for providing us with experimental data. I.D. is grateful to S. Damjanovic and H. Specht for very illuminating and fruitful discussions, in particular, on the role of the experimental acceptance. This work has been supported in part by the RFBR grants 06-02-16864, 06-02-17051. References [1] G. Agakichiev et al. (CERES), Phys. Rev. Lett. 75, 1272 (1995); Phys. Lett. B422, 405 (1998); Eur. Phys. J. C41, 475 (2005). [2] D. Adamova et al. (CERES), Phys. Rev Lett. 91, 042301 (2003); 96, 152301 (2006); nucl-ex/0611022. [3] R. Arnaldi et al. (NA60), Phys. Rev. Lett. 96, 162302 (2006); S. Dam- janovic et al. (NA60), Eur. Phys. J. C49, 235 (2007), nucl-ex/0609026 and Nucl. Phys. A783, 327 (2007), nucl-ex/0701015. [4] D. Trnka et al., Phys. Rev. Lett. 94, 192303 (2005). [5] M. Naruki et al. (KEK), Phys. Rev. Lett. 96, 092301 (2006), nucl-ex/0504016. [6] R. Muto et al. (KEK), Phys. Rev. Lett. 98, 042501 (2007), nucl-ex/0511019. [7] A. Kozlov (PHENIX), nucl-ex/0611025. [8] M. Kotulla (CBELSA/TAPS), nucl-ex/0609012. [9] I. Tserruya, Nucl. Phys. A774, 415 (2006). [10] R. Pisarski, Phys. Lett. B110, 155 (1982). [11] M. Harada and K. Yamawaki, Phys. 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0704.1082

Superconductivity and magnetic order in CeRhIn$_{5}$; spectra of coexistence

Superconductivity and magnetic order in CeRhIn5; spectra of coexistence. J.V. Alvarez and Felix Yndurain Departamento de F́ısica de la Materia Condensada, Universidad Autónoma de Madrid, 28049 Madrid, Spain (Dated: November 14, 2018) We discuss the fixed-point Hamiltonian and the spectrum of excitations of a quasi-bidimensional electronic system supporting simultaneously antiferromamagnetic ordering and superconductivity. The coexistence of these two order parameters in a single phase is possible because the magnetic order is linked to the formation of a metallic spin density wave, and its order parameter is not associated to a spectral gap but to an energy shift of the paramagnetic bands. This peculiarity entails several distinct features in the phase diagram and the spectral properties of the model, which may have been observed in CeRhIn5. Apart from the coexistence, we find an abrupt suppression of the spin density wave when the superconducting and magnetic ordering temperatures are equal. The divergence of the cyclotron mass extracted from de Haas-van Alphen experiments is also analyzed in the same framework. PACS numbers: 71.10.Pm,74.50.+r,71.20.Tx The interplay between magnetism and superconduc- tivity is a recurrent area of research in condensed matter physics. This interest has being activated in the last years due to the experimental findings of their coexis- tence in materials based on Ce, particularly in the 1-1- 5 CeMIn5 family [1] [2]. A prominent member of this family is CeRhIn5, which grows in tetragonal form, al- ternating CeIn3 and RhIn2 planes along the c crystallo- graphic axis. The structural anisotropy, induces quasi- bidimensionality in the electronic bonding and the Fermi surface, as evidenced in a series of de Haas-van Alphen measurements and band structure calculations [3][4][5]. At ambient pressure CeRhIn5 becomes an antiferromag- net (AFM) below TN ∼ 3.8K [1], with a small staggered magnetization aligned in the ab plane. Within the stan- dard Doniach’s Kondo lattice paradigm, applying pres- sure in a weak AFM heavy-fermion system opens a route to very interesting effects. As the pressure increases this theoretical scenario predicts: i) A reduction of TN due to Kondo compensation. ii) The eventual suppression of the AFM order in a quantum critical point (QCP), which alike other heavy-fermion compounds, would be respon- sible of the anomalies in the metallic phase. iii) The set- ting of unconventional superconductivity. However, all calorimetric [6], NQR [7], transport [8] and susceptibil- ity [9] measurements provide a consistent picture for the pressure-temperature phase diagram (presented schemat- ically in Fig. 1) in conflict with the aforementioned the- oretical scenario. Surprisingly, TN first increases with pressure and it only starts to decrease for pressures higher than 0.7GPa. Superconductivity shows up before TN has gone down to zero, i.e. AFM and SC coexist. Finally, AFM disappears abruptly at Pc = 1.9GPa exactly when TN = TSC , in a first order transition and before a QCP could have taken place. Despite the lack of a QCP in the pressure-temperature phase diagram, the metallic phase still might be understood in the framework of a quan- 0 0.5 1 1.5 2 2.5 3 P(GPa) T_N_Park T_SC_Park T_SC_Chen T_N_Chen T_N_Flouquet T_SC_Flouquet AFM+SC FIG. 1: Schematic experimental pressure-temperature phase diagram of CeRhIn5 at zero-magnetic field adapted from ref- erences [6], [9] and [11]. tum criticality if, changing another experimental knob, one could find a QCP nearby in the phase diagram. The natural choice is using a magnetic field to quench the superconductivity, and in that way, continue the point Pc(H = 0) into a line of first order transitions down to zero temperature, ending with a QCP at H∗c2. A new surprise appeared on this type of experiments [6, 10]. For pressures higher than Pc, the AFM reenters applying an in-plane magnetic field Hm < Hc2. Besides, trans- port measurements suggest [1, 8] that magnetic order in CeRhIn5 may not be associated to a gap in the single- particle spectrum. Actually, the resistivity as a function of the temperature does not show the conventional mini- mum characteristic of a metal-insulator transition at any temperature. The small anomaly observed in the resistiv- ity close to TN seems related to a change on the scattering mechanism when the AFM sets in. http://arxiv.org/abs/0704.1082v1 In vivid contrast with quantum critical and quasi-one- dimensional systems, the understanding of the individual AFM or SC states rely on simple but accurate mean-field theories. In this Letter, we propose that a basic compre- hension of the microscopic coexistence of AFM and SC and the first-order transition between these two phases can also be achieved within a mean-field scenario. We dis- cuss this phenomenon at the microscopic level in terms of a quasi-bidimensional model of interacting electrons proposed by one of us [12, 13, 14]. We will enumerate some implications of that model, discussing to which ex- tent can be related to the phenomenology observed in CeRhIn5. The model Hamiltonian contains the terms that nat- urally establish superconductivity and antiferromag- netism. H = Hk +HV +HU (1) ε(k)c kσckσ (2) (k+q)↑ (−k−q)↓ c−k↓ck↑ (3) k′−σckσck′−σ (4) Where U is a Hubbard on-site repulsion, and Vkq the effective interaction in the Cooper channel, which we will take to be attractive. Only s-wave pairing will be con- sidered throughout this work. Superconductivity with an order parameter having a symmetry related to Vkq is favored by HV but the presence of a Hubbard repul- sion establish a competition, which in terms of the gap is given by ∆ = ∆2 −∆1, such that: ∆2 = V −k−σ〉 (5) ∆1 = U −k−σ〉 (6) where the prime restricts the summation to states with an energy (measured from the FL) smaller than a cut-off energy Ec. For those states we assume a very weak mo- mentum dependence of Vkq . In the absence of magnetic order and setting a constant density of states at the FL (i.e. far from a logarithmic divergence) , the competition between terms favoring and disfavoring the SC is clearly shown in the McMillan-like formula for the critical tem- perature. TSC = 1.13Ec exp (V − U∗)D(EF ) where U∗ = U 1+UD(EF ) ln(W/Ec) FIG. 2: Calculated electronic structure of paramagnetic CeRhIn5. (a) Density of states and (b) band structure. The arrow indicates a saddle point singularity at the X point of the Brillouin zone. The inset shows the variation of the sad- dle point energy along the z R-X-R direction (it crosses the Fermi Level at approximately the (π/a, 0, 0.6π/c) point). An essential ingredient of the model is the quasipar- ticle dispersion relation ε(k) in the paramagnetic phase taken to set the Fermi level (FL) very close to the sad- dle point of a 2D system of itinerant electrons. Based on first principles calculations, Hall et al.[3] find at the FL a sharp peak in the Density of States characteris- tic of a 2D van Hove logarithmic singularity. We have also performed a full first principles calculation of the electronic structure of tetragonal paramagnetic CeRhIn5 using the VASP code [15]. The generalized gradient ap- proximation of Perdew et al.[16] for the exchange and correlation was adopted. The results for the experimen- tal lattice parameters [17] are reported in Figure 2. In Figure 2(a) we observe, like Hall et al.[3], a sharp peak in the density of states near the Fermi level. In addi- tion, in Figure 2(b) we display the band structure at the vicinity of the Fermi level. We find a saddle point very close to the Fermi level whose dispersion along the z- direction (R-X-R in the standard notation) is about 0.3 eV indicating the two-dimensional character of the sad- dle point singularity. Under these conditions we have, besides the Van Hove singularity in the density of states that favors superconductivity and magnetic order, an im- portant kinematic restriction, namely ε(k) = ε(k + Q), where Q = (π/a, π/a) is the vector connecting two equiv- alent saddle points within the Brillouin zone. The above restriction determines the gapless nature of the SDW like in the CDW case proposed by Rice and Scott [18]. The SDW order parameter is given by, γσ = U kσck+Q−σ〉 (8) with γσ = −γ−σ = γ. The resulting quadratic Hamil- tonian is solved by a Bogoliubov transformation and the SC and SDW order parameters obtained selfconsistently. The Hamiltionian eigenvalues are obtained by solving: εk − Ek −γ↑ −∆ 0 −γ↑ ε(k+Q) − Ek 0 −∆ −∆ 0 ε−k − Ek γ↓ 0 −∆ γ↓ ε−(k+Q) − Ek The four solutions are E1 = −E−, E2 = −E+, E3 = E−, E4 = E+ where E±(k) = ε(k)2 +∆2 ± γ. No- tice that for a 1D or 2D nested system ε(k) = −ε(k+Q) and therefore E±(k) = ± ε(k)2 +∆2 + γ2. The system is solved self-consistently for an effective half bandwidth at ambient pressure of W0 and the phase diagram for V = 4W0, Ec = 0.7W0 , n = 0.92 electrons and U = 2.25W0 , U = 2.50W0 and U = 2.75W0, is presented in Fig. 3. To simulate the effect of the pres- sure we have considered a linear variation of the band- width with the pressure and a bandwidth independent electron-electron interaction U. These assumptions have being found to be reasonable with a first principles calcu- lation using the SIESTA code [19] taking Ni as a bench- mark. Also, for simplicity, a constant V interaction and hole concentration independent of pressure are assumed. The results of our model agree qualitatively with the ex- perimental findings depicted in Fig. 1and Fig. 3(d). We indeed find a competition between SDW and SC but, as seen experimentally, they can coexist in a non negligi- ble region of the phase diagram. In addition, the SDW disappears abruptly when the two critical temperatures became equal, i.e., the SDW transition temperature can not be lower than the superconducting one. This numeri- cal result is similar to the analytical finding by Bilbro and McMillan [20] concerning superconductivity and marten- sitic transformation in A15 compounds. The proximity of the FL to a saddle point is an ingredi- ent of model (1) necessary for the formation of the metal- lic SDW. We ask ourselves whether in CeRhIn5 there is experimental evidence for such a proximity. A recent de Haas-van Alphen study [4] shows a divergence in the cy- clotron mass at a pressure Pc2 ∼ 2.4GPa, accompanied with a change in the quasi-2D Fermi surface. Those ex- periments were performed for values of magnetic fields and pressures in which the system is AFM. Following Park et al. Pc2 is very close to the pressure at which TN would extrapolate to zero in absence of SC (see upper inset in Fig. 4 ). The cyclotron mass in a 2D elec- tronic system mc = h̄ 2/2π(∂A(E)/∂E) where A is the 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 0 5 10 15 20 25 (a) (b) (c) (d) FIG. 3: Phase diagram (temperature versus pressure) ob- tained using the model Hamiltonian (1) and a two dimensional band structure with the FL close to a saddle point. The vari- ables are in units of the half bandwidth W0. TN and TSC stand for the SDW and SC critical temperatures respectively. Panels (a), (b) and (c) are the calculated results for the pa- rameters given in the text and U = 2.25W0 , U = 2.50W0 and U = 2.75W0 respectively. The inset in panel (c) indicates the metallic DOS in the three (SDW, SDW+SC and SC) different regimes. The shaded area indicates occupied levels. Panel (d) represents the experimental results of ref. [9] area enclosed in the isoenergetic contour line E(k) = E. Close to the 2D Van Hove singularity one expects the cyclotron mass to diverge logarithmically. Actually, the precise functional form has been computed in Ref. [21] and found to be mc = mc0 C +D ln |EF − Evhs| In a tetragonal crystal structure C and D are numbers nearly independent of the pressure and mc0 is the cy- clotron mass at the bottom of the band. The divergence is driven by the denominator in the argument of the log- arithm which in our model is: EF (γ) − Evhs ∼ γ(P ) ∼ TN(P ). In other words, in model (1), mc is enhanced as the AFM vanishes because the FL in the SDW ap- proaches the saddle point at X (see the inset in Fig. 4). To elucidate if the experimental results are compatible with this argument and with expression (9), we have ex- tracted the pressure dependence of TN , fitting the ex- perimental data in Ref ([6]) (see upper inset in Fig. 4 to a cubic polynomial law in (Pc2 − P ) in the range of pressures between P = 0.65 and P = 2.4 GPa. We do not attempt to justify physically this fit here, since our only goal is extracting an analytic expression for TN (P ) for the range of pressures of interest, to insert it in (9). The results are presented in the main panel of Fig. 4. Our model reproduces reasonably well these experimen- tal findings. Within this model, we expect an anomaly in the spe- cific heat at TN , which is not associated to a SDW spec- 0 0.5 1 1.5 2 2.5 3 P(Gpa) 0 0.5 1 1.5 2 2.5 3 P(Gpa) FIG. 4: Cyclotron mass (mc) as a function of pressure. Tri- angles are the experimental values from Ref. [4] showing the mc divergence close to P ∼ 2.4 GPa, where the magnetic order disappears. The solid line is a fit to the theoretical model. Close to a Van Hove singularity the cyclotron mass diverges logarithmically as the difference between EF and the Evhs vanishes (see lower inset). According to the model this energy difference is proportional to TN and its pressure de- pendence can be extracted, for instance, from Ref. [6](see upper inset). tral gap but to the entropy released when the magnetic order disappears. The electronic part of the specific heat Ce−V = −2β Ei(k) ∂Ei(k) × (10) Ei(k) + ε(k)2 +∆2 where i=3,4. The second term inside the parenthesis gives the SC anomaly at TC and the third term gives the antiferromagnetic anomaly at TN . The SC anomaly is much weaker in the coexisting phase, because the FL lies in a depression of the density of states (see central graph on the panel (c) of Fig. 3) created by the under- lying SDW, while in the purely SC phase the FL is very close to a divergence in the density of states (right graph). Remarkably, this enhancement has been also observed in calorimetric measurements on CeRhIn5 [6] To summarize: beyond the detailed boundary shape in the phase diagram of CeRhIn5 we have identified two unequivocal, clear-cut features of the phenomenology of CeRhIn5 in the discussed model, namely; coexistence and abrupt disappearance of AFM when TN = TSC . The es- sential ingredient in the model is the metallic SDW, fa- vored for the proximity of Fermi level to a Van Hove logarithmic singularity in the density of states. The gapless nature of the SDW implies the lack of a metal- insulator transition at, or close to, TN as shown by resis- tivity measurements. The kinematic conditions needed for the metallic SDW to be formed seem to be present in CeRhIn5 as shown by the logarithmic divergence of the cyclotron mass. We appreciate very much discussions with G. Gomez- Santos, H. Suderow and S. Vieira. Financial support of the Spanish Ministry of Science ( Ramon y Cajal contract and Grant BFM2003-03372) is acknowledged. [1] H. Hegger, C. Petrovic, E.G. Moshopoulou, M.F. Hund- ley, J.L. Sarrao, Z. Fisk and J.D. Thompson, Phys. Rev. Lett. 84, 4986, (2000). [2] M. Nicklas, V.A. Sidorov, H. A. Borges, P.G. Pagliuso, J.L. Sarrao and J.D. Thompson, Phys. Rev. B 70, 020505(R) (2004). [3] D. Hall, E. Palm, T. Murphy, S.W. Tozer, C. Petrovic, E. Miller-Ricci, L. Peabody, CharisQayHuei Li, U. Alver, R. G. Goodrich, J. L. Sarrao, P.G. Pagliuso, J. M. Wills, Z. Fisk. Phys. Rev. B 64, 064506 (2001). [4] H. Shishido, R. Settai, H. Harima, Y.Ōnuki, J. Phys. Soc. Jpn. 74, 1103 (2005). [5] S. Fujimori et al. Phys. Rev. B 67, 144507 (2003). [6] G. Knebel, D. Aoki, D. Braithwaite, B. Salce and J. Flouquet,cond-mat 0512078. [7] T. Mito, S. Kawasaki, Y. Kawasaki, G.-q. Zheng, Y. Ki- taoka, D. Aoki, Y. Haga and Y.Ōnuki, Phys. Rev. Lett. 90, 077004, (2003). [8] A. Llobet, J.S. Gardner, E.G. Moshopoulou, J.M. Mignot, M. Nicklas, W. Bao, N.O. Moreno, P.G. Pagliuso, I.N. Goncharenko, J.L. Sarrao, J.D. Thomp- son, Phys. Rev. B 69, 024403 (2004). [9] G. F. Chen, K. Matsubayashi, S. Ban, K. Deguchi, N.K. Sato, Phys. Rev. Lett. 97, 017005, (2006). [10] T. Park, F. Ronning, H.Q. Yang, M.B. Salamon, R. Movshovich, J.L.Sarrao, J.D.Thompsom, Nature 440, 65, (2006). [11] T.Park, J.L. Sarrao and J.D. Thompson, J. of Magn. and Magn. Mater. (in press). [12] F. Yndurain, Solid St. Commun. 81, 939 (1991). [13] F. Yndurain, Phys. Rev. B. 51, 8494 (1995). [14] F. Yndurain, in New trends in Magnetism, Magnetic Ma- terials and their Applications. Plenum Press, New York, (1994). [15] G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169 (1996). [16] J.P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). [17] E. G. Moshopoulou, Z. Fisk, J. L. Sarrao and J.D. Thompson, J. Solid Stat. Chem. , 158 25 (2001). [18] T.M. Rice and G. K. Scott, Phys. Rev. Lett. 35, 120 (1975). [19] J.M. Soler, E. Artacho, J.D.Gale, A.Garcia, J.Junquera, P.Ordejon and D.Sanchez-Portal, J. Phys. Condens. Mat- ter 14, 2745 (2002). [20] G. Bilbro and W.L. McMillan, Phys. Rev. B 14, 1887 (1976). [21] M.A. Itskovsky and T. Maniv, Phys. Rev. B 72, 075124 (2005). http://arxiv.org/abs/cond-mat/0512078

0704.1084

Vacuum Energy and Renormalization on the Edge

arXiv:0704.1084v1 [hep-th] 9 Apr 2007 Vacuum Energy and Renormalization on the Edge M. Asorey1, D. Garćıa-Álvarez2, J. M. Muñoz-Castañeda1 1 Departamento de F́ısica Teórica. Facultad de Ciencias. Universidad de Zaragoza, 50009 Zaragoza. Spain 2 Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom E-mail: [email protected] Abstract. The vacuum dependence on boundary conditions in quantum field theories is analysed from a very general viewpoint. From this perspective the renormalization prescriptions not only imply the renormalization of the couplings of the theory in the bulk but also the appearance of a flow in the space of boundary conditions. For regular boundaries this flow has a large variety of fixed points and no cyclic orbit. The family of fixed points includes Neumann and Dirichlet boundary conditions. In one-dimensional field theories pseudoperiodic and quasiperiodic boundary conditions are also RG fixed points. Under these conditions massless bosonic free field theories are conformally invariant. Among all fixed points only Neumann boundary conditions are infrared stable fixed points. All other conformal invariant boundary conditions become unstable under some relevant perturbations. In finite volumes we analyse the dependence of the vacuum energy along the trajectories of the renormalization group flow providing an interesting framework for dark energy evolution. On the contrary, the renormalization group flow on the boundary does not affect the leading behaviour of the entanglement entropy of the vacuum in one-dimensional conformally invariant bosonic theories. PACS numbers: 11.10.Hi,11.25.Hf Keywords: Renormalization group, boundary conditions, vacuum energy. http://arxiv.org/abs/0704.1084v1 Vacuum Energy and Renormalization on the Edge 2 1. Introduction The emergence of the dark energy as one of the basic ingredients of the current standard cosmological scenario, and the absence of an even vague understanding of its possible origin, opens a window to the analysis of all possible mechanisms that generate background energy (see e.g. [1] for a review of recent proposals). The main problem is that the apparent value of the dark energy is very tiny compared with any physical energy scale. A second problem is that in a generic quantum field theory there is generation of vacuum energy and any renormalization prescription requires a fine tuning, which is not very convincing without the quantisation of the gravitational interaction. The guess that dark energy might change with the evolution of the Universe can be understood even if dark energy is just vacuum energy. The finite corrections due to finite size of the causal Hubble domain decrease as the Universe continues to expand. The aim of this paper is to analyse the variation of these finite size corrections under of renormalization group on the space of boundary conditions for scalar field theories in flat space, although the results are generalisable for more general backgrounds. The dependence of the vacuum energy on the boundary conditions [2] is well known since the discovery of Casimir effect [3] (see [4, 5] and [6] and references therein for recent revisions). However, boundaries might also be considered as a source of new, although peculiar, interactions and therefore can undergo renormalization [7, 8]. The renormalization of boundary conditions modifies the critical behaviour of the theory [9, 10, 11]. In systems with boundaries or defects, the boundary RG flow induces a dynamical behaviour on the boundaries. The dynamics of D-branes in string theory emerges in this way [12]. The renormalization group flow is analysed from a global viewpoint in the most general framework for boundary conditions of scalar field theories introduced in Ref. [13]. In particular, we consider the possible existence of topological transitions [14] induced by the renormalization of boundary conditions or cyclic orbits in the boundary RG flow [15]. The dependence of the finite size corrections to the vacuum energy and vacuum entanglement entropy [16, 17] under the boundary RG flow is analysed from a very general perspective. 2. Boundary conditions in Field Theory The action which governs the dynamics of scalar field theory in a bounded domain Ω of flat space consists of two different of terms, S(φ) = SB(φ) + Sb(φ). The first one SB(φ) = g dDx |φ̇|2 − |∇φ|2 − V (|φ|2) is defined in terms of the values of the fields in the bulk. The second term Sb(ϕ) = dD−1x |ϕ̇|2+ ϕ∗∂nϕ+ ∗)ϕ−|∇ϕ|2 Vacuum Energy and Renormalization on the Edge 3 depends only on the values of the fields at the boundary ∂Ω † . g denotes the metric induced on the boundary by the bulk flat metric, and ∂n is the normal derivative at the boundary ϕ = φ| ∂nϕ = ∂nφ|∂Ω. (3) The presence of the boundary term Sb allows the generation of local classical equations of motion without requiring any specific type of boundary conditions [19, 20]. Indeed, the gradient term V = 1 |∇φ|2 (4) can be rewritten as φ†∆φ+ φ† ∂nφ (5) where ∆ is the Laplace-Beltrami operator ∆ = − ∂µ∂µ . In the quantum theory the Laplace-Beltrami operator must have a real spectrum in order to have a selfadjoint Hamiltonian H = 1 ∆+m2. (6) for the free field theory (The inclusion of interactions does not changes the picture [21]). This means that the classical fields must satisfy boundary conditions which make the operator ∆ selfadjoint. The complete set of boundary conditions which satisfy this requirement [13] are in one-to-one correspondence with the group of unitary operators of the boundary Hilbert space L2(∂Ω, C ). For any unitary operator U ∈ L2(∂Ω, C ), the fields satisfying the boundary condition ϕ− i ∂nϕ = U (ϕ+ i ∂nϕ) (7) define a domain where ∆ is a selfadjoint operator. In the case of open strings, the corresponding conformal 1+1 dimensional scalar field theories is defined on the space interval Ω = [0, 1] ⊂ IR and there is a large variety of admissible boundary conditions described by the unitary group M = U(2). The unitary matrices define Dirichlet, Neumann and periodic boundary conditions, which in string theory correspond to a string attached to a D-brane background, free open and closed string theories, respectively. For higher N-dimensional target spaces, or N-component strings, the set of boundary conditions becomes M = U(2N) which includes matrices which interpolate between one single closed string or N disconnected strings [13]. The topology change is described in this picture by a simple change of boundary conditions in L2(∂Ω, C N) [14]. † We will assume that the boundary is regular and smooth. See e.g. [18] for the peculiarities associated to the presence of irregular boundaries Vacuum Energy and Renormalization on the Edge 4 If the spectrum of eigenvalues of the unitary operator U does not include the value ±1 (i.e. ±1 /∈ SpU) the boundary condition (7) can be rewritten as ∂nϕ = −i ϕ (9) which means that only the boundary values of the fields at the boundary can have an arbitrary value ϕ whereas its normal derivative is determined by U and ϕ. The corresponding operator mappings from unitary into selfadjoint operators A± = −i are the celebrated Cayley transforms. The inverse Cayley transform I∓ iA± I± iA± recovers the unitary operator U from their selfadjoint Cayley transforms A±. The condition of ∆ being selfadjoint is necessary but not sufficient to guarantee the unitarity of the corresponding quantum field theory. Indeed, in the case of free field theory the Hamiltonian (6) must be selfadjoint. This requires that the spectrum of ∆ +m2 must be not only real but also positive which restricts the set of admissible boundary conditions to a subset M of L2(∂Ω, C ). Because of the existence of the boundary term in (5) the Hamiltonian H (6) is not selfadjoint if the spectrum of the unitary operator U intersects the following domain of phase factors S1m = {e2αi;−π < α ≤ π, 0 < α < − arctan m2, or < −α < π − arctan m2 }. In any other case, −m2 is a lower bound for the spectrum of the operator ∆ and H is selfadjoint. One possible source of unitarity loss is the existence of edge estates with large negative eigenvalues of operator ∆. The consistency of the quantum field theory imposes, thus, a very stringent condition on the type of acceptable boundary conditions, even in the case of massive theories in order to prevent this type of pathological behaviour of vacuum energy. For real scalar fields there is a further condition. U has to satisfy a CP symmetry preserving condition U † = U∗, U = UT . (12) The usual Neumann and Dirichlet boundary conditions U = ±I satisfy this condition. In general, for BT A2 the condition requires that A1 = A 1 , A2 = A 2 , A1B ∗ +BA 2 = 0 (14) BB† + A1A 1 = I, A2A TB∗ = I (15) Vacuum Energy and Renormalization on the Edge 5 In particular, the quasi-periodic condition ϕ(L) = M−1ϕ(0), ∂nϕ(L) = M∂nϕ(0) is also compatible if M = M t = M∗. In the case of one single real massless scalar the set of compatible boundary conditions has two connected components: M0 given by the operators of the form Uβ = cos β I+ i sin β σy, and M1 given by Uα = cosα σz + sinα σx . (16) M0 includes Neumann (β = 0) and Dirichlet (β = π) conditions; and M1 contains the quasi-periodic boundary conditions ϕ(L) = tan ϕ(0); ∂nϕ(L) = ∂nϕ(0) (17) which include periodic (α = π ) and antiperiodic (α = −π ) boundary conditions. 3. Boundary Conditions and Renormalization Group Since boundary conditions appear more naturally in the Schrödinger picture of field theory and the theory is plagued of ultraviolet singularities some doubts were raised about their relevance for the quantum field theory. The pioneer work of Symanzik [21] confirmed the consistence of the standard picture even in presence of bulk renormalizable interactions (see [22] for an explanation of a recent controversy [23]). Moreover, there is a renormalization of the very boundary conditions because the boundary terms are the source of new interactions. The renormalization group can be defined in the continuum approach by = Λ[φ][φ(x)− ξΛ(x)] (18) by means of a fluctuating field ξΛ with short range fluctuations of order . This implies that the boundary condition ∂nϕ = Aϕ (19) is renormalised to ∂nϕΛ = AΛϕΛ, (20) since = Λ[φ]+1[∂nφ(xb)− ∂nξΛ(xb)] = AΛ[φ]+1φ(xb) = AΛφΛ with AΛ = ΛA. For more general boundary conditions the continuum renormalization group is given by Λ∂ΛUΛ = Λ − UΛ t ∂tUt = t − Ut Vacuum Energy and Renormalization on the Edge 6 for Λ = Λ0 e t. Fixed points correspond, therefore, to self-adjoint boundary conditions U † = U . In particular, Dirichlet and Neumann (U = ∓ I) are renormalization group fixed points. For mixed boundary conditions the RG flows from Dirichlet (UV) toward Neumann (IR) conditions. U = e2i arctan e I. (24) Critical exponents can be identified with the eigenvalues of the matrix Uc at the fixed points. Since Uc is also hermitian all critical exponents are either 1 or −1 and there is no room for cyclic orbits. It is well known, however, that some quantum systems with singular boundaries and singular interactions [15, 24] exhibit cyclic renormalization group flows. Moreover, some topological field theories (e.g. Russian doll models) present a similar behaviour [25]. In scalar field theories, this phenomenon simply does not occur for regular boundaries. For the same reasons topological transitions do not occur for finite scale transformations since the flip of eigenvalues from −1 to +1 requires a change in the parameter t of the flow from −∞ to ∞ as in (24)). 4. Conformal Invariance and boundary conditions In 1+1 dimensions the theory of massless scalar fields is formally conformal invariant. However, boundary conditions might break this symmetry [9, 10, 11]. Conformal invariance is only preserved if the boundary conditions are stable under the boundary renormalization group flow. The fixed points can easily be identified. For a complex scalar field, besides the above mentioned fixed points, which correspond to Dirichlet, Neumann and pseudo-periodic boundary conditions and obviously are conformal invariant, there are fixed points corresponding to quasi-periodic boundary conditions (17). They also preserve the conformal symmetry. In 1+1 dimensions this exhausts the whole set of conformal invariant boundary conditions. Any other boundary condition flows toward one of these fixed points. The most stable fixed point corresponds to Neumann conditions because all its critical exponents are +1. The most unstable is that of Dirichlet conditions since all critical exponents −1. This is compatible with the fact that the neighbourhood of Dirichlet boundary conditions is plagued of singularities Periodic, quasi-periodic and pseudo-periodic fixed points present relevant and irrelevant perturbations with critical exponents ±1, respectively. Negative values label the possible instabilities. Implications of these results for string theory are well known. Periodic boundary conditions, appear as attractors of systems with quasi-periodic and pseudo-periodic conditions which stresses the stability of closed string theory vacuum. For open strings the (stable) attractor points are standard free strings (Neumann). Any other boundary condition flow toward one of those fixed points. Notice that the absence of topological transitions in the boundary renormalization group flow is a consequence of the fact that all relevant perturbations are always associated with -1 critical exponents. Vacuum Energy and Renormalization on the Edge 7 In higher dimensions (D > 1) conformal invariance requires, even in the massless case m = 0, that Neumann boundary conditions have to be modified in order to preserve conformal invariance with a term ∂nϕ = D − 1 K ϕ, (25) proportional to the extrinsic curvature K of the boundary. In the case of singular boundaries some more interesting boundary renormalization group flows arise (see e.g. [18] for a review): fixed points and cyclic orbits of the boundary renormalization group flow can appear [15, 25, 24] and conformal invariance can be partially broken to a discrete subgroup Z [24]. 5. Vacuum energy and boundary conditions The infrared properties of quantum field theory are very sensitive to boundary conditions [26]. In particular, physical properties of the quantum vacuum state like the vacuum energy may exhibit a very strong dependence on the type of boundary conditions. This can be explicitly shown in the simple case of a massless field defined on a finite one-dimensional interval [0, L]. For pseudoperiodic boundary conditions defined by the unitary operator Uθ = cos θ σx − sin θ σy : ϕ(L) = eiθϕ(0) (26) the Casimir vacuum energy (see e.g. Ref. [5] and references therein) is given by +n− 1  (27) The vacuum energy dependence on θ is in this case relatively smooth. The only cuspidal point at θ = 0 corresponds to periodic boundary conditions. A completely regular behaviour is obtained for Robin boundary conditions U = e2αiI : ∂nϕ(0) = tanαϕ(0), ∂nϕ(L) = tanαϕ(L) (28) which smoothly interpolate between Dirichlet (α = π ) and Neumann (α = π) conditions when α is restricted to the interval α ∈ [π , π] [27, 28, 29] . Finally, the Casimir energy for quasi-periodic boundary conditions [30] is also dependent on the choice of the parameter α. Two particularly interesting cases are given by α = 0, UZ = σz; ϕ(L) = 0, ∂nϕ(0) = 0 and α = π, U ′Z = σz; ϕ(0) = 0, ∂nϕ(L) = 0 which correspond to a Zaremba (mixed) boundary conditions: one boundary is Dirichlet and the other Neumann. Vacuum Energy and Renormalization on the Edge 8 6. Vacuum entanglement entropy The dependence of vacuum energy on boundary conditions seems to suggest that many other observables may suffer the same effect. In particular, one may wonder whether or not the entropy of the system is dependent on the type of conditions that constrain the values of the fields at the boundary. The entropy of the field theory at finite temperature scales with the volume of physical space. Only in quantum gravity or string theory the entropy can scale with the area of black hole horizon. However, in field theory it is possible Figure 1. Information loss by integration over the fluctuations of the fields inside the domain ω to generate a mixed state from the pure vacuum state Ψ0 by integrating out the fluctuating modes in a bounded domain ω of the physical space Ω (see Figure 1) Ψ∗0Ψ0. (30) The entropy of this state Sω = −Tr ρω log ρω, although ultravioletly divergent, provides a measure of the degree of entanglement of the vacuum state. In the case of a free massless real scalar field theory in one-dimensional spaces (D = 1) this entropy scales logarithmically with the size lω of ω and the ultraviolet cut-off ǫ introduced to split apart the domain ω and its complement Ω\ω , (31) and in D = 2 dimensions it scales linearly with the perimeter Rω of ω Sω = c2 − γ (32) and in D > 2 dimensions as the volume of the boundary of ω Sω = cDVωǫ 1−D. (33) In particular in three-dimensional spaces it scales with the area of the boundary of ω like in the presence of a blackhole [16, 17]. Although the coefficients of the leading terms c2, cD in (32) and (33) have been explicitly computed, they are not universal because they obviously depend on the choice of the UV cutoff ǫ. On the contrary, the coefficient Vacuum Energy and Renormalization on the Edge 9 c1 = 1/3 of the logarithmic term in (31) is universal and does coincide with one third of the central charge of the corresponding conformal field theory. Similarly, the finite γ term in (32) is also universal in D = 2 dimension and is related to a degree of topological entanglement [31]. It is remarkable that in D = 1 the coefficient c1 = 1/3 is also independent of the choice of boundary condition in Ω. This in contrast with what happens for the finite size corrections to vacuum energy. The coefficient of the 1/L term is also proportional to the central charge but in that case the corresponding factor is very sensitive to the type of boundary conditions imposed at the boundary of Ω. The above results indicate that whereas the Casimir energy is closely related with the infrared properties of the conformal theory which are sensitive to the boundary conditions, the entanglement entropy is rather associated to the behaviour at the interface between ω and its complement Ω\ω which do not depend on the choice of boundary conditions at the edge of the physical space. 7. Conclusions The description of regular boundary conditions in terms of unitary matrices provides a very useful framework for the description of the boundary renormalization group flow and the breaking of conformal invariance due to boundary effects. Neumann conditions turn out to be the only boundary conditions which are absolutely stable under RG flow. All other boundary conditions may have some relevant perturbations which are the source of RG instabilities. However, the global structure of the flow does not permit topological transitions. The finite size corrections to vacuum energy are very sensitive to the choice of boundary conditions which discriminate between the different fixed points of the renormalization group flow. On the contrary, the leading contribution to entanglement entropy of the vacuum is insensitive, for one-dimensional massless scalar field theories, to the change of boundary conditions. In D=2 dimensions the same property holds for the finite correction to the entanglement entropy of massless scalar theories. This fact, is very relevant for the implementation of quantum codes with topological stability [31]. However, these properties do not hold for the leading terms contributing to the entanglement entropy. Acknowledgements We thank E. Elizalde, J.G. Esteve, S. Odintsov and G. Sierra for interesting discussions on closely related subjects. This work is partially supported by CICYT (grant FPA2004-02948) and DGIID-DGA (grant2006-E24/2). Vacuum Energy and Renormalization on the Edge 10 References [1] T. Padmanabhan, Phys. Rept. 380 (2003) 235 [2] D. Deutsch, P. Candelas, Phys. Rev. D20 (1979) 3063 [3] H. B. G. Casimir, Proc. K. ned. Akad. Wet. 51(1948) 793 [4] K. A. Milton, The Casimir Effect: Physical Manifestations of the Zero Point Energy, World Sci., Singapore (2001) [5] M. Bordag, U. Mohideen, V. M. Mostepanenko, Phys. Rep. 353 (2002) 1 [6] M. Asorey, D. Garćıa-Alvarez and J.M. Muñoz-Castañeda, J. Phys. A 39 (2006) 6127 [7] I. G. Moss, Class. Quant. Grav. 6 (1989) 759 [8] S. D. Odintsov, Class. Quant. Grav., 7(1990)445 [9] I. Affleck, Nucl. Phys. (Proc. Suppl.) B58 (1997) 35 [10] J. B. Zuber and V. B. Petkova, Arxive preprint [hep-th/0103007] [11] J. Cardy, In Encyclopedia of Mathematical Physics, Eds. J.-P. Françoise, G. L. Naber and T. S. Tsun, Academic Press (2006) [12] J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724 [13] M. Asorey, A. Ibort and G. Marmo, Int. J. Mod. Phys. A 20 (2005) 1001 [14] A. P. Balachandran, G. Bimonte, G. Marmo and A. Simoni, Nucl. Phys. B 446 (1995) 299 [15] S. D. Glazek and K. G. Wilson, Phys. Rev. Lett. 89 (2002) 230401 [16] L. Bombelli, R.K. Koul, J. Lee and R. Sorkin, Phys. Rev. D 34 (1986) 373 [17] M. Srednicki, Phys. Rev. Lett. 71 (1993) 666 [18] I. Tsutsui and T. Fülöp, Arxive preprint [quant-ph/0312028] [19] M. Bordag, H. Falomir, E. M. Santangelo and D. V. Vassilevich, Phys.Rev. D65 (2002) 064032 [20] A. A. Saharian, Phys. Rev. D69 (2004) 085005 [21] K. Symanzik, Nucl. Phys. B190 (1981) 1 [22] E. Elizalde, J. Phys. A 36 (2003) L567 [23] N.Graham, R.L.Jaffe, V.Khemani, M.Quandt, O. Schroeder and H.Weigel, Nucl.Phys. B677 (2004) 379-404 [24] M. Asorey, J.G. Esteve and G. Sierra, In preparation (2006) [25] A. LeClair, J. M. Roman and G. Sierra, Phys. Rev. B69 (2004) 20505 [26] M.Asorey, J. Geom. Phys. 11(1993)94 [27] A. Romeo and A. A. Saharian, J. Phys. A35 (2002) 1297 [28] L. C. Albuquerque, R. M. Cavalcanti, J.Phys. A37 (2004) 7039 [29] B. Mintz, C. Farina, P. A. Maia Neto and R. B. Rodrigues, J. Phys. A39 (2006) 11325 [30] M. Asorey, unpublished (1994) [31] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96 (2006) 110404

0704.1085

Gauge Mediation in String Theory

arXiv:0704.1085v2 [hep-th] 11 Apr 2007 CALT-68-2642 UT-07-12 Gauge Mediation in String Theory Teruhiko Kawano,1 Hirosi Ooguri,1,2 and Yutaka Ookouchi2 1Department of Physics, University of Tokyo, Tokyo 113-0033, Japan 2California Institute of Technology, Pasadena, CA 91125, USA Abstract We show that a large class of phenomenologically viable models for gauge mediation of supersymmetry breaking based on meta-stable vacua can be realized in local Calabi-Yau compactifications of string theory. April, 2007 http://arxiv.org/abs/0704.1085v2 1. Introduction The use of meta-stable vacua in supersymmetric model building has attracted much attention lately, especially after the discovery [1] that generic supersymmetric field theories in four dimensions such as the supersymmetric QCD with massive flavors have meta- stable vacua with broken supersymmetry. In [2], realistic models of direct mediation were constructed using superpotentials without U(1)R symmetry. Though explicit breaking of the U(1)R symmetry generates meta-stable vacua, there is a range of parameters where one can make them sufficiently long lived, while satisfying the phenomenological constraints on the masses of the gauginos, the gravitino, and the scalars without artificially elaborate constructions. The models can also avoid producing Landau poles in standard model gauge interactions below the unification scale. Recently beautiful realizations of these models in string theory, including a natural mechanism to generate small parameters of these models, were found in [3]. Gauge mediation models were also constructed using meta-stable vacua with similar phenomenological benefits [4,5]. Related ideas have been explored in [6-10]. Accepting the possibility that our universe may be in a meta-stable state allows us to circumvent the theoretical constraints due to the Nelson-Seiberg theorem on R-symmetry [11] and the Witten index [12] and gives us greater flexibility in model building, as emphasized in [13] among other recent papers. Among the models constructed recently based on meta-stable vacua, the ones dis- cussed in [4] are particularly simple. In this paper, we will show that they have ultra-violet completions in supersymmetric quiver gauge theories which can be realized in string com- pactifications. Moreover, our construction can be naturally generalized to a large class of quiver gauge theories, providing a basis for the speculation in [4] that “gauge mediation may be a rather generic phenomenon in the landscape of possible supersymmetric theo- ries.” In this paper, we will demonstrate the idea by explicitly working out one example: a model based on type IIB superstrings compactified on the A4-fibered geometry [14]. We will also give an outline of generalizations of this construction to a large class of quiver gauge theories. Detailed analysis of meta-stable vacua in these models will be given in a separate paper [15]. 2. The Model The model we will consider in this paper is realized in string theory compactified on the local Calabi-Yau manifold described by the equation, x2 + y2 + (z + ti(w)) = 0, ti(w) = 0, ti(w)− ti+1(w) = µi(w − xi), (x, y, z, w) ∈C (2.1) Since ti’s are functions of w, this gives the A4 singularity fibered over w ∈C. In particular, there exist four two-cycles S2 on which D branes can be wrapped. The low energy limit of D5 branes wrapping the two-cycles S2 and extending along the four uncompactified dimensions is the A4 quiver gauge theory with the gauge group U(N1)×U(N2)×U(N3)× U(N4) with the adjoint chiral multiplets Xi=1,2,3,4 for the four gauge group factors and the bi-fundamental chiral multiplets (Q12, Q21), (Q23, Q32), and (Q34, Q43). This quiver gauge theory can also be realized on intersecting brane configuration with NS5 and D4 branes, as expected from the T-duality between the An singularity and NS5 branes [16]. N 1 N 2 N 3 N 4 Q 12,21 Q 23,32 Q 34,43 X1 X2 X3 X 4 Fig. 1: A4 quiver diagram From the Calabi-Yau singularity (2.1), one can read off the superpotential of the quiver theory as [17,18] WA4 = tr (Qi+1,iXiQi,i+1 −Qi,i+1Xi+1Qi+1,i) + (Xi − xi) 2. (2.2) Note that the dimensionful parameters µi and xi are the moduli of the Calabi-Yau manifold given by (2.1), namely they are closed string moduli. The dynamical scales Λi=1,···,4 of the four gauge group factors are also closed string moduli, related to the sizes of the S2’s. These closed string moduli are frozen and can be regarded as parameters of the low energy theory. Let us suppose that µi are sufficiently larger than Λi so that we can integrate out all the adjoints Xi to obtain the effective superpotential Weff = mitrQi i+1Qi+1 i − tr (Qi i+1Qi+1 i) trQ21Q12Q23Q32 + trQ32Q23Q34Q43, (2.3) where mi = ci − ci+1, µ̃i = 2µiµi+1 µi + µi+1 (i = 1, 2, 3). This quiver gauge theory can be used as a gauge mediation model as follows. We identify the bi-fundamentals (Q34, Q43) as messenger fields. One way to incorporate the standard model sector would be to identify a subgroup of the U(N4) gauge group with the standard model gauge group or a GUT gauge group. Alternatively, we can replace the 4th node of the quiver diagram of Fig. 1 carrying the U(N4) gauge group with a string theory construction of the standard model. For example, if the standard model is realized on intersecting branes, messengers can be open strings connecting the 3rd node carrying the U(N3) gauge group to the standard model branes. 1 In the following, we will denote the bi-fundamental fields (Q34, Q43) as (f, f̃) to distinguish them from the rest of the quiver gauge theory and to emphasize their role as the messengers. The rest of the quiver gauge theory is treated as a hidden sector, where supersymmetry is broken dynamically. To use the result of [1], let us assume that the ranks of the gauge group factors satisfy N2 + 1 ≤ N1 +N3 < N2 (2.4) and that Λ1,Λ3,Λ4 ≪ Λ2 ≪ µi. In this case, one can identify the gauge group SU(Nc) of the model of [1] with SU(N2) ⊂ U(N2) of the quiver theory. Since the metastable vacuum can be found near the origin of the meson fields M11 ∼ Q12Q21, M33 ∼ Q32Q23, the terms tr (Q12Q21) , tr (Q32Q23) tr (Q12Q21Q32Q23) in the superpotential (2.3) are irrelevant in our discussion below, if the masses µi of the adjoints satisfy the following bounds [4,5], µ̃1,2 ≤ min m1,2Λ2, . (2.5) In this range of the parameters, the hidden sector and its interaction with the mes- senger sector is described by the superpotential, W = mtrQ12Q21 +mtrQ32Q23 + trQ32Q23f f̃ +m3tr f f̃ − . (2.6) 1 In this case, we can still use the effective potential (2.3) to describe the interaction of the messengers and the hidden sector, but we should set 1/µ̃4 = 1/(2µ3) since we do not have the adjoint field X4. Here, we set the mass parameters m1 = m2 = m, for simplicity. Consider the case when N1 = N2 = 3 and N3 = 1 so that the Landau pole problem can easily be avoided. The resulting model is a variant of the models proposed in [4]. The model [4] has the global symmetry U(4) × U(1)mess, where U(4) is the flavor symmetry of the ISS model and U(1)mess acts on the messengers (f, f̃). The meta-stable vacuum spontaneously breaks the U(4) symmetry, giving rise to Nambu-Goldstone bosons, when m1 ≃ m2. In our model, the would-be Nambu-Goldstone bosons are eaten by the gauge symmetry. This difference is not important in the low energy analysis of supersymmetry breaking effects. Let us discuss phenomenological constraints on the parameters in (2.6). We will focus on the following part of the superpotential (2.6), Wmess = M33f f̃ +m3 f f̃ , (2.7) where M33 = Q32Q23/Λ2 is neutral under the U(N3) = U(1) gauge group. We have dropped the irrelevant quartic term (f f̃)2 because the messengers (f, f̃) are weakly inter- acting at energies above the electroweak scale, if the mass parameter µ̃3 is large enough. The F -component of the meson superfield M33 develops the vacuum expectation value and breaks supersymmetry [1]. The supersymmetric mass and the soft supersymmetry breaking mass of the messenger fields (f, f̃) are then given by Wmess ≃ m3 + θ f f̃ . (2.8) Following the analysis in [4,5], we find that all the phenomenological requirements for the messenger sector can be satisfied, for example, in the following range of parameters, Λ2 ≃ 10 11GeV, m ≃ 108GeV, m3 ≃ 10 7GeV, µ1 ≥ µ2 ≥ 10 13GeV, µ3 ≃ 10 18GeV. (2.9) 3. Generalization We found that both the messenger sector and the hidden sector of the models proposed in [4] can be realized in the A4 quiver gauge theory. This construction naturally suggests the following generalization. Consider a quiver diagram which can be separated into two disjoint diagrams Γ1 and Γ2 by cutting at one node, which we denote by a. If the scale Λa associated to the gauge group on the a-node is sufficiently low, and if superpotential interactions between them are small, we have effectively two separate quiver gauge theories for phenomena much above the scale Λa, one associated to Γ1 and another associated to Γ2, which are weakly interacting with each other through the a-node. If supersymmetry is broken in the sector Γ1, it can be communicated to the sector Γ2 by the gauge mediation mechanism. The beauty of the quiver gauge theory construction is that, because of the presence of bi-fundamental and adjoint fields on links and nodes, an effective superpotential of the form (2.8) is naturally generated when supersymmetry is broken in a part of the diagram connected to the a-node. It follows trivially that any quiver theory that is vector like with adjustable mass terms has meta-stable supersymmetry breaking vacua in some range of its parameter. All one has to do is to identify a part of the diagram where supersymmetry can be broken using a known mechanism, for example as in [1] or its variant [19], and to have its effect communicated to the rest of the diagram by messengers. One can also consider the scenario where the quiver theory associated to a sub-diagram Γ2 has a supersymmetric vacuum with dynamically generated small scales, which can be used to set parameters of the theory associated to another sub-diagram Γ1, where supersymmetry is broken. The supersymmetry breaking effect can then be communicated back to the sub-diagram Γ2. This would give a string theory realization of the idea of [20]. These and other mechanisms of supersymmetry breaking will be explored further in [15]. These supersymmetry breaking quiver gauge theories can be coupled to the messenger sector. In fact, as in the case of the A4 model discussed in the previous section, the mes- senger sector itself can be included in quiver theories. If the messenger sector is attached at the end of the quiver diagram, the effective low energy superpotential always takes the form (2.3). Thus, one can see that the models in [4] and their generalizations are robust and naturally appear in this large class of string compactifications. Acknowledgments We thank D. Berenstein, M. Dine, R. Kitano, J. Marsano, C. S. Park, N. Seiberg, M. Shigemori, and T. Watari for discussions. H.O. thanks the hospitality of the high energy theory group at the University of Tokyo at Hongo. H.O. and Y.O. are supported in part by the DOE grant DE-FG03-92-ER40701. The research of H.O. is also supported in part by the NSF grant OISE-0403366 and by the 21st Century COE Program at the University of Tokyo. Y.O. is also supported in part by the JSPS Fellowship for Research Abroad. The research of T.K. was supported in part by the Grants-in-Aid (#16740133) and (#16081206) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. References [1] K. Intriligator, N. Seiberg and D. Shih, “Dynamical SUSY breaking in meta-stable vacua,” JHEP 0604, 021 (2006), arXiv:hep-th/0602239. [2] R. Kitano, H. Ooguri and Y. Ookouchi, “Direct mediation of meta-stable supersym- metry breaking,” Phys. Rev. D 75, 045022 (2007), arXiv:hep-ph/0612139. [3] R. Argurio, M. Bertolini, S. Franco and S. Kachru, “Metastable vacua and D-branes at the conifold,” arXiv:hep-th/0703236. [4] H. Murayama and Y. Nomura, “Gauge mediation simplified,” arXiv:hep-ph/0612186. [5] H. Murayama and Y. Nomura, “Simple scheme for gauge mediation,” arXiv:hep- ph/0701231. [6] A. Amariti, L. Girardello and A. Mariotti, “On meta-stable SQCD with adjoint matter and gauge mediation,” arXiv:hep-th/0701121. [7] M. Dine and J. Mason, “Gauge mediation in metastable vacua,” arXiv:hep-ph/0611312. [8] C. Csaki, Y. Shirman and J. Terning, “A simple model of low-scale direct gauge mediation,” arXiv:hep-ph/0612241. [9] O. Aharony and N. Seiberg, “Naturalized and simplified gauge mediation,” JHEP 0702, 054 (2007), arXiv:hep-ph/0612308. [10] D. Shih, “Spontaneous R-symmetry breaking in O’Raifeartaigh models,” arXiv:hep- th/0703196. [11] A. E. Nelson and N. Seiberg, “R symmetry breaking versus supersymmetry breaking,” Nucl. Phys. B 416, 46 (1994), arXiv:hep-ph/9309299. [12] E. Witten, “Dynamical Breaking Of Supersymmetry,” Nucl. Phys. B 188, 513 (1981). [13] K. Intriligator, N. Seiberg and D. Shih, “Supersymmetry breaking, R-symmetry break- ing and metastable vacua,” arXiv:hep-th/0703281. [14] F. Cachazo, S. Katz and C. Vafa, “Geometric transitions and N = 1 quiver theories,” arXiv:hep-th/0108120. [15] T. Kawano, H. Ooguri, Y. Ookouchi and C.S. Park, in preparation [16] H. Ooguri and C. Vafa, “Two-dimensional black hole and singularities of CY mani- folds,” Nucl. Phys. B 463, 55 (1996), arXiv:hep-th/9511164. [17] C. Vafa, “Superstrings and topological strings at large N ,” J. Math. Phys. 42, 2798 (2001), arXiv:hep-th/0008142. [18] F. Cachazo, B. Fiol, K. A. Intriligator, S. Katz and C. Vafa, “A geometric unification of dualities,” Nucl. Phys. B 628, 3 (2002), arXiv:hep-th/0110028. [19] H. Ooguri and Y. Ookouchi, ‘Landscape of supersymmetry breaking vacua in geomet- rically realized gauge theories,” Nucl. Phys. B 755, 239 (2006), arXiv:hep-th/0606061. [20] M. Dine, J. L. Feng and E. Silverstein, “Retrofitting O’Raifeartaigh models with dynamical scales,” Phys. Rev. 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0704.1087

EPR, Bell, Schrodinger's cat, and the Monty Hall Paradox

EPR, Bell, Schrodinger’s cat, and the Monty Hall Paradox Doron Cohen Department of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel The purpose of this manuscript is to provide a short pedagogical explanation why “quantum collapse” is not a metaphysical event, by pointing out the analogy with a “classical collapse” which is associated with the Monty Hall Paradox. This manuscript constitutes a short self-contained version of some selected sections taken from “lecture notes in Quantum Mechanics” [quant-ph/0605180](∼250p). In particular section 47.4 regarding the notion of collapse has attracted some attention. In this section I suggest to use the “Monty Hall Paradox” as a pedagogical introduction to the discussion of “quantum collapse”. From my experience it is the most effective way to convince students and other non-experts that “quantum collapse” is not a metaphysical event. [8] Quantum States ====== [8.1] Is the world classical? (EPR, Bell) We would like to examine whether the world we live in is “classical” or not. The notion of classical world includes mainly two ingredients: (i) realism (ii) determinism. By realism we means that any quantity that can be measured is well defined even if we do not measure it in practice. By determinism we mean that the result of a measurement is determined in a definite way by the state of the system and by the measurement setup. We shall see later that quantum mechanics is not classical in both respects: In the case of spin 1/2 we cannot associate a definite value of σ̂y for a spin which has been polarized in the σ̂x direction. Moreover, if we measure the σ̂y of a σ̂x polarized spin, we get with equal probability ±1 as the result. In this section we would like to assume that our world is ”classical”. Also we would like to assume that interactions cannot travel faster than light. In some textbooks the latter is called ”locality of the interactions” or ”causality”. It has been found by Bell that the two assumptions lead to an inequality that can be tested experimentally. It turns out from actual experiments that Bell’s inequality are violated. This means that our world is either non-classical or else we have to assume that interactions can travel faster than light. If the world is classical it follows that for any set of initial conditions a given measurement would yield a definite result. Whether or not we know how to predict or calculate the outcome of a possible measurement is not assumed. To be specific let us consider a particle of zero spin, which disintegrates into two particles going in opposite directions, each with spin 1/2. Let us assume that each spin is described by a set of state variables. state of particle A = xA1 , x 2 , ... (1) state of particle B = xB1 , x 2 , ... The number of state variables might be very big, but it is assumed to be a finite set. Possibly we are not aware or not able to measure some of these “hidden” variables. Since we possibly do not have total control over the disintegration, the emerging state of the two particles is described by a joint probability function ρ xA1 , ..., x 1 , ... . We assume that the particles do not affect each other after the disintegration (“causality” assumption). We measure the spin of each of the particles using a Stern-Gerlach apparatus. The measurement can yield either 1 or −1. For the first particle the measurement outcome will be denoted as a, and for the second particle it will be denoted as b. It is assumed that the outcomes a and b are determined in a deterministic fashion. Namely, given the state variables of the particle and the orientation θ of the apparatus we have a = a(θA) = f(θA, x 1 , x 2 , ...) = ±1 (2) b = b(θB) = f(θB, x 1 , x 2 , ...) = ±1 http://arxiv.org/abs/0704.1087v1 where the function f() is possibly very complicated. If we put the Stern-Gerlach machine in a different orientation then we will get different results: a′ = a(θ′A) = f θ′A, x 1 , x 2 , ... = ±1 (3) b′ = b(θ′B) = f θ′B , x 1 , x 2 , ... We have following innocent identity: ab+ ab′ + a′b− a′b′ = ±2 (4) The proof is as follows: if b = b′ the sum is ±2a, while if b = −b′ the sum is ±2a′. Though this identity looks innocent, it is completely non trivial. It assumes both ”reality” and ”causality” This becomes more manifest if we write this identity as a(θA)b(θB) + a(θA)b(θ B) + a(θ A)b(θB)− a(θ′A)b(θ′B) = ±2 (5) The realism is reflected by the assumption that both a(θA) and a(θ A) have definite values, though it is clear that in practice we can measure either a(θA) or a(θ A), but not both. The causality is reflected by assuming that a depends on θA but not on the distant setup parameter θB. Let us assume that we have conducted this experiment many times. Since we have a joint probability distribution ρ, we can calculate average values, for instance: 〈ab〉 = xA1 , ..., x 1 , ... θA, x 1 , ... θB , x 1 , ... Thus we get that the following inequality should hold: |〈ab〉+ 〈ab′〉+ 〈a′b〉 − 〈a′b′〉| ≤ 2 (7) This is called Bell’s inequality. Let us see whether it is consistent with quantum mechanics. We assume that all the pairs are generated in a singlet (zero angular momentum) state. It is not difficult to calculate the expectation values. The result is 〈ab〉 = − cos(θA − θB) ≡ C(θA − θB) (8) we have for example C(0o) = −1, C(45o) = − 1√ , C(90o) = 0, C(180o) = +1. (9) If the world were classical the Bell’s inequality would imply |C(θA − θB) + C(θA − θ′B) + C(θ′A − θB) + C(θ′A − θ′B)| ≤ 2 (10) Let us take θA = 0 o and θB = 45 o and θ′A = 90 o and θ′B = −45o. Assuming that quantum mechanics holds we get 2 > 2 (11) It turns out, on the basis of celebrated experiments that Nature has chosen to violate Bell’s inequality. Furthermore it seems that the results of the experiments are consistent with the predictions of quantum mechanics. Assuming that we do not want to admit that interactions can travel faster than light it follows that our world is not classical. ====== [8.2] The four Postulates of Quantum Mechanics The 18th century version classical mechanics can be derived from three postulates: The three laws of Newton. The better formulated 19th century version of classical mechanics can be derived from three postulates: (1) The state of classical particles is determined by the specification of their positions and its velocities; (2) The trajectories are determined by a minimum action principle. (3) The form of the Lagrangian of the theory is determined by symmetry considerations, namely Galilei invariance in the non-relativistic case. See the Classical Mechanics book of Landau and Lifshitz for details. Quantum mechanically requires four postulates: Two postulates define the notion of quantum state, while the other two postulates, in analogy with classical mechanics, are about the laws that govern the evolution of quantum me- chanical systems. [The rest of this section can be found in the lecture notes]. ====== [8.3] What is a Pure State ”Pure states” are states that have been filtered. The filtering is called ”preparation”. For example: we take a beam of electrons. Without ”filtering” the beam is not polarized. If we measure the spin we will find (in any orientation of the measurement apparatus) that the polarization is zero. On the other hand, if we ”filter” the beam (e.g. in the left direction) then there is a direction for which we will get a definite result (in the above example, in the right/left direction). In that case we say that there is full polarization - a pure state. The ”uncertainty principle” tells us that if in a specific measurement we get a definite result (in the above example, in the right/left direction), then there are different measurements (in the above example, in the up/down direction) for which the result is uncertain. The uncertainty principle is implied by the first postulate. ====== [8.4] What is a Measurement In contrast with classical mechanics, in quantum mechanics measurement only has meaning in a statistical sense. We measure ”states” in the following way: we prepare a collection of systems that were all prepared in the same way. We make the measurement on all the ”copies”. The outcome of the measurement is an event x̂ = x that can be characterized by a distribution function. The single event has no statistical meaning. For example, if we measured the spin of a single electron and get σ̂z = 1, it does not mean that the state is polarized ”up”. In order to know if the electron is polarized we must measure a large number of electrons that were prepared in an identical way. If only 50% of the events give σ̂z = 1 we should conclude that there is no definite polarization in the direction we measured! ====== [8.5] Random Variables A random variable is an object that can have any numerical value. In other words x̂ = x is an event. Let’s assume, for example, that we have a particle that can be in one of five sites: x = 1, 2, 3, 4, 5. An experimentalist could measure Prob(x̂ = 3) or Prob(p̂ = 3(2π/5)). Another example is a measurement of the probability Prob(σ̂z = 1) that the particle will have spin up. The collection of values of x is called the spectrum of values of the random variable. We make the distinction between random variables with a discrete spectrum, and random variables with a continuous spectrum. [The rest of this section can be found in the lecture notes]. ====== [8.6] Quantum Versus Statistical Mechanics Quantum mechanics stands opposite classical statistical mechanics. A particle is described in classical statistical mechanics by a probability function (for presentation purpose we treat x̂ and p̂ as having discrete spectrum): ρ(x, p) = Prob{x̂ = x, p̂ = p} (12) The expectation value of a random variable  = A(x̂, p̂) is calculated using the definition: 〈Â〉 = ρ(x, p)A(x, p) ≡ trace(ρA) (13) In particular we can write: ρ(x, p) = 〈 δ(p̂− p) δ(x̂ − x) 〉 (14) From the definition of the expectation value follows the linear relation 〈αÂ+ βB̂〉 = α〈Â〉+ β〈B̂〉 for any pair of observables. This linear relation is a trivial result of classical probability theory. It assumes that the joint probability function Eq.(12) can be defined. But in quantum mechanics we cannot define a “quantum state” using a joint probability function, as implied by the observation that our world is not “classical”. For this reason, we have to use a more sophisticated approach. Loosely speaking one may say that Quantum Mechanics takes Eq.(14) as the definition of ρ, and use the linear relation of the expectation values as a second postulate in order to deduce Eq.(13). [The rest of this section an be found in the lecture notes]. [47] Theory of Quantum Measurements ====== [47.4] Measurements, the notion of collapse In elementary textbooks the quantum measurement process is described as inducing “collapse” of the wavefunction. Assume that the system is prepared in state ρinitial = |ψ〉〈ψ| and that one measures P̂ = |ϕ〉〈ϕ|. If the result of the measurement is P̂ = 1 then it is said that the system has collapsed into the state ρfinal = |ϕ〉〈ϕ|. The probability for this “collapse” is given by the projection formula Prob(ϕ|ψ) = |〈ϕ|ψ〉|2. If one regard ρ(x, x′) or ψ(x) as representing physical reality, rather than a probability matrix or a probability amplitude, then one immediately gets into puzzles. Recalling the EPR experiment this world imply that once the state of one spin is measured at Earth, then immediately the state of the other spin (at the Moon) would change from unpolarized to polarized. This would suggest that some spooky type of “interaction” over distance has occurred. In fact we shall see that the quantum theory of measurement does not involve any assumption of spooky “collapse” mechanism. Once we recall that the notion of quantum state has a statistical interpretation the mystery fades away. In fact we explain (see below) that there is “collapse” also in classical physics! To avoid potential miss-understanding it should be clear that I do not claim that the classical “collapse” which is described below is an explanation of the the quantum collapse. The explanation of quantum collapse using a quantum measurement (probabilistic) point of view will be presented in a later section. The only claim of this section is that in probability theory a correlation is frequently mistaken to be a causal relation: “smokers are less likely to have Alzheimer” not because cigarettes help to their health, but simply because their life span is smaller. Similarly quantum collapse is frequently mistaken to be a spooky interaction between well separated systems. Consider the thought experiment which is known as the “Monty Hall Paradox”. There is a car behind one of three doors. The car is like a classical ”particle”, and each door is like a ”site”. The initial classical state is such that the car has equal probability to be behind any of the three doors. You are asked to make a guess. Let us say that you peak door #1. Now the organizer opens door #2 and you see that there is no car behind it. This is like a measurement. Now the organizer allows you to change your mind. The naive reasoning is that now the car has equal probability to be behind either of the two remaining doors. So you may claim that it does not matter. But it turns out that this simple answer is very very wrong! The car is no longer in a state of equal probabilities: Now the probability to find it behind door #3 has increased. A standard calculation reveals that the probability to find it behind door #3 is twice large compared with the probability to find it behind door #2. So we have here an example for a classical collapse. If the reader is not familiar with this well known ”paradox”, the following may help to understand why we have this collapse (I thank my colleague Eitan Bachmat for providing this explanation). Imagine that there are billion doors. You peak door #1. The organizer opens all the other doors except door #234123. So now you know that the car is either behind door #1 or behind door #234123. You want the car. What are you going to do? It is quite obvious that the car is almost definitely behind door #234123. It is also clear the that the collapse of the car into site #234123 does not imply any physical change in the position of the car. ====== [47.5] Quantum measurements, Schroedinger’s cat What do we mean by quantum measurement? In order to clarify this notion let us consider a system and a detector which are prepared independently as ψa|a〉 ⊗ |q = 0〉 (15) As a result of an interaction we assume that the detector correlates with the system as follows: ÛmeasurementΨ = ψa|a〉 ⊗ |q = a〉 (16) We call such type of unitary evolution ”ideal measurement”. If the system is in a definite a state, then it is not affected by the detector. Rather, we gain information on the state of the system. One can think of q as representing a memory device in which the information is stored. This memory device can be of course the brain of a human observer. Form the point of view of the observer, the result at the end of the measurement process is to have a definite a. This is interpreted as a ”collapse” of the state of the system. Some people wrongly think that ”collapse” is something that goes beyond unitary evolution. But in fact this term just makes over dramatization of the above unitary process. The concept of measurement in quantum mechanics involves psychological difficulties which are best illustrated by considering the ”Schroedinger’s cat” experiment. This thought experiment involves a radioactive nucleus, a cat, and a human being. The half life time of the nucleus is an hour. If the radioactive nucleus decays it triggers a poison which kills the cat. The radioactive nucleus and the cat are inside an isolated box. At some stage the human observer may open the box to see what happens with the cat... Let us translate the story into a mathematical language. A time t = 0 the state of the universe (nucleus⊗cat⊗observer) is Ψ = | ↑= radioactive〉 ⊗ |q = 1 = alive〉 ⊗ |Q = 0 = ignorant〉 (17) where q is the state of the cat, and Q is the state of the memory bit inside the human observer. If we wait a very long time the nucleus would definitely decay, and as a result we will have a definitely dead cat: UwaitingΨ = | ↓= decayed〉 ⊗ |q = −1 = dead〉 ⊗ |Q = 0 = ignorant〉 (18) If the observer opens the box he/she would see a dead cat: UseeingUwaitingΨ = | ↑= decayed〉 ⊗ |q = −1 = dead〉 ⊗ |Q = −1 = shocked〉 (19) But if we wait only one hour then UwaitingΨ = | ↑〉 ⊗ |q = +1〉+ | ↓〉 ⊗ |q = −1〉 ⊗ |Q = 0 = ignorant〉 (20) which means that from the point of view of the observer the system (nucleus+cat) is in a superposition. The cat at this stage is neither definitely alive nor definitely dead. But now the observer open the box and we have: UseeingUwaitingΨ = | ↑〉 ⊗ |q = +1〉 ⊗ |Q = +1 = happy〉 + | ↓〉 ⊗ |q = −1〉 ⊗ |Q = −1 = shocked〉 We see that now, form the point of view of the observer, the cat is in a definite(!) state. This is regarded by the observer as “collapse” of the superposition. We have of course two possibilities: one possibility is that the observer sees a definitely dead cat, while the other possibility is that the observer sees a definitely alive cat. The two possibilities ”exist” in parallel, which leads to the ”many worlds” interpretation. Equivalently one may say that only one of the two possible scenarios is realized from the point of view of the observer, which leads to the ”relative state” concept of Everett. Whatever terminology we use, ”collapse” or ”many worlds” or ”relative state”, the bottom line is that we have here merely a unitary evolution. ====== [47.6] Measurements, formal treatment In this section we describe mathematically how an ideal measurement affects the state of the system. First of all let us write how the U of a measurement process looks like. The formal expression is Ûmeasurement = P̂ (a) ⊗ D̂(a) (22) where P̂ (a) = |a〉〈a| is the projection operator on the state |a〉, and D̂(a) is a translation operator. Assuming that the measurement device is prepared in a state of ignorance |q = 0〉, the effect of D̂(a) is to get |q = a〉. Hence ÛΨ = P̂ (a) ⊗ D̂(a) ψa′ |a′〉 ⊗ |q = 0〉 ψa|a〉 ⊗ D̂(a)|q = 0〉 = ψa|a〉 ⊗ |q = a〉 (23) A more appropriate way to describe the state of the system is using the probability matrix. Let us describe the above measurement process using this language. After ”reset” the state of the measurement apparatus is σ(0) = |q=0〉〈q=0|. The system is initially in an arbitrary state ρ. The measurement process correlates that state of the measurement apparatus with the state of the system as follows: Ûρ⊗ σ(0)Û † = P̂ (a)ρP̂ (b) ⊗ [D̂(a)]σ(0)[D̂(b)]† = P̂ (a)ρP̂ (b) ⊗ |q=a〉〈q=b| (24) Tracing out the measurement apparatus we get ρsystem = P̂ (a)ρpreparationP̂ (a) = (a) (25) Where pa is the trace of the projected probability matrix P̂ (a)ρP̂ (a), while ρ(a) is its normalized version. We see that the effect of the measurement is to turn the superposition into a mixture of a states, unlike unitary evolution for which ρsystem = Usystem ρ preparation U † system . So indeed a measurement process looks like a non-unitary process: it turns a pure superposition into a mixture. A simple example is in order. Let us assume that the system is a spin 1/2 particle. The spin is prepared in a pure polarization state ρ =| ψ〉〈ψ | which is represented by the matrix ρab = ψaψ | ψ1 |2 ψ1ψ∗2 1 | ψ2 |2 where 1 and 2 are (say) the ”up” and ”down” states. Using a Stern-Gerlech apparatus we can measure the polarization of the spin in the up/down direction. This means that the measurement apparatus projects the state of the spin using P (1) = and P (2) = leading after the measurement to the state ρsystem = P (1)ρpreparationP (1) + P (2)ρpreparationP (2) = | ψ1 |2 0 0 | ψ2 |2 Thus the measurement process has eliminated the off-diagonal terms in ρ and hence turned a pure state into a mixture. It is important to remember that this non-unitary non-coherent evolution arise because we look only on the state of the system. On a universal scale the evolution is in fact unitary. 8 Quantum States 47 Theory of Quantum Measurements

0704.1089

Representative Ensembles in Statistical Mechanics

arXiv:0704.1089v1 [cond-mat.stat-mech] 9 Apr 2007 Representative Ensembles in Statistical Mechanics V.I. Yukalov Bogolubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia Abstract The notion of representative statistical ensembles, correctly representing statistical systems, is strictly formulated. This notion allows for a proper description of statis- tical systems, avoiding inconsistencies in theory. As an illustration, a Bose-condensed system is considered. It is shown that a self-consistent treatment of the latter, using a representative ensemble, always yields a conserving and gapless theory. Key words: Statistical systems; representative statistical ensembles; Bose-condensed systems. PACS numbers: 05.30.Ch, 05.30.Jp, 05.70.Ce, 64.10.+h, 67.40.Db http://arxiv.org/abs/0704.1089v1 1 Introduction Statistical systems are characterized by statistical ensembles. It is crucially important that the given statistical system be correctly represented by a statistical ensemble. In other words, the chosen statistical ensemble must be representative for the considered statistical system. This necessitates a thorough definition of what, actually, a statistical system is, requiring an accurate enumeration of all its basic features. The usage of a nonrepresentative ensemble, incorrectly representing the considered statistical system, may lead, and often does lead, to inconsistencies in the theoretical description of the system. The necessity of defining a statistical ensemble that would correctly represent the given statistical system was, first, emphasized already by Gibbs [1], who stressed that all additional conditions and constraints, imposed on the system, must be taken into account. The prob- lem of a proper representation of equilibrium statistical systems by equilibrium statistical ensembles was discussed by ter Haar [2,3] and also analized in the review article [4]. The aim of the present paper is to formalize the notion of representative statistical en- sembles by giving precise mathematical definitions and to generalize this notion for arbitrary systems, whether equilibrium or nonequilibrium. The application of the notion is illustrated by systems with Bose-Einstein condensate, when the global gauge symmetry is broken. It is shown that employing a representative ensemble for a Bose-condensed system results in the theory enjoying conservation laws and having no gap in the spectrum of collective excitations. Throughout the paper, the system of units will be used with the Planck and Boltzmann constants set to unity, h̄ = 1, kB = 1. 2 Representative Ensembles Let us, first, recall several general preliminary definitions that are necessary for precisely defining the basic notion of a representative statistical ensemble. Physical system is a collection of objects characterized by their typical features distin- guishing this collection from other systems. For example, a collection of particles can be characterized by their Hamiltonian, that is, by their energy operator. Statistical system is a many-body physical system, whose typical features are compli- mented by all additional constraints and conditions which are necessary for uniquely de- scribing the statistical properties of the system. Statistical systems are characterized by statistical ensembles. Statistical ensemble is a pair {F , ρ̂(t)} composed by the space of microstates F and a statistical operator ρ̂(t) on that space. The space of microstates can be the Fock space or its appropriate subspace. The statis- tical operator ρ̂(t), generally, is a function of time t. To give ρ̂(t) implies to define its form ρ̂(0) at the initial time t = 0 and to specify the evolution operator Û(t) such that ρ̂(t) = Û(t)ρ̂(0)Û+(t) . (1) Therefore, a statistical ensemble can be defined as a triplet {F , ρ̂(0), Û(t)} ←→ {F , ρ̂(t)} . The knowledge of a statistical ensemble allows one to find statistical averages. Statistical average for an operator Â(t) on F is < Â(t) > = TrF ρ̂(t)Â(0) = TrF ρ̂(0)Â(t) . (2) Here the Heisenberg representation of the operator Â(t) is assumed, for which Â(t) = Û+(t)Â(0)Û(t) . (3) Representative ensemble is a statistical ensemble equipped with all additional constraints and conditions that are necessary for a unique representation of the given statistical system. Additional constraints and conditions for statistical systems are usually formulated as conditions on statistical averages for some specified condition operators Ĉi(t), where i = 1, 2, . . .. These operators do not need to be necessarily the integrals of motion, but they are supposed to be Hermitian. Statistical condition is a prescribed equality for the statistical average of a condition operator, Ci(t) = < Ĉi(t) > = Trρ̂(0)Ĉi(t) . (4) Here and in what follows, the trace operation is assumed to be over the appropriate space of microstates F . Let us consider, first, an equilibrium statistical system, for which the statistical operator does not depend on time, ρ̂(t) = ρ̂(0) ≡ ρ̂ . (5) The explicit form of the statistical operator follows from the principle of minimal information [5]. The latter presumes the conditional maximization of the Gibbs entropy S = −Trρ̂ ln ρ̂ (6) under the statistical conditions (4), among which one usually distinguishes the definition of the internal energy E = Trρ̂Ĥ (7) and the normalization condition Trρ̂ = 1 . (8) The information functional is I[ρ̂] = −S + λ0 (Trρ̂− 1) + β Trρ̂Ĥ −E Trρ̂Ĉi − Ci , (9) where λ0 ≡ lnZ − 1 is the Lagrange multiplier preserving the normalization condition (8), β is the inverse temperature, which is the Lagrange multiplier for condition (7), and βνi are the Lagrange multipliers related to statistical conditions (4). The minimization of the information functional (9) yields the statistical operator e−βH , (10) corresponding to the grand canonical ensemble with the grand Hamiltonian H ≡ Ĥ + νiĈi . (11) The most customary expression for the grand Hamiltonian (11) is H = Ĥ − µN̂ , where µ is the chemical potential and N̂ is the number-of-particle operator. However, the general form of the grand Hamiltonian is given by Eq. (11), in which any condition operators can be involved. Thus, an equilibrium representative ensemble is described by the statistical operator (10) with the grand Hamiltonian (11). The evolution operator for an equilibrium system is Û(t) = e−iHt , (12) which commutes with the statistical operator (10), because of which ρ̂(t) = [H, ρ̂(t)] = 0 . The general way of obtaining the evolution equations for arbitrary nonequilibrium sys- tems is through the extremization of action functionals [6]. In our case, this extremization has to be accomplished under the prescribed statistical conditions (4). Let the system Hamiltonian be a functional of the field operators ψ(x, t) and ψ†(x, t), that is, Ĥ = Ĥ [ψ], where ψ = ψ(x, t). The system Lagrangian is L̂[ψ] ≡ ψ†(x, t) i ψ(x, t) dx− Ĥ[ψ] . (13) The action functional, or effective action, under the prescribed statistical conditions (4), takes the form A[ψ] ≡ L̂[ψ]− νiĈi(t) dt , (14) where νi are the Lagrange multipliers guaranteeing the validity of the given statistical con- ditions. The action functional is defined so that to be a self-adjoint operator, A+[ψ] = A[ψ] . (15) Similarly to Eq. (11), the grand Hamiltonian in the Heisenberg representation is H [ψ] = Ĥ [ψ] + νiĈi(t) . (16) Then the effective action (14) can be rewritten as A[ψ] = ψ†(x, t) i ψ(x, t) dx−H [ψ] dt . (17) The extremization of the action functional, requiring that δA[ψ] = 0 , δA[ψ] = δA[ψ] δψ(x, t) δψ(x, t) + δA[ψ] δψ†(x, t) δψ†(x, t) , yields the evolution equations δA[ψ] δψ†(x, t) = 0 , δA[ψ] δψ(x, t) = 0 . (18) These equations are the Hermitian conjugated forms of each other. From Eqs. (17) and (18), it is evident that the evolution equations for the field operators can be represented as ψ(x, t) = δH [ψ] δψ†(x, t) and its Hermitian conjugated. This should be equivalent to the Heisenberg equation of motion ψ(x, t) = [ψ(x, t), H [ψ]] , that is, to the Heisenberg representation for the field operator ψ(x, t) = Û+(t)ψ(x, 0)Û(t) . Hence, the evolution operator satisfies the Schrödinger equation Û(t) == H [ψ(x, 0)] Û(t) . (20) In this way, a nonequilibrium representative ensemble is the set of the given space of microstates F , initial statistical operator ρ̂(0), and of the evolution operator Û(t) defined by Eq. (20). An equilibrium representative ensemble is, of course, just a particular case of the general nonequilibrium ensemble. 3 Bose-Condensed Systems To illustrate the explicit construction of a representative ensemble, let us consider a system with Bose-Einstein condensate. Such systems possess a variety of interesting properties, as can be inferred from review works [7–10]. Moreover, theoretical description of these systems is known to confront the notorious difficulty of defining a self-consistent approach. The theory of Bose-condensed systems is based on the Bogolubov idea [11-14] of breaking the global gauge symmetry by means of the famous Bogolubov shift for field operators. The condensate wave function, introduced in the course of this shift, has to satisfy the minimum of the related thermodynamic potential, which is the stability condition necessary for mak- ing the system stable and the theory conserving and self-consistent. At the same time, the spectrum of collective excitations, according to the Hugenholtz-Pines theorem [15], has to be gapless. The notorious problem is the appearance of the contradiction between the above two requirements, when the theory is either nonconserving or gapful. This contradiction does not arise only in the lowest orders with respect to particle interactions, when one uses the Bogolubov approximation at low temperatures [11,12] or the quasiclassical approxima- tion at high temperatures [16]. However, this contradiction immediately arises as soon as the interaction strength is not asymptotically weak and one has to invoke a more elaborate approximation. This problem of conserving versus gapless approximations was first empha- sized by Hohenberg and Martin [17] and recently covered comprehensively by Andersen [9]. The problem is caused by the usage of nonrepresentative ensembles, which renders the sys- tem unstable [18]. Here we show that employing a representative ensemble never yields the above contradiction, always resulting in a self-consistent theory, being both conserving and gapless. We consider a system with the Hamiltonian ψ†(r) ψ(r) dr+ ψ†(r)ψ†(r′)Φ(r− r′)ψ(r′)ψ(r) drdr′ , (21) in which the field operators ψ(r) = ψ(r, t) satisfy the Bose commutation relations, U = U(r, t) is an external field, and Φ(r) = Φ(−r) is an interaction potential. For describing a Bose-condensed system with broken global gauge symmetry, the Bogolubov shift [13,14] has to be done through the replacement ψ(r, t) −→ ψ̂(r, t) ≡ η(r, t) + ψ1(r, t) , (22) where η(r, t) is the condensate wave function and ψ1(r, t) is the field operator of noncon- densed particles. The latter field variables are assumed to be orthogonal to each other, η∗(r, t)ψ1(r, t) dr = 0 . (23) It is necessary to emphasize that the Bogolubov shift (22) realizes unitary nonequivalent operator representations [18,19]. Accomplishing the Bogolubov shift (22) in Hamiltonian (21), as well as in all operators of observables, we get the algebra of observables defined on the Fock space F(ψ1) generated by the field operators ψ†1(r) (see details in Refs. [5,18,19]). The condensate function is normalized to the number of condensed particles |η(r, t)|2dr . (24) The Bogolubov shift (22) is only rational when the number of condensed particles (24) is macroscopic, which means that the limit is not zero, where N is the total number of particles. The latter is given by the average N = < N̂ > (25) for the number-of-particle operator ψ̂†(r)ψ̂(r) dr , (26) in which the Bogolubov shift (22) is again assumed. The statistical averaging in Eq. (25) and everywhere below is over the Fock space F(ψ1). Substituting the Bogolubov shift (22) into Hamiltonian (21) gives in the latter the terms linear in ψ1, because of which the average < ψ1 > can be nonzero. This, however, would result in the nonconservation of quantum numbers, e.g., of spin or momentum. Therefore, one has to impose the constraint for the conservation of quantum numbers, < ψ1(r, t) > = 0 . (27) Defining the self-adjoint condition operator Λ̂(t) ≡ λ(r, t)ψ 1(r, t) + λ ∗(r, t)ψ1(r, t) dr , (28) in which λ(r, t) is a complex function, we may represent constraint (27) as the quantum conservation condition < Λ̂(t) > = 0 . (29) In this way, there are three statistical conditions. The first condition is the normalization (24) for the number of condensed particles. Condition (24) can be represented in the standard form (4) by defining the operator N̂0 ≡ 1̂ |η(r, t)|2dr , (30) in which 1̂ is the unity operator in the Fock space F(ψ1). Then Eq. (24) reduces to the statistical condition N0 = < N̂0 > . (31) The second condition is the normalization (25) for the total number of particles. Equivalently, instead of normalization (25), we may consider the normalization condition N1 = < N̂1 > , N̂1 ≡ 1(r)ψ1(r) dr (32) for the number of uncondensed particles N1 = N − N0. And the third condition is the conservation condition (29). Respectively, the effective action, which is now a functional of the two field variables, η(r, t) and ψ1(r, t), with taking account of the statistical conditions (29), (31), and (32), becomes A[η, ψ1] = L̂+ µ0N̂0 + µ1N̂1 + Λ̂ dt . (33) Here L̂ = L̂[ψ̂] is the Lagrangian (13) under the Bogolubov shift (22) and Λ̂ = Λ̂(t) from Eq. (28). The quantities µ0, µ1, and λ(r, t) are the Lagrange multipliers guaranteeing the validity of the corresponding statistical conditions. Introducing the grand Hamiltonian H [η, ψ1] ≡ Ĥ − µ0N̂0 − µ1N̂1 − Λ̂ , (34) in which Ĥ = Ĥ[ψ̂], with shift (22), and the effective Lagrangian L[η, ψ1] ≡ η∗(r, t) i η(r, t) + ψ 1(r, t) i ψ1(r, t) dr−H [η, ψ1] , (35) for the action functional (33), we get A[η, ψ1] = L[η, ψ1] dt . (36) The evolution equations follow from the extremization of the action functional (36), that is, from the variations δA[η, ψ1] δη∗(r, t) = 0 (37) δA[η, ψ1] 1(r, t) = 0 . (38) These equations, as is clear from Eqs. (35) and (36), are equivalent to the equations of motion η(r, t) = δH [η, ψ1] δη∗(r, t) ψ1(r, t) = δH [η, ψ1] 1(r, t) . (40) One has to substitute here Hamiltonian (34) under the Bogolubov shift (22). Accomplishing the variation in Eq. (39), we get η(r, t) = + U − µ0 η(r)+ Φ(r− r′) |η(r′)|2η(r) + X̂(r, r′) dr′ , (41) where the time dependence in the right-hand side, for short, is not explicitly shown, U = U(r, t), and the notation for the correlation operator X̂(r, r′) ≡ ψ†1(r′)ψ1(r′)η(r) + ψ ′)η(r′)ψ1(r)+ +η∗(r′)ψ1(r ′)ψ1(r) + ψ ′)ψ1(r ′)ψ1(r) (42) is used. The variation in Eq. (40) gives ψ1(r, t) = + U − µ1 ψ1(r)+ Φ(r− r′) |η(r′)|2ψ1(r) + η∗(r′)η(r)ψ1(r′) + η(r′)η(r)ψ†1(r′) + X̂(r, r′) dr′ . (43) To get an equation for the condensate wave function, we have to take the statistical average of Eq. (41). For this purpose, we introduce the normal density matrix ρ1(r, r ′) ≡ < ψ†1(r′)ψ1(r) > , (44) the anomalous density matrix σ1(r, r ′) ≡ < ψ1(r′)ψ1(r) > , (45) and their diagonal elements, giving the density of noncondensed particles ρ1(r) ≡ ρ1(r, r) = < ψ†1(r)ψ1(r) > (46) and the anomalous average σ1(r) ≡ σ1(r, r) = < ψ1(r)ψ1(r) > . (47) The quantity |σ1(r)| can be interpreted as the density of paired particles [19]. The total density of particles ρ(r) = ρ0(r) + ρ1(r) (48) consists of the condensate density ρ0(r) ≡ |η(r)|2 (49) and the density of noncondensed particles (46). Averaging Eq. (41), we find the equation for the condensate wave function η(r, t) = + U − µ0 η(r)+ Φ(r− r′) ρ(r′)η(r) + ρ1(r, r ′)η(r′) + σ1(r, r ′)η∗(r′)+ < ψ ′)ψ1(r ′)ψ1(r) > dr′ . (50) Equations (43) and (50) are the basic equations of motion for the field variables η(r, t) and ψ1(r, t). These equations, according to Eqs. (39) and (40), are generated by the variation of the grand Hamiltonian (34). The latter, in agreement with Eq. (20), also defines the evolution operator Û(t), which satisfies the Schrödinger equation Û(t) = H [η(r, 0), ψ1(r, 0)]Û(t) . Thus, the representative ensemble for a Bose-condensed system is the triplet {F(ψ1), ρ̂(0), Û(t)} . It is important to stress that the so defined representative ensemble possesses a principal feature making it different from the standardly used ensemble having the sole Lagrange multiplier µ0 ≡ µ1. But then the normalization condition (24) cannot be guaranteed. Then the evolution equation for the condensate wave function is not a result of a variational procedure. For an equilibrium system, this means that the number of condensed particles N0 does not provide the minimum of a thermodynamic potential, which implies the system instability. All notorious inconsistencies in theory, manifesting themselves in the lack od conservation laws or in the appearance of an unphysical gap in the spectrum, are caused by the usage of nonrepresentative ensembles. 4 Green Functions The equations of motion (43) and (50) allow us to derive the evolutional equations for the Green functions. To this end, we shall use the compact notation denoting the set {rj, tj} by the sole letter j, so that the dependence of functions on the spatial and temporal variables looks like f(12 . . . n) ≡ f(r1, t1, r2, t2, . . . , rn, tn) . The product of the differentials drjdtj will be denoted as d(j), so that d(12 . . . n) ≡ drj dtj . We shall employ the Dirac delta function δ(12) ≡ δ(r1 − r2) δ(t1 − t2) . For the interaction potential, we shall use the retarded form Φ(12) ≡ Φ(r1 − r2)δ(t1 − t2 + 0) . (51) The matrix Green function G(12) = [Gαβ(12)] is a 2 × 2 matrix, with α, β = 1, 2, and with the following elements: G11(12) ≡ −i < T̂ψ1(1)ψ†1(2) > , G12(12) ≡ −i < T̂ψ1(1)ψ1(2) > , G21(12) ≡ −i < T̂ψ†1(1)ψ 1(2) > , G22(12) ≡ −i < T̂ψ 1(1)ψ1(2) > , (52) where T̂ is the time-ordering operator. Let us introduce the operator K̂j ≡ − + U(j)− µ1 , (53) the condensate effective potential V (12) ≡ δ(12) Φ(13)|η(3)|2d(3) + Φ(12)η(1)η∗(2) , (54) and let us rewrite the correlation operator (42) in the form X̂(12) = ψ 1(2)ψ1(2)η(1) + ψ 1(2)η(2)ψ1(1) + η ∗(2)ψ1(2)ψ1(1) + ψ 1(2)ψ1(2)ψ1(1) . (55) We also define the matrix correlation function X(123) = [Xαβ(123)] with the elements: X11(123) ≡ − < T̂ X̂(12)ψ†1(3) > , X12(123) ≡ − < T̂ X̂(12)ψ1(3) > , X21(123) ≡ − < T̂X̂+(12)ψ†1(3) > , X22(123) ≡ − < T̂X̂+(12)ψ1(3) > . (56) From the equations of motion (43) and (50), we find the equations − K̂1 G11(12)− V (13)G11(32) d(3)− Φ(13) [η(1)η(3)G21(32) + iX11(132)] d(3) = δ(12) , − K̂1 G12(12)− V (13)G12(32) d(3)− Φ(13) [η(1)η(3)G22(32) + iX12(132)] d(3) = 0 , − K̂1 G21(12)− V ∗(13)G21(32) d(3)− Φ(13) [η∗(1)η∗(3)G11(32) + iX21(132)] d(3) = 0 , − K̂1 G22(12)− V ∗(13)G22(32) d(3)− Φ(13) [η∗(1)η∗(3)G12(32) + iX22(132)] d(3) = δ(12) . (57) The self-energy Σ(12) = [Σαβ(12)] is a matrix whose elements are defined by the relations [Σ11(13)G11(32) + Σ12(13)G21(32)] d(3) = V (13)G11(32) d(3) + Φ(13) [η(1)η(3)G21(32) + iX11(132)] d(3) , [Σ11(13)G12(32) + Σ12(13)G22(32)] d(3) = V (13)G12(32) d(3) + Φ(13) [η(1)η(3)G22(32) + iX12(132)] d(3) , [Σ21(13)G11(32) + Σ22(13)G21(32)] d(3) = V ∗(13)G21(32) d(3) + Φ(13) [η∗(1)η∗(3)G11(32) + iX21(132)] d(3) , [Σ21(13)G12(32) + Σ22(13)G22(32)] d(3) = V ∗(13)G22(32) d(3) + Φ(13) [η∗(1)η∗(3)G12(32) + iX22(132)] d(3) . (58) Let us introduce the matrix condensate propagator C(12) = [Cαβ(12)], with the elements C11(12) ≡ −iη(1)η∗(2) , C12(12) ≡ −iη(1)η(2) , C21(12) ≡ −iη∗(1)η∗(2) , C22(12) ≡ −iη∗(1)η(2) , (59) The latter have the properties C11(21) = C22(12) , C11(11) = C22(11) , C12(21) = C12(12) , C21(21) = C21(12) , C 11(12) = −C22(12) , C∗12(12) = −C21(12) . (60) The binary Green function is a matrix B(123) = [Bαβ(123)], B(123) = C(12)G(23)− iρ0(2)G(13) +X(123) , (61) whose elements are B11(123) ≡ C11(12)G11(23) + C11(22)G11(13) + C12(12)G21(23) +X11(123) , B12(123) ≡ C11(12)G12(23) + C11(22)G12(13) + C12(12)G22(23) +X12(123) , B21(123) ≡ C22(12)G21(23) + C22(22)G21(13) + C21(12)G11(23) +X21(123) , B22(123) ≡ C22(12)G22(23) + C22(22)G22(13) + C21(12)G12(23) +X22(123) , (62) and where ρ0(1) ≡ |η(1)|2. With Eqs. (59) and (61), relations (58), defining the self-energy, can be rewritten in the matrix form Σ(13)G(32) d(3) = i Φ(13)B(132) d(3) . (63) Then the equations of motion (57) acquire the matrix representation τ̂3 i − K̂1 G(12)− Σ(13)G(32) d(3) = δ(12) , (64) in which the delta function δ(12) in the right-hand side is assumed to be factored with the unity matrix 1̂ = [δαβ ] and τ̂3 ≡ is a Pauli matrix. Introducing the inverse propagator G−1(12) ≡ τ̂3 i − K̂1 δ(12)− Σ(12) (65) allows us to transform Eq. (64) into G−1(13)G(32) d(3) = δ(12) . (66) An equivalent representation, following from Eq. (66), is G(13)G−1(32) d(3) = δ(12) . (67) For the self-energy, using Eq. (63), we have Σ(12) = i Φ(13)B(134)G−1(42) d(34) . (68) The equations for the Green functions are to be complimented by the equation for the condensate wave function (50), which, introducing one more anomalous average ξ(12) ≡ < ψ†1(2)ψ1(2)ψ(1) > , (69) can be represented as η(1) = + U(1)− µ0 η(1)+ Φ(12) [ρ(2)η(1) + ρ1(12)η(2) + σ1(12)η ∗(2) + ξ(12)] d(2) . (70) It is the equations for the Green functions and the equation for the condensate func- tion, which become mutually incompatible in the standard approach, while employing the representative ensemble renders the theory self-consistent in any approximation. 5 Theory Self-Consistency One usually confronts inconsistencies in theory considering a uniform equilibrium Bose- condensed system. Then, in any given approximation, one gets either a nonconserving theory, that is, an unstable system, or one finds an unphysical gap in the spectrum, which, actually, again corresponds to an unstable system [9,18]. To analyze this problem, we pass now to the case of an equilibrium uniform system, when U = 0. Then we use the Fourier transform for the Green function G(12) = G(k, ω)ei(k·r12−ωt12) (2π)4 in which r12 ≡ r1 − r2 , t12 ≡ t1 − t2 . By their definition in Eq. (52), the Green function elements possess the properties G11(21) = G22(12) , G12(21) = G12(12) , G21(21) = G21(12) . (71) Therefore the corresponding Fourier transforms satisfy the relations G11(−k,−ω) = G22(k, ω) , G12(−k,−ω) = G12(k, ω) , G21(−k,−ω) = G21(k, ω) . (72) Assuming that the system is isotropic, one has Gαβ(−k, ω) = Gαβ(k, ω) (73) for all α, β. Combining Eqs. (72) and (73), we find G11(k,−ω) = G22(k, ω) , G12(k,−ω) = G12(k, ω) , G21(k,−ω) = G21(k, ω) . (74) Also, for a uniform equilibrium system, one has [14] the equality G21(k, ω) = G12(k, ω) . (75) Fourier-transforming the self-energy Σ(12) = Σ(k, ω)ei(k·r12−ωt12) (2π)4 and, similarly, the inverse propagator (65), we have for the latter G−1(k, ω) = τ̂3ω − + µ− Σ(k, ω) . (76) Then Eq. (67) reduces to G−1(k, ω)G(k, ω) = 1̂ , (77) where 1̂ = [δαβ ]. From Eqs. (76) and (77), it follows that G−1(k, ω), hense, also Σ(k, ω), have the same symmetry properties as G(k, ω). In particular, Σαβ(−k, ω) = Σαβ(k, ω) , Σ12(k,−ω) = Σ12(k, ω) , Σ21(k,−ω) = Σ21(k, ω) , Σ21(k, ω) = Σ12(k, ω) , Σ11(k,−ω) = Σ22(k, ω) . (78) The matrix equation (77), explicitly, is the system of equations ω − k + µ1 − Σ11 G11 − Σ12G21 = 1 , + µ1 − Σ11 G12 − Σ12G22 = 0 , −ω − k + µ1 − Σ22 G21 − Σ21G11 = 0 , −ω − k + µ1 − Σ22 G22 − Σ21G12 = 1 , (79) where, for short, Gαβ = Gαβ(k, ω) and Σαβ = Σαβ(k, ω). The solutions to these equations G11(k, ω) = ω + k2/2m+ Σ11(k, ω)− µ1 D(k, ω) G12(k, ω) = − Σ12(k, ω) D(k, ω) , (80) with the denominator D(k, ω) ≡ ω − k − Σ11(k, ω) + µ1 + Σ22(k, ω)− µ1 + Σ212(k, ω) . (81) The solutions for Σ21(k, ω) and G22(k, ω) are defined by the symmetry properties (72) to (75). The excitation spectrum is given by the poles of the Green functions, that is, by the zero of denominator (81), D(k, εk) = 0 . (82) Equation (82) can be represented as [Σ11(k, εk)− Σ22(k, εk)]± ω2k − Σ212(k, εk) , (83) with the notation [Σ11(k, εk) + Σ22(k, εk)]− µ1 . (84) Denominator (81) enjoys the property D(k,−ω) = D(k, ω) . Consequently, if εk is a solution of Eq. (82), then −εk is also its solution, which is in agreement with the form of Eq. (83). For an equilibrium uniform system, the Bogolubov shift (22) is equivalent to the sepa- ration of the zero-momentum term in the expansion of the field operator over plane waves. The shift itself has meaning only under the normalization condition (24), in which N0 ∼ N , that is, the zero-momentum state is macroscopically occupied. The latter becomes possible when the single particle spectrum touches zero. Therefore, the necessary condition for the existence of Bose-Einstein condensate is εk = 0 . (85) This is to be complimented by the stability condition Re εk ≥ 0 , Im εk ≤ 0 . (86) This condition should be kept in mind when choosing the sign plus in front of the square root in spectrum (83). Taking limit (85) for spectrum (83), we notice that, according to properties (78), Σ11(k, 0) = Σ22(k, 0) . (87) By using perturbation theory for a stable system, one can show [15] that in all orders of the theory Σαβ(0, 0) ≥ 0 . (88) Then the necessary condition (85) yields the expression for the chemical potential µ1 = Σ11(0, 0)− Σ12(0, 0) , (89) which is the Hugenholtz-Pines relation [15]. On the other hand, we have Eq. (70) for the condensate wave function. For an equilibrium uniform system, with no external potential U , all densities do not depend on the spatial and temporal variables, ρ0(r) = ρ0 , ρ1(r) = ρ1 , σ1(r) = σ1 , ρ(r) = ρ . (90) The condensate wave function reduces to the constant η(r, t) = η = ρ0 . (91) Then we substitute into Eq. (70) the Fourier transforms for the interaction potential Φ(r) = ik·r dk (2π)3 for the normal density matrix (44), ρ1(r1, r2) = ik·r12 (2π)3 and for the anomalous density matrix (45), σ1(r1, r2) = ik·r12 (2π)3 Similarly, the Fourier transform for the anomalous average (69) is ξ1(r1, r2) = ik·r12 (2π)3 As a result, Eq. (70) gives µ0 = ρΦ0 + nk + σk + (2π)3 . (92) Generally, expressions (92) and (89) do not coincide with each other, their difference being µ0 − µ1 = ρΦ0 + nk + σk + (2π)3 −Σ11(0, 0) + Σ12(0, 0) . (93) This is the general expression for the difference between the Lagrange multipliers µ0 and µ1 for an arbitrary equilibrium uniform Bose-condensed system. Usually, one does not distinguish between the Lagrange multipliers µ0 and µ1, which implies setting µ0 − µ1 → 0. However, as is evident from Eq. (93), there is no any reason for requiring that this quantity be zero. As an illustration, we may resort to the Hartree- Fock-Bogolubov approximation, in which ξk = 0 and Σ11(0, 0) = (ρ+ ρ0)Φ0 + (2π)3 , Σ12(0, 0) = ρ0Φ0 + (2π)3 Relation (89) then yields µ1 = ρΦ0 + (nk − σk)Φk (2π)3 . (94) The difference of the chemical potentials (93) becomes µ0 − µ1 = 2 (2π)3 , (95) which, certainly, is nonzero [20]. In this way, the introduction of the additional Lagrange multiplier makes the theory completely self-consistent. All inconsistencies that often arise in other works, such as the appearance of a gap in the spectrum, system instability or a distortion of the phase transition order, are caused by neglecting the difference between the multiplier µ1 and the multiplier µ0. It is worth emphasizing that the introduction of the Lagrange multiplier µ0 for preserving the normalization condition (24), from the mathematical point of view is strictly necessary. In other case, the employed ensemble would not be representative, hence, could not correctly describe the Bose-condensed system with broken gauge symmetry. Acknowledgement I am grateful for many useful discussions to M. Girardeau, R. Graham, H. Kleinert, and E.P. Yukalova. References [1] J.W. Gibbs, Collected Works (Longmans, New York, 1931), Vol. 2. [2] D. ter Haar, Elements of Statistical Mechanics (Reinhart, New York, 1954). [3] D. ter Haar, Rep. Prog. Phys. 24, 304 (1961). [4] V.I. Yukalov, Phys. Rep. 208, 395 (1991). [5] V.I. Yukalov, Statistical Green’s Functions (Queen’s University, Kingston, 1998). [6] H. Kleinert, Path Integrals (World Scientific, Singapore, 2004). [7] P.W. Courteille, V.S. Bagnato and V.I. Yukalov, Laser Phys. 11, 659 (2001). [8] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Clarendon, Oxford, 2003). [9] J.O. Andersen, Rev. Mod. Phys. 76, 599 (2004). [10] K. Bongs and K. Sengstock, Rep. Prog. Phys. 67, 907 (2004). [11] N.N. Bogolubov, J. Phys. (Moscow) 11, 23 (1947). [12] N.N. Bogolubov, Moscow Univ. Phys. Bull. 7, 43 (1947). [13] N.N. Bogolubov, Lectures on Quantum Statistics (Gordon and Breach, New York, 1967), Vol. 1. [14] N.N. Bogolubov, Lectures on Quantum Statistics (Gordon and Breach, New York, 1970), Vol. 2. [15] N.M. Hugenholtz and D. Pines, Phys. Rev. 116, 489 (1959). [16] N. Prokofiev, O. Ruebenacker and B. Svistunov, Phys. Rev. A 69, 053625 (2004). [17] P.C. Hohenberg and P.C. Martin, Ann. Phys. 34, 291 (1965). [18] V.I. Yukalov, Phys. Rev. E 72, 066119 (2005). [19] V.I. Yukalov, Laser Phys. 16, 511 (2006). [20] V.I. Yukalov and E.P. Yukalova, Laser Phys. Lett. 2, 506 (2005).

0704.1090

Bouncing Universe with Quintom Matter

Bouncing Universe with Quintom Matter Yi-Fu Caia∗, Taotao Qiua†, Yun-Song Piaob, Mingzhe Lic, Xinmin Zhanga Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918-4, Beijing 100049, P. R. China College of Physical Sciences, Graduate School of Chinese Academy of Sciences, YuQuan Road 19A, Beijing 100049, China Fakultät für Physik, Universität Bielefeld, D-33615 Bielefeld, Germany The bouncing universe provides a possible solution to the Big Bang singularity problem. In this paper we study the bouncing solution in the universe dominated by the Quintom matter with an equation of state (EoS) crossing the cosmological constant boundary. We will show explicitly the analytical and numerical bouncing solutions in three types of models for the Quintom matter with an phenomenological EoS, the two scalar fields and a scalar field with a modified Born-Infeld action. I. INTRODUCTION A bouncing universe with an initial contraction to a non-vanishing minimal radius, then subsequent an expanding phase provides a possible solution to the singularity problem of the standard Big Bang cosmology. For a successful bounce, it can be shown that within the framework of the standard 4-dimensional Friedmann-Robertson-Walker (FRW) cosmology with Einstein gravity the null energy condition (NEC) is violated for a period of time around the bouncing point. Moreover, for the universe entering into the hot Big Bang era after the bouncing, the EoS of the matter content w in the universe must transit from w < −1 to w > −1. The Quintom model [1], proposed to understand the behavior of dark energy with an EoS of w > −1 in the past and w < −1 at present, has been supported by the observational data[2]. Quintom is a dynamical model of dark energy. It differs from the cosmological constant, Quintessence, Phantom, K-essence and so on in the determination of the cosmological evolution. A salient feature of the Quintom model is that its EoS can smoothly cross over w = −1. In the recent years there has been a lot of proposals for the Quintom-like models in the literature. In this paper we study the bouncing solution in the universe dominated by the Quintom matter and working with three specific models we will show explicitly the analytical and numerical solutions of the bounce. We will start with a detailed examination on the necessary conditions required for a successful bounce. During the contracting phase, the scale factor a(t) is decreasing, i.e., ȧ(t) < 0, and in the expanding phase we have ȧ(t) > 0. At the bouncing point, ȧ(t) = 0, and around this point ä(t) > 0 for a period of time. Equivalently in the bouncing cosmology the hubble parameter H runs across zero from H < 0 to H > 0 and H = 0 at the bouncing point. A successful bounce requires around this point, Ḣ = −4πGρ(1 + w) > 0 . (1) From (1) one can see that w < −1 in a neighborhood of the bouncing point. After the bounce the universe needs to enter into the hot Big Bang era, otherwise the universe filled with the matter with an EoS w < −1 will reach the big rip singularity as what happens to the Phantom dark energy[3]. This requires the EoS of the matter to transit from w < −1 to w > −1. In this paper, we study the bouncing solutions in the Quintom models. The paper is organized as follows. In section II, we present the analytical and numerical solutions for different types of models of the Quintom matter. Specifically we consider three models: i)a phenomenological Quintom fluid with a parameterized EoS crossing the cosmological constant boundary; ii)the two-field models of Quintom matter with one being the quintessence-like scalar and another the phantom-like scalar; iii) a single scalar with a Born-Infeld type action. III is the summary of the paper. ∗ [email protected] † [email protected] http://arxiv.org/abs/0704.1090v1 II. BOUNCING SOLUTION IN THE PRESENCE OF QUINTOM MATTER A. A phenomenological Quintom model We start with a study on the possibility of obtaining the bouncing solution in a phenomenological Quintom matter described by the following EoS: w(t) = −r − . (2) In (2) r and s are parameters and we require that r < 1 and s > 0. One can see from (2) that w runs from negative infinity at t = 0 to the cosmological constant boundary at t = and then crosses this boundary. Assuming that the universe is dominated by the matter with the EoS given by (2), we solve the Friedmann equation and obtain the corresponding evolution of hubble parameter H(t) and scale factor a(t) as follows, H(t) = (1 − r)t2 + s , (3) a(t) = (t2 + 3(1−r) . (4) Here we choose t = 0 as the bouncing point and normalize a = 1 at this point. One can see that our solution provides a picture of the universe evolution with contracting for t < 0, and then bouncing at t = 0 to the expanding phase for t > 0. In Fig. 1 we plot the evolution of the EoS, the hubble parameter and the scale factor. -10 -8 -6 -4 -2 0 2 4 6 8 10 -10 -8 -6 -4 -2 0 2 4 6 8 10 -10 -8 -6 -4 -2 0 2 4 6 8 10 FIG. 1: Plot of the evolution of the EoS w, the hubble parameter H and the scale factor a as a function of the cosmic time t. Here in the numerical calculation we have taken r = 0.6 and s = 1. One can see from Fig. 1 that a non-singular bouncing happens at t = 0 with the hubble parameter H running across zero and a minimal non-vanishing scale factor a. At the bouncing point w approaches negative infinity. B. Two-field Quintom model Having presented the bouncing solution with the phenomenological Quintom matter, we now study the bounce in the scalar field models of Quintom matter. However it is not easy to build a Quintom model theoretically. The No-Go theorem proven in Ref. [4] (also see Ref. [1, 5, 6, 7, 8, 9]) forbids the traditional scalar field model with a lagrangian of general form L = L(φ,∇µφ∇µφ) to have its EoS cross over the cosmological constant boundary. Therefore, to realize a viable Quintom field model in the framework of Einstein’s gravity theory, it needs to introduce extra degree of freedom to the conventional theory with a single scalar field. The simplest Quintom model involves two scalars with one being the Quintessence-like and another the Phantom-like [1, 10]. This model has been studied in detail later on in the literature. In the recent years there have been a lot of activities in the theoretical study on Quintom-like models such as a single scalar with high-derivative [11, 12], vector field[13], extended theory of gravity[14] and so on, see e.g. [15]. In this section we consider a two-field Quintom model with the action given by ∂µφ1∂ µφ1 − ∂µφ2∂µφ2 − V (φ1, φ2) , (5) where the metric is in form of (+,−,−,−). Here the field φ1 has a canonical kinetic term, but φ2 is a ghost field. In the framework of FRW cosmology, we can easily obtain the energy density and the pressure of this model, + V , p = − V , (6) and the Einstein equations are given by + V ) , (7) φ̈1 + 3Hφ̇1 + = 0 , (8) φ̈2 + 3Hφ̇2 − = 0 . (9) From Eq. (1), we can see that a bouncing solution requires φ̇2 = φ̇2 + 2V when H crosses zero; and the Quintom behavior requires φ̇2 = φ̇2 when w crosses −1. These constraints can be easily satisfied in the parameter space of this model. In Fig. 2 and Fig. 3, we show the bouncing solution for two different type of potentials. In Fig. 2 we take V (φ1, φ2) = V1e M2 + V2e M2 . In the numerical calculation we normalize the dimensional parameters such as V1, V2, φ1 and φ2 by a mass scale M which we take specifically to be 10 −2Mpl. And the hubble parameter is normalized with M . One can see from this figure the non-singular behavior of the Hubble parameter and the scale factor for a bounce. The EoS w crosses over the w = −1 and approaches to negative infinity at the bouncing point. And due to the oscillatory behavior of the field φ1 in the evolution, the EoS w is also oscillating around bouncing point. In Fig. 3, we take V (φ) = 1 2 + V0φ2 −2. This model also provides a bouncing solution, however the detailed evolution of the universe differs from the one shown in Fig. 2. Fig. 3 shows that the EoS of the Quintom matter will approach w = 1 asymptotically. C. A single scalar with high-derivative terms In this section we consider a class of Quintom models described by an effective lagrangian with higher derivative operators. Starting with a canonical scalar field with the lagrangian L = 1 µφ − V (φ). This type of models has been considered as a candidate for dark energy, however as shown by the No-Go theorem it does not give w crossing -1. As an effective theory we know that the lagrangian should include more operators, especially if these operators involve the term ✷φ, as pointed in Ref. [11] it will give rise to an EoS across w = −1. A connection of this type of Quintom theory to the string theory has been considered in Ref. [16] and [12]. In this paper we take the string-inspired model in [16] for the detailed study on the bouncing solution, where the action is given by −V (φ) 1− α′∇µφ∇µφ+ β′φ✷φ . (10) -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 -1.0 -0.5 0.0 0.5 1.0 FIG. 2: The plots of the evolutions of the EoS w, hubble parameter H and the scale factor a. In the numerical calculation we choose V (φ1, φ2) = V1e M2 + V2e M2 with parameters: V1 = 15, V2 = 1, λ1 = −1.0, λ2 = 1.0, and for the initial conditions φ1 = 0.5, φ̇1 = 0.1, φ2 = 0.3, φ̇2 = 4. 0.6 0.7 0.8 0.9 1.0 1.1 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 FIG. 3: The same plots as Fig. 2 with different potential and model parameters V (φ) = 1 2 + V0φ2 , m = 2, V0 = 0.4, and for the initial conditions φ1 = 2, φ̇1 = 3, φ2 = 1, φ̇2 = 2. This is a generalized version of “Born-Infeld” action[17, 18] with the introduction of the β′ term. To the lowest order, the Box-operator term φ✷φ is equivalent to the term ∇µφ∇µφ when the tachyon is on the top of its potential. However when the tachyon rolls down from the top of the potential, these two terms exhibit different dynamical behavior. The two parameters α′ and β′ in (10) could be arbitrary in the case of the background flux being turned on [19]. One interesting feature of this model is that it provides the possibility of its EoS w running across the cosmological constant boundary. In the analytical and numerical studies below to make two parameters (α′, β′) dimensionless, it is convenient to redefine α = α′M4 and β = β′M4 where M is an energy scale of the effective theory of tachyon. From (10) we obtain the equation of motion for the scalar field φ: ) + α∇µ( ) +M4Vφf + ✷φ = 0 , (11) where f = 1− α′∇µφ∇µφ+ β′φ✷φ and Vφ = dV/dφ. Correspondingly, the stress energy tensor of the model is given by Tµν = gµν [V f − ∇ρφ)] + ∇µφ∇νφ+ )∇νφ+ )∇µφ . (12) Technically, it is very useful to define a parameter ψ ≡ ∂L = − βφV to solve (11) and (12). In the framework of a flat FRW universe filled with a homogenous scalar field φ, we have the equations of motion in form of φ̈+ 3Hφ̇ = 4M4ψ2 V 2 − φ̇2 , (13) ψ̈ + 3Hψ̇ = (2α+ β)( V Vφ − (2α− β) φ̇2 − ψ̇φ̇ . (14) Moreover, the energy density and the pressure of this field can be written as ρ = − φ̇2 − ψ̇φ̇− V 2 − , (15) p = − φ̇2 − ψ̇φ̇+ V 2 + . (16) From Eq.(1), one can see that a successful bounce requires: V 2 + φ̇2 − ψ̇φ̇ < 0 . (17) We will show below that (17) can be satisfied easily for our model. In the numerical study on the bouncing solution, we constrain the parameters α and β so that the model in (10) when expanding the derivative terms in the square root to the lowest order gives rise to a canonical kinetic term for the scalar field φ [16], i.e., α+ β > 0. In Fig. 4 and Fig. 5 we show the bounce solution for different potentials. In Fig. 4, we take V (φ) = V0e −λφ2/M2 with λ being a dimensionless parameter. One can see from this figure the scale factor initially decreases, then passes through its minimum and increases. Moreover, away from the bouncing point in the expanding phase, the EoS of the scalar field crosses w = −1 and approaches w = −0.6, which gives rise to a possible inflationary phase after the bouncing. In the numerical calculation we take the energy scale M to be 10−2Mpl, and the hubble parameter is normalized with M In Fig. 5, we consider the model with potential V (φ) = V0/φ and then show another example of the bouncing solution1. Here the energy scale M is chosen to be 10−2Mpl as well. One can see from this figure the clear picture of the bouncing, however the detailed evolution of the universe differs from the one shown in Fig. 4. After entering the expanding phase, the EoS w crosses the cosmological constant boundary and approaches w = 1 , which is equivalent to the EoS of the radiation. III. CONCLUSION AND DISCUSSIONS In this paper we have studied the possibility of obtaining a non-singular bounce in the presence of the Quintom matter. In the literature there have been a lot of efforts in constructing the bouncing universe, for instance, the Pre Big Bang scenario[21], and the Ekpyrotic scenario[22]. In Refs. [23, 24, 25] and [26, 27] the authors have considered models with the modifications of gravity with the high order terms. In general these models modify the 4-dimensional 1 A Born-Infeld lagrangian with this potential provides a scaling solution, see Ref. [20]. -1 0 1 2 3 4 5 6 7 0 1 2 3 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 FIG. 4: The plots of the evolution of the EoS w, the hubble parameter H and the scale factor a. Here in the numerical calculation we take the potential V (φ) = V0e , α = −0.2, β = 2, λ = 2, V0 = 5, and the initial values are φ = 1, φ̇ = 3, H = −1, and ψ = −80. 1.0 1.5 2.0 2.5 3.0 -3 -2 -1 0 1 2 3 -2 -1 0 1 w=1/3 FIG. 5: The plots of the evolutions of the EoS w, hubble parameter H and the scale factor a. In the numerical calculation we choose the potential as V (φ) = V0 , α = −0.2, β = 2, V0 = 0.7, and for the initial conditions φ = 10, φ̇ = −3, H = −1, ψ = −40. Einstein gravity. However, the models we consider for bounce universe in this paper are restricted to be within the standard 4-dimensional FRW framework. Recently two papers [28, 29] have studied the possibilities of having a bounce universe with the ghost condensate. In the original formulation the ghost condensate[30] will not be able to give EoS crossing w = −1. The authors of these papers[28, 29] have considered a generalized model of ghost condensate[31] and shown the bouncing solutions. In this paper we have studied the general issue of obtaining a bouncing universe with the Quintom matter. Our results show that a universe in the presence of the Quintom matter will avoid the problem of the Big Bang singularity. 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Vernov, arXiv:hep-th/0701184. http://arxiv.org/abs/astro-ph/0612728 http://arxiv.org/abs/astro-ph/0703202 http://arxiv.org/abs/astro-ph/0609040 http://arxiv.org/abs/0704.0666 http://arxiv.org/abs/astro-ph/0608165 http://arxiv.org/abs/gr-qc/0609039 http://arxiv.org/abs/gr-qc/0612026 http://arxiv.org/abs/astro-ph/0607126 http://arxiv.org/abs/hep-th/0701016 http://arxiv.org/abs/hep-th/0010108 http://arxiv.org/abs/hep-th/0610274 http://arxiv.org/abs/hep-th/0702165 http://arxiv.org/abs/hep-th/0702154 http://arxiv.org/abs/hep-th/0701184 Introduction Bouncing solution in the presence of Quintom matter A phenomenological Quintom model Two-field Quintom model A single scalar with high-derivative terms Conclusion and discussions Acknowledgments References

0704.1091

On the weight structure of cyclic codes over $GF(q)$, $q>2$

I. Mullayeva On the weight structure of cyclic codes over GF (q) , q > 2 . Abstract The interrelation between the cyclic structure of an ideal, i.e., a cyclic code over Galois field GF (q) , q > 2 , and its classes of proportional elements is considered. This relation is used in order to define the code’s weight structure. The equidistance conditions of irreducible nonprimitive codes over GF(q) are given. Besides that, the minimum distance for some class of nonprimitive cyclic codes is found. The relation of proportionality for elements of algebra An , consisting of polynomials in x over Galois field GF (q) , modulo polynomial xn − 1 , is the equivalence relation [1]. Therefore An falls into several disjoint sub- sets and every such subset contains all elements which are proportional to each other. These subsets will be called the classes of proportionality. Let z(x) 6= 0 be some vector of An . If α1 = 1, α2, . . . , αq−1 are all dif- ferent elements of the multiplicative group GF (q)∗ of the field GF (q) , then the following q − 1 vectors α1z(x), α2z(x), . . . , αq−1z(x) (1) are some different and proportional to each other elements of An . The set of vectors (1) is closed under the multiplication by the elements of the group GF (q)∗ . Hence the set (1) represents some class of proportional elements, which will be denoted by Pz(x) . Because of an arbitrary choice for z(x) , every nonzero class consists of q− 1 elements of the form (1) . Consequently An contains (q n − 1)/(q − 1) different nonzero classes. Evidently all elements of one class have the same period [2, 3] or the same order [8]. Clearly, the supporting sets [2] of vectors, entering into the same class of proportionality, are similar too. Hence, the Hamming weight is also the same for all vectors of one class. Thus, we can say that any proportionality class Pz(x) , Pz(x) ⊂ An , has its order, its supporting set and its Hamming weight. Obviously, any proportionality class of An is characterizied by its unique monic polynomial. Now consider an ideal J , J ⊂ An , i.e., some cyclic code over GF (q) , having the following generator g(x) = (xn − 1)/h(x) [7], where h(x) is some parity-check polynomial of degree m , having some order n , n = ord(h(x)) [8]. Below we suppose that q > 2 and gcd(n, q) = 1 . It’s also known [3, 4, 10] that any ideal is partitioned into several disjoint subsets, that is cycles, under the multiplication of ideal’s vectors http://arxiv.org/abs/0704.1091v1 by x . On the other hand, some ideal J , as a subspace of An , consists of (qm−1)/(q−1) nonzero proportionality classes. Obviously, the existence of two different partitions into some disjoint subsets of any ideal assumes a certain dependence between proportionality classes and cycles of ideal. Further, any ideal J ⊂ An is the direct sum of minimal ideals [2, 8] Ji, (2) where Ji is some minimal ideal, having an irreducible parity-check poly- nomial hi(x) of degree mi and of order ni , ni = ord(hi(x)) , 1 ≤ i ≤ t . This implies that the following polynomial h(x) = hi(x), (3) is the parity-check polynomial of J and n , n = lcm(n1, n2, . . . , nt) , is the order of h(x) [8]. It should be stressed that under the condition gcd(n, q) = 1 the polynomial (3) has no repeated factors [8]. Remark 1 . It is worth mentioning that the number n can be some number of either the primitive form n = qm − 1 or of the nonprimitive form n 6= qm−1 . In the first case, we have some cyclic primitive code and the second case corresponds to a certain cyclic nonprimitive code [11]. Let us stress that n 6= qm − 1 if and only if ni 6= q mi − 1 , 1 ≤ i ≤ t . But if there is at least one primitive polynomial among the polynomials hi(x) , 1 ≤ i ≤ t , then n = q m − 1 . Furthermore, applying the theory of linear recurring sequences [8, 12] to elements of an ideal, we obtain that every element of some ideal J ⊂ An is characterized by its unique minimal polynomial. Denote by C the set of all elements of J , having the same minimal polynomial c(x) . The set C is either some minimal ideal Ji , 1 ≤ i ≤ t , or a certain subset of all such elements of J , whose characteristic polynomial of the smallest degree coincides with c(x) . In the general case, the polynomial c(x) is equal to the product of some k , 1 ≤ k ≤ t , polynomials from t different prime divisors of h(x) . Thus, c(x) = hij (x), 1 ≤ k ≤ t. (4) This means that any element of C has the same period or the same order nc = ord(c(x)) , nc ≤ n , nc|n . Lemma 1 . For the set C , C ⊂ J , having some minimal polynomi- al c(x) in terms of (4) , the following equality takes place nc · sc = Rc · (q − 1), (5) where sc is the number of all cycles, and Rc is the number of all pro- portionality classes of C . Proof. The set C is closed under two different operations. The first operation is the cyclic shift of vector and the second one is the multipli- cation of vectors by elements of group GF (q)∗ . Hence equality (5) can be obtained by the counting of the number of all elements, belonging to C , via the two different ways. The lemma is proved. Theorem 1 . Any cycle {z(x)} of ideal (2) , having some period nz , nz|n , consists of dz subsets. The first element of each such subset is proportional to z(x) . Every such subset contains rz nonproportional to each other vectors, i.e., nz = rz · dz , dz|(q − 1) . And the number rz is the index of the subgroup, belonging to GF (q)∗ , of order dz in the group of the roots of unity, having the least possible order. Proof. Let rz , 1 ≤ rz ≤ nz , is the smallest natural number such that the following equality holds xrz · z(x) = α · z(x)mod(xnz − 1), (6) where α is some element of GF (q)∗ . Then the following rz vectors of cycle {z(x)} z(x), x · z(x), ..., xr −1 · z(x) (7) are some non–proportional to each others vectors because, assuming the inverse, we should be able to decrease the number rz , but it is impossible. Hence the set of elements (7) belongs to the following rz classes of proportionality Pz(x), Pxz(x), . . . , ..., Pxrz−1z(x). (8) Since xnzz(x) = z(x) in the ring An and also, considering (6) , we see that Pz(x) = Pxrz z(x) = Pxnz z(x) . This means that the cycle {z(x)} belongs to the classes (8) and every such class contains dz vectors, dz = nz/rz , 1 < dz ≤ q − 1 . In terms of equality (6) the following different vectors z(x), xrzz(x), x2rzz(x), . . . , x(dz−1)rzz(x) of class Pz(x) can be represented as α0z(x), αz(x), . . . , αdz−1z(x) . This implies that α0 = 1, α, α2, . . . , αdz−1 are the different elements of group GF (q)∗ . Since xnzz(x) = xrzdzz(x) = αdzz(x) = z(x) , we see that αdz = 1 . This yields that dz is the order of element α in GF (q) ∗ . Consequently, dz|q − 1 , 1 ≤ dz ≤ q − 1 . Finally, under the condition gcd(n, q) = 1 the polynomial xn − 1 has n different roots in GF (qh) field, where h is the multiplicative or- der of q modulo n [8]. Denote by E(n) the multiplicative group of n - th roots of unity over GF (q) . Let ξ ∈ E(n) be some n - th primitive root of unity. Then the following set of elements ξ0 = 1, ξ, ξ2, . . . , ξnz−1, . . . , ξn−1 represents the group E(n) . Since nz|n , we have E(nz) ⊂ E(n) , where E(nz) is the multiplicative group of nz -th roots of unity. Moreover, taking into account the isomorphism of the groups, having the same or- der [9], we can state that the subgroup {α} , {α} ⊂ GF (q)∗ , of order dz belongs to E(nz) because dz|nz . As mentioned above, nz is the period of z(x) , so that nz is the small- est divisor of n such that the following congruence xrz ≡ αmod(xnz −1) takes place. Hence E(nz) is the smallest group of n - th roots of uni- ty, which contains {α} . Since α = ξrz , we see that the following el- ements ξrzdz = 1, ξrz , ξ2rz , . . . , ξ(dz−1)rz represent the subgroup {α} in the group E(nz) . Besides that, the decomposition of E(nz) relative to the subgroup {α} consists of rz different cosets. Thus, the number rz is the index of subgroup {α} in the group of roots of unity, having the smallest possible order. The theorem is proved. Remark 2 . Notice that when a parity-check polynomial of code is some primitive polynomial of degree m and of order n = qm−1 , m > 1 , then r = R = (qm − 1)/(q − 1) and d = q − 1 . (see [2], [7]). Corrollary 1 . The period nz , nz = rz·dz ,of element z(x) , z(x) ∈ J , equals dz , dz|(q − 1) , 1 ≤ dz ≤ (q − 1) ,if and only if the cycle {z(x)} is contained in one class of proportionality. Corollary 2 . All code words of any cyclic code, havinq some length n over GF (q) , fall into some equal-weight subsets and every such subset includes all proportional to each other cycles. Besides that, consider some minimal ideal J , J ⊂ An , having an irreducible nonprimitive parity-check polynomial h(x) of degree m and of order n , n 6= qm−1 , i.e., some irreducible code of nonprimitive length. Remark 3 . The degree m of the polynomial h(x) coincides with the multiplicative order h of the number q modulo n [8]. Also, the order n of the polynomial h(x) is some divisor of qh−1 . This means that the order n can change in the following limits 1 < n < qh−1 , n 6= q − 1 . If 1 < n < q − 1 , i.e., n 6= q − 1 , then some minimal ideal J , J ⊂ An , of dimention one contains only one nonzero class of proportyonality. Con- sequently, n = d , 1 < d < q − 1 , d|q − 1 . Theorem 2 . Any cycle of minimal ideal J , J ⊂ An , having some parity-check polynomial h(x) of degree m , m > 1 , and of order n , n 6= qm − 1 , is contained in r proportionality classes, 1 ≤ r ≤ R , r|R , R = (qm − 1)/(q − 1) . Every such class consists of d , 1 ≤ d ≤ (q − 1) , different vectors of cycle, and d = gcd(q − 1, n), r = n/d. (9) Proof. All elements of minimal ideal J , J ⊂ An , have the same order n , n = ord(h(x)) . Applying the theorem 1 to some element f(x) , f(x) ∈ J , we have n = rf ·df , df |(q−1) , 1 ≤ rf ≤ n , 1 ≤ df ≤ (q−1) , and also the following equality xrf · f(x) = γ · f(x), (10) where γ is some element of GF (q)∗ . Evidently, if either rf = 1 , i.e., df = n , or rf = n and df = 1 , then equalities (9) take place. Hence, below we suppose that 1 < rf < n , 1 < df < q − 1 , and therefore q − 1 < n < qm − 1 . Taking into account (9) , we have df ≤ d . Let us show that the strong unequality df < d is impossible. Indeed, if df < d , then the subgroup {γ} , where γ is the element from equality (10) , belongs to some group of n -th roots of unity, having the order d , because df |d . Since d|(q−1) , we see that the subgroup {γ} belongs to GF (q)∗ . This means that the cycle {f(x)} is contained in one class of proportionality, i. e., rf = 1 . But this fact contradicts to the condition rf > 1 . This implies that the strong unequality df < d is impossible. Hence df = d . Because of an arbitrary choice of f(x) we can conclude that equalities (9) take place for any element of J . The theorem is proved [6]. Remark 4 . It is necessary to note that the theorem 2 is valid only for irreducible codes of non–primitive length except Reed-Solomon codes of length n = q − 1 as it was shown above. In the case of irreducible codes of primitive length n , n = qm− 1 , m > 1 , the theorem 2 will be valid if and only if gcd(m, q − 1) = 1 . Indeed, when the last condition takes place, then gcd((qm − 1)/(q − 1), q − 1) = 1 [7]. Thus, d = q − 1 = gcd(qm − 1, q − 1) that is d = gcd(n, q − 1) . It follows that the theorem 2 holds. Remark 5 . Notice that under the condition gcd(m, q − 1) = 1 the number r from (9) has no divisors of (q−1) except 1 , so gcd(r, q − 1) = 1 . This means that gcd(r, d) = 1 . Hence, considering the fact that r|R and also, taking into account (9) and the following equality s = R(q−1)/n , we have r = gcd(R, n) . Besides that, if either one from the two numbers n and q− 1 does not contain multiple prime divisors or the same prime divisors of these numbers have the same degrees under the decomposi- tion of both n and q − 1 , then the following equalities r = gcd(R, n) , gcd(r, d) = 1 also take place. Corollary 3 . Both the number r and the number d are the same numbers of all irreducible divisors of polynomial xn − 1 over GF (q) , having the same order. Corollary 4 . The number R , R = (qm−1)/(q−1) , of proportional- ity classes, of some irreducible code K ,having some length n , n 6= (qm− 1) , n = d · r , over GF (q) field, consists of some v different subsets. And every such subset contains r equal-weight proportionality classes, i.e., R = v · r . Besides that, every subset includes b equal-weight cycles, b = (q − 1)/d , 1 < b ≤ (q − 1) . So that the number of all cycles for K equals s = v · b and gcd(r, b) = 1 . Proof. According to the theorem 2 any cycle of code K is contained in r , r|R , proportionality classes. Therefore the number v = R/r gives us the common quantity of different subsets of J , each of which consists of r classes, i.e., R = v · r . The number b , b = (q−1)/d , is the number of all different equal-weight cycles, contained in every such subset, which consists of some r classes. Hence the number of all cycles for K is equal to s = v · b . Since d = gcd(q− 1, n) we see that gcd(r, b) = 1 . Actually, assuming the inverse, we would have been able to decrease the number d , but it’s impossible. The corollary is proved. Corollary 5 . The irreducible nonprimitive code K is some equidis- tant code if s = b .Besides that, the last equation is equivalent to the following ones: r = R or gcd(s, R) = 1 . Remark 6 . Note that the condition s = b was obtained in [14, 15], but only for some subclass of irreducible nonprimitive codes and under the following additional restriction gcd(b,m) = 1 . Corollary 6 . The weight of any element, belonging to some irre- ducible nonprimitive code K of length n over GF (q) , is multiple of the number d , d = gcd(q − 1, n) . Proof. The weight of any element z(x) , z(x) ∈ K , of order n , n = ord(h(x)) , is equal to the number of such j , 0 ≤ j ≤ n − 1 , for which the polynomial xj · z(x) has the following degree n−1 . According to the theorem 2 , the number of such polynomials for the cycle z(x) , having degree n − 1 , is equal to wr · d , where wr is the number of polynomials, having degree n − 1 , among the first r cyclic shifts of z(x) , and d = gcd(q − 1, n) . The corollary is proved. In addition, consider some ideal J , J ⊂ An , of the form (2) , having the parity-check polynomial h(x) in terms of (3) . Theorem 3 .If the following condition gcd(h, q− 1) = 1 , where h is the multiplicative order of number q mod n , takes place, then any cycle of set C , C ⊂ J , having some minimal polynomial of the form (4) , is contained in rc proportionality classes, rc|Rc , and every such class includes dc , dc|q − 1 , elements of cycle, that is nc = rc · dc , nc = ord(c(x)) , where Rc is the number of all proportionality classes of set C , and rc = gcd(Rc, nc), dc = gcd(q − 1, nc). (11) Proof. It is sufficient to consider the case k = 2 because the general case can be obtained by the induction. Thus assume that c(x) = h1(x)·h2(x) , where hi(x) is of degree mi and of order ni , ni = q mi − 1 , 1 ≤ i ≤ 2 , is a certain prime miltiplier of c(x) . It is known [8], that the number mi , 1 ≤ i ≤ 2 , equals either h or some divisor of this number. Hence, taking into account the theorem 2 , and also remarks 4 and 5 , we have ni = ri ·di , where ri = gcd(Ri, ni) , di = gcd(q−1, ni) , gcd(ri, q−1) = 1 , 1 ≤ ri ≤ Ri , 1 ≤ di ≤ q − 1 , and Ri = (q mi − 1)/(q − 1) , where Ri is the number of all proportionality classes of minimal ideal Ji , 1 ≤ i ≤ 2 . Therefore the order nc , nc = lcm(n1, n2) = n1 · n2/gcd(n1, n2) of the polynomial c(x) can be rewritten as nc = r1 d1 · r2 d2/gcd(r1d1 · r2d2). (12) Since gcd(ri, q − 1) = 1 , we have gcd(ri, d1 · d2) = 1 , 1 ≤ i ≤ 2 . Thus gcd(r1r2, d1d2) = 1 . Hence gcd(gcd(r1, r2), gcd(d1, d2)) = 1 so that (12) may be represented in the following form nc = (r1 r2/gcd(r1, r2)) · (d1 d2/gcd(d1, d2)). (13) Thus, nc = lcm(r1, r2) · lcm(d1, d2) . Now by rc and dc we denote lcm(r1, r2) and lcm(d1, d2) , respectively. Thus nc = rc·dc and gcd(rc, dc) = 1 . Since dc|q− 1 and gcd(rc, q− 1) = 1 , we obtain dc = gcd(q− 1, nc) . Considering (5) , it follows that nc|(Rc · (q − 1)) . Hence we have rc = gcd(Rc, nc) . Consequently the order nc of any element of set C is equal to the product of two relatively prime numbers, i.e., nc = rc · dc , where rc = lcm(r1, r2) = gcd(Rc, nc) and dc = lcm(d1, d2) = gcd(q − 1, nc) . Furthermore, applying the theorem 1 to some element a(x) ∈ C of period nc = ra · da , we have xra · a(x) = θa(x), (14) where θ ∈ GF (q)∗ is some element of order da , and ra is the smallest natural number such that equality (14) takes place. Notice that the subgroup {θ} has the order da , da < dc . If dc = 1 , then da = 1 and nc = ra = rc = gcd(Rc, nc) , so that equalities (11) hold. For this reason below we suppose that dc > 1 . If under this condition the number rc is equal to one, then nc = dc= gcd(q − 1, nc) and the theorem is valid. Therefore below we suppose that both rc > 1 and dc > 1 . Evidently, da ≤ dc . Now let us show that the inequality da < dc is not possible. Indeed, if da < dc , then we come to the following conclusion. The subgroup {θ} , where θ is the element from equality (14) , belongs to some subgroup of GF (q)∗ , having the order dc , because da|dc . Since dc is some divisor of nc , then, considering the uniqueness of subgroups, having the same order, the subgroup {θ} belongs to some group of dc - th roots of unity. This implies that the smallest group of roots of unit, containing {θ} , has an order, which either less or equals dc . Thus, both the period of a(x) and the order of c(x) must be either less or equal to dc . This yields that the order of c(x) must be some divisor of q−1 . But this fact contradicts the condition rc > 1 . Hence the inequality da < dc is impossible so that da = dc and ra = rc . Because of an arbitrary choice of a(x) equalities (11) take place for any element of set C . The theorem is proved. Corollary 7 . The order nc of reducible factor c(x) of polynomial xn − 1 , having some degree m over GF (q) , is some divisor of num- ber qm − 1 , if gcd(h, q − 1) = 1 , where h is the multiplicative order of number q mod n . Remark 7 . In terms of condition gcd(h, q − 1) = 1 , where h is the multiplicative order of number q mod n , the theorem 3 is valid for cyclic codes of both the primitive and the nonprimitive length. Also, taking into consideration the remark 5 , the order of any reducible factor of the polynomial (xn − 1) over GF (q) of degree m , is some divisor of the number (qm − 1) , if gcd (h, q − 1) = 1 . Theorem 4 . Any cycle of set C , C ⊂ J , having some minimal polynomial c(x) of the type (4) and of order nc = lcm(n1, n2, . . . , nk) , where ni 6= q mi − 1 , 1 ≤ i ≤ k , is contained in rc , rc|Rc , 1 ≤ rc ≤ Rc , classes of proportionality and every such class includes dc , 1 ≤ dc ≤ q − 1 , elements of cycle, where Rc is the number of all proportionality classes of C , and dc = gcd(q − 1, nc), rc = nc/dc. (15) Proof. It is sufficient to assume that k = 2 because the general case can be obtained by the induction. This implies that c(x) = h1(x)h2(x) , where hi(x) is some prime divisor of equality (3) , having some degree mi , and of order ni , ni 6= q mi − 1 , 1 ≤ i ≤ 2 . This yields that the theorem 2 holds for the minimal ideal Ji , 1 ≤ i ≤ 2 . Evidently, if one of the two numbers, i.e., either dc or rc is equal to one, then equalities (15) hold. For this reason below we assume that both rc > 1 and dc > 1 . Let z(x) be some vector of set C . According to the theorem 1 some cycle {z(x)} ⊂ C is contained in rz classes and every such class includes dz different elements of this cycle. Thus nc = rz dz , 1 ≤ dz ≤ q − 1 , dz|q−1 , 1 ≤ rz ≤ nc . And in addition, the following equality takes place xrz · z(x) = β z(x), (16) where β is some element of GF (q)∗ and the subgroup {β} , {β} ⊂ GF (q)∗ , has the order dz . Evidently, dz ≤ dc . Let us show that the number dz can not be smaller than dc . Assume the inverse, i.e. let dz be less than dc . Since at least one of the two numbers ether r1 or r2 is not equal to one, we see that at least one of the numbers ni , 1 ≤ i ≤ 2 , is more than q−1 , as was established in the theorem 2. This means that nc = lcm(n1, n2) > (q − 1). (17) Further, since dz|dc , we see that the subgroup {β} of order dz belongs to some subgroup of GF (q)∗ , having the order dc , where β is the element from equality (16) . Due to the uniqueness of groups, having the same order, the subgroup {β} belongs to the group of dc -th roots of unity because dc|nc . It follows that the smallest group of the roots of unity, which contains the subgroup {β} , has the order less or equals to dc . This implies that both the period of z(x) and the order of c(x) is some divisor of q − 1 . But this fact contradicts to (17) . It follows that our assumption is not true, so that dz = dc , and rz = rc . Besides that, since the number dc is the same number for every cycle of set C , we see that every subset, consisting of rc proportionality class- es, contains the same number of cycles, which is equal to bc = q − 1/dc . Moreover, since dc = gcd(q − 1, nc) , we obtain gcd(bc, rc) = 1 . Hence, taking into account the following equality sc = Rc(q − 1)/nc , which fol- lows from (5) , we see that rc is some divisor of Rc and sc = vcbc , where vc is equal to Rc/rc . The theorem is proved. Corollary 7 . (Equidistance signs of the subset C , C ⊂ J , having some minimal polynomial of the form (4) .) All vectors of the subset C , C ⊂ J , having some minimal polynomial c(x) of order nc = lcm(n1, n2, ..., nk) , ni 6= q mi − 1 , 1 ≤ i ≤ k , k > 1 , have the same weight if at least one of the following conditions holds: 1. sc = bc , 1 < bc ≤ q − 1 , 2. rc = Rc , 3. gcd(sc, Rc) = 1 . Corollary 8 . The order of the reducible factor c(x) of the polynomial xn−1 over GF (q) , having some degree m , is some divisor of the number qm− 1 , if the decomposition of the polynomial c(x) into prime multiples does not contain any primitive polynomial. Remark 8 . Note that corollaries (6) and (8) show us in what cases corollary 3.4 from [8] takes place for some reducible polynomial of the degree m over GF (q) . Also, consider some cyclic code, having the following parity-check polynomial h(x) = hi(x), (18) where hi(x) is an irreducible polynomial over GF (q) , q > 2 , of degree mi and of order ni = (q mi − 1)/bi , bi|q − 1 , 1 ≤ bi ≤ q − 1 , 1 ≤ i ≤ t , gcd(mi, mj) = 1 and gcd(bi, bj) = 1 , provided i 6= j , 1 ≤ i, j ≤ t , so that the order n of h(x) equals n = lcm(n1, n2, ..., nt) . It is worth mentioning that in [14] and [15] the following cases of polynomial (18) have been considered. Namely t = 1 and t = 2 , and besides that, with some additional restrictions, which can be omitted. Also, some particular case of polynomial (18) , that is provided bi = 1 for all i , 1 ≤ i ≤ t , was obtained in [16]. But there are some unnecessary restrictions in this paper too. Also, there are some mistakes in that paper. Namely, the order of the product for some two polynomials from (18) was defined incorrectly in [16]. Finally, using the results obtained above, we have found the minimal distance of code, having the parity-check polynomial (18) , (see [17]). Denote by M the set of degrees for the polynomials hi(x) , 1 ≤ i ≤ t , from the eq. (18) that is M = (m1, m2, ..., mt) . Let the number m denote the degree of polynomial h(x) , i. e., m = m1+m2+ ...+mt . By Mj,k , 1 ≤ j ≤ C t , 1 ≤ k ≤ t − 1 , denote j ,s k -subset of M , where Ckt is the binomial coefficient. Thus Mj,k = (mj1, mj2 , ..., mjk) , 1 ≤ j ≤ C t , 1 ≤ k ≤ t − 1 . At last, denote by mj,k the sum of degrees, from the subset Mj,k , so that mj,k = mj1 +mj2 + ... +mjk , 1 ≤ j ≤ C t , 1 ≤ k ≤ t− 1 . It is obvious that 1 ≤ mjk < m . Let us remark that provided k = 1 the number mj,k = mj,1 = mj and 1 ≤ j ≤ t because C t = t . Theorem 5 .The minimal distance of code, having the parity-check polynomial (18) , has the following form dmin = q m−1 − qmj,1−1 −− qmj,2−1 − ...−− qmj,t−1−1, t ≥ 2, dmin = q m−1(q − 1)/b, t = 1, 1 ≤ b ≤ q − 1, dmin = q m−1(q − 1), t = 1, b = 1. In conclusion I should like to express my sincere gratitude to L.A.Bassaligo, M.I. Boguslavskii and E.T. Akhmedov for helping me to correct some mistakes in the original version of my paper. References [1] Van der Waerden B.L. Algebra, vol. 1 , 1976 . [2] Macwilliams F. J., Sloane N. J. A. The theory of error-correcting codes, North-Holland, Amsterdam, Sixth Printing, 1988 . [3] Nili H. Matrixschaltungen zur codierung und decodierung von Grup- pen codes, Archiv der elektrishen ubertragung, Band 18 , Het. 9 , 1964 , S. 555− 564 . [4] Macwilliams F.J. The structure and properties of binary cyclic al- phabets, Bell System Tech. J., 44 , 303− 332 , 1965 . [5] Mullayeva Iren I. Dependence between cyclic structure of ide- al J , J ⊂ An and proportionality classes of algebra An = GF (q)[x]/(Xn−1) , AZNIINTI, Deposited scient. works, N 1(5) , p. 15 , Baku, Azerb., 1994 . [6] Mullayeva I.I. Interdependence between cyclic and weight structure of codes over GF(q) and its classes of proportyonality. RAS, Prob- lems of Information Transmition, v. 41 , No. 2 , 2005 . [7] Peterson W. W., Weldon E.J. Error - correcting codes, 2 nd, ed., M.I.T. Press, Cambridge, Mass., 1972 . [8] Lidl R., Niderreiter H. Finite fields, Addison-Wesley Publ. Com., Massachusetts, 1983 . [9] Kurosch. Higher algebra, 1962 . [10] Elspas B. The theory of autonomous linear sequential networks, IRE Transactions on circuit theory, CT- 6 , N 1 , March ( 1959 ), 45−60 . [11] Richard E. Blahut. Theory and practice of error control codes, Addison-Wesley Publ. Com., Massachusetts, 1984 . [12] Zierler N. Linear recurring sequences, J. Soc. Ind. Appl. Math., 7 ( 1959 ), N 1 , 45− 60 . [13] Oganesyan S., Tairyan V., Yagdzyan V. Decomposition of cyclic codes into equel-weight classes, Probl. of Control and Inform. Th., 3 ( 1974 ), N 2 , 117− 125 . [14] Clark W. E. Equidistant cyclic codes over GF(q), Discrete Math., 17 ( 1977 ), N 2 , 139− 141 . [15] Oganesyan S.,Yagdzyan V., Tairyan V. On a class of optimal cyclic codes, Proc. 2 nd Int. Symp. on Inform. Theory, Tsahkadzor, Arme- nia, USSR, 1971 , Budapest: Acad. Kiado, 1973 , 219− 224 . [16] Tenengolts G. M. A new class of cyclic error-corecting codes, Proc. 3 nd Confr. on Coding and Transm. Inform., Soviet Un., Ugegorod, 1967 , 18− 28 . [17] Mullayeva Iren I. Equidistance criterion of cyclic subset for ideal J , of algebra An = GF (q)[x]/(x n−1) , consisting of polynomials mod- ulo xn − 1 over Galois field GF (q) , q > 2 , AZNIINTI, Deposited scient. works, N2(6) , p. 16 , Baku, Azerbaijan, 1994 .

0704.1092

Simplifying additivity problems using direct sum constructions

Simplifying additivity problems using direct sum constructions Motohisa Fukuda1, Michael M. Wolf2 1Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge 2 Max-Planck-Institute for Quantum Optics, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany. (Dated: August 20, 2021) We study the additivity problems for the classical capacity of quantum channels, the minimal output entropy and its convex closure. We show for each of them that additivity for arbitrary pairs of channels holds iff it holds for arbitrary equal pairs, which in turn can be taken to be unital. In a similar sense, weak additivity is shown to imply strong additivity for any convex entanglement monotone. The implications are obtained by considering direct sums of channels (or states) for which we show how to obtain several information theoretic quantities from their values on the summands. This provides a simple and general tool for lifting additivity results. I. INTRODUCTION A central question in classical and quantum information theory is, how much information can be transmitted through a given noisy channel. For classical channels the maximal asymptotically achievable rate—the capacity—was derived in the seminal work of Shannon [1]. For quantum channels, however, the matter is complicated by the existence of entan- glement and the possibility of exploiting it in the encoding to protect information against decoherence. If one excludes this possibility, a capacity formula for the transmission of clas- sical information through quantum channels was proven by Holevo [2] and Schumacher and Westmoreland [3] (HSW). Since then, considerable effort was devoted to the question whether (or in which cases) entangled inputs can lead to rates beyond the HSW capacity. This issue—the additivity problem for the HSW capacity—is still undecided, although for sev- eral classes of channels additivity has been shown to be true, i.e., entanglement does not seem to help in any case (see, e.g., [4, 5, 6, 7] and references therein). Instead, other additiv- ity problems appeared which are similar in spirit but concern very different quantities like the minimal output entropy and the entanglement of formation, an entanglement measure for bipartite states for which in addition strong super-additivity has been conjectured. A major conceptional insight was then gained in [8, 9, 10, 11] where it was shown that all these additivity problems are globally equivalent in the sense that if additivity holds for one of these quantities in general, then it does so for all of them. Here ‘in general’ means that it has to be true for arbitrary pairs of channels (or states), a condition we will call strong additiv- In this work we present a further conceptional simplifica- tion of these and related additivity problems. We show that strong additivity is implied by weak additivity, meaning ad- ditivity for arbitrary pairs of equal channels or states. More- over, based on [12] we argue that it suffices to consider pairs of identical unital channels only. This observation may be a small step on a notorious path but it might guide future research as it for instance underlines recent attempts to un- derstand the asymptotic structure of tensor powers of unital channels [13]. Moreover, one may think of other additivity questions than the ones stated above for which our techniques could be of use. In particular, we think of regularized quanti- ties (like quantum capacities (cf. [14, 15]) or certain entangle- ment measures) for which weak additivity holds by definition. Our main tool is the use of direct sums of channels or states. For the latter case similar constructions appeared in [16, 17, 18]. We begin with a discussion of direct sum chan- nels. This will contain more than what is needed for the sub- sequent additivity results as we think that these tools might be of independent interest. II. DIRECT SUMS OF QUANTUM CHANNELS We consider direct sums of channels, i.e., completely posi- tive and trace preserving maps of the form T = ⊕iTi, where each Ti is a channel in its own right. Our aim in this section is to express information theoretic functionals of T in terms of their values for the Ti’s. The definition of the quantities ap- pearing in the following proposition will be given in the proof. Proposition 1 (Direct sums) Consider a direct sum T = ⊕ni=1Ti, n ∈ N of arbitrary finite dimensional channels. Then 1. Minimal output α-Renyi entropy (α ≥ 1): Smin,α = min Smin,α(Ti) , (1) 2. Coherent information: = max J(Ti) , (2) 3. Mutual information: = max S({λi}) + λiI(Ti), (3) = log 2I(Ti), (4) where {λi} is a probability distribution and S({λi}) its entropy. 4. HSW capacity: = max S({λi}) + λiχ(Ti) (5) = log 2χ(Ti). (6) http://arxiv.org/abs/0704.1092v2 Remark: Let us briefly comment on the interpretation of the above formulas. Concerning the HSW capacity, classical information can either be sent through the channelsTi or it can be encoded in the choice of blocks i = 1, . . . , n. Eq.(5) shows exactly the competition between these two ways of communi- cating classical information. For the quantum mutual infor- mation, which gives the entanglement assisted capacity [19], we obtain the same interpretation (note that the ’2’ comes from the fact that we take log in base 2). The coherent infor- mation is related (via regularization) to the quantum capacity [20]. In this case encoding information in the choice of blocks is not possible—this would be purely classical as all the co- herences get lost. Similarly, for the minimal output entropies the minimum is obtained by putting all the weight into the least noisy channel. Proof. 1. The α-Renyi entropy is defined as Sα(ρ) = log tr[ρα] = log ‖ρ‖α, (7) for 0 ≤ α ≤ ∞. Here ‖ · ‖p is the Schatten p-norm. When α = 1 the functional is defined by its limit which is the minimal output entropy Smin(T ) = infρ S(T (ρ)) with S(ρ) = −trρ log ρ the von Neumann entropy. Let us consider this case first. As the direct sum ⊕iTi erases the off-diagonal blocks so that all possible outputs can be obtained upon block-diagonal inputs we can restrict to ρ = ⊕iρ̃i. Here ρ̃i is not necessarily normalized so that the weights tr[ρ̃i] =: λi form a probability distribution. Writing ρi := ρ̃i/λi and using the concavity of von Neumann entropy we get ⊕i Ti(λiρi) λiS(Ti(ρi)). (8) This leads to Eq.(1) when α = 1. For α > 1 the minimization of Sα(T (ρ)) amounts to a maximization of ‖T (ρ)‖α and the result follows from convexity of ‖ · ‖α in a similar way. 2. The coherent information is defined as J(T ) = sup T (ρ) T ⊗ id(Ψ) , (9) where Ψ is a purification of ρ such that ρ = trBΨ. Since T and T ⊗ id erase the off-diagonal blocks we can replace ρ and Ψ by their diagonal blocks: ⊕iλiρi and ⊕iλiΨi. Here, Ψi is an extension of ρi. Since the conditional entropy S(ρAB)−S(ρA) is concave in ρAB [21] considering a convex decomposition of each Ψi into pure states shows Eq.(2) in a similar way as above. 3. The mutual information defined as I(T ) = sup S(ρ) + S T (ρ) T ⊗ id(Ψ) is concave in ρ so the maximum will be achieved by a block diagonal ρ = ⊕iλiρi for T = ⊕iTi. To see this, let V = ⊕j exp{2πij/n}Ij and average V kρV ∗k over k = 1, . . . , n. Take a purification Ψ of ρ = ⊕iλiρi, and then replace Ψ by its diagonal blocks: ⊕iλiΨi as before. However, each Ψi is a purification of ρi in this case. Indeed, suppose ρi = pij |ij〉〈ij|, where {|ij〉}j is an orthonormal basis in the ith subspace. Then, Ψ is λiλkpijpkl|ij〉〈kl| ⊗ |ij〉〈kl|, and its ith diagonal block λiΨi is pijpil|ij〉〈il| ⊗ |ij〉〈il| = λi|Ψi〉〈Ψi|. Here, |Ψi〉 = pij |ij〉 ⊗ |ij〉. Exploiting this together with S(λρ) = λ(S(ρ) − logλ) then gives Eq.(3). Eq.(4) fol- lows then from determining the optimal λi via Lagrange mul- tipliers in the following way. The maximization problem of S({λi})+ λici for a probability distribution {λi} amounts then to maximizing S({λi}) + λici + Λ λi − 1 , (11) where Λ is the Lagrange multiplier. Taking partial derivatives we obtain for extremal {λ̃i}: − log λ̃i − + ci + Λ = 0 ∀i (12) λ̃i − 1 = 0. (13) Hence (12) shows ci − log λ̃i is a constant, say, C for ∀i, and by (13) we get C = log ci . Therefore S({λ̃i}) + λ̃ici = λ̃iC = log 2ci . (14) As this is lower bounded by mini[I(Ti)] it must be the maxi- 4. The HSW capacity χ is given by χ(T ) = sup S(T (ρ))−HT (ρ), (15) where HT (ρ) = inf pkρk=ρ pkS(T (ρk)). (16) Here, HT (ρ) is the convex closure of the output entropy; {pk} is a probability distribution and ρk are density matrices. The r.h.s. of Eq.(15) for a fixed average input state ρ is a constraint HSW capacity which we will denote by χ(T, ρ). Since the inputs can again be assumed to be block-diagonal we have: χ (⊕iTi,⊕iλiρi) = S(⊕iλiTi(ρi))− λiHTi(ρi) = S({λi}) + λiχ(Ti, ρi). (17) The first equality is explained by the fact that since the von Neumann entropy is concave there is an optimal decompo- sition of ⊕iλiρi for which each state has its support in one of the diagonal blocks. The second equality comes from S(λiρi) = λi(S(ρi) − logλi). Taking the supremum over all states {ρi} then leads to Eq.(5). Again, Eq.(6) is obtained by using Lagrange multipliers as above. In fact, for unital channels (6) has been obtained in [30]. III. SIMPLIFYING ADDITIVITY PROBLEMS Let us now turn to the additivity conjectures and exploit Prop.1 in order to show that in several cases weak additivity (for equal channels or states) implies strong additivity (i.e., for different ones). Proposition 2 (Reduction for channels) The following (in-) equalities hold for arbitrary pairs of different channels T1 and T2 iff they hold for arbitrary equal pairs T1 = T2. 1. Smin,α(T1 ⊗ T2) = Smin,α(T1) + Smin,α(T2), for any α ≥ 1. 2. χ(T1 ⊗ T2) = χ(T1) + χ(T2). 3. HT1⊗T2(ρ) ≥ HT1(ρ1) +HT2(ρ2) for all states ρ with respective subsystems ρ1, ρ2. 4. HT1⊗T2(ρ1⊗ρ2) = HT1(ρ1)+HT2(ρ2) for all product states ρ1 ⊗ ρ2. Remark: The conjectured equality in 1. is the additiv- ity of the minimal output entropy when α = 1 [22], and it becomes the multiplicativity of maximal output p-norms for p = α > 1. This was conjectured to be true for all α ∈ [1,+∞] before a counterexample was found [23] ruling out all values α > 4.79. The equation in 2. is the conjec- tured additivity of the HSW capacity, which gives the clas- sical capacity as long as entangled states are not allowed to be used in the encoding [2, 3]. The additivity would show that the HSW capacity itself is the unconstrained classical ca- pacity of quantum channels. The conjectures 3. and 4. are called strong superadditivity and additivity of the convex clo- sure of the output entropy. When T1, T2 are partial traces they become strong superadditivity and additivity of entanglement of formation, respectively, which we discuss in greater detail below. We note that Prop.2 remains valid in the case where ‘arbi- trary channels’ refers to a restricted set of channels which is closed under direct sums and tensor products. Proof. 1. Let σ1 and σ2 be optimal output states for T1 and T2 respectively. Then, form the following two channels: T ′1(ρ) = T1(ρ)⊗ σ2, T ′2(ρ) = σ1 ⊗ T2(ρ). (18) It is not difficult to see that T1 ⊗ T2 and T ′1 ⊗ T ′2 share the additivity property. Hence we can assume that T1 and T2 have the same optimal output: Smin,α(T1) = Smin,α(T2). If we apply first weak additivity and then Prop. 1.1. we obtain: Smin,α(((T1 ⊕ T2)⊗ (T1 ⊕ T2))) (19) = 2Smin,α(T1 ⊕ T2) = Smin,α(T1) + Smin,α(T2). (20) On the other hand, if we first apply Prop. 1.1. and then weak additivity, we obtain that (19) and thus (20) is upper bounded by Smin,α(T1 ⊗ T2). The converse inequality is trivial. 2. Consider T1 ⊕ T2 T1 ⊕ T2 = 2 log 2χ(T1) + 2χ(T2) = log 22χ(T1) + 22χ(T2) + 2χ(T1)+χ(T2)+1 This follows from first applying weak additivity and then the proposition 1.4. On the other hand, applying them in reverse order we have T1 ⊕ T2 = log 2χ(T1⊗T1) + 2χ(T2⊗T2) + 2 · 2χ(T1⊗T2) = log 22χ(T1) + 22χ(T2) + 2χ(T1⊗T2)+1 Together they prove the claimed equality. For 3. we obtain by weak superadditivity, HT1⊗T2(ρ) = H(T1⊕T2)⊗(T1⊕T2)(0⊕ ρ⊕ 0⊕ 0) ≥ HT1⊕T2(ρ1 ⊕ 0) +HT1⊕T2(0⊕ ρ2) = HT1(ρ1) +HT2(ρ2). (21) Here, ρ1, ρ2 are reduced states of ρ. This proves 3. and the statement 4. follows in a similar way when replacing ρ by a product state. Proposition 3 (Unital channels) Proving one of the conjec- tures in proposition 2 for all pairs of identical unital channels would show the conjecture is true for arbitrary channels. Proof. In [12] a unital channel T̃ is constructed for a given channel T so that these two channels T̃ and T share the fol- lowing additivity properties: additivity of minimal output α- Renyi entropy, and strong superadditivity and additivity of the convex closure of the output entropy. Hence these conjectures can be restricted to products T̃1 ⊗ T̃2 for all channels T1, T2. As for the HSW, we have the same reduction but for a differ- ent reason (See the remark below). Finally, for the above two unital channels T̃1, T̃2 we can construct the direct sum T̃1⊕ T̃2 which is again a unital channel. Then the result follows from the proof of proposition 2. Remark: We explain local relation between minimal out- put entropy and the HSW capacity, which was implicitly writ- ten but not clear in [12]. Since the unital extension T̃ sort of mixes up outputs of T we have the following formula. χ(T̃1 ⊗ T̃2) = log d1d2 − Smin(T̃1 ⊗ T̃2), (22) where d1, d2 are the dimensions of the output spaces of T̃1 and T̃2 respectively. Hence the additivity of HSW capacity is equivalent to the additivity of the minimal output entropy for products of those extensions T̃1 ⊗ T̃2 by Eq.(22). Hence the additivity conjecture of the HSW capacity can also be re- stricted to products T̃1 ⊗ T̃2 for all channels T1, T2 by using global equivalence [8, 9, 10, 11]. Finally, we will discuss additivity issues for entanglement measures. The one already mentioned is the entanglement of formation which was introduced in [24]. Since then the fol- lowing conjectures have been considered: EF (ρ) ≥ EF (ρ1) + EF (ρ2) (23) EF (ρ1 ⊗ ρ2) = EF (ρ1) + EF (ρ2). (24) In fact, both are again globally equivalent to the additivity of the HSW capacity and the minimal output entropy. Moreover, additivity would imply that EF equals an important opera- tionally defined entanglement measure, the entanglement cost Ec, since Ec(ρ) = limn→∞ EF (ρ ⊗n) [25]. The entanglement of formationEF (ρ) is the convex closure of output entropy HT (ρ) when T is a partial trace. Following a similar strategy as above we will now show that strong additivity in the sense of Eq.(24) is again implied by weak additivity (i.e., Eq.(24) with ρ1 = ρ2). In fact, this will not only hold for EF but for any convex entanglement monotone [24, 26, 27]. The main reason behind is that every such functional satisfies [28]: f(⊕iλiρi) = λif(ρi), (25) where {λi} is a probability distribution and ρi are states as before. Proposition 4 (Convex entanglement monotones) [29] Suppose f is a convex entanglement monotone which is weakly additive, i.e., f(ρ1 ⊗ ρ2) = f(ρ1) + f(ρ2) for all ρ1 = ρ2. Then f is strongly additive in the sense that this holds also for all ρ1 6= ρ2. Proof. Let ρ = 1 (ρ1 ⊕ ρ2). Then f(ρ⊗ ρ) = 2f(ρ) = f(ρ1) + f(ρ2). (26) Here, we applied the weak additivity and then (25). Applying them in reverse order we get f(ρ⊗ ρ) = i,j=1 f(ρi ⊗ ρj) (f(ρ1) + f(ρ2) + f(ρ1 ⊗ ρ2)). (27) Using similar ideas, it has recently been shown that for reg- ularized entanglement measures like Ec or the asymptotic rel- ative entropy of entanglement, monotonicity (i.e., essentially Eq.(25)) and strong additivity are equivalent [18]. Acknowledgement M.F. would like to thank his supervisor Y.M.Suhov for constant encouragement and numerous discus- sions. 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0704.1093

Modulational instability in nonlocal Kerr-type media with random parameters

Modulational instability in nonlocal Kerr-type media with random parameters E.V. Doktorov∗ and M.A. Molchan† B.I. Stepanov Institute of Physics, 68 F. Skaryna Ave., 220072 Minsk, Belarus Modulational instability of continuous waves in nonlocal focusing and defocusing Kerr media with stochastically varying diffraction (dispersion) and nonlinearity coefficients is studied both analyt- ically and numerically. It is shown that nonlocality with the sign-definite Fourier images of the medium response functions suppresses considerably the growth rate peak and bandwidth of insta- bility caused by stochasticity. Contrary, nonlocality can enhance modulational instability growth for a response function with negative-sign bands. PACS numbers: 42.25.Dd, 42.70.-a, 42.65.Jx I. INTRODUCTION Modulational instability (MI) in nonlinear media is a destabilization mechanism which produces a self-induced breakup of an initially continuous wave into localized (solitary wave) structures. This phenomenon was pre- dicted in plasma [1, 2], nonlinear optics [3, 4], fluids [5] and atomic Bose-Einstein condensates [6, 7, 8]. MI of continuous waves can be used to generate ultra-high repetition-rate trains of soliton-like pulses [9, 10, 11]. It is common knowledge that MI is absent in the defocusing Kerr medium and presents as the long-wave instability with a finite bandwidth in the focusing Kerr medium [12]. The above results were obtained for media with de- terministic parameters. Contrary, in realistic media the characteristic parameters are not constants, as a rule, but fluctuate randomly around their mean values. It was shown in the setting of nonlinear optics that stochastic inhomogeneities in a Kerr-type medium extend the do- main of MI of continuous waves, as compared with deter- ministic systems, over the whole spectrum of modulation wavenumbers, even for the defocusing regime [13, 14, 15]. A comprehensive review of MI of electromagnetic waves in inhomogeneous and in discrete media is given in Ref. [16]. Another important aspect of a class of realistic nonlin- ear media is concerned with their nonlocality. Nonlocal- ity is typically a result of underlying transport processes such as heat conduction in thermal nonlinear media [17], diffusion of atoms in a gas [18], long-range electrostatic interaction in liquid crystals [19], charge carrier transfer in photorefractive crystals [20, 21], and many-body in- teraction in Bose-Einstein condensates [22]. Nonlocality can prevent the collapse of self-focused beams [23, 24] and dramatically alter interaction between dark solitons [25]. MI in deterministic nonlocal Kerr-type media was stud- ied in Refs. [26, 27], and it was shown that nonlocality does not produce MI in the defocusing case for small and moderate values of the product “modulation amplitude ∗Electronic address: [email protected] †Electronic address: [email protected] × nonlocality parameter”. In the present paper we unite the two above lines of study of nonlinear media and analyze MI in nonlocal me- dia with stochastic parameters. Since nonlocality spreads out localized excitations, it is reasonable to expect a par- tial suppression of the stochasticity-induced MI gain. In- deed, we demonstrate that the aforementioned situation with MI in local stochastic media with the sign-definite Fourier images of the response functions changes drasti- cally, if nonlocality is taken into account. Namely, both the growth rate peaks and bandwidths of instability are considerably decreased. On the other hand, there can be an “anomalous” behavior of nonlocality when the Fourier image of the response function of a nonlocal medium al- lows for sign-negative bands. In this case the MI gain of a nonlocal medium can exceed that of a local stochastic medium for some values of the modulation wavenumber. We adopt the nonlocal nonlinear Schrödinger equation with random coefficients as a model to reveal peculiar- ities of MI of continuous waves. The results obtained are illustrated by the white noise model for parameter fluctuations and by response functions of several types. II. MODEL The propagation of an optical beam along the z axis in a nonlocal medium with random parameters is governed by the nonlinear Schrödinger equation iuz + d(z)uxx + g(z)u dx′R(x− x′)|u|2(x′, z) = 0. (2.1) Here x is the transverse coordinate, u(x, z) is the complex envelope amplitude and we use the standard dimension- less variables. The group velocity dispersion (or diffrac- tion) coefficient d(z) and nonlinearity coefficient g(z) are considered as stochastic functions which fluctuate around their mean values d0 (d0 > 0) and g0 (g0 ≷ 0): d(z) = d0(1 +md(z)), g(z) = g0(1 +mg(z)). (2.2) Here md and mg are independent zero-mean random pro- cesses of the Gaussian white-noise type, 〈md〉 = 〈mg〉 = 0, 〈md(z)md(z′)〉 = 2σ2dδ(z − z′), http://arxiv.org/abs/0704.1093v1 mailto:[email protected] mailto:[email protected] 〈mg(z)mg(z′)〉 = 2σ2gδ(z − z′)〉, and the angle brackets stand for the expectation with respect to the distribution of the processes md(z) and mg(z). The integral in equation (2.1) represents the field- intensity dependent change of the refractive index char- acterized by the normalized symmetric response function R(x), dxR(x) = 1. The delta-function response function R(x) = δ(x) corresponds to the local limit of the model. We will discriminate between the focusing (g0 > 0) and defocusing (g0 < 0) media. Eq. (2.1) possesses the homogeneous plane wave solu- u0 = A exp dz′g(z′) , (2.3) where A is a real amplitude. Now we perform the linear stability analysis of the solution (2.3). Assume that u(x, z) = (A+ v(x, z)) exp dz′g(z′) (2.4) is a perturbed solution of Eq. (2.1) with v(x, z) being a small complex modulation. Substituting Eq. (2.4) into Eq. (2.1) and linearizing about the plane wave (2.3), we get a linear equation for v(x, z): ivz + d(z)vxx + 2g(z)A dx′R(x− x′)Re v(x′, z) = 0. (2.5) After decomposing v into real and imaginary parts, v = r(x, z) + is(x, z), and performing the Fourier transforms ρ(k, z) = dx r(x, z)eikx, σ(k, z) = dx s(x, z)eikx, R̂(k) = dxR(x)eikx, Eq. (2.5) is converted to a system of linear equations for ρ and σ: d(z)k2 d(z)k2 + 2g(z)A2R̂ 0 (2.6) If we were deal with the deterministic system with the parameters d0 and g0, Eq. (2.6) would be the main object to study MI [26]. However, MI induced by the random fluctuations is not captured by the analysis of the first moments 〈ρ〉 and 〈σ〉 [15], and it is necessary to compute the modulational intensity growth given by the higher- order moments. III. THE SECOND-ORDER MOMENT MI GAIN We consider the second moments 〈ρ2〉, 〈ρσ〉 and 〈σ2〉 as constituents of the column vector X(2) = 〈ρ2〉, 〈ρσ〉, 〈σ2〉 . (3.1) The moment 〈ρσ〉 is added to close the equations for the second-order moments. Then we should calculate z- evolution of the vector X(2). Its first component gives 〈ρ2〉 = 2〈ρzρ〉 = d0k2〈ρσ〉 + d0k2〈md(z)ρσ〉, in accordance with Eqs. (2.2) and (2.6). For decoupling of the mean 〈md(z)ρσ〉 we apply the Furutsu-Novikov formula [28, 29] 〈md(z)ρσ〉 = dyσ2dB(z − y) . (3.2) Here B(z − y) = δ(z − y) for the white-noise Gaussian random process, while the functional derivative (δ/δmd) is calculated from Eq. (2.6). Indeed, writing ρ(z) as the integral ρ(z) = (1/2)k2 dyd(y)σ(y) (and the similar integral for σ(z)) and accounting for the explicit repre- sentation (2.2) of d(z) in terms of md gives δ(ρσ) σ + ρ 2(σ2 − ρ2). Therefore, 〈md(z)ρσ〉 = (1/2)σ2dd0k2(〈σ2〉 − 〈ρ2〉) and finally 〈ρ2〉 = d0k2〈ρσ〉+ 4(〈σ2〉 − 〈ρ2〉). Just in the same way we can calculate z-derivatives of the other components of the vector X(2). As a result, we obtain the evolution equation (d/dz)X(2) = M (2)X(2) with the 3× 3 matrix M (2) of the form M (2) = 4 d0k 2R̂ − 1 2 −σ2dd20k4 16σ2gg 4R̂2 + σ2dd 2R̂− 1 . (3.3) Eigenvalues of M (2) with positive real parts lead to in- stabilities, and the largest positive value determines the MI gain G2(k). The eigenvalues λj are easily found from Eq. (3.3) but they are too cumbersome to be reproduced here explicitly. Below we separately analyze the cases of the defocusing (g0 < 0) and focusing (g0 > 0) nonlin- earities. Following [26], we will use for illustration the Gaussian response function RG(x) = , R̂G(k) = exp (3.4) and the exponential one Re(x) = , R̂e(k) = 1 + a2k2 (3.5) as examples of the response functions with the sign- definite Fourier images, as well as the rectangular re- sponse function Rr(x) = for |x| ≤ a, 0 for |x| > a, R̂r(k) = sin(ak) (3.6) whose Fourier transform has negative-sign bands. Here a is the nonlocality parameter, a → 0 means R(x) → δ(x) and R̂(k) → 1. A. Defocusing nonlinearity For the defocusing nonlinearity g0 < 0 we obtain one real eigenvalue λ1 and two complex conjugate ones λ2 and λ3. Numerical analysis shows that λ1 is positive for all k2, while λ2 and λ3 have negative real parts for the Gaussian and exponential response functions. Let us re- mind that there is no MI for g0 < 0 for local deterministic Kerr media, while randomness of the coefficients d(z) and g(z) completely destroys stability of the continuous wave solution. This situation considerably changes for nonlo- cal media. Indeed, Fig. 1 clearly shows that nonlocality with the sign-definite response functions suppresses both the growth rate peak of G2(k) ≡ λ1 and MI bandwidth, the latter being practically finite. When the nonlocality parameter a grows, the suppression effect becomes more pronounced. Somewhat different situation takes place 0 1 2 3 4 0 1 2 3 4 FIG. 1: Defocusing media. Plots of the MI gain G2(k) for local stochastic medium (solid line), nonlocal stochastic media with the Gaussian (dash-dotted line) and exponential (dashed line) response functions. Here d0 = 2, |g0|A2 = 1, σ2d = σ2g = 0.1. Upper panel: a = 1; lower panel: a = for the rectangular response function (3.6). For suffi- ciently high nonlocality, MI gain maximum for a given wavenumber k can exceed the corresponding value of G2 for a local random medium (Fig. 2). Besides, the MI bandwidth becomes strictly finite in this limit. B. Focusing nonlinearity In the case of the focusing nonlinearity (g0 > 0) a local deterministic medium produces the long-wave instability with a finite bandwidth. Stochasticity of medium param- eters extends the bandwidth to the whole spectrum of modulation wavenumbers. Calculation of eigenvalues of the matrix M2 (3.3) with g0 > 0 demonstrates that non- locality suppresses the MI gain and bandwidth for media with both sigh-definite (Fig. 3) and sign-indefinite (Fig. 0 1 2 3 4 FIG. 2: Defocusing media with the rectangular response func- tion. Plots of the MI gain G2(k) for local stochastic medium (solid line), nonlocal stochastic media with a = 2 (dashed line), a = 6 (dash-dotted line), and a = 10 (dotted line). Other parameters are the same as in Fig. 1. 0 1 2 3 4 0 1 2 3 4 FIG. 3: Focusing media. Plots of the MI gain G2(k) for a lo- cal deterministic medium (solid line), local stochastic medium (dotted line), nonlocal stochastic media with the Gaussian (dash-dotted line) and exponential (dashed line) response functions. Here d0 = 2, g0A 2 = 1, σ2d = 0.1, σ g = 0.2. Upper panel: a = 1; lower panel: a = 2.5. 4) response functions. Notice that stronger nonlocality is needed for focusing media to achieve a reduction of the MI gain, as compared with defocusing ones. Besides, maximum positions of the MI gains shift toward smaller wavenumbers k under nonlocality growth, producing fi- nite bandwidth. 0 0.5 1 1.5 2 2.5 3 FIG. 4: Focusing media. Plots of the MI gain G2(k) for a local deterministic medium (solid line), local stochastic medium (dotted line), nonlocal stochastic media with the rectangular response function: a = 2 (dash-dotted line), a = 8 (dashed line). Other parameters are the same as in Fig. 3. IV. HIGHER-ORDER MOMENTS The second-order moments (3.1) do not provide an analysis of the MI gain in stochastic media with sufficient detail. In particular, it is important to see fluctuations of the exponential growth of the modulation amplitude. More deep insight into the problem demands to account for higher-order moments X(2n) = < ρ(2n−j)σj > , j = 0, . . . , 2n. (4.1) In this section we study the interplay of nonlocality and exponential growth of the higher momentsX(2n) in virtue of stochasticity. As before, applying the Furutsu-Novikov formula (3.2), we obtain a matrix M (2n) in the form M (2n) = d0k 2A(2n) + 2R̂ (4.2) B(2n) + d20k 4σ2dC (2n) + 16g20A 4R̂2σ2gD (2n). Non-zero entries of the matrices A(2n), B(2n), C(2n) and D(2n) are written as j,j+1 = n− j,j−1 = j; C jj = − (n+2nj−j2), j,j+2 = j + 1 , (4.3) j,j−2 = D j,j−2 = j(j − 1), j = 0, . . . , 2n. Then the maximal real part of roots of the characteristic polynomial det |M (2n) − λI| will give nG2n(k). Since all the matrix elements of M (2n) are real and the character- istic polynomial is of the odd degree, at least one of the eigenvalues of M (2n) is real and the others are mutually complex conjugate. In what follows we will consider the 4-th and 6-th moments. A. Defocusing nonlinearity In Fig. 5 we show the results of calculating MI gains G2, G4 and G6 for both the exponential and Gaussian re- sponse functions and compare them with the same curves for local stochastic media obtained in [15]. It is seen that nonlocality suppresses the higher-order moments as well. Notice that in defocusing media positions of MI gain maxima for moments of different orders coincide [15] (they are deterministic rather than random). Nonlocality does not disturb this property. Fig. 6 demonstrate simi- lar curves for the rectangular response function for differ- ent values of the nonlocality parameter a. It is seen that for sufficiently high a the medium demonstrates prac- tically coinciding distributions of higher-moment growth rates, their maxima being shifted to shorter wavelengths. Evidently, higher-order moments for the rectangular re- sponse function manifest the same “anomalous” enhance- ment of the growth rate in a narrow region of modulation wavenumbers, as compared with the local stochastic case. 0 1 2 3 4 0 1 2 3 4 FIG. 5: Defocusing media. Plots of the MI gains G6 (solid line), G4 (dashed line), and G2 (dash-dotted line) for a lo- cal stochastic medium (upper three curves) and for nonlocal stochastic media (lower three curves). Here d0 = 2, a 2 = 1, |g0|A2 = 1, σ2d = σ2g = 0.1. Upper panel: exponential re- sponse function; lower panel: Gaussian response function. B. Focusing nonlinearity For the focusing media the higher-order MI gains demonstrate much the same behavior as for the defo- cusing ones for both sign-definite and sign-indefinite re- 0 1 2 3 4 5 0 1 2 3 0 0.5 1 1.5 FIG. 6: Defocusing nonlocal stochastic media with the rect- angular response function. Plots of the MI gains G6 (solid line), G4 (dashed line), and G2 (dash-dotted line) for d0 = 2, |g0|A2 = 1, σ2d = σ2g = 0.1. Upper panel: a = 2; middle panel: a = 6; lower panel: a = 8. sponse functions. Fig. 7 shows the MI gains for local and nonlocal stochastic media for the Gaussian response function. Curves for the exponential and rectangular re- sponse functions are qualitatively the same. With in- creasing the nonlocality parameter a, curves for MI gains of different orders become closer one the other, so high nonlocality smoothes fluctuations of the modulation am- plitude growth. 0 0.5 1 1.5 2 2.5 3 FIG. 7: Focusing media. Plots of the MI gains G6 (solid line), G4 (dashed line), and G2 (dash-dotted line) for a lo- cal stochastic medium (upper three curves) and for nonlo- cal stochastic medium with the Gaussian response function (lower three curves). Here d0 = 2, a = 2, g0A 2 = 1, d = σ g = 0.1. V. CONCLUSION Within the limits of the linear stability analysis, we have investigated the MI of a homogeneous wave in a nonlocal nonlinear Kerr-type medium with random pa- rameters. For the case of the white-noise model of pa- rameter fluctuations, we derived the equations which gov- ern the dependence of the MI gain on the modulation wavenumber. As was expected from physical motiva- tions, nonlocality causes considerable suppression of the stochasticity-induced MI growth rate for media with the sign-definite Fourier images of the response functions. At the same time, nonlocal media with the sign-indefinite Fourier images of the response functions can display a somewhat different behavior leading to an increase, as compared with local media, of the MI gain for some do- mains of modulation wavenumbers. Acknowledgments The authors are very grateful to F. Abdullaev and J. Garnier for constructive comments. [1] G.A. Askar’yan, Zh. Eksp. Teor. Fiz. 42, 1576 (1962) [Sov. Phys. JETP 15, 1088 (1962)]. [2] T. Taniuti and H. Washini, Phys. Rev. Lett. 21, 209 (1968). [3] V.I. Bespalov and V.I. Talanov, Pis’ma Zh. Eksp. Teor. Fiz. 3, 471 (1966) [JETP Lett. 3, 307 (1966)]. [4] L.A. Ostrovskii, Zh. Eksp. Teor. Fiz. 51, 1189 (1966) [Sov. Phys. JETP 24, 797 (1967)]. [5] T.B. Benjamin and J.E. Feir, J. Fluid Mech. 27, 417 (1967). [6] B. Wu and Q. Niu, Phys. Rev. A 64, 061603 (2001). [7] V.V. Konotop and M. Salerno, Phys. Rev. A 65, 021602(R) (2001). [8] L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. Lett. 91, 080405 (2003). [9] K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986). [10] G. Millot, E. Seve, S. Wabnitz, and M. Haelterman, J. Opt. Soc. Am. B 15, 1266 (1998). [11] K.E. Steeker, G.B. Partridge, A.G. Truscott, and R.J. Hulet, Nature 417, 150 (2002). [12] Yu.S. Kivshar and G.P. Agraval, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, San Diego, 2003. [13] F.Kh. Abdullaev, S.A. Darmanyan, S. Bishoff, and H.P. Sørensen, J. Opt. Soc. Am. B 14, 27 (1997). [14] M. Karlsson., J. Opt. Soc. Am. B 15, 2269 (1998). [15] J. Garnier and F.Kh. Abdullaev, Physica D 145, 65 (2000). [16] F.Kh. Abdullaev, S.A. Darmanyan, and J. Garnier, Progr. Opt. 44, 303 (2002). [17] A. Dreischuh, G.G. Paulus, F. Zacher, F. Grasbon, and H. Walther, Phys. Rev. E 60, 6111 (1999). [18] D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993). [19] C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (1993); 92,113902 (2004). [20] M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. 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0704.1094

SUBARU HDS Observations of a Balmer-Dominated Shock in Tycho's Supernova Remnant

Draft version October 28, 2018 Preprint typeset using LATEX style emulateapj v. 04/21/05 SUBARU HDS OBSERVATIONS OF A BALMER-DOMINATED SHOCK IN TYCHO’S SUPERNOVA REMNANT ∗ Jae-Joon Lee, Bon-Chul Koo, John Raymond, Parviz Ghavamian, Tae-Soo Pyo, Akito Tajitsu, Masahiko Hayashi Draft version October 28, 2018 ABSTRACT We present an Hα spectral observation of a Balmer-dominated shock on the eastern side of Tycho’s supernova remnant using SUBARU Telescope. Utilizing the High Dispersion Spectrograph (HDS), we measure the spatial variation of the line profile between preshock and postshock gas. Our observation clearly shows a broadening and centroid shift of the narrow-component postshock Hα line relative to the Hα emission from the preshock gas. The observation supports the existence of a thin precursor where gas is heated and accelerated ahead of the shock. Furthermore, the spatial profile of the emission ahead of the Balmer filament shows a gradual gradient in the Hα intensity and line width ahead of the shock. We propose that this region (∼ 1016 cm) is likely to be the spatially resolved precursor. The line width increases from ∼ 30 km s−1 up to ∼ 45 km s−1 and its central velocity shows a redshift of ∼ 5 km s−1 across the shock front. The characteristics of the precursor are consistent with a cosmic ray precursor, although a possibility of a fast neutral precursor is not ruled out. Subject headings: ISM:supernova remnants – ISM: individual (Tycho, G120.1+1.4) – Shock Waves – line: profiles 1. INTRODUCTION Balmer-dominated filaments are the signature of non-radiative shocks propagating into partially neu- tral medium (Chevalier & Raymond 1978). The Hα line profile is composed of two distinctive components (narrow and broad) representing the velocity distri- bution of preshock and postshock gases, respectively (Chevalier et al. 1980). High resolution spectroscopic ob- servations of several of these shocks have revealed that the width of the narrow component is unusually large (30 ∼ 50 km s−1) for ambient neutral hydrogen, and it was proposed that the gas was heated in a precursor thin enough (. 1017 cm) to avoid complete ionization of hydrogen (see Ghavamian et al. 2001; Sollerman et al. 2003, and references therein).6 Two likely candidates are cosmic-ray (CR) and fast neutral precursors (Smith et al. 1994; Hester et al. 1994; Ghavamian et al. 2001). Both scenarios predict significant Doppler shifts of preshock gas. No clear indication of such a shift of the Hα nar- row component is reported (but see Lee et al. 2004). A careful comparison of postshock and preshock line pro- files with high spectral resolution is crucial for confirming the existence of such a precursor. The blast wave of Tycho’s SNR, the historical ∗BASED ON DATA COLLECTED AT SUBARU TELESCOPE, WHICH IS OPERATED BY THE NATIONAL ASTRONOMI- CAL OBSERVATORY OF JAPAN 1 Astronomy Program, Department of Physics and Astronomy, Seoul National University, Seoul 151-742, Korea 2 [email protected] 3 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 4 Department of Physics and Astronomy, Johns Hopkins Univer- sity, 3400 North Charles Street, Baltimore, MD 21218 5 Subaru Telescope, National Astronomical Observatory of Japan, 650 North A’ohōkū Place, Hilo, HI 96720 6 We refer to this precursor explicitly as a “thin precursor”, to avoid potential confusion with a photoionization precursor which will be shortly introduced. remnant of the 1572 supernova, has been known to exhibit Balmer-dominated emission (van den Bergh 1971; Kamper & van den Bergh 1978). The re- gion of the brightest Hα emission (Knot g from Kamper & van den Bergh 1978) is located along the north-eastern edge of the remnant. Faint, diffuse optical emission extends ∼1 pc ahead of the Balmer-dominated filaments (Ghavamian et al. 2000, G00 hereafter). This feature has been identified as a photoionization precur- sor (PIP) produced by photoionization of the preshock gas by He II λ304Å emission from behind the blast wave (G00). The estimated temperature of the PIP is ∼ 1.2 × 104 K, which is not high enough to explain the observed width of the Balmer narrow component. The existence of Hα emission from the PIP makes Tycho an unique target for the study of nature of the thin precursor, as we can investigate the change of the line profile across this precursor by comparing the line emission from the PIP (preshock) and that of Knot g (postshock). In this Letter, we present high resolution (Echelle) long slit Hα spectra of Tycho Knot g and its PIP using the SUBARU High Dispersion Spectrograph (Noguchi et al. 2002). Our observations reveal the line broadening and the Doppler shift of the the narrow com- ponent, providing strong evidence for the existence of a cosmic ray or fast neutral precursor. 2. OBSERVATIONS AND RESULTS The spectrosopic observation was carried out on 2004 October 1. Longslit Echelle spectroscopy of the Hα line was performed using order-blocking filters (HDS stan- dard setup “stdHa”). This gives a spectral coverage of 6540 Å∼ 6690 Å over the 60′′ of slit length. The slit was centered at α(2000), δ(2000) = (00h 25m 56.s5, 64◦ 09′ 28′′) with position-angle of 76◦ (measured E of N), covering both Knot g and its PIP simultaneously (Fig. 1(a)). A total of 3×30 minutes of source expo- sure was obtained, and the same amount of exposure for http://arxiv.org/abs/0704.1094v1 2 Lee et al. nearby sky. The spectrum was binned by 2 along the slit direction and 4 along the dispersion direction before the readout. The pixel scale after the binning is 0.27′′ pixel−1 (9.3 × 1015 cm pixel−1 at a distance of 2.3 kpc (Kamper & van den Bergh 1978)) and 0.08 Å respec- tively. The slit width was 2′′, which gives velocity resolu- tion of 17 km s−1. The seeing was 0.′′5±0.′′1. The process- ing of the SUBARU data included a typical CCD prepro- cessing (including overscan correction and flat fielding) and two-dimensional spectral extraction. A wavelength calibration solution is obtained from the spectrum of a Th-Ar lamp. The source spectrum was sky-subtracted, and normalized using the spectra of standard stars. The uncertainty in the wavelength calibration is estimated to be around 0.2 km s−1 at the wavelength of Hα. In Fig. 1(b), we present the fully processed two dimen- sional spectrum of Hα line. The bright patch at the bot- tom with three distinct local emission peaks corresponds to Knot g, and the faint emission extending to the top of the image corresponds to the PIP. The average Hα spectrum of Knot g and that of PIP are shown together in Fig. 2. The velocity width of the Knot g narrow com- ponent line is clearly larger than that of the PIP Hα line. In addition, the velocity centroid of the Knot g narrow component is slightly redshifted (5.5± 0.6 km s−1) rela- tive to that of the PIP Hα line. And as clearly seen in Fig. 2(b), the Knot g spectrum shows a very broad (∼ 1,000 km s−1), faint Hα line. The small peak near 900 km s−1 is the [N II] λ6583.4 Å line from the PIP. The spectrum of the PIP is well fitted by a single Gaus- sian, and yields a FWHM of 33.8± 0.8 km s−1 and cen- troid velocity of −35.8 ± 0.6 km s−1 (in LSR frame).7 The measured FWHM of [N II] λ6583.4 Å in the PIP is ∼ 23 km s−1. If the broadening were purely thermal, the widths of the lines whould be inversely proportional to the square root of their atomic masses. The expected width of the [N II] line in this case is 1/ 14 = 0.267 of Hα. As this is significantly narrower than what is observed, a significant amount of nonthermal broaden- ing is suggested. If we simply assume that the observed line widths are a convolution of thermal and nonther- mal broadenings, the estimated thermal temperature is ∼ 13, 000 K, consistent with 12, 000 K of G00. It is also possible that the large line width is due to a residual of Galactic Hα emission. An adequate fit to the Knot g spectrum requires three Gaussian components. They have velocity widths of 45.3 ± 0.9 km s−1 (narrow), 108 ± 4 km s−1 (intermedi- ate) and 931± 55 km s−1 (broad), with central velocities of −30.3± 0.2, −25.8± 0.8 and 29± 18 km s−1, respec- tively. This result confirms that the Hα narrow com- ponent line of Knot g is redshifted and broadened rela- tive to that of the PIP. The broad component (FWHM∼ 1, 000 km s−1) should correspond to the previously re- ported ∼ 2, 000 km s−1 component (Ghavamian et al. 2001). The relatively narrower width of our observation is probably due to the insensitivity of our spectroscopic configuration to the very broad line, although the pos- sibility of temporal variation (e.g., by crossing a density 7 throughout this paper, we give line widths corrected for instru- mental broadening and velocity in LSR frame. jump) does exist. The abrupt density increase might be possible if the shock is propagating into the edge out- skirts of the dense cloud (Lee et al. 2004). The existence of the intermediate width component has already been reported by G00. It may be produced when slow pro- tons picked up by the postshock magnetic field undergo a secondary charge exchange. Alternatively, it might be an artifact of the assumption of Gaussian distributions, which would not necessarily be appropriate if the mo- tions are non-thermal. The characteristics of the narrow component Hα line of Knot g are consistent with G00, except that the ve- locity centroid of the line in our data is significantly dif- ferent from theirs (vLSR = −30.3± 0.2 from our data vs. −53.9 ± 1.3 km s−1 from G00). We have carefully as- sessed our wavelength calibration, but couldn’t find any significant source of error. The wavelengths of night sky lines observed near Hα matched well with those of VLT UVES night-sky emission line catalog (Hanuschik 2003) within error of 0.02 Å (∼ 1 km s−1). The Hα spectrum of blank sky is also consistent with nearby WHAM spectra (Haffner et al. 2003). Furthermore, the central velocity of [N II] λ6583.4 Å emission line from the PIP is con- sistent with that of Hα. Although we believe that our wavelength calibration solution is quite secure, the dis- crepancy with the previous observation should be con- firmed by an independent observation. For the rest of this paper, we mainly concentrate on the line width and the relative variation of central velocity. 3. LOCATION OF THE SHOCK FRONT AND DISCOVERY OF A THIN (1016 CM) PRECURSOR Fig. 3 shows the spatial variation of the Hα line profile along the slit. In the top panel, we separately plot the integrated intensities of the representative Hα of narrow and broad components, together with their sum (refer to the figure caption for the velocity range of each com- ponent). In the middle and bottom panels, we plot the central velocities and FWHMs at each slit position from the single Gaussian fit to the full profile. In all three panels, the x-axis represents the pixel offset from the southwest end of the slit. Although three Gaussian com- ponents are actually required to adequately fit the Hα profile of Knot g, the non-uniqueness of the multiple- component fitting can complicate the interpretation of the different emission line components. Therefore, we plot the fit result from a single Gaussian and describe our interpretation of this fit in the region of Knot g as following: First, the central velocities closely match those of the narrow component. Fitting with multiple compo- nents leaves the fitted line centroids unchanged to within the errors. Second, the FWHMs basically trace the spa- tial variation of the narrow component width, but the presence of the intermediate and the broad components contribute significantly to the width. When fit with three Gaussian components, the large widths of the Hα nar- row component in Knot g from the single profile models (60 ∼ 80 km s−1) are reduced to 40 ∼ 50 km s−1. Inspection of Fig. 3 reveals that there exists only one location (pixel offset 27, marked as a dotted vertical line) where the intensity and width of the Hα line exhibits an abrupt jump. The jump is most noticeable for the broad component, where the flux is virtually zero toward the di- rection of ambient medium (i.e., toward positive pixel off- SUBARU HDS Observation of Tycho 3 6559 6560 6561 6562 6563 6564 A(htgnelevaW (a) (b) Fig. 1.— (a) Hα image of Tycho Knot g with position of the 2′′ ×60′′ slit overlaid. (b) The fully-processed Hα 2-d spectrum (only the western part of the slit is shown). The bright emission knot at the bottom is Knot g. sets). The central velocity of the narrow component also shows rapid change around this location. The FWHM also steeply increases behind this point, but this might be largely due to the sudden appearance of the broad component. The fact that the intensity of the broad component, which is associated with the postshock gas, shows a significant jump at this location suggests that it corresponds to the location of the shock front. The Hα intensity in the PIP region is nearly constant along the slit, but shows an rapid rise just before the shock front, i.e., from pixel offset 32 to 27 in Fig. 3. The intensity of the Hα line in the PIP region is about 10% of the observed peak value in Knot g, and it reaches about half the peak very near the shock. The line width also increases from 30 to 45 km s within this region. We propose that the steep increase of the Hα intensity accompanied with line broadening corresponds to a thin precursor where gas is heated and accelerated. Unlike the line broadening, the observed velocity centroids shift slightly behind the shock instead of the precursor region. This does not necessarily indicate that the broadening and the Doppler shift takes at different region as this is likely due to geometrical projection effects. The Hα intensity profile gives an e-folding thickness of 1.4 ± 0.4 pixel (after accounting for seeing), which corresponds to (1.4± 0.4)× 1016 cm at a distance of 2.3 kpc. The observed emission of Knot g shows a few local peaks indicating possible substructure, e.g., a collection of shock tangencies seen in projection along the line of sight. This leads to a possibility that the ‘precursor’ is simply the results of geometric projection of fainter Balmer-dominated filament lying just ahead of Knot g. If we assume that this filament has a line profile simi- lar to that of Knot g, i.e., have a same broad-to-narrow ratio, then a detectable amount of broad component is expected. The flux profile of broad component plotted in Fig. 3 would show similar gradual increase in the pre- cursor region, which is not seen. Also, no hint of a broad component is found in a summed spectra of precursor region, which is expected to show up. Examining the archival Chandra observation of Tycho’s SNR (the anal- ysis is presented by Warren et al. 2005), we did not find evidence of X-ray emission extending ahead of Knot g. Therefore, there is no strong supporting evidence in favor of a projection, although we cannot rule out the possibility. 4. NATURE OF THE THIN PRECURSOR As the shock is nearly tangential to our line of sight, the actual amount of bulk gas acceleration could be much -200 -100 0 100 Knot g : observed Knot g : fit (sum) Knot g : fit (individual) PIP : observed -2000 -1500 -1000 -500 0 500 1000 Velocity (km/s) [NII] Knot g : observed Knot g : fit (broad) PIP : observed Fig. 2.— (a) Hα spectrum of Knot g and the PIP. The spectrum of Knot g is fitted with three Gaussian components, and two nar- rowest components are displayed (thin dashed) together with their sum (thin solid). (b) Same as (a) except the full observed velocity range is presented and the spectra are binned. The Knot g broad component from the above fit is displayed as a thin solid line. larger than the observed Doppler shift. The shock an- gle can be inferred from the radial velocity shift of the broad component of Knot g relative to the narrow com- ponent (Chevalier et al. 1980), which is measured to be ∼ 60 km s−1. As our observation could be insensitive to this broad component, using this value is rather conser- vative. On the other hand, Ghavamian et al. (2001) re- ported a redshift of ∼ 130 km s−1, but their field of view is different from ours. We take these two values as limits and give shock angle of 2◦−5◦ assuming a shock velocity of 2,000 km s . We consider 5 km s to be the repre- sentative redshift of the narrow component compared to unperturbed medium, which gives actual acceleration of 60 ∼ 130 km s−1. The line width of the narrow component at the shock front (∼ 45 km s−1) corresponds to a neutral hydrogen temperature of 40, 000 K, if the line broadening is purely thermal, or lower, if there is a significant non-thermal broadening such as a wave motion. Neutral hydrogen atoms and protons may have similar velocity distribu- tions due to their charge exchange interactions. But that of electrons can be different, which would depend on the heating mechanism in the precursor. In the following, we estimate the electron temperature in the thin precur- sor (TP) from the observed intensity increase of factor 5. Since the observed spectrum is an integrated emis- sion along the line of sight where a significant contribu- tion from PIP region is expected, the actual emissivity increase within the precursor should be much greater. We assume that regions of the PIP and the TP are represented by two concentric shells with thicknesses of δPIP ≃ 1016 cm and δTP ≃ 1 pc (G00), respectively, and that both have an inner diameter of the Tycho (5.4 pc). Then the ratio of path length through each shell along the tangential direction of the shock front is ≃ 18. This implies that the emissivity increase in the TP could be as large as a factor of 90. This value should be regarded 4 Lee et al. total narrow broad 0 10 20 30 40 50 60 pixel offset Fig. 3.— (top) Integrated fluxes along the slit position of Hα spectrum for a given velocity range. The thin solid line shows emission in the range −60 < vLSR < 10 km s −1, rep- resentative of Hα narrow component, while the thick solid line shows emission in the range −400 < vLSR < −120 km s −1 and +100 < vLSR < +350 km s −1, for the broad component. The thick dashed line shows the summed intensity for both components. (−400 < vLSR < +350 km s −1). Middle and bottom panels: cen- tral velocities and FWHMs from fits with a single Gaussian. The proposed location of shock front is marked as a vertical dotted line. as an upper limit as it is likely that the filament is nearly flat and tangent to the line of sight. The collisional ex- citation rate of Hα at Te ∼ 40, 000K is 2, 000 times greater than that of 12, 000K which is the temperature of the PIP region (G00). This value greatly exceeds the emissivity increase of 90, and implies a few possibilities. The electron temperature can be less than 40, 000K, ei- ther if observed Hα line width has significant nonthermal broadening, or if Te is intrinsically less than Tp. On the other hand, the estimated emissivity increase might be explained if the emissivity increase due to high Te is sup- pressed by ionization of neutrals. Since the existence of Blamer-dominated filaments requires a significant frac- tion of neutral hydrogen atoms to survive within precur- sor, this possibility is less favored. The two likely candidates for this precursor are fast neutral and CR precursors. The momentum and en- ergy carried by upstreaming fast neutrals can be large enough to explain the observed heating and accelera- tion in the precursor, but the available model calcula- tions do not predict significant heating by these neutrals (Lim & Raga 1996; Korreck 2005). Furthermore, It is difficult to reproduce the small range of FWHMs ob- served for narrow component Hα lines from shocks of dif- ferent velocities (Smith et al. 1994; Hester et al. 1994). The observed characteristics of the precursor are consis- tent with a CR precursor. CR acceleration in the shock does require a precursor (e.g., Blandford & Eichler 1987) and there has been growing evidence of CR acceleration in young SNRs including Tycho itself (e.g., Warren et al. 2005). Significant heating and acceleration are expected to happen in the CR precursor (Blandford & Eichler 1987). Interaction of CR particles in the upstream gen- erates Alfv́en waves and significant amplification of mag- netic field has been suggested (Bell & Lucek 2001). Al- though a measurement of the preshock magnetic field is hardly available, a high value (40µG) of preshock magnetic field is suggested for Tycho (Völk et al. 2002), which may explain the line width of the narrow compo- nent. To conclude, our SUBARU observation clearly has shown the line broadening and the Doppler shift between the preshock gas and postshock gas. This strongly sug- gests the existence of the thin precursor. Furthermore, the precursor itself is likely to be resolved. Given the lack of observational constraints on CR ion acceleration in SNR shocks, the tentatively measured width (∼ 1016 cm) of the thin precursor whose primary candidate is the CR precursor is promising. Follow-up observation, such as a high resolution imaging with HST, would clearly resolve the structures of the precursor. We thank to the anonymous referee for valuable com- ments. This work was supported by the Korea Research Foundation (grant No. R14-2002-058-01003-0). JJL has been supported in part by the BK 21 program. REFERENCES Bell, A. R. & Lucek, S. G. 2001, MNRAS, 321, 433 Blandford, R. & Eichler, D. 1987, Phys. Rep., 154, 1 Chevalier, R. A., Kirshner, R. P., & Raymond, J. C. 1980, ApJ, 235, 186 Chevalier, R. A. & Raymond, J. C. 1978, ApJ, 225, L27 Ghavamian, P., Raymond, J., Hartigan, P., & Blair, W. P. 2000, ApJ, 535, 266 Ghavamian, P., Raymond, J., Smith, R. C., & Hartigan, P. 2001, ApJ, 547, 995 Haffner, L. M., Reynolds, R. J., Tufte, S. L., Madsen, G. J., Jaehnig, K. P., & Percival, J. W. 2003, ApJS, 149, 405 Hanuschik, R. W. 2003, A&A, 407, 1157 Hester, J. J., Raymond, J. C., & Blair, W. P. 1994, ApJ, 420, 721 Kamper, K. W. & van den Bergh, S. 1978, ApJ, 224, 851 Korreck, K. 2005, Ph.D. Thesis Lee, J., Koo, B., & Tatematsu, K. 2004, ApJ, 605, L113 Lim, A. J. & Raga, A. C. 1996, MNRAS, 280, 103 Noguchi, K., Aoki, W., Kawanomoto, S., Ando, H., Honda, S., Izumiura, H., Kambe, E., Okita, K., Sadakane, K., Sato, B., Tajitsu, A., Takada-Hidai, T., Tanaka, W., Watanabe, E., & Yoshida, M. 2002, PASJ, 54, 855 Smith, R. C., Raymond, J. C., & Laming, J. M. 1994, ApJ, 420, Sollerman, J., Ghavamian, P., Lundqvist, P., & Smith, R. C. 2003, A&A, 407, 249 Völk, H. J., Berezhko, E. G., Ksenofontov, L. T., & Rowell, G. P. 2002, A&A, 396, 649 van den Bergh, S. 1971, ApJ, 168, 37 Warren, J. S., Hughes, J. P., Badenes, C., Ghavamian, P., McKee, C. F., Moffett, D., Plucinsky, P. P., Rakowski, C., Reynoso, E., & Slane, P. 2005, ApJ, 634, 376

0704.1095

Orbits of tori extended by finite groups and their polynomial hulls: the case of connected complex orbits

ORBITS OF TORI EXTENDED BY FINITE GROUPS AND THEIR POLYNOMIAL HULLS: THE CASE OF CONNECTED COMPLEX ORBITS V.M. GICHEV Abstract. Let V be a complex linear space, G ⊂ GL(V ) be a compact group. We consider the problem of description of polynomial hulls cGv for orbits Gv, v ∈ V , assuming that the identity component of G is a torus T . The paper contains a universal construction for orbits which satisfy the inclusion Gv ⊂ TCv and a characterization of pairs (G, V ) such that it is true for a generic v ∈ V . The hull of a finite union of T -orbits in TCv can be distinguished in closTCv by a finite collection of inequalities of the type |z1| s1 . . . |zn| sn ≤ c. In particular, this is true for Gv. If powers in the monomials are independent of v, Gv ⊂ TCv for a generic v, and either the center of G is finite or TC has an open orbit, then the space V and the group G are products of standard ones; the latter means that G = SnT , where Sn is the group of all permutations of coordinates and T is either Tn or SU(n) ∩ Tn, where Tn is the torus of all diagonal matrices in U(n). The paper also contains a description of polynomial hulls for orbits of isotropy groups of bounded symmetric domains. This result is already known, but we formulate it in a different form and supply with a shorter proof. Introduction Let V be a finite-dimensional complex linear space and G ⊂ GL(V ) be a compact subgroup of GL(V ). We consider the problem of description of polynomially convex hulls for orbits Ov = Gv, v ∈ V . The polynomially convex hull (or polynomial hull) Q̂ of a compact set Q ⊂ V is defined as Q̂ = {z ∈ V : |p(z)| ≤ sup |p(ζ)| for all p ∈ P(V )},(0.1) where P(V ) is the algebra of all holomorphic polynomials on V . It is usually difficult to find Q̂. For Q = Gv, the answer is known if G is an isotropy group of a bounded symmetric domain in Cn. Paper [9] contains a description of G-invariant polynomially convex compact sets, including hulls of orbits (Q ⊂ V is polynomially convex if Q̂ = Q); it continues paper [10] and uses results of [8]. On the other hand, it is known that an orbit of a compact linear group is polynomially convex if and only if the complex orbit GCv is closed and Gv is its real form ([7]). The cases G = U(2), SU(2) were considered in [1], [4]. The problem of determination of polynomial hulls of orbits admits the following natural generalization: given a homogeneous space M of a compact group G, describe maximal ideal spaces MA of G-invariant closed subalgebras A of C(M), where C(M) is the Banach algebra of all 1991 Mathematics Subject Classification. Primary 32E20; Secondary 32M15, 32M05. Key words and phrases. Polynomial hulls, bounded symmetric domains. The author was partially supported by RFBR Grants 06-08-01403 and 06-07-89051. http://arxiv.org/abs/0704.1095v1 2 V.M. GICHEV continuous complex-valued functions on M with the sup-norm. If A is generated by a finite-dimensional invariant subspace, then MA can be realized as the polynomial hull of an orbit. Paper [6] contains a description of MA for bi-invariant algebras on compact groups and partial results on spherical homogeneous spaces. Maximal ideal spaces for U(n)-invariant algebras on spheres in C n are described in [11]. In this paper we consider orbits Gv of groups G = FT , where F ⊆ G is a finite subgroup and T is a torus, such that GCv = TCv. Let t ⊆ gl(V ) be the Lie algebra of T and set tR = it, TR = exp(tR). Suppose that v ∈ V has a trivial stable subgroup in T and let X ⊂ TRv be finite. The hull of Y = TX admits a simple description. If X = {v}, then Ŷ = T̂ v is the closure of T exp(CT )v, where CT is a cone in tR. If TC is closed, then Ŷ = T exp(QX)v, where QX ⊆ tR is a convex polytope (the convex hull of the inverse image of X for the mapping ξ → exp(ξ)v, ξ ∈ tR). Any segment in QX corresponds to an analytic strip or an annulus in Ŷ . In general, Ŷ is the union of T̂ u, where u runs over exp(QX)v. Also, Ŷ is distinguished in closTCv by a finite family of monomial inequalities of the type |z1|s1 . . . |zn|sn ≤ c,(0.2) where c ≥ 0 and s = (s1, . . . , sn) ∈ Rn depend on v and X . Vectors s correspond to normals of faces of CT +QX . Thus, the problem of determination of Ĝv is not difficult if Gv ⊂ TCv. The latter is equivalent to the assumption that the complex orbit GCv is connected. In Example 3.4, we give a construction for orbits which satisfy this condition; here is a sketch. The group G = FT acts on the space V = C(K), where K is a finite F -invariant subset of t∗: F acts naturally on C(K), t = t∗∗ is naturally embedded into C(K), and T = exp(t) acts on C(K) by multiplication. If v ∈ C(K) is an F -invariant function, then Gv ⊂ TCv. According to Theorem 3.5, each connected complex orbit can be realized in this way. Further, we describe pairs (V,G) such Gv ⊂ TCv for a generic v ∈ V.(0.3) By Theorem 4.3, under the additional assumption that the complex linear span of TCv coincides with V , this happens if and only if the group GCZ, where Z is the centralizer of G in GL(V ), has an open orbit in V . There are two extreme cases: (A) Z ⊆ GC; (B) G has a finite center. An example for (A) is the group G = SnTn acting in Cn, where Tn is the torus of all diagonal matrices in U(n) and Sn is the group of all permutations of coordinates. Replacing Tn with SU(n)∩Tn, we get an example for (B). Example 4.4 contains a construction for pairs (V,G) that satisfy (0.3). Theorem 4.5 states that the construction is a universal one. In Theorem 4.10, we determine pairs which satisfy (0.3) and the following condition: vectors s in (0.2) are independent of v.(0.4) The paper also contains a description of hulls Ĝv for G = Aut0(D), where D is a bounded symmetric domain in the canonical realization and Aut0(D) is the stable subgroup of zero, which coincides with the group of all linear automorphisms of D. These hulls have already been described: the final step was done in paper [9], which essentially used [10], partial results appear in [14] and [8]. Most of them use the technique of Jordan triples and Jordan algebras. We use Lie theory, in particular, an explicit construction of paper [15] for a maximal abelian subspace a. A compact group acting in a Euclidean space is called polar if there exists a subspace ORBITS OF TORI EXTENDED BY FINITE GROUPS AND THEIR POLYNOMIAL HULLS: THE CASE OF CONNECTED COMPLEX ORBITS3 (a Cartan subspace) such that each orbit meets it orthogonally. The group G is polar in the ambient linear space d, and a is the Cartan subspace for G. Real polar representations are classified in paper [3]; they are orbit equivalent (i.e., have the same orbits) to isotropy representations of Riemannian symmetric spaces. If D is a polydisc Dn ⊂ Cn, where D is the unit disc in C, then G = SnTn; the polynomial hulls Ĝv are determined by the inequalities µk(z) ≤ µk(v),(0.5) where k = 1, . . . , n and µk are defined by µk(z) = max{|zσ(1) . . . zσ(k)| : σ ∈ Sn}.(0.6) The general case can be reduced to this one in the following way. Any bounded symmetric domain D ⊂ d of rank n admits an equivariant embedding of Cn to d, which induces an embedding of Dn to D, such that Rn ⊂ Cn is the maximal abelian subspace a, and, for any v ∈ a , the hull of Aut0(D)v is the orbit of the hull of Aut0(D n)v. Each µk(z) has a unique continuation to a K-invariant function on d. The extended functions determine hulls by the same inequalities. Moreover, they are plurisubharmonic and can be treated as products of singular values of z ∈ d or as norms of exterior powers of adjoint operators in suitable spaces. The subsystem of long roots of the restricted root system (i.e., the root system for a) has type nA1; this defines the above embedding Cn → d. Furthermore, this makes it possible to determine hulls in terms of the adjoint representation (Theorem 5.7). Thus, there is no need to consider different types of domains separately. The reduction to the case of a torus extended by a finite group, which is de- scribed above, is contained in Section 5 (in papers [9], [14], the problem is also reduced to this case by another method). It does not use essentially the results of the previous sections (only Proposition 3.2, in proof of Theorem 5.7). These extensions satisfy conditions (0.3) and (0.4); in addition, they possess the property that the complexified groups have open orbits. According to Theorem 4.10, any group with these properties is the product of groups SnT n acting in Cn; it admits a natural realization as a group of automorphisms of a bounded symmetric domain (Corollary 5.3). The following simple examples illustrate the case Gv 6⊆ TCv and show that condition (0.3) is essential. Let G = SnT n, and let ǫ1, . . . , ǫn be the standard base in Cn. Then Ĝǫ1 is the closure the union of discs Dǫk, k = 1, . . . , n. Set H = SnT, where T acts by z → eitz, t ∈ R, z ∈ Cn. Then Ĥǫ1 = Ĝǫ1. For v = ǫ1 + ǫ2, Ĝv is the closure of the union of bidiscs but Tn contains no proper torus T such that Ĝv = Ĥv for H = SnT . However, for any subgroup F ⊆ Sn which acts transitively on 2-sets and H = FTn we have Ĝv = Ĥv. 1. Preliminaries We keep the notation of Introduction, in particular, (0.1) and (0.6). Linear spaces are supposed to be finite dimensional and complex unless the contrary is explicitly stated. ”Generic” means ”in some open dense subset”. Throughout the paper, we use the following notation: D and T are the open unit disc and the unit circle in C, respectively; V denotes a complex linear space (except for Section 5); 4 V.M. GICHEV if V is equipped with a linear base identifying it with Cn, then Tn is the group of all diagonal unitary transformations; Zn2 consists of all transformations in T n with eigenvalues ±1; ǫ1, . . . , ǫn is the standard base in C n and Rn; Rn+ is the set of vectors in R n with positive entries; SK denotes the group of all permutations of a finite setK; ifK = {1, . . . , n}, then SK = Sn; C(K) is the algebra of all complex-valued functions on K; 1 is the identity of C(K); G ⊂ GL(V ) is a compact group whose identity component is a torus T (except for Section 5); t ⊂ gl(V ) is the Lie algebra of T , tR = it, tC = t+ tR; TR = exp(tR), TC = exp(tC); C∗ = TC = C \ {0}; Ť = Hom(T,T) is the dual group to T ; Aut(D) is the group of all holomorphic automorphisms of a domain D ⊂ V , Aut0(D) = Aut(D) ∩GL(V ); coneX denotes the least convex cone which contains the set X ; convX is the convex hull of X ; closX is the closure of X ; spanFX is the linear span of X over the field F = C,R,Q. Clearly, exp is bijective on tR and TR ∼= TC/T . The differentiating at the identity e defines an embedding of Ť into the dual space t∗: χ → −ideχ, where χ ∈ Ť . This is a lattice in the vector group t∗, moreover, T ∼= t/L, where L is the dual lattice to Ť in t. For χ ∈ Ť , let Vχ = {v ∈ V : gv = χ(g)v for all g ∈ T } be the corresponding isotypical component of V . Then ⊕Vχ.(1.1) We assume that V is equipped with a G-invariant inner product 〈 , 〉. Then de- composition (1.1) is orthogonal. Let spec(v) denote the spectrum of v ∈ V (the set of χ ∈ Ť such that the χ-component of v is nonzero); for X ⊆ V , spec(X) = ∪x∈X spec(x). We say that T has a simple spectrum if dimVχ ≤ 1(1.2) for all χ ∈ Ť . If (1.2) is true, then there exists a unique (up to scaling factors) orthogonal base in V which agree with (1.1) and a unique maximal torus Tn in GL(V ) which contains T . In what follows, we assume that (1.2) holds; we shall see in the next section that such assumption is not restrictive. Thus, we may fix an identification V = Cn = C(K),(1.3) where K = {1, . . . , n}. If F is a subgroup of SK , then C(K)F denotes the set of all F -invariant functions on K; clearly, 1 ∈ C(K)F . Further, (C∗)n is the multiplicative group of all invertible functions in C(K), Tn consists of functions with values in T, and (Tn)C = (C∗)n. The Lie algebra of Tn is realized as iRn ⊂ Cn. ORBITS OF TORI EXTENDED BY FINITE GROUPS AND THEIR POLYNOMIAL HULLS: THE CASE OF CONNECTED COMPLEX ORBITS5 The embedding T → Tn induces embeddings of the Lie algebra and the fundamental group: t → iRn, π1(T ) → iZn ⊂ iRn, respectively. Let Γ be the image of π1(T ). Then spanR Γ = t; moreover, t∩ iZn = Γ and t/Γ = T . The dual mapping Ťn → Ť , which is defined by the restriction of characters e−i〈x,y〉, where x ∈ iZn, to t, is the orthogonal projection πt : iZ n → t. Thus, Γ is a subgroup of finite index in Ť = πtiZ n. Vectors in spanQ Ť are called rational. The image of t in iR n can be distinguished by linear equations with integer coefficients. Hence, clos(TCv), for a generic v ∈ V , is the set of all solutions to a finite number of equalities with holomorphic monomials. Thus, Y ⊂ TC implies Ŷ ⊂ clos(TCv). Set CT = t R ∩ clos(−Rn+),(1.4) The cone iCT is dual to cone(specV ) ⊆ t∗ ⊆ iRn. If −ξ ∈ clos(Rn+), then ι = limt→+∞ exp(tξ) is an idempotent in C(K) such that the multiplication by the complementary idempotent 1− ι is a projection onto spanC(spec(ξ)). Set IT = {limt→+∞ exp(tξ) : ξ ∈ CT }.(1.5) Clearly, IT is finite and contains 1. Lemma 1.1. The closure of exp(CT ) is equal to IT exp(CT ). Proof. Due to the evident inclusion clos(exp(CT )) ⊇ IT exp(CT ), it is sufficient to prove that the set ST = IT exp(CT ) is closed. Clearly, ST is an abelian semigroup. The cone CT is polyhedral; hence, it is finitely generated: CT = cone{ξ1, . . . , ξm}, where R+ξk are the extreme rays of CT , k = 1, . . . ,m. Obviously, IT is a finite semigroup, which is generated by the idempotents limt→+∞ exp(tξk). Thus, the cor- respondence (e−t1 , . . . , e−tm) → exp(t1ξ1 + . . . , tmξm) defines a mapping of (0, 1]m onto exp(CT ), which continuously extends to [0, 1] m. It follows that its image is closed and coincides with ST . � Note that there is a natural one-to-one correspondence between IT and the set of faces of CT . 2. Hulls of finite unions of T -orbits in a TC-orbit Let v ∈ TC. If v = χ∈Ť vχ, where vχ ∈ Vχ, g ∈ TC, and u = gv, then χ∈Ť χ(g)vχ. Since χ(g) 6= 0 for all g ∈ G and χ ∈ Ť , we get u ∈ TCv =⇒ spec(u) = spec(v);(2.1) dim (Vχ ∩ spanC Tv) ≤ 1 for all v ∈ V and χ ∈ Ť .(2.2) Thus, the assumption that T has a simple spectrum in V is not restrictive in the problem of description of polynomial hulls of orbits Gv such that Gv ⊂ TCv. Clearly, Dn = T̂n in L(V ). For each x ∈ CT and any polynomial p on L(V ), the holomorphic function f(ζ) = p(exp(ζx)) is bounded in the halfplane Π : Re ζ ≥ 0. Hence, exp(Π) is contained in T̂ . On the other hand, if z ∈ Dn ∩ TC, then z = t exp(x) for some t ∈ T and x ∈ CT (the polar decomposition). By Lemma 1.1, T̂ = clos(Dn ∩ TC) = T clos(exp(CT )) = TIT exp(CT ). 6 V.M. GICHEV If v ∈ (C∗)n, then (C∗)nv = (C∗)n, and the mapping z → zv is a linear nondegen- erate transformation of Cn. Therefore, v ∈ (C∗)n =⇒ T̂ v = T̂ v = TIT exp(CT )v.(2.3) For an arbitrary v ∈ V = Cn, set CvT = {ξ ∈ tR : ξk ≤ 0 if vk 6= 0, k = 1, . . . , n}. Applying (2.3) to spanC(spec(v)) = C nv, we get T̂ v = T clos(exp(CvT )v.(2.4) Clearly, CvT depends only on spec(v). For s ∈ Rn and z ∈ (C∗)n, set νs(z) = |zk|sk . If sk ≥ 0, then the k-th factor in (C∗)n can be replaced with C (i.e., νs continuously extends to this product). It is well known that for any holomorphically convex T -invariant set U ⊆ TC, the set log(U ∩ TR) ⊆ tR is convex. In particular, this is true for sets of g ∈ TC such that gv ∈ T̂X, where X ⊂ TCv, v ∈ V . Nevertheless, it is convenient to have an explicit construction of an analytic strip (or an annulus, if it is periodic) in a TC-orbit, which corresponds to a segment that joins two points in tR; it is contained in the following lemma. Set S = {z ∈ C : 0 ≤ Im z ≤ 1}. Lemma 2.1. Let v ∈ Cn and u ∈ TRv. Then, there exists ξ ∈ tR such that λ(z) = exp(zξ)v(2.5) is a holomorphic mapping λ : S → TCv which satisfies conditions λ(∂S) ⊆ Tv ∪ Tu, λ(0) = v, λ(1) = u. If the stable subgroup of v in TR is trivial, then ξ is unique. Proof. These properties hold for ξ ∈ tR such that exp(ξ)v = u; such a ξ exists, since exp is a bijection tR → TR. The last assertion is clear. � If ξ ∈ CT , then (2.5) defines an analytic halfplane in T̂ v; for Γ-rational ξ, λ is periodic and defines an analytic disc in T̂ v. Together with Lemma 2.1 this gives a characterization of hulls for finite unions of T -orbits in TC. Suppose that X ⊂ TRv is finite and the stable subgroup of v in T is trivial. Then, the inverse to the mapping x → exp(x)v is well defined. Let us denote it by logv and set QX = conv(logv X),(2.6) PX = QX + CT .(2.7) The set PX is a convex polyhedron, which is unbounded if CT 6= 0. Hence, there exists a finite set NX ⊂ Rn and, for each s ∈ NX , real numbers cs such that PX = {x ∈ tR : 〈x, s〉 ≤ cs for all s ∈ NX}.(2.8) The set NX consists of vectors orthogonal to faces of PX , whose projections into spanR PX look outside of it; clearly, it is not unique in general. ORBITS OF TORI EXTENDED BY FINITE GROUPS AND THEIR POLYNOMIAL HULLS: THE CASE OF CONNECTED COMPLEX ORBITS7 Proposition 2.2. Let v ∈ (C∗)n. Suppose that Y ⊂ TCv is a finite union of T -orbits (including Tv), and set X = TRv ∩ Y . Then X is finite and Ŷ = clos (T exp(PX)v)(2.9) = clos {z ∈ (C∗)n : νs(z) ≤ ecsνs(v), s ∈ NX}(2.10) u∈exp(QX )v T̂ u(2.11) = T exp(PX)IT v,(2.12) where QX , PX ,NX are as above and IT is defined in (1.5). Proof. Due to the polar decomposition, the set Tu∩TRv, for each u ∈ TCv, is non- void and consists of a single point. Hence, X is finite and Y = TX . The inclusion (expQX)v ⊆ Ŷ follows from Lemma 2.1 and Phragmén–Lindelöf Principle. The in- clusion exp(CT )u ⊆ T̂ u is true for any u ∈ Cn. Since it holds for all u ∈ T exp(QX), the left-hand side of (2.9) includes the right-hand side. If z = exp(ξ)v, where ξ ∈ tC, then zk = e ξkvk, k = 1, . . . , n; due to (2.8), this implies that the right-hand side of (2.9) coincides with (2.10). According to (2.3), the right-hand side of (2.9) and (2.11) intersect TCv by the set T exp(PX)v = exp(QX)T exp(CT )v; clearly, it is dense in (2.11). Since QX is compact, the set (2.11) is closed. The compactness of QX , the above equality, and Lemma 1.1 imply that (2.12) is closed; hence, it is the same as the right-hand side of (2.9). Each of the sets (2.9)–(2.12) includes Y . Thus, it remains to prove that (2.12) is polynomially convex. If x ∈ tR \ PX , then there exists s ∈ Rn such that sup{〈y, s〉 : y ∈ PX} < 〈x, s〉 .(2.13) Since QX is compact, the linear functional on t in the right-hand side of (2.13) must be nonnegative on CT . According to (1.4), we may assume that s ∈ closRn+. It follows that (2.13) holds in a neighborhood of s in closRn+. Thus, s can be assumed rational (hence, integer) with strictly positive entries. Then, p(z) = zs11 . . . z a holomorphic polynomial such that |p | separates exp(x)v and T clos(exp(PX)v). Therefore, Ŷ ∩ TCv = T exp(PX)v. For any ι ∈ IT , the projection z → ιz commutes with T . This makes it possible to apply the above arguments to the vector ιv, the set ιX , and to the restriction of T to ιCn. Consequently, ι̂Y ∩ TCιv = T exp(PιX)ιv = ιT exp(PX)v(2.14) (clearly, ι exp(PX)v = exp(PιX)ιv). By (1.5), ιY ⊆ Ŷ , hence, ι̂Y ⊆ Ŷ ; on the other hand, ιŶ ⊆ ι̂Y since p◦ι is a polynomial on Cn for any polynomial p on ιCn. Thus, ιŶ = ι̂Y = Ŷ ∩ ιCn. Together with (2.14), this implies the polynomial convexity of (2.12). � If T = Tn, then Proposition 2.2 follows from the well-known characterization of polynomially convex Reinhardt domains. Corollary 2.3. For any v ∈ (C∗)n, the orbit TCv is closed in Cn if and only if Tv is polynomially convex, and this is equivalent to CT = 0. Then, Ŷ = T exp(QX)v for all Y,X as above. 8 V.M. GICHEV Proof. The orbit TCv is closed if and only if the convex hull of spec(v) = spec(Cn) contains 0 in its relative interior (see, for example, [13, Proposition 6.15]). Since T ⊆ GL(n,C), the set spec(Cn) is generating in t∗. Hence, TCv is closed if and only if CT = 0; by (2.4), this is equivalent to T̂ v = Tv. Then, Ŷ = T exp(QX)v by (2.9) and (2.1). � There is a version of the first assertion for an arbitrary compact linear group G: a GC-orbit is closed if and only if it contains a polynomially convex G-orbit ([7, Theorem 1 and Theorem 5]). For a torus T , all T -orbits in TCv are simultaneously polynomially convex or non-convex, but this is not true if G is not abelian. 3. Finite extensions of T that keep a TC-orbit In this section, we consider the case where the set X defined in the previous section is an orbit of a finite group F which normalizes T and keeps the TC-orbit. We assume that T ⊆ G, T is a torus, G is a subgroup of GL(V ), F is a finite subgroup of G, and G = FT = TF, F ∼= G/T,(3.1) Gv ⊆ TCv,(3.2) v ∈ (C∗)n ⊂ Cn = V.(3.3) By (3.1), T is normal in G. Clearly, (3.2) is equivalent to Fv ⊆ TCv and to the connectedness of GC. Here is an illustrating example. Example 3.1. Let G = Aut0(D 2) be the group of linear automorphisms of the bidisc D2 ⊂ C2. Clearly, G = FT , where F = S2 is generated by the transposition τ of the coordinates, T = T2, TC = (C∗)2, and TCv = (C∗)2 for any v that lies outside the coordinate lines. Thus, (3.2) holds for all v ∈ (C∗)2 (however, (3.2) fails for any v 6= 0 in C2 \ (C∗)2). The hull Ĝv can be distinguished by the inequalities max{|z1|, |z2|} ≤ max{|v1|, |v2|},(3.4) |z1z2| ≤ |v1v2|.(3.5) Clearly, (3.4) and (3.5) define a polynomially convex set. Let z1, z2 > 0 (a generic T -orbit evidently contains such a point z). Then, z and τz can be joined by an analytic strip with the boundary in Tz ∪ Tτz: λz(s) = (z 2 ), s ∈ S. Set q = ln z1 and let z1 > z2. Then, the strip can be written in the form λz(s) = (e −sz1, e sz2), 0 ≤ Re s ≤ q. It is periodic with the period 2πi and defines a τ -invariant annulus in Ĝv with τ - fixed points ( z1z2, z1z2) and (− z1z2,− z1z2). As z2 → 0, the annulus tends to a couple of discs: (e−sz1, 0) and (0, e −sz1), where Re s > 0, 0 ≤ Im s ≤ 2π (the circle Re s = q , 0 ≤ Im s ≤ 2π collapses to zero). Let z ∈ Ĝv ∩ R2. Then Ĝv contains a bidisc D2z. It intersects R2 by a rectangle, which is symmetric with respect to the coordinate axes. If z lies on an axis, then the rectangle degenerates into a segment. Let v1 > v2 > 0. The union of these rectangles with vertices in the set Q of real points of the annulus, which joins v and τv, is a curvilinear octagon. It degenerates into a pair of segments if v2 = 0 and into a square if v1 = v2 (see [10, Fig. 2] for the 3-dimensional case). In the logarithmic coordinates in the ORBITS OF TORI EXTENDED BY FINITE GROUPS AND THEIR POLYNOMIAL HULLS: THE CASE OF CONNECTED COMPLEX ORBITS9 first quadrant, Q is a segment. Also, note that all nontrivial TC-orbits are not closed. � In [2], Björk found a typical situation where analytic annuli appear in the maxi- mal ideal space MA of a commutative Banach algebra A which admits a nontrivial action of T by automorphisms: this happens if T -invariant functions on MA do not separate distinct T-orbits. In [7], it was noted that analytic strips and/or an- nuli appear in Ĝv if the stable subgroup of v in GC does not coincide with the complexification of the stable subgroup of v in G. Proposition 3.2. The hulls Ĝv for orbits of G = Aut0(D n) = SnT n are distin- guished by inequalities (0.5), where µk are defined by (0.6). Proof. The approximation by decreasing sequences of hulls makes it possible to reduce the proposition to the case of a generic v in (0.5). Then, applying to v = (v1, . . . , vn) a suitable transformation in T n, we may assume that v1 > v2 > · · · > vn > 0.(3.6) Moreover, we may use Proposition 2.2 with X = Snv, CT = − closRn+ (we keep the notation of Proposition 2.2). Since X , QX , CT , PX , and µk are Sn-invariant, Sn is transitive on X , by (0.6), (0.5), and (2.10), it is sufficient to prove that the vectors ξk = r=1 ǫr, k = 1, . . . , n, correspond to the faces of PX that meet at v, are orthogonal to them, and look outside of PX . Set η1 = ǫ2 − ǫ1, . . . , ηn−1 = ǫn − ǫn−1, ηn = −ǫn. Then {−ηk}nk=1 is a base in Rn, which is dual to the base {ξk}nk=1. We claim that the cone of the polyhedron PX at the vertex v is generated by {ηk}nk=1. This implies the assertion above (note that both cones are simplicial). If τ ∈ Sn is a transposition (k, j), then v − τv = (vk − vj)(ǫk − ǫj). If σ, κ ∈ Sn then v − σκv = (v − κv) + (κv − σκv). Furthermore, Sn is generated by transpositions (k, k + 1), and vk − vk+1 > 0 by (3.6), where k = 1, . . . , n− 1. Therefore, vectors η1, . . . , ηn−1 generate the cone of QX at v. Since −ǫk = r=0 ηn−r and CT is generated by −ǫk, k = 1, . . . , n, this proves the proposition. � Property (3.2) implies spanC Gv = spanC T Cv. Hence, we may assume that (1.2) is valid. Then, a generic ξ ∈ t has a simple spectrum. Any f ∈ F permutes eigenvalues and eigenspaces. Thus, assuming (1.2) and identifying V with C(K) in accordance with (1.3), we get that each element of F is a composition of a permutation of K and a multiplication by a function on K. Further, (3.3) implies that the stable subgroup of v in TC is trivial. Hence, TRv ∼= TC/T ∼= tR, where the identification of TRv and tR is realized by ξ → exp(ξ)v, ξ ∈ tR. Lemma 3.3. Let G ⊂ GL(V ), a subgroup F ⊆ G, and v ∈ V satisfy (3.1)– (3.3). Then TCv contains a G-invariant T -orbit. Moreover, there exists a mapping f → tf , F → T , such that F̃ = {tff : f ∈ F} is a subgroup of G which has a fixed point in TC and satisfies (3.1)–(3.3). Proof. The group F naturally acts on TRv ∼= TC/T ∼= tR. Any g ∈ F is a composi- tion of σ ∈ SK and a multiplication by a function in C(K). Since t acts on C(K) by multiplication on linear functions and σ induces a linear transformation in tC, the induced action of F on tR is affine. Since F is finite, it has a fixed point in tR. 10 V.M. GICHEV Hence, TCv contains a G-invariant T -orbit. Let us fix a point u in it and define tf by tffu = u; the choice is unique due to (3.3). Taken together with (3.1), this implies that F̃ is a group, which obviously satisfies the lemma. � According to Lemma 3.3, we may assume without loss of generality that fv = v for all f ∈ F.(3.7) In the following example we give a construction (associated with a given finite group F ) for orbits with property (3.2). Example 3.4. Let t be a real linear space, t∗ be the dual space to t, L be a lattice in t, and L∗ ⊂ t∗ be the dual lattice to L. Set λx(y) = y(x), where x ∈ t, y ∈ t∗. Let K be a finite subset of L∗ that generates L∗ as a subgroup of the vector group t∗. Then t∗ = spanR K,(3.8) L = {x ∈ t : λx(K) ⊂ Z}.(3.9) Further, let F be a finite subgroup of GL(t) which keeps K. Set V = C(K). The mapping λ : x → iλx (3.10) is an embedding t → V , which has a natural extension to tC. Set exp(x) = e2πiλx .(3.11) Clearly, L = ker exp. Hence, exp defines an embedding of T = t/L and TC into the group (C∗)n: TC = exp(tC) ⊆ (C∗)n. The group TC acts on C(K) by multiplication. The inclusion v ∈ C(K)F is the same as (3.7); it implies (3.2). Furthermore, if v ∈ (C∗)n, then spanC Tv = V.(3.12) Indeed, the space spanC T is a subalgebra of C(K), which separates points of the finite set K. Hence, it coincides with C(K). � Theorem 3.5. Let a group G ⊂ GL(V ), a finite subgroup F ⊆ G, a torus T , and a vector v ∈ V satisfy (3.1)–(3.3), (3.7), and (3.12). Then V,G, F, T, v can be realized as in Example 3.4, where v ∈ (C∗)n ∩ C(K)F .(3.13) Conversely, if V,G, F, T, v are as in Example 3.4 and v satisfies (3.13), then (3.1)– (3.3), (3.7), and (3.12) are true. Proof. The group F acts in t and t∗ by the adjoint action. Let K ⊂ t∗ be the collection of all weights for the representation of T in V ; clearly, K is F -invariant. It follows from (3.12) and (3.2) that the weights are multiplicity free. This defines an equivariant linear isomorphism between V and C(K), where the group T acts by multiplication. Thus, λ and exp are well defined by (3.10) and (3.11). According to (3.7) and (3.3), (3.13) is true; (3.8) holds since T ⊂ GL(V ) is compact and acts effectively on V (note that the annihilator of spanR K in t acts trivially due to ORBITS OF TORI EXTENDED BY FINITE GROUPS AND THEIR POLYNOMIAL HULLS: THE CASE OF CONNECTED COMPLEX ORBITS11 (3.11) and (3.13)). Let us define L by (3.9). Then L = ker exp by (3.11). Hence, L is a lattice in t and the group L∗ generated by K is the dual lattice in t∗. The converse was proved in Example 3.4. � 4. Finite extensions of T which keep generic TC-orbits In what follows, we use the setting of Example 3.4. Let Z denote the centralizer of G in GL(V ). We assume that (C∗)n acts in V = C(K) by multiplication. Lemma 4.1. Z = C(K)F ∩ (C∗)n. Proof. Since λ(t) separates points of K, Z ⊆ (C∗)n. The multiplication by u ∈ C(K) commutes with F if and only if u is F -invariant. � In general, condition (3.2) does not hold for a generic vector v. Hence, there is a natural problem: describe V and G such that generic orbits satisfy (3.2). The following proposition contains a simple criterion. Proposition 4.2. Let V,G be as in Example 3.4. Then G satisfies (3.2) for a generic v ∈ V if and only if C(K) = λ(tC) + C(K)F .(4.1) In this case, each TC-orbit in (C∗)n intersects C(K)F . Proof. It follows from Lemma 4.1 that the right-hand side of (4.1) is the tangent space at 1 to the set TC Z. Clearly, G̃C = ZGC is a group, TCZ is the identity component of G̃C, and the right-hand side of (4.1) is the tangent space to G̃C1. Hence, (4.1) holds if and only if G̃C1 is open. Moreover, this is equivalent to the equality TCZ = exp(λ(tC) + C(K)F ) = (C∗)n. Therefore, each TC-orbit in (C∗)n intersects C(K)F , i.e., contains an F -fixed point. Thus, (4.1) implies (3.2) for v ∈ (C∗)n. Let (3.2) hold and letW be an F -invariant neighborhood of 1. IfW is sufficiently small, then the condition log 1 = 0 defines a branch of log in W . We may assume that logW is convex and symmetric. This makes it possible to define roots in W : r = exp . For v ∈ W 12 and f ∈ F , set gf = , where r = cardF , and g = f∈F gf . Then gv is F -fixed. If (3.2) holds for v, then gf ∈ TC for all f ∈ F ; hence, gv ∈ TCv. Consequently, for all v ∈ W , TCv intersects C(K)F . Since Z keeps this property of orbits, it follows that TCZ has a nonempty interior. This implies (4.1). � Theorem 4.3. Let G ⊂ GL(V ) be a semidirect product of a torus T and a finite subgroup F , and let Z be the centralizer of G in GL(V ). Suppose that spanC Tv = V for some v ∈ V . Then the following conditions are equivalent: (i) Gv ⊂ TCv for a generic v ∈ V ; (ii) GCZv is open in V for a generic v ∈ V . Proof. By Theorem 3.5, we may use the construction of Example 3.4. According to Lemma 4.1, (ii) is equivalent to (4.1), and the assertion follows from Proposition 4.2. 12 V.M. GICHEV We shall give a constructive description of these spaces and groups. Set C0(K) = u ∈ C(K) : u(q) = 0 Sometimes, we identify points in K with their characteristic functions. Example 4.4. Let V = Cn = C(K), where K = {1, . . . , n}, let F be a subgroup of Sn, and K = K1 ∪ · · · ∪Kp(4.2) be the partition of K into F -orbits. For k ∈ {1, . . . , p}, set Vk = C(Kk). Then V = V1 ⊕ · · · ⊕ Vp. Set t0k = C0(Kk) ∩ iRn, = exp(t0 ) ⊂ C(Kk), where exp is defined by (3.11). Set t0 = t01 ⊕ · · · ⊕ t0p, T 0 = exp(t0) = T 01 × · · · × T 0p . Let T be an F -invariant torus such that T 0 ⊆ T ⊆ Tn(4.3) and set G = FT . Then generic GC-orbits satisfy (3.2). The group G is irreducible if and only if F is transitive on K; in general, F -orbits in K define G-irreducible components of V . There are two extreme cases in (4.3). (A) If T = Tn, then there is one open orbit (C∗)n of the groupGC = FTC, which evidently satisfies (3.2). If F is nontrivial, then there exist degenerate orbits that do not satisfy (3.2); moreover, if F is transitive onK, then all non-open GC-orbits, except for zero, are nontrivial finite unions of TC-orbits. (B) If T = T 0, then generic orbits are closed. They have codimension p and are distinguished by equations zr = ck, where ck ∈ C∗, k = 1, . . . , p. Note that (A) and (B) are invariant under the Cartesian product (the group F need not be the product of groups Fk of irreducible components but must have the same orbits in K as F1 × · · · × Fp). In terms of Example 3.4: in (A), t = Rn, the mapping λ : tC → C(K) is surjective, K = {ǫ1, . . . , ǫn}; in (B), t = iRn ∩ C0(K), λ(tC) = C0(K), and the set K is the projection of {ǫ1, . . . , ǫn} into t∗ = t. In both cases, K is the set of all vertices of a regular simplex. � Theorem 4.5. Let V,G be as in Theorem 4.3 and let (i) hold. Then V,G can be realized as in Example 4.4. Furthermore, (1) V,G are of type (A) if and only if GC has an open orbit, (2) (B) is equivalent to the assumption that the center of G is finite, (3) if G is irreducible, then either (A) or (B) holds. Let C(K)F+ be the cone of all nonnegative functions in C(K) Lemma 4.6. Let G and V be as in Example 3.4. Then, the orbit GCv is closed for a generic v ∈ V if and only if R ∩ C(K)F+ = 0.(4.4) ORBITS OF TORI EXTENDED BY FINITE GROUPS AND THEIR POLYNOMIAL HULLS: THE CASE OF CONNECTED COMPLEX ORBITS13 Proof. Clearly, GCv is closed if and only if TCv is closed. Let v ∈ (C∗)n. By Proposition 2.2 and Corollary 2.3, TCv is not closed if and only if CT 6= 0. Since CT is F -invariant by (1.4), it contains f∈F fu for each u ∈ CT . Thus, CT = 0 is equivalent to (4.4). � Proof of Theorem 4.5. Suppose that G is irreducible or, equivalently, F is transi- tive. Then Z = C∗ 1 according to Lemma 4.1. If 1 ∈ λ(tC), then TC ⊇ Z and TCv is open for a generic v ∈ V by Theorem 4.3. If 1 /∈ λ(tC), then (4.4) is true; by Lemma 4.6, TCv is closed for a generic v ∈ V . By Proposition 4.2, a generic TC-orbit intersects C∗1. Consequently, we have codimGCv = 1.(4.5) Let 1 ∈ TC ∩ Z. The orthogonal projection of 1 into the tangent space T1TC1 is F -fixed. Hence, it is proportional to 1; since 1 /∈ λ(tC), this implies 1 ⊥ λ(tC)1. Therefore, T1T C1 coincides with the tangent space to the hypersurface z1 . . . zn = 1 at 1; since the monomial on the left is an eigenfunction of TC, this group keeps it. Due to (4.5), TC1 coincides with this hypersurface. Then, T = Tn ∩ SU(n), and any TC-orbit that intersects Z is a hypersurface z1 . . . zn = c, for some c ∈ C∗. This implies tC = C0(K) and T = T Thus, the theorem is proved for all irreducible G. The projection onto each irreducible component keeps the property (3.2) for generic orbits since it commutes with G. Hence, (i) holds for all irreducible components. They correspond to F - orbits Kk in the partition (4.2). Let t , k = 1, . . . , p , be defined as in Example 4.4. According to the arguments above, λ (t|Kk) ⊇ λ(t0k) for all k. If x ∈ t, then the averaging Ax = 1 f∈F fx, r = cardF, distinguishes the F -fixed component of x (i.e., Ax ∈ C(K)F ∩ t and x− Ax ∈ t0); since t is F -invariant, it contains both components. By Lemma 4.1, if G has a finite center, then λ (t|Kk) = λ(t0k) for all k. It follows that t ⊆ t0 = t01 ⊕ · · · ⊕ t0p. On the other hand, (ii) and Lemma 4.1 imply codim t ≤ dimC(K)F = p. Hence, the inclusion above is in fact the equality. Thus, we get (B) assuming that G has a finite center. The converse is true since t0 does not contain a nontrivial F -fixed element. The same arguments show that any F -invariant torus T includes T 0 if (i) is true. This proves that V,G admit the realization of Example 4.4; (1) and (2) are clear. � Corollary 4.7. Let G be as in Theorems 4.5 and 4.3. Then G contains a closed subgroup G0 such that (1) each connected component of G contains a connected component of G0, (2) G0 has a finite center, (3) generic orbits of (G0)C are closed, (4) Gv ∩ TRv = G0v ∩ (T 0)Rv for a generic v ∈ V . Proof. By Theorem 4.5 and (4.3), G ⊇ T 0, where T 0 is as in (B). Clearly, F normalizes T 0. Hence, G0 = FT 0 is a group, which satisfies the corollary. � 14 V.M. GICHEV Proposition 2.2 makes it possible to find Ĝv for G as above. If T = Tn, then T ⊃ Zn2 and generic T -orbits intersect Rn+; hence, we may assume v ∈ Rn+. Then T̂ v ∩ Rn is a parallelepiped Πv = conv{(±v1, . . . ,±vn)}. Clearly, Πv = Zn2Π+v , where Π+v = Πv ∩ closRn+. Since Rn+ = TRv, we may use Proposition 2.2 with X = Fv, CT = − closRn+, PX = conv(Fv) − Rn+: Ĝv = ∪u∈exp(Qv)Dnu = T ∪u∈exp(Qv) Πu = T ∪u∈exp(Qv) Π+u , where Qv = convFv. For the description in the form (2.10), one has to know normal vectors to faces of convFv. Since F may be an arbitrary subgroup of Sn, they need not be proportional to rational vectors (for example, this is true for the cyclic subgroup of order 3 in S3). We shall describe the situation where they are locally independent of v; since they depend on v continuously, this is equivalent to the condition that they are rational. Note that the vector which joins two points in tR as in Lemma 2.1 is rational if and only if the strip reduces to an annulus. In Example 4.4, F need not be the product of groups corresponding to the irreducible components; we shall see that F possesses this property in the case under consideration. Let U be a real vector space and F ⊂ GL(U) be a finite group. Set Cu = cone(u− Fu); this is the cone at the vertex u of the polytope conv(Fu) (which may be degenerate). We say that Cu is locally independent of u if, for a generic u ∈ U , Cu = Cw for all w that are sufficiently close to u. Lemma 4.8. Let U be a real vector space and F be a finite subgroup of GL(U). Suppose that Cu is locally independent of u. Then F is generated by reflections in hyperplanes in U . Proof. We may assume without loss of generality that U is equipped with an inner product and that F ⊆ O(U). Let R+(u − fu), f ∈ F , be an extreme ray of Cu. The equality Cu = Cw for w in a neighborhood of u implies that this ray does not change near u. Hence, dim(1 − f)U = 1. Since f is orthogonal and nontrivial, it is a refection in a hyperplane. The stable subgroup of a generic u ∈ U is trivial (hence, F acts freely on a generic orbit) and each vertex of conv(Fu) can be joined with u by a chain of edges. Applying the above arguments repeatedly to u, fu, etc., we get that F is generated by reflections in hyperplanes. � For any g ∈ Zn2Sn and k = 1, . . . , n, gǫk = ± ǫσ(k) for some σ ∈ Sn. The mapping f → σ is a natural homomorphism Zn2Sn → Sn, which we denote by φ. Lemma 4.9. Let F be a transitive subgroup of Sn acting in R n by permutations of coordinates and let a group H ⊆ Zn2Sn be generated by reflections in hyperplanes in Rn. If φ(H) = F , then F = Sn. Proof. Let ρ be a reflection in a hyperplane in Rn. If ρ ∈ Zn2Sn = BCn, then it is conjugate to a reflection in a wall of the Weyl chamber that is distinguished by the inequalities x1 > · · · > xn > 0. Hence, φ(ρ) is a transposition if it is nontrivial. Since F = φ(H), F is generated by transpositions. It remains to note that any subgroup of Sn, which is generated by transpositions, coincides with Sn if it is transitive on {1, . . . , n} (consider the graph with the vertices {1, . . . , n} and edges corresponding to transpositions and note that inclusions (k, l) ∈ F , (l,m) ∈ F imply (k,m) ∈ F ; this makes it possible to use the induction). � ORBITS OF TORI EXTENDED BY FINITE GROUPS AND THEIR POLYNOMIAL HULLS: THE CASE OF CONNECTED COMPLEX ORBITS15 We say that a pair (V,G) is standard if it is isomorphic to (A) or (B) in Ex- ample 4.4 with F = SK . The product of pairs (Vk, Gk), k = 1, . . . ,m, is the pair k=1 Vk, k=1 Gk). Theorem 4.10. Let G = FT be a compact subgroup of GL(n,C), where T ⊆ Tn is a torus and F is a subgroup of Sn. Suppose that Gv ⊂ TCv for a generic v ∈ V (1) either T = Tn or the center of G is finite, (2) for a generic v ∈ Cn, Ĝv can be distinguished in closTCv by a family of inequalities |z1|s1 . . . |zn|sn ≤ ρs(v), where ρs(v) ≥ 0 and vector s = (s1, . . . , sn) runs over a certain finite subset of Rn which is independent of v. Then (V,G) is isomorphic to the product of standard pairs. Moreover, if G is irreducible, then (V,G) is standard. Proof. Let G be irreducible. Then F is transitive and (V,G) are as in (A) or as in (B) by Theorem 4.5. Suppose that (B) is the case. It follows from (2) and Proposition 2.2 that the polytope QX ⊂ tR, where X = Gv ∩ TRv, for a generic v, satisfies the assumption of Lemma 4.8. Therefore, F is generated by reflections (we may assume that F ⊂ O(tR)). They extend to reflections in hyperplanes in tR+R1 = Rn if we assume that they fix 1. Then, Lemma 4.9 implies F = Sn. The case (A) can be reduced to (B): it is sufficient to replace Tn with T = SU(n) ∩ Tn since F evidently keeps T and to note that (2) remains true due to Proposition 2.2. Thus, (V,G) is standard. Let the center of G be finite. According to Theorem 4.5, T may be identified with the group T 0 in Example 4.4. In particular, GCv is closed for a generic v and CT = 0 due to Proposition 2.2. By Proposition 4.2, generic orbits contain F -fixed points. Applying the arguments above (which did not use the assumption that G is irreducible), we get that the cones at the vertices of the convex polytope QX , X = Gv ∩ TR ⊂ tR, are locally independent of v. Clearly, the same is true for its projection into each space t0k corresponding to an irreducible component Vk of V = C n. This implies that all irreducible components are standard. Thus, Fk = S(Kk), where k = 1, . . . , p and K = K1 ∪ · · · ∪Kp is the partition of K into F -orbits. Due to Theorem 4.5, it is sufficient to prove that F = F1 × · · · × Fp.(4.6) By Lemma 4.8, F |t0 is generated by reflections in hyperplanes in t0; the condition that they keep real F -invariant functions on K uniquely defines their extension to Rn. Hence, F is generated by reflections in Rn. A permutation which induces a reflection in a hyperplane in Rn is a transposition of a pair of coordinates; this pair is necessarily contained in only one of the sets Kk, k = 1, . . . , p. This proves (4.6). If T = Tn, then T is a product of tori in irreducible components. Thus, the case T = Tn follows from the above case, since the assumptions of the theorem hold true for the group T 0 if they hold for T in (4.3) in Example 4.4. � 5. Hulls of isotropy orbits of bounded symmetric domains We start with a preliminary material on hermitian symmetric spaces following [15] but adapting the exposition to our purpose in order to be as self contained as 16 V.M. GICHEV possible. For a subset X of a Lie algebra g, z(X) = {z ∈ g : [z,X ] = 0} is the centralizer of X . Let G be a simple real noncompact Lie group with a finite center, K be its maximal compact subgroup, and g, k be their Lie algebras, respectively. If the center z = z(k) of k is nontrivial, then g is called hermitian. Then k = z(z) and dim z = 1 (note that K is irreducible in g/k). Let c be a Cartan subalgebra of k. Then c is also a Cartan subalgebra of g and z ⊆ c. There exists k ∈ z such that ad(k) has eigenvalues 0,±i (it is unique up to a sign; ker ad(k) = k). Then κ = eπ ad(k) is the Cartan involution which defines the Cartan decomposition g = k⊕ d,(5.1) where k, d are eigenspaces for 1,−1, respectively. Furthermore, j = ad(k) is a complex structure in d. This defines the structure of a hermitian symmetric space of noncompact type in D = G/K. These spaces can be realized as bounded symmetric domains in Cn with K = Aut0(D). Any irreducible bounded symmetric domain admits such a realization. Let ∆ ⊆ ic∗ be the root system of gC. Each α ∈ ∆ corresponds to an sl2-triple hα, eα, fα such that ihα ∈ c. Thus, α(hα) = 2, [eα, fα] = hα, and [h, eα] = α(h)eα, [h, fα] = −α(h)fα(5.2) for all h ∈ cC. We identify cC and (c∗)C equipping g with an Ad(K)-invariant sesquilinear inner product and normalize it by the condition max{|α| : α ∈ ∆} = 2.(5.3) Then short roots must have length 1 (note that G2 has no real hermitian form). The set ∆∨ = {hα : α ∈ ∆} is the dual root system. The above normalization implies hα = α for long roots and hα = 2α for short ones. Since ad(h), h ∈ c, has eigenvalues 0 and α(h), where α ∈ ∆, we get α(ik) = 0,±1, i.e., ik is a microweight (of ∆∨). For s = 0,±1, set ∆s = {α ∈ ∆ : α(ik) = s}.(5.4) Since k⊕ id is a compact real form of gC and spanR{ihα, eα − fα, i(eα + fα)} is the su(2)-subalgebra corresponding to a root α ∈ ∆, we have d = spanR{eα + fα, i(eα − fα) : α ∈ ∆1}.(5.5) Set sα = spanR{ihα, eα + fα, i(eα − fα)}. Then sα is an sl(2,R)-subalgebra of gC α ∈ ∆±1 ⇐⇒ sα ⊆ g.(5.6) Let E be a maximal subset of pairwise orthogonal long roots in ∆1. Set α∈E hα, e = α∈E eα, f = α∈E fα; α∈E ⊕ sα. Let α, β ∈ E, α 6= β. Since α, β are long and orthogonal, ±α± β /∈ ∆. Hence, α, β ∈ E, α 6= β =⇒ [sα, sβ] = 0.(5.7) It follows that h, e, f is an sl2-triple and s is a subalgebra of g. Set θ = e π ad(e−f),(5.8) a = spanR θE.(5.9) ORBITS OF TORI EXTENDED BY FINITE GROUPS AND THEIR POLYNOMIAL HULLS: THE CASE OF CONNECTED COMPLEX ORBITS17 Here is the standard realization of root systems Bn and Cn: Bn = {±ǫk ± ǫl, ±ǫm : k, l,m = 1, . . . , n, k < l}; Cn = {±ǫk ± ǫl, ±2ǫm : k, l,m = 1, . . . , n, k < l}. Then Cn = B n , but Cn does not satisfy (5.3). These systems have microweights; up to the action of the Weyl group, they are: Bn : ǫ1; (ǫ1 + · · ·+ ǫn). There are no other irreducible root systems which have microweights and contain roots of different lengths. Also, Bn and Cn have the sameWeyl group BCn = Z Lemma 5.1. The space a is a maximal abelian subspace of d. Proof. A straightforward calculation with 2-matrices shows that θh = e + f. By (5.7), θhα = eα + fα for all α ∈ E.(5.10) It follows from (5.5) that a ⊆ d. Moreover, a is abelian due to (5.7). Set Ξ = ∆∩E⊥. We claim that Ξ ⊆ ∆0.(5.11) Indeed, a root in ∆1 ∩ Ξ must be short. This may happen only in Bn or Cn, since G2 and F4 have no microweights and other irreducible root systems have no roots of different lengths. In Bn, k is a short root and all other short roots are orthogonal to k. Hence, they do not belong to ∆1. In Cn, E = {2ǫ1, . . . 2ǫn}; then Ξ = ∅. Since ∆−1 = −∆1, this proves (5.11). Set b = E⊥ ∩ c and m = spanC{eα, fα : α ∈ Ξ}. It follows from (5.11) that m ⊆ kC. Clearly, z(E) = cC ⊕ m. The space m is θ-invariant, because θ fixes roots in Ξ. Due to (5.9), we get z(a) = θz(E) = bC ⊕ aC ⊕m Since bC ⊕m ⊆ kC, this implies z(a) ∩ d = a. � The projection of θ∆ into a is the restricted root system ∆a (it is also the set of roots for ad(a) in g). The group W = {Ad(g) : g ∈ K, Ad(g)a = a}|a, acting in a, is the Weyl group of a. In what follows, we denote by v the complexification of a with respect to the complex structure j (thus, v ⊂ d). The set θE is a base in v; enumerating it, we identify v with Cn. Set t = spanR iE, T = exp t, H = WT . The torus T = T n is a maximal compact subgroup in the group exp s ⊆ G. Proposition 5.2. The following assertions hold: (1) ∆a is a root system of type BCn or Cn; (2) the pair (v, H) is standard with T = Tn. Proof. (1). Let ∆a \ θE contain a long root α. Then α = 12 (α1 + α2 + α3 + α4) for some α1, . . . , α4 ∈ θE, since |α|2 = 2 and 〈α, β〉 = 0,±1 for all β ∈ E due to the normalization (5.3) (note that α, β generate A2 if 〈α, β〉 6= 0). Roots α, α1, . . . , α4 generate D4, since only A4 and D4 among irreducible systems of rank 18 V.M. GICHEV 4 consist of roots of equal length, but A4 does not contain an orthogonal base. Since 〈ik, β〉 = 1 for all β ∈ E, the projection of iθk into spanR D4 is a microweight ω such that 〈ω, αk〉 = 1, k = 1, . . . , 4, but D4 has no microweight with this property (in the realization above, D4 = B4 ∩ C4 and the microweights are either ±ǫk or (±ǫ1± ǫ2± ǫ3± ǫ4)). Thus, E ∪ (−E) is the set of all long roots in ∆a. According to the classification of irreducible root systems, only Cn and BCn = Bn ∪ Cn has the property that linearly independent long roots are mutually orthogonal. (2). The maximal compact subgroup of the group corresponding to s is Tn. Hence, T = Tn ⊃ Zn2 . Systems Cn and BCn have the same Weyl group W = BCn. Therefore, H = WT = SnT Let D be a bounded symmetric domain in a complex linear space d (may be, reducible) and v ⊆ d be the complex linear span of a maximal abelian subspace in d (thus, we identify d with the corresponding space in the Cartan decomposition (5.1), which is induced by the Cartan involutions in irreducible components). Let Aut00(v, D) denote the subgroup of all linear transformations in Aut(D) which keep v and each irreducible component of D. Corollary 5.3. Let F be a subgroup of Sn, G = FT n ⊂ GL(n,C). Then G satisfies condition (2) of Theorem 4.10 if and only if (V,G) is isomorphic to a pair (v,Aut00(v, D)) for a bounded symmetric domain D. Proof. All pairs (Cn, SnT n) appear as (v,Aut00(v, D)) for matrix balls D. It re- mains to combine Theorem 4.10 and Proposition 5.2. � It is possible now to describe hulls of K-orbits in d (with respect to the complex structure j) it terms of Proposition 3.2. The key point is that K is polar in d: each K-orbit meets a orthogonally (i.e., a is a Cartan subspace). This is true, since all maximal abelian subspaces are conjugate in d by K, ad(a) is symmetric if a ∈ d and, for a generic a ∈ a, ker ad(a) = a; hence, [a, g] = a⊥.(5.12) We may include the linear base in v into a base in d as the first n vectors of the latter. Then z1, . . . , zn are coordinates in v and linear functions in d. The functions µk in (0.6) admit a K-invariant extension to d: µk(z) = sup{|(gz)1 . . . (gz)k| : g ∈ K},(5.13) where k = 1, . . . , n. The following lemma shows that (5.13) is an extension indeed. Lemma 5.4. For z ∈ v, (0.6) and (5.13) coincide. Proof. It follows from (5.12) that any critical point of the linear function Re z1 on the orbit Kz belongs to a. If the lemma is not true, then there exist z ∈ v and k ∈ {1, . . . , n} such that |(gz)k| > |zk|. Transformations in Sn and T reduce the problem to the case z1 > · · · > zn > 0 and k = 1, but then the assumption implies that Re z1 attains its maximal value on Kz outside of a. � Proposition 5.5. For any v ∈ d, K̂v = {z ∈ d : µk(z) ≤ µk(v), k = 1, . . . , n}. Proof. Due to (5.13), each µk is a supremum of absolute values of holomorphic polynomials. Hence, the right-hand side is polynomially convex. Thus, it includes K̂v. The inverse inclusion holds, since each K-orbit intersects v by an H-orbit and hulls of H-orbits are distinguished in v by the same inequalities according to Proposition 3.2 and Lemma 5.4. � ORBITS OF TORI EXTENDED BY FINITE GROUPS AND THEIR POLYNOMIAL HULLS: THE CASE OF CONNECTED COMPLEX ORBITS19 The functions µk can be written in more invariant terms. To do it, note that the Weyl group of ∆a has the form Z 2Sn in the base θE by (5.9); thus, zk = αk(z), k = 1, . . . , n, where αk ∈ θE and z ∈ a. Therefore, zk are eigenvalues of ad(z) in the subspace generated by the corresponding root vectors. The problem is to distinguish this subspace (in fact, we use a slightly different version). After that, functions µk can be defined as norms of some operators according to the following lemma (this observation was used in [12] in another context). Lemma 5.6. Let V be a Euclidean space and A be a symmetric nonnegative oper- ator in V with eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λm ≥ 0, where m = dim V . Let A∧k be its natural extension to the k-th exterior power V ∧k = V . Then ‖A∧k‖V ∧k = λ1 . . . λk, where ‖ ‖k is the operator norm with respect to the inner product in V ∧k. Proof. The norm of a nonnegative symmetric operator is equal to its maximal eigenvalue. � Let v ∈ g be semisimple and π(v) denote the projection onto ker ad(v) along other eigenspaces of ad(v) (note that π(v) is a function of ad(v), since it is the residue at zero of the resolvent of ad(v)). Set a(v) = ad([v, [v, k]])π(v) ad(k), pk(v) = ‖a(v)∧k‖g∧k , k = 1, . . . , n. The space d is a(v)-invariant and ker a(v) ⊇ k. We assume that g is equipped with some K-invariant inner product, which extends the inner product in d. It follows from the calculation below that a(v) is symmetric and has range ad(k)a. Let n = dim a be the rank of the symmetric space D. It is equal to the codimension of a generic K-orbit in d. Theorem 5.7. For any v ∈ d, K̂v = {z ∈ d : pk(z) ≤ pk(v), k = 1, . . . , n}. Proof. It is sufficient to prove the assertion for a generic v ∈ d. Clearly, pk are K-invariant. Hence, we may assume v ∈ a. Then, by (5.9) and (5.10), vα(eα + fα), where vα ∈ R. According to (5.2) and (5.4), [k, v] = α∈E ivα(eα − fα). Thus, [v, [v, k]] = 2iv2αhα due to (5.7). Also, (5.7) implies that ad(ihα) keeps v and has eigenvalues 0,±2i in it for each α ∈ E. Therefore, v is ad([v, [v, k]])-invariant and its eigenvalues are ±4v2αi, α ∈ E. Since ad(k)g = d and π(v)d = a for a generic v ∈ a, the space v is a(v)-invariant; moreover, a(v)g = ad(k)a ⊆ v. Thus, a(v) has eigenvalues 0,±4v2α in g. According to Lemma 5.6 and (0.6), pk(v) = 4 kµ2k(v)(5.14) for v ∈ v and k = 1, . . . , n. Since pk and µk are K-invariant, (5.14) holds for all v ∈ d. The theorem follows from Proposition 5.5. � 20 V.M. GICHEV Corollary 5.8. Functions pk, k = 1, . . . , n, are plurisubharmonic in d with respect to the complex structure j = ad(k). Proof. By (5.14) and (5.13), pk(z) = 4 k sup{|(gz)21 . . . (gz)2k| : g ∈ K}. The right-hand side is plurisubharmonic, since the functions z2 are j-holomorphic and j is K-invariant. � One can get the same functions pk by replacing g with d, endowed with the complex structure j, and a(v) with ad([v, jv])(π(v) + π(jv)). Acknowledgement. I thank Anton Pankratiev for helpful comments. References [1] J.T. Anderson, Polnomial hulls of sets invariant under an action of the special unitary group, Can. J. Math., v. XL, (1988) no. 5, 1256-1271. [2] J.-E. Björk, Compact groups operating on Banach algebras, Math. Ann., v. 205 (1973), no. 4, 281–297. [3] J. Dadok, Polar coordinates induced by actions of compact Lie groups, Transactions Amer. Math. Soc., 288 (1985), 125-137. [4] A. Debiard, B. Gaveau, Equations de CauchyRiemann sur SU(2) et leurs enveloppes dholo- morphie, Can. J. Math., v. 38 (1986), 1009-1024. [5] T. Gamelin, Uniform Algebras, Prentice-Hall, Englewood Cliffs, N. J., 1969. [6] V.M. Gichev, Maximal ideal spaces of invariant function algebras on compact groups, preprint, 46 pp., available at http://arXiv.org/abs/math/0603449 [7] V.M. Gichev, I.A. Latypov, Polynomially Convex Orbits of Compact Lie Groups, Transfor- mation Groups, v. 6 (2001) no. 4, 321-331. [8] J. Faraut, L. Bouattour, Enveloppes polynomiales densembles compacts invariants, Math. Nachrichten 266 (2004), 20-26. [9] W. Kaup, Bounded symmetric domains and polynomial convexity, Manuscr. Math. 114 (2004), No.3, 391-398. [10] W. Kaup, D. Zaitsev, On the CR-structure of compact group orbits associated with bounded symmetric domains, Invent. Math., 153 (2003), 45-104. [11] J. Kane, Maximal ideal spaces of U-algebras, Illinois J. Math., v.27 (1983), No 1, 1-13. [12] B. Kostant, On convexity, the Weyl group and the Iwasawa decimposition, Ann. Sci. Ecole. Norm. Super., 6 (1973), 413–455. [13] V.L. Popov, E.B. Vinberg, Invariant theory, Itogi Nauki i Tekhniki, Sovr. Probl. Mat. Fund. Napravl., v. 55, VINITI, Moscow 1989, pp. 137–309 (in Russian); English transl.: Algebraic Geometry IV, Encyclopaedia of Math. Sciences, v. 55, Springer-Verlag, Berlin 1994, pp. 123- [14] C. Sacre, Enveloppes polynomiales de compacts, Bull. Sci. math. 116 (1992), 129-144. [15] J.A. Wolf, Fine Structure of Hermitian Symmetric Spaces, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969-1970), Pure Appl. Math., Vol. 8. New York: Dekker 1972, pp. 271-357. Omsk Branch of Sobolev Institute of Mathematics, Pevtsova 13, 644099, Omsk, Rus- E-mail address: [email protected] http://arXiv.org/abs/math/0603449 Introduction 1. Preliminaries 2. Hulls of finite unions of T-orbits in a TC-orbit 3. Finite extensions of T that keep a TC-orbit 4. Finite extensions of T which keep generic TC-orbits 5. Hulls of isotropy orbits of bounded symmetric domains Acknowledgement References

0704.1096

Formation of a sonic horizon in isotropically expanding Bose-Einstein condensates

YITP-07-16, OCU-PHYS-264, AP-GR-40 Formation of a sonic horizon in isotropically expanding Bose-Einstein condensates Yasunari Kurita Osaka City University Advanced Mathematical Institute, Osaka 558-8585, Japan Takao Morinari Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan We propose a simple experiment to create a sonic horizon in isotropically trapped cold atoms within currently available experimental techniques. Numerical simulation of the Gross-Pitaevskii equation shows that the sonic horizon should appear by making the condensate expand. The ex- pansion is triggered by changing the interaction which can be controlled by the Feshbach resonance in real experiments. The sonic horizon is shown to be quasi-static for sufficiently strong interaction or large number of atoms. The characteristic temperature that is associated with particle emission from the horizon, which corresponds to the Hawking temperature in an ideal situation, is estimated to be a few nK. PACS numbers: 03.75.Kk, 03.75.Hh, 04.62.+v, 05.30.Jp I. INTRODUCTION For the exploration of cosmology and gravitational physics, it is necessary to have a deep understanding of quantum filed theory in curved spacetime: It is widely believed that everything except for the spacetime it- self should originate from quantum fluctuations in the early Universe. Quantum effects on curved spacetime, such as the Hawking radiation, give us theoretical sup- port for black hole thermodynamics. However, it is ex- tremely hard to verify such quantum effects experimen- tally. For instance, the Hawking radiation is thermal radiation emitted from a dynamically formed stationary black hole [1]. However, the characteristic temperature of the thermal radiation, the Hawking temperature, is on the order of several tens of nanokelvins at most, which is much lower than the cosmic microwave background radi- ation temperature. So detecting thermal radiation from a real black hole is almost impossible. One way to circumvent this difficulty is to make use of artificial black holes [2][3]. Unruh showed in his seminal paper[4] that excitations in a supersonic flow corresponds to a scalar field equation on a curved spacetime includ- ing a horizon. Since the phenomenon of the Hawking radiation can be separated from gravitational physics, it is possible to detect the corresponding phenomenon in a fluid system with sonic horizon [4]. The basic idea is to identify fluid flow with curved spacetime and excitation modes with fields on the curved spacetime. A black hole event horizon corresponds to a sonic horizon in a fluid. For the purpose of investigating the quantum effects, a quantum fluid should be considered. As such a quantum fluid, Bose-Einstein condensates (BEC) in trapped cold atoms [5, 6] are one of the most suitable systems [7]-[9]. A crucial advantage is that one can control scattering length between atoms by making use of the Feshbach resonance[10]. In fact, that experimental technique was used in observing jets and bursts in a collapsing conden- sate, which is called “Bose-Novae” [11]. An remarkable explanation of burst and jet phenomena in Bose-Novae was proposed in [12][13], based on quantum field theory of particle creation and structure formation in cosmolog- ical spacetime. In order to verify the Hawking effect in fluid analogy, it is necessary to create a stationary sonic horizon because it is a phenomenon on a dynamically formed stationary black hole. Although several possibilities have been dis- cussed so far [7, 8], it seems difficult to realize exactly sta- tionary sonic horizon in cold atoms. However, if one can make a quasi-static horizon for high frequency modes, particle emission from the horizon is also expected. In this paper, we numerically demonstrate that a quasi- static horizon is realized without introducing new exper- imental techniques beyond currently available ones. We consider an expanding BEC driven by a sudden change of the interaction. Numerically solving the Gross-Pitaevskii (GP) equation, we show that a quasi-static horizon ap- pear. We note that there have been made great efforts to cre- ate cosmological geometry using expanding BEC[14]-[21]. In these papers, the analogue models with specific cos- mological metrics such as Friedmann-Robertson-Walker (FRW) metric or de Sitter metric were discussed and the effects of particle creation in these cosmological space- times were investigated. However, in this paper, we do not intend to obtain any cosmological analogue model with well-known analytic metric. But we try to obtain dynamically formed quasi-static sonic horizon. Further- more, the sonic horizon should be formed in hydrody- namic regime of the condensate because the spacetime analogy is only valid in such regime. The appearance of a horizon due to expansion of a condensate was no- ticed in the previous works, for example in [15], and its formation itself is not surprising. But it is non-trivial whether the condensate flow at the horizon is in the hy- drodynamic regime, or not. In this paper, we show that the quasi-static sonic horizon will appear in the hydrody- namic regime of the condensate by changing the atomic interaction instantaneously. http://arxiv.org/abs/0704.1096v3 II. ANALOGUE SPACETIME IN BEC In the coherent state path integral formulation, the action of bosons is given by i~φ̄∂tφ− ∇φ̄ · ∇φ − Vextφ̄φ U0(φ̄φ) , (1) where Vext is the confining potential and U0 = 4π~ with a the s-wave scattering length. For φ and Vext, spatial and time dependences are implicit. The saddle point equation for this action leads to the GP equation: i~∂tΨ = ∇2 + Vext + U0|Ψ|2 Ψ. (2) This GP equation governs the dynamics of the conden- sate whose order parameter is given by Ψ. Now we consider hydrodynamical approximation. We denote the bosonic field φ as φ = ρ0 + ρe i(ϕ0+ϕ), where√ ρ0 and ϕ0 are the amplitude and the phase of Ψ, re- spectively. (Namely, Ψ = ρ0 exp(iϕ0).) The fields ρ and ϕ describe the non-condensate part of the bosonic field. If the density gradient is sufficiently smooth over the scale determined by the local healing length ξ(r, t) ≡ ~/(2mρ0U0) 1/2, or, in other words if the conditions, |ξ∇ρ0/ρ0|2 ≪ 1 and |ξ∇ρ/ρ|2 ≪ 1, (3) are satisfied, hydrodynamical approximation is justified. (The condition (3) shall be examined later.) Under the above condition, the equation for ρ is ρ = −~(ϕ̇+ v0 · ∇ϕ)/U0, (4) where v0 = ∇ϕ0 is the background fluid velocity, and the effective action for ϕ is Seff = (ϕ̇+ v0 · ∇ϕ)2 − (∇ϕ)2 Taking variation with respect to ϕ, we find that the field equation for ϕ has the form of a propagating wave equa- tion. Also, the equation for the field ϕ can be expressed as ∂µ( −ggµν∂ν)ϕ = 0, where gµν is the inverse of the following matrix: gµν ∝ −(c2s − v02) −v0 −v0 1 , (6) with cs = ρ0U0/m and g = detgµν . Thus, the equa- tion is equivalent to an equation for a massless field on a curved spacetime determined by the metric (6) with cs the speed of ”light.” Note that in order to interpret the quantity cs as a velocity, U0 must be positive because, for negative U0, cs becomes pure imaginary. Hereafter we consider positive U0, which leads to an effective space- time with Lorentzian signature. For the excitation modes of ϕ whose frequencies, say ω, are much higher than the frequency ωBEC, which is associated with the condensate motion, the condensate will be quasi-static. (For moderate changes of the inter- action, ωBEC turns out to be the trapping harmonic po- tential frequency ωho, as shall be discussed below .) The analogy between fields on the curved spacetime and ex- citation modes on the fluid flow is meaningful only when the conditions (3) are satisfied. The latter condition in Eq.(3) turns out to be ω2 ≪ (cs/ξ)2, by using Eq.(4). Thus, the frequency ω has an upper limit. The former condition in Eq.(3) is satisfied in the regions far from the edge of the condensate. (In contrast, if one is very close to the edge, zero-point oscillations become domi- nant, and so the former condition in (3) is not satisfied.) If there exists intermediate region for ω of ωBEC ≪ ω ≪ cs/ξ, (7) then the hydrodynamical approximation is justified and the condensate is quasi-static for excitation modes. Note that those modes are associated with particle emission from the horizon if the hydrodynamical flow has a dy- namically formed sonic horizon. The necessary condition for the existence of the intermediate region (7) is ξωBEC ≫ 1. (8) In the following, we mainly consider condensate satisfy- ing the above condition. III. FORMATION OF SONIC HORIZON Now we investigate sonic horizon formation in an ex- panding BEC trapped in isotropic harmonic potential, Vext = mω 2/2, where r is the radial coordinate. Ini- tially, we set the condensate in a ground state with an initial atomic interaction ai. At t = 0, the atomic inter- action is changed suddenly from ai to af (> ai), which makes the condensate expand. Then, formation of sonic horizon can be expected. The reason is as follows: The sound velocity is proportional to square root of the con- densate density and a decreasing function of r. In con- trast, the fluid velocity is an increasing function of r and the condensate expands fast around its edge whereas v0(r = 0) = 0 due to the boundary condition. Therefore, at an intermediate radius, v0 exceeds cs and the fluid flow is transonic. It has a surface satisfying cs = |v0| which is called a sonic horizon. We should note that the sonic horizon corresponds to a horizon in the analogue space- time defined by the metric (6). We also note that the subsonic region is around the center of the condensate and inside of the sonic horizon. In general, if a fluid has a static sonic horizon and a proper quantum state for an excitation field is realized, then it is theoretically predicted that the horizon will emit thermal radiation of the quantum field. As will 0 5 10 15 20 25 30 af/ai ai=50a0 ai=200a0 ai=800a0 FIG. 1: Times tc in each simulation are shown in units of ω−1ho . The horizontal axis is the ratio of af to ai. be discussed in Appendix A, if the sonic horizon in the expanding condensate is quasi-static for the field ϕ, the horizon will emit thermal radiation into the center of the condensate. The temperature characterizing the thermal emission (Hawking temperature) is given by the following formula: Tpc = ∂r(v0 − cs)|rH , (9) where rH is the horizon radius and kB is the Boltzmann’s constant[4][24]. From the above expression, it is found that the Hawking temperature is determined by gradi- ent of fluid and sound velocity at the horizon. Thus, it is important to investigate the velocity gradients at the horizon. In the derivation of the formula (9), it is as- sumed that the dynamically formed horizon is static, but in actual experiments, this assumption is not satisfied ex- actly. Therefore, the spectrum of the particle is not fully given by the single Planck’s distribution function, but rather given by a superposition of the Planck’s distribu- tion functions with slightly different temperatures. Even if this is the case, the energy scale of the particle creation emitted from the dynamically formed horizon is on the order of Tpc. We have simulated the expansion of the condensate by solving numerically (using the Crank-Nicolson scheme) the time-dependent GP equation. The initial ground- state wave function is obtained by solving the GP equa- tion using the steepest descent method for an initial s- wave scattering length ai and the number of atoms N . We have computed cs and the radial velocity of the con- densate via (Ψ∗Ψ)U0/m, (10) v0 = ~ [Ψ ∗∂rΨ− (∂rΨ∗)Ψ] /(2mi|Ψ|2), (11) and searched for parameter sets leading to |v0| > cs. In the following, we assume that the condensate con- sists of N = 105 Rb atoms. (The values of atomic inter- action given below are those in the case of N = 105. If 0 2 4 6 8 10 12 14 FIG. 2: Sound velocity cs(solid line) and the fluid velocity v0(dashed line) versus r at t̃ = 0.4 in the case of ai = 200a0 and af = 5ai are shown in units of (~ωho/m) 1/2. The healing length ξ(dotted line) is shown as well in units of aho = (~/mωho) N = 105/n with an integer n, then ai and af should be multiplied by n.) The initial atomic interaction is assumed to be ai = 50a0, 200a0 and 800a0 where a0 = 0.53 × 10−10m is the Bohr radius. The follow- ing change of the atomic interaction has been simulated: af/ai = 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, 25, 30. Just after t = 0, the condensate begins to expand in the trapping potential and the expansion is accelerated for a while. At some time, say t = tc, the expansion turns to be decelerated. Figure 1 shows tc as a function of af/ai. It is seen that tc does not depend on the initial strength of the interaction. For t̃ := ωhot < π/2, the condensate continues to expand, and at t̃ ≃ π/2, the condensate starts to collapse. Therefore, we turn off the trapping potential at t̃ = π/4 and make the condensate expand freely in order to keep the horizon for a while. As far as we have investigated, sonic horizon always appears in the sence of the surface where v0 exceeds cs. As an example of a sonic horizon, we show Fig. 2 which is the snapshot at t̃ = 0.4 in the case of ai = 200a0 and af = 5ai. We see that, around r = 7aho, the fluid velocity exceeds the sound velocity, and the sonic horizon exists there. In this case, we find that, at t̃ ≡ ωhot = 0.11, the horizon appears. Fig. 3 shows the time dependence of the radius of the horizon, say rH , and the velocity gradient at the horizon ∂r(v0 − cs)|rH . If we keep the trapping potential for a long time, an oscillating behavior of the condensate is observed. The period of the oscillation is about π in units of ω−1ho and we find ωBEC ≃ ωho, within a moderate change of the interaction. This oscillation is just like an oscillation of a droplet confined in a harmonic potential. Therefore, the condition (8) can be rewritten as ≫ 1. (12) FIG. 3: Time dependence of ∂r(v0−cs)rH (solid line) in units of ωho and the position of the sonic horizon rH (dashed line) in units of aho = (~/mωho) in the case of ai = 200a0 and af = 5ai. 0 5 10 15 20 25 30 af/ai ai=50a0 ai=200a0 ai=800a0 FIG. 4: cs/ξωho as a function of af/ai is shown for each initial scattering length. Now, we are interested in sonic horizon where (12) is sat- isfied. The intermediate region (7) exists if, for example, the following inequality is satisfied: ≥ 22.5. (13) For this choice of the lower bound, there exists a re- gion of ω satisfying both conditions of ω ≥ 10ωho and (cs/ωξ) 2 ≥ 5. The condition (13) ensures hydrodynamic flow and quasi-static nature of the condensate. Fig. 4 shows cs/ξωho as a function of af/ai. We define horizon life time as the time interval during which the condition (13) continues to be satisfied at the horizon. The horizon life time is shown in Fig. 5. As far as we have investi- gated, the condensate flow satisfying (13) at sonic horizon appears only when af ≥ 5ai for ai = 50a0, af ≥ 4ai for ai = 200a0 and af ≥ 3ai for ai = 800a0. The Hawking temperature at t = tc and t̃ = 0.79 (just 0 5 10 15 20 25 30 af/ai ai=50a0 ai=200a0 ai=800a0 FIG. 5: Horizon life time as a function of af/ai is shown in units of ω−1ho . 0 5 10 15 20 25 30 af/ai ai=50a0 ai=200a0 ai=800a0 FIG. 6: Hawking temperature at t = tc in units of nK. In the evaluation, we assume ωho = 1400 Hz. after turning off the trapping potential) are shown in Fig. 6 and Fig. 7, respectively. In the evaluation, we assume the frequency ωho = 1400 Hz. The Hawking temperature at t = tc depends on the ratio af/ai almost linearly. In contrast, for af/ai ≥ 9, the Hawking temperature at t̃ = 0.79 does not depend on the ratio so much. From the simulations, the temperature is expected to be a few nK. For this spherically symmetric trap, one may concern the three-body recombination loss of condensed atoms. Now, we check the effect of three-body losses for the given peak density. This effects may be taken into account by incorporating the imaginary term describing the inelastic process in the GP equation [22] i~∂tΨ = ∇2 + Vext + U0|Ψ|2 Ψ− i~ K3|Ψ|4Ψ, where K3 denotes three-body recombination loss-rate co- efficient. Then, the three-body loss is proportional to the 0 5 10 15 20 25 30 af/ai ai=50a0 ai=200a0 ai=800a0 FIG. 7: Hawking temperature at t̃ = 0.79 in units of nK, when just after the trapping potential is turned off. In the evaluation, we assume ωho = 1400 Hz. cube of the atomic density |Ψ|2d3r = −K3 |Ψ|6d3r, which implies that the three-body loss rate is given by R3 ≡ K3 |Ψ|6d3r/ |Ψ|2d3r. For the value of K3, we assume K3 = 2 × 10−28cm6/s, according to [23]. Of course, high atomic density causes many inelastic pro- cesses and gives high atomic loss rate. In our numerical simulation, the upper limit of the loss rate can be esti- mated by use of the peak density asR3 ≤ 3×10s−1, where the total atomic number was set to be N = |Ψ|2d3r = 105. Then, the three-body loss can be ignored because we consider the time scale of ≤ 10 ms. In the above evaluation for Hawking temperature, hori- zon lifetime and R3, we have assumed that the trapping frequency is ωho = 1400 Hz. Note that ωho is the en- ergy scale of the system. Therefore, a large value of ωho is plausible to increase the characteristic tempera- ture for the particle emission, though the time evolution process becomes rapid for large ωho. If lower frequency is assumed, lower temperature, longer horizon lifetime and fewer three-body loss rate would be expected. As an example, Fig. 8 shows ωho-dependence of the Hawk- ing temperature and the horizon lifetime in the case of ai = 200a0 and af = 10ai. IV. BOGOLIUBOV SPECTRUM In the above numerical simulations, we assume there is no dynamical instability. Now, we check whether there is dynamical instability or not, within Gaussian approx- imation. For that purpose, we study the Bogoliubov-de Gennes equations: the second quantized field equations 0 200 400 600 800 1000 1200 1400 horizon life time FIG. 8: ωho-dependence of TH and horizon lifetime in the case of ai = 200a0 and af = 10ai. The Hawking temperature is shown in units of nK and horizon lifetime is in units of ω−1ho . for the excitation fields δφ and δφ are given by i~∂tδφ = ∇2 + Vext + 2U0|Ψ|2 δφ+ U0Ψ −i~∂tδφ = ∇2 + Vext + 2U0|Ψ|2 δφ+ U0 The excitation spectrum is computed by performing the Bogoliubov transformation: uα (r) bαe −iEαt/~ − vα (r) b†αeiEαt/~ ,(14) u∗α (r) b iEαt/~ − v∗α (r) bαe−iEαt/~ .(15) The energy spectrum Eα is calculated by diagonalizing the skew symmetric matrix, which is carried out by using a routine in LAPACK. For the parameter values taken above, we find that all eigenvalues do not have the imag- inary parts within numerical errors. Therefore, within Gaussian approximation, there is no dynamical instabil- ity. In addition, we find that there is no level crossing. V. SUMMARY To summarize, we have proposed an experiment to create a quasi-static sonic horizon using an expanding BEC. It has been shown that the dynamically formed quasi-static sonic horizon is in hydrodynamic regime as it should be to discuss analogy with curved spacetime in BEC. Under suitable choices of the interaction parameter and the confining potential, the characteristic tempera- ture of the particle emission is expected to be a few nK for sufficiently strong confining potential. Large num- ber of atoms or strong atomic interaction improves the quasi-static nature of the horizon. Of course, other effect such as cosmological particle creation can occur in this expanding BEC setup, as dis- cussed in [14]-[21]. In this paper, we have focused on how to make dynamically formed quasi-static sonic hori- zon in the hydrodynamic regime of the condensate flow. In order to investigate cosmological particle creation ef- fect and other excitations arising from depletion, we need a different numerical simulation scheme. The result will be reported in a future publication. Furthermore, it is interesting to investigate numeri- cally the behavior of negative frequency modes with pos- itive norm which seem to be related to Hawking effect as was discussed in [26][27]. This point shall be investigated in a future publication. Acknowledgments Y.K. thanks Hideki Ishihara, Ken-ichi Nakao, and Makoto Tsubota for useful discussions. The authors thank Michikazu Kobayashi and Takashi Uneyama for useful comments on numerical simulations. Y.K. was partially supported by the Yukawa memorial founda- tion. This work was also supported by the 21st Century COE ”Center for Diversity and Universality in Physics” and ”Constitution of wide-angle mathematical basis fo- cused on knots” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. The numerical calculations were carried out on Altix3700 BX2 at YITP in Kyoto University. APPENDIX A: PARTICLE CREATION PHENOMENON Here we focus on spherically symmetric quantum fluc- tuations by symmetry. At t ≤ 0, the fluid velocity v0 = 0, and the metric of the initial static effective spacetime is ds2 ∝ −c2sdt2 + dr2 + r2dΩ2S2 , (A1) where dΩ2S2 is the element of solid angle on the unit sphere S2. After the increase of the interaction, the ef- fective spacetime evolves dynamically as the BEC starts to expand. Then, the sonic horizon is formed as was shown by the above numerical simulation. If the effec- tive spacetime is static, we can introduce a following time coordinate: τ = t + v0dr/(c s − v20), and the effective spacetime metric becomes ds2 ∝ −(c2s − v20)dτ2 + c2sdr c2s − v20 + r2dΩ2S2 . (A2) From this expression, it is found that the horizon is lo- cated at the surface where the condition cs = |v0| is sat- isfied. A new coordinate v is introduced as v ≡ τ + r∗ where r∗ ≡ csdr/(c s − v20), which is a coordinate char- acterizing ingoing light-like (null) rays in the effective spacetime. We assume here that the initial state of the quantum field ϕ is the vacuum state for the static observer in the initial effective spacetime. Under the time evolution of the effective spacetime caused by the expansion of the condensate, the creation and annihilation operators for the field ϕ also evolve, and particle creation occurs. Now we consider an observer who moves along his or her outgoing geodesic with proper time λ, crossing the horizon at λ = 0. Hereafter, we term the observer geodesic observer. If we assume that the horizon is lo- cated at r = rH , the proper time λ is related to the coor- dinate v there via λ ≈ −λ0e− v, where cH ≡ cs(rH), α ≡ 2cH∂r(v0 − cs)|r=rH and λ0 is a constant. The ingo- ing mode functions ϕω = e −iωv have λ-dependence near the horizon as ϕω ≈ exp ln(−λ) . (A3) Initially, the state is the vacuum for the static observer and therefore the geodesic observer would see no exci- tation at short distance, because there will be no much higher positive frequency excitations than those deter- mined by the time scale of the dynamical expansion of the BEC. If we ignore the short distance cut-off deter- mined by the healing length, or equivalently, if the lat- ter condition in Eq.(3) is ignored, this λ-dependence of the ingoing mode functions implies that the particle cre- ation from the horizon into the inside of the condensate has thermal spectrum with the temperature given by (9). Furthermore, even if the short distance cut-off is taken into account, it is known that the result does not change in principle [25]. Therefore, the particle emission from horizon will oc- cur in the case of expanding condensate, even where the subsonic region is inside of the horizon. [1] S.W. Hawking, Nature 248, 30 (1974); S.W. Hawking, Commun. Math. Phys. 43, 199 (1975). [2] Artificial Black Holes, edited by M. Novello, M. Visser, and G. Volovik (World Scientific, 2002). [3] C. Barcelo, S. Liberati and M. Visser, Living Rev. Rel. 8, 12 (2005) [4] W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981). [5] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E.A. Cornell, Science 269, 198 (1995). [6] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995). [7] L. J. Garay, J. R. Anglin, J. I. Cirac and P. Zoller, Phys. Rev. Lett. 85, 4643 (2000) [8] L. J. Garay, J. R. Anglin, J. I. Cirac and P. Zoller, Phys. Rev. A 63, 023611 (2001) [9] C. Barcelo, S. Liberati and M. Visser, Class. Quantum Grav. 18, 1137 (2001). [10] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn, and W. Ketterle, Nature 392, 151 (1998); P. Courteille, R.S. Freeland, D.J. Heinzen, F.A. van Abeelen, and B.J. Verhaar, Phys. Rev. Lett. 81, 69 (1998); J.L. Roberts, N.R. Claussen, J.P. Burke, C.H. Greene, E.A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 81, 5109 (1998). [11] E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell, and C. E. Wieman, Nature 412, 295 (2001). [12] E. A. Calzetta and B. L. Hu, Phys. Rev. A 68, 043625 (2003) [13] E. A. Calzetta and B. L. Hu, cond-mat/0208569 [14] C. Barcelo, S. Liberati and M. Visser, Int.J.Mod. Phys. D 12 1641 (2003). [15] C. Barcelo, S. Liberati and M. Visser, Phys. Rev. A 68 053613 (2003). [16] P. O. Fedichev and U. R. Fischer, Phys. Rev. Lett. 91, 240407 (2003). [17] P. O. Fedichev and U. R. Fischer, Phys. Rev. A. 69, 033602 (2004). [18] P. O. Fedichev and U. R. Fischer, Phys. Rev. D 69, 064021 (2004). [19] U. R. Fischer and R. Schützhold, Phys. Rev. A 70, 063615 (2004). [20] J. E. Lidsey, Class. Quantum Grav. 21, 777 (2004). [21] S. E. C. Weinfurtner, gr-qc/0404063. [22] Yu. Kagan, A.E. Muryshev, and G.V. Shlyapnikov, Phys. Rev. Lett. 81 933 (1998). [23] H. Saito and M. Ueda, Phys. Rev. A 65, 033624 (2002). [24] For derivation details of the particle creation, see, for example, T. Jacobson, gr-qc/0308048. [25] S. Corley and T. Jacobson, Phys. Rev. D 54, 1568 (1996). [26] U. Leonhardt, T. Kiss and P. Öhberg, J. Opt. B 5 S 42 (2003). [27] U. Leonhardt, T. Kiss and P. Öhberg, Phys. Rev. A 67, 033602 (2003) http://arxiv.org/abs/cond-mat/0208569 http://arxiv.org/abs/gr-qc/0404063 http://arxiv.org/abs/gr-qc/0308048

0704.1097

Zero-temperature resistive transition in Josephson-junction arrays at irrational frustration

Zero-temperature resistive transition in Josephson-junction arrays at irrational frustration Enzo Granato Laboratório Associado de Sensores e Materiais, Instituto Nacional de Pesquisas Espaciais, 12245-970 São José dos Campos, SP Brazil We use a driven Monte Carlo dynamics in the phase representation to determine the linear re- sistivity and current-voltage scaling of a two-dimensional Josephson-junction array at an irrational flux quantum per plaquette. The results are consistent with a phase-coherence transition scenario where the critical temperature vanishes. The linear resistivity is nonzero at any finite tempera- tures but nonlinear behavior sets in at a temperature-dependent crossover current determined by the thermal critical exponent. From a dynamic scaling analysis we determine this critical exponent and the thermally activated behavior of the linear resistivity. The results are in agreement with earlier calculations using the resistively shunted-junction model for the dynamics of the array. The linear resistivity behavior is consistent with some experimental results on arrays of superconducting grains but not on wire networks, which we argue have been obtained in a current regime above the crossover current. PACS numbers: 74.81.Fa, 74.25.Qt, 75.10.Nr Most theoretical investigations of the vortex-glass phase in superconductors have considered model systems where there is a combined effect of quenched disorder and frustration1. However, in artificial Josephson-junction arrays, frustration without disorder can in principle be introduced by applying an external magnetic field on a perfect periodic array of weakly coupled supercon- ducting grains2,3,4 and similarly on superconducting wire networks5,6. The frustration parameter f , the number of flux quantum per plaquette, is given by f = φ/φo, the ratio of the magnetic flux through a plaquette φ to the su- perconducting flux quantum φo = hc/2e. It can be tuned by varying the strength of the external field. Frustration effects can be viewed as resulting from a competition be- tween the underlying periodic pinning potential of the array and the natural periodicity of the vortex lattice7. At a rational value of f , the ground state is a commensu- rate pinned vortex lattice leading to discrete symmetries in addition to the continuous U(1) symmetry of the su- perconducting order parameter. The resistive transition is only reasonably well understood for simple rational values of f . At irrational values of f , the resistive behavior is much less understood since the vortex lattice is now incom- mensurate with the periodic array. In early Monte Carlo (MC) simulations8 the ground state was found to con- sist of a disordered vortex pattern lacking long range or- der which could be regarded as a some sort of vortex- glass state without quenched disorder. Glassy-like be- havior was indeed observed in these simulations suggest- ing a possible superconducting (vortex-glass) transition at finite temperatures. However, some arguments also suggested that the critical temperature should vanish7,9. Simulations of the current-voltage scaling using the resis- tively shunted-junction model for the dynamics of the ar- ray found that the behavior was consistent with an equi- librium resistive transition where the critical temperature vanishes10, similar to the resistive transition described by the the gauge-glass model in two dimensions1,11, but with different values for the correlation-length critical expo- nent ν. The linear resistivity is nonzero at any finite tem- peratures but nonlinear behavior sets in at a crossover current with a temperature dependence determined by the exponent ν. This zero-temperature transition leads to slow relaxation dynamics where the correlation length diverges as a power law and the relaxation time diverges exponentially as the temperature vanishes. Simulations of the relaxation dynamics12 found a be- havior analogous to relaxation in supercooled liquids with a characteristic dynamic crossover temperature rather than an equilibrium transition temperature, which is not inconsistent with the zero-temperature transition sce- nario. On the other hand, a systematic study by MC simulations13 of a sequence of rational values of f con- verging to the irrational frustration, using the vortex rep- resentation, found two phase transitions at finite temper- atures, a vortex-order transition weakly dependent on f and a vortex pinning transition at much lower tem- peratures varying with f , which should correspond to the resistive transition. These results are in qualitative agreement with MC simulations using the phase repre- sentation of the same model14 but different ground states were found. More recently, MC simulations for the the specific heat and relaxation dynamics found an intrinsic finite- size effect15. The corresponding scaling analysis sug- gested a zero-temperature transition with a critical ex- ponent ν consistent with the value obtained initially from current-voltage scaling10. However, a study of the low-temperature configurations for frustrations close the irrational value by MC simulations in the vortex representation16, find two phase transitions consistent with earlier work13. On the experimental side, some results on arrays of su- http://arxiv.org/abs/0704.1097v1 perconducting grains at irrational frustration2,3 are con- sistent with the scenario of the zero-temperature resis- tive transition but on wire networks5,6, resistivity scaling showed evidence of a transition at finite temperature. Re- cently, resistivity scaling suggesting a finite temperature transition was also observed in arrays of superconducting grains4. In view of these conflicting results, it seems useful to further investigate the current-voltage scaling for the ar- ray at irrational frustration by studying both the non- linear and linear resistivity with an improved method17 taking into account the long relaxation times. In fact, as found recently, current-voltage scaling turned out to be quite reliable in determining the phase-coherence transi- tion even for a model with quenched disorder, such as the three-dimensional XY-spin glass model17,18. The main question is therefore, if the array at irrational frustration displays an equilibrium phase-coherence transition at a nonzero critical temperature into a state with vanishing linear resistivity or its critical temperature vanishes and the linear resistivity is finite at nonzero temperatures. In this work, we investigate the resistivity scaling of Josephson-junction arrays at a irrational frustration f = 5)/2, a golden irrational, using a driven MC dy- namics in the phase representation introduced recently17. The results are consistent with a phase-coherence tran- sition scenario where the critical temperature vanishes, Tc = 0. The linear resistivity is finite at nonzero temper- atures but nonlinear behavior sets in at a temperature- dependent crossover current determined by the thermal critical exponent ν. The results agree with earlier simula- tions using the resistively shunted-junction model for the dynamics of the array10. However, with the present MC method we are able to reach much lower temperatures and current densities, improving the analysis of resistiv- ity scaling and the estimate of the critical exponent ν. We also argue that the finite-temperature transition found in resistivity measurements on wire networks5,6have been obtained in a current regime above the crossover current. We consider a two-dimensional Josephson-junction square array described by the Hamiltonian H = −Jo cos(θi − θj −Aij)− J (θi − θi+x) (1) The first term gives the Josephson-coupling energy be- tween nearest neighbor grains where line integral of the vector potential Aij is constrained to ij Aij = 2πf around each plaquette. The second term represents the effects of an external driving current density J applied in the x direction. When J 6= 0, the total energy is un- bounded and the system is out of equilibrium. The lower- energy minima occur at phase differences θi−θi+x which increases with time t, leading to a net phase slippage rate proportional to < d(θi − θi+x)/dt >, corresponding to the voltage Vi,i+x. We take the frustration parameter f equals an irrational number, f = (3− 5)/2, related to the Golden Ratio Φ = (1+ 5)/2 as f = 1− 1/Φ. In the numerical simulations we use periodic (fluctuating twist) boundary conditions on lattices of linear sizes L and cor- responding rational approximations Φ = Fn+1/Fn, where Fn are Fibonacci numbers (13, 21, 34, 55), with L = Fn. To study the current-voltage scaling, we use a driven MC dynamics method17. The time dependence is ob- tained by identifying the MC time as the real time t and we set the unit of time dt = 1, corresponding to a com- plete MC pass through the lattice. Periodic (fluctuat- ing twist) boundary conditions are used19. This bound- ary condition adds new dynamical variables, uα (α = x and y), corresponding to a uniform phase twist between nearest-neighbor sites along the principal axis directions x̂ and ŷ. A MC step consists of an attempt to change the local phase θi and the phase twists uα by fixed amounts, using the Metropolis algorithm. If the change in en- ergy is ∆H , the trial move is accepted with probability min{1, exp(−∆H/kT )}. The external current density J in Eq. 1 biases these changes, leading to a net voltage (phase slippage rate) across the system given by (θ1,j − θL+1,j − uxL), (2) in arbitrary units. The main advantage of this MC method compared with the Langevin dynamics used earlier10 is that in principle much longer time scales can be accessed which allows one to obtain reliable data at much lower temperatures and current densities. We have determined the electric field E = V/L and nonlinear re- sistivity ρ = E/J as a function of the driving current density J for different temperatures T and different sys- tem sizes. We used typically 2×105 MC steps to reach the nonequilibrium steady state at finite current and equal time steps to perform time averages with and additional average over 4− 6 independent runs. We have also determined the linear resistivity, ρL = limJ−>0E/J , from equilibrium MC simulations. As any transport coefficient, this quantity can be obtained from equilibrium fluctuations and therefore can be calculated in absence of an imposing driving current (J = 0). From Kubo formula, the linear resistivity (resistance in two dimensions) is given in terms of the equilibrium voltage autocorrelation as dt〈V (t)V (0)〉. (3) Since the total voltage V is related to the phase difference across the system ∆θ(t) by V = d∆θ(t)/dt, we find more convenient to determine ρL from the long-time equilib- rium fluctuations11 of ∆θ(t) as 〈(∆θ(t) −∆θ(0))2〉, (4) which is valid for sufficiently long times t. To insure that only equilibrium fluctuations are considered, the calcu- lations were performed in two steps. First, simulations using the exchange MC method (parallel tempering)20 T=0.3 T=0.275 T=0.25 T=0.225 T=0.20 T=0.175 T=0.15 FIG. 1: Nonlinear resistivity E/J at different temperatures T for system size L = 55. were used to obtain equilibrium configurations of the sys- tems at different temperatures21. This method is known to reduce significantly the critical slowing down near the transition allowing fully equilibration in finite small sys- tem sizes. These configurations were then used as ini- tial states for the driven MC dynamics process described above, with J = 0, in order to obtain the ρL. The ini- tial states are similar to the low-temperature states ob- tained previously13,16 including thermal excitations. In the parallel-tempering method20, many replicas of the system with different temperatures are simulated simul- taneously and the corresponding configurations are al- lowed to be exchanged with a probability satisfying de- tailed balance. The equilibration time can be measured as the average number of MC steps required for each replica to travel over the whole temperature range. We used typically 4× 106 (parallel tempering) MC steps for equilibration which is much larger than the estimated equilibration time in systems with up to 100 replicas. Subsequent MC simulations for the linear resistivity ob- tained from Eq. 4 were performed using 2 × 103 time averages for 2× 105 MC steps which is much larger than the equilibrium relaxation time. Fig. 1a shows the nonlinear resistivity E/J as a func- tion of temperature for the largest system size. At small current densities J , the nonlinear resistivity E/J tends to a constant value, corresponding to the linear resis- tivity ρL, which decreases rapidly with decreasing tem- perature. For increasing J , the resistivity cross over to a nonlinear behavior at a characteristic current density Jnl, which also decreases with decreasing temperature. To verify that the nonzero values approached at low cur- rents in Fig. 1 correspond indeed to the linear resistivity ρL, we show in Fig. 2 the temperature dependence of ρL obtained without current bias from Eq.(4) for dif- ferent system sizes. ρL decreases with system size but approaches nonzero values for the largest system size. These values are in agreement with the corresponding values at the lowest current in Fig. 1. Since the be- 3 4 5 6 7 L=13 L=21 L=34 L=55 FIG. 2: Temperature dependence of the linear resistivity for different system sizes. havior of the ρL for the largest system size on the log- linear plot in Fig. 2 is a straight line, it indicates an activated Arrhenius behavior, where the linear resistiv- ity decreases exponentially with the inverse of tempera- ture with a temperature-independent energy barrier, es- timated as Eb ∼ 1.07. Such activated behavior suggests that the linear resistivity can be very small at low tem- peratures but nevertheless remains finite for all tempera- tures T > 0 and therefore there is no resistive transition at finite temperatures. However, as will be described below, the system behaves as if a resistive transition occurs at zero temperature, corresponding to a phase- coherence transition where the critical temperature van- ishes, Tc = 0. The behavior of the linear resistivity can be related to the equilibrium relaxation time for phase fluctuations. Since the voltage is the rate of change of the phase, a nonzero ρL requires measurements over a time scale τ ∝ 1/ρL, corresponding to the relaxation time for phase fluctuations. Thus, we expect that τ should also have an activated behavior, increasing exponentially with the inverse of temperature. To verify this behavior, we have in addition calculated the relaxation time τ for different temperatures from the autocorrelation function of phase fluctuations C(t) as C(0)2 dtC(t) (5) using MC simulations with J = 0. The starting con- figurations were taken from equilibrium configurations obtained21 with the parallel tempering MC method20. The results shown on the log-linear plot in Fig. 3 are in- deed consistent with an activated behavior of τ with an energy barrier Eb = 1.18 in reasonable agreement with the value obtained for the linear resistivity in Fig. 2. The behavior in Figs. 1, 2 and 3 has the main features associated with a phase transition that only occurs at zero temperature, Tc = 0, similar to the two-dimensional gauge glass model of disordered superconductors1,11. In this case the correlation length ξ is finite for T > 0 but it increases with decreasing temperature as ξ ∝ T−ν , with ν a critical exponent. The divergent correlation length near the transition determines both the linear an nonlin- ear resistivity behavior leading to current-voltage scaling sufficiently close to the critical temperature and suffi- ciently small driving current. To understand in detail the behavior of the linear ρL and nonlinear resistivity ρ we need a scaling theory for the resistive behavior. If the data satisfy such scaling behavior for different driving currents and temperatures, the critical temperature and critical exponents of the underlying equilibrium transi- tion at J = 0 can then be determined from the best data collapse. A detailed scaling theory has been described in the context of the current-voltage characteristics of vortex-glass models1 but the arguments should also ap- ply to the present case. The basic assumption is the existence of a second order phase transition. Measurable quantities should then scale with the diverging correla- tion length ξ ∝ |T −Tc|−ν and relaxation time τ near the critical point. The nonlinear resistivity E/J should then satisfy the scaling form1 τ = g±( ), (6) in two-dimensions, where g±(x) is a scaling function. The + and − signs correspond to T > Tc and T < Tc, respec- tively. If Tc 6= 0, then to satisfy such scaling form, the nonlinear resistivity curves on the log-log plot in Fig. 1 should have a positive curvature at small J , with E/J de- creasing with decreasing J to a temperature dependent value for T > Tc while for T < Tc, the curvature should be negative, with E/J vanishing in the limit J → 0. The data in Fig. 1 do not show a change in curvature even for the lowest temperature, already suggesting the possibil- ity of a resistive transition at much lower temperatures or at Tc = 0. However, a full scaling analysis of the data is required to show that a transition indeed occur with Tc = 0. If Tc = 0, then the correlation length ξ ∝ T−ν and the linear resistivity ρL are both finite at T > 0. One can then consider the behavior of the dimensionless ratio E/JρL which should satisfy the scaling form T 1+ν ) (7) where g is a scaling function with g(0) = 1. A crossover from linear behavior, when g(x) ∼ 1, to nonlinear behav- ior, when g(x) >> 1, occurs when x ∼ 1 which leads to a characteristic current density at which nonlinear behav- ior sets in decreasing with temperatures as a power law, Jnl ∝ T/ξ ∝ T 1+ν . The scaling form in Eq. (7)contains a single critical exponent ν and does not depend on the par- ticular form assumed for the divergence of the relaxation time τ . However, for sufficiently low temperatures, the relaxation process is expected to be thermally activated1 with τ ∝ exp(Eb/kT ). This corresponds formally to a dynamic exponent z → ∞, if power-law behavior is as- sumed for the relaxation time τ ∝ ξz . From the scal- ing form of Eq.(6), the linear resistivity should scale as 3 4 5 6 7 τ � e E =1.18(2) FIG. 3: Temperature dependence of the relaxation time τ of phase fluctuations for system size L = 55. ν = 1.4 T=0.30 T=0.275 T=0.25 T=0.225 T=0.20 T=0.175 T=0.15 J / T 1 + ν FIG. 4: Scaling plot of the nonlinear resistivity in Fig. 1 for ν = 1.4. ρL ∝ 1/τ and therefore it is also expected to have an activated behavior, τ ∝ exp(−Eb/kT ). In general, the energy barrier Eb also scales with the correlation length as Eb ∝ ξψ , which leads to a temperature-dependent bar- rier Eb ∝ T−ψν . A pure Arrhenius behavior corresponds to ψ = 0. The behavior of the nonlinear and linear resis- tivity in Figs 1, 2 and the relaxation time in Fig. 3 are quite consistent with these predictions from the scaling theory of a zero-temperature transition. If there is a zero-temperature transition, as suggested by the behaviors in Figs. 1, 2 and 3, then the data for the nonlinear resistivity should satisfy the scaling form of Eq.(7), if finite-size effects are negligible, and the best data collapse provides an estimate of the critical expo- nent ν. We expect that finite-size effects are negligible for the largest system size L = 55 in Fig. 1 since at this length scale the behavior of the linear resistivity is roughly independent of the size as can be seen from Fig. 2. Fig. 4 shows that indeed the data for the largest system size satisfy this scaling form with ν ∼ 1.4± 0.2. The nonlinear resistivity should also satisfy the ex- pected finite-size behavior in smaller system sizes when the correlation length ξ approaches the system size L. According to finite-size scaling, the scaling function in Eq. (7), should also depend on the dimensionless ratio L/ξ and so to account for finite-size effects the nonlinear resistivity should satisfy the scaling form = ḡ( T 1+ν , L1/νT ). (8) The scaling analysis of the whole nonlinear resistivity data is rather complicated in this case since the scal- ing function depends on two variables. To simplify the analysis22 we first estimate the temperature and finite-size behavior of the crossover current density Jnl where nonlinear behavior sets in as the value of J where E/JρL = C, a constant. Then, from Eq. (8), the finite- size behavior of Jnl can be expressed in the scaling form (1+ν)/ν = ¯̄g(L1/νT ). (9) The best data collapse according to the scaling in Eq. (9) provides an alternative estimate of the critical exponent ν. Fig. 5 shows that indeed the values of Jnl for different system sizes and temperatures satisfy this scaling form with ν ∼ 1.4, in agreement with the estimate obtained for the largest system in Fig. 4 size using Eq. (7). In addition to the standard finite-size effects, which oc- cur when the correlation length is comparable to the sys- tem size, already taken into account in the scaling form of Eq. (8), there are also intrinsic finite-size effects15 resulting from the rational approximations used for the irrational value of f . Since we use rational approxima- tions Φ = Fn+1/Fn, where Fn are Fibonacci numbers (13, 21, 34, 55), with the system size set to L = Fn, this amounts essentially to have different values of the frus- tration, fL = 1 − 1/Φ, for different system sizes which will only converge to the correct value f = (3 − in the infinite-size limit. We have assumed that such ef- fects are negligible in the above scaling analysis but they should affect our estimate of the critical exponent ν. In principle, this intrinsic effect could be taken into account within the zero-temperature transition scenario by allow- ing for a size-dependent critical temperature Tc(L) in the scaling analysis15. Alternatively, we could regard it as a crossover from the critical behavior at the true irrational frustration (infinite-size limit) to a phase with an addi- tional small frustration δf = fL − f which should act as a relevant perturbation. In this case, the scaling func- tion in Eq. (7) should also depend on the dimensionless ratio ξ2δf and again a scaling analysis with more than one variable is required. However, our present numeri- cal data is not sufficiently accurate to separate this effect from standard finite-size effects. The present results for the linear and nonlinear resis- tivity of the array at irrational frustration obtained by the driven MC dynamics agree with earlier simulations of the current-voltage scaling using the resistively shunted- junction model for the dynamics of the array10, where a zero-temperature resistive transition was suggested and 1 2 3 4 5 6 ν = 1.4 L=13 L=21 L=34 L=55J n 1 / ν FIG. 5: Finite-size scaling plot of the crossover current den- sity Jnl with ν = 1.4, for different system sizes L. the critical exponent was estimated as ν = 0.9(2). Al- though the later model is expected to be a more realistic description for the dynamics of the array, the value of the static critical exponent ν should be the same for both models. In general, the dynamic exponent z may de- pend on the particular dynamics but since the relaxation time τ is found to diverge exponentially for decreasing temperature it corresponds to z → ∞ for both dynam- ics. The present estimate of ν = 1.4(2), however, should be more reliable since it considers much lower tempera- tures and current densities and larger system size. In- terestingly, similar behavior for the resistive transition has been found both numerically and experimentally for two-dimensional disordered superconductors in a mag- netic field described as a gauge-glass model1,11 but with different value for critical exponent ν ∼ 2. It should be noted however that the actual ground state at irra- tional frustration (without disorder) can be quite differ- ent, as the self similar structure which has already been proposed5,23. As would be expected, the different nature of ground state leads to the different values of the critical exponent ν. Although the above scaling analysis is consistent with a zero temperature transition, on pure numerical grounds the data in Figs 1 and 2 can not complete ruled out a vortex-order or a phase-coherence transition at tem- peratures much lower than T = 0.15. In fact, phase- coherence transitions were found in MC simulations using the Coulomb-gas presentation13 at temperatures as low as T ∼ 0.03 for the sequence of rational approximations fL of the irrational f but since they show considerable variation with fL it is not clear if it will remain nonzero in the large size limit. However, the lowest temperature in Figs 1 and 2 is already much smaller than the apparent freezing temperature Tf ∼ 0.25 observed in earlier MC simulations8. Below Tf , a nonzero Edwards-Anderson order parameter q(t) =< ~Si > 2, was observed, where ~S = (cos θ, sin θ) and the average was taken over the sim- ulation times t. Although this could suggest a diverging relaxation time τ ∝ q(t)dt near a finite temperature Tc ∼ Tf , such long relaxation time can also result from a zero-temperature transition (Tc = 0) as suggested by the above scaling analysis since in this case τ diverges expo- nentially with decreasing temperature, τ ∝ exp(Eb/kT ), as shown in Fig. 3. For low enough temperatures, τ will eventually be larger than any simulation or experimental measuring time scale and an apparent (time dependent) freezing transition could occur depending on the partic- ular dynamics and system size. Some experimental results on arrays of superconduct- ing grains at irrational frustration2,3 are consistent with the scenario of a zero-temperature resistive transition since even at the lowest temperatures a zero-resistance state was not observed in these experiments. On the other hand, current-voltage scaling analysis of experi- mental data on wire networks5,6 was found to be con- sistent with a resistive transition at finite temperature. We note, however, that although the equilibrium behav- ior of wire networks can be described by the same model of Eq. 1, the nonlinear dynamical behavior may be quite different since the nodes of the network are connected by continuous superconducting wires, instead of weak links, leading to additional larger energy barriers for vor- tex motion, not included in the model, and consequently larger phase-coherence length ξ and relaxation time τ when compared with weak links24. In this case, the characteristic crossover current to the linear resistivity regime at low temperatures due to thermal fluctuations, Jnl ∝ kT/ξ, expected in the zero-temperature transition scenario, may only occur at current scales too small to be detected experimentally. Thus the resistive behavior is observed in a current regime at higher currents where it follows the mean-field theory result25 where a vortex- glass transition is possible at finite temperatures. How- ever, the zero-temperature resistive could in principle be observed in specially prepared wire networks in the weak coupling regime where the additional energy barrier for vortex motion can be minimized26. Other effects, such as weak disorder, which is inevitably present in both ex- perimental systems, should also be considered. It could provide a possible explanation for the finite-temperature resistive transition observed recently in arrays of super- conducting grains4. In conclusion, we have investigated the resistivity scal- ing of Josephson-junction arrays at a irrational frustra- tion using a driven MC dynamics17. The results are con- sistent with a phase-coherence transition scenario where the critical temperature vanishes, Tc = 0. The linear resistivity is finite at nonzero temperatures but nonlin- ear behavior sets in at a crossover current determined by the thermal critical exponent ν. The results agree with earlier simulations using the resistively shunted- junction model for the dynamics of the array10 and more recent MC simulations taking into account the intrinsic finite-size effect15. Although we have only studied the array at a particular value of irrational frustration, the golden mean, we believe that the conclusion of a zero- temperature phase-coherence transition should be valid for all irrationals but possibly with different values of the thermal critical exponent ν. The main advantage of studying the golden mean value is that it is considered the farthest from the low-order rationals and so intrinsic finite-effects should be smaller. However, other irrational frustrations have also been studied numerically15,23 and experimentally5. The resistive behavior probes mainly the phase-coherence of the system and since we find that phase coherence is only attained at zero temperature, we can not address directly the question of the existence of a vortex-order transition at finite temperatures. In fact, vortex order does not require long-range phase coherence. Therefore, a vortex-order transition at zero temperature or at finite temperature is consistent with the present work. However, in view of the results for the supercooled relaxation12 suggesting an analogy to structural glasses such transition may be expected at finite temperature and in fact is consistent with MC simulations indicating a first-order vortex transition13,14,16. Thus, the inter- esting possibility arises where the array undergoes two transitions for decreasing temperature, a finite-resistance vortex-order transition at finite temperature and a su- perconducting transition only at zero temperature. This phase transition scenario and the predicted behavior of the linear and nonlinear resistivity provides an interesting experimental signature for a Josephson-junction array at irrational frustration. This work was supported by FAPESP (grant 03/00541- 0) and computer facilities from CENAPAD-SP. 1 R.A. Hyman, M. Wallin, M.P.A. Fisher, S.M. Girvin, and A.P. Young, Phys. Rev. B 51, 15304 (1995); D.S. Fisher, M.P.A. Fisher, and D.A. Huse, Phys. Rev. B 43 130 (1991). 2 J.P. Carini, Phys. Rev. B 38, 63 (1988). 3 H. S. J. Zant, H. A. Rijken, and J. E. Mooij, J. Low Temp. Phys. 82, 67 (1991). 4 I.C Baek, Y.J.Y Yu, and M.Y Choi, Phys. Rev. B 69, 172501 (2004). 5 F.Yu, N.E. Israeloff, A.M. Goldman, and R. Bojko, Phys. Rev. Lett. 68, 2535 (1992). 6 X.S. Ling, H.J. Lezec, M.J. Higgins, J.S. Tsai, J. Fujita, H. Numata, Y. Nakamara, Y. Ochiai, C. Tang, P.M. Chaikin, and S. Bhattacharya, Phys. Rev. Lett. 76, 2989 (1996). 7 S. Teitel and C. Jayaprakash, Phys. Rev. Lett. 51, 1999 (1983). 8 T. Halsey, Phys. Rev. Lett. 55, 1018 (1985). 9 M.Y. Choi and D. Stroud, Phys. Rev. B 35, 7109 (1987). 10 E. Granato, Phys. Rev. B 54, 9655 (1996). 11 E. Granato, Phys. Rev. B 58, 11161 (1998). 12 B. Kim and S.J. Lee, Phys. Rev. Lett. 78, 3709 (1997). 13 P. Gupta, S. Teitel, M.J.P. Gingras, Phys. Rev. Lett. 80, 105 (1998). 14 C. Denniston and C. Tang, Phys. Rev. B 60, 3163 (1999). 15 S.Y. Park, M.Y. Choi, B.J. Kim, G.S. Jeon, and J.S. Chung, Phys. Rev. Lett. 85, 3484 (2000). 16 S.J. Lee, J.R. Lee, and B. Kim, Phys. Rev. Lett. 88, 025701 (2002). 17 E. Granato, Phys. Rev. B 69, 144203 (2004). 18 L.W. Lee and A.P. Young, Phys. Rev. Lett. 22, 227203 (2003). 19 W.M. Saslow, M. Gabay, and W.-M. Zhang, Phys. Rev. Lett. 68, 3627 (1992). 20 K. Hukushima and K. Nemoto, J. Phys. Soc. Jpn. 65, 1604 (1996); E. Marinari and G. Parisi, Europhys. Lett. 19, 451 (1992). 21 E. Granato, unpublished. 22 C. Wengel and A.P. Young, Phys. Rev. B 56, 5918 (1997) 23 M.R. Kolahchi, Phys. Rev. B 59, 9569 (1999). 24 H. S. J. Zant, M.N. Webster, J. Romijn, and J.E. Mooij, Phys. Rev. B 42, 2647 (1990). 25 G. Parisi, J. Phys. A 27, 7555 (1994). 26 M. Giroud, O. Buisson, Y.Y. Wang, and B. Pannetier, J. Low Temp. Phys. 87, 683 (1992).

0704.1098

Protostellar clusters in intermediate-mass (IM) star forming regions

Astronomy & Astrophysics manuscript no. 7297 c© ESO 2021 September 15, 2021 Letter to the Editor Protostellar clusters in intermediate mass (IM) star forming regions A. Fuente1, C. Ceccarelli2, R. Neri3, T. Alonso-Albi1, P. Caselli4,5, D. Johnstone6,7 , E.F. van Dishoeck8, and F. Wyrowski9 1 Observatorio Astronómico Nacional (OAN), Apdo. 112, E-28803 Alcalá de Henares (Madrid), Spain e-mail: [email protected] 2 Laboratoire d’Astrophysique de l’Observatoire de Grenoble, BP 53, 38041 Grenoble Cedex 9, France 3 Institute de Radioastronomie Millimétrique, 300 rue de la Piscine, 38406 St Martin d’Heres Cedex, France 4 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy 5 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 0213 6 Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 1A1, Canada 7 National Research Council of Canada, Herzberg Institute of Astrophysics, 5071 West Saanich Road, Victoria, BC V9E 2E7, Canada 8 Leiden Observatory, PO Box 9513, 2300 RA Leiden, Netherlands 9 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany Received February 14, 2007; accepted March 29, 2007 ABSTRACT Context. The transition between the low density groups of T Tauri stars and the high density clusters around massive stars occurs in the intermediate-mass (IM) range (M∗∼2–8 M⊙). High spatial resolution studies of IM young stellar objects (YSO) can provide important clues to understand the clustering in massive star forming regions. Aims. Our aim is to search for clustering in IM Class 0 protostars. The high spatial resolution and sensitivity provided by the new A configuration of the Plateau de Bure Interferometer (PdBI) allow us to study the clustering in these nearby objects. Methods. We have imaged three IM Class 0 protostars (Serpens-FIRS 1, IC 1396 N, CB 3) in the continuum at 3.3 and 1.3mm using the PdBI. The sources have been selected with different luminosity to investigate the dependence of the clustering process on the luminosity of the source. Results. Only one millimeter (mm) source is detected towards the low luminosity source Serpens–FIRS 1. Towards CB 3 and IC1396 N, we detect two compact sources separated by ∼0.05 pc. The 1.3mm image of IC 1396 N, which provides the highest spatial resolution, reveal that one of these cores is splitted in, at least, three individual sources. Key words. stars:formation–stars: individual (Serpens–FIRS 1, IC 1396 N, CB 3) 1. Introduction Low and high mass stars (M∗>8 M⊙) are formed in different regimes. While low mass stars can be formed isolated or in loose associations, high mass stars are always found in tight clusters. Intermediate-mass young stellar objects (IMs) (protostars and Herbig Ae/Be [HAEBE] stars with M∗ ∼ 2 - 8 M⊙) constitute the link between low- and high-mass stars. In particular the transi- tion between the low density groups around T Tauri stars and the dense clusters around massive stars occurs in these objects. Testi et al. (1998,1999) studied the clustering around HAEBE stars using optical and near-infrared (NIR) images and concluded that transition occurs smoothly from Ae to Be stars. Thus, these stars are key objects to study the onset of clustering. Thus far, clustering has only been studied at infrared and op- tical wavelengths because of the limited spatial resolution and sensitivity of the mm telescopes. Thus, the earliest stages of the cluster formation were hidden to the observers. The sub- arcsecond angular resolution provided by the new A configu- ration of the PdBI allows, for the first time, to study clustering at mm wavelengths with a similar sensitivity and spatial resolu- tion to the NIR studies. In this Letter, we present interferomet- ric continuum observations of the IM protostars Serpens-FIRS 1 (precursor of a Ae star) and CB 3 (precursor of a Be star) aimed Send offprint requests to: A. Fuente to study the clustering phenomena in the early Class 0 phase. We also use the data at highest spatial resolution towards IC 1396 N reported in this special issue by Neri et al. (Paper II, hereafter). 1.1. Serpens-FIRS 1 Serpens-FIRS 1 is a 46 L⊙ Class 0 source located in a very active star forming region. Previous mid-IR and NIR studies show that the population of YSOs is strongly clustered, with the Class I sources more clustered than the Class II ones (Kaas et al. 2004). The sub-clusters of Class I sources are located in a NW-SE oriented ridge following the distribution of dense cores in the molecular cloud with a subclustering spatial scale of 0.12 pc (see Fig. 1). The Class II stars are located surrounding the molecular cores with a subclustering spatial scale of 0.25 pc. Adopting a distance of 310 pc, the YSOs density in the sub-clusters ranges from 360–780 pc−2. Several high angular resolution mm studies have been made in the Serpens molecular cloud (Testi & Sargent 1998, William & Myers 1999, Hogerheijde et al. 1999, Testi et al. 2000). We have imaged at higher spatial resolution a region of 0.04 pc around the intense mm-source FIRS 1. http://arxiv.org/abs/0704.1098v1 2 A. Fuente et al.: Protostellar clusters in intermediate mass (IM) star forming regions Table 1. Millimeter flux densities, sizes, spectral indexes and masses Position Peak Gaussian width1 Int. Intensity Mass2 Size3 α4 Sensitivity5 Sampled area6 (mJy/beam) (”) (mJy) (M⊙) (AU) (M⊙) r(pc) Serpens-FIRS 1 1.3mm 18:29:49.80 01:15:20.41 273(1) 0.50”×0.63” 357 0.1 65 1.57 0.01 0.02 3.3mm 18:29:49.80 01:15:20.41 63(0.5) 0.80”×1.73” 71 0.04 CB 3-1 1.3mm 00:28:42.60 56:42:01.11 20(1) 0.36”×0.48” 34 0.62 600 2.52 0.04 0.16 3.3mm 00:28:42.60 56:42:01.11 2.0(0.5) 0.88”×1.27” 2.9 0.32 CB 3-2 1.3mm 00:28:42.20 56:42:05.11 10(1) 0.31”×0.43” 13 0.24 330 1.87 0.04 0.16 3.3mm 00:28:42.20 56:42:05.11 2.1(0.5) 0.80”×1.00” 2.1 0.32 1 Half-power width of the fitted 2-D elliptical Gaussian 2 Mass estimated using the 1.3mm fluxes and assuming Td=100 K and κ1.3mm=0.01 g −1 cm2 3 Deconvolved source size at 1.3mm 4 1.3mm/3.3mm spectral index 5 5×rms mass sensitivity derived from the 1.3mm image assuming Td=100 K and κ1.3mm=0.01 g −1 cm2 6 Radius (HPBW/2) of the PdBI primary beam at the source distance Fig. 1. Dust continuum mosaic (contours and grey scale) of the Serpens main core as observed with the IRAM 30m telescope. The location of the Class II (blue filled squares), flat (red crosses) and Class I sources (red empty circles) is indicated (adapted from Kaas et al. 2004). In the inset, we show the 3mm and 1.3mm (small inset) continuum images observed with the PdBI. Note that only one compact core is detected in this region down to a spatial scale of less than 100 AU. The dashed circle marks a region of 0.2 pc radius around FIRS 1. 1.2. IC 1396 N IC 1396 N is a ∼300 L⊙ source located at a distance of 750 pc (Codella et al. 2001). A total population of ∼30 YSOs has been found in this region (Getman et al. 2007, Nisini et al. 2001). These YSOs present an elongated spatial distribution with an age gradient towards the center of the Class I/0 system. The Class III sources are located in the outer rim of the globule, the Class II sources are congregated in the bright ionized rim and the Class I/0 objects are located towards the dense molecular clump (see Fig. 2). The average density of YSOs in the globule is ∼200 pc−2. We have mapped a region of 0.1 pc around the Class 0/I system. 1.3. CB 3 CB 3 is a large globule (930 L⊙) located at 2.5 Kpc from the Sun (Codella & Bachiller 1999). A strong submillimeter source is observed in the central core (see Fig. 3 and Huard et al. 2000). Deep NIR images of the region show ∼40 NIR sources, from which at least 22 are very red, indicative of pre-main sequence stars (Launhardt et al. 1998). Up to our knowledge, there are no mid-IR and/or X-ray studies in this region. Then, the census of YSOs is not complete in this IM source. We have mapped a region of 0.32 pc around the submillimeter source. 2. Observations The observations were made on January and February, 2006. The spectral correlator was adjusted to cover the entire RF pass- bands (580 MHz) for highest continuum sensitivity. The overall flux scales for each epoch and for each frequency band were set on 3C454.3 and MWC349 (for CB 3), and 1749+096 (for Serpens–FIRS 1). The resulting continuum point source sensi- tivities (5×rms) were estimated to 2.00 mJy at 237.571 GHz and 0.5 mJy at 90.250 GHz for CB 3 and 40.00 mJy at 237.571 GHz and 7.0 mJy at 90.250 GHz for Serpens–FIRS 1. The corre- sponding synthesized beams adopting uniform weighting were 0.4′′ × 0.3′′ at 237.571 GHz and 1.0′′ × 0.8′′ at 90.250 GHz for CB 3 and 0.6′′ × 0.4′′ at 237.571 GHz and 1.7′′ × 0.7′′ at 90.250 GHz for Serpens–FIRS 1. (See Paper II for IC 1396 N.) 3. Results In Table 1 we present the coordinates, sizes and mm fluxes of the compact cores detected in Serpens–FIRS 1 and CB 3. The results towards IC 1396 N are presented in Paper II. Only 1 mm-source is detected in Serpens–FIRS 1 down to a separation of less than 100 AU. The other targets turned out to be multiple sources. We have detected 2 mm-sources towards CB 3 and 4 mm-sources towards IC 1396 N. A. Fuente et al.: Protostellar clusters in intermediate mass (IM) star forming regions 3 Fig. 2. On the left, we show the 5′×5′ Spitzer IRAC 3.6 µm image towards IC 1396 N (adapted from Getman et al. 2007). The location of the globule is marked by the green contour and the Class III (yellow triangles), Class II (red circles) and blue squares (Class 0/I) sources are indicated. On the right, we show the 3mm (up) and 1.3mm (down) continuum images observed with the PdBI. In the 3mm image we also indicate the Class III (black triangles), Class II (red circles) and Class 0/I (filled blue squares) sources. The 4 compact sources towards IC 1396 N are grouped in 2 sub-clusters separated by 0.05 pc which are spatially coinci- dent with the sources named BIMA 2 and BIMA 3 by Beltrán et al. (2002). The projected distance between these sub-clusters is similar to that found by Hunter et al. (2007) between the mm sub-clusters in the massive star forming region NGC 6336 I. This distance is also similar to the distance between the stars form- ing the Trapezium in Orion (from 5000 to 10000 AU). Thus it is a typical distance between the IM and massive stars in the same cloud. Our high angular resolution observations reveal that BIMA 2 is itself composed of 3 compact cores embedded in a more extended component (see Fig. 2). These 3 compact cores are new mm detections and constitute the first sub-cluster of Class 0 IM sources detected thus far. In CB 3 we have detected 2 mm-sources separated by 0.06 pc (see Table 1 and Fig. 3). These compact cores are new detec- tions and the separation between them is similar to that between BIMA 2 and BIMA 3 in IC 1396 N. In fact, the structure of the globule CB 3 resembles much that of IC 1396 N but the angu- lar resolution of our observations prevent us from resolving any possible sub-cluster of compact cores in this more distant source. Note that the masses of CB 3-1 and CB 3-2 are similar to that of the sub-cluster BIMA 2 (Paper II). The number of detections is limited by the sensitivity of our observations. In Table 1 we show the point source mass sensitivity assuming a dust temperature of 100 K (typical for hot cores and circumstellar disks around luminous Be stars) and κ1.3mm=0.01 g −1 cm2 for each target. It is possible that we miss a population of weak Class 0/I sources in CB 3 where the mass sensitivity is poor (0.04 M⊙). However, the sensitiv- ity in Serpens–FIRS 1 (0.01 M⊙) and IC1396 N (0.007 M⊙) is good enough to detect disks around early Be stars that usually have masses of ∼0.01 M⊙ (see e.g Fuente et al. 2003, 2006). We should have also detected massive disks (∼0.1 M⊙) around Herbig Ae and T Tauri stars although the dust temperature is lower, Td=15–56 K (Natta et al. 2000). But there is still the possibility of the existence of HAEBE or T Tauri stars with weak circumstellar disks that are not detected in our mm im- ages. Another possibility is that we are missing a population of hot corinos (we refer as “hot corino” to the warm material (∼100 K) around a low mass Class 0 protostar regardless of its chemical composition) with masses below the values reported in Table 1. Our sensitivity is good enough to detect a hot corino similar to IRAS 16293–2422 A and B (L∼10 L⊙) at the dis- tance of our sources (see Bottinelli et al. 2004). Thus the possi- ble “missed” hot corinos should correspond to lower luminosity protostars. Finally, we can be missing a population of dense and cold cores. Assuming a dust temperature of 10 K, these compact cold cores should have masses of less than 0.17, 0.12 and 0.7 M⊙ in Serpens–FIRS 1, IC 1396 N and CB 3 respectively. These masses are not large enough to form new IM stars. 4 A. Fuente et al.: Protostellar clusters in intermediate mass (IM) star forming regions 4. Discussion Testi et al. (1999) studied the clustering around a large sample of HAEBE stars. In order to quantify the concept, they introduced the parameter Nk, defined as the number of stars in a radius of 0.2 pc, the typical cluster radius. They showed that rich clus- ters are only found around the most massive stars, although the parameter Nk is highly variable. Some Be stars are born quite isolated, while others have Nk >70. For our sources this number is 22 (Launhardt et al. 1998, but the census is not complete), 29 (from Fig. 1) and 28 (Getman et al 2007; Nisini et al. 2001) in CB3, Serpens and IC 1396 N respectively, where all previously known YSOs (Class 0, I, II and III) in the regions are considered. Our maps show 2 sources in CB 3 on a 0.3 pc scale, 1 source in Serpens-FIRS 1 on a 0.04 pc scale, and 4 sources in IC 1396 N on a 0.1 pc scale. Defining Nmm as the number of mm sources in a radius of 0.2 pc, we can estimate Nmm from our observations and provide a revised value for the total number of YSOs at this scale. In Serpens our interferometric observations do not add any new mm source to previous data. We have observed the most in- tense mm clump in Fig, 1, the most likely to be a multiple source, and only found 1 compact source. Based on the 30m map shown in Fig. 1 and assuming that all the clumps host only one source we estimate Nmm∼7 from a total of 29 YSOs. In CB 3, our data add 2 new mm sources (Nmm=2) to the previous census of YSOs based on NIR studies. In IC 1396 N, we estimate Nmm=4–16. The upper limit has been calculated assuming a constant den- sity of mm sources in the region. Usually, the Class 0/I stars are not uniformly distributed in the clouds, but grouped in sub- clusters that are coincident with the peak of dense cores. Thus the value of Nmm is very likely close to 4 and we assumed this number hereafter. Since BIMA 2 and BIMA 3 were previously detected in the X-rays surveys by Getman et al. (2007), we only add two new sources (due to the multiplicity of BIMA 2) to the total number of YSOs in this region. Summarizing, the total number of YSOs is now 29, 24 and 30 for Serpens–FIRS 1, CB 3 and IC 1396 N respectively. While Serpens–FIRS 1 is an extraordinarily rich cluster compared with the clusters around Ae stars reported by Testi et al. (1999), CB 3 and IC 1396 N do not seem to become one of the crowded clusters (Nk∼70) detected by these authors around Be stars. However, this conclusion might not be true. The interferometer is only sensitive to dense and compact cores and provides a bi- ased vision of the star forming regions. In fact our interferomet- ric observations accounts for less than 1% of the total interstellar mass in the studied globules, i.e., ∼ 10, 58 and 64 M⊙ are missed in Serpens–FIRS 1, CB 3 and IC 1396 N respectively (Alonso- Albi et al. 2007). One possibility is that this mass is in the form of many weak hot corinos which could eventually become low mass stars. The fate of these hot corinos is, however, linked to the evolution of the IM protostar that is progressively dispers- ing and warming the surrounding material (Fuente et al. 1998). Another possibility is that the ”missed” mass is in the form of an extended and massive envelope. This envelope (if not totally dis- persed by the IM star) could produce new stars in a forthcoming star formation event. 5. Summary We have searched for clustering at mm wavelengths in 3 IM star forming regions. We have detected 1, 2 and 4 compact cores in Serpens–FIRS 1, CB 3 and IC 1396 N respectively. The compact cores are not distributed uniformly but grouped in sub-clusters separated by ∼0.05 pc. Such a separation is a typical distance Fig. 3. Dust continuum emission at 850 µm as observed with SCUBA towards CB 3. In the inset, we show the 3mm contin- uum image observed with PdBI. Note that two compact cores are detected towards the single-dish peak. for both IM and massive stars within the same cloud. We have used our mm observations to complete the census of YSOs in these regions and compare them with the clusters found by Testi et al. (1999) in the more evolved HAEBE stars. Serpens–FIRS 1 seems to belong to an extraordinarily rich cluster. The density of YSOs in the high luminosity sources IC 1396 N and CB 3 is consistent with the density found in the clusters around Be stars although our sources are not found between the most crowded regions. The large amount of interstellar gas and dust in the stud- ied regions suggest that new star formation events are still pos- sible. Acknowledgements. We are grateful to Phil Myers for his careful reading of the manuscript. A.F. is grateful for support from the Spanish MEC and FEDER funds under grant ESP 2003-04957, and from SEPCT/MEC under grant AYA 2003-07584. References Alonso-Albi, T. et al. 2007, in preparation Beltrán, M. T., Girart, J. M., Estalella, R., Ho, P. T. P., & Palau, A. 2002, ApJ, 573, 246 Bottinelli, S., Ceccarelli, C., Neri, R. et al. 2004, ApJ, 617, L69 Codella, C., & Bachiller, R. 1999, A&A, 350, 659 Codella, C., Bachiller, R., Nisini, B., Saraceno, P., & Testi, L. 2001, A&A, 376, Davis, C. J., Matthews, H. E., Ray, T. P., Dent, W. R. F., & Richer, J. S. 1999, MNRAS, 309, 141 Fuente, A., Martı́n-Pintado, J., Bachiller, R., Neri, R., & Palla, F. 1998, A&A, 334, 253 Fuente, A., Rodrı́guez-Franco, A., Testi, L., Natta, A., Bachiller, R., & Neri, R. 2003, ApJ, 598, L39 Fuente, A., Alonso-Albi, T., Bachiller, R., Natta, A., Testi, L., Neri, R., & Planesas, P. 2006, ApJ, 649, L119 Getman, K.V., Feigelson, E.D., Garmire, G., Broos, P., Wang, J., 2007, ApJ 654, Hogerheijde, M. R., van Dishoeck, E. F., Salverda, J. M., & Blake, G. A. 1999, ApJ, 513, 350 Huard, T.L., Weintraub, D.A., & Sandell, G., 2000, A&A 362, 635 Hunter, T. R., Brogan, C. L., Megeath, S. T., Menten, K. M., Beuther, H., & Thorwirth, S. 2006, ApJ, 649, 888 Kaas, A.A., Olofsson, G., Bontemps, S., et al., 2004, A&A 421, 623 A. Fuente et al.: Protostellar clusters in intermediate mass (IM) star forming regions 5 Launhardt, R., Henning, T., Klein, R., 1998, in: Yun J.L., Liseau R. (eds), ASP Conf. Ser. 132, Star Formation with the Infrared Space Observatory, San Francisco: ASP, p. 119 Natta, A., Grinin, V., & Mannings, V. 2000, Protostars and Planets IV, 559 Neri, R., Fuente, A., Ceccarelli, et al., 2007, this issue. Nisini, B., et al. 2001, A&A, 376, 553 Testi, L., & Sargent, A. I. 1998, ApJ, 508, L91 Testi, L., Palla, F., & Natta, A. 1998, A&AS, 133, 81 Testi, L., Palla, F., & Natta, A., 1999, A&A 342, 515 Testi, L., Sargent, A. I., Olmi, L., & Onello, J. S. 2000, ApJ, 540, L53 Williams, J. P., & Myers, P. C. 1999, ApJ, 518, L37 Introduction Serpens-FIRS 1 IC 1396 N CB 3 Observations Results Discussion Summary

0704.1099

The Epps effect revisited

The Epps effect revisited Bence Tóth∗1,2 János Kertész2,3 December 2, 2008 1 ISI Foundation - Viale S. Severo, 65 - I-10133 Torino, Italy 2 Department of Theoretical Physics, Budapest University of Technology and Economics - Budafoki út. 8. H-1111 Budapest, Hungary 3 Laboratory of Computational Engineering, Helsinki University of Tech- nology - P.O.Box 9203, FI-02015, Finland Abstract We analyse the dependence of stock return cross-correlations on the sampling frequency of the data known as the Epps effect: For high resolution data the cross-correlations are significantly smaller than their asymptotic value as observed on daily data. The former description implies that changing trading frequency should alter the characteristic time of the phenomenon. This is not true for the em- pirical data: The Epps curves do not scale with market activity. The latter result indicates that the time scale of the phenomenon is con- nected to the reaction time of market participants (this we denote as human time scale), independent of market activity. In this paper we give a new description of the Epps effect through the decomposition of cross-correlations. After testing our method on a model of generated random walk price changes we justify our analytical results by fitting the Epps curves of real world data. 1 Introduction 1979 Epps reported results showing that stock return correlations decrease as the sampling frequency of data increases [1]. Since his discovery the phenomenon has been detected in several studies of different stock markets [2, 3, 4] and foreign exchange markets [5, 6]. ∗E-mail: [email protected] http://arXiv.org/abs/0704.1099v2 Cross-correlations between the individual assets are the main factors in classical portfolio management thus it is important to understand and give their accurate description on different time scales. This is especially so, since today the time scale in adjusting portfolios to events occuring may be in the order of minutes. Considerable effort has been devoted to uncover the phenomenon found by Epps [7, 8, 9, 10, 11, 12]. However most of the works aim to construct a better statistical measure for co-movements in prices in order to exclude bias of the estimator by microstructure effects [13, 14, 15, 16, 17, 18, 19, 20], only a few searching for the description of the microstructure dynamics. Up to now two main factors causing the effect have been revealed: The first one is a possible lead-lag effect between stock returns [21, 22, 23] which can appear mainly between stocks of very different capitalisation and/or for some functional dependencies between them. In this case the maximum of the time-dependent cross-correlation function can be found at non zero time lag, resulting in increasing cross-correlations as the sampling time scale gets into the same order of magnitude as the characteristic lag. This factor can be easily understood, morever, in a recent study [23] we showed that through the years this effect becomes less important as the characteristic time lag shrinks, signalising an increasing efficiency of stock markets. As the Epps effect can also be found for the case when no lead-lag effect is present, in the following we will focus only on other possible factors. The second, more important factor is the asynchronicity of ticks in case of different stocks [7, 8, 21, 24]. Empirical results [7] showed that taking into account only the synchronous ticks reduces to a great degree the Epps effect, i.e. measured correlations on short sampling time scale increase. Naturally one would expect that for a given sampling frequency growing activity de- creases the asynchronicity, leading to a weaker Epps effect. Indeed Monte Carlo experiments showed an inverse relation between trading activity and the correlation drop [7]. However, the analysis of empirical data showed [25] that the explanation of the effect solely by asynchronicity is not satisfactory. After eliminating the effect of changing asymptotic cross-correlations through the years (scaling with the asymptotic value), the curves of cross-correlation as a function of sampling time scale tend to collapse to one curve and surprisingly we do not find a measurable reduction of the characteristic time of the Epps effect, while the trading frequency grew by a factor of ∼ 5−10 in the period. These results will be discussed further in details in Section 2.2. The characteristic time of market phenomena can usually be split up into three kinds of market time scales: the frequency of trading on the market (which we will denote as activity), market periodicities and the reaction time of traders to news, events. In Ref. [25] we showed that the characteristic time of the Epps effect does not scale with changing market activity (this we will discuss in Section 2.2), which points out that the characteristic time of the Epps effect can not be determined solely by the market activity causing asyn- chronicity. Market periodicities in high frequency data are the different types of patterns, which can be found in intraday data, as well as on broader time scales (see e.g. Refs. [26] and [27]). Market periodicities and intraday struc- ture do not have a role in our results since we are averaging them out. Hence we believe that the characteristic time of the Epps effect is the outcome of a human time scale present on the market: The time that market participants need to react to certain pieces of news. There are several studies in the litera- ture about reaction time. The issue is connected both to behavioural finance questions and to market efficiency. 1970, Fama defined an efficient market as one in which prices fully reflect all available information [28]. This response to information in practice can not happen instantaneously. There are several results reporting that prices incorporate news within five to fifteen minutes after news announcements [29, 30, 31, 32, 33]. More recent studies showed similar results on the time that traders needed to react to news [34, 35, 36]. Supposing that the Epps effect is possibly the outcome of a human time scale present on the market motivated us to separate the terms in the cross- correlation function, in order to study their behaviour one by one. In this paper we suggest an analytic decomposition of the cross-correlation function of asynchronous events using time lagged correlations. As a second step we demonstrate the efficiency of the formalism on the example of generated data. Finally we describe and fit the empirically observed dependence of the cross- correlations. We find that the origin of the independence of the characteristic time of the Epps effect on the trading frequency is the presence of a human time scale in the time lagged autocorrelation functions. Ref. [12] already called the attention to the importance of lagged cross-influences of stock returns in explaining the Epps-effect. Using a somewhat different formalism, here we investigate thoroughly this relationship. The paper is built up as follows: in Section 2 we present the data used and discuss the problems of the former descriptions. Section 3 the decompo- sition of the cross-correlation coefficient, Section 4 shows a simulation model demonstrating the idea. In Section 5 we present the assumptions concerning real data and show fits and statistics for the Epps curves. We finish the paper with a discussion. 2 Empirical analysis 2.1 Data and methodology In our analysis we used the Trade and Quote (TAQ) Database of the New York Stock Exchange (NYSE) for the period of 4.1.1993 to 31.12.2003, con- taining tick-by-tick data. The data used was adjusted for dividends and splits. We computed the logarithmic returns of stock prices: rA∆t(t) = ln pA(t) pA(t −∆t) , (1) where pA(t) stands for the price of stock A at time t. The prices were deter- mined using previous tick estimator on the high frequency data, i.e. prices are defined constant between two consecutive trades. The time dependent cross-correlation function C ∆t (τ) of stocks A and B is defined by ∆t (τ) = rA∆t(t)r ∆t(t + τ) rA∆t(t) rB∆t(t + τ) . (2) The notion 〈· · · 〉 stands for the time average over the considered period: 〈r∆t(t)〉= T −∆t r∆t(i), (3) where time is measured in seconds and T is the time span of the data. The standard deviation σ of the returns reads as: 〈r∆t(t)2〉−〈r∆t(t)〉 , (4) both for A and B in (2). We computed correlations for each day separately and averaged over the set of days, this way avoiding large overnight returns and trades out of the market opening hours. For pairs of stocks with a lead–lag effect the function C ∆t has a peak at non-zero τ. The equal-time cross-correlation coefficient is naturally: ρA/B∆t ≡C ∆t (τ = 0). In our notations the Epps effect means the decrease of ρ∆t as ∆t decreases (see Figure 1). Since the prices are defined as being constant between two consecutive trades, the ∆t time scale of the sampling can be chosen arbitrarily. As stated above, we do not want to discuss the Epps effect originated from the lead-lag effect in the correlations. Thus we consider only pairs of 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 ∆t [sec] Figure 1: The cross-correlation coefficient as a function of sampling time scale for the period 1993–2003 for the Coca-Cola Pepsi pair. Several hours are needed for the correlation to reach its asymptotic value. stocks where the latter effect is negligible, i.e., for which the price changes are highly correlated with the peak position of C ∆t of Equation (2) being at τ ≈ 0. The results shown in this paper can be generalised for all stock pairs (though in case of an empirical study one can never fit all data). To illus- trate our results we will present results for some stock pairs and in Section 5 we will show statisitcs for a broader set of data. The stocks mentioned in the paper are the following: Avon Products, Inc. (AVP), Caterpillar Inc. (CAT), Colgate-Palmolive Company (CL), E.I. du Pont de Nemours & Com- pany (DD), Deere & Company (DE), The Walt Disney Company (DIS), The Dow Chemical Company (DOW), General Electric Co. (GE), International Business Machines Corp. (IBM), Johnson & Johnson (JNJ), The Coca-Cola Company (KO), 3M Company (MMM), Motorola Inc. (MOT), Merck & Co., Inc. (MRK), PepsiCo, Inc. (PEP), Pfizer Inc. (PFE), The Procter & Gamble Company (PG), Sprint Nextel Corp. (S), Vodafone Group (VOD), Wal-Mart Stores Inc. (WMT). 2.2 Time evolution of the characteristic time Previous studies claimed the asynchronicity of ticks for different stocks as the main cause of the Epps effect [7, 8]. It is natural to assume that, for a given sampling frequency, increasing trading activity should enhance synchronicity, leading to a weaker Epps effect. To study the trading frequency dependence of the cross-correlation drop, we computed the Epps curve separately for different years. In Figure 2 the cross-correlation coefficients can be seen as a function of the sampling time scale for the years 1993, 1997, 2000 and 2003 for three example stock pairs. 0 2000 4000 6000 8000 ∆t [sec] 0 2000 4000 6000 8000 ∆t [sec] 0 2000 4000 6000 8000 ∆t [sec] Figure 2: The Epps curves for the CAT/DE (top), KO/PEP (middle) and MRK/JNJ (bottom) pairs for the years 1993, 1997, 2000 and 2003. The asymptotic value of the cross-correlations varies in time. It is known that cross-correlation coefficients are not constant through the years. The asymptotic values of cross-correlations (long sampling time scale) depend on the economical situation, the state of the economic sectors that the pairs of stocks belong to, and several other factors. We need to take this into account and try to extract the effect of changing asymptotic cross-correlations from the Epps phenomenon. In order to get comparable curves, we scaled the cross-correlation curves with their asymptotic value: The latter was defined as the mean of the cross-correlation coefficients for the sampling time scales ∆t = 6000 seconds through ∆t = 9000 seconds, and the cross-correlations were divided by this value. Figure 3 shows the scaled curves for the same years and pairs as Figure 2. The frequency of trades changed considerably in the last two decades: Trading activity has grown almost monotonically, as it can be seen in Figure 4. This would infer the diminution of the Epps effect and a much weaker decrease of the correlations as sampling frequency is increased. However, after scaling with the asymptotic cross-correlation value, the curves give a reasonable data collapse and no systematic trend can be seen. Surprisingly, as it can be seen in Table 1, a rise of the trading frequency by a factor of ∼ 5−10 does not lead to a measurable reduction of the characteristic time of the Epps effect (where we define the characteristic time the time scale for which the cross-correlation reaches the 1− e−1 rate of its asymptotic value). These results show that explaining the Epps effect merely as a result of the asynchronicity of ticks is not satisfactory. It is also important to mention that not even the changing tick sizes for the stocks (likely to change the arrival rate of price changes) alter the characteristic time of the effect. Table 1: The characteristic time of the Epps effect for the years 1993, 1997, 2000 and 2003 measured in seconds for the stocks pairs: CAT/DE, KO/PEP and MRK/JNJ (characteristic time was defined as the time scale for which the cross-correlation value reaches the 1− e−1 rate of its asymptotic value). No clear trends can be seen in the charactersitic time while the activity is growing rapidly. CAT/DE KO/PEP MRK/JNJ 1993 940 920 800 1997 620 760 420 2000 1320 1040 880 2003 700 800 1060 0 2000 4000 6000 8000 ∆t [sec] 0 2000 4000 6000 8000 ∆t [sec] 0 2000 4000 6000 8000 ∆t [sec] Figure 3: The Epps curves scaled with the asymptotic cross-correlation values for the CAT/DE (top), KO/PEP (middle) and MRK/JNJ (bottom) pairs for the years 1993, 1997, 2000 and 2003. The scaled curves give a reasonable data collapse in spite of the considerably changing trading frequency, showing that the characteristic time of the Epps effect does not change with growing activity. 1994 1996 1998 2000 2002 yearsa Figure 4: The average intertrade time for the years 1993 to 2003 for some example stocks (CAT, DE, KO, PEP, MRK, JNJ). The activity was growing almost monotonically. 3 Decomposition of the cross-correlations In this section we show calculations for the relation between the value of cross-correlations on different time scales. We connect the cross-correlation on a certain time scale (∆t) to lagged autocorrelations and cross-correlations on smaller time scales (∆t0). Returns in a certain time window ∆t are mere sums of returns in smaller, non-overlapping windows ∆t0, where ∆t is a multiple of ∆t0: r∆t(t) = ∆t/∆t0 r∆t0(t −∆t + s∆t0). (5) Using this relationship the time average of the product of returns on the large time scale (∆t) can be written in terms of the averages on the short time scale (∆t0) in the following way: rA∆t(t)r ∆t(t) T −∆t rA∆t(i)r ∆t(i) = T −∆t ∆t/∆t0 rA∆t0(i−∆t + s∆t0) ∆t/∆t0 rB∆t0(i−∆t +q∆t0) ∆t/∆t0 ∆t/∆t0 rA∆t0(i−∆t + s∆t0)r ∆t0(i−∆t +q∆t0) . (6) We can see that on the right side of Equation (6) the lagged time average of return products appear on the short time scale, ∆t0, i.e., the non-trivial part of the lagged cross-correlations. Naturally in the case of A = B, we get the relation for r∆t(t) In order to apply Equation (6), we need to have information about the lagged autocorrelation and cross-correlation functions. Writing out the sum in Eq. (6) we get: rA∆t(t)r ∆t(t) x=− ∆t∆t0 rA∆t0(t)r ∆t0(t + x∆t0) , (7) and similarly rA∆t(t) x=− ∆t∆t0 rA∆t0(t)r ∆t0(t + x∆t0) rB∆t(t) x=− ∆t∆t0 rB∆t0(t)r ∆t0(t + x∆t0) . (8) Since the mean of returns is 1-2 orders of magnitude smaller than the second moments in the correlation function, we can omit the expressions rA∆t(t) rB∆t(t + τ) and 〈r∆t(t)〉 in Equation (2). As these terms are of second order, this can even be done in case of slight price trends. Hence Equation (2) becomes: ρA/B∆t = rA∆t(t)r ∆t(t) rA∆t(t) rB∆t(t) . (9) For simplicity, we introduce decay functions to describe lagged correla- tions: (x∆t0) = rA∆t0(t)r ∆t0(t + x∆t0) rA∆t0(t)r 〉 , (10) defined for both positive and negative x values, and similarly f (x∆t0) and (x∆t0). Thus the correlation can be written in the following form: ρA/B∆t = x=− ∆t∆t0 (x∆t0) rA∆t0(t)r ∆t0(t) x=− ∆t∆t0 (x∆t0) rA∆t0(t) )−1/2 x=− ∆t∆t0 (x∆t0) rB∆t0(t) )−1/2 . (11) Hence ρA/B∆t = x=− ∆t∆t0 (x∆t0) x=− ∆t∆t0 (x∆t0) )−1/2 x=− ∆t∆t0 (x∆t0) )−1/2 ρA/B∆t0 . (12) This way we obtained an expression of the cross-correlation coefficient for any sampling time scale, ∆t, by knowing the coefficient on a shorter sampling time scale, ∆t0, and the decay of lagged autocorrelations and cross- correlations on the same shorter sampling time scale (given that ∆t is multiple of ∆t0). Our method is to measure the correlations and fit their decay func- tions on a certain short time scale and compute the Epps curve using the above formula. 4 Model calculations In this section we demonstrate the decomposition process on computer gen- erated time series which should mimic two correlated return series. We will demonstrate the Epps effect and see how the decomposition works for these controlled cases. Our aim is to show that in case of generated “price” series, the decomposition process leads to a very good description of the time scale dependence of the cross-correlation coefficients. More discussion and details on the analytic treatment of the model can be found in Ref. [38]. To mimic some properties of financial data, we simulate two correlated but asynchronous price time series. As a first step we generate a core random walk with unit steps up or down in each second with equal possibility (W (t)). Second we sample the random walk, W (t), twice independently with waiting times drawn from an exponential distribution. This way we obtain two time series (pA(t) and pB(t)), which are correlated since they are sampled from the same core random walk, but the steps in the two walks are asynchronous. The core random walk is: W (t) = W (t −1)+ ε(t), where ε(t) is ±1 with equal probability (and W (0) is set high in order to avoid negative values). The two price time series are determined by pA(ti) = W (ti) if ti = ∑ik=1 Xk p(ti −1) otherwise pB(ti) = W (ti) if ti = ∑ik=1Yk p(ti −1) otherwise . (14) from the core random walk, where, i = 1,2, · · · , and Xk and Yk are drawn from an exponential distribution: P(y) = λe−λy if y ≥ 0 0 y < 0 with parameter λ = 1/60. A snapshot as an example of the generated time series with exponentially distributed waiting times can be seen on Figure 5. As a next step we create the logarithmic return time series (rA∆t(t) and rB∆t(t)) of p A(t) and pB(t) as defined by Equation (1). In case of the random walk model of price changes we know that rA∆t(t) rB∆t(t) = 0 without having to make any assumptions. Of course having a random walk model, the autocorrelation function of the steps is zero for all non-zero time lags: (x∆t0) = f (x∆t0) = δx,0, (16) 0 100 200 300 400 10000 10010 10020 pA(t) pB(t) Figure 5: A snapshot of the model with exponentially distributed waiting times. The original random walk is shown with lines (black), the two sampled series with dots and lines (red) and triangles and lines (blue). rA∆t(t) rA∆t0(t) rB∆t(t) rB∆t0(t) . (17) Hence the cross-correlation can be written in the following form: ρA/B∆t = x=− ∆t∆t0 (x∆t0) rA∆t0(t)r ∆t0(t) rA∆t0(t) rB∆t0(t) )−1/2 x=− ∆t∆t0 (x∆t0) ρA/B∆t0 . (18) In the model case we set the smallest time scale ∆t0 = 1 time step. It can be shown [38] that in the case of λ ≪ 1 (small density of ticks) the exact analytical expression for the cross-correlations is identical to (18) with an exponential decay function: (x∆t0) = e−λ∆t0|x|, (19) where λ is the parameter of the original exponential distribution used for sampling. Further results and exact computations of the cross-correlations for the model can be found in Ref. [38]. Figure 6 shows the computed cross-correlations of the generated time series on several sampling time scales and the computed cross-correlations using Equation (18) and the exponential decay function (19). The two curves are in very good agreement showing that the decomposition procedure is able to well capture the Epps effect for generated time series. 0 5000 measured correlations analytically computed correlations ∆t [simulation steps] Figure 6: The measured and the computed cross-correlation coefficients using exponential decay function as a function of sampling time scale for the simu- lated time series with exponentially distributed waiting times. The analytic fit is in very good agreement with the Epps curve. 5 Application of the theory to the data In this section we discuss the properties of the decay functions in case of real world data, and inserting them into Equation (12) we derive analytical fits for the measured Epps curves. 5.1 Decay functions As discussed, we measure the equal-time cross-correlations and the decay of cross and autocorrelations on a certain short sampling time scale and from these we obtain the value of equal-time cross-correlations on larger sampling time scales. To do this, in case of the toy model, we had the possibility of using the smallest time scale available in the generated data as ∆t0, i.e., the resolution being one simulation step. When studying data from real world markets, one has to make restrictions. As being the highest resolution commonly used in financial analysis, it would be plausible to choose windows of one second as ∆t0. However on this time scale one is only able to measure noise, no valid correlations and decay functions can be found. Thus we had to use less dense data for the smallest sampling time scale: in the results shown below we set ∆t0 = 120 seconds. Using this resolution we get an acceptable signal-to-noise ratio and we hope not to lose too much information compared to higher frequencies. To avoid new parameters in the model we use the raw decay functions in the formula (12), without fitting them. Since it is an empirical approach to determine the decay functions for real data, we have to distinguish the signal from the noise in the decay functions. Concerning the sensitivity from the input (decay function) we observed that the results are quite robust against little changes in the input functions, however the noise in the tail can cause significant deviations. According to this we take into account the decay functions for correlations only for short time lags. For the decay of the cross- correlations we take into account the function only up to the time lag where the decaying signal reaches zero for the first time, for larger lags we assume it to be zero. For the decay of autocorrelations we take into the account the function only up to the time lag where after the negative overshoot of the beginning it decays to zero from below for the first time, for larger lags we assume it to be zero. Figure 7 shows an example of the decay functions in case of the stock pair KO/PEP (for other pairs the decay functions are very much similar). The plot shows the decay functions up to the time lags of 1000 seconds, with a vertical line showing how long we take the empirical decays into account. We can see that the time lag for which the decay functions disappear is in the order of a few minutes. In fact in case of all stock pairs studied we found the decay disappearing after 5–15 minutes. 5.2 Fits Inserting the empirical decay of lagged autocorrelations and cross-correlations on the short time scale into the formula of Equation (12), we compare the computed and the measured Epps curves. Figures 8, 9 and 10 show these plots for a few example stock pairs. One can see, that the fits are able to describe the change of cross-correlation 0 200 400 600 800 1000 x∆t0 [sec] 0 200 400 600 800 1000 x∆t0 [sec] -1000 -750 -500 -250 0 250 500 750 1000 x∆t0 [sec] Figure 7: Top: The decay of lagged autocorrelations for KO. Middle: The decay of lagged autocorrelations for PEP. Bottom: The decay of lagged cross- correlations for KO/PEP pair. A vertical lines show the threshold up to which we take the decays into account, for larger lags we assume them to be zero. Sampling time scale is ∆t0 = 120 seconds on all three plots. with increasing sampling time scale. Note, that as it has been shown in 0 2000 4000 6000 8000 measured correlations analytically computed correlations ∆t [sec] 0 2000 4000 6000 8000 measured correlations analytically computed correlations ∆t [sec] Figure 8: The measured and the analytically computed cross-correlation co- efficients as a function of sampling time scale for the pairs CAT/DE and KO/PEP. Note that using only the autocorrelations and cross-correlations measured on the smallest time scale (∆t0 = 120 seconds) we are able to give reasonable fits to the cross-correlations on all time scales. Details on the goodness parameter of the measured and computed correlations can be found in Table 2. 0 2000 4000 6000 8000 measured correlations analytically computed correlations ∆t [sec] 0 2000 4000 6000 8000 measured correlations analytically computed correlations ∆t [sec] Figure 9: The measured and the analytically computed cross-correlation co- efficients as a function of sampling time scale for the pairs WMT/S and GE/MOT. Details on the goodness parameter of the measured and com- puted correlations can be found in Table 2. Section 3, in the analytical formula only the autocorrelations and cross- correlations on the smallest time scale (∆t0) and the decay functions are 0 2000 4000 6000 8000 measured correlations analytically computed correlations ∆t [sec] 0 2000 4000 6000 8000 measured correlations analytically computed correlations ∆t [sec] Figure 10: The measured and the analytically computed cross-correlation coefficients as a function of sampling time scale for the pairs MRK/JNJ and GE /IBM. Details on the goodness parameter of the measured and computed correlations can be found in Table 2. taken into account as input to compute the cross-correlations on all other time scales, no additional parameters are used. To show a broader set of results, we introduce a goodness parameter for the agreement between the measured and the analytically determined Epps curves. We define the goodness parameter as the absolute error between the measured and the analytically computed points: g(∆t) = 100 |ρmeasured∆t −ρ computed ρmeasured∆t . (20) Table 2 shows the maximum, the mean and the median of the goodness parameters for a broader set of stocks. The results show that the absolute mean error is very low, with a maximum around 7 percents and both a mean and a median around 2 percents. It is important to mention, that the maximal error is usually found for high frequency scales, for longer time scales and especially for the asymptotic correlation value the aggrement is very good. These results show that the growing cross-correlations with decreasing sampling frequency are due to finite time decay of the lagged autocorrelations and cross-correlations in the high frequency sampled data. The finite decay of the cross-correlations on the short time scale (∆t0) is not caused by difference in the capitalisation of the two stocks or func- tional dependencies between them. Instead, it is an artifact of the market Table 2: The maximum, the mean and the median of the goodness parameters a broader set of stocks. The results show that the absolute mean error is low. Note that the maximal absolute error in general occurs for high frequency scales. stock pair max [%] mean [%] median [%] CAT/DE 4.81 1.26 0.94 KO/PEP 10.67 2.46 1.23 WMT/S 7.66 2.32 1.74 GE/MOT 6.43 3.29 3.44 MRK/JNJ 5.26 1.51 0.95 GE/IBM 3.90 1.57 1.12 PG/CL 6.05 1.81 1.24 MRK/PFE 4.76 1.42 1.32 AVP/CL 10.37 7.75 9.53 DD/DOW 8.84 2.05 1.49 DD/MMM 5.76 2.17 1.93 MOT/VOD 9.73 2.57 1.82 DIS/GE 5.78 1.54 0.97 average 6.92 2.4 2.1 microstructure. Reaction to a certain piece of news is usually spread out on an interval of a few minutes for the stocks [29, 30, 31, 32, 33, 34, 35, 36, 39, 40] due to human trading nature, thus not scaling with activity, with ticks being distributed more or less randomly. This means that correlated returns are spread out for this interval (asynchronously), causing non zero lagged cross- correlations on the short time scale and thus the Epps effect. This way, as stated by Ref. [7], the asynchronicity is indeed important in describing the Epps effect but only in promoting the lagged correlations. (Even in case of completely synchronous, but randomly spread ticks we could have the finite decay of lagged correlations on short time scale, and hence the Epps effect.) 6 Discussion In our study we examined the causes of the Epps effect, the dependence of stock return cross-correlations on sampling time scale. We showed that explaining the effect solely through asynchronicity of price ticks is not sat- isfactory. When scaling the Epps curves with their asymptotic value for different years, we get a reasonable data collapse and a growing activity of the order ∼ 5−10 does not affect the characteristic time of the Epps effect. The main point of our calculations is that we connected the cross-correlations on longer time scales to the lagged autocorrelations and cross-correlations on any shorter time scale. We demonstrated the idea of these calculations on a random walk asynchronous model of prices, getting a very good agreement with the cross-correlation curves. Assuming the time average of stock returns to be zero we were able to decompose the expression for the cross-correlation coefficient deriving an an- alytical formula of the cross-correlations on any time scale, given the decay of the autocorrelations and cross-correlations on a certain short time scale. With this analytical formula we were able to give fits to the Epps curves of real stock pairs getting acceptable results. The fits show that the Epps effect is caused by the finite time decay of the lagged correlations in the high frequency sampled data. The reason for the characteristic time not chang- ing with growing activity is a human time scale present in the phenomenon, which does not scale with the changing inter-tick time. The finite decay of lagged correlations on the short time scale is due to market microstructure properties: different actors on the market have different time horizons of in- terest resulting in the reactions to certain pieces of news being spread out for a time interval of a few minutes. The correlated returns ranging over this interval cause the finite time decay of lagged correlations on the short time scale resulting in the Epps effect. Our results do not contradict to the earlier observations on the importance of asynchronicity in the Epps-effect, however, its role has been put into a new perspective. Acknowledgments Support by OTKA T049238 is acknowledged. References [1] T.W. Epps, Journal of the American Statistical Association 74, 291-298 (1979) [2] G. Bonanno, F. Lillo, R.N. Mantegna, Quantitative Finance 1, 1-9 (2001) [3] A. Zebedee, A closer look at co-movements among stock returns, San Diego State University, working paper (2001) [4] M. Tumminello, T. Di Matteo, T. Aste, R.N. Mantegna, Eur. Phys. J. B. 55, 209-217 (2007) [5] M. Lundin, M. Dacorogna, U. A. Müller, Correlation of high-frequency financial time series. In P. Lequeux (Ed.), Financial Markets Tick by Tick. Wiley & Sons. [6] J. Muthuswamy, S. Sarkar, A. Low, E. Terry, Journal of Futures Markets 21(2), 127-144 (2001) [7] R. Renò, International Journal of Theoretical and Applied Finance 6(1), 87-102 (2003) [8] O. V. Precup, G. Iori, Physica A 344, 252-256 (2004) [9] O. V. Precup, G. Iori, European Journal of Finance (2006) [10] J. Kwapień, S. Drożdż, J. Speth, Physica A 337, 231-242 (2004) [11] L. Zhang, Estimating Covariation: Epps Effect, Microstructure Noise working paper (2006) [12] M. Potters, J.-P. Bouchaud, L. Laloux, Acta Physica Polonica B, 36, 9, (2005) [13] N. Shephard, O.E. Barndorff-Nielsen, University of Oxford, Department of Economics, Economics Se- ries Working Papers No. 240. (2005) Available at: http://www.economics.ox.ac.uk/Research/wp/pdf/paper240.pdf [14] F. Corsi, F. Audrino, University of St. Gallen, Department of Eco- nomics, Discussion Paper No. 2007-02 (2007) Available at SSRN: http://ssrn.com/abstract=957997 [15] T. Hayashi, N. Yoshida, Bernoulli 11(2), 359-379 (2005) [16] M. Martens, Working Paper, University of Rotterdam (2003) [17] T.G. Andersen, T. Bollerslev, P.F. Christoffersen, F.X. Diebold, in G. Elliot, C.W.J. Granger, and Allan Timmermann (eds.), Handbook of Economic Forecasting, Amsterdam: North-Holland, 778-878 (2006) [18] T. Andersen, T. Bollerslev, International Economic Review, 39, 885-905 (1998) [19] F. de Jong, T. Nijman, Journal of Empirical Finance 4, 259-277 (1997) [20] C. Ball, W. Torous, Journal of Empirical Finance 7, 373-388 (2000) http://www.economics.ox.ac.uk/Research/wp/pdf/paper240.pdf http://ssrn.com/abstract=957997 [21] A. Lo, A. C. MacKinlay, Rev. Finance Stud 3, 175-205 (1990) [22] L. Kullmann, J. Kertész, K. Kaski, Phys. Rev. E 66, 026125 (2002) [23] B. Tóth, J. Kertész, Physica A 360 505-515 (2006) [24] A. Lo, A. C. MacKinlay, Journal of Econometrics 45, 181-211 (1990) [25] B. Tóth, J. Kertész, Physica A 383, 54-58 (2007) [26] R.A. Wood, T.H. McInish, J.K. Ord, Journal of Finance 40(3), 723-739 (1985) [27] K. Chan, Y.P. Chung, H. Johnson, The Journal of Financial and Quan- titative Analysis, 30, 3, 329-346 (1995) [28] E. Fama, Journal of Finance 25, 383-417 (1970) [29] L. Dann, D. Mayers, R. Raab, Journal of Financial Economics 4, 3-22 (1977) [30] J. Patell, M. Wolfson, Journal of Financial Economics 13, 223-252 (1984) [31] R. Jennings, L. Starks, Journal of Accounting Research 23, 336-350 (1985) [32] M. Barclay, R. Litzenberger, Journal of Financial Economics 21, 71-99 (1988) [33] S. Kim, J. Lin, M. Slovin, Journal of Financial and Quantitative Analysis 32, 507-524 (1997) [34] J. Busse, C. Green, Journal of Financial Economics 65, 415-437 (2002) [35] T. Chordia, R. Roll, A. Subrahmanyam, Journal of Financial Economics 76, 271-292 (2005) [36] T. Chordia, R. Roll, A. Subrahmanyam, Journal of Financial Economics 87, 249-268 (2008) [37] P.Ch. Ivanov, A. Yuen, B. Podobnik, Y. Lee, Phys. Rev. E 69, 056107 (2004) [38] Bence Tóth, János Kertész, Bálint Tóth, Proceedings of SPIE Vol. 6601 (Fluctuations and Noise 2007) [39] M.M. Dacorogna, R. Gençay, U.A. Müller, R.B. Olsen, O.V. Pictet, An Introduction to High-Frequency Finance, Academic Press, 2001 [40] A. Almeida, C. Goodhart, R. Payne, The Journal of Financial and Quan- titative Analysis, 33, 383-408 (1998) Introduction Empirical analysis Data and methodology Time evolution of the characteristic time Decomposition of the cross-correlations Model calculations Application of the theory to the data Decay functions Fits Discussion

0704.1100

Transitive powers of Young-Jucys-Murphy elements are central

TRANSITIVE POWERS OF YOUNG-JUCYS-MURPHY ELEMENTS ARE CENTRAL I.P. GOULDEN∗ AND D.M. JACKSON† ABSTRACT. Although powers of the Young-Jucys-Murphy elements Xi = (1 i) + (2 i) + · · · + (i − 1 i), i = 1, . . . , n, in the symmetric group Sn acting on {1, . . . , n} do not lie in the centre of the group algebra of Sn, we show that transitive powers, namely the sum of the contributions from elements that act transitively on [n], are central. We determine the coefficients, which we call star factorization numbers, that occur in the resolution of transitive powers with respect to the class basis of the centre of Sn, and show that they have a polynomiality property. These centrality and poly- nomiality properties have seemingly unrelated consequences. First, they answer a question raised by Pak [P] about reduced decompositions; second, they explain and extend the beautiful symmetry result discovered by Irving and Rattan [IR]; and thirdly, we relate the polynomiality to an existing polynomiality result for a class of double Hurwitz numbers associated with branched covers of the sphere, which therefore suggests that there may be an ELSV-type formula (see [ELSV]) associated with the star factorization numbers. 1. INTRODUCTION AND BACKGROUND We begin with an account of the main theorem of this paper and its relationship to the enumer- ation of a class of ramified covers of the sphere, a question that arises in algebraic geometry. 1.1. Young-Jucys-Murphy elements and the Main Theorem. The Young-Jucys-Murphy elements in the group algebra CSn of the symmetric group Sn on [n] := {1, . . . , n}, are given by Xi = (1 i) + (2 i) + · · ·+ (i− 1 i), i = 1, . . . , n, where X1 = 0 (see, e.g., [VO] for a detailed description and further references). Let Z(n) denote the centre of CSn, n ≥ 1. Then the algebra generated by Z(1), . . . , Z(n) is called the Gel’fand-Tsetlin algebra, and one of the key results described in [VO] is the fact that this algebra is also generated by X1, . . . ,Xn, despite the fact that Xn is clearly not contained in Z(n) for any n > 2. We define a linear operator T on CSn by T(σ1 · · · σr) = σ1 · · · σr, if the group generated by the permutations σ1, · · · , σr acts transitively on [n], and T(σ1 · · · σr) = 0 otherwise. The subject of this paper is T Xrn, for an arbitrary non-zero integer r, which we call a transitive power of Xn. It is straightforward matter to apply T to Xrn = i∈[n−1](i n) , since the only products not annihilated are those containing at least one occurrence of (i n) as a factor for every i ∈ [n − 1]. It follows immediately from the Principle of Inclusion-Exclusion that a transitive power can be written explicitly as T Xrn = γ⊆[n−1] (−1)|γ|Xn(γ) where Xn(γ) = j∈γ(j n). Date: April 5, 2007. ∗Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada.; [email protected]. †Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada.; [email protected]. http://arxiv.org/abs/0704.1100v1 2 I. P. GOULDEN AND D. M. JACKSON Our main result, Theorem 1.1, is that the transitive powers of Xn, unlike powers, are contained in Z(n). Moreover, since a basis for Z(n) is given by the set of all Kα where Kα = π, and Kα is the conjugacy class of Sn (naturally) indexed by the partition α of n, then Theorem 1.1 expresses T Xrn as an explicit linear combination of the Kα. We use the following notation and terminology for partitions. If α1, · · · , αk are positive integers with 1 ≤ α1 ≤ · · · ≤ αk and α1 + · · · + αk = n, then α = (α1, . . . , αk) is a partition of n with k parts, and we write α ⊢ n and l(α) = k, for n, k ≥ 0. Let α \ αj denote the partition obtained by removing the single part αj from α, for any j = 1, . . . , k. Let α ∪m denote the partition obtained by inserting a single new part equal to m into α (placed in the appropriate ordered position). Let 2α = (2α2, . . . , 2αk), and aα = aα1 · · · aαk for any indeterminates a1, a2, . . .. Let P denote the set of all partitions, including the empty partition ε, which it a partition of 0 with 0 parts. If α has fj parts equal to j for each j ≥ 1, then we also use (1f12f2 · · · ) to denote α, and we write |Aut α| = j≥1 fj! In the statement of our main result we use the notation pi ≡ pi(α) to denote the i-th power sum of the parts of the partition α, i ≥ 1, and qi ≡ qi(α) := pi+ p1− 2, i ≥ 2, and we define ξ2j and ξ by 2j := log (ξ(x)) , where ξ(x) := 2x−1sh where sh and ch denote, respectively, hyperbolic sine and cosine. Theorem 1.1 (Main Theorem). For r ≥ 0, TXrn is contained in the centre Z(n) of CSn. Moreover, the resolution of TXrn with respect to the class basis of Z(n) is TXrn = ag(α)Kα where the range of summation on the right hand side is restricted by the condition n + m − 2 + 2g = r, with m = l(α), and ag(α) is a polynomial in the parts of α given by ag(α) = (n+m− 2 + 2g)!α1 · · ·αmQg(α), where Qg := ξ2βq2β |Aut β| , g ≥ 0. For example, the explicit expressions for small genera g = 0, . . . , 5 are Q0 = 1, Q1 = q2, Q2 = (−2q4 + 5q22) , Q3 = (16q6 − 42q4 2 + 35q23) , 3 · 2710! (−144q8 + 320q6 2 + 84q42 − 420q4 22 + 175q24) , 3 · 2812! (768q10 − 1584q8 2 − 704q6 4 + 1760q6 22 + 924q42 2 − 1540q4 23 + 385q25) , 1.2. Background. 1.2.1. Minimal factorizations into star transpositions. We now turn our attention temporarily to a another point of view. The transpositions (1 a), for a = 2, . . . , n, are called star transpositions in Sn, with the distinguished element 1 (it appears in each transposition) referred to as the pivot element. An ordered factorization (τ1, . . . , τr) of σ ∈ Sn into star transpositions is said to be transitive if the group generated by τ1, . . . , τr acts transitively on [n]. For a transitive factorization of σ ∈ Kα into r star transpositions, a result in [GJ0] implies that r = n + m − 2 + 2g for some non-negative integer g, where α has m parts. Thus r ≥ n + m − 2, and we refer to transitive factorizations into n+m− 2 star transpositions as minimal. Pak [P] enumerated minimal factorizations (he called them reduced decompositions) into star transpositions for permutations fixing the pivot element 1, with exactly m other cycles, each of TRANSITIVE POWERS OF YOUNG-JUCYS-MURPHY ELEMENTS ARE CENTRAL 3 length k ≥ 2. More recently, Irving and Rattan [IR] generalized Pak’s result by considering min- imal factorizations of arbitrary permutations into star transpositions, and proved the following elegant result. Theorem 1.2 ([IR]). For each permutation σ ∈ Kα with α = (α1, . . . , αm), α1+· · ·+αm = n and m,n ≥ 1, the number of transitive factorizations of σ into n+m− 2 star transpositions is (n +m− 2)! α1 · · ·αm. Because of the apparent asymmetry of these factorizations (i.e., the pivot element 1 appears in every factor), the fact that Theorem 1.2 is constant on conjugacy classes is particularly surprising (we shall refer to this fact as the centrality property of Theorem 1.2). The proofs given in [P] and [IR] are bijective, involving restricted words and plane trees. In terms of factorizations into star transpositions, the number ag(α) given by the Main Theorem clearly can be interpreted as the number of transitive factorizations of each σ ∈ Kα into n +m − 2 + 2g star transpositions, with pivot element n. We shall call ag(α) a star factorization number. Thus Theorem 1.2 is precisely the case g = 0 of Theorem 1.1 (the necessary relabelling of the pivot element is justified by the centrality of these results). The investigation described in this paper answers Pak’s [P] question about an explicit expression for the general case. It was motivated by Irving and Rattan’s paper, in our attempt to determine whether the centrality of their remarkable result for star factorizations with a minimum number of factors persisted for star factorizations with an arbitrary number of factors. 1.2.2. Connections with algebraic geometry. The connection to algebraic geometry is made through Hurwitz’s encoding [H] of an n-sheeted branched cover of the sphere in terms of transpositions that represent the sheet transitions at the elementary branch points. In this context, the transi- tivity of the factorizations corresponds to the connectedness of the cover. From this perspective, the coefficient ag(α) in Theorem 1.1 counts genus g branched covers of the sphere in which the branching over the point 0 is specified by α, and there are n + m − 2 + 2g other simple branch points, each of which corresponds to a transition between sheet number n (the pivot sheet) and an- other sheet. For the corresponding transitive factorizations into star transpositions, we therefore also refer to g as the genus of the factorization (e.g., Theorem 1.2 counts genus 0 factorizations). For further details about branched covers, see, for example, [GJ0], [GJVn], [GJV] and [H]. The double Hurwitz number H (n),α is equal to the number of genus g branched covers of the sphere in which the branching over the points 0 and ∞ is specified by (n) and α, respectively, together with m− 1 + 2g other simple branch points. A scaling of this double Hurwitz number to bg(α) := α1 · · ·αmH (n),α gives the number of transitive factorizations of each σ ∈ Kα into m − 1 + 2g transpositions and a single n-cycle. There is a striking similarity between Theorem 1.1 and the following result, in which the notation q̂i := pi − 1, i ≥ 1 is used. Theorem 1.3 ([GJV]). For r ≥ 0, the resolution of Kr (1n−2 2) Kn with respect to the class basis of the centre Z(n) of CSn is (1n−2 2)Kn = bg(α)Kα, 4 I. P. GOULDEN AND D. M. JACKSON where the range of summation on the right hand side is restricted by the condition m − 1 + 2g = r, with m = l(α), and bg(α) is a polynomial in the parts of α given by bg(α) = (m− 1 + 2g)!n m−2+2g α1 · · ·αm Q̂g(α), where Q̂g := ξ2β q̂2β |Aut β| , g ≥ 0. This is a restatement of Theorem 3.1 in [GJV], which gives a formula for the double Hurwitz number H (n),α , since Kr 1n−22 Kn = T(K 1n−22 Kn) (each term in Kn acts transitively on [n]). 1.2.3. Two relationships between Theorems 1.1 and 1.3. To explore a more direct relationship between Theorems 1.1 and 1.3, we now give two expressions for ag(α) in terms of the bh(γ)’s. The first is very simple and expresses ag(α), which enumerates factorizations in Sn, directly in terms of bg(α ∪ 1 n−1), which enumerates factorizations in S2n−1. Corollary 1.4. For g ≥ 0 and α a partition of n with m parts, we have ag(α) = n!(2n− 1)n+m−3+2g bg(α ∪ 1 n−1). Proof. In the notation of Theorems 1.1 and 1.3, clearly qi(α) = q̂i(α∪1 n−1), so Qg(α) = Q̂g(α∪1 n−1). The result follows immediately from Theorems 1.1 and 1.3. � The second expresses ag(α) as a linear combination of bg−h(α), 0 ≤ h ≤ g, each of which enumerates factorizations in Sn. Corollary 1.5. For g ≥ 0 and α, a partition of n with m parts, we have ag(α) = bg−h(α) nm−2+2g−2h n+m− 2 + 2g n− 1 + 2h ) n−1∑ (−1)j (n− 1)− j )n−1+2h Proof. In the notation of Theorems 1.1 and 1.3, clearly qi(α) = q̂i(α)+n−1. Then from Theorems 1.1 and 1.3, and (1), we have Qg(α)x 2g = exp ξ2jq2j(α)x  = ξ(x)n−1 Q̂g(α)x But, for h ≥ 0, we have (using the notation [A]B to denote the coefficient of A in B) [x2h]ξ(x)n−1 = [xn−1+2h] 2 − e− (−1)j (n− 1)− j )n−1+2h (n − 1 + 2h)! and, together with (2), this gives Qg(α) = Q̂g−h(α) (n− 1 + 2h)! (−1)j (n − 1)− j )n−1+2h The result follows immediately from Theorems 1.1 and 1.3. � TRANSITIVE POWERS OF YOUNG-JUCYS-MURPHY ELEMENTS ARE CENTRAL 5 1.3. Outline. In Section 2, we introduce a generating series for the number of transitive factoriza- tions into star transpositions in arbitrary genus, and prove that it is the unique formal power series solution of a linear partial differential equation that we call the Join-cut Equation for this class of factorizations. The proof is based on a join-cut analysis of these factorizations, since the left-most factor σ either joins two cycles of the product π of the remaining factors to form one cycle or cuts one cycle of π into two, depending on whether the two elements moved by σ are, respectively, in different cycles of π, or in the same cycle. This approach has been applied previously where the factors are arbitrary transpositions, for the genus 0 case in [GJ0], and for arbitrary genus in [GJVn] and [GJV]. In Section 3, we solve the Join-cut Equation to obtain the generating series for transitive factor- izations into star transpositions in arbitrary genus. Then, by determining the coefficients in this generating series, we prove Theorem 1.1 (and hence also give a new proof of Theorem 1.2). In Section 4, we pose some questions that arise from this investigation, but that we have been unable to resolve. 2. THE JOIN-CUT EQUATION Let SA denote the symmetric group on an arbitrary set A. For an arbitrary set A of size n containing 1 (for convenience, we shall consider star transpositions with pivot element 1), let K denote the set of all permutations in SA in which 1 lies on a cycle of length i and the remaining cycle-lengths in the disjoint cycle representation are given by the parts of α, where α ⊢ n − i, for n ≥ i ≥ 1. It is straightforward to determine, independently of the choice of A, that (3) |K(i)α | = (i− 1)!|Kα| = (n− 1)! α1 · · ·αk|Aut α| where α = (α1, . . . , αk). Consider a fixed permutation σ ∈ K α in Sn, and let cg(i, α) be the number of transitive factorizations of σ into n + k − 1 + 2g star transpositions (this number is constant for each such σ because of the symmetry of elements 2, . . . , n; note that σ lies in the conjugacy class Kα∪i, which has m = k + 1 cycles). Let Ψ denote the generating series (4) Ψ(t, u, x; z,y) := n≥i≥1, k,g≥0 un+k−1+2g (n+ k − 1 + 2g)! x2gzi α⊢n−i, l(α)=k |K(i)α |cg(i, α)yα. The following result is the Join-cut Equation for the set of transitive factorizations into star trans- positions. It states that Ψ is annihilated by the partial differential operator (5) ∆ := i,j≥1 ∂zi+j i,j≥1 jzi+j ∂zi∂yj Theorem 2.1 (Join-cut Equation). The generating series Ψ = Ψ(t, u, x; z,y) is the unique formal power series solution of ∆Ψ = 0, with initial condition Ψ(t, 0, x; z,y) = z1t. Proof. Fix a triple (k, g, i) of integers with k, g ≥ 0 and i ≥ 1 to be other than (0, 0, 1). Also fix a partition α with l(α) = k and a permutation σ ∈ K α in Sn, where α ⊢ n − i. Consider a transitive factorization (τ1, . . . , τr) of σ into star transpositions, where r = n+ k − 1 + 2g. For this factorization, we let π = τ2 · · · τr = τ1σ, and τ1 = (1 a). There are cg(i, α) such factorizations of σ, and we obtain a recurrence equation for cg(i, α) by considering the following case analysis for these factorizations which is based on the left-most factor τ1. 6 I. P. GOULDEN AND D. M. JACKSON Case 1: τ1 6= τj for any j = 2, . . . , r. In this case, the element a is a fixed point in π, and (τ2, . . . , τr) is not a transitive factorization of π. But, if we let π′ ∈ S[n]\{a}, whose disjoint cycle representation is obtained by removing the one-cycle containing a from the disjoint cycles of π, then (τ2, . . . , τr) is a transitive factorization of π′. But σ is obtained from π′ by inserting a immediately before 1 in the cycle of π′ containing 1. This implies that π′ ∈ K (i−1) α in S[n]\{a}. Note that the transitive factorization (τ2, . . . , τr) of π ′ has r− 1 = (n− 1) + k− 1 + 2g factors and that this is reversible, so we conclude that the number of such factorizations is cg(i− 1, α), the contribution from this case. Case 2: τ1 = τj for some j = 2, . . . , r. In this case, (τ2, . . . , τr) is a transitive factorization of π, since for a product of star transpositions in Sn to be transitive, it is necessary and sufficient that each of (1 2), . . . , (1n) appears at least once as a factor (as observed in [IR]). There are two subcases, based on which disjoint cycles of π contain elements 1 and a. Subcase 2(a): 1 and a appear on the same cycle of π. In this subcase, that cycle of π is cut into two cycles in σ, one containing 1, and the other containing a. Consequently, for each factorization (τ1, . . . , τr) of σ, we obtain a factorization of π in this subcase by selecting a to be any element on the k cycles of σ not containing 1. We account for the choices of a on these cycles as follows: Suppose the cycles are indexed so they have lengths α1, . . . , αk (the cycles are all non-empty, so they are distinguishable, even if their lengths are equal). If a is on the jth such cycle, of length αj , then there are αj choices of a, and the cycle of π containing 1 has length i + αj , so we have π ∈ K (i+αj ) in Sn. Since the transitive factorization (τ2, . . . , τr) of π has r− 1 = n+(k− 1)− 1+2g factors and this is reversible, we conclude that there are cg(i+αj , α\αj) such factorizations, giving a total contribution from this subcase of j=1 αjcg(i+ αj , α \ αj). Subcase 2(b): 1 and a appear on different cycles of π. In this subcase, these cycles of π are joined into a single cycle of σ, containing both 1 and a. Consequently, for each factorization (τ1, . . . , τr) of σ, we obtain a factorization of π in this subcase by selecting a to be any other ele- ment on the cycle of σ containing 1. We account for these i − 1 choices of a as follows: Suppose that the cycle of σ containing 1, in cyclic order, is (1 ji−1 . . . j1) (i.e., so σ(1) = ji−1, σ(jt) = jt−1, for t = 2, . . . , i − 1, and σ(j1) = 1). If a = jm, then π has disjoint cycles (1 ji−1 . . . jm+1) (containing 1) and (jm . . . j1), together with all the cycles of σ not containing 1, so we have π ∈ K (i−m) α∪m in Sn, and the transitive factorization (τ2, . . . , τr) of π has r− 1 = n+ (k+1)− 1+ 2(g− 1) factors. Since this is reversible, we conclude that there are cg−1(i −m,α ∪m) such factorizations, giving a total contribution from this subcase of m=1 cg−1(i−m,α ∪m). Adding together the contributions from these disjoint cases, we obtain the linear recurrence equation cg(i, α) = cg(i− 1, α) + αjcg(i+ αj , α \ αj) + cg−1(i−m,α ∪m), for k, g ≥ 0, i ≥ 1 (except for the simultaneous choices k = g = 0 and i = 1) and α with l(α) = k. The partial differential equation follows by multiplying this recurrence equation by ntn u n+k−2+2g (n+k−2+2g)! xgzi|K α |yα, and summing over the above range of k, g, i, α. The initial condition follows from the fact that there is a single, empty factorization with no factors, of the single permutation (with 1 as a fixed point) in S1. Thus we have c0(1, ε) = 1. � TRANSITIVE POWERS OF YOUNG-JUCYS-MURPHY ELEMENTS ARE CENTRAL 7 3. A PROOF OF THEOREM 1.1 3.1. An explicit solution to the Join-cut Equation. The next result gives the explicit solution of the Join-cut Equation in terms of the series ξ defined in (1) and W ≡ W (t, u, x : z) where zℓξ(ℓux)ξ(ux) ℓ−2uℓ−1tℓ. Theorem 3.1. Let Z := t ∂ W (t, u, x : z) and Y := ξ(ux)2u2W (t, u, x : y). Then Ψ = ZeY . Proof. It is a straightforward matter to show that the Join-cut Equation with the given boundary condition has a unique solution. The remainder of the proof is a verification that ∆ annihilates Ψ and that the boundary condition is satisfied. The operator ∆ is a linear combination of four differential operators. It is straightforward to obtain the four expressions for the application of each of these operators to Ψ. Let x̂ := ux for brevity. Then the expressions are: ℓzℓξ(x̂) ℓ−3uℓ−2tℓ (ℓ− 1)ξ(ℓx̂)ξ(x̂) + ℓx̂ξ′(ℓx̂)ξ(x̂) + (ℓ− 2)x̂ξ(ℓx̂)ξ′(x̂) ymξ(x̂) m−1umtm (m+ 1)ξ(mx̂)ξ(x̂) +mx̂ξ′(mx̂)ξ(x̂) +mx̂ξ(mx̂)ξ′(x̂) e−Y t izi+1ξ(ix̂)ξ(x̂) i−2ui−1ti+1 i+ 1 + mymξ(mx̂)ξ(x̂) mum+1tm i,j≥1 ∂zi+j i,j≥1 (i+ j)ziyjξ (i+ j)x̂ ξ(x̂)i+j−2ui+j−1ti+j, e−Y x2 i,j≥1 jzi+j ∂ziyj i,j≥1 ijzi+jξ(ix̂)ξ(jx̂)ξ(x̂) i+j−2ui+jti+j zℓξ(x̂) ℓ−2uℓ−2tℓSℓ,(6) where, with r := exp , we have i,j≥1, i+j=ℓ (ri − r−i)(rj − r−j) = (ℓ− 1)(rℓ + r−ℓ)− 2 rℓ−1 − r−ℓ+1 r − r−1 Now let θ := 1 x̂, and substituting this expression for Sℓ in (6), we obtain the revised fourth ex- pression e−Y x2 i,j≥1 jzi+j ∂ziyj zℓξ(x̂) ℓ−2uℓ−2tℓ (ℓ− 1)ch(ℓθ)− (ℓ− 1)θ sh(θ) Combining these four expressions, and recalling the definition (5) of the partial differential opera- tor ∆, we have (7) e−Y ∆Φ = zℓξ(x̂) ℓ−3uℓ−2tℓTℓ + ℓ,m≥1 zℓymξ(x̂) ℓ+m−3uℓ+m−1tℓ+mUℓ,m, where Tℓ, for ℓ ≥ 1 and Uℓ,m, for ℓ,m ≥ 1, are explicit polynomials in hyperbolic cosines and hyperbolic sines of multiples of θ, and in θ, using (1). It is readily shown, using the addition formulae for hyperbolic sines and cosines that Tℓ = 0 for ℓ ≥ 1 and, similarly, that Uℓ,m = 0 for 8 I. P. GOULDEN AND D. M. JACKSON ℓ,m ≥ 1. Thus, from (7), we have ∆Φ = 0. But ξ(0) = 1, so Φ|u=0 = z1t and we conclude from Theorem 2.1 and the uniqueness of the solution of the Join-cut Equation that Ψ = Φ, giving the result. � 3.2. An expression for the coefficients of Ψ. It is now straightforward to determine the coeffi- cients in the generating series Ψ, and thus obtain a proof of Theorem 1.1. Proof of Theorem 1.1. Suppose that α is a partition of n − i with k parts. Then for all n ≥ i ≥ 1, k, g ≥ 0, Theorem 3.1 and (1) gives ([A]B denotes the coefficient of A in B) [tnun+k−1+2gx2gziyα]Ψ = |Aut α| [u2gx2g]ξ(iux)ξ(ux)i−2 l(α)∏ ξ(αjux)ξ(ux) |Aut α| [x2g]ξ(x)n−2ξ(ix) l(α)∏ ξ(αjx) |Aut α| [x2g] exp ξ2jq2j(α ∪ i)x so, together with (3) and (4), this gives cg(i, α) = (n+ k − 1 + 2g)! α1 · · ·αk i ξ2βq2β(α ∪ i) |Aut β| But this is symmetric in α1, . . . , αk, i, and the result follows immediately by renaming α ∪ i as α, which has m = k + 1 parts. � 4. FURTHER QUESTIONS The following questions arise in the light of the results of this paper: (1) Is it possible to find a simple proof of the centrality in Theorem 1.1, without evaluating the class coefficients ag(α)? This might follow from a decomposition for Young-Jucys-Murphy elements, or from a more elementary argument in the symmetric group. (2) Is it possible to give a direct proof of Corollary 1.4 or 1.5 – i.e., to establish these relation- ships between ag(α) and bg(α) without appealing, as we have, to the explicit formulae? This would be particularly interesting, since bg(α), as defined, is clearly central. Such a proof might involve Young-Jucys-Murphy elements, or a more elementary argument in the symmetric group, or the geometry of branched covers. Presumably such a proof would contain a solution to Question 1 above. (3) In [GJV], the polynomiality of bg(α) (in the parts of α) in Theorem 1.3, was the basis for a conjectured ELSV-type formula for H (n),α , involving a Hodge integral over some, unspec- ified, moduli space. Does the polynomiality of ag(α) in Theorem 1.1 also lead to a similar ELSV-type formula when ag(α) is rescaled as a covering number? ACKNOWLEDGEMENTS The work of both authors was supported by Discovery Grants from NSERC. We thank John Irving, Igor Pak, Amarpreet Rattan and Ravi Vakil for helpful comments. TRANSITIVE POWERS OF YOUNG-JUCYS-MURPHY ELEMENTS ARE CENTRAL 9 REFERENCES [ELSV] T. Ekedahl, S. Lando, M. Shapiro and A. Vainstein, Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math. 146 (2001), 297–327. [GJ0] I. P. Goulden and D. M. Jackson, Transitive factorizations into transpositions and holomorphic mappings on the sphere, Proc. Amer. Math. Soc., 125 (1997), 51–60. [GJ1] I. P. Goulden and D. M. Jackson, A proof of a conjecture for the number of ramified coverings of the sphere by the torus, J. Combinatorial Theory (A), 88 (1999), 246–258. [GJ3] I. P. Goulden and D. M. Jackson, The number of ramified coverings of the sphere by the double torus, and a general form for higher genera, J. Combinatorial Theory (A), 88 (1999), 259–275. [GJVn] I. P. Goulden, D. M. Jackson and A. Vainshtein, The number of ramified coverings of the sphere by the torus and surfaces of higher genera, Annals of Combinatorics, 4 (2000), 27–46. [GJV] I. P. Goulden, D. M. Jackson, R. Vakil, Towards the geometry of double Hurwitz numbers, Adv. Math. 198 (2005), 43–92. [H] A. Hurwitz, Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), 1–60. [IR] J. Irving and A. Rattan, Factorizations of permutations into star transpositions, math.CO/0610640. [P] I. Pak, Reduced decompositions of permutations in terms of star transpositions, generalized Catalan numbers and k-ary trees, Disc. Math. 204 (1999), 329 – 335. [VO] A. M. Vershik and A. Yu. Okounkov, A new approach to the representation theory of the symmetric groups. II, math.RT/0503040, v3. http://arxiv.org/abs/math/0610640 http://arxiv.org/abs/math/0503040 1. Introduction and background 1.1. Young-Jucys-Murphy elements and the Main Theorem 1.2. Background 1.3. Outline 2. The Join-cut Equation 3. A proof of Theorem ?? 3.1. An explicit solution to the Join-cut Equation 3.2. An expression for the coefficients of 4. Further questions Acknowledgements References

0704.1101

On the S_n-module structure of the noncommutative harmonics

ON THE Sn-MODULE STRUCTURE OF THE NONCOMMUTATIVE HARMONICS. EMMANUEL BRIAND, MERCEDES ROSAS, AND MIKE ZABROCKI Abstract. Using a noncommutative analog of Chevalley’s decomposi- tion of polynomials into symmetric polynomials times coinvariants due to Bergeron, Reutenauer, Rosas, and Zabrocki we compute the graded Frobenius characteristic for their two sets of noncommutative harmon- ics with respect to the left action of the symmetric group (acting on variables). We use these results to derive the Frobenius series for the enveloping algebra of the derived free Lie algebra in n variables. In honor of Manfred Schocker (1970-2006). The authors would also like to acknowledge the contributions that he made to this paper. 1. Introduction A central result of Claude Chevalley [3] decomposes the ring of polyno- mials in n variables (as graded representation of the symmetric group Sn) as the tensor product of the symmetric polynomials times the coinvariants of Sn (i.e., polynomials modulo symmetric polynomials with no constant term). The coinvariants of the symmetric group can also be defined as its har- monics (the polynomials annhilated by all symmetric polynomial differen- tial operators with no constant term). They admit as a basis the famous Schubert polynomials of Schubert calculus, that play an important role in algebraic combinatorics, see for instance [6]. The space of invariant polynomials in noncommutative variables was in- troduced in 1936 by Wolf [16] where she found a noncommutative version of the fundamental theorem of symmetric functions. This space has been stud- ied from a modern perspective in [13, 1, 2]. On the other hand, two sets of noncommutative harmonics for the symmetric group were introduced in [1] that translated into two noncommutative analogues of Chevalley decompo- sition for the ring of polynomials in noncommuting variables. The question of decomposing as Sn–modules both kinds of noncommutative harmonics was left open. This is the starting point in our investigations. We begin the present work with the computation of the graded Frobenius characteristic of noncommutative harmonics. We then use these calculations Expanded version of a paper to appear in Journal of Combinatorial theory, series A, http://www.elsevier.com/locate/jcta. Emmanuel Briand is supported by a contract Juan de la Cierva, MEC. Mercedes Rosas is supported by a contract Ramón y Cajal, MEC. Mike Zabrocki is supported by NSERC. http://arxiv.org/abs/0704.1101v2 2 EMMANUEL BRIAND, MERCEDES ROSAS, AND MIKE ZABROCKI to derive the Frobenius series for the enveloping algebra of the derived free Lie algebra in n variables, A′n. This last computation is achieved by using the existence of an isomorphism of GLn(Q)–modules between the space of polynomials in noncommutative variables, and the tensor product of the space of commuting polynomials with A′n. Such an isomorphism is presented explicitly in the last section. We conclude this introduction with some basic definitions and results that we will be using in the following sections. Let Sn denote the sym- metric group in n letters. Denote by Q[Xn] = Q[x1, x2, . . . , xn] the space of polynomials in n commuting variables and by Q〈Xn〉 = Q〈x1, x2, . . . , xn〉 the space of polynomials in n noncommutative variables. The space of symmetric polynomials in n variables will be denoted by Symn and the space of noncommutative polynomials which are invariant under the canonical action of the symmetric group Sn will be denoted by NCSymn. Given any polynomial f(Xn) ∈ Q[Xn], the notation f(∂Xn) represents the polynomial turned into an operator with each of the variables replaced by its corresponding derivative operator. Analogous notation will also hold for f(Xn) ∈ Q〈Xn〉 except that there are two types of differential operators acting on words in noncommutative variables. The first is the Hausdorff derivative, ∂x, whose action on a word w is defined to be the sum of the subwords of w with an occurrence of the letter x deleted. The second deriv- ative is the twisted derivative, dx, which is defined on w to be w ′ if w = xw′, and 0 otherwise. Both derivations are extended to polynomials by linearity. It is interesting to remark (as does Lenormand in [8], section Séries comme opérateurs) that these two operations are dual to the shuffle and concate- nation products respectively, with respect to a scalar product where the noncommutative monomials are self dual. That is, 〈∂xf, g〉 = 〈f, x⊔⊔g〉, and 〈dxf, g〉 = 〈f, xg〉. Following [1], we introduce the following two sets of noncommutative ana- logues of the harmonic polynomials. The canonical action of the symmetric group endow them with the structure of Sn–modules. MHarn = {f ∈ Q〈Xn〉 : p(∂Xn)f(Xn) = 0 for all p ∈ Mn} NCHarn = {f ∈ Q〈Xn〉 : p(dXn)f(Xn) = 0 for all p ∈ Mn} where Mn = {p ∈ NCSymn with p(0) = 0}. We are now ready to state the two decompositions of Q〈Xn〉 as the ten- sor product (over Q) of its invariants times its coinvariants that we have described. Proposition 1 ([1], Theorems 6.8 and 8.8). As graded Sn–modules, Q〈Xn〉 ≃ MHarn ⊗ Symn, Q〈Xn〉 ≃ NCHarn ⊗NCSymn. NONCOMMUTATIVE HARMONICS 3 2. The Frobenius characteristic of noncommutative harmonics In this section we compute the Frobenius characteristic of both kinds of noncommutative harmonics. This section is based of the observation that the graded Frobenius series for each of the Sn–modules appearing in Proposition 1 is either known or can be deduced from the existence of the isomorphisms described there. The expressions for Frobenius images and characters will require a little use of symmetric function notation and identities. We will follow Macdonald [9] for the notation of the sλ Schur, hλ homogeneous, eλ elementary and pλ power sums bases for the ring of symmetric functions Sym, that we identify with Q[p1, p2, p3, . . .]. For convenience we will make use of some plethystic notation. For a symmetric function f , f [X] represents the symmetric function eval- uated at an unspecified (possibly infinite) alphabet X. Then, f [X(1 − q)] is the image of f under the algebra automorphism sending the power sum symmetric function pk to (1 − q k)pk[X]. Similarly, f is the image of the symmetric function f under the inverse automorphism (sending the power sum pk to pk/(1− q In our calculations, we use the Kronecker product ⊙ of symmetric func- tions. This operation on symmetric functions corresponds, under the Frobe- nius map, to the inner tensor product of representations of the symmetric group (tensor product of representations with the diagonal action on the ten- sors). It can also be defined directly on symmetric functions by the equation pλ⊙ pµ = δλ,µ i ni(λ)!i ni(λ) pλ where ni(λ) is the number of parts of size i in λ, and then extended by bilinearity. We introduce the notations (q; q)k = (1− q)(1− q 2) · · · (1− qk), {q; q}k = (1− q)(1− 2q) · · · (1− k q). Then qd/{q; q}d is the generating function for the set partitions with length d and qd/(q; q)d is the generating function for partitions with length d, [15]. Finally, since Symn and NCSymn are made of graded copies of the trivial Sn-module we conclude that FrobSn(NCSymn) = hn [X] {q, q}d FrobSn(Symn) = hn [X] (q; q)n = hn [X] (q; q)d In the following lemma we compute the graded Frobenius characteristic for the module Q〈Xn〉. 4 EMMANUEL BRIAND, MERCEDES ROSAS, AND MIKE ZABROCKI Lemma 2 (The Frobenius characteristic of Q〈Xn〉). FrobSn(Q〈Xn〉) = {q, q}d h(n−d,1d)[X]. Proof. For each monomial xi1 · · · xir , we define its type ∇(xi1 · · · xir) to be the set partition of [r] = {1, 2, . . . , r} such that a and b are in the same part of the set partition if and only if ia = ib in the monomial. For a set partition A with at most n parts, we will let MA equal the Sn submodule of Q〈Xn〉 spanned by all monomials of type A. As Sn–module, Q〈Xn〉 ≃ A : ℓ(A)=d where the second direct sum is taken over all set partitions A with d parts. Fix a set partition A, and let d be the number of parts of A, and x~i = xi1xi2 ...xir be the smallest monomial in lex order in M A. It involves only the variables x1, x2, . . . , xd. The representation M A is the representation of Sn induced by the action of the subgroup Sd×S 1 ≃ Sd on the subspace Q[Sd] · x~i. The representation Q[Sd] · x~i of Sd is isomorphic to the regular representation. We use the rule for a representation R of Sd induced to Sn, FrobSn(R ↑ ) = hn−d[X]FrobSd(R), and conclude that the Frobenius characteristic of MA is h(n−d,1d)[X]. Hence the graded Frobenius characteristic of Q〈Xn〉 is FrobSn(Q〈Xn〉) = A:ℓ(A)=d q|A|h(n−d,1d)[X] = {q, q}d h(n−d,1d)[X]. We are now able to compute the Frobenius characteristic for MHarn and NCHarn. Theorem 3 (The Frobenius characteristic of the noncommutative harmon- ics). FrobSn(MHarn) = (q; q)n {q, q}d h(n−d,1d)[X] FrobSn(NCHarn) = {q, q}d {q, q}d h(n−d,1d)[X]. Proof. This follows since FrobSn(MHarn ⊗ Symn) = FrobSn(MHarn) ⊙ FrobSn(Symn). Since hn[X] is the unity for the Kronecker product on symmetric functions of degree n, and since FrobSn(Symn) = hn[X]/(q; q)n, we conclude that FrobSn(MHarn)/(q; q)n = FrobSn(Q〈Xn〉). We can now solve for FrobSn(MHarn). NONCOMMUTATIVE HARMONICS 5 A similar argument demonstrates the formula for FrobSn(NCHarn). We have from Proposition 1 and Lemma 2, {q, q}d h(n−d,1d)[X] = FrobSn(Q〈Xn〉) = FrobSn(NCHarn)⊙FrobSn(NCSymn) {q, q}d hn[X] ⊙FrobSn(NCHarn) {q, q}d FrobSn(NCHarn). From this equation we can solve for FrobSn(NCHarn). � As a corollary, we obtain the generating functions for the graded dimen- sions of these spaces. Corollary 4 (The Hilbert series of the noncommutative harmonics). dimq(MHarn) = (q; q)n 1− nq dimq(NCHarn) = (1− nq) {q,q}d Proof. After Theorem 3, FrobSn(MHarn) = (q; q)n FrobSn(Q〈Xn〉) FrobSn(NCHarn) = {q, q}d FrobSn(Q〈Xn〉) This implies dimq(MHarn) = (q; q)n dimq(Q〈Xn〉) dimq(NCHarn) = {q, q}d dimq(Q〈Xn〉) since the Hilbert series of a graded Sn–module is obtained by coefficient extraction from the graded Frobenius characteristic (of the coefficient of p(1n)[X]/n! in the expansion in power sum symmetric functions). Last, the Hilbert series of Q〈Xn〉 is The graded dimensions of MHarn for 2 ≤ n ≤ 5 are listed in [14] as sequences A122391 through A122394. The sequences of graded dimensions of NCHarn for 3 ≤ n ≤ 8 are listed in [14] as sequences A122367 through A122372. 6 EMMANUEL BRIAND, MERCEDES ROSAS, AND MIKE ZABROCKI 3. Non–commutative harmonics and the enveloping algebra of the derived free Lie algebra Let Ln be the canonical realization of the free Lie algebra inside the ring of polynomials in noncommuting variables Q〈Xn〉. More precisely, Ln is the linear span of the minimal set of polynomials in Q〈Xn〉 that includes Q and the variables Xn, and is closed under the bracket operation [x, y] = xy − yx. Let L′n = [Ln,Ln] be the derived free Lie algebra. Remark that Ln = L n ⊕ QXn, where QXn denotes the space of linear polynomials. The enveloping algebra A′n of Ln can be realized as a subalgebra of Q〈Xn〉 as follows (see [12] 1.6.5): A′n = ker ∂x. More explicitly, A′n is the subalgebra of Q〈Xn〉 generated by all the brackets under concatenation. In [1] it was established that there is an isomorphism of vector spaces between MHarn and A n ⊗ Hn. In this section we will show the following result. Theorem 5. As Sn–modules, MHarn ≃ A n ⊗Hn. The proposition will be established by comparing the Frobenius image of MHarn (known from Theorem 3) to FrobSn(A n ⊗ Hn), which is equal to FrobSn(A n)⊙FrobSn(Hn). We will determine FrobSn(A n) in Theorem 8 below. An intermediate step will make use the following Theorem due to V. Drensky. Proposition 6 (Drensky, [5] Theorem 2.6). As GLn(Q)–modules (and con- sequently as Sn–modules), Q〈Xn〉 ≃ Q[Xn]⊗A Drensky proved Proposition 6 by exhibiting an explicit isomorphism be- tween these two representations. We will look at it in the next section. For now, we will provide a non–constructive proof of the theorem. Before, we need to introduce some notation. It is known that Q〈Xn〉 is the universal enveloping algebra (u.e.a) of the free Lie algebra, Ln. Using the Poincaré-Birkhoff-Witt theorem, a linear basis for Q〈Xn〉 is given by decreasing products of elements of Ln. Since we can choose an ordering of the elements of Ln so that the space of linear polynomials is smallest and decreasing products of linear polynomials are isomorphic to Q[Xn] (as a vector space), we note that as vector spaces Q〈Xn〉 = u.e.a.(Ln) = u.e.a(QXn ⊕ L n) ≃ Q[Xn]⊗A To distinguish between the commutative elements of Q[Xn] and the non- commutative words of Q〈Xn〉, we will place a dot over the variables (as in ẋi) to indicate the commutative variables. NONCOMMUTATIVE HARMONICS 7 Let [n] = {1, 2, . . . , n} and let [n]r denote the words of length r in the alphabet of the numbers 1, 2, . . . , n. A word w ∈ [n]r is called a Lyndon word if w < wkwk+1 · · ·wr for all 2 ≤ k ≤ r where< represents lexicographic order on words. Every word w ∈ [n]r is equal to a unique product w = ℓ1ℓ2 · · · ℓk such that ℓ1 ≥ ℓ2 ≥ · · · ≥ ℓk and each ℓi is Lyndon (e.g. Corollary 4.4 of [12]). Let ℓ be a Lyndon word of length greater than 1. We say that ℓ = uv is the standard factorization of ℓ if v is the smallest nontrivial suffix in lexicographic order. It follows that u and v are Lyndon words and u < v. For a Lyndon word ℓ, if ℓ is a single letter a then define Pa = xa ∈ Q〈Xn〉. If ℓ = uv is the standard factorization of ℓ, then Pℓ = [Pu, Pv ]. For any w ∈ [n]r with Lyndon decomposition w = ℓ1ℓ2 · · · ℓk, define Pw = Pℓ1Pℓ2 · · ·Pℓk . The set {Pw}w∈[n]r forms a basis for the noncommutative polynomials of degree r ([12], Theorem 5.1). The elements Pw with Lyndon decomposition w = ℓ1ℓ2 · · · ℓk such that each Lyndon factor has degree at least 2 are a basis of A′n. Proof. To prove that Q〈Xn〉 and Q[Xn] ⊗ A n are isomorphic as GLn(Q)– modules, we use the fact that two polynomial GLn(Q)–modules with the same character are isomorphic (see for instance the notes by Kraft and Procesi, [7]). The character of a GLn(Q)–module is the trace of the action of the diagonal matrix diag(a1, a2, . . . , an). A basis for Q[Xn] ⊗A n are the elements ẋ α ⊗ Pℓ1 · · ·Pℓk with ℓ1 ≥ ℓ2 ≥ · · · ≥ ℓk and |ℓi| ≥ 2. The action of the diagonal matrix diag(a1, a2, . . . , an) on this basis element is the same as the action on the noncommutative polynomial xα11 x 2 · · · x n Pℓ1Pℓ2 · · ·Pℓk (in both cases: multiplication by α1+m1 α2+m2 2 · · · a αn+mn n where mi is the number of occurrences of i in the word ℓ1ℓ2 · · · ℓk). By the Poincaré-Birkhoff-Witt theorem, these polynomials form a basis for Q 〈Xn〉, hence the trace of the action of diag(a1, a2, . . . , an) acting on Q 〈Xn〉 and Q[Xn]⊗A n are equal. Since their characters are equal, we conclude that they are isomorphic as GLn(Q) modules. � The GLn(Q)–character of Q[Xn] is , and the GLn(Q)–character of Q〈Xn〉 is 1−(a1+a2+···+an) . Therefore, the existence of a GLn(Q)-module isomorphism between Q〈Xn〉 and Q[Xn]⊗A n implies the following result. Corollary 7 (The GLn(Q)–character of A charGLn(Q)(A n)(a1, a2, · · · , an) = (1− a1) · · · (1− an) 1− (a1 + a2 + · · · + an) (−1)ie(i,1k−i)(a1, a2, . . . , an). 8 EMMANUEL BRIAND, MERCEDES ROSAS, AND MIKE ZABROCKI Moreover this last sum is equal to sshape(T )(a1, a2, . . . , an) where the sum is over all standard tableaux T such that the smallest integer which does not appear in the first column of T is odd. By Schur-Weyl duality, the above formula also describes the decompo- sition of the subspace of multilinear polynomials (i.e. with distinct occur- rences of the variables) of A′n. That is, if n is the number of variables, the the multilinear polynomials of degree n will be an Sn-module with Frobenius image equal to i=2(−1) ie(i,1n−i)[X]. This decomposition was considered in the papers [4], [10], [11] where an expression was given degree by degree up to n = 7. The expansion of this formula in the Schur basis provided in the Theorem agrees with the computations in those papers. We can derive a formula for the Frobenius characteristic of A′ by using a similar technique. Theorem 8 (The Frobenius characteristic of A′n). FrobSn(A {q; q}d h(n−d,1d)[X(1− q)]. Proof. For any symmetric function f [X] of degree n, we have that f [X]⊙ hn In particular, since FrobSn(Q[Xn]) = hn , we conclude that FrobSn(Q〈Xn〉) = FrobSn(A n ⊗Q[Xn]) = FrobSn(A n)⊙ hn = FrobSn(A This implies that if we make the plethystic substitution X→X(1 − q) into both sides of this equation and using Lemma 2 we arrive at the stated formula. � We can now prove Theorem 5. Proof. From Theorem 3 we know the Frobenius image of MHarn, we com- pare this to FrobSn(A n ⊗Hn) = FrobSn(A n)⊙FrobSn(Hn), {q; q}d h(n−d,1d)[X(1 − q)]⊙ hn (q; q)n = (q; q)n {q; q}d h(n−d,1d)[X] = FrobSn(MHarn). NONCOMMUTATIVE HARMONICS 9 Since the two Sn–modules have the same Frobenius image, we conclude that they must be isomorphic. � 4. An explicit isomorphism between Q〈Xn〉 and Q[Xn]⊗A Let V be a finite–dimensional vector space over Q. Let S(V ) and T (V ) be its symmetric algebra and tensor algebra respectively. There exists a unique embedding ϕ of GL(V )–modules of S(V ) into T (V ) such that ϕ(v1v2 · · · vr) = vσ(1) ⊗ vσ(2) ⊗ · · · ⊗ vσ(r) for all r ≥ 0, v1, v2, . . . , vr ∈ V. Its image is the subspace of the symmetric tensors. In the case V = i=1Qxi, we have S(V ) = Q[Xn] and T (V ) = Q〈Xn〉. Then the embed- ding ϕ and the inclusion A′n ⊂ Q〈Xn〉 induce a map of GLn(Q)–modules Φ : Q[Xn] ⊗ A n −→ Q〈Xn〉 characterized by Φ(f ⊗ a) = ϕ(f)a for all f ∈ Q[Xn] and all a ∈ A n. Then, Proposition 9 (Drensky, [5] Theorem 2.6). The map Φ is a GLn(Q) equi- variant isomorphism from Q[Xn]⊗A n to Q〈Xn〉. Indeed, Drensky showed that given an arbitrary homogeneous basis of G of A′n, the elements Φ(m⊗ g) for m monomial and g ∈ G, are a basis of A ([5] Lemma 2.4). We refine Drensky’s proof by considering for G the bracket basis {Pw}w∈[n]r of A n (introduced before the proof of Proposition 6) and the shuffle basis (see below) to realize Q[Xn] in Q〈Xn〉. We show that the elements Φ(m⊗g) form a basis Q〈Xn〉 (the hybrid basis) that is triangularly related and expands positively in the bracket basis of Q〈Xn〉 (Theorem 10 below). We follow the book of Reutenauer [12] for the classical definitions and results used in this section. The bracket basis Pw has been introduced in the previous section (before the proof of Proposition 6). Before presenting the hybrid basis we introduce another classical basis of Q〈Xn〉: the shuffle basis. The shuffle basis of Q〈Xn〉. Consider two monomials, xi1xi2 · · · xir and xj1xj2 · · · xjr′ in Q〈Xn〉. For a subset S = {s1, s2, . . . , sr} ⊆ [r + r and the complement subset T = {t1, t2, . . . , tr′} = [r + r ′]\S, we let xi1xi2 · · · xir⊔⊔Sxj1xj2 · · · xjr′ := w be the unique monomial in Q〈Xn〉 of length r+r ′ such that ws1ws2 · · ·wsr = xi1xi2 · · · xir and wt1wt2 · · ·wtr′ = xj1xj2 · · · xjr′ . 10 EMMANUEL BRIAND, MERCEDES ROSAS, AND MIKE ZABROCKI The shuffle of any two monomials is defined as u⊔⊔v = S⊆[|u|+|v|] |S|=|u| u⊔⊔Sv. This shuffle of monomials is then extended to a bilinear operation on any two elements of Q〈Xn〉 The shuffle product is a commutative and associative operation on Q〈Xn〉. Let w be a word in [n]r and let w = ℓ 2 · · · ℓ be the factorization of w into decreasing products of Lyndon words ℓ1 > ℓ2 > · · · > ℓk. For a Lyndon word ℓ = i1i2 · · · ir, let Sℓ be the corresponding monomial in Q〈Xn〉, that is Sℓ = xi1xi2 · · · xir . If w is not a single Lyndon word then define i1!i2! · · · ik! ⊔⊔ · · · ⊔⊔S The set {Sw}w∈[n]r forms a basis for the noncommutative polynomials of degree r ([12], Corollary 5.5). It is interesting to note that the bracket basis Pw and the shuffle basis Sw are dual with respect to the scalar product where the noncommutative monomials are self-dual. The hybrid basis of Q〈Xn〉. We are now ready to introduce the hybrid basis. Given a word w ∈ [n]r with a factorization into decreasing products of Lyndon words w = ℓ 2 · · · ℓ , then let ℓj1 , ℓj2 , · · · , ℓjr be the Lyndon words of length 1 in this decomposition and set M(w) = x ⊔⊔ij1 ⊔⊔ij2 ⊔⊔ · · · ⊔⊔x ⊔⊔ijr = ij1 !ij2 ! · · · ijr !S Observe that M(w) is the image under the embedding ϕ of the monomial X(w) = ẋ · · · ẋ . For all of the remaining Lyndon words ℓa1 , ℓa2 , . . . , ℓak−r with length greater than 1 we define the Lie portion of the word to be L(w) = P iak−r . We will define the hybrid elements to be Hw := M(w)L(w) = Φ(X(w) ⊗ L(w)). The result of this section is: Theorem 10. The noncommutative polynomials Hw are triangularly related to and expand positively in the Pu basis. Precisely, for w of length r, Hw = r!Pw + terms cuPu with u lexicographically smaller than w. As a consequence, the set {Hw}w∈[n]r is a basis for the noncommutative polynomials of Q〈Xn〉 of degree r. We require a few facts about Lyndon words and the lexicographic ordering which can be found in [12]. (1) If u and v are Lyndon words and u < v then uv is a Lyndon word. ([12], (5.1.2)) NONCOMMUTATIVE HARMONICS 11 (2) If u < v and u is not a prefix of v, then ux < vy for all words x, y. ([12], Lemma 5.2.(i)) (3) If w = ℓ1ℓ2 · · · ℓk with ℓ1 ≥ ℓ2 ≥ · · · ≥ ℓk then ℓk is the smallest (with respect to the > order) nontrivial suffix of w. ([12], Lemma 7.14) (4) If ℓ′ < ℓ, are both Lyndon words, then ℓ′ℓ < ℓℓ′ (follows from (1)). As a consequence, for ℓ1 ≥ ℓ2 ≥ · · · ≥ ℓk, ℓ1ℓ2 · · · ℓk ≥ ℓσ(1)ℓσ(2) · · · ℓσ(k) for any permutation σ ∈ Sk with equality if and only ℓi = ℓσ(i) for all 1 ≤ i ≤ k. Proof. To see that (4) holds consider a weakly decreasing product of Lyndon words ℓ1ℓ2 · · · ℓk. If id → σ (1) → σ(2) → · · · → σ is a chain in the weak right order then we have just shown that ℓσ(i)(1)ℓσ(i)(1) · · · ℓσ(i)(k) ≥ ℓσ(i+1)(1)ℓσ(i+1)(2) · · · ℓσ(i+1)(k) with equality if and only if the two Lyndon factors which were transposed are equal. Therefore there exists a chain of words one greater than or equal to the next with ℓ1ℓ2 · · · ℓk on one end and ℓσ(1)ℓσ(2) · · · ℓσ(k) on the other. � Theorem 10 will be established after the following lemma. Lemma 11. Let w be a word and ℓ1ℓ2 · · · ℓr the decomposition of w into a decreasing product of Lyndon words. Let ℓ be a Lyndon word such that ℓ = af1f2 · · · fk with a one of the variables, each fi a Lyndon word and fi ≥ fi+1 and fk ≥ ℓ1. Let u = ℓ1 · · · ℓdℓℓd+1 · · · ℓr where ℓd > ℓ ≥ ℓd+1 or d = 0 and ℓ ≥ ℓ1. Then PℓPw = Pu + terms cvPv where v is lexicographically smaller than u and cv ≥ 0. Proof. Assume that r = 1, and we have that either ℓ ≥ ℓ1 and PℓPℓ1 = Pℓℓ1 and we are done, or ℓ < ℓ1 and PℓPℓ1 = Pℓ1Pℓ + [Pℓ, Pℓ1 ]. In this case Pℓ1Pℓ = Pℓ1ℓ. By (1) we know that ℓℓ1 is Lyndon. Moreover, ℓℓ1 is its standard factorization (this follows from (3), since the nontrivial suffixes of ℓℓ1 are all suffixes of f1f2 · · · fkℓ1, which is a nonincreasing product of Lyndon words). Therefore Pℓℓ1 = [Pℓ, Pℓ1 ] and PℓPℓ1 = Pℓ1ℓ + Pℓℓ1 . By (4), ℓℓ1 < ℓ1ℓ so the triangularity relation holds. Now for an arbitrary r > 1 we have the same two cases. Either ℓ ≥ ℓ1 and PℓPℓ1Pℓ2 · · ·Pℓr = Pℓw, or ℓ < ℓ1 and PℓPℓ1Pℓ2 · · ·Pℓr = Pℓ1PℓPℓ2 · · ·Pℓr + [Pℓ, Pℓ1 ]Pℓ2 · · ·Pℓr . Our induction hypothesis holds for PℓPℓ2 · · ·Pℓr since fk ≥ ℓ1 ≥ ℓ2, hence PℓPℓ2 · · ·Pℓr = Pu′ + v′<u′ c Pv′ where u ′ = ℓ2 · · · ℓdℓℓd+1 · · · ℓr. Moreover, Pℓ1Pu′ = Pℓ1u′ = Pu since ℓ1 ≥ ℓ2 and Pℓ1Pv′ = Pℓ1v′ since ℓ1 ≥ any Lyndon prefix of v′. 12 EMMANUEL BRIAND, MERCEDES ROSAS, AND MIKE ZABROCKI Since [Pℓ, Pℓ1 ] = Pℓℓ1 by (3), and ℓ1 ≥ ℓ2, we have by the induction hypothesis that Pℓℓ1Pℓ2 · · ·Pℓr = Pu′′ + v′′<u′′ c Pv′′ where u′′ = ℓ2 · · · ℓd′ℓℓ1ℓd′+1 · · · ℓr with ℓd′ > ℓℓ1 ≥ ℓd′+1. In order to justify the induction step we also need to have that u′′ < u. This follows from (4) since u′′ is a permutation of the factors of u and ℓ1 > ℓ and ℓ lies to the left of ℓ1 in u We are now in a position to prove Theorem 10. Proof. Hw is defined as the product M(w)L(w) where M(w) is a a shuffle of monomials. It expands as M(w) = c̃bxb with c̃b = r! and where each monomial in M(w) has the same number of x1s, x2s, etc. We fix one such monomial xb that as follows by indexing its letters backwards: xb = xikxik−1 · · · xi1 . We define inductively words w[k], . . . , w[1], w[0] as follows: w[k] := w and w[j − 1] is the word obtained from w[j] by removing one of its Lyndon factors of length 1 equal to xij . Remark that L(w[j]) = L(w) for all j. Then we establish by induction on j that xijxij−1 · · · xi1L(w) = Pw[j] + terms cvPv with v lexicographically smaller than w[j] by applying Lemma 11 with xij for ℓ and w[j] for w. � References [1] N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, to appear in Canad. J. Math. [2] N. Bergeron and M. Zabrocki, The Hopf algebra of symmetric functions in non- commutative variables is free and cofree. Preprint, arXiv: math.CO/0509265. [3] C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782. [4] V. Drensky, Lattices of varieties of associative algebras. Serdica 8 (1982), 20–31. [5] V. 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Reutenauer, Free Lie algebras. London Mathematical Society Monographs. New Series, 7. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. http://arxiv.org/abs/math/0509265 http://www.math.unibas.ch/~kraft/Papers/KP-Primer.pdf NONCOMMUTATIVE HARMONICS 13 [13] M. Rosas, B. Sagan, Symmetric Functions in Noncommuting Variables. Transactions of the American Mathematical Society, 358:1 (2006), 215–232. [14] N. J. A. Sloane, The Online Encyclopedia of Integer Sequences, published electron- ically at http://www.research.att.com/~njas/sequences/. [15] H. Wilf, Generatingfunctionology. Second edition. Academic Press, Inc., Boston, MA, 1994. [16] M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626–637. Emmanuel Briand and Mercedes Rosas, Universidad de Sevilla, Sevilla, Spain Mike Zabrocki, York University, Toronto, Canada http://www.research.att.com/~njas/sequences/ 1. Introduction 2. The Frobenius characteristic of noncommutative harmonics 3. Non–commutative harmonics and the enveloping algebra of the derived free Lie algebra 4. An explicit isomorphism between Q"426830A Xn "526930B and Q[Xn] An'. The shuffle basis of Q"426830A Xn "526930B . The hybrid basis of Q"426830A Xn "526930B . References