16434321
Operator realization of the SU(2) WZNW model
Decoupling the chiral dynamics in the canonical approach to the WZNW model requires an extended phase space that includes left and right monodromy variables M and M. Earlier work on the subject, which traced back the quantum group symmetry of the model to the Lie-Poisson symmetry of the chiral symplectic form, left some open questions: 
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- How to reconcile the necessity to set MM−1 = 1 (in order to recover the monodromy invariance of the local 2D group valued field g = uu) with the fact the M and M obey different exchange relations? 
 
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- What is the status of the quantum symmetry in the 2D theory in which the chiral fields u(x−t) and u(x + t) commute? 
 
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- Is there a consistent operator formalism in the chiral (and the extended 2D) theory in the continuum limit? 
 
 
 
 
We propose a constructive affirmative answer to these questions for G = SU(2) by presenting the quantum fields u and u as sums of products of chiral vertex operators and q-Bose creation and annihilation operators.
118403024
Canonical approach to the quantum WZNW model
The canonical approach to the chiral SU(n) WZNW model with a monodromy independent r{matrix is reviewed. Taking the quantum group symmetry of the model (which re ects its classical Poisson{Lie symmetry) as a guiding principle, we derive a complete set of exchange relations in the enlarged chiral phase space that includes the Borel components M of the monodromy matrix. Regarded as new dynamical variables the elements of M in the left and right sectors cannot be identi ed: their Poisson brackets have opposite signs. This is a technical reason why the canonical reduction of the pair of chiral models to a single 2{dimensional theory that involves the left and right movers' elds does not respect the quantum group symmetry. A simple modi cation of the Poisson brackets which does lead to an SLq(n) invariant model is proven unacceptable as a substitute for the 2D theory. As a way out we suggest a weak form of the monodromy and quantum group invariance of the extended 2D theory (involving mean values in a physical subspace of the tensor product of chiral state spaces). 1 1 Introduction It took some ve years after the multivalued action for the Wess{Zumino{Novikov{Witten (WZNW) model was written down [1] (during which its conformal current algebra treatment was developed [2, 3, 4]) before a canonical r{matrix approach to the theory was undertaken [5, 6, 7]. It aimed to understand the classical origin of (just discovered [8{12]) quantum group symmetries of rational conformal eld theories (for a review and further references { see [13]). This development, reaching a peak in the early 1990ies [14{19], still continues to attract attention [20, 21]. The basic eld of the theory is a map g(= g(t; x)) from the cylinder (t 2 IR, x 2 IR=2 Z) to a simple compact Lie group G. To make the discussion concrete we shall work out in this paper the case G = SU(n). It was demonstrated in [14] (with some details of the proof still unpublished, [18]) that the 2{dimensional (2D) symplectic form (2) of the classical WZNW theory can be written as a sum of two (symplectic) chiral parts, and 
, corresponding to the splitting2 g = u u ; u = u(x t) ; u = u(x+ t) : (1.1) This is achieved by extending the phase space of u (and the one of u) by the monodromy degrees of freedom, where the monodromyM is de ned by the twisted periodicity condition u(x+ 2 ) = u(x)M; u(x+ 2 ) = M 1 u(x); M 2 G : (1.2) (We are working throughout the paper with equal time relations and will be skipping the time argument of u and u.) In e ect, is a (closed) 2{form in the space of triples (u(x);M+;M ) where M+(M ) is an upper (lower) triangular matrix in SL(n; I C) (for G = SU(n)) such that the diagonal elements of M+ and M 1 coincide and M =M+M 1 (diag M+ diag M = 1) : (1.3) Furthermore, (and 
) can be chosen in such a way that the Poisson brackets (PB) of a pair of u's ( u's) are monodromy independent. Such PB correspond (as anticipated in [6] and argued in [17]) to simple exchange relations for the quantized chiral elds that are manifestly quantum group symmetric. They are to be contrasted with the rather complicated exchange relations for u and u derived recently [21] from di erent principles. 2We are using throughout the space{like Lorentzian metric, 11 = 1 = 00, resulting, in particular, in interchanging the equations (@t + @x)u = 0 = (@t @x) u for u and u as compared to [14]. 2 It is usually presumed (and often stated) that the quantum group shifts in the left and right sectors have the form u! uT 1; M ! TMT 1; T 2 SLq(n) (1.4) and u! T u so that the 2D eld (1.1) is SLq(n) invariant. This is, however, inconsistent with the canonical quantization scheme in which u and u commute3. The quantum groups in the two sectors turn out to be di erent; already the classical PB of the u and u monodromies have opposite signs. We indicate the ow of a simple minded attempt to cure the di culty and suggest an alternative way out that mimics the inde nite metric space approach to a local gauge theory. The paper is organized as follows. We start in Section 2 with a concise review of classical PB for the chiral WZNWmodel making public some results of [18]. We point out on the way that the left and right sectors only decouple if we double the monodromy degrees of freedom regarding the monodromy matrices M and M in and 
 as independent. Section 3 is devoted to a derivation of the quantum exchange relations by expressing them in terms of the pair of (fundamental representations of) quantum group R matrices. Respecting the symmetry of the classical theory reduces the ambiguity in quantization to a single deformation parameter q which is determined by the ratio of eigenvalues of the braid operator. In Section 4 we analyse the problem of combining the left and right sector in a 2D quantum theory. 2 Canonical formalism and chiral PB The information contained in an action integral A in D space{time dimensions can be decoded from an associated closed D + 1 form (for a general discussion { see [14], Sec.2). We shall review the realization of this approach in the WZNW model where the presence of a multivaluedWess{Zumino term in the 2D action A makes it particularly appropriate. The multivaluedness of A is translated into a cohomological property: the canonical 3{ form ! = 1 4 tr(d [ 1 2k j j dtdx+ i(j1 dt j0 dx) g 1 dg ] k3 (g 1dg)3) (2.1) is closed (one readily checks that d! = 0) but not exact. The \coupling constant" k (to be identi ed with the level of the a ne Kac{Moody algebra generated by the chiral components of the current j) has to be a positive integer in order to ensure the single valuedness of the eld theoretic "partition function" eiA [1]. 3The contradiction was pointed out in a 1995 Schladming lecture by Anton Alekseev. 3 Rather than recalling the path from the WZNW action integral to the form ! we shall justify (2.1) by deriving from it the equations of motion. There are two ways to do that yielding the same answer. The canonical variational approach consists in contracting ! with a vertical vector eld, @=@j or tr(gX@=@g);X 2 G, and then pulling back the resulting 2{form to the base space setting s (g 1dg) = g 1@ g dx g 1@tgdt+g 1@g dx etc. We thus obtain the equations of motion in the rst order formalism: s (!b @=@j ) = 1 k j + ig 1@ g dt dx = 0 ) j = ikg 1@ g (2.2a) s (!btr(gX @=@g)) = trn k[g 1@0g; g 1@1g] + i@ j Xo dt dx = 0 ) @ j = 1 ik [j0; j1] : (2.2b) (In deriving (2.2b) we have used the relation g 1dgbtr(gX @=@g) = X:) The second way requires regarding the forms dj and g 1dg as having both horizontal and vertical components so that, e.g., 1 3 tr(g 1dg)3b @0 ^ @1 = tr([g 1@0g; g 1@1g]g 1dg), and setting the contraction of ! with the bivector eld @0 ^ @1 equal to zero (then the coe cient to dj gives (2.2a), the coe cient to g 1dg reproduces (2.2b)). Taking the curl of (2.2a) and combining the result with (2.2b) we arrive at the left invariant chiral current @0j1 @1j0(= @0j1 + @1j0) = 1 ik [j0; j1]) (@0 @1) j = 0; j := 1 2(j0 j1) = ikg 1@+g (2.2c) (@+ := 1 2(@0 + @1)). The right invariant counterpart j of j is j = ik(@ g)g 1; 2@ = @1 @0 ) @+j = g(@ j)g 1 = 0 : (2.2d) These equations are in fact the starting point of [1] (suggested by the non{abelian boson{ fermion correspondence). The symplectic form (2) of the theory is obtained by integrating (2.1) in x (over the circle) for constant time (dt = 0): (2) = 1 4 Z trni dj0 g 1 dg (ij0 + kg 1@g)(g 1dg)2o dx : (2.3) The general solution g of (2.2) is given by the factorized expression (1.1) while the periodicity of g; g(t; x + 2 ) = g(t; x) is re ected into the twisted periodicity condition (1.2) which introduces the monodromy degree of freedom. If we insert (1.1) into (2.3), using the rst order equations for u and u to write j0(= j0) = ik g 1@tg = ik u 1 (@ u) u 1 u 1@u u; @ = @ @x ; (2.4) 4 we can express (2) as a sum of two chiral symplectic forms on the price of incorporating in the chiral phase spaces the boundary values u< and u< of u and u at x = and the monodromy (or, more precisely, its Gauss decomposition in the complexi ed group GC). The result can be written as (2) = (u;M) + 
( u; M ) ; (2.5) with M and M treated as independent dynamical variables (to be identi ed after the reduction to (2)) and (= (u;M)) = k 4 tr Z @(u 1 du)u 1 du dx + dMM 1 u 1 < du< + k 4 (M) ; (2:6) (= ( u; M)) = k 4 tr Z d u u 1 @(d u u 1)dx+ d M M 1 d u< u 1 < k 4 ( M) : (2: 6) The 2 form that is added (subtracted) to ( 
) is assumed to only depend on the monodromy and is designed to render the chiral forms closed: = 1 2 tr(dMM 1 K dMM 1); K = K(M) ; (2:7) here K is a skew symmetric linear map of the complexi ed Lie algebra GC into itself, tr(XKY ) = tr((KX)Y ), restricted by d = 13 tr(dMM 1)3 () d = 0 = d 
) : (2:8) There is no global solution of (2.8) on the (compact) group manifoldG. One can, however, construct such solutions on a dense open neighbourhood of the identity of the complexi ed group GC using, in e ect, the Gauss{like decomposition (1.3): M =M+M 1 ; M 2 B GC; Mdiag + Mdiag = 1 ; (2:9) here B+(B ) are Borel subgroups of GC, whose Lie algebras are generated (for the standard r{matrices (2.12) below) by the Cartan elements hi 2 GC and by the raising (respectively, lowering) operators e (e ) where runs through the positive roots of GC. (For G = SU(n) B are the subgroups of SL(n; I C) of upper and lower triangular matrices.) Theorem 2.1 ([14] Sec.3, [18]) There is a one{to{one correspondence between the constant solutions r of the classical Yang{Baxter equation (YBE) [r 12; r 13] + [r 12; r 23] + [r 13; r 23] = 0 (2.10a) 5 of the form r ij = rij Cij; rij = rji; Cij = Cji; tr2C12 2 X= X for any X 2 GC (2.10b) and the skew symmetric maps K(M): GC! GC, and hence the 2{forms (2.7) satisfying (2.8), such that the resulting chiral symplectic form (2.6) yields PB involving the above (monodromy independent) r : 1 u (x1); 2 u (x2) = k 1 u (x1) 2 u (x2)(r+ 12 (x21) + r 12 (x12)); xij = xi xj : (2.11) Here is the step function satisfying (x) + ( x
120723238
Generalized Euler angles as intrinsic coordinates for nonlinear spinors
Among the nonlinear spinor representations of the pseudo‐orthogonal rotation groups presented in a preceding work, those ones are considered, whose carrier spaces are isomorphic with group spaces. The general element of such groups is written as a product of one‐parameter subgroups, allowing, for the SO(ν,ν) and SO(ν+1,ν) cases, to write explicitly the nonlinear spinor components in terms of independent intrinsic coordinates satisfying the Cartan’s quadratic constraints.
120683606
Exchange relations and correlation functions for a quantum particle on the SU2 group manifold
The SO4 invariant quantum dynamics of a point particle moving on the 3 sphere (or equivalently, of the relative motion of the spherical top) is considered. Quantum exchange relations for different times are derived with an ‘‘R matrix’’ depending on the time difference and on the conserved angular momentum. Their implications for correlation functions are spelled out. The chiral exchange relations of Alekseev and Faddeev [Commun. Math. Phys. 141, 413–422 (1991)] are also extended to different times.
3223149
Chiral zero modes of the SU(n) WZNW model
We define the chiral zero modes’ phase space of the G = SU(n) Wess-Zumino-Novikov-Witten (WZNW) model as an (n − 1)(n + 2)dimensional manifold Mq equipped with a symplectic form Ωq involving a Wess-Zumino (WZ) term ρ which depends on the monodromy M and is implicitly defined (on an open dense neighbourhood of the group unit) by dρ(M) = 1 3 tr (MdM) . (∗) This classical system exhibits a Poisson-Lie symmetry that evolves upon quantization into an Uq(sln) symmetry for q a primitive even root of 1 . For each (non-degenerate, constant) solution of the classical Yang-Baxter equation (CYBE) we write down explicitly a ρ(M) satisfying Eq.(∗) and invert the form Ωq , thus computing the Poisson bivector of the system. The resulting Poisson brackets (PB) appear as the classical counterpart of the exchange relations of the quantum matrix algebra studied previously in [31]. We argue that it is advantageous to equate the determinant D of the zero modes’ matrix (aα) to a pseudoinvariant under permutations polynomial in the SU(n) weights, rather than to adopt the familiar convention D = 1 . A finite dimensional ”Fock space” operator realization of the factor algebra Mq/Ih , where Ih is an appropriate ideal in Mq for qh = −1 , is briefly discussed.
124797224
Dimensions for conformal spinors
SummaryThe notion of dimension for conformal spinor fields is reanalysed, with special emphasis on the relation occurring between dilatations inM3,1 andSO4,2 rotations inM4,2. Necessary and sufficient conditions for scale invariance of some Lagrangians, describing eight-componentSO4,2 spinor fields inM4,2, are given. New homogeneity conditions forSO4,2 spinor fields are proposed in order to obtain in Minkowski space a scale-invariant Dirac-like Lagrangian for a doublet of canonical four-component spinor fields.RiassuntoSi analizza la nozione di dimensione per campi spinoriali conformi, con speciale riguardo alla relazione che intercorre fra dilatazioni inM3,1 e rotazioni diSO4,2 inM4,2. Si danno condizioni necessarie e sufficienti per l’invarianza di scala di alcuni lagrangiani, descriventi campi spinoriali diSO4,2 ad otto componenti inM4,2. Si propongono nuove condizioni di omogeneità per i campi spinoriali diSO4,2 allo scopo di ottenere nello spazio di Minkowski un lagrangiano invariante di scala del tipo di Dirac per un doppietto di campi spinoriali canonici a quattro componenti.РезюмеЗаново анализируется понятие размерностм для конформных спинорных оояей, причем, особое внимание уделяется соотношению между расширениями вM3,1 иSO4,2 вращениями вM4,2. Приводятся необходимые и достаточные условия для масштабной инвариантности некоторых Лагранжианов, описывающих восьми-компонентныеSO4,2 спинорные поля вM4,2. Предлатаются новые условия однородности дляSO4,2 спинорных полей, чтобы получить в пространстве Минковского масштабно-инвариантный Лагранжиан, дираковското типа, для дублета канонических четырех-компонентных спинорных полей.
199514134
SURVEY AND ALIGNMENT FOR THE SYNCHROTRON LIGHT SOURCE ELETTRA
ELETTRA is a third generation light Source currently under construction in Trieste, Italy, which will be commissioned in 1993. As a high brilliance machine, ELETTRA is designed for extremely small emittance, which causes a high sensitivity to alignment errors. First results and the general alignment method, as well as the instruments, survey tools and applied software, are described.