Fuzzballs with internal excitations
arXiv:0704.0690v1  [hep-th]  5 Apr 2007
ITFA-2007-09
Fuzzballs with internal excitations
Ingmar Kanitscheider, Kostas Skenderis and Marika Taylor
Institute for Theoretical Physics, University of Amsterdam,
Valckenierstraat 65, 1018XE Amsterdam, The Netherlands
ingkanit, skenderi, taylor@science.uva.nl
ABSTRACT
We construct general 2-charge D1-D5 horizon-free non-singular solutions of IIB super-
gravity on T 4 and K3 describing fuzzballs with excitations in the internal manifold; these
excitations are characterized by arbitrary curves. The solutions are obtained via dualities
from F1-P solutions of heterotic and type IIB on T 4 for the K3 and T 4 cases, respectively.
We compute the holographic data encoded in these solutions, and show that the internal
excitations are captured by vevs of chiral primaries associated with the middle cohomology
of T 4 or K3. We argue that each geometry is dual to a specific superposition of R ground
states determined in terms of the Fourier coefficients of the curves defining the supergravity
solution. We compute vevs of chiral primaries associated with the middle cohomology and
show that they indeed acquire vevs in the superpositions corresponding to fuzzballs with in-
ternal excitations, in accordance with the holographic results. We also address the question
of whether the fuzzball program can be implemented consistently within supergravity.
http://arxiv.org/abs/0704.0690v1
Contents
1 Introduction 4
2 Fuzzball solutions on T 4 9
2.1 Chiral null models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 The IIA F1-NS5 system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Dualizing further to the D1-D5 system . . . . . . . . . . . . . . . . . . . . . 12
3 Fuzzball solutions on K3 13
3.1 Heterotic chiral model in 10 dimensions . . . . . . . . . . . . . . . . . . . . 13
3.2 Compactification on T 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 String-string duality to P-NS5 (IIA) on K3 . . . . . . . . . . . . . . . . . . 16
3.4 T-duality to F1-NS5 (IIB) on K3 . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5 S-duality to D1-D5 on K3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 D1-D5 fuzzball solutions 21
5 Vevs for the fuzzball solutions 26
5.1 Holographic relations for vevs . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2 Application to the fuzzball solutions . . . . . . . . . . . . . . . . . . . . . . 30
6 Properties of fuzzball solutions 33
6.1 Dual field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.2 Correspondence between geometries and ground states . . . . . . . . . . . . 34
6.3 Matching with the holographic vevs . . . . . . . . . . . . . . . . . . . . . . 35
6.4 A simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.5 Selection rules for curve frequencies . . . . . . . . . . . . . . . . . . . . . . . 40
6.6 Fuzzballs with no transverse excitations . . . . . . . . . . . . . . . . . . . . 41
7 Implications for the fuzzball program 43
A Conventions 46
A.1 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
A.2 Duality rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
B Reduction of type IIB solutions on K3 49
B.1 S-duality in 6 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
B.2 Basis change matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
C Properties of spherical harmonics 54
D Interpretation of winding modes 55
E Density of ground states with fixed R charges 59
1 Introduction
Over the last few years an interesting new proposal for the gravitational nature of black hole
microstates has emerged [1, 2, 3]; see also [4, 5, 6], and [7]. According to this proposal there
should exist non-singular horizon-free geometries associated with the black hole microstates.
These so-called fuzzball geometries should asymptotically approach the original black hole
geometry and should generically differ from each other around the horizon scale. In this
scenario the black hole provides only an average statistical description of the physics and
thus longstanding issues such as the information loss paradox would be resolved. The
underlying physics of the black hole would not be conceptually different from that of a
star, with the temperature and entropy being of statistical origin. Given the importance
of understanding black hole physics and its implications for quantum gravity, this proposal
should be developed, explored and tested where possible.
Many issues need to be addressed to implement the fuzzball proposal at a quantitative
and precise level. The proposal requires the existence of exponential numbers of horizon-
free non-singular solutions for each black hole. So the most basic of questions is whether
one can find such a number of solutions with the required properties and moreover what
precisely are the required properties for any given black hole. Moreover one would like to
show quantitatively how black hole properties emerge upon coarse-graining; for this one
needs to know the precise relationship between geometries and microstates.
Much of the recent work on this proposal has been focused on constructing fuzzball ge-
ometries for certain supersymmetric black holes with macroscopic horizons; for a summary
of progress in this direction see [8]. The method of construction here uses crucially super-
symmetry and known classifications of supersymmetric solutions: one looks for non-singular
horizon-free supersymmetric solutions with the correct charges to match those of the black
hole.
This method however has a number of limitations. One is that the supersymmetric
classifications are not sufficiently restrictive for cases of interest and thus one needs a specific
ansatz to make progress. To date many of the fuzzball geometries constructed are rather
atypical; for example, they have angular momenta much larger than those of a typical black
hole microstate. Whilst families of typical geometries are presumably contained in the
supersymmetric classification, finding an ansatz to construct families rather than isolated
examples is not easy.
Another key issue is that one does not know precisely what is the relationship between
a given geometry and the black hole microstates. This in turn means that one does not
know whether one has constructed the correct geometries to describe the black hole. Nor
does one know whether one has enough geometries to account for the black hole entropy
upon geometric quantization, using the methods of [9]. For example, in cases where the
dual theory has distinct Higgs and Coulomb branches, one needs to determine whether a
given fuzzball geometry describes Higgs or Coulomb branch physics. More importantly,
one would like to see explicitly how black hole properties emerge upon coarse graining; to
understand how to do such a computation properly the precise relation between the fuzzball
geometries and microstates is crucial.
To address this issue, we have advocated and developed the use of AdS/CFT methods.
That is, the supersymmetric black holes of interest admit a dual CFT description and
the fuzzball geometries therefore have a decoupling limit which is asymptotically AdS.
One can therefore use well-developed techniques of AdS/CFT, in particular Kaluza-Klein
holography [10], to extract field theory data from the geometry and diagnose precisely what
the geometry describes.
It is worth emphasizing at this point that the AdS/CFT correspondence both motivates
and supports the fuzzball picture. The gravity/gauge theory dictionary relates a given
asymptotically AdS geometry to either a deformation of the CFT or the CFT in a non-trivial
vacuum characterized by the expectation values of gauge invariant operators. Conversely,
one expects that for any stable state of the CFT (such as the BPS states) there exists
an asymptotically AdS solution, whose asymptotics encode the vevs of gauge invariant
operators in that state. If the field theory is in a pure state, there is no entropy and
one does not expect the corresponding geometry to have a horizon, and hence entropy.
AdS/CFT thus implies that the field theory in a given pure stable (black hole) state should
have a geometric dual with no horizon; there is however no guarantee that the geometry
should be well-described by supergravity alone, i.e. weakly curved everywhere1.
In our recent papers [11, 12], we have discussed in some detail the case of the D1-D5
system, for which (some) fuzzball geometries were constructed in [1]. Since this is a 2-charge
system, there is no macroscopic horizon: the naive geometry is singular, with the horizon
believed to form on taking into account α′ corrections. Whilst this is not a macroscopic
black hole system, there are a number of reasons to explore this case fully before moving
on to supersymmetric macroscopic black holes.
1There are additional subtleties in low dimensional quantum field theories due to the strong infrared
fluctuations. More properly one should view a given fuzzball geometry as dual to a wavefunction on the
Higgs branch of the field theory, but it seems in any case likely that such wavefunctions would be localized
around specific regions in the large N limit and thus that this issue does not play a key role at infinite N .
Firstly, one can obtain all fuzzball geometries in this system by dualities from known
solitonic solutions of F1-P systems. Thus one should be able to account for all the entropy,
and show how the average black hole description emerges. Moreover, the dual description
of this black hole is the simplest and best understood: the black hole entropy arises from
the degeneracy of the Ramond ground states of the dual (4, 4) CFT. This is an ideal system
in which to address the question of what is the precise correspondence between geome-
tries and microstates, and moreover how the properties of given microstates determine and
characterize the fuzzball geometries.
In the original work of [1], only a subset of the 2-charge fuzzball geometries were con-
structed using dualities from F1-P solutions. Recall that the D1-D5 system on T 4 is related
by dualities to the type II F1-P system, also on T 4, whilst the D1-D5 system on K3 is re-
lated to the heterotic F1-P system on T 4; the exact duality chains needed will be reviewed in
sections 2 and 3. Now the solution for a fundamental string carrying momentum in type II
is characterized by 12 arbitrary curves, eight associated with transverse bosonic excitations
and four associated with the bosonization of eight fermionic excitations on the string [13].
The corresponding heterotic string solution is characterized by 24 arbitrary curves, eight
associated with transverse bosonic excitations and 16 associated with charge waves on the
string.
In the work of [1], the duality chain was carried out for type II F1-P solutions on T 4 for
which only bosonic excitations in the transverse R4 are excited. That is, the solutions are
characterized by only four arbitrary curves; in the dual D1-D5 solutions these four curves
characterize the blow-up of the branes, which in the naive solutions are sitting in the origin
of the transverse R4, into a supertube. In this paper we carry out the dualities for generic
F1-P solutions in both the T 4 and K3 cases, to obtain generic 2-charge fuzzball solutions
with internal excitations. Note that partial results for the T 4 case were previously given
in the appendix of [3]; we will comment on the relation between our solutions and theirs
in section 2. The general solutions are then characterized by arbitrary curves capturing
excitations along the compact manifold M4, along with the four curves describing the
blow-up in R4. They describe a bound state of D1 and D5-branes, wrapped on the compact
manifold M4, blown up into a rotating supertube in R4 and with excitations along the part
of the D5-branes wrapping the M4.
The duality chain that uses string-string duality from heterotic on T 4 to type II on K3
provides a route for obtaining fuzzball solutions that has not been fully explored. One of
the results in this paper is to make explicit all steps in this duality route. In particular, we
work out the reduction of type IIB on K3 and show how S-duality acts in six dimensions.
These results may be useful in obtaining fuzzball solution with more charges.
In our previous work [11, 12], we made a precise proposal for the relationship between
the 2-charge fuzzball geometries characterized by four curves F i(v) and superpositions of R
ground states: a given geometry characterized by F i(v) is dual to a specific superposition of
R vacua with the superposition determined by the Fourier coefficients of the curves F i(v).
In particular, note that only geometries associated with circular curves are dual to a single
R ground state (in the usual basis, where the states are eigenstates of the R-charge). This
proposal has a straightforward extension to generic 2-charge geometries, which we will spell
out in section 6, and the extended proposal passes all kinematical and accessible dynamical
tests, just as in [11, 12].
In particular, we extract one point functions for chiral primaries from the asymptotically
AdS region of the fuzzball solutions. We find that chiral primaries associated with the middle
cohomology of M4 acquire vevs when there are both internal and transverse excitations;
these vevs hence characterize the internal excitations. Moreover, there are selection rules
for these vevs, in that the internal and transverse curves must have common frequencies.
These properties of the holographic vevs follow directly from the proposed dual super-
positions of ground states. The vevs in these ground states can be derived from three point
functions between chiral primaries at the conformal point. Selection rules for the latter,
namely charge conservation and conservation of the number of operators associated with
each middle cohomology cycle, lead to precisely the features of the vevs found holographi-
cally.
To test the actual values of the kinematically allowed vevs would require information
about the three point functions of all chiral primaries which is not currently known and
is inaccessible in supergravity. However, as in [12], these vevs are reproduced surprisingly
well by simple approximations for the three point functions, which follow from treating
the operators as harmonic oscillators. This suggests that the structure of the chiral ring
may simplify considerably in the large N limit, and it would be interesting to explore this
question further.
An interesting feature of the solutions is that they collapse to the naive geometry when
there are internal but no transverse excitations. One can understand this as follows. Ge-
ometries with only internal excitations are dual to superpositions of R ground states built
from operators associated with the middle cohomology of M4. Such operators account for
a finite fraction of the entropy, but have zero R charges with respect to the SO(4) R sym-
metry group. This means that they can only be characterized by the vevs of SO(4) singlet
operators but the only such operators visible in supergravity are kinematically prevented
from acquiring vevs. Thus it is consistent that in supergravity one could not distinguish
between such solutions: one would need to go beyond supergravity to resolve them (by, for
instance, considering vevs of singlet operators dual to string states).
This brings us to a recurring question in the fuzzball program: can it be implemented
consistently within supergravity? As already mentioned, rigorously testing the proposed
correspondence between geometries and superpositions of microstates requires information
beyond supergravity. Furthermore, the geometric duals of superpositions with very small or
zero R charges are not well-described in supergravity. Even if one has geometries which are
smooth supergravity geometries, these may not be distinguishable from each other within
supergravity: for example, their vevs may differ only by terms of order 1/N , which cannot
be reliably computed in supergravity.
The question of whether the fuzzball program can be implemented in supergravity could
first be phrased in the following way. Can one find a complete basis of fuzzball geometries,
each of which is well-described everywhere by supergravity, which are distinguishable from
each other within supergravity and which together span the black hole microstates? On
general grounds one would expect this not to be possible since many of the microstates
carry small quantum numbers. We quantify this discussion in the last section of this paper
in the context of both 2-charge and 3-charge systems.
To make progress within supergravity, however, it would suffice to sample the black hole
microstates in a controlled way. I.e. one could try to find a basis of geometries which are
well-described and distinguishable in supergravity and which span the black hole microstates
but for which each basis element is assigned a measure. In this approach, one would deal
with the fact that many geometries are too similar to be distinguished in supergravity
by picking representative geometries with appropriate measures. In constructing such a
representative basis, the detailed matching between geometries and black hole microstates
would be crucial, to correctly assign measures and to show that the basis indeed spans all
the black hole microstates.
The plan of this paper is as follows. In section 2 we determine the fuzzball geometries
for D1-D5 on T 4 from dualizing type II F1-P solutions whilst in section 3 we obtain fuzzball
geometries for D1-D5 on K3 from dualizing heterotic F1-P solutions. The resulting so-
lutions are of the same form and are summarized in section 4; readers interested only in
the solutions may skip sections 2 and 3. In section 5 we extract from the asymptotically
AdS regions the dual field theory data, one point functions for chiral primaries. In section
6 we discuss the correspondence between geometries and R vacua, extending the proposal
of [11, 12] and using the holographic vevs to test this proposal. In section 7 we discuss
more generally the implications of our results for the fuzzball proposal. Finally there are a
number of appendices. In appendix A we state our conventions for the field equations and
duality rules, in appendix B we discuss in detail the reduction of type IIB on K3 and ap-
pendix C summarizes relevant properties of spherical harmonics. In appendix D we discuss
fundamental string solutions with winding along the torus, and the corresponding duals
in the D1-D5 system. In appendix E we derive the density of ground states with fixed R
charges.
2 Fuzzball solutions on T 4
In this section we will obtain general 2-charge solutions for the D1-D5 system on T 4 from
type II F1-P solutions.
2.1 Chiral null models
Let us begin with a general chiral null model of ten-dimensional supergravity, written in
the form
ds2 = H−1(x, v)dv(−du +K(x, v)dv + 2AI(x, v)dxI) + dxIdxI ; (2.1)
e−2Φ = H(x, v); B(2)uv =
(H(x, v)−1 − 1); B(2)vI = H(x, v)
−1AI(x, v).
The conventions for the supergravity field equations are given in the appendix A.1. The
above is a solution of the equations of motion provided that the defining functions are
harmonic in the transverse directions, labeled by xI :
�H(x, v) = �K(x, v) = �AI(x, v) = (∂IA
I(x, v) − ∂vH(x, v)) = 0. (2.2)
Solutions of these equations appropriate for describing solitonic fundamental strings carry-
ing momentum were given in [14, 15]:
H = 1 +
|x− F (v)|6 , AI = −
QḞI(v)
|x− F (v)|6 , K =
Q2Ḟ (v)2
Q|x− F (v)|6 , (2.3)
where F I(v) is an arbitrary null curve describing the transverse location of the string, and
Ḟ I denotes ∂vF
I(v). More general solutions appropriate for describing solitonic strings with
fermionic condensates were discussed in [13]. Here we will dualise without using the explicit
forms of the functions, thus the resulting dual supergravity solutions are applicable for all
choices of harmonic functions.
The F1-P solutions described by such chiral null models can be dualised to give cor-
responding solutions for the D1-D5 system as follows. Compactify four of the transverse
directions on a torus, such that xi with i = 1, · · · , 4 are coordinates on R4 and xρ with
ρ = 5, · · · , 8 are coordinates on T 4. Then let v = (t−y) and u = (t+y) with the coordinate
y being periodic with length Ly ≡ 2πRy, and smear all harmonic functions over both this
circle and over the T 4, so that they satisfy
�R4H(x) = �R4K(x) = �R4AI(x) = 0, ∂iA
i = 0. (2.4)
Thus the harmonic functions appropriate for describing strings with only bosonic conden-
sates are
H = 1 +
|x− F (v)|2 ; Ai = −
dvḞi(v)
|x− F (v)|2 ; (2.5)
Aρ = −
dvḞρ(v)
|x− F (v)|2 ; K =
dv(Ḟi(v)
2 + Ḟρ(v)
|x− F (v)|2 .
Here |x−F (v)|2 denotes
i(xi−Fi(v))2. Note that neither Ḟi(v) nor Ḟρ(v) have zero modes;
the asymptotic expansions of AI at large |x| therefore begin at order 1/|x|3. Closure of the
curve in R4 automatically implies that Ḟi(v) has no zero modes. The question of whether
Ḟρ(v) has zero modes is more subtle: since the torus coordinate x
ρ is periodic, the curve
Fρ(v) could have winding modes. As we will discuss in appendix D, however, such winding
modes are possible only when the worldsheet theory is deformed by constant B fields. The
corresponding supergravity solutions, and those obtained from them by dualities, should
thus not be included in describing BPS states in the original 2-charge systems.
The appropriate chain of dualities to the D1−D5 system is
T5678→
D5y5678
NS5y5678
NS5y5678
 , (2.6)
to map to the type IIA NS5-F1 system. The subsequent dualities
NS5y5678
NS5y5678
D5y5678
 (2.7)
result in a D1-D5 system. Here the subscripts of Dpa1···ap denote the spatial directions
wrapped by the brane. In carrying out these dualities we use the rules reviewed in appendix
A.2. We will give details of the intermediate solution in the type IIA NS5-F1 system since
it differs from that obtained in [3].
2.2 The IIA F1-NS5 system
By dualizing the chiral null model from the F1-P system in IIB to F1-NS5 in IIA one obtains
the solution
ds2 = K̃−1[−(dt−Aidxi)2 + (dy −Bidxi)2] +Hdxidxi + dxρdxρ
e2Φ = K̃−1H, B
ty = K̃
−1 − 1, (2.8)
µ̄i = K̃
−1Bµ̄i , B
ij = −cij + 2K̃
−1A[iBj]
C(1)ρ = H
−1Aρ, C
tyρ = (HK̃)
−1Aρ, C
µ̄iρ = (HK̃)
−1Bµ̄i Aρ,
ijρ = (λρ)ij + 2(HK̃)
−1AρA[iBj], C
ρστ = ǫρστπH
−1Aπ,
where
K̃ = 1 +K −H−1AρAρ, dc = − ∗4 dH, dB = − ∗4 dA, (2.9)
dλρ = ∗4dAρ, Bµ̄i = (−Bi, Ai),
with µ̄ = (t, y). Here the transverse and torus directions are denoted by (i, j) and (ρ, σ)
respectively and ∗4 denotes the Hodge dual in the flat metric on R4, with ǫρστπ denoting the
Hodge dual in flat T 4 metric. The defining functions satisfy the equations given in (2.4).
The RR field strengths corresponding to the above potentials are
iρ = ∂i(H
−1Aρ), F
tyiρ = K̃
−1∂i(H
−1Aρ),
µ̄ijρ = 2K̃
−1Bµ̄
∂j](H
−1Aρ), F
iρστ = ǫρστπ∂i(H
−1Aπ), (2.10)
= K̃−1
6A[iBj∂k](H
−1Aρ) +Hǫijkl∂
l(H−1Aρ)
Thus the solution describes NS5-branes wrapping the y circle and the T 4, bound to funda-
mental strings delocalized on the T 4 and wrapping the y circle, with additional excitations
on the T 4. These excitations break the T 4 symmetry by singling out a direction within
the torus, and source multipole moments of the RR fluxes; the solution however has no net
D-brane charges.
Now let us briefly comment on the relation between this solution and that presented in
appendix B of [3]2. The NS-NS sector fields agree, but the RR fields are different; in [3] they
are given as 1, 3 and 5-form potentials. The relation of these potentials to field strengths
(and the corresponding field equations) is not given in [3]. As reviewed in appendix A.2, in
the presence of both electric and magnetic sources it is rather natural to use the so-called
democratic formalisms of supergravity [16], in which one includes p-form field strengths with
2We thank Samir Mathur for discussions on this issue.
p > 5 along with constraints relating higher and lower form field strengths. Any solution
written in the democratic formalism can be rewritten in terms of the standard formalism,
appropriately eliminating the higher form field strengths. If one interprets the RR forms
of [3] in this way, one does not however obtain a supergravity solution in the democratic
formalism; the Hodge duality constraints between higher and lower form field strengths are
not satisfied. Furthermore, one would not obtain from the RR fields of [3] the solution
written here in the standard formalism, after eliminating the higher forms.
2.3 Dualizing further to the D1-D5 system
The final steps in the duality chain are T-duality along a torus direction, followed by S-
duality. When T-dualizing further along a torus direction to a F1-NS5 solution in IIB, the
excitations along the torus mean that the dual solution depends explicitly on the chosen
T-duality cycle in the torus. We will discuss the physical interpretation of the distinguished
direction in section 4. In the following the T-duality is taken along the x8 direction, resulting
in the following D1-D5 system:
ds2 =
5 f̃1
[−(dt−Aidxi)2 + (dy −Bidxi)2] + f1/21 f
5 dxidx
i + f
5 dxρdx
e2Φ =
f5f̃1
f5f̃1
µ̄i =
ABµ̄i
f5f̃1
, (2.11)
ij = λij +
2AA[iBj]
f5f̃1
αβ = −ǫαβγf
5 Aγ , B
α8 = f
5 Aα,
C(0) = −f−11 A, C
ty = 1− f̃−11 , C
µ̄i = −f̃
ij = cij − 2f̃
1 A[iBj], C
tyij = λij +
f5f̃1
(cij + 2A[iBj]),
µ̄ijk
f5f̃1
cjk], C
= −ǫαβγf−15 A
γ , C
tyα8 = f
5 Aα,
αβγ8 = ǫαβγf
5 A, C
ijα8 = (λα)ij + f
5 Aαcij , C
ijαβ = −ǫαβγ(λ
ij + f
γcij),
where
f5 ≡ H, f̃1 = 1 +K −H−1(AαAα + (A)2), f1 = f̃1 +H−1(A)2,
dc = − ∗4 dH, dB = − ∗4 dA, Bµ̄i = (−Bi, Ai), (2.12)
dλα = ∗4dAα, dλ = ∗4dA.
Here µ̄ = (t, y) and we denote A8 as A with the remaining Aρ being denoted by Aα where
the index α runs over only 5, 6, 7. The Hodge dual over these coordinates is denoted by
ǫαβγ . Explicit expressions for these defining harmonic functions in terms of variables of the
D1-D5 system will be given in section 4.
The forms with components along the torus directions can be written more compactly
as follows. Introduce a basis of self-dual and anti-self dual 2-forms on the torus such that
ωα± =
(dx4+α± ∧ dx8 ± ∗T 4(dx4+α± ∧ dx8)), (2.13)
with α± = 1, 2, 3. These forms are normalized such that
ωα± ∧ ωβ± = ±(2π)4V δα±β± , (2.14)
where (2π)4V is the volume of the torus. Then the potentials wrapping the torus directions
can be expressed as
B(2)ρσ = C
tyρσ =
2f−15 A
α−ωα−ρσ , (2.15)
ijρσ =
(λij)
α− + f−15 A
α−cij
ωα−ρσ ,
C(4)ρστπ = ǫρστπf
with ǫρστπ being the Hodge dual in the flat metric on T
4. Note that these fields are
expanded only in the anti-self dual two-forms, with neither the self dual two-forms nor the
odd-dimensional forms on the torus being switched on anywhere in the solution. As we
will discuss later, this means the corresponding six-dimensional solution can be described
in chiral N = 4b six-dimensional supergravity. The components of forms associated with
the odd cohomology of T 4 reduce to gauge fields in six dimensions which are contained in
the full N = 8 six-dimensional supergravity, but not its truncation to N = 4b.
3 Fuzzball solutions on K3
In this section we will obtain general 2-charge solutions for the D1-D5 system on K3 from
F1-P solutions of the heterotic string.
3.1 Heterotic chiral model in 10 dimensions
The chiral model for the charged heterotic F1-P system in 10 dimensions is:
ds2 = H−1(−dudv + (K − 2α′H−1N (c)N (c))dv2 + 2AIdxIdv) + dxIdxI
B̂(2)uv =
(H−1 − 1), B̂(2)vI = H
−1AI , (3.1)
Φ̂ = −1
lnH, V̂ (c)v = H
−1N (c),
where I = 1, · · · , 8 labels the transverse directions and V̂ (c)m are Abelian gauge fields, with
((c) = 1, · · · , 16) labeling the elements of the Cartan of the gauge group. The fields are
denoted with hats to distinguish them from the six-dimensional fields used in the next
subsection. The equations of motion for the heterotic string are given in appendix A.1; here
again the defining functions satisfy
�H(x, v) = �K(x, v) = �AI(x, v) = (∂IA
I(x, v)− ∂vH(x, v)) = �N (c) = 0. (3.2)
For the solution to correspond to a solitonic charged heterotic string, one takes the following
solutions
H = 1 +
|x− F (v)|6 , AI = −
QḞI(v)
|x− F (v)|6 , N
(c) =
q(c)(v)
|x− F (v)|6 ,
Q2Ḟ (v)2 + 2α′q(c)q(c)(v)
Q|x− F (v)|6 , (3.3)
where F I(v) is an arbitrary null curve in R8; q(c)(v) is an arbitrary charge wave and ḞI(v)
denotes ∂vFI(v). Such solutions were first discussed in [14, 15], although the above has
a more generic charge wave, lying in U(1)16 rather than U(1). In what follows it will be
convenient to set α′ = 1
These solutions can be related by a duality chain to fuzzball solutions in the D1-D5
system compactified on K3. The chain of dualities is the following:
Het,T 4
NS5ty,K3
NS5ty,K3
D5ty,K3
(3.4)
The first step in the duality is string-string duality between the heterotic theory on T 4 and
type IIA on K3. Again the subscripts of Dpa1···ap denote the spatial directions wrapped by
the brane. To use this chain of dualities on the charged solitonic strings given above, the
solutions must be smeared over the T 4 and over v, so that the harmonic functions satisfy
�R4H = �R4K = �R4AI = �R4N
(c) = ∂iA
i = 0 (3.5)
where i = 1, · · · , 4 labels the transverse R4 directions. Note that although the chain of
dualities is shorter than in the previous case there are various subtleties associated with it,
related to the K3 compactification, which will be discussed below.
3.2 Compactification on T 4
Compactification of the heterotic theory on T 4 is straightforward, see [18, 19] and the review
[20]. The 10-dimensional metric is reduced as
Ĝmn =
gMN +GρσV
(1) ρ
(1) σ
(1) ρ
M Gρσ
(1) σ
N Gρσ Gρσ
 , (3.6)
where V
(1) ρ
M with ρ = 1, · · · 4, are KK gauge fields. (Recall that the ten-dimensional quan-
tities are denoted with hats to distinguish them from six-dimensional quantities.) The
reduced theory contains the following bosonic fields: the graviton gMN , the six-dimensional
dilaton Φ6, 24 Abelian gauge fields V
M ≡ (V
(1) ρ
M , V
M ρ, V
(3) (c)
M ), a two form BMN and an
O(4, 20) matrix of scalars M . Note that the index (a), (b) for the SO(4, 20) vector runs
from (1, · · · , 24). These six-dimensional fields are related to the ten-dimensional fields as
Φ6 = Φ̂−
ln detGρσ ;
M ρ = B̂
Mρ + B̂
(1) σ
V̂ (c)ρ V
(3) (c)
M ; (3.7)
(3) (c)
M = V̂
M − V̂
(1) ρ
HMNP = 3(∂[M B̂
L(a)(b)F (V )
with the metric gMN and V
(1) ρ
M defined in (3.6). The matrix L is given by
0 −I20
 , (3.8)
where In denotes the n× n identity matrix. The scalar moduli are defined via
M = ΩT1
G−1 −G−1C −G−1V T
−CTG−1 G+ CTG−1C + V TV CTG−1V T + V T
−V G−1 V G−1C + V I16 + V G−1V T
Ω1, (3.9)
where G ≡ [Ĝρσ ], C ≡ [12 V̂
σ + B̂
ρσ ] and V ≡ [V̂ (c)ρ ] are defined in terms of the
components of the 10-dimensional fields along the torus. The constant O(4, 20) matrix Ω1
is given by
I4 I4 0
−I4 I4 0
. (3.10)
This matrix arises in (3.9) as follows. In [18, 20] the matrix L was chosen to be off-diagonal,
but for our purposes it is useful for L to be diagonal. An off-diagonal choice is associated
with an off-diagonal intersection matrix for the self-dual and anti-self-dual forms of K3,
but this is an unnatural choice for our solutions, in which only anti-self-dual forms are
active. Thus relative to the conventions of [18, 20] we take L → ΩT1 LΩ1, which induces
M → ΩT1MΩ1 and F → ΩT1 F . The definitions of this and other constant matrices used
throughout the paper are summarized in appendix B.2.
These fields satisfy the equations of motion following from the action
−ge−2Φ6 [R+ 4(∂Φ6)2 −
H23 −
F (V )
MN (LML)(a)(b)F (V )
(b)MN
tr(∂MML∂
MML)], (3.11)
where α′ has been set to 1/4 and κ26 = κ
10/V4 with V4 the volume of the torus.
The reduction of the heterotic solution to six dimensions is then
ds2 = H−1
−dudv +
K −H−1(1
(N (c))2 + (Aρ)
dv2 + 2Aidx
+ dxidx
Buv =
(H−1 − 1), Bvi = H−1Ai, Φ6 = −12 lnH (3.12)
V (a)v =
2H−1Aρ,H
−1N (c)
, M = I24,
where i = 1, · · · , 4 runs over the transverse R4 directions and ρ = 5, · · · , 8 runs over the
internal directions of the T 4. Thus the six-dimensional solution has only one non-trivial
scalar field, the dilaton, with all other scalar fields being constant.
3.3 String-string duality to P-NS5 (IIA) on K3
Given the six-dimensional heterotic solution, the corresponding IIA solution in six dimen-
sions can be obtained as follows. Compactification of type IIA on K3 leads to the following
six-dimensional theory [17]:
6 [R′ + 4(∂Φ′6)
2 − 1
tr(∂MM
′L∂MM ′L)] (3.13)
F ′(V )
MN (LM
′L)(a)(b)F
′(V )
(b)MN
B′2 ∧ F ′2(V )(a) ∧ F ′2(V )(b)L(a)(b).
The field content is the same as for the heterotic theory in (3.11); note that in contrast to
(3.7) there is no Chern-Simons term in the definition of the 3-form field strength, that is,
H ′MNP = 3∂[MB
The rules for string-string duality are [17]:
Φ′6 = −Φ6, g′MN = e−2Φ6gMN , M ′ =M, V
M = V
H ′3 = e
−2Φ6 ∗6 H3; (3.14)
these transform the equations of motion derived from (3.11) into ones derived from the
action (3.13).
Acting with this string-string duality on the heterotic solutions (3.12) yields, dropping
the primes on IIA fields:
ds2 = −dudv + (K −H−1((N (c))2/2 + (Aρ)2))dv2 + 2Aidxidv +Hdxidxi,
Hvij = −ǫijkl∂kAl, Hijk = ǫijkl∂lH, Φ6 =
lnH, (3.15)
V (a)v =
2H−1Aρ,H
−1N (c)
, M = I24,
with ǫijkl denoting the dual in the flat R
4 metric. This describes NS5-branes on type IIA,
wrapped on K3 and on the circle direction y, carrying momentum along the circle direction.
3.4 T-duality to F1-NS5 (IIB) on K3
The next step in the duality chain is T-duality on the circle direction y to give an NS5-F1
solution of type IIB on K3. It is most convenient to carry out this step directly in six
dimensions, using the results of [22] on T-duality of type II theories on K3× S1.
Recall that type IIB compactified on K3 gives d = 6, N = 4b supergravity coupled to 21
tensor multiplets, constructed by Romans in [23]. The bosonic field content of this theory
is the graviton gMN , 5 self-dual and 21 anti-self dual tensor fields and an O(5,21) matrix of
scalars M which can be written in terms of a vielbein M−1 = V TV . Following the notation
of [30] the bosonic field equations may be written as
RMN = 2P
PQ +HrMPQH
∇MPnrM = QMnmPmrM +QMrsPnsM +
HnMNPHrMNP , (3.16)
along with Hodge duality conditions on the 3-forms
∗6Hn3 = Hn3 , ∗6Hr3 = −Hr3 , (3.17)
In these equations (m,n) are SO(5) vector indices running from 1 to 5 whilst (r, s) are
SO(21) vector indices running from 6 to 26. The 3-form field strengths are given by
Hn = GAV nA ; H
r = GAV rA, (3.18)
where A ≡ {n, r} = 1, · · · , 26; GA = dbA are closed and the vielbein on the coset space
SO(5, 21)/(SO(5) × SO(21)) satisfies
V T ηV = η, V =
 , η =
0 −I21
 . (3.19)
The associated connection is
dV V −1 =
2P rn Qrs
 , (3.20)
where Qmn and Qrs are antisymmetric and the off-diagonal block matrices Pms and P rn
are transposed to each other. Note also that there is a freedom in choosing the vielbein;
SO(5) × SO(21) transformations acting on H3 and V as
V → OV, H3 → OH3, (3.21)
leave G3 and M−1 unchanged. Note that the field equations (3.16) can also be derived from
the SO(5, 21) invariant Einstein frame pseudo-action [21]
tr(∂M−1∂M)− 1
GAMNPM−1ABG
, (3.22)
with the Hodge duality conditions (3.17) being imposed independently.
Now let us consider the T-duality relating a six-dimensional IIB solution to a six-
dimensional IIA solution of (3.13); the corresponding rules were derived in [22]. Given
that the six-dimensional IIA supergravity has only an SO(4, 20) symmetry, relating IIB to
IIA requires explicitly breaking the SO(5, 21) symmetry of the IIB action down to SO(4, 20).
That is, one defines a conformal frame in which only an SO(4, 20) subgroup is manifest and
in which the action reads
R+ 4(∂Φ)2 +
tr(∂M−1∂M)
∂l(a)M−1
(a)(b)
∂l(b)
GAMNPM−1ABG
. (3.23)
The SO(5, 21) matrix M−1 has now been split up into the dilaton Φ, an SO(4, 20) vector
l(a) and an SO(4, 20) matrix M−1
(a)(b)
, and we have chosen the parametrization
M−1AB = Ω
e−2Φ + lTM−1l + 1
e2Φl4 −1
e2Φl2 (lTM−1)(b) +
e2Φl2(lTL)(b)
e2Φl2 e2Φ −e2Φ(lTL)(b)
(M−1l)(a) +
e2Φl2(Ll)(a) −e2Φ(Ll)(a) M−1(a)(b) + e
2Φ(Ll)(a)(l
TL)(b)
(3.24)
where l2 = l(a)l(b)L(a)(b), L(a)(b) was defined in (3.8) and Ω3 is a constant matrix defined in
appendix B.2.
The fields Φ, l(a) and M−1 and half of the 3-forms can now be related to the IIA fields
of section 3.3 by the following T-duality rules (given in terms of the 2-form potentials bA)
[22]:
g̃yy = g
yy , b̃
yM + b̃
g−1yy gyM , (3.25)
g̃yM = g
yy ByM , b̃
MN + b̃
g−1yy (BMN + 2(gy[MBN ]y)),
g̃MN = gMN − g−1yy (gyMgyN −ByMByN ), l̃(a) = V (a)y ,
Φ̃ = Φ− 1
log |gyy|, M̃−1(a)(b) =M
(a)(b)
(a)+1
M − g
y gyM ), (1 ≤ (a) ≤ 24),
Here y is the T-duality circle, the six-dimensional index M excludes y and IIB fields are
denoted by tildes to distinguish them from IIA fields. The other half of the tensor fields,
that is
(b̃1yM − b̃26yM ), (b̃1MN − b̃26MN ), b̃
(a)+1
MN , b̃
(a)+1
, can then be determined using the Hodge
duality constraints (3.17).
We now have all the ingredients to obtain the T-dual of the IIA solution (3.15) along
y ≡ 1
(u−v). The IIA solution is expressed in terms of harmonic functions which also depend
on the null coordinate v, and thus one needs to smear the solutions before dualizing. Note
that it is the harmonic functions (H,K,AI , N (c)) which must be smeared over v, rather than
the six-dimensional fields given in (3.15), since it is the former that satisfy linear equations
and can therefore be superimposed.
The Einstein frame metric and three forms are given by
ds2 =
[−(dt−Aidxi)2 + (dy −Bidxi)2)] +
HK̃dxidx
GAtyi = ∂i
, GAµ̄ij = −2∂[i
, (3.26)
GAijk = ǫijkl∂
lnA + 6∂[i
AjBk]
where
(H +K + 1, 04) , n
−2Aρ,−
2N (c),H −K − 1
, (3.27)
K̃ = 1 +K −H−1(1
(N (c))2 + (Aρ)
2), dB = − ∗4 dA, Bµi = (−Bi, Ai).
Recall that n = 1, · · · , 5 and r = 6, · · · , 26 and ∗4 denotes the dual on flat R4; µ̄ = (t, y).
The SO(4, 20) scalars are given by
, l(a) =
2H−1Aρ,H
−1N (c)
, M = I24. (3.28)
The SO(5, 21) scalar matrix M−1 = V TV in (3.24) can then conveniently be expressed in
terms of the vielbein
V = ΩT3
H−1K̃ 0 0
H3K̃)−1(A2ρ +
(N (c))2)
HK̃−1 −
HK̃−1l(b)
l(a) 0 I24
Ω3. (3.29)
3.5 S-duality to D1-D5 on K3
One further step in the duality chain is required to obtain the D1-D5 solution in type IIB,
namely S duality. However, in the previous section the type II solutions have been given in
six rather than ten dimensions. To carry out S duality one needs to specify the relationship
between six and ten dimensional fields. Whilst the ten-dimensional SL(2, R) symmetry is
part of the six-dimensional symmetry group, its embedding into the full six-dimensional
symmetry group is only defined once one specifies the uplift to ten dimensions. The details
of the dimensional reduction are given in appendix B, with the six-dimensional S duality
rules being given in (B.16); the S duality leaves the Einstein frame metric invariant, and
acts as a constant rotation and similarity transformation on the three forms GA and the
matrix of scalars M respectively. The S-dual solution is thus
ds2 =
f5f̃1
[−(dt−Aidxi)2 + (dy −Bidxi)2)] +
f5f̃1dxidx
i, (3.30)
GAtyi = ∂i
f5f̃1
, GAµ̄ij = −2∂[i
f5f̃1
GAijk = ǫijkl∂
lmA + 6∂[i
f5f̃1
AjBk]
(f5 + F1)
, (3.31)
(f5 − F1),−2Aα,−
2N (c), 2A5
((f5 − F1),−2Aα− , 2A) .
Here the index α = 6, 7, 8. Note that the specific reduction used here, see appendix B,
distinguished A5 from the other Aρ and N
(c). A different embedding would single out
a different harmonic function, and hence a different vector, and it is thus convenient to
introduce (A,Aα−) to denote the choice of splitting more abstractly. Also as in (2.12) it is
convenient to introduce the following combinations of harmonic functions:
f5 = H, f̃1 = 1 +K −H−1(A2 +Aα−Aα−), (3.32)
F1 = 1 +K, f1 = f̃1 +H
−1A2.
The vielbein of scalars is given by
V = ΩT4
f−11 f̃1 0 0 0 0
f̃−11 f1 −GAF1 (
f1f̃1)
−1A −GA kγ
−FA 0
f−15 f1 0 0
FA 0 −1
f−15 F (k
1 −Fkγ
0 0 f−15 k
γ 0 I22
Ω4, (3.33)
where to simplify notation quantities (F,G) are defined as
F = (f1f5)
−1/2, G = (f1f̃1f
−1/2. (3.34)
We also define the 22-dimensional vector kγ as
kγ = (03,
2Aα−). (3.35)
Here γ = 1, · · · , b2 where the second Betti number is b2 = 22 for K3. Using the reduction
formulae (B.13) and (B.14), the six-dimensional solution (3.30), (3.33) can be lifted to ten
dimensions, resulting in a solution with an analogous form to the T 4 case (2.11). We will
thus summarize the solution for both cases in the following section.
4 D1-D5 fuzzball solutions
In this section we will summarize the D1-D5 fuzzball solutions with internal excitations, for
both the K3 and T 4 cases. In both cases the solutions can be written as
ds2 =
[−(dt−Aidxi)2 + (dy −Bidxi)2] + f1/21 f
5 dxidx
i + f
e2Φ =
f5f̃1
f5f̃1
µ̄i =
ABµ̄i
f5f̃1
, (4.1)
ij = λij +
2AA[iBj]
f5f̃1
, B(2)ρσ = f
γωγρσ, C
(0) = −f−11 A,
ty = 1− f̃−11 , C
µ̄i = −f̃
i , C
ij = cij − 2f̃
1 A[iBj],
tyij = λij +
f5f̃1
(cij + 2A[iBj]), C
µ̄ijk =
f5f̃1
cjk],
tyρσ = f
γωγρσ, C
ijρσ = (λ
ij + f
γcij)ω
ρσ, C
ρστπ = f
5 Aǫρστπ,
where we introduce a basis of self-dual and anti-self-dual 2-forms ωγ ≡ (ωα+ , ωα−) with
γ = 1, · · · , b2 on the compact manifold M4. For both T 4 and K3 the self-dual forms are
labeled by α+ = 1, 2, 3 whilst the anti-self-dual forms are labeled by α− = 1, 2, 3 for T
4 and
α− = 1, · · · 19 for K3. The intersections and normalizations of these forms are defined in
(2.13), (2.14) and (B.4). The solutions are expressed in terms of the following combinations
of harmonic functions (H,K,Ai,A,Aα−)
f5 = H; f̃1 = 1 +K −H−1(A2 +Aα−Aα−); f1 = f̃1 +H−1A2;
kγ = (03,
2Aα−); dB = − ∗4 dA; dc = − ∗4 df5; (4.2)
dλγ = ∗4dkγ ; dλ = ∗4dA; Bµ̄i = (−Bi, Ai),
where µ̄ = (t, y) and the Hodge dual ∗4 is defined over (flat) R4, with the Hodge dual
in the Ricci flat metric on the compact manifold being denoted by ǫρστπ. The constant
term in C
ty is chosen so that the potential vanishes at asymptotically flat infinity. The
corresponding RR field strengths are
i = −∂i
f−11 A
tyi = (f1f̃1f
f25∂if̃1 + f5A∂iA−A2∂if5
µ̄ij = (f
5 f1f̃1)
(f5∂j]f̃1 + f5A∂j]A−A2∂j]f5) + 2f̃1f25∂[iB
ijk = −ǫijkl(∂
lf5 − f−11 A∂lA)− 6f
1 ∂[i(AjBk]) (4.3)
+(f25f1f̃1)
6A[iBj(f5∂k]f̃1 + f5A∂k]A−A2∂k]f5)
iρσ = f
1 A∂i(f
γ)ωγρσ,
iρστπ = ǫρστπ∂i(f
5 A), F
tyijk = ǫijklf̃
1 f5∂
l(f−15 A),
µ̄ijkl = −ǫijklf5f̃
m(f−15 A),
tyiρσ = f̃
1 ∂i(k
γ/f5)ω
ρσ, F
µ̄ijρσ = 2f̃
∂j](f
γ)ωγρσ,
ijkρσ =
6f̃−11 A[iBj∂k](f
γ) + ǫijklf5∂
l(f−15 k
ωγρσ.
It has been explicitly checked that this is a solution of the ten-dimensional field equations
for any choices of harmonic functions (H,K,Ai,A,Aα−) with ∂iAi = 0. Note that in the
case of K3 one needs the identity (B.15) for the harmonic forms to check the components
of the Einstein equation along K3.
We are interested in solutions for which the defining harmonic functions are given by
H = 1 +
|x− F (v)|2 ; Ai = −
dvḞi(v)
|x− F (v)|2 , (4.4)
A = −Q5
dvḞ(v)
|x− F (v)|2 ; A
α− = −Q5
dvḞα−(v)
|x− F (v)|2 ,
dv(Ḟ (v)2 + Ḟ(v)2 + Ḟα−(v)2)
|x− F (v)|2 .
In these expressions Q5 is the 5-brane charge and L is the length of the defining curve in
the D1-D5 system, given by
L = 2πQ5/R, (4.5)
where R is the radius of the y circle. Note that Q5 has dimensions of length squared and is
related to the integral charge via
Q5 = α
′n5 (4.6)
(where gs has been set to one). Assuming that the curves (Ḟ(v), Ḟα− (v)) do not have zero
modes, the D1-brane charge Q1 is given by
dv(Ḟ (v)2 + Ḟ(v)2 + Ḟα−(v)2), (4.7)
and the corresponding integral charge is given by
, (4.8)
where (2π)4V is the volume of the compact manifold. The mapping of the parameters from
the original F1-P systems to the D1-D5 systems was discussed in [1] and is unchanged here.
The fact that the solutions take exactly the same form, regardless of whether the compact
manifold is T 4 or K3, is unsurprising given that only zero modes of the compact manifold
are excited.
The solutions defined in terms of the harmonic functions (4.4) describe the complete
set of two-charge fuzzballs for the D1-D5 system on K3. In the case of T 4, these describe
fuzzballs with only bosonic excitations; the most general solution would include fermionic
excitations and thus more general harmonic functions of the type discussed in [13]. Solutions
involving harmonic functions with disconnected sources would be appropriate for describing
Coulomb branch physics. Note that, whilst the solutions obtained by dualities from super-
symmetric F1-P solutions are guaranteed to be supersymmetric, one would need to check
supersymmetry explicitly for solutions involving other choices of harmonic functions.
In the final solutions one of the harmonic functions A describing internal excitations is
singled out from the others. In the original F1-P system, the solutions pick out a direction
in the internal space. For the type II system on T 4, the choice of Aρ singles out a direction
in the torus whilst in the heterotic solution the choice of (Aρ, N
(c)) singles out a direction
in the 20d internal space. Both duality chains, however, also distinguish directions in the
internal space. In the T 4 case one had to choose a direction in the torus, whilst in the
K3 case the choice is implicitly made when one uplifts type IIB solutions from six to ten
dimensions. In particular, the uplift splits the 21 anti-self-dual six-dimensional 3-forms into
19 + 1 + 1 associated with the ten-dimensional (F (5), F (3),H(3)) respectively.
When there are no internal excitations, the final solutions must be independent of the
choice of direction made in the duality chains but this does not remain true when the original
solution breaks the rotational symmetry in the internal space. A is the component of the
original vector along the direction distinguished in the duality chain, whilst Aα− are the
components orthogonal to this direction. When there are no excitations along the direction
picked out by the duality, i.e. A = 0, the solution considerably simplifies, becoming
ds2 =
(f1f5)1/2
[−(dt−Aidxi)2 + (dy −Bidxi)2] + f1/21 f
5 dxidx
i + f
e2Φ =
, B(2)ρσ = f
γωγρσ, C
ty = 1− f−11 , C
µ̄i = −f
ij = cij − 2f
1 A[iBj], C
tyρσ = f
γωγρσ, C
ijρσ = (λ
ij + f
γcij)ω
In this solution the internal excitations induce fluxes of the NS 3-form and RR 5-form along
anti-self dual cycles in the compact manifold (but no net 3-form or 5-form charges). By
contrast the excitations parallel to the duality direction induce a field strength for the RR
axion, NS 3-form field strength in the non-compact directions and RR 5-form field strength
along the compact manifold (but again no net charges).
Let us also comment on the M4 moduli in our solutions. The solutions are expressed in
terms of a Ricci flat metric on M4 and anti-self dual harmonic two forms. The forms satisfy
ωγρσω
δρσ = Dǫ δdγǫ ≡ δγδ , (4.9)
where the intersection matrix dδγ and the matrix D
δ relating the basis of forms and dual
forms are defined in (B.4) and (B.6) respectively. The latter condition on D
ǫ arose from
the duality chain, and followed from the fact that in the original F1-P solutions the internal
manifold had a flat square metric. Thus, the final solutions are expressed at a specific point
in the moduli space of M4 because the original F1-P solutions have specific fixed moduli.
It is straightforward to extend the solutions to general moduli: one needs to change
f̃1 = 1 +K −H−1(A2 +Aα−Aα−) → 1 +K −H−1(A2 + 12k
γkδDǫ δdγǫ), (4.10)
with kγ as defined in (4.2), to obtain the solution for more general D
Given a generic fuzzball solution, one would like to check whether the geometry is indeed
smooth and horizon-free. For the fuzzballs with no internal excitations this question was
discussed in [3], the conclusion being that the solutions are non-singular unless the defining
curve F i(v) is non-generic and self-intersects. In the appendix of [3], the smoothness of
fuzzballs with internal excitations was also discussed. However, their D1-D5 solutions were
incomplete: only the metric was given, and this was effectively given in the form (3.30)
rather than (4.1). Nonetheless, their conclusion remains unchanged: following the same
discussion as in [3] one can show that a generic fuzzball solution with internal excitations is
non-singular provided that the defining curve F i(v) does not self-intersect and Ḟi(v) only
has isolated zeroes. In particular, if there are no transverse excitations, F i(v) = 0, the
solution will be singular as discussed in section 6.6.
One can show that there are no horizons as follows. The harmonic function f5 is clearly
positive definite, by its definition. The functions (f1, f̃1) are also positive definite, since
they can be rewritten as a sum of positive terms as
f5f̃1 =
|x− F |2
dvḞ 2
|x− F |2
(4.11)
dv(Ḟ (v))2 + (Ḟα−(v))2
|x− F |2
dvdv′
(Ḟ(v)− Ḟ(v′))2 + (Ḟα−(v)− Ḟα−(v′))2
|x− F (v)|2 |x− F (v′)|2
and a corresponding expression for f5f1. Note that in the decoupling limit only the terms
proportional to Q25 remain, and these are also manifestly positive definite. Given that the
defining functions have no zeroes anywhere, the geometry therefore has no horizons.
Now let us consider the conserved charges. From the asymptotics one can see that the
fuzzball solutions have the same mass and D1-brane, D5-brane charges as the naive solution;
the latter are given in (4.6) and (4.8) whilst the ADM mass is
(Q1 +Q5), (4.12)
where Ly = 2πR, Ω3 = 2π
2 is the volume of a unit 3-sphere, and 2κ26 = (2κ
2)/(V (2π)4)
with 2κ2 = (2π)7(α′)4 in our conventions. The fuzzball solutions have in addition angular
momenta, given by
J ij =
dv(F iḞ j − F jḞ i). (4.13)
These are the only charges; the fields F (1) and F (5) fall off too quickly at infinity for the
corresponding charges to be non-zero. One can compute from the harmonic expansions
of the fields dipole and more generally multipole moments of the charge distributions. A
generic solution breaks completely the SO(4) rotational invariance in R4, and this symmetry
breaking is captured by these multipole moments.
However, the multipole moments computed at asymptotically flat infinity do not have a
direct interpretation in the dual field theory. In contrast, the asymptotics of the solutions in
the decoupling limit do give field theory information: one-point functions of chiral primaries
are expressed in terms of the asymptotic expansions (and hence multipole moments) near
the AdS3 ×S3 boundary. Thus it is more useful to compute in detail the latter, as we shall
do in the next section.
5 Vevs for the fuzzball solutions
In the decoupling limit all of the fuzzball solutions are asymptotic to AdS3 × S3 × M4,
where M4 is T 4 or K3. Therefore one can use AdS/CFT methods to extract holographic
data from the geometries. In particular, the asymptotics of the six-dimensional solutions
near the AdS3 × S3 boundary encode the vevs of chiral primary operators in the dual field
theory.
The precise relationship between asymptotics and vevs is however rather subtle. A
systematic method for extracting vevs from asymptotically AdS ×X solutions (with X an
arbitrary compact manifold) was only recently constructed, in [10], building on earlier work
[24, 25, 26, 27, 28], see also the review [29]. This method of Kaluza-Klein holography was
then applied to the case of asymptotically AdS3×S3 solutions of d = 6, N = 4b supergravity
coupled to nt tensor multiplets in [11, 12] and in what follows we will make use of many of
the results derived there.
For fuzzball solutions on K3, the relevant solution of six-dimensional N = 4b super-
gravity coupled to 21 tensor multiplets was given explicitly in (3.30). For the case of T 4,
we obtained the solution in ten dimensions, but there is a corresponding six-dimensional
solution of N = 4b supergravity coupled to 5 tensor multiplets. This solution is of exactly
the same form as the K3 solution given in (3.30), but with the index α− = 1, 2, 3. Thus
in what follows we will analyze both cases simultaneously. As mentioned earlier, the T 4
solution reduces to a solution of d = 6, N = 4b supergravity rather than a solution of
d = 6, N = 8 supergravity because forms associated with the odd cohomology of T 4 (and
hence six-dimensional vectors) are not present in our solutions.
5.1 Holographic relations for vevs
Consider an AdS3 × S3 solution of the six-dimensional field equations (3.16), such that
ds26 =
(−dt2 + dy2 + dz2) + dΩ23
; (5.1)
G5 = H5 ≡ go5 =
Q1Q5(rdr ∧ dt ∧ dy + dΩ3),
with the vielbein being diagonal and all other three forms (both self-dual and anti-self dual)
vanishing. In what follows it is convenient to absorb the curvature radius
Q1Q5 into an
overall prefactor in the action, and work with the unit radius AdS3 × S3. Now express the
perturbations of the six-dimensional supergravity fields relative to the AdS3×S3 background
gMN = g
MN + hMN ; G
A = goA + gA; (5.2)
V nA = δ
A + φ
nrδrA +
φnrφmrδmA ;
V rA = δ
A + φ
nrδnA +
φnrφnsδsA.
These fluctuations can then be expanded in spherical harmonics as follows:
hµν =
hIµν(x)Y
I(y), (5.3)
hµa =
(hIvµ (x)Y
a (y) + h
(s)µ(x)DaY
I(y)),
h(ab) =
(ρIt(x)Y It
(y) + ρIv
(x)DaY
b (y) + ρ
(s)(x)D(aDb)Y
I(y)),
haa =
πI(x)Y I(y),
gAµνρ =
3D[µb
(x)Y I(y),
gAµνa =
(b(A)Iµν (x)DaY
I(y) + 2D[µZ
(A)Iv
(x)Y Iva (y));
gAµab =
(A)I(x)ǫabcD
cY I(y) + 2Z(A)Ivµ D[bY
gAabc =
(−ǫabcΛIU (A)I(x)Y I(y));
φmr =
φ(mr)I(x)Y I(y),
Here (µ, ν) are AdS indices and (a, b) are S3 indices, with x denoting AdS coordinates and
y denoting sphere coordinates. The subscript (ab) denotes symmetrization of indices a and
b with the trace removed. Relevant properties of the spherical harmonics are reviewed in
appendix C. We will often use a notation where we replace the index I by the degree of the
harmonic k or by a pair of indices (k, I) where k is the degree of the harmonic and I now
parametrizes their degeneracy, and similarly for Iv, It.
Imposing the de Donder gauge condition DAhaM = 0 on the metric fluctuations re-
moves the fields with subscripts (s, v). In deriving the spectrum and computing correlation
functions, this is therefore a convenient choice. The de Donder gauge choice is however not
always a convenient choice for the asymptotic expansion of solutions; indeed the natural
coordinate choice in our application takes us outside de Donder gauge. As discussed in [10]
this issue is straightforwardly dealt with by working with gauge invariant combinations of
the fluctuations.
Next let us briefly review the linearized spectrum derived in [30], focusing on fields
dual to chiral primaries. Consider first the scalars. It is useful to introduce the following
combinations which diagonalize the linearized equations of motion:
4(k + 1)
(5r)k
I + 2(k + 2)U
I ), (5.4)
σkI =
12(k + 1)
(6(k + 2)Û
I − π̂
The fields s(r)k and σk correspond to scalar chiral primaries, with the masses of the scalar
fields being
s(r)k
= k(k − 2), (5.5)
The index r spans 6 · · · 5 + nt with nt = 5, 21 respectively for T 4 and K3. Note also that
k ≥ 1 for s(r)k; k ≥ 2 for σk. The hats (Û (5)kI , π̂kI ) denote the following. As discussed in
[10], the equations of motion for the gauge invariant fields are precisely the same as those
in de Donder gauge, provided one replaces all fields with the corresponding gauge invariant
field. The hat thus denotes the appropriate gauge invariant field, which reduces to the de
Donder gauge field when one sets to zero all fields with subscripts (s, v). For our purposes
we will need these gauge invariant quantities only to leading order in the fluctuations, with
the appropriate combinations being
I = πI2 +Λ
2ρI2(s); (5.6)
2 = U
2 − 12ρ
2(s);
ĥ0µν = h
h1±αµ h
Next consider the vector fields. It is useful to introduce the following combinations which
diagonalize the equations of motion:
h±µIv =
(C±µIv −A
(C±µIv +A
). (5.7)
For general k the equations of motion are Proca-Chern-Simons equations which couple
(A±µ , C
µ ) via a first order constraint [30]. The three dynamical fields at each degree k have
masses (k − 1, k + 1, k + 3), corresponding to dual operators of dimensions (k, k + 2, k + 4)
respectively; the operators of dimension k are vector chiral primaries. The lowest dimension
operators are the R symmetry currents, which couple to the k = 1 A±αµ bulk fields. The
latter satisfy the Chern-Simons equation
Fµν(A
±α) = 0, (5.8)
where Fµν(A
±α) is the curvature of the connection and the index α = 1, 2, 3 is an SU(2)
adjoint index. We will here only discuss the vevs of these vector chiral primaries.
Finally there is a tower of KK gravitons with m2 = k(k + 2) but only the massless
graviton, dual to the stress energy tensor, will play a role here. Note that it is the com-
bination Ĥµν = ĥ
µν + π
0goµν which satisfies the Einstein equation; moreover one needs the
appropriate gauge covariant combination ĥ0µν given in (5.6).
Let us denote by (O
) the chiral primary operators dual to the fields (s
I , σ
respectively. The vevs of the scalar operators with dimension two or less can then be
expressed in terms of the coefficients in the asymptotic expansion as
i ]1;
I ]2; (5.9)
2[σ2I ]2 −
2aIij
i ]1[s
Here [ψ]n denotes the coefficient of the z
n term in the asymptotic expansion of the field
ψ. The coefficient aIij refers to the triple overlap between spherical harmonics, defined in
(C.5). Note that dimension one scalar spherical harmonics have degeneracy four, and are
thus labeled by i = 1, · · · 4.
Now consider the stress energy tensor and the R symmetry currents. The three dimen-
sional metric and the Chern-Simons gauge fields admit the following asymptotic expansions
ds23 =
g(0)µ̄ν̄ + z
g(2)µ̄ν̄ + log(z
2)h(2)µ̄ν̄ + (log(z
2))2h̃(2)µ̄ν̄
+ · · ·
dxµ̄dxν̄ ;
A±α = A±α + z2A±α
+ · · · (5.10)
The vevs of the R symmetry currents J±αu are then given in terms of terms in the asymptotic
expansion of A±αµ as
J±αµ̄
g(0)µ̄ν̄ ± ǫµ̄ν̄
A±αν̄ . (5.11)
The vev of the stress energy tensor Tµ̄ν̄ is given by
〈Tµ̄ν̄〉 =
g(2)µ̄ν̄ +
Rg(0)µ̄ν̄ + 8
1g(0)µ̄ν̄ +
(5.12)
where parentheses denote the symmetrized traceless combination of indices.
This summarizes the expressions for the vevs of chiral primaries with dimension two or
less which were derived in [12]. Note that these operators correspond to supergravity fields
which are at the bottom of each Kaluza-Klein tower. The supergravity solution of course
also captures the vevs of operators dual to the other fields in each tower. Expressions for
these vevs were not derived in [12], the obstruction being the non-linear terms: in general
the vev of a dimension p operator will include contributions from terms involving up to p
supergravity fields. Computing these in turn requires the field equations (along with gauge
invariant combinations, KK reduction maps etc) up to pth order in the fluctuations.
Now (apart from the stress energy tensor) none of the operators whose vevs are given
above is an SO(4) (R symmetry) singlet. For later purposes it will be useful to review
which other operators are SO(4) singlets. The computation of the linearized spectrum in
[30] picks out the following as SO(4) singlets:
τ0 ≡ 1
π0; t(r)0 ≡ 1
φ5(r)0, (5.13)
along with φ0i(r) with i = 1, · · · , 4. Recall ψ0 denotes the projection of the field ψ onto
the degree zero harmonic. The fields (τ0, t(r)0) are dual to operators of dimension four,
whilst the fields φ0i(r) are dual to dimension two (marginal) operators. The former lie in
the same tower as (σ2, s(r)2) respectively, whilst the latter are in the same tower as s(r)1. In
total there are (nt + 1) SO(4) singlet irrelevant operators and 4nt SO(4) singlet marginal
operators, where nt = 5, 21 for T
4 and K3 respectively.
Consider the SO(4) singlet marginal operators dual to the supergravity fields φi(r).
These operators have been discussed previously in the context of marginal deformations
of the CFT, see the review [32] and references therein. Suppose one introduces a free
field realization for the T 4 theory, with bosonic and fermionic fields (xiI(z), ψ
I(z)) where
I = 1, · · · , N . Then some of the marginal operators can be explicitly realized in the
untwisted sector as bosonic bilinears
∂xiI(z)∂̄x
I(z̄); (5.14)
there are sixteen such operators, in correspondence with sixteen of the supergravity fields.
The remaining four marginal operators are realized in the twisted sector, and are associated
with deformation from the orbifold point.
5.2 Application to the fuzzball solutions
The six-dimensional metric of (3.30) in the decoupling limit manifestly asymptotes to
ds2 =
(−dt2 + dy2) +
+ dΩ23
. (5.15)
where
dv(Ḟ (v)2 + Ḟ(v)2 + Ḟα−(v)2). (5.16)
Note that the vielbein (3.33) is asymptotically constant
V o = ΩT4
I2 0 0 0
Q1/Q5 0 0
Q5/Q1 0
0 0 0 I22
Ω4, (5.17)
but it does not asymptote to the identity matrix. Thus one needs the constant SO(5, 21)
transformation
V → V (V o)−1, G3 → V oG3. (5.18)
to bring the background into the form assumed in (5.1).
The fields are expanded about the background values, by expanding the harmonic func-
tions defining the solution in spherical harmonics as
f5kIY
k (θ3)
, K =
f1kIY
k (θ3)
, (5.19)
k≥1,I
(AkI)iY
k (θ3)
, A =
k≥1,I
(AkI)Y Ik (θ3)
Aα− =
k≥1,I
Y Ik (θ3)
The polar coordinates here are denoted by (r, θ3) and Y
k (θ3) are (normalized) spherical
harmonics of degree k on S3 with I labeling the degeneracy. Note that the restriction k ≥ 1
in the last three lines is due to the vanishing zero mode, see [12]. As in [12], the coefficients
in the expansion can be expressed as
f5kI =
L(k + 1)
dv(CIi1···ikF
i1 · · ·F ik), (5.20)
f1kI =
L(k + 1)Q1
Ḟ 2 + Ḟ2 + (Ḟα−)2
CIi1···ikF
i1 · · ·F ik ,
(AkI)i = −
L(k + 1)
dvḞiC
i1···ikF
i1 · · ·F ik ,
(AkI) = −
Q1L(k + 1)
dvḞCIi1···ikF
i1 · · ·F ik ,
Aα−kI = −
Q1L(k + 1)
dvḞα−CIi1···ikF
i1 · · ·F ik .
Here the CIi1···ik are orthogonal symmetric traceless rank k tensors on R
4 which are in one-
to-one correspondence with the (normalized) spherical harmonics Y Ik (θ3) of degree k on S
Fixing the center of mass of the whole system implies that
(f11i + f
1i) = 0. (5.21)
The leading term in the asymptotic expansion of the transverse gauge field Ai can be written
in terms of degree one vector harmonics as
(A1j)iY
(aα−Y α−1 + a
α+Y α+1 ), (5.22)
where (Y α−1 , Y
1 ) with α = 1, 2, 3 form a basis for the k = 1 vector harmonics and we have
defined
aα± =
e±αij(A1j)i, (5.23)
where the spherical harmonic triple overlap e±αij is defined in C.6. The dual field is given by
B = −
(aα−Y α−1 − aα+Y
1 ). (5.24)
Now given these asymptotic expansions of the harmonic functions one can proceed to expand
all the supergravity fields, and extract the appropriate combinations required for computing
the vevs defined in (5.9), (5.11) and (5.12). Since the details of the computation are very
similar to those in [12], we will simply summarize the results as follows. Firstly the vevs of
the stress energy tensor and of the R symmetry currents are the same as in [12], namely
〈Tµ̄ν̄〉 = 0; (5.25)
aα±(dy ± dt). (5.26)
The vanishing of the stress energy tensor is as anticipated, since these solutions should be
dual to R vacua. As in [12], however, the cancellation is very non-trivial. The vevs of the
scalar operators dual to the fields (s
I , σ
I ) are also unchanged from [12]:
2f51i); (5.27)
6(f12I − f52I));
2(−(f12I + f52I) + 8aα−aβ+fIαβ).
The internal excitations of the new fuzzball solutions are therefore captured by the vevs of
operators dual to the fields s
I with r > 6:
(5+nt)1
2(A1i);
(6+α−)1
2Aα−1i ; (5.28)
(5+nt)2
6(A2I);
(6+α−)2
6Aα−2I .
Here nt = 5, 21 for T
4 and K3 respectively, with α− = 1, · · · , b2− with b2− = 3, 19 respec-
tively. Thus each curve (F(v),Fα− (v)) induces corresponding vevs of operators associated
with the middle cohomology of M4. Note the sign difference for the vevs of operators which
are related to the distinguished harmonic function F(v).
6 Properties of fuzzball solutions
In this section we will discuss various properties of the fuzzball solutions, including the
interpretation of the vevs computed in the previous section.
6.1 Dual field theory
Let us start by briefly reviewing aspects of the dual CFT and the ground states of the R
sector; a more detailed review of the issues relevant here is contained in [12]. Consider
the dual CFT at the orbifold point; there is a family of chiral primaries in the NS sector
associated with the cohomology of the internal manifold, T 4 or K3. For our discussions
only the chiral primaries associated with the even cohomology are relevant; let these be
labeled as O(p,q)n where n is the twist and (p, q) labels the associated cohomology class. The
degeneracy of the operators associated with the (1, 1) cohomology is h1,1. The complete set
of chiral primaries associated with the even cohomology is then built from products of the
(Opl,qlnl )
nlml = N, (6.1)
where symmetrization over the N copies of the CFT is implicit. The correspondence between
(scalar) supergravity fields and chiral primaries is 3
σn ↔ O(2,2)(n−1), n ≥ 2; (6.2)
s(6)n ↔ O
(0,0)
(n+1)
, s(6+α̃)n ↔ O
(1,1)
(n)α̃
, α̃ = 1, · · · h1,1, n ≥ 1.
Spectral flow maps these chiral primaries in the NS sector to R ground states, where
hR = hNS − jNS3 +
jR3 = j
, (6.3)
where c is the central charge. Each of the operators in (6.1) is mapped by spectral flow to
a (ground state) operator of definite R-charge
(O(pl,ql)nl )
(OR(pl,ql)nl )
ml , (6.4)
jR3 =
(pl − 1)ml, j̄R3 = 12
(ql − 1)ml.
Note that R operators which are obtained from spectral flow of those associated with the
(1, 1) cohomology have zero R charge.
3As discussed in [12], the dictionary between (σn, s
n ) and (O
(2,2)
(n−1)
(0,0)
(n+1)
) may be more complicated,
since their quantum numbers are indistinguishable, but this subtlety will not play a role here.
6.2 Correspondence between geometries and ground states
In [11, 12] we discussed the correspondence between fuzzball geometries characterized by a
curve F i(v) and R ground states (6.4) with (pl, ql) = 1± 1. The latter are related to chiral
primaries in the NS sector built from the cohomology common to both T 4 and K3, namely
the (0, 0), (2, 0), (0, 2) and (2, 2) cohomology.
The following proposal was made in [11, 12] for the precise correspondence between
geometries and ground states; see also [33]. Given a curve F i(v) we construct the corre-
sponding coherent state in the FP system and then find which Fock states in this coherent
state have excitation number NL equal to nw, where n is the momentum and w is the wind-
ing. Applying a map between FP oscillators and R operators then yields the superposition
of R ground states that is proposed to be dual to the D1-D5 geometry.
This proposal can be straightforwardly extended to the new geometries, which are char-
acterized by the curve F i(v) along with h1,1 additional functions (F(v),Fα− (v)). Consider
first the T 4 system, for which the four additional functions are F ρ(v). Then the eight
functions F I(v) ≡ (F i(v), F ρ(v)) can be expanded in harmonics as
F I(v) =
(αIne
−inσ+ + (αIn)
∗einσ
), (6.5)
where σ+ = v/wR9. The corresponding coherent state in the FP system is
, (6.6)
where
is a coherent state of the left moving oscillator âIn, satisfying â
= αIn
Contained in this coherent state are Fock states, such that
(âInI )
mI |0〉 , N =
nImI . (6.7)
Now retain only the terms in the coherent state involving these Fock states, and map the
FP oscillators to CFT R operators via the dictionary
(â1n ± iâ2n) ↔ OR(±1+1),(±1+1)n ; (6.8)
(â3n ± iâ4n) ↔ OR(±1+1),(∓1+1)n ;
âρn ↔ O
R(1,1)
(ρ−4)n.
The dictionary for the case of K3 is analogous. Here one has four curves F i(v) describing
the transverse oscillations and twenty curves F α̃(v) describing the internal excitations. The
oscillators associated with the former are mapped to operators associated with the universal
cohomology as in (6.8) whilst the oscillators associated with the latter are mapped to
operators associated with the (1, 1) cohomology as
âα̃n ↔ O
R(1,1)
α̃n . (6.9)
This completely defines the proposed superposition of R ground states to which a given
geometry corresponds. Note that below we will suggest that a slight refinement of this
dictionary may be necessary, taking into account that one of the internal curves is dis-
tinguished by the duality chain. For the distinguished curve the mapping may include a
negative sign, namely ân ↔ −OR(1,1)n ; this mapping would explain the relative sign between
the vevs found in (5.28) associated with the distinguished curve F and the remaining curves
Fα respectively.
Note that there is a direct correspondence between the frequency of the harmonic on
the curve and the twist label of the CFT operator. The latter is strictly positive, n ≥ 1,
and thus in the dictionary (6.8) there are no candidate CFT operators to correspond to
winding modes of the curves (F(v),Fα− (v)). In the case of T 4 such candidates might be
provided by the additional chiral primaries associated with the extra T 4 in the target space
of the sigma model, discussed in [34]. However the latter is related to the degeneracy of
the right-moving ground states in the dual F1-P system, rather than to winding modes.
For K3 all chiral primaries have been included (except for the additional primaries which
appear at specific points in the K3 moduli space). Thus one confirms that winding modes
of the curves (F(v),Fα− (v)) should not be included in constructing geometries dual to the
R ground states. As discussed in appendix D these winding modes may describe geometric
duals of states in deformations of the CFT.
6.3 Matching with the holographic vevs
In this section we will see how the general structure of the vevs given in (5.28) can be
reproduced using the proposed dictionary. The holographic vevs take the form
O(1,1)α̃kI
dvḞ α̃CIi1···ikF
i1 · · ·F ik . (6.10)
Thus the vevs of the operators O(1,1)α̃kI are zero unless the curve F α̃(v) is non-vanishing and
at least one of the F i(v) is non-vanishing. Moreover, the dimension one operators will
not acquire a vev unless the transverse and internal curves have excitations with the same
frequency. Analogous selection rules for frequencies of curve harmonics apply for the vevs
of higher dimension operators.
These properties of the vevs follow directly from the proposed superpositions, along
with selection rules for three point functions of chiral primaries. The superposition dual to
a given set of curves is built from the R ground states
ORI |0〉 =
(OR(pl,qq)nl )ml |0〉, (6.11)
l nlml = N and I labeling the degeneracy of the ground states. So this superposition
can be denoted abstractly as |Ψ) =
I aIORI |0〉 with certain coefficients aI . In particular,
if the curve F α̃(v) = 0 the superposition does not contain any R ground states built from
OR(1,1)α̃n operators. Moreover, if there are no transverse excitations, the superposition will
contain only states with zero R charge.
Now consider evaluating the vev of a dimension k operator O(1,1)α̃k in such a superposition.
This is determined by three point functions between this operator and the chiral primary
operators occurring in the superposition. More explicitly, the operator vev is related to
three point functions via
(ΨNS |O(1,1)α̃k |ΨNS) =
a∗IaJ 〈(OI)†(∞)O
(1,1)
α̃k (µ)(O
J )(0)〉. (6.12)
Here OI is the NS sector operator which flows to ORI in the R sector and |ΨNS) is the flow
of the superposition back to the NS sector, namely
I aIOI |0〉. The quantity µ is a mass
scale. Note we are evaluating the relevant three point function in the NS sector, and have
hence flowed the ground states back to NS sector chiral primaries. We would get the same
answer by flowing the operator whose vev we wish to compute, O(1,1)
, into the Ramond
sector and computing the three point function there. Recall that the R charges of these
operators are related by the spectral flow formula (6.3) as jNS3 = j
N . In particular,
NS sector chiral primaries built only from operators associated with the middle cohomology
all have the same R charges, namely 1
There are two basic selection rules for the three point functions (6.12). Firstly, as usual
one has to impose conservation of the R charges. Secondly, a basic property of such three
point functions is that they are only non-zero when the total number of operators O(1,1)α̃ with
a given index α̃ in the correlation function is even 4. From a supergravity perspective one can
see this selection rule arising as follows. One computes n-point correlation functions using
n-point couplings in the three dimensional supergravity action, with the latter following
from the reduction of the ten-dimensional action on S3×M4. Since a (1, 1) form integrates
4Note that this selection rule was used for the computation of three point functions of single particle
operators in the orbifold CFT in [35].
to zero over M4, the three dimensional action only contains terms with an even number of
fields sα̃ associated with a given (1, 1) cycle α̃ on M4. Therefore non-zero n-point functions
must contain an even number of operators O(1,1)α̃ , and so do corresponding multi-particle
3-point functions obtained by taking coincident limits.
Expressed in terms of cohomology, allowed three point functions contain an even number
of (1, 1)α̃ cycles labeled by α̃. Thus in single particle correlators one can have processes
such as O(0,0) +O(1,1)α̃ → O
(1,1)
α̃ and O
(1,1)
α̃ +O
(1,1)
α̃ → O(2,2), but processes such as O(0,0) +
O(1,1)α̃ → O(0,0) which involve an odd number of α̃ cycles are kinematically forbidden.
This kinematical selection rule for (1, 1) cycle conservation immediately explains why the
operator O(1,1)α̃k can only acquire a vev when the curve F α̃(v) is non-vanishing: only then
does the ground state superposition contain operators OR(1,1)α̃ such that the selection rule
can be satisfied.
One can also easily see why the operator only acquires a vev if there are transverse
excitations as well. All Ramond ground states associated with the middle cohomology have
zero R charge, with the corresponding chiral primaries in the NS sector having the same
charge jNS3 =
N . Thus a superposition involving only O(1,1) operators has a definite R
charge, and a charged operator cannot acquire a vev. Including transverse excitations means
that the superposition of Ramond ground states contains charged operators, associated with
the universal cohomology, and does not have definite R charge. Therefore a charged operator
can acquire a vev.
Thus, to summarize, the proposed map between curves and superpositions of R ground
states indeed reproduces the principal features of the holographic vevs. Using basic selection
rules for three point functions we have explained why the operators O(1,1)
acquire vevs only
when the curve F α̃(v) is non-zero and when there are excitations in R4. We will see below
that using reasonable assumptions for the three point functions we can also reproduce the
selection rules for vevs relating to frequencies on the curves. Before discussing the general
case, however, it will be instructive to consider a particular example.
6.4 A simple example
Consider a fuzzball geometry characterized by a circular curve in the transverse R4 and one
additional internal curve, with only one harmonic of the same frequency:
F 1(v) =
cos(2πn
); F 2(v) =
sin(2πn
); F(v) = µB
cos(2πn
), (6.13)
where µ =
Q1Q5/R and the D1-brane charge constraint (5.16) enforces
(A2 + 1
B2) = 1. (6.14)
The corresponding dual superposition of R ground states is then given by
|Ψ) =
Cl(OR(2,2)n )l(O
R(1,1)
−l |0〉 , (6.15)
− l)!l!
with the operators being orthonormal in the large N limit. In the case that either A or B
are zero the superposition manifestly collapses to a single term. In the general case, this
superposition gives the following for the expectation values of the R charges:
Ψ|jR3 |Ψ
Ψ|j̄R3 |Ψ
C2l l; (6.16)
N/n−1
− 1)!
− (l + 1))!
A2(l+1)(
−(l+1)) =
Evaluating (5.26) for (6.13) gives
A2(dy ± dt), (6.17)
and thus the integrated R charges defined in our conventions as
〈j3〉 =
; 〈j̄3〉 =
, (6.18)
agree with those of the superposition of R ground states.
The kinematical properties also match between the geometry and the proposed super-
position. In particular, when B 6= 0 the SO(2) symmetry in the 1-2 plane is broken: the
harmonic functions (K,A) depend explicitly on the angle φ in this plane. The asymptotic
expansions of these functions involve charged harmonics, and therefore charged operators
acquire vevs characterizing the symmetry breaking. More explicitly, the relevant terms in
(5.20) are
f1kI ∝
dv(A2 +B2 sin2(
))CIi1···ikF
i1 · · ·F ik ; (6.19)
AkI ∝
dvB sin(
))CIi1···ikF
i1 · · ·F ik .
Now the symmetric tensor of rank k and SO(2) charge in the 1-2 plane of ±m behaves as
((F 1)2 + (F 2)2)k−m(F 1 ± iF 2)m = (µA
)ke±2πinm
L . (6.20)
Note that m is related to (j3, j̄3) via m = j3 + j̄3. Thus, when B 6= 0, harmonics in the
expansion of f1 with charges |m| = 2 are excited, and terms with |m| = 1 are excited in the
expansion of A. Following (6.10) the latter implies that the dimension k operators O(1,1)
1(km)
only acquire vevs when their SO(2) charge m in the 1-2 plane is ±1. In particular using
(5.28) the vevs of the dimension one operators are
〈O(1,1)
1(1±1)〉 = ∓i
µAB, (6.21)
where the normalized degree one symmetric traceless tensors are
2(F 1 ± iF 2).
These properties are implied by the superposition (6.15). The latter is a superposition
of states with different R charge, and therefore it does break the SO(2) symmetry, with
the symmetry breaking being characterized by the vevs of charged operators. Moreover
following (6.12) the vev of O(1,1)
1(km)
is given by
C∗l Cl′〈(O(2,2)n )l(O
(1,1)
−l|O(1,1)
1(km)
(µ)|(O(2,2)n )l
(O(1,1)1n )
−l′〉. (6.22)
For the dimension one operators, charge conservation reduces this to
C∗l±1Cl〈(O(2,2)n )l±1(O
(1,1)
∓1−l|O(1,1)
1(1±1)(µ)|(O
(2,2)
l(O(1,1)1n )
−l〉. (6.23)
Thus there are contributions only from neighboring terms in the superposition. Computing
the actual values of these vevs is beyond current technology: one would need to know three
point functions for single and multiple particle chiral primaries at the conformal point.
However, as in [12], the behavior of the vevs as functions of the curve radii (A,B) can be
captured by remarkably simple approximations for the correlators, motivated by harmonic
oscillators. Suppose one treats the operators as harmonic oscillators, with the operator
O(1,1)
1(11)
destroying one O(1,1)1n and creating one O
(2,2)
n . For harmonic oscillators such that
[â, â†] = 1 the normalized state with p quanta is given by |p〉 = (â†)p/
p!|0〉 and therefore
â†|p〉 =
p+ 1|p+ 1〉. Using harmonic oscillator algebra for the operators gives
〈(O(2,2)n )l+1(O
(1,1)
−1−l|O(1,1)
1(11)
(µ)|(O(2,2)n )l(O
(1,1)
−l〉 ≈ µ
− l)(l + 1). (6.24)
Then the corresponding vev in the superposition |Ψ) is
〈O(1,1)
1(11)
〉Ψ = µ
N/n−1
c∗l+1cl
− l)(l + 1) = µN
AB, (6.25)
which has exactly the structure of (6.21). Given that such simple approximations (and
factorizations) of the correlators reproduce the structure of the vevs so well, it would be
interesting to explore whether this relates to simplifications in the structure of the chiral
ring in the large N limit.
Next consider the vevs of dimension k operators. Using charge conservation and (1, 1)
cycle conservation in (6.22) implies that only operators with m odd can acquire a vev. To
reproduce the holographic result, that vevs are non-zero only when m = ±1, requires the
assumption that only nearest neighbor terms in the superposition contribute to one point
functions. This would follow from a stronger selection rule for (1, 1) cycle conservation, that
the number of (1, 1) cycles in the in and out states differ by at most one. In particular,
multi-particle processes such as (O(1,1)ãn )3 + O
(1,1)
α̃n → (O
(2,2)
3 would be forbidden. The
selection rules for holographic vevs suggest that there is indeed such cycle conservation,
and it would be interesting to explore this issue further.
Let us now return to the comment made below (6.9), that one may need to include a
minus sign in the dictionary for the distinguished curve. Such a minus sign would intro-
duce factors of (−1)N/n−l into the superposition (6.15), and thence an overall sign in the
vevs of the associated operators O(1,1)
1(kI)
. This would naturally account for the relative sign
difference between the vevs associated with the distinguished curve and those associated
with the remaining curves. It is not conclusive that one needs such a minus sign without
knowing the exact three point functions and hence vevs. However such a sign change for
oscillators associated with the direction distinguished by the duality would not be surpris-
ing. Recall that under T-duality of closed strings right moving oscillators associated with
the duality direction switch sign, whilst the left moving oscillators and oscillators associated
with orthogonal directions do not.
6.5 Selection rules for curve frequencies
Selection rules for charge and (1, 1) cycles are sufficient to reproduce the general structure
of the vevs. In the particular example discussed above, these rules also implied the selection
rules for the curve frequencies: operators acquire vevs only when the transverse and internal
curves have related frequencies.
Here we will note how, with reasonable assumptions, one can motivate the selection rules
for frequencies in the general case. Consider the computation of the vev of a dimension one
operator O(1,1)α̃1 for a general superposition |Ψ) using (6.12). Using the selection rules for
charge and (1, 1) cycles, the contributions to (6.12) involve only certain pairs of operators
(OI ,OJ ). Their SO(2) charges must differ by (±1/2,±1/2) and they must differ by an
odd number of O(1,1)α̃ operators.
Now let us make the further assumption that there are contributions to (6.12) only from
pairs of operators (OI ,OJ ) which differ by only one term, the relevant operators taking
the form
OJ = O(p,q)n OJ̃ , (6.26)
with OJ̃ being the same for in and out states, but the single operator O(p,q)n differing
between in and out states. Thus we are assuming that the relevant three point functions
factorize, with the non-trivial part of the correlator arising from a single particle process.
This is indeed the structure of the three point functions arising in our example. Only
nearest neighbor terms in the superposition contribute in the computation of the vev of the
dimension one operator in (6.23). Moreover the m = ±1 charge selection rule for the vevs of
higher dimension operators immediately follows from restricting to nearest neighbor terms
in the three-point functions. Note further that this factorization structure is present in the
orbifold CFT computation of the three point functions. The operator O(1,1)α̃1 ≡ O
(1,1)
α̃1 I
is the identity operator in (N −1) copies of the CFT and thus only acts non-trivially in one
copy of the CFT.
Consider the case of the vev of the operator with SO(2) charges (1/2, 1/2); it would
take the form
I,J ,Ĩ
a∗IaJNĨ
〈(O(2,2)n )†(∞)O
(1,1)
α̃1 (µ)(O
(1,1)
α̃n )(0)〉 (6.27)
+〈(O(1,1)n )†(∞)O
(1,1)
α̃1 (µ)(O
(0,0)
α̃n )(0)〉
where NĨ is the norm of OĨ . Analogous expressions would hold for the dimension one
operators with other charge assignments. Such a factorization would immediately explain
the frequency selection rule found in the holographic vevs obtained from supergravity (6.10).
The superposition contains operators of the form (6.26) with both (p, q) = (1, 1) and (p, q) 6=
(1, 1) only when the internal curve and the transverse curves share a frequency. Extending
these arguments to vevs of higher dimension operators would be straightforward, and would
imply selection rules for curve frequencies.
6.6 Fuzzballs with no transverse excitations
Consider the case where the fuzzball geometry has only internal excitations, F i(v) = 0.
Then the corresponding dual superposition of ground states can involve only states built
from the operators OR(1,1)αn . Any such state will be a zero eigenstate of both jR3 and j̄R3 .
Furthermore, such ground states associated with the middle cohomology account for a finite
fraction of the entropy of the D1-D5 system. In the case of K3 the total entropy behaves
S = 2π
, (6.28)
with c = 24N . The ground states associated with the middle cohomology account for a
central charge c = 20N . In the case of T 4 the entropy behaves as (6.28) with c = 12N . The
states associated with the universal cohomology account for c = 4N , the odd cohomology
accounts for another c = 4N and the middle cohomology accounts for the final c = 4N .
Now let us consider the properties of the corresponding fuzzball geometry. When there
are no transverse excitations and no winding modes of the internal curves, the SO(4)
symmetry in R4 is unbroken, and the defining harmonic functions (4.4) reduce to
H = 1 +
; K =
; (6.29)
with Ai = 0 and where Q1 is defined in (5.16). The solutions manifestly all collapse to the
standard (singular) D1-D5 solution and so, whilst one would need an exponential number
of geometries (upon quantization) to account for dual ground states build from operators
associated with the middle cohomology, one has only one singular geometry. Therefore the
relevant fuzzball solutions are not visible in supergravity: one needs to take into account
higher order corrections.
One can understand this from several perspectives. Firstly, as discussed above, R ground
states associated with the middle cohomology have zero R charge; they do not break the
SO(4) symmetry. A geometry which is asymptotically AdS3 × S3 for which the SO(4)
symmetry is exact can be characterized by the vevs of SO(4) singlet operators. The only
such operators in supergravity are the stress energy tensor, and the scalar operators listed
in (5.13). Since the vev of the stress energy tensor must be zero for the D1-D5 ground
states, the geometry would have to be distinguished by the vevs of the singlet operators
given in (5.13).
Our results imply that these operators do not acquire vevs, and therefore within su-
pergravity (without higher order corrections) geometries dual to different R ground states
associated with the middle cohomology cannot be distinguished. The reason is the follow-
ing. The SO(4) singlet operators dual to supergravity fields are related to chiral primaries
by the action of supercharge raising operators; they are the top components of the multi-
plets. Thus these SO(4) singlet operators cannot acquire vevs in states built from the chiral
primaries. SO(4) singlet operators associated with stringy excitations would be needed to
characterize the different ground states.
A heuristic argument based on the supertube picture also indicates that geometries
dual to these ground states are not to be found in the supergravity approximation. The
geometries with transverse excitations in R4 can be viewed as a bound state of D1-D5
branes, blown up by their angular momentum in the R4. Indeed, the characteristic size of
the fuzzball geometry is directly related to this angular momentum. The simplest example,
related to a circular supertube, is to take a geometry characterized by a circular curve; this
is obtained by setting B = 0 in (6.13). The characteristic scale of the geometry is
rc ∼ gsµ/n, (6.30)
where gs is the string coupling and µ has dimensions of length, whilst the (dimensionless)
angular momentum behaves as j12 = N/n, and thus rc ∼ gsµ(j12/N). Hence the size of
the D1-D5 bound state increases linearly with the angular momentum. A general fuzzball
geometry will of course not be as symmetric but nonetheless the characteristic scale averaged
over the R4 is still related to the total angular momentum. In our previous paper [12] we
noted that fuzzball geometries dual to vacua for which the R charge is very small are not
well described by supergravity. Here we have found that this implies that an exponential
number of geometries dual to a finite fraction of the Ramond ground states, with strictly
zero R charge, cannot be described at all in the supergravity approximation.
7 Implications for the fuzzball program
In this section we will consider the implications of our results for the fuzzball program,
focusing in particular on whether one can find a set of smooth weakly curved supergravity
geometries which span the black hole microstates.
We have seen in the previous sections that the geometric duals of superpositions of R
vacua with small or zero R charge are not well-described in supergravity. The natural basis
for R ground states (6.4) uses states of definite R-charges, and it is therefore straightforward
to work out the density of ground states with given R-charges, dN,j3,j̄3 , with the total number
of ground states being given by dN =
N,j3,j̄3
dN,j3,j̄3 . This computation is discussed in
appendix E with the resulting density in the large N limit being
dN,j1,j2
4(N + 1− j)31/4
2π(2N − j)√
N + 1− j
cosh2(
N+1−j ) cosh
N+1−j )
, (7.1)
where j1 = (j3 + j̄3) and j
2 = (j3 − j̄3) and j = |j1| + |j2|. The key feature is that the
number of states with zero R charge differs from the total number of R ground states given
in (E.16) only by a polynomial factor:
dN,0,0 ∼= dN/N. (7.2)
The geometries dual to such ground states are unlikely to be well-described in supergravity,
and therefore the basis of black hole microstates labeled by R charges is not a good basis
for the geometric duals. This argument reinforces the discussion of [12], where we showed
in detail that the geometric duals of specific states (in this basis) must be characterized by
very small vevs which cannot be reliably distinguished in supergravity; they are comparable
in magnitude to higher order corrections.
The geometries that are smooth in supergravity correspond to specific superpositions
of the R charge eigenstates, for which some vevs are atypically large. The natural basis
for the field theory description of the microstates is thus not the natural basis for the
geometric duals. This issue is likely to persist in other black hole systems. For example, the
microstates of the D1-D5-P system are also most naturally described as (j3, j̄3) eigenstates,
with a relation analogous to (7.2) holding, so the number of states with zero R-charge is
suppressed only polynomially compared to the total number of black hole microstates. Just
as in the 2-charge system discussed here, the geometric duals are related to supertubes
whose radii depend on the R-charges. States or superpositions of states which have small
or zero R-charges are unlikely to be well-described by supergravity solutions. Thus a given
smooth supergravity geometry should be described by a specific superposition of the black
hole microstates. Identifying the specific superpositions for known 3-charge geometries is
an open and important question.
The issue is whether there exist enough such geometries, well-described and distinguish-
able in supergravity, to span the entire set of black hole microstates. It seems unlikely
that a basis exists which simultaneously satisfies all three requirements. Firstly, on general
grounds microstates with small quantum numbers will not be well-described in supergrav-
ity. Even when considering superpositions that are well described by supergravity, to span
the entire basis, one will have to include superpositions which can only be distinguished by
these small vevs. I.e. in choosing a basis of geometries for which some vevs are sufficiently
large for the supergravity description to be valid one will find that some of these geometries
cannot be distinguished among themselves in supergravity.
We have already seen several examples of this problem in the 2-charge system. Let us
parameterize the curves as
F i(v) = µ
(αine
2πinv/L + (αin)
∗e−2πinv/L); (7.3)
F β̃(v) = µ
(αβ̃ne
2πinv/L + (αβ̃n)
∗e−2πinv/L),
where µ =
Q1Q5/R and β̃ runs from 1 to h
1,1(M4). The D1-brane charge constraint (4.7)
limits the total amplitude of these curves as
) = 1. (7.4)
Thus in general increasing the amplitude in one mode, to make certain quantum numbers
large, decreases the amplitudes in the others. Moreover, the amplitude in a given mode is
bounded via |αn|2 ≤ 1/n2, and is thus is intrinsically very small for high frequency modes,
which sample vacua with large twist labels in the CFT. Note also that the vevs of R-charges
are given in terms of
jij = iN
n(αin(α
∗ − αjn(αin)∗) (7.5)
As we have seen, to be describable in supergravity, geometries must have transverse R4
excitations, and thus some large R-charges, requiring jij ≫ 1. Combining (7.5) and (7.4) one
sees that this restricts the amplitudes of the internal excitations, and thus of the sampling
of the black hole microstates associated with the middle cohomology of M4.
Another way to understand the limitations of supergravity is to go back to the F1-P
system where the corresponding state is the coherent state |{αin}, {α
m}). These states form
a complete basis of states, so we know that there is an F1-P geometry corresponding to
every 1/2 BPS state. However, only when all αin, α
m are large are the geometries well-
described and distinguishable within supergravity. Indeed, the amplitudes αin, α
m are also
the root mean deviations of the distribution around the mean (which is described by the
classical curve), so only for large αin, α
m is the classical string that sources the supergravity
solution a good approximation of the quantum state. Putting it differently, when some of the
amplitudes are small the difference in the solutions for different amplitudes is comparable
with the error in the solutions due to the approximation of the source by a classical string,
so one cannot reliably distinguish them within this approximation.
If one could not find a basis of distinguishable supergravity geometries spanning the mi-
crostates, one might ask whether a sufficiently representative basis exists. That is, suppose
one chooses a single representative of the indistinguishable geometries, and assigns a mea-
sure to this geometry. Then is the corresponding basis of weighted geometries sufficiently
representative to obtain the black hole properties? In the 2-charge system, the now com-
plete set of fuzzball geometries along with the precise mapping between these geometries
and R vacua allows these questions to be addressed at a quantitative level and we hope to
return to this issue elsewhere.
Acknowledgments
The authors are supported by NWO, KS via the Vernieuwingsimpuls grant “Quantum
gravity and particle physics” and IK, MMT via the Vidi grant “Holography, duality and
time dependence in string theory”. This work was also supported in part by the EU contract
MRTN-CT-2004-512194. KS and MMT would like to thank both the 2006 SimonsWorkshop
and the theoretical physics group at the University of Crete, where some of this work was
completed. The computer algebra package GRTensor was used to verify that our solutions
satisfy the supergravity field equations.
A Conventions
The following table summarises the indices used throughout the paper. In some cases an
index is used more than once, with different meanings, in separate sections of the paper.
Index Range Usage
(m,n) 0, · · · , 9 10d sugra fields
(M,N) 0, · · · , 5 6d sugra fields
(µ, ν) 0, 1, 2 3d fields
(a, b) 1, 2, 3 S3 indices
(i, j) 1, 2, 3, 4 R4 indices
(ρ, σ) 1, 2, 3, 4 M4 indices
(µ̄, ν̄) 0, 1 2d fields
(α, β) 1, 2, 3 SU(2) vector index
(γ, δ) 1, · · · , b2 H2(M4)
(α̃, β̃) 1, · · · , h1,1 H1,1(M4)
(I, J) 1, · · · , 8 SO(8) vector
((c), (d)) 1, · · · , 16 heterotic vector fields
((a), (b)) 1, · · · , 24 SO(4, 20) vector
(A,B) 1, · · · , 26 SO(5, 21) vector
(m,n) 1, · · · , 5 SO(5) vector
(r, s) 6, · · · , (nt + 1) SO(nt) vector
A.1 Field equations
The equations of motion for IIA supergravity are:
e−2Φ(Rmn + 2∇m∇nΦ−
H(3)mpqH
(3)pq
F (2)mpF
2 · 3!F
mpqrF
(4)pqr
(F (2))2 +
(F (4))2) = 0, (A.1)
4∇2Φ− 4(∇Φ)2 +R− 1
(H(3))2 = 0,
dH(3) = 0, dF (2) = 0, ∇mF (2)mn −
H(3)pqrF
(4)npqr = 0,
∇m(e−2ΦH(3)mnp)−
F (2)qr F
(4)qrnp − 1
2 · (4!)2 ǫ
npm1···m4n1···n4F
m1···m4F
n1···n4 = 0,
dF (4) = H(3) ∧ F (2), ∇mF (4)mnpq −
3! · 4!ǫ
npqm1···m3n1···n4H
m1···m3F
n1···n4 = 0.
The corresponding equations for type IIB are:
e−2Φ(Rmn + 2∇m∇nΦ−
H(3)mpqH
(3)pq
F (1)m F
F (3)mpqF
(3)pq
4 · 4!F
mpqrsFn
(5)pqrs
Gmn((F
(1))2 +
(F (3))2) = 0,
4∇2Φ− 4(∇Φ)2 +R− 1
(H(3))2 = 0,
dH(3) = 0, ∇m(e−2ΦH(3)mnp)− F (1)m F (3)mnp −
F (3)mqrF
(5)mqrnp = 0, (A.2)
dF (1) = 0, ∇mF (1)m +
H(3)pqrF
(3)pqr = 0,
dF (3) = H(3) ∧ F (1), ∇mF (3)mnp +
H(3)mqrF
(5)mqrnp = 0,
dF (5) = d(∗F (5)) = H(3) ∧ F (3),
where the Hodge dual of a p-form ωp in d dimensions is given by
(∗ωp)i1···id−p =
ǫi1···id−pj1···jpω
j1···jp
p , (A.3)
with ǫ01···d−1 =
√−g. The RR field strengths are defined as
F (p+1) = dC(p) −H(3) ∧C(p−2). (A.4)
The equations of motion for the heterotic theory are:
4∇2Φ− 4 (∇Φ)2 +R− 1
(H(3))2 − α′(F (c))2 = 0,
e−2ΦH(3)mnr
Rmn + 2∇m∇nΦ− 1
H(3)mrsH(3)nrs − 2α′F (c)mrF (c)nr = 0,
e−2ΦF (c)mn
e−2ΦH(3)nrsF (c)rs = 0.
mn with (c) = 1, · · · 16 are the field strengths of Abelian gauge fields V (c)m ; we consider
here only supergravity backgrounds with Abelian gauge fields. This restriction means that
the gauge field part of the Chern-Simons form in H3,
H(3) = dB(2) − 2α′ω3(V ) + · · · , (A.5)
does not play a role in the supergravity solutions, nor does the Lorentz Chern-Simons term
denoted by the ellipses.
A.2 Duality rules
The T-duality rules for RR fields were derived in [36] by reducing type IIA and type IIB
supergravities on a circle and relating the respective RR potentials in the 9-dimensional
theory. However, for calculations involving magnetic sources, it is more convenient to work
with T-duality rules for RR field strengths, since potentials can only be defined locally. In
the following we will rederive the T-duality rules in terms of RR field strengths.
It is slightly easier although not necessary to use the democratic formalism of IIA and
IIB supergravity introduced in [16]. In this formalism one includes p-form field strengths
for p > 5 with Hodge dualities relating higher and lower-form field strengths being imposed
in the field equations. This formalism is natural when both magnetic and electric sources
are present; moreover there is no need for Chern-Simons terms in the field equations. The
RR part of the (pseudo)-action is simply
SRR = −
2κ210
(F (q))2, (A.6)
where q = 2, 4, 6, 8 is even in the IIA case and q = 1, 3, 5, 7, 9 is odd in the IIB case. The
field strengths are defined as F (q) = dC(q−1) − H(3) ∧ C(q−3) for q ≥ 3 and Fq = dC(q−1)
for q < 3. The Hodge duality relation between higher and lower form field strengths in our
conventions is
∗F (q) = (−1)⌊
⌋F (10−q), (A.7)
where ⌊n⌋ denotes the largest integer less or equal to n.
Now to compactify on a circle the ten-dimensional metric can be parameterized as
ds2 = e2ψ(dy −Aµdxµ)2 + ĝµνdxµdxν , (A.8)
where y denotes the compact direction, and 9-dimensional quantities will be denoted as
hatted. An economic way to derive the T-duality rules for the field strengths is the following.
Choose the vielbein to be
ey = eψ(dy −Aµdxµ); eµ = êµ, (A.9)
where µ denotes a tangent space index, and êµ is the 9-dimensional vielbein. Now reduce
the field strengths (in the tangent frame) as
F̂ (q)µ
= F (q)µ
, F̂ (q−1)µ
= F (q)µ
y. (A.10)
The corresponding 9-dimensional action for the field strengths is given by
SRR = −
2κ210
eψF̂ 2q . (A.11)
Since ψIIA = −ψIIB under T-duality, one can read from this action the transformation
rules for field strengths in 10d:
(q+1)
y = e
, (A.12)
F̃ (q+1)µ
= eψF (q+2)µ
Here q even defines IIB fields in terms of IIA fields and q odd defines IIA in terms of IIB.
Note that the field strengths on both sides are in the tangent frame. Given the T-duality
rules for NSNS fields
eψ̃ = e−ψ, Ãµ = B
yµ , B̃
ym = Am, (A.13)
B̃(2)mn = B
mn + 2A[mB
, Φ̃ = Φ− ψ,
with the metric gmn invariant, one can easily convert (A.12) back into
F (q)m1...mq = F
(q+1)
m1...mqy
− q(−1)qB(2)
(q−1)
m2...mq]
+ q(q − 1)B(2)
(q−1)
m3...mq]y
F (q)m1...mq−1y = F
(q−1)
m1...mq−1
− (q − 1)(−1)qA[m1F
(q−1)
m2...mq−1]y
. (A.14)
Strictly speaking, this gives the duality rules in the democratic formalism. However we can
obtain the usual rules by simply dropping the (p > 5)-form field strengths as long as we
make sure to self-dualise F (5) in each IIB solution.
The S duality rules for type IIB are
τ̃ = −1
, B̃(2) = C(2), C̃(2) = −B(2),
F̃ (5) = F (5), G̃mn = |τ |Gmn, (A.15)
where τ = C(0) + ie−Φ.
B Reduction of type IIB solutions on K3
The reduction of type IIB on K3 is very similar to the reduction of type IIA, which was
discussed in some detail in [37]. In the following we will use the reduction of the NS-NS
sector fields given in [37], and derive the reduction of the type IIB RR fields. Let us first
review the reduction of the NS-NS sector. Starting from the ten-dimensional action
SNS =
2κ210
e−2Φ̂(R̂+ 4(∂Φ̂)2 − 1
Ĥ23 )
, (B.1)
where ten-dimensional fields are denoted by hats, the corresponding six-dimensional field
equations can be derived from the action [37]
−ge−2Φ
R+ 4(∂Φ)2 − 1
H23 +
tr(∂M−1∂M)
, (B.2)
where the six-dimensional fields are defined as follows. Firstly the 10-dimensional 2-form
potential is reduced as
B̂(2)(x, y) = B2(x) + b
γ(x)ω
2 (y), (B.3)
where (x, y) are six-dimensional and K3 coordinates respectively and the two forms ω
2 with
γ = 1, · · · 22 span the cohomology H2(K3,R). The 2-forms ωγ2 transform under an O(3, 19)
symmetry, with a metric defined by the 22-dimensional intersection matrix
dγδ =
(2π)4V
2 ∧ ω
2, (B.4)
where (2π)4V is the volume of K3. A natural choice for dγδ is
dγδ =
0 −I19
 , (B.5)
corresponding to a diagonal basis for the 3 self-dual and 19 anti-self dual two forms of K3.
Furthermore, there is a matrix Dδ γ defined by the action of the Hodge operator
∗K34 ω
2 = ω
γ , (B.6)
which is dependent on the K3 metric and satisfies
ǫ = δ
δdǫζD
γ = dδζ . (B.7)
The SO(4, 20) matrix of scalars M−1
(a)(b)
was derived in [37] to be
M−1 = ΩT2
e−ρ + bγbδdγǫD
eρb4 1
eρb2 1
eρb2bγdγδ + b
γdγǫD
eρb2 eρ eρbγdγδ
eρb2bγdγδ + b
γdγǫD
ρbγdγδ e
ρbǫdǫγb
ζdζδ + dγǫD
Ω2, (B.8)
with b2 ≡ bγbδdγδ. Here ρ is the breathing mode of K3, e−ρ = 1(2π)4V
∗41. The six-
dimensional dilaton is related to the 10-dimensional dilaton via Φ = Φ̂ + ρ/2.
The dimensional reduction of the NS sector makes manifest only an SO(4, 20) subgroup
of the full SO(5, 21) symmetry. Including the reduction of the RR sector should thus give
the equations of motion following from the six-dimensional string frame action, which for
IIB was given in (3.23)
R+ 4(∂Φ)2 +
tr(∂M−1∂M)
∂l(a)M−1
(a)(b)
∂l(b)
GAMNPM−1ABG
and in which only an SO(4, 20) subgroup of the total SO(5, 21) symmetry is manifest; recall
that M−1AB here is an SO(5, 21) matrix, with M
(a)(b)
being SO(4, 20). Note that the six-
dimensional coupling is related to the ten-dimensional coupling via (2π)4V (2κ26) = 2κ
where (2π)4V is the volume of K3.
Following the same steps as [37] the RR potentials can be reduced as
Ĉ(0)(x, y) = C0(x), Ĉ
(2)(x, y) = C2(x) + c
(0,2)
2 (y), (B.9)
Ĉ(4)(x, y) = C4(x) + c
(2,4)
(x) ∧ ωγ2 (y) + c(0,4)(x)(e
ρ ∗K3 1)(y),
where ∗K3 denotes the Hodge dual in the K3 metric and the corresponding field strengths
F̂ (1)(x, y) = F1(x), (B.10)
F̂ (3)(x, y) = dC2(x)− C0(x)H3(x) +
(0,2)
(x)− C0(x)dbγ(x)
ω2(y) ≡ F3 +Kγ1 ∧ ω
Ĥ(3)(x, y) = dB2(x) + db
γ(x) ∧ ωγ2 (y) ≡ H3 + db
γ ∧ ωγ2 ,
F̂ (5)(x, y) = dC4(x)− C2(x) ∧H3(x) +
(2,4)
(x)− C2(x)dbγ(x)− cγ(0,2)(x)H3(x)
∧ ωγ2 (y)
dc(0,4)(x)− cγ0,2(x)dbδ(x)dγδ
∧ (eρ(x) ∗K3 1)(y)
≡ F5 +Kγ3 ∧ ω
2 + F̃1 ∧ eρ ∗K3 1.
The reduction of the potentials thus gives two three form field strengths H3 and F3, 3 self-
dual and 19 anti-self dual three form field strengths K
3 and 46 scalars b
γ , c
(0,2)
, c(0,4) and
C0. After splitting the three forms H3 and F3 into their self-dual and anti-self-dual parts,
we obtain 5 self-dual and 21 anti-self dual tensors in total, as described in [38].
It is then straightforward to obtain the map relating six and ten-dimensional fields by
inserting the expressions (B.9) and (B.10) into the ten-dimensional field equations (A.2).
The additional RR scalars are contained in
l(a) = ΩT2
c̃(0,4)
(0,2)
, (B.11)
with Ω2 defined in the appendix B.2 and the shifted fields defined as
(0,2)
(0,2)
− C0bγ , (B.12)
c̃(0,4) = c(0,4) − bγcδ(0,2)dγδ +
b2C0.
The fields Φ, l(a) and the SO(4, 20) matrix M−1 given in (B.8) can be recombined into
the SO(5, 21) matrix M−1 = V TV , with the latter conveniently expressed in terms of the
vielbein
V = ΩT4
e−Φ 0 0 0 0
−eΦ(C0c(0,4) − 12c
(0,2)
) eΦ −eΦc̃(0,4) −eΦC0 eΦc̃γ(0,2)dγδ
e−ρ/2C0 0 e
−ρ/2 0 0
eρ/2c(0,4) 0
eρ/2b2 eρ/2 eρ/2bγdγδ
Ṽδγc
(0,2)
0 Ṽδγb
γ 0 Ṽγδ
Ω4. (B.13)
Here the SO(3, 19) vielbein Ṽαβ is defined by dαβD
γ = ṼαβṼβγ , c
(0,2)
(0,2)
(0,2)
dγδ and
the matrix Ω4 is defined in the appendix B.2. The six-dimensional tensor fields are related
to the ten-dimensional fields as
H13 =
(1 + ∗6)H3, Hα++13 = −
(Ṽ K3)
α+ , (B.14)
H53 = −
e−ρ/2
(1 + ∗6)F3, H63 = −
e−ρ/2
(1− ∗6)F3,
3 = −
(Ṽ K3)
α− , H263 =
(1− ∗6)H3.
Here α+ = 1, 2, 3 and α− = 4, · · · 22, labeling the self dual and anti-self dual forms respec-
tively. Note that using formulas (B.13) and (B.14) to lift a six-dimensional solution to ten
dimensions requires a specific choice of six-dimensional vielbein.
The solutions we find have D
= dγδ ; this implies the identity
2 )ρσ(ω
α−β− , (B.15)
where (ρ, τ) are K3 coordinates and gρτ is the K3 metric. As discussed in [39], a choice of
ǫ fixes the complex structure completely and implies (ω
2 )ρσ(ω
ρσ = Dǫ δdγǫ. Varying
this identity with respect to the metric results in (B.15).
B.1 S-duality in 6 dimensions
Given the map between 10-dimensional and 6-dimensional fields, we can now obtain the
action of S-duality on 6-dimensional fields as part of the SO(5, 21) symmetry:
G3 → OSG3, M−1 → OSM−1OTS , (B.16)
where
(OS)ij =
0 0 −1
0 I3 0
1 0 0
, (OS)rs =
0 0 1
0 I19 0
−1 0 0
, (B.17)
Moreover one can perform an SO(5) × SO(21) transformation to bring the vielbein of the
S-dual solution back to the form used by the 10-dimensional lift. Including this transfor-
mation, H3 and V transform as
H3 → OGH3, V → OGV OTS , (B.18)
(OG)ij =
C0 0 −eΦ̂
0 I3 0
eΦ̂ 0 C0
, (OG)rs =
C0 0 −eΦ̂
0 I19 0
eΦ̂ 0 C0
, (B.19)
where τ = C0 + ie
−Φ̂, Φ̂ = Φ− ρ/2 is the 10-dimensional dilaton and the fields C0 and eΦ̂
are the original ones taken before the S-duality.
B.2 Basis change matrices
In defining six-dimensional supergravities there are implicit choices of constant SO(p, q)
matrices. When discussing the compactification from the ten to six dimensions, the most
convenient choices for these matrices are certain off-diagonal forms, see for example [15,
17, 19, 20, 21, 22]. When one is interested in specific solutions of the six-dimensional
supergravity equations, such as AdS3 × S3 solutions, and deriving the spectrum in such
backgrounds, it is rather more convenient to use diagonal choices for these matrices, see for
example [30, 31]. In this paper we both compactify from ten to six dimensions, and expand
six-dimensional solutions about a given background. We therefore find it most convenient to
use diagonal choices for the constant matrices. To use previous results on compactification
and T-duality, we need to apply certain similarity transformations. For the most part these
may be implicitly written in terms of basis change matrices, so that compactification and
duality formulas remain as simple as possible. Thus let us define matrices Ω1 and Ω2 for
SO(4, 20), and Ω3 and Ω4 for SO(5, 21) via:
(vρ −wρ)
(vρ +wρ)
, ΩT3
(v − w)
(v + w)
,(B.20)
(v − w)
(v + w)
, ΩT4
(v1 − w1)
(v2 − w2)
(v2 + w2)
(v1 + w1)
where ρ = 1, · · · 4, (c) = 1, · · · 16, (a) = 1, · · · 24, α = 1, 2, 3 and α− = 1, · · · 19. These satisfy
the conditions:
0 −I4 0
−I4 0 0
0 0 −I16
ΩT1 =
0 −I20
 , (B.21)
σ1 0 0
0 I3 0
0 0 −I19
ΩT2 =
0 −I20
σ1 0 0
0 I4 0
0 0 −I20
ΩT3 =
0 −I21
σ1 0 0 0
0 σ1 0 0
0 0 I3 0
0 0 o −I19
ΩT4 =
0 −I21
Here σ1 is the Pauli matrix
C Properties of spherical harmonics
Scalar, vector and tensor spherical harmonics satisfy the following equations
�Y I = −ΛkY I , (C.1)
�Y Iva = (1− Λk)Y Iva , DaY Iva = 0,
�Y It
= (2− Λk)Y It(ab), D
aY It
k(ab)
where Λk = k(k+2) and the tensor harmonic is traceless. It will often be useful to explicitly
indicate the degree k of the harmonic; we will do this by an additional subscript k, e.g.
degree k spherical harmonics will also be denoted by Y Ik , etc. � denotes the d’Alambertian
along the three sphere. The vector spherical harmonics are the direct sum of two irreducible
representations of SU(2)L × SU(2)R which are characterized by
ǫabcD
bY cIv± = ±(k + 1)Y Iv±a ≡ λkY Iv±a . (C.2)
The degeneracy of the degree k representation is
dk,ǫ = (k + 1)
2 − ǫ, (C.3)
where ǫ = 0, 1, 2 respectively for scalar, vector and tensor harmonics. For degree one vector
harmonics Iv is an adjoint index of SU(2) and will be denoted by α. We use normalized
spherical harmonics such that
Y I1Y J1 = Ω3δ
I1J1 ;
Y aIvY Jva = Ω3δ
IvJv ;
Y (ab)ItY Jt
= Ω3δ
ItJt, (C.4)
where Ω3 = 2π
2 is the volume of a unit 3-sphere. We define the following triple integrals as
Y IY JY K = Ω3aIJK ; (C.5)
(Y α±1 )
1 = Ω3e
αij ; (C.6)
D Interpretation of winding modes
In the fundamental string supergravity solutions (2.1) the null curves describing the motion
of the string along a torus direction xρ (whose periodicity is 2πRρ) could have winding modes
such that Fρ(v) = wρRρv/Ry, with wρ integral. Consider now the correspondence with
quantum string states. Such winding modes are not consistent with both supersymmetry
and momentum and winding quantization for a string propagating in flat space, with no B
field. Recall that the zero modes of a worldsheet compact boson field can be written as
X(σ+, σ−) = x+
+ nR)σ+ +
− nR)σ− ≡ x+ w̃σ+ + wσ−, (D.1)
where R is the radius and (p, n) are the quantized momentum and winding respectively; note
that we define σ± = (τ ± σ). BPS left-moving states with no right-moving excitations have
w = 0 and hence α′p = −nR2. However the latter condition has no solutions at generic
radius and so states with winding along the torus directions cannot be BPS. Therefore
winding modes should not be included to describe the F1-P states and corresponding dual
D1-D5 ground states of interest here.
Now consider switching on constant B
ρv ≡ bρ on the worldsheet. The constant B field
shifts the momentum charges, and thus there are BPS left-moving states with winding
around the torus directions. To be more precise, following the discussion of [12], one can
describe a string with left-moving excitations using a null lightcone gauge. The relevant
terms in the worldsheet fields are then
V = wvσ−; U = wuσ− + w̃uσ+ +
a−n e
−inσ− ; (D.2)
XI = δIρw
Iσ− +
−inσ− ,
where winding modes are included only along torus directions, labeled by ρ. The L0 con-
straint implies
wvwu = (wρ)2 + 2
|n|aInaI−n ≡ (wv)2|∂VXI |20, (D.3)
where |A|0 denotes the projection onto the zero mode. The momentum and winding charges
are given by
dσ(∂τX
m +B(2)mn∂σX
n); Wm =
dσ∂σX
m, (D.4)
respectively, where α′ = 2. Requiring no winding in the time direction and no momentum
along the xρ directions imposes w̃u = wu + wv and wρ = bρw
v . The conserved momentum
and winding charges are then
PM = 1
(1 + |∂VXI |20 + b2ρ), (|∂VXI |20 − b2ρ), 0
; WM = wv(0, 1, 0, bρ). (D.5)
Note that the integral quantized momentum charge py along the y direction is therefore
py = Ry(w
u − (wv)−1(wρ)2). (D.6)
Now consider the solitonic string supergravity solution (2.1) with defining curves F I(v)
where F ρ(v) = bρv + F̄
ρ(v), with F̄ ρ(v) having no zero mode. The ADM charges of this
solitonic string were computed in [15], and are given by
PMADM = kQ
(1 + |∂vF I |20), |∂vF I |20, 0, bρ
, (D.7)
where the effective Newton constant is k = Ω3Ly/2κ
6. When bρ = 0 these charges match
the worldsheet charges (D.5) provided that wv = 2kQ as in [15] but when bρ 6= 0 they do
not quite agree with the worldsheet charges. The reason is that in the supergravity solution
ρv approaches zero at infinity, but to match with the constant B
ρv background on the
worldsheet, B
ρv should approach bρ at infinity. This can be achieved via a constant gauge
transformation Aρ → Aρ − bρ, combined with a coordinate shift u→ u+ 2bρxρ. The ADM
charges of this shifted background indeed exactly match the worldsheet charges (D.5). The
harmonic functions Aρ then take the form
Aρ = −bρH −
|x− F |2 , (D.8)
where in the latter expression |x−F |2 denotes
i−F i(v))2; the harmonic function has
been smeared over the T 4 and the y circle. Note that when F i(v) = 0 the supergravity
solution collapses to
ds2 = H−1dv(−du +Kdv) + dxIdxI ; K = (1 +
Q|∂vF ρ|20
), (D.9)
e−2Φ = H ≡ (1 + Q
); B(2)uv =
(H−1 − 1); B(2)vρ = −bρ.
This is the naive SO(4) invariant F1-P solution, with an additional constant B field. Finally
let us note that one can similarly switch on winding modes for the curves q(c)(v) charac-
terizing the charge waves in the heterotic solution (3.1) by including constant A
v on the
worldsheet.
Now let us consider solutions in the D1-D5 system, and the interpretation of including
winding modes of the internal curves. In particular, it is interesting to note that the general
SO(4) invariant solutions include harmonic functions
A = ao +
; Aα− = aα−o +
, (D.10)
in addition to the harmonic functions (H,K) given in (6.29). The non-constant terms in
these harmonic functions are related to the winding modes of the internal curves, with the
quantities aα̃ = (a, aα−) being given by
a = −Q5
dvḞ(v); aα− = −Q5
dvḞα−(v). (D.11)
Following the duality chain, these constants are given by aα̃ = −Q5bα̃ where for the T 4
case bα̃ ≡ B(2)ρv = bρ and for the K3 case bα̃ ≡ (B(2)ρv = bρ, A(c)v = b(c)). The constant terms
(ao, a
o ) are related to the boundary conditions at asymptotically flat infinity, as we will
discuss below.
When these functions (A,Aα−) are non-zero, the geometry generically differs from the
naive D1-D5 geometry. The functions (f1, f̃1) appearing in the metric behave as
f̃1 = 1 +
− (1 + Q5
(ao +
)2 + (aα−o +
f1 = 1 +
− (1 + Q5
(aα−o +
. (D.12)
In the decoupling limit these functions become
f̃1 → r−2(Q1 −Q−15 (a2 + aα−aα−)) ≡
; f1 → r−2(Q1 −Q−15 (aα−aα−)) ≡
, (D.13)
and thus (ao, a
o ) drop out. Note that q̃1 corresponds to the conserved momentum charge
in the F1-P system (D.6). Substituting the decoupling region functions into (4.1), one finds
that the near horizon region of the solution is AdS3×S3×M4, supported by both F (3) and
H(3) flux:
ds2 =
(−dt2 + dy2) +
q1Q5(
+ dΩ23) +
ds2M4 ; (D.14)
e2Φ =
Q5q̃1
tyr = −
= 2q−11 q̃1Q5;
tyr = 2aQ
1 r, H
= −2a.
The field strengths F (1) and F (5) vanish, but there are non-vanishing potentials:
B(2)ρσ =
2Q−15 a
α−ωα−ρσ , C
(0) = −q−11 a, C(4)ρστπ = Q
5 aǫρστπ; (D.15)
tyαβ = a(1 + q̃
2)ǫαβ, C
αβρσ = 2
2ǫαβa
α−ωα−ρσ , C
tyρσ =
2Q−15 a
α−ωα−ρσ ,
where ǫ is a 2-form such that dǫ is the volume form of the unit 3-sphere. The conserved
charges therefore include Chern-Simons terms; using the equations of motion (A.2) one
finds that they are given by
D5 : Q5 =
(F (3) +H(3)C(0));
D1 : q̃1 =
S3×M4
(∗F (3) +H(3) ∧ C(4)); (D.16)
D3 : aα− =
S3×ωα−
B(2) ∧ (F (3) +H(3)C(0));
NS5 : a = −1
H(3),
where we drop terms which do not contribute to the charges. The curvature radius of the
AdS3 × S3 is l = (q1Q5)1/4, and the three-dimensional Newton constant is
8πV4Ω3
(q1Q5)
3/4, (D.17)
with the volume of M4 being (2π)4V and 2κ210 = (2π)
7(α′)4. Then using [40, 41] the central
charge of the dual CFT is
(α)′4
q̃1Q5 ≡ 6ñ1n5 (D.18)
where the integral charges (ñ1, n5) are given by
Q5 = α
′n5; q̃1 =
(α′)3ñ1
. (D.19)
Now consider the relation between this system and the F1-P system discussed previously.
The conserved charges here are (Q5, q̃1, a, a
α−), which correspond to the winding, momen-
tum along the y circle and winding along the internal manifold in the original system. The
fact that (a, aα−) measure NS5-brane and D3-brane charges in the final system is consistent
with the duality chains from the F1-P systems: applying the standard duality rules along
the chains given in (2.6),(2.7) and (3.4), one indeed finds that the original winding charges
become NS5-brane and D3-brane charges.
Finally let us comment on the constant terms in the harmonic functions, (ao, a
These clearly determine the behavior of the solution at asymptotically flat infinity: the B
field and RR potentials at infinity depend on them. Now consider how these constant terms
can be described in the CFT. In the context of the pure D1-D5 system it was noted in
[12] that (infinitesimal) constant terms in the harmonic functions (f1, f5) can be reinstated
by making (infinitesimal) irrelevant deformations of the CFT by SO(4) singlet operators.
See also [42] for a related discussion in the context of the AdS5/CFT4 correspondence.
It seems probable that a similar interpretation would hold here: the (nt − 1) parameters
(ao, a
o ) (where nt = 5, 21 for T
4 and K3 respectively) would be related to the parameters
of deformations of the CFT by irrelevant SO(4) singlet operators. In total taking into
account these (nt − 1) zero modes, plus the two constant terms in the (f1, f5) harmonic
functions, one gets (nt+1) parameters. This agrees exactly with the count of the number of
irrelevant SO(4) singlet operators5. How to describe these deformations in the field theory
beyond the infinitesimal level is not known, however.
E Density of ground states with fixed R charges
In this appendix we will derive an asymptotic formula for the number of R ground states with
given R charges. Our derivation follows closely that of [43] for the density of fundamental
string states with a given mass and angular momentum. In fact, we will consider the case of
K3, so the relevant counting is precisely that of the density of left moving heterotic string
states with a given excitation level N and (commuting) angular momenta (j12, j34) in the
5Such deformations may also be related to the attractor flow of moduli; this idea is currently being
developed by Kyriakos Papadodimas and collaborators.
transverse R4. For this purpose we can consider the following Hamiltonian
(a)=1
+ λ1j
1 + λ2j
2, (E.1)
where (λ1, λ2) are Lagrange multipliers and
j1 = j12 = −i
n−1(α1−nα
n −α2−nα1n); j2 = j34 = −i
n−1(α3−nα
n−α4−nα3n). (E.2)
Here the oscillators satisfy the standard commutation relations, namely
n , α
nδn+mδ
(a)(b). In [43] the partition function was computed in the case λ2 = 0, and thus
the partition function of interest here can be computed by generalizing their results. The
first step is to diagonalize the Hamiltonian by introducing combinations
a12n =
(α1n + iα
n); b
(α1n − iα2n) (E.3)
and analogously (a34n , b
n ). Then the Hamiltonian takes the form
(a)=5
n + (n− λ1)(a12n )†a12n + (n+ λ1)(b12n )†b12n (E.4)
+(n− λ2)(a34n )†a34n + (n+ λ2)(b34n )†b34n
The partition function Z = Tr(e−βH) is then
(1− wn)−20(1− c1wn)−1(1− c−11 wn)−1(1− c2wn)−1(1− c
(E.5)
with w = e−β and c1 = e
βλ1 , c2 = e
βλ2 . To estimate the asymptotic density of states, one
as usual expresses the partition function in terms of modular functions and then uses the
modular transformation properties. Here one needs the Jacobi theta function
θ1(z|τ) = 2f(q2)q1/4 sin(πz)
(1− 2q2n cos(2πz) + q4n), (E.6)
f(q2) =
(1− q2n), q = eiπτ , (E.7)
and the modular transformation property
| − 1
) = eiπ/4
τeiπz
2/τθ1(z|τ) (E.8)
Rewriting the partition function in terms of the modular functions, applying this modular
transformation and then taking the high temperature limit results in
Z(β, λ1, λ2) = Cβ
12e4π
2/β λ1λ2
sin(πλ1) sin(πλ2)
, (E.9)
with C a constant. From this expression one can extract the density of states with level N
and angular momenta (j1, j2) by expanding
Z(w, k1, k2) =
dN,j1,j2w
Neik1j
1+ik2j
, (E.10)
where k1 = −iβλ1 and k2 = −iβλ2, and projecting out the dN,j1,j2 . Integrating over (k1, k2)
can be done exactly, since
dkeiky
sinh(πk/β)
cosh2(βy/2)
, (E.11)
resulting in the following contour integral over a circle around w = 0 for dN,j1,j2 :
dN,j1,j2 = C
β14e4π
2/β 1
cosh2(βj1/2) cosh2(βj2/2)
. (E.12)
Assuming N is large the integral can be approximated by a saddle point evaluation, with
the saddle point defined by the solution of
= N + 1− j1 tanh(1
j1β)− j2 tanh(1
j2β). (E.13)
For small angular momenta, which is the case of primary interest here, the solution is
β ∼= 2π/
N + 1. For (
∣) = O(N) the stationary point is at
N + 1− |j1| − |j2|
. (E.14)
Note that
∣ ≤ N . This latter stationary point is equally applicable to small angular
momenta, and thus one can write the asymptotic density of states as
dN,j1,j2
4(N + 1− j)31/4
2π(2N − j)√
N + 1− j
cosh2(
N+1−j ) cosh
N+1−j )
, (E.15)
where j =
∣. The constant of proportionality is fixed by the state with j1 = N ,
j2 = 0 being unique. Note that the commuting generators (j3, j̄3) of (SU(2)L, SU(2)R)
respectively are related to the rotations in the 1-2 and 3-4 planes via j3 =
(j1 + j2) and
j̄3 =
(j1 − j2). The total number of states at level N is
dN ∼=
N27/4
exp(4π
N), (E.16)
and thus the density of states with zero angular momenta differs from the total number of
states only by a factor of 1/N ; the exponential growth with N is the same.
References
[1] O. Lunin and S. D. Mathur, “AdS/CFT duality and the black hole information para-
dox,” Nucl. Phys. B 623, 342 (2002) [arXiv:hep-th/0109154].
[2] O. Lunin, S. D. Mathur and A. Saxena, “What is the gravity dual of a chiral primary?,”
Nucl. Phys. B 655 (2003) 185 [arXiv:hep-th/0211292].
[3] O. Lunin, J. Maldacena and L. Maoz, “Gravity solutions for the D1-D5 system with
angular momentum,” arXiv:hep-th/0212210.
[4] O. Lunin and S. D. Mathur, “Metric of the multiply wound rotating string,” Nucl.
Phys. B 610, 49 (2001) [arXiv:hep-th/0105136].
[5] V. Balasubramanian, J. de Boer, E. Keski-Vakkuri and S. F. Ross, “Supersymmetric
conical defects: Towards a string theoretic description of black hole formation,” Phys.
Rev. D 64, 064011 (2001) [arXiv:hep-th/0011217].
[6] J. M. Maldacena and L. Maoz, “De-singularization by rotation,” JHEP 0212 (2002)
055 [arXiv:hep-th/0012025].
[7] S. D. Mathur, “The fuzzball proposal for black holes: An elementary review,”
arXiv:hep-th/0502050.
[8] I. Bena and N. P. Warner, “Black holes, black rings and their microstates,” arXiv:hep-
th/0701216.
[9] V. S. Rychkov, “D1-D5 black hole microstate counting from supergravity,” JHEP 0601
(2006) 063 [arXiv:hep-th/0512053].
[10] K. Skenderis and M. Taylor, “Kaluza-Klein holography,” JHEP 0605, 057 (2006)
[arXiv:hep-th/0603016].
[11] K. Skenderis and M. Taylor, “Fuzzball solutions for black holes and D1-brane-D5-brane
microstates,” Phys. Rev. Lett. 98, 071601 (2007) [arXiv:hep-th/0609154].
[12] I. Kanitscheider, K. Skenderis and M. Taylor, “Holographic anatomy of fuzzballs,”
JHEP 0704 (2007) 023 [arXiv:hep-th/0611171].
[13] M. Taylor, “General 2 charge geometries,” JHEP 0603 (2006) 009 [arXiv:hep-
th/0507223].
[14] C. G. Callan, J. M. Maldacena and A. W. Peet, “Extremal Black Holes As Fundamental
Strings,” Nucl. Phys. B 475, 645 (1996) [arXiv:hep-th/9510134].
[15] A. Dabholkar, J. P. Gauntlett, J. A. Harvey and D. Waldram, “Strings as Solitons and
Black Holes as Strings,” Nucl. Phys. B 474, 85 (1996) [arXiv:hep-th/9511053].
[16] E. Bergshoeff, R. Kallosh, T. Ortin, D. Roest and A. Van Proeyen, “New formulations
of D = 10 supersymmetry and D8 - O8 domain walls,” Class. Quant. Grav. 18, 3359
(2001) [arXiv:hep-th/0103233].
[17] A. Sen, “String String Duality Conjecture In Six-Dimensions And Charged Solitonic
Strings,” Nucl. Phys. B 450 (1995) 103 [arXiv:hep-th/9504027].
[18] A. Sen, “Strong - weak coupling duality in four-dimensional string theory,” Int. J. Mod.
Phys. A 9 (1994) 3707 [arXiv:hep-th/9402002].
[19] J. Maharana and J. H. Schwarz, “Noncompact symmetries in string theory,” Nucl.
Phys. B 390, 3 (1993) [arXiv:hep-th/9207016].
[20] D. Youm, “Black holes and solitons in string theory,” Phys. Rept. 316, 1 (1999)
[arXiv:hep-th/9710046].
[21] E. Bergshoeff, H. J. Boonstra and T. Ortin, “S Duality And Dyonic P-Brane Solutions
In Type II String Theory,” Phys. Rev. D 53 (1996) 7206 [arXiv:hep-th/9508091].
[22] K. Behrndt, E. Bergshoeff and B. Janssen, “Type II Duality Symmetries in Six Di-
mensions,” Nucl. Phys. B 467 (1996) 100 [arXiv:hep-th/9512152].
[23] L. J. Romans, “Selfduality For Interacting Fields: Covariant Field Equations For Six-
Dimensional Chiral Supergravities,” Nucl. Phys. B 276 (1986) 71.
[24] S. de Haro, S. N. Solodukhin and K. Skenderis, “Holographic reconstruction of space-
time and renormalization in the AdS/CFT correspondence,” Commun. Math. Phys.
217, 595 (2001) [arXiv:hep-th/0002230].
[25] M. Bianchi, D. Z. Freedman and K. Skenderis, “How to go with an RG flow,” JHEP
0108, 041 (2001) [arXiv:hep-th/0105276].
[26] M. Bianchi, D. Z. Freedman and K. Skenderis, “Holographic renormalization,” Nucl.
Phys. B 631, 159 (2002) [arXiv:hep-th/0112119].
[27] I. Papadimitriou and K. Skenderis, “AdS / CFT correspondence and geometry,”
arXiv:hep-th/0404176.
[28] I. Papadimitriou and K. Skenderis, “Correlation functions in holographic RG flows,”
JHEP 0410, 075 (2004) [arXiv:hep-th/0407071].
[29] K. Skenderis, “Lecture notes on holographic renormalization,” Class. Quant. Grav. 19
(2002) 5849 [arXiv:hep-th/0209067].
[30] S. Deger, A. Kaya, E. Sezgin and P. Sundell, “Spectrum of D = 6, N = 4b supergravity
on AdS(3) x S(3),” Nucl. Phys. B 536, 110 (1998) [arXiv:hep-th/9804166].
[31] G. Arutyunov, A. Pankiewicz and S. Theisen, “Cubic couplings in D = 6 N = 4b su-
pergravity on AdS(3) x S(3),” Phys. Rev. D 63 (2001) 044024 [arXiv:hep-th/0007061].
[32] J. R. David, G. Mandal and S. R. Wadia, “Microscopic formulation of black holes in
string theory,” Phys. Rept. 369, 549 (2002) [arXiv:hep-th/0203048].
[33] L. F. Alday, J. de Boer and I. Messamah, “The gravitational description of coarse
grained microstates,” JHEP 0612, 063 (2006) [arXiv:hep-th/0607222].
[34] F. Larsen and E. J. Martinec, “U(1) charges and moduli in the D1-D5 system,” JHEP
9906, 019 (1999) [arXiv:hep-th/9905064].
[35] A. Jevicki, M. Mihailescu and S. Ramgoolam, “Gravity from CFT on S**N(X): Sym-
metries and interactions,” Nucl. Phys. B 577, 47 (2000) [arXiv:hep-th/9907144].
[36] E. Bergshoeff, C. M. Hull and T. Ortin, “Duality in the type II superstring effective
action,” Nucl. Phys. B 451, 547 (1995) [arXiv:hep-th/9504081].
[37] M. J. Duff, J. T. Liu and R. Minasian, “Eleven-dimensional origin of string / string
duality: A one-loop test,” Nucl. Phys. B 452, 261 (1995) [arXiv:hep-th/9506126].
[38] P. K. Townsend, “A New Anomaly Free Chiral Supergravity Theory From Compacti-
fication On K3,” Phys. Lett. B 139 (1984) 283.
[39] P. S. Aspinwall, “K3 surfaces and string duality,” arXiv:hep-th/9611137.
[40] J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization of
Asymptotic Symmetries: An Example from Three-Dimensional Gravity,” Commun.
Math. Phys. 104 (1986) 207.
[41] M. Henningson and K. Skenderis, “The holographic Weyl anomaly,” JHEP 9807, 023
(1998) [arXiv:hep-th/9806087].
[42] K. Skenderis and M. Taylor, “Holographic Coulomb branch vevs,” JHEP 0608, 001
(2006) [arXiv:hep-th/0604169].
[43] J. G. Russo and L. Susskind, “Asymptotic level density in heterotic string theory and
rotating black holes,” Nucl. Phys. B 437, 611 (1995) [arXiv:hep-th/9405117].
