#### Black-hole entropy from supergravity superstrata states

Iosif Bena
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Masaki Shigemori
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Nicholas P. Warner
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CEA Saclay
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F-
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Gif sur Yvette
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France
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University of Southern California
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Los Angeles, CA 90089, U.S.A
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Yoshida-Ushinomiya-cho, Sakyo-ku,
Kyoto 606-8501, Japan
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Department of Physics and Astronomy
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Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502 Japan
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Yukawa Institute for Theoretical Physics, Kyoto University
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Hakubi Center, Kyoto University
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Institut de Physique Theorique
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Open Access, c The Authors
There are, by now, several arguments that superstrata, which represent D1-D5P bound states that depend upon arbitrary functions of two variables and that preserve four supersymmetries, exist in string theory, and that their gravitational back-reaction results in smooth horizonless solutions. In this paper we examine the shape and density modes of the superstratum and give strong evidence that the back-reacted supergravity solution allows for fluctuation modes whose quantization reproduces the entropy growth of black holes as a function of the charges. In particular, we argue that the shape modes of the superstratum that lie purely within the non-compact space-time directions account for at least 1/6 of the entropy of the D1-D5-P black hole and propose a way in which the rest of the entropy could be captured by superstratum fluctuations. We complete the picture by conjecturing a relationship between bound states of multiple superstrata and momentum excitations of different twisted sectors of the dual CFT.
1 Introduction 2 3 4
Towards the full black-hole entropy
The shape modes of the superstratum
The MSW counting of black-hole entropy
5.3 In search of the lost 5/6ths
Structure of three-charge states in CFT
Black-hole microstate structure
Representing black hole microstates with superstrata
The D1-D5 CFT and the visible sector
Adding momentum: the three-charge states
Details of the perturbative momentum states
The supergravity back-reaction and holography
Black-hole microstate structure
The prototypical example of a string theory black hole whose entropy can be accounted
for microscopically is the D1-D5-P black hole. If one considers the various ways in which a
combination of N1 D1 and N5 D5 branes can carry NP units of momentum (in the regime
of parameters where the back-reaction of these branes is not important and the physical
picture of the momentum-carrying excitations is clear), one finds that the corresponding
entropy is given by 2N1N5NP , which exactly matches the Bekenstein-Hawking entropy
of the black hole that these branes form in the regime of parameters where their
backreaction is important. Since the original work of [1, 2], such entropy-matching calculations
have been extended to many other families of supersymmetric, or merely extremal black
holes, and even near-extremal black holes. The matching of the entropies has proven
In 1996, the perturbative counting of black-hole microstates at vanishing string
coupling in [2] represented the first real progress on the microstate problem in many years.
However, this work opened up a whole new set of questions. In particular, it remained to
understand how one particular black-hole microstate manifests itself in the finite-coupling
regime in which the classical black-hole solution exists and has a large horizon area. For a
long time it had been thought that all the microstates at weak coupling develop a horizon
and are indistinguishable from the classical black-hole solution (except perhaps in a
Plancksize region around the singularity) [3, 4]. This intuition was challenged by the construction
of several families of fully back-reacted solutions that have the same charges and mass as
the black hole, but differ from the classical black-hole solution at the scale of the horizon
and, in particular, are smooth and horizonless [5, 6]. Such solutions are called microstate
geometries, because, via the AdS/CFT correspondence, one can map them onto states
of the dual CFT. However, despite having many properties indicating that they belong
to the typical sector of the black-hole microstates, these solutions have an entropy that is
parametrically lower than the black-hole entropy [7], which is presumably related to the
fact that these solutions have a lot of symmetry.
If one is to try to reproduce the black hole entropy from supergravity one should
therefore find solutions with less symmetry, and the first step in this direction was the
construction of three-charge solutions that contain an wiggly supertube [8]. These solutions
are parametrized by an arbitrary continuous function and hence can have an infinite number
of continuous parameters [9]. The entropy of these solutions grows with the charges as
N 5/4 [9], which is more than all other known supergravity solutions, but is still less than the
black hole entropy growth, N 3/2. In [10] we have furthermore argued that if one relaxes one
more symmetry one can construct smooth horizonless superstratum solutions that depend
on arbitrary continuous functions of two variables, and it is the purpose of this paper to
argue that the perturbative semi-classical quantization of superstrata yields a
black-holelike entropy growth, and that in the fully back-reacted regime all the three-charge black-hole
entropy might be reproduced by space-time fluctuation modes of the superstrata.
In parallel with our efforts, there have also been several relatively-recent developments
that support this general approach. First amongst these is Mathurs tightening [1113]
of Hawkings result to show that information can only be recovered if there are O(1)
corrections to the semi-classical physics outside black holes. That is, in order to solve the
information problem, we need to make some O(1) changes at the horizon scale. This
discussion can be taken to a new level by asking whether these changes result in a firewall for
an incoming observer, as argued by [1423, 23, 24] or rather whether the quantum
superposition of these states can result in a smooth infall experience for macroscopic infalling
observers [16, 2527]. However, finding a mechanism that can support such O(1) changes in
the structure at the horizon scale is notoriously difficult essentially because the horizon
is null, any massive object must fall in, while any massless wave packet will dilute to
nothing after several horizon-crossing times. The only time-independent way to support such
a structure within supergravity is to place magnetic fluxes on topologically non-trivial
cycles [28, 29], and this is precisely the mechanism that underpins all the known BPS [5, 6, 30]
and near-extremal [18, 31] microstate geometries. Furthermore, as we have argued in [32],
this mechanism extrapolates well beyond the regime of validity of supergravity, and can
manifest itself either via brane polarization [33] or via non-Abelian effects.
As explained in [32], there are two separate issues that one must address in order to
understand the microstate structure of black holes and the effect that this structure has
at the horizon scale. The first is how one can make changes at the horizon scale and we
now know [28] that the geometric transition discovered in five dimensions [5, 6] provides
the only way to replace the horizon with horizonless time-independent structure thereby
making the O(1) corrections. Such geometric transitions will therefore be an essential part
of any string-based resolution of black-holes. The microstate structure itself, whatever its
ultimate form, can then be supported by the canvas provided the geometric transition
to large microstate geometries.
The second issue is to determine the extent to which this microstate structure can be
captured by semi-classical geometries. This paper will advance the latter goal by arguing
that there is indeed a class of microstate geometries, called superstrata, that can achieve
the second goal at least with sufficient fidelity to obtain the correct charge-dependence of
the BPS black-hole entropy.
The superstratum is a smooth, horizonless soliton (a microstate geometry) that is 18 -BPS
(preserving 4 supersymmetries), depends on several arbitrary functions of two variables
and has the same charges as the D1-D5-P black hole. The existence of this object was
conjectured in [10] (building on earlier work in [34]) by arguing that a certain combination
of branes, Kaluza-Klein monopoles (KKMs) and momentum preserves the same
supersymmetries as the D1-D5-P black hole irrespective of its orientation, and hence one can
glue these branes into a supersymmetric configuration that depends on functions of two
variables. Furthermore, since the superstratum locally resembles a D1-D5 supertube with
a KKM dipole charge, the fully back-reacted superstratum solution should be smooth and
hence be a microstate geometry. Even though there is not yet an explicit construction
of a generic fully back-reacted superstratum, one can find further evidence for their
existence by analyzing string emission from the D1-D5-P system [3537], or by constructing
supergravity solutions that depend of two different functions of two different variables [38],
which could be thought of as limits of the more general superstratum solution.
There are several ways by which one might realize the construction of a superstratum.
The first way is via a double supertube transition [10, 34, 39]: one combines the D1 branes
with some momentum to give a D1-P supertube (D1s with traveling waves on them) and,
at the same time, one combines some D5 branes with some momentum to obtain a D5-P
supertube (D5s with traveling waves on them). One must do this in such a manner that
the D1-profile lies entirely within the D5-profile. Next one executes a second supertube
transition by locally puffing out the D1-D5 system using a Kaluza-Klein monopole and
the result is a D1-D5-P bound state. Since supertube transitions give the configuration
an arbitrary profile and the second transition can, in principle, be done independently
and locally on each D1-D5 segment, it seems plausible [10] that two supertube transitions
could give rise to a smooth superstratum solution that can be parametrized by functions
The second way to think of a superstratum is to begin with a D1-D5 supertube with
KKM dipole charge (parametrized by several arbitrary functions of one variable) and start
adding momentum to it. Again, for each original configuration, given by the Lunin-Mathur
geometry [4042] one expects to be able to add a general wave profile along the common
D1-D5 direction, and hence to obtain a configuration that depends on functions of two
variables. Thus, every mode of the original D1-D5 supertube will act as a momentum carrier,
and therefore the number of carriers over which one can distribute a given momentum is
the number of modes of the D1-D5 supertube. This suggests that such excitations should
describe a moduli space of D1-D5 supertubes, and each such modulus should be able to
A third perspective on superstrata comes from the fact that they describe bubbled
microstate geometries. Indeed, the single, circular, unexcited superstratum is identical to a
D1-D5 supertube geometry and this geometry, in the near-tube limit, is, up to orbifolding,
the maximally-symmetric geometry global AdS3 S3 [40]. More generally, multiple
superstrata are expected to describe geometries with topological 3-cycles held up by
cohomological fluxes. Changing the shapes of the superstrata corresponds to changing the shapes
of these cycles and letting these shape changes depend upon the compact circle in AdS3.
On a single superstratum, the modes transform under the isometries SL(2, R)L SL(2, R)R
SU(2)L SU(2)R. If the structure is to carry momentum then supersymmetry requires
that this momentum be either purely left-moving or purely right-moving and so BPS
fluctuations can only excite half the modes. As we will discuss in section 4, within the D1-D5
CFT, the left-moving excitations in the space-time directions are correlated with fermionic
excitations that only carry SU(2)L quantum numbers.1 It is this that places restrictions
on the BPS modes and thus upon the perturbative shape fluctuations. This perturbative
approach to superstrata has been developed in [35, 36] and very simple, restricted classes
of fully back-reacted solutions were described in [38].
Representing black hole microstates with superstrata
The problem with the quantization of the superstratum is that we do not know its action
and so we cannot start from first principles and quantize. On the other hand we do know
the perturbative description of the D1-D5-P microstates that give the black-hole entropy
and we know the field theory dual of the AdS3 S3 solution corresponding to the unexcited
superstratum. From these observations we can reverse engineer precisely which states
of the superstratum will be visible within supergravity. Our ultimate goal is to argue that
the modes of the D1-D5-P system will, in supergravity, give rise to geometric modes whose
1This observation also has interesting implications for future work: near-BPS and non-BPS solutions
have long been obtained by exciting both left-moving and right-moving momentum [4346] and so we expect
generic shape fluctuations to be a natural way to access such non-BPS solutions.
semi-classical quantization will reproduce the exact black-hole entropy:
We will, however, start far more conservatively with what we believe can be
substantiated with a high level of confidence, namely, that the semi-classical quantization of the
space-time shape modes of a single superstratum can lead to an entropy count of, at least,
This differs from (1.1) by a factor of 16 because, as we will discuss, the perturbative
space-time shape modes of a single superstratum must involve only one sixth of the
complete set of perturbative BPS modes. More precisely, these BPS space-time shape modes
describe a sector of the CFT with central charge c = N1N5 corresponding to half of the
bosonized fermions in the D1-D5 CFT. The remaining part of the CFT, with central charge
c = 5N1N5, arises from the other half of the bosonized fermions and the original bosonic
excitations of the D1-D5 CFT. These states correspond to corrections to the internal
metric and fields on the T 4 upon which the D5 branes are compactified. We will examine
the extent to which this other five-sixths of the BPS states will be visible within
supergravity and argue that in the fully back-reacted regime the modes that contain internal
torus fluctuations will have an energy gap that is parametrically larger than that of the
typical black hole microstates. We suggest that these internal torus modes will be pushed
on the Coulomb branch and will become visible as transverse supergravity modes of the
The important point here is that, whatever the ultimate status of the internal T 4
excitations, the arguments based upon group theory and perturbation theory allow us to
assert with considerable confidence that the shape modes of a single superstratum can, at
N1N5NP as a function of N1N5NP .
least, recover the correct entropy growth S
It is also possible to estimate the entropy of superstrata by starting from the original
argument [10] that they can be constructed as momentum-carrying fluctuations of the
D1D5 supertube. This construction appears to allow all the shape modes of the supertube
to be promoted to momentum carriers.2 We will argue in section 5 that the dimension of
the moduli space of these shape modes is 4N1N5, which would imply that the entropy of
a superstratum will come from distributing NP units of momentum over 4N1N5 bosonic
carriers and their fermionic superpartners, and this would reproduce exactly the black-hole
entropy (1.1). This construction appears to be at odds with the perturbative analysis that
gives the entropy (1.2). It is possible that the 4N1N5 shape modes are not independent and
unobstructed moduli. It is also quite possible, as we will also discuss in section 5, that the
extra shape modes that go beyond the perturbative analysis of section 4 will only emerge
in the fully back-reacted superstratum solution. We therefore hope that an complete and
explicit superstratum solution will clarify whether the space-time modes of the superstrata
will reproduce all the black-hole entropy or only 1 of it.
2This also agrees with the physics of certain explicit solutions that can be thought of as singular limits
of the superstratum solution [47, 48].
In formulating the entropy-counting arguments above we have taken it as given that
adding momentum charge to a BPS system of branes will always lead to transverse shape
modes once the supergravity back-reaction is included. We will also assume the converse:
semi-classical quantization of such supergravity shape modes will recover a full description
of the Hilbert space of the original perturbative momentum modes. This is certainly true
of the F1-P system, since this is simply the quantization of the fundamental string [49] and
it is also true of momentum modes on many systems of branes. We do not believe that
there is much danger in assuming that this is a universal result.3
There are two frequently-expressed concerns about any program, as the one advanced
here, that involves obtaining the black-hole entropy by counting supergravity solutions.
The first is that classical supergravity modes only correspond to coherent quantum states
and that the states that contribute to the entropy cannot be geometric. The second is that
it is possible that the fluctuations that contribute primarily to the entropy may have very
small scales, and hence the corresponding solutions will have structure below the Planck
scale and will not be therefore correctly described by supergravity.
The first concern might equally be raised as an objection to considering the vibrational
motion of a diatomic molecule to be that of a spring. Obviously this is a dramatic classical
simplification of a complex quantum system and the real motions of a diatomic molecule
are intrinsically quantum phenomena. However, approximating the chemical bond by a
classical harmonic oscillator and semi-classically quantizing this oscillator gives an
excellent description of the quantum states and the vibrational spectrum because the spring
isolates the essential physical degrees of freedom that govern the system. It is in this spirit
that we believe that microstate geometries and their semi-classical quantization will
describe sufficiently many microstates of black holes and give a valuable description of their
thermodynamics: while the quantum mechanical states of a black hole are manifestly not
geometric, and only very few of them have classical descriptions, the important insight
coming from microstate geometries is that this allows us to identify the degrees of freedom
at strong coupling that need to be quantized in order to capture the essential underlying
physics of the black-hole microstates.
The second concern is more serious in that the entropy might be coming primarily
from a sector in which the supergravity approximation is failing. There are two reasonable
ways around this issue. First, we know that exactly the same issue arises in other instances
of adding momentum modes to branes, as with the fundamental string, and yet there is
no problem with the semi-classical quantization of states. The reason why there is no
difficulty is precisely because such states are based upon well-understood systems of objects
that make sense in string theory. Thus the easiest answer to the second concern is that we
may ultimately have to broaden the scope of the semi-classical quantization and go beyond
3Strictly speaking, this must hold for the momentum added to the unique ground state of the system
and does not apply to the momentum carried by the ground state itself. We are always concerned with
the former. For example, a straight supertube [8] carries a fixed amount of angular (longitudinal)
momentum coming from the crossing of electric and magnetic worldvolume fluxes. However, any change in the
momentum on top of that leads to transverse fluctuation of the supertube shape and of the back-reacted
smooth microstate geometries, whose scales, by definition, lie comfortably above the Planck
length, and include microstate solutions. The latter are defined [32] to be horizonless,
physical limits of smooth geometries that have the same mass, charge and angular momentum
as a given black hole, but can have singularities that either correspond to fundamental
( 12 -BPS) D-brane sources or can be patch-wise dualized into a smooth solution.
It is also possible that smooth microstate geometries will resolve these issues without
needing to introduce stringy singularities. Indeed, one important realization in the study
of microstate geometries was that if one wants to construct a solution that has the same
charges as a five-dimensional three-charge black hole with a macroscopically-large horizon
area, one must use scaling solutions [5355]. In these solutions the size of the bubbles
appears to shrink to zero size from the perspective of the metric of the auxiliary
fourdimensional base-space that is used to construct the solutions, but, in fact, the bubbles
remain finite once the supergravity back-reaction is taken into account. In the scaling limit,
these bubbles descend down a very long AdS throat that resembles, more and more, that
of the corresponding black hole. Hence, it is possible that adding a third charge to what
appear to be very stringy two-charge microstates will expand the physical length scales
and result in smooth fluctuating solutions at the bottom of a very long throat.
Returning to our main goal, we wish to describe the detailed structure of the semi-classical
superstratum in terms of the D1-D5 CFT. We therefore begin in section 2 by reviewing
the D1-D5 CFT and in section 3 we describe the two-charge ( 14 -BPS) states of the D1-D5
system and how they correspond to supertube profiles. In section 4 we add momentum to
the system and relate the three-charge ( 18 -BPS) states to profiles of the superstratum. We
initially adopt a rather conservative approach by focussing on the details of the microstate
structure that we are confident can be reproduced by quantizing the supergravity modes.
In particular, we focus on the space-time shape modes of the superstratum and how they
can be matched to perturbative modes of a particular sector of the D1-D5 CFT. This
allows us to reproduce the correct charge growth of the black-hole entropy, albeit with a
smaller overall coefficient. In section 5 we adopt a less conservative view of the possible
modes that a superstratum can have, which is closer to the original arguments for the
existence of superstrata [10] and to the physics of certain singular limits of superstratum
solutions [47, 48]. This allows us to use a counting argument similar to that of Maldacena,
Strominger and Witten [56] to reproduce exactly the entropy of the three-charge black
hole, and to obtain the correct overall coefficient as well. We then discuss several ways
in which the liberal and conservative approaches to superstrata can be reconciled, and in
particular we suggest in section 6 that bound states of multiple superstrata may be a key
ingredient in relating all the states of the CFT to bulk supergravity solutions. Section 7
contains our concluding remarks.
The D1-D5 CFT and the visible sector
The easiest way to quantize the two-charge system is in the F1-P frame where the states are
simply those of the perturbative string. However, for the superstratum, we are going to need
the detailed description in the D1-D5 duality frame where there are N5 D5 branes wrapped
S1 and N1 D1 branes wrapped on the common S1. Let R be the radius of the S1
and v the corresponding coordinate. For fixed v, the moduli space of the configurations is
the same as that of N1 D0 branes inside N5 D4 branes and so it may be identified with
the moduli space of N1 instanton sector of SU(N5) Yang-Mills. The dimension of this
moduli space is 4N1N5. These moduli can be made into functions of v and thus, in the
perturbative regime, one has a CFT with 4N1N5 bosons on this S1. However, the D1-D5
system has 8 supersymmetries, which extend the CFT to an N = (4, 4) SCFT. There are
thus 8N1N5 free fermions that split into 4N1N5 left-movers and 4N1N5 right-movers.4
To be more precise, the underlying field theory is the N = (4, 4) superconformal sigma
model whose target space is the orbifold, (T 4)N /SN , where N N1N5 and SN is the
permutation group on N elements.5 There are thus 4N free bosons and 4N free fermions.
Following [36, 59] the bosons will be labeled, X(ArA)(z, z), where r = 1, . . . , N , is the copy
index of the T 4 and A, A = 1, 2 are spinorial indices for the SO(4)I = SU(2)1SU(2)2 of the
, = , transform as doublets of fixed helicity on the T 4 and as doublets of different
helicities under the R-symmetry, SO(4)R = SU(2)L SU(2)R. Note that the fermions
transforming in the (2, 1) and (1, 2) of the R-symmetry are left-moving and right-moving,
respectively. The T 4 is, of course, the compactification manifold of the D5s and, as usual
in theories on D-branes, the R-symmetry is generated by rotations in the (non-compact)
spatial directions transverse to all the branes, that is, in the space-time directions.
In the fully back-reacted D1-D5 geometry, the near-brane limit is global AdS3 S3 T 4
and the symmetry outside the T 4 is SL(2, R)L SU(2)L SL(2, R)R SU(2)R. These
symmetries correspond to the left-moving and right-moving (finite) conformal invariance
and R-symmetry via the holographic duality.
By construction, the excitations of the bosons, X(ArA), only involve motions in the
compactified (T 4) directions, whereas the fermionic excitations carry polarizations
(Rcharge) that are visible within the six-dimensional space-time. To understand what portion
of the fermion Hilbert space is visible from the space-time, it is convenient to bosonize the
fermions by defining the currents
algebra. Each such algebra may be viewed as being generated by a single boson.
If one sums over r, the currents
4For more details on the D1-D5 CFT, see, for example, [36, 5759].
5This is the description of the CFT at the free orbifold point.
generate the level N , SU(2)R SU(2)L current algebra of the R-symmetry. Because of the
pseudo-reality of the fermions [36, 59], the standard angular momentum operators, J and
JL3 = J 12 = J 21 ,
J R3 = J12 = J21 ,
J + = J 11 ,
J + = J11 ,
J = J 22 ;
J = J22 .
For each value of r, the currents K(ArB) and K(ArB) also generate level 1, SU(2)1 current
completely orthogonal sets of operators that commute with one another7 and similarly for
excitations that are purely visible from the space-time with no component of this chiral
algebra creating an excitation on the torus. Conversely, the K(ArB) and K(ArB) represent the
chiral algebras that are visible only from the T 4 and invisible from the space-time. Thus the
perturbative excitations that are visible from the six-dimensional space-time form Hilbert
spaces, Hst, that can be characterized by the representations of, and excitations created
by, the conformal field theory:
where the J(r) and J(r) generate these level 1 current algebras. This theory has central
charge c = N = N1N5. Similarly, the CFT that lies purely on the internal directions has
c = 5N = 5N1N5 and is generated by the bosons, XAA , and the currents K(r) and K(r).
We will denote the internal Hilbert spaces by Hint and think of the states of the D1-D5
theory as being decomposed into a sums of the products of the form
H = Hst Hint .
The back-reaction of the fermionic and bosonic modes of the D1-D5 CFT will result
in shape and charge-density modes of the corresponding supergravity solution. Conversely,
we will argue, in the next section, that the semi-classical quantization of the corresponding
families of BPS microstate geometries will lead to the states of the D1-D5 CFT. Indeed
this is precisely what holographic field theory on AdS3 S3 suggests. Moreover, because of
the split into c = N = N1N5 and c = 5N = 5N1N5 sectors detailed above, we expect that
sector of the CFT while the remaining c = 5N1N5 sector will be visible from semi-classical
quantization of internal modes of the D1-D5 system.
We now substantiate this view by revisiting the geometry and semi-classical structure
of the two-charge system and argue how this will be modified via the addition of the third
charge via momentum modes.
6For the full internal SU(2) symmetry current, we must include the contribution from the bosonic field
of SU(2)L SU(2)1 and then the Js and Ks generate these two SU(2)s.
n1 strings of length 1, n2 strings of length 2, and so on, and the total length of the system if N .
The two-charge states
The two-charge states of the D1-D5 system are the Ramond-Ramond (RR) ground states of
the CFT and preserve half the CFT supercharges, or eight supersymmetries (note that these
states are called 14 -BPS states, relative to the 32 supercharges of type IIB superstring before
One can spectrally flow these states to the NS sector to obtain chiral primary fields and the
RR ground states can viewed as being created by chiral primaries acting on the
maximallyspinning RR ground state, |0i, with JL3 = J R3 = N2 [60, 61]. Spectral flow takes the
The chiral primaries of the D1-D5 CFT can be obtained from the twist fields of the
SN orbifold, and these fields are labeled by the conjugacy classes of SN . The conjugacy
classes of SN are in one-to-one correspondence with the partitions of N , which are given
by collections of non-negative integers {nk}k1 satisfying
N =
k1
It is useful to imagine these as describing a collection of effective strings. Namely, one
associates the conjugacy class {nk}k1 with n1 effective strings of length 1, n2 effective
strings of length 2, and so on. The total length of all the effective strings is N . See
figure 1. The effective string of length k represents a twist field that intertwines k copies of
by chiral primaries and so involves no intertwining of CFTs. It thus corresponds to the
partition with n1 = N and all other nk = 0.
The holographic dual of the maximally-spinning state is a single, maximally-spinning,
perfectly circular supertube in an R
2 plane. In the near-supertube limit this geometry
is exactly global AdS3 S3. The chiral primaries carry R-symmetry, by definition, and
also have T 4 indices. In the effective string picture, we may view the effective strings as
now the T 4 structure.8 The partition (3.1) is now refined according to
carrying R-symmetry and T 4 indices coming from fermion zero modes. We will focus here
on the R-charge since it is visible from six-dimensional space-time and we will suppress for
8For a more detailed description of the geometries dual to effective strings that carrying T 4 indices,
N =
where nk = 0, 1, 2, . . . is the number of effective strings with length k and SU(2)L SU(2)R
spin (, ). The maximally-spinning state |0i with JL3 = J R3 = N2 corresponds to the
Introducing twist fields generates excitations in the shape and density modes and the
bulk geometry dual to a generic two-charge state of the form (3.2) is the Lunin-Mathur
geometry [40] which is D1-D5 supertube with KKM dipole charge and an arbitrary
profile, or shape. (For a more detailed dictionary see [63].) The Lunin-Mathur geometry
is completely regular [41] and parametrized by arbitrary functions of one variable, f i(w)
(i = 1, 2, 3, 4), describing the profile of the D1-D5 supertube in the R4 transverse to the
D1-D5 world-volume. The SO(4) vector index i of the f i(w) in R4 is simply a pair of
expanded in Fourier series as
kZ
k6=0
of the R4. The AdS/CFT dictionary for the two-charge states [40]10 is that the number of
In the bulk viewpoint, the constraint (3.2) is nothing other than the requirement that the
supertube carries N1 units of D1-brane charge.
In this way one can substantiate the idea that semi-classical quantization of the D1-D5
profiles yields a description of the states of the D1-D5 system [40, 65]. For the two-charge
system, the profiles for the typical states have curvatures of order the Planck scale and so
one must appeal to the idea of microstate solutions [32] discussed in the Introduction, to
argue that while the supergravity approximation is not strictly valid, supergravity is
capturing the essential semi-classical degrees of freedom that underlie the microstate structure.
On the other hand, adding the third charge to the system means that there can be deep
scaling solutions [5355] in which the underlying structures remain macroscopic but lie
at the bottom of long AdS throats. This means that the supergravity approximation can
remain valid over a large range of excitations and that the semi-classical description of
smooth low-curvature geometries may be enough to account for the entropy.
This dictionary (3.4) is in complete accord with the idea that the effective strings
carry SU(2)L SU(2)R charges and they must represent visible microstates in the dual
sixdimensional spacetime. As we argued above, the effective strings arise from twist fields that
10For a precise dictionary and its subtleties, see [63].
intertwine k copies of CFT, with k = 1, . . . , N . The fact that these fields carry R-charges,
i.e., space-time angular momenta, means that they have polarizations directed into the
space-time and so describe fluctuations in space-time. Indeed, acting with these twist fields
changes the length and spins of effective strings and, by the AdS/CFT dictionary (3.4),
corresponds to changing the shape of the back-reacted D1-D5 supertube. We may look
on these twist fields as providing a Landau-Ginzburg description of the shape modes of
the D1-D5 system. It should be stressed that these shape modes correspond to supertube
profiles in the R4 transverse to the D1-D5 world-volume. There will be similar shape modes
in the T 4 directions but in this paper we focus on the space-time shape modes.
charge system is, of course, obvious in the F1-P duality frame where one is simply describing
shape modes of a fundamental string. Indeed, one can go from the F1-P modes to the
description of the D1-D5 modes by a suitable set of duality transformations. However, we
need to work in the D1-D5 frame and see that the states in this frame are also represented
by shape modes because we are now going to add a third charge to the system and it
is easiest to understand what this entails if the new third charge is a momentum charge
and not some other brane charge. By showing that the D1-D5 states involve shapes as a
function of one variable we are now going to see that the D1-D5-P states are obtained by
giving these D1-D5 shape modes an extra dependence on another direction.
The correspondence between the quantization of shape modes and the states of the
two
Adding momentum: the three-charge states
Adding the momentum
As we have seen, the two-charge ( 14 -BPS) states of the D1-D5 system can be mapped onto
the RR ground states of the CFT on the common S1 of the D1 and D5 branes. The
three-charge ( 18 -BPS) states are obtained simply if we keep the Ramond ground states in
the right-moving sector, thereby preserving half of the right-moving supersymmetries, but
supersymmetries. (The choice of the left/right sector to break/preserve supersymmetry is
purely conventional and we could have done it in the other way around.) The eigenvalue
L0 c/24, of the corresponding 18 -BPS state. It was this construction that originally led
to the perturbative counting of BPS microstates [2] and the microscopic description of the
entropy (1.1). As we saw above, the 14 -BPS shape modes along the profile in the spatial R4
and these may be thought of as choices of Ramond ground states or as the states generated
Just as for fundamental strings, adding momentum to any system of branes is expected
to involve excitations transverse to the branes (see footnote 3). In the fully back-reacted
supergravity solution, these momentum states are reflected in a non-trivial profile that
sources the solution. Conversely, the quantization of that profile yields a semi-classical
description of the momentum states of the system. If we assume that these are also true in
the current situation, adding momentum to the D1-D5 system means that the back-reacted
supergravity solution will now not only have a profile in the spatial R4, parametrized by w,
but that such a profile will now also depend upon v, the coordinate along the S1 common
to the D1 and the D5 branes11 Thus one obtains shape modes that depend upon functions
of two variables and these functions will provide a semi-classical description of all the states
of the D1-D5 system.
In particular, if we focus on the perturbative states visible within the space-time and
described by Hst then these shape modes are captured by the space-time shape modes of
a generic, single superstratum. We therefore expect that the two-charge profile functions,
Put differently, we can take a Landau-Ginzburg perspective in which the D1-D5 modes
are created by chiral primaries and these, considered as Landau-Ginzburg fields, become
momentum carriers simply through their descendant states within the left-moving Hilbert
space. Thus we see how a generic perturbative BPS excitation can give rise to a double
D1-D5 system, or unexcited superstratum.
Details of the perturbative momentum states
The connection between perturbative CFT states and the supergravity shape modes can
S3, which is the dual of the maximally-rotating RR ground state.
be made very explicit. In the near-superstratum limit the geometry is simply AdS3
of the superstratum are simply Fourier modes of supergravity fields on the S3 and thus
correspond to representations of the SU(2)L SU(2)R. While the two-charge D1-D5 shape
modes carry quantum numbers of both SU(2)L and SU(2)R, the momentum-carrying BPS
operators that excite those states carry only the quantum numbers of SU(2)L and hence
adding momentum does not involve changing the D1-D5 shape modes that transform under
SU(2)R. In particular, consider the maximally-spinning D1-D5 solution whose near-brane
geometry is AdS3 S3. The generic D1-D5 ground states can be thought of as fluctuation
modes on the S3. In the NS sector, they are the chiral primary states and have quantum
numbers under SU(2)L SU(2)R given by (`, m; `, m) = (`, `; `, `). Note that these D1-D5
supertube shape modes on the S3 are very special, in that the quantum numbers are
constrained to satisfy ` = m, ` = m and, furthermore, |` `| is equal to the spin of the
fields that exist in the theory. For a fixed spin field the Fourier modes are determined by
one quantum number and hence correspond to one-dimensional shape modes on the S3.
In contrast, the BPS momentum carrying modes, which are of the form (any, chiral) in
the NS sector, allow more general excitations under SU(2)L, while the SU(2)R quantum
numbers remain unchanged. So, the generic 18 -BPS mode will have SU(2)L SU(2)R
11In general, the geometries dual to CFT states that are exact eigenstates of the momentum operator
P L0 L0 are v-independent, while coherent states, which are not a precise eigenstate of P , are
vdependent [36]. We are concerned with the latter because we are interested in the traveling waves on the
supertube along v and their classical description is given by coherent states.
quantum numbers (`, m; `, `). Since we now have m independent of `, these will generate
intrinsically two-dimensional shape modes on the S3.
A particular subset of the BPS states involve arbitrary excitations created by operators
the associated left-moving CFT in (2.5) reflect purely space-time modes and will be visible
in the perturbative space-time shape modes of the superstratum.
To make this more precise, one can easily describe the complete set of two-charge
supertube shape and density modes within supergravity and express the result in terms
of exact supergravity solutions in six dimensions. One can also realize the action of the
superconformal algebra on the geometry and, in particular, implement the action of the
currents (2.3) in terms of rotations on the supergravity solutions. In this way one can,
at the linearized level, generate the linearized supergravity solutions with shape modes
in the (`, m; `, `) representations by starting with the D1-D5 shape modes (`, `; `, `) that
correspond to chiral primaries in CFT. Realizing this procedure has been one of the major
goals of [3537]. The fact that BPS equations of the six-dimensional supergravity are
essentially linear means that knowing the linearized solutions is almost enough to construct
the fully back-reacted solutions [47]. This observation was exploited to significant effect
in [37, 38]. To construct the fully back-reacted BPS fluctuations of the superstratum and
show that there is indeed an intrinsically two-dimensional BPS shape modes in space-time
one simply needs to take the special fluctuating modes considered in [38] and use the
current algebra action, as in [37, 66], to find the generic supergravity modes and then try
to compute the fully back-reacted solution using [47].
The foregoing procedure of rotating supertube fluctuation modes by the generators
to obtain the fluctuations with quantum numbers of the form (`, m; `, `). Put differently,
this is equivalent to a rather trivial statement that acting on a chiral primary by the
generators of the finite Lie algebras SL(2, R)L SU(2)L only gives the descendant of a
chiral primary but certainly does not yield generic 18 -BPS states that are descendants of the
non-chiral primaries. It therefore seems, at first sight, that the procedure we have outlined
only generates an extremely small subset of the general momentum-carrying states, which
However, this is not exactly what we are doing: we are not simply rotating a complete,
known classical BPS state. Instead we are using rotations to generate all the individual
fluctuating modes of some of the fields but discarding all of the rest of the rotated solution.
We then take arbitrary linear combinations of those modes as seeds to generate new classical
solutions using the linear BPS system replete with its sources that depend non-linearly on
the fluctuating modes. In this way we construct the most general, fully back-reacted
fluctuating supergravity solution. In the quantum theory, classical solutions can be regarded as
coherent quantum states and so taking such classical linear combinations amounts to
taking tensor products of the corresponding quantum states. The products of descendants of
chiral primaries generically yield the descendant of non-chiral primaries [67, 68]. Therefore,
if we complete the fully back-reacted supergravity solution based on linear combinations
of modes, they will represent the descendants of the non-chiral primaries.
Thus the process of feeding a general superposition of classical fluctuations into the
complete BPS system will certainly generate the most general exact classical BPS states
and we claim that this will also give a semi-classical description of the most general BPS
quantum state. Indeed, precisely this sort of result was established in [68] where it was
shown that the space of supergravity fluctuations in a finite neighborhood of the AdS3 S3
background precisely reproduced the elliptic genus of the CFT (ref. [68] is when the internal
manifold is K3; for T 4, see [69]).
It is important to note that the result of [68, 69] was only established using a
perturbative supergravity gas around a solution that lay outside the black-hole regime and so one
may quite reasonably doubt the applicability of this result within microstate geometries
that look like black holes. However, to make a microstate geometry that looks like a black
hole one does not simply use small perturbations of AdS3 S3: one must incorporate the
back-reaction of the momentum to obtain deep, scaling microstate geometries in which the
topological cycles descend a long AdS2 throat. We will discuss this further in the next
section, but here we want to note that AdS3 S3 represents a good local model of
individual topological bubbles and it is expected that their fluctuations will give the microstate
structure only when these bubbles are located at the bottom of a deep, scaling throat. All
we therefore need from [68, 69] is the result that the that semi-classical quantization of
supergravity modes on AdS3 S3 captures the quantum CFT states locally. It is then
expected that these states generate the correct microstate structure of a black hole when
they are located deep within a scaling solution and greatly red-shifted as a result.
Before concluding this section we want to return to the other classical modes that live
on the internal T 4 and whose semi-classical quantization should give rise to Hint in (2.6).
Indeed, one of the points emphasized in [35, 36] is that all the perturbative excitations
of D1-D5 system will be visible within the ten-dimensional supergravity description of the
superstratum. The left-moving c = N theory (2.5) whose states lie in Hst will indeed
be visible within the space-time of the effective six-dimensional theory but the remaining
modes, lying in Hint and described in terms of the other c = 5N part of the full CFT, will be
also visible as perturbative fluctuations of geometry and fluxes in the full ten-dimensional
solution. Thus, even though the space-time shape modes of the superstratum will only
lead to an entropy (1.2), one might hope that the internal supergravity modes should lead
to the full accounting for the entropy (1.1).
However, as we will now describe, there is a subtlety in the supergravity back-reaction
that suggests that only the space-time shape modes will have sufficient resolution to capture
a large enough section of the Hilbert space of the D1-D5-P system.
The supergravity back-reaction and holography
One of the important features of the CFT dual of black-hole microstates is the fact that the
CFT can have an energy gap as low as Egap c1
N11N5 . This can be viewed as coming
from the scaling dimensions of the longest twist operators or from the longest-wavelength
momentum excitations of the longest effective strings. For a long time it was a puzzle as to
how such fractionation, and the energy gap in particular, could emerge from fluctuations
of smooth microstate geometries. Such a match is crucial if the semi-classical quantization
of supergravity is to reproduce the perturbative states of the CFT with sufficient fidelity
To understand the holographic description of the correct Egap, one should first recall
that the only way to construct microstate geometries whose charges correspond to a
fivedimensional black hole with a finite horizon area is to use deep, scaling BPS geometries
have a very long AdS throat that is smoothly capped off by bubbles, or homology cycles.
The energy gap of these solutions then emerges holographically [53] by taking the
longestwavelength fluctuation of the microstate geometry and red-shifting it according to the
depth of the throat. The depth of the throat is typically a free classical parameter in the
microstate geometry however semi-classical quantization of such geometries sets the throat
depth and thus fixes the energy gap [7, 55, 70]. It was thus one of the triumphs of the
microstate geometry program that this correctly reproduced the energy gap of the dual
CFT. The simplest microstate geometries, in which the holographic energy gap was first
computed, can then be viewed as containing unexcited superstrata and so the semi-classical
quantization of the superstratum will reproduce the correct energy levels.
Thus, in the holographic dual, modes of with energy Egap N11N5
time fluctuations whose wavelengths are of order the diameter of throat of the BPS black
hole.12 If there is only a handful of bubbles or superstrata, then this wavelength is set
by the longest wavelength fluctuation of homology cycles that spread across the throat. If
there are a lot of bubbles or superstrata then this wavelength should be thought of as the
longest wavelength collective mode of all the bubbles and superstrata.
This result relies upon the crucial structure of the warp factors in the metric. In the
IIB formulation, the ten-dimensional metric takes the form:
ds120 = 2
= Z3Z1Z2
For BPS solutions, the base metric, ds24, is hyper-Kahler and ambi-polar; the deep, scaling
solutions come from taking limits in which a cluster of two-cycles in this base appear to
scale to zero size. In the physical metric (4.1) the warp factor (Z1Z2) 21 modifies this so that
the cluster of cycles limits to a finite size determined by Q1Q2 in the spatial directions of
the base. In the full ten-dimensional metric, the two-cycles are lifted to three-cycles via the
v fiber and their volume also involves Q3. The important point is that the area of the
throat scales with Q3/2 and so, as a result of the warp factor, the longest wavelength mode
that fits across the throat scales as Q1/2. The red-shift of the deep throat then gives an
additional factor of Q3/2 to obtain Egap Q2 [53]. On the other hand the warp factors
12This should, of course be defined as the area of the throat to some suitable power. Alternatively, for
a microstate geometry where the throat is capped off, this scale can also be defined by the diameter of all
in the T 4 directions are O(Q0) = O(1) and so the T 4 does not expand to the typical size
of the throat. This suggests that fluctuations around the T 4 will develop the wrong energy
gap, ET4 gap Q3/2.
Thus it seems that the supergravity fluctuations of the superstratum in the
spacetime directions do give rise to the correct spectrum of microstates but the supergravity
fluctuations on the T 4 will lead to a rather coarse sampling of the microstate structure. It
is possible that our supergravity analysis of the T 4 fluctuations is too simplistic and we
will return to these issues in section 5 where we will conjecture how the T 4 modes may
ultimately be accounted for in the supergravity back-reaction.
To finish this rather conservative analysis based upon perturbation theory, we want to
reiterate two important conclusions from our discussion. First, and most important, is that
whatever the ultimate outcome is on the holography of the T 4 modes, we have provided
a good match between the supergravity shape modes and the perturbative microstate
structure at least for the states in Hst, with central charge c = N1N5. Thus quantizing
the superstratum should, at least, reproduce (1.2) and thus obtain the correct growth in
entropy with N1N5NP . This is already huge progress. In particular, since these microstate
geometries describe a macroscopic fraction of the black-hole entropy, this means that all
the typical states that contribute to the black-hole entropy will have a finite transverse
size. Hence the entire system will not be surrounded by a horizon and thus we will have
established the fuzzball proposal for BPS black holes in string theory.
The other thing we want to stress is that we have studied the perturbative properties
of a single, round superstratum and our work and conclusions so far are based upon this
rather conservative but fairly detailed correspondence. In section 5 and section 6 we will
argue that superstrata that have more complicated shapes, and possibly split into bound
states of multiple superstrata will in fact be able to capture the full black-hole entropy.
Towards the full black-hole entropy
Our conservative counting of superstrata entropy in section 4 was based on the description
of the maximally-spinning supertube in the dual D1-D5 CFT and on the fact that in this
CFT the left-moving (supersymmetric) fermions are charged under SU(2)L but do not
carry SU(2)R angular momentum, and hence only a fraction of the shape modes of the
supertube will be able to carry momentum. In this section, we will be slightly bolder and
discuss how the missing shape modes might re-emerge and account for the full entropy
of the D1-D5-P black hole.
The shape modes of the superstratum
From the perspective of the original argument for the existence of the superstratum [10]
and from the perspective of supergravity solutions that describe certain superstratum
components, the restriction on the possible shape modes encountered in section 4.2 appears
Indeed, if one constructs the superstratum by gluing together 16-supercharge
plaquettes that preserve the D1-D5-P Killing spinors irrespective or their orientation [10], there
appears to be no restriction on the possible shapes of the resulting object, and hence
the general superstratum solution might be expected to be determined by four arbitrary
continuous functions of two variables.
This picture is further supported by the explicit construction of supersymmetric
solutions that have all the charges and dipole charges of superstrata except one (the KKM
dipole moment), and depend also on four arbitrary continuous functions of two
variables [48]. These solutions are dubbed supersheets. Recall that, as mentioned in section 1.2,
the first way to get a superstratum is to use a supertube transition to puff out D1 branes
and momentum into a D1-P supertube and D5 branes and momentum into a D5-P
supertube (first stage), and then to use a second supertube transition to puff out again the
resulting (boosted and rotated) D1-D5 system into a KKM dipole charge (second stage).
Because supersheets do not have a KKM dipole moment, they must be describing the first
stage of this bubbling process and, consequently, represent singular supergravity solutions.
The solution is expected to become a smooth superstratum once the KKM dipole moment
is added and it was shown in [10] that adding the KKM dipole is compatible with
supersymmetry. If the circle wrapped by the KKM dipole charge is small, this will only affect
the solution in the immediate vicinity of the supersheets and hence one might reasonably
expect that the KKM will not upset the shape and the supersymmetry.
Based on the foregoing arguments, we are going to assume in the rest of section 5 that
a suitably generic superstratum can be given four independent shape functions. However,
before proceeding on this assumption, we wish to raise several issues that might lead to
restrictions on the BPS shape modes and limit such modes to those described in section 4.
First, it was noted in [10] that adding a KKM monopole requires the orientation of
the KKM to be properly aligned with the underlying compactification circles, a fact that
also was manifest in [38] and leads, potentially, to restrictions on the orientations of the
solutions. Nevertheless, it is unclear whether this condition leads to significant restrictions
on the moduli space.
Another issue is that the shape modes outlined in [10] were based upon brane
configurations that were not fully back-reacted and the description of shape modes was based
upon the local geometry of the solution. In the fully back-reacted superstratum some
of the directions necessarily pinch off to make the smooth underlying topological cycles.
Moreover, the directions that get pinched off are typically those upon which the shape
modes depend. For a smooth solution the shape modes must therefore be required to die
off as they approach these pinch-off points. This may well lead to restrictions on the
allowed BPS modes that can be smoothly excited on a superstratum and some of these
restrictions were encountered and analyzed in [38]. It remains to be seen what the full
range of allowable smooth shape modes can be for a single cycle but it may be only the
modes considered in section 4.
Finally, there is an interesting intermediate ground between the two extremes of four
shape modes and the modes of section 4. It is possible that some of the shape modes
have been suppressed by focusing on a single topological cycle and, in particular, on the
scale-invariant AdS3 S3 near-superstratum limit. The missing degrees of freedom could
then emerge either as one restores the asymptotic flatness or adds more structure so as
to introduce a scale. In the same vein, it may be that when one tries to make a KKM
resolution of a BPS supersheet of arbitrary shape, it is possible that one may not be able to
do it with a single topological bubble but that it will require several such bubbles and that
the combination of the modes on such a multi-bubble solution can lead to more functions
of two variables. We will pursue this idea further in section 6.
The MSW counting of black-hole entropy
As we have argued, it is possible that once the full non-perturbative superstratum is
constructed, the original picture of the BPS superstratum [10] could prove correct in terms of
predicting the number of shape modes. We will therefore examine what this would mean
for the superstratum and in particular we will argue that that such fluctuation modes
reproduce all the entropy of the three-charge black hole.
To see how this comes about, it is useful to recall the second way to get a
superstratum by starting with a D1-D5 supertube with KKM dipole charge and subsequently adding
momentum to it. Then the counting is very similar to the Maldacena-Strominger-Witten
(MSW) counting of the entropy of four-dimensional black holes [56]: one argues that the
number of momentum carriers on a superstratum is equal to the dimension of the moduli
space of deformations of the D1-D5 supertubes and then derives the entropy by counting
the ways of distributing the momentum amongst these moduli. At first glance the number
of supertube moduli is infinite, since an arbitrary shape can be decomposed into an infinite
Fourier series with arbitrary components. However, the quantization of the shapes of the
supertubes reduces the range of the Fourier modes and hence renders the dimension finite.
As we explained in section 2, this can be seen from the dictionary to the dual D1-D5 CFT,
which restricts the length of the maximal effective string on the boundary (which
corresponds to the Fourier mode of the round supertube) to N1N5, and since there are four
functions determining the embedding of the supertube in spacetime this corresponds to a
moduli space dimension 4N1N5.13
There is another way to figure out that the dimension of the moduli space of spacetime
deformations of two-charge supertubes is 4N1N5. As we explained in section 3, these
supertubes can be dualized to fundamental strings carrying momentum, and the entropy
of this system comes from the various ways of splitting a given amount of momentum, NP ,
among different fractionated momentum carriers that carry momentum quantized in units
of 1/N1 [1, 71]. This entropy is given by the number of possible ways of writing
much as in equation (3.1). Upon taking into account the fact that the fundamental string
has eight species of bosonic momentum carriers (corresponding to its 8 transverse
directions) and their fermionic partners, the number of partitions reproduces the entropy of the
13More precisely, because of the constraint (3.2) imposed on the 4N1N5 Fourier modes, the moduli space
dimension is 4N1N5 1, but this difference is negligible for the entropy counting.
N1NP =
k1
two charge system. The dimension of the moduli space of these configurations is given by
the number of modes carrying momentum that can be excited, and for one species alone
this number is given by the maximal value of k, which is the product of its two charges:
N1NP . Hence, the dimension of the moduli space of oscillations that will become D1-D5
supertube oscillations in the transverse four-dimensional space is again 4N1N5.
One can also argue that the dimension of the supertube moduli space is of order N1N5
by considering the maximally-spinning (round) supertube and counting its entropy `a la
Marolf and Palmer [5052]. This supertube has angular momentum J = N1N5, and if
one tries to change its shape the angular momentum becomes smaller. One can use the
Born-Infeld action describing this supertube to quantize the possible deformations of the
maximally-spinning supertube and find that this entropy comes from integer partitions
of N1N5 J . This counting therefore implies that the dimension of the moduli space
of a supertube with angular momentum J is equal to N1N5 J (again for each bosonic
mode). Strictly speaking, this counting is only valid in the vicinity of the
maximallyspinning supertube configuration (when N1N5 J
N1N5), but if one extrapolates it to a
supertube with zero angular momentum one finds again the dimension of the moduli space
of transverse oscillations to be 4N1N5.
In the foregoing discussion, we only counted the dimension of the moduli space of the
supertube fluctuations in the transverse non-compact R
4 directions (label them 1234) and
not the internal T 4 directions (label them 6789). This restriction can be justified by a
supersymmetry analysis similar to the one in [10]. As mentioned above, the first way to
get a superstratum is to first puff out D1 branes and momentum, P, into a D1-P supertube
4 and, simultaneously, puff out D5 branes and P into a D5-P supertube inside
R4. If the resulting D1-profile lies entirely within the D5-profile, it is locally the same as
the D1-D5 system which can be puffed out again into a KKM dipole charge. However,
we could have puffed them out into a curve inside T64789. For example, D1(5) and P(5)
can be puffed out into D1(6) and P(6) dipoles, where the numbers in the parentheses
denote the directions along which the object is extending. Correspondingly, D5(56789)
and P(5) can be puffed out into D3(789) and F1(6) dipoles (dissolved as fluxes inside
the D5 worldvolume). However, it is an straightforward algebraic exercise [10] to show
that these puffed-out charges cannot undergo a second supertube transition. Therefore,
interestingly, the second supertube transition is kinematically (supersymmetrically) allowed
only if the first transition is in the transverse R
4 directions. This holds true even if the
internal manifold is not T 4 but K3, because there is no difference between T 4 and K3 in
Hence, the dimension of the moduli space of bosonic fluctuations of D1-D5 supertubes
in the transverse space is 4N1N5. Much as for the MSW black-hole entropy calculation,
this dimension gives the number of bosonic modes that carry momentum, and one expects
by supersymmetry that there should be an equal number of fermionic momentum
carriers. As we explained above, there is a tension between the perturbative analysis of these
modes (described in section 4) which indicates that only N1N5 of these modes can carry
momentum supersymmetrically, and the original argument for the existence of superstrata
and the solutions of [47, 48], which suggests that all the four bosonic modes, and hence all
their four fermionic partners as well, can carry momentum supersymmetrically.
If there really are four bosonic modes and four fermionic counterparts then they will
give a semi-classical description of momentum-carrying states with c = 6N1N5, and the
entropy of the superstrata is given by the possible ways of carrying NP units of momentum:
which reproduces exactly the Bekenstein-Hawking entropy of the three-charge black hole.
Since this entropy comes entirely from spacetime modes and their fermionic partners, this
entropy count also reproduces the entropy of the D1-D5-P black hole if one replaces the
We have thus argued that the shape modes of the superstratum have the capacity
to describe a full set of semi-classical microstates of a black hole and while this would
represent a very happy state of affairs, there are some words of caution to be made. First,
as we explained at the end of section 5.1, adding a KKM monopole and pinching off circles
to make topological cycles could potentially restrict the shape modes [10]. Second, we
have argued that one should think of the 4N1N5 spatial shape modes of the superstratum
as independent moduli just as those of the MSW string and hence can independently
be assigned momentum states. It remains unclear whether these moduli are sufficiently
independent and unobstructed. Indeed, these excitations have to satisfy the constraint (3.2)
and this restricts the size and degeneracies of the putative moduli space. This constraint
will be modified once one adds momentum and previously indistinguishable CFT states
become distinguishable. Thus the independence of, and restrictions upon, the supertube
moduli remain unclear but as we have seen, it is conceivable that the complete set of shape
modes can capture the complete BPS black-hole entropy.
In search of the lost 5/6ths
The analysis of section 4 starts from a single round supertube, corresponding to a state
of the D1-D5 CFT in which the long effective string of length N1N5 is split into N1N5
effective strings each of length one, and considers adding supersymmetric (left-moving)
momentum perturbatively on this object. The left-moving momentum modes are only
charged under SU(2)L but not under SU(2)R, which implies that only the modes that give
one sixth of the central charge of all the modes that one might have hoped to promote
to momentum carriers are in fact supersymmetric. Moreover, in the original discussion of
the superstratum [10] it was pointed out that, while it seemed plausible that the shape
modes could be excited independently in the two directions of the superstratum surface,
this independence was not established rigorously. So the most conservative conclusion of
the perturbative analysis of section 4 is that the space-time modes of superstrata are still
given by functions of two variables, as argued in [10], but that these modes only give 1
of the entropy of the black hole.
It is important to examine the tension between the results of section 4 and the
arguments of the previous subsection. Indeed, the results of section 4 indicate that 5/6 of the
modes that give rise to the black-hole entropy should appear as semi-classical fluctuations
on the internal T 4 and only 1/6 of these modes are visible in space-time. This suggests that
we should simply be looking at the full supergravity solution in ten dimensions and the
shape modes on the T 4 in particular. On the other hand the arguments we presented above
suggest that all the modes that carry the black hole entropy can be visible as superstratum
space-time modes. We thus appear to be in danger of over-counting.
One possible solution to this tension could be that the restrictions on the
supersymmetric momentum carriers coming from the perturbative analysis are valid only in the vicinity
of the maximally-spinning supertube configuration in the free orbifold limit, and that far
away from that point in the CFT moduli space these restrictions will be lifted.14 Indeed,
the supersheets of [48] and other singular solutions that have black hole charges and carry
momentum with both SU(2)L and SU(2)R angular momentum [47] can be thought of as
limits of superstrata solutions in which one has turned off the KKM dipole charge. This
can be done by making the radius of the second supertube transition very small, which can
be achieved by taking the number of KKMs to be very large.15 From the perspective of the
dual CFT, the number of KKMs is the length of the effective strings, and increasing this
number brings one very far away from the state we considered in section 4, where there are
N length-one effective strings carrying J R3 = O(N ) as a whole, towards the sector where
there are a few long effective strings of length O(N ) carrying J R3 = O(1). Incidentally,
this is also the sector where the black-hole entropy lives, so if the superstratum counting
that gives the entropy (5.2) is correct, this entropy comes exactly from where it should
come. Starting with this sector with J R3 = O(1), one has a large degree of freedom to
increase/decrease J R3 by creating short effective strings and making them carry the desired
R 2
still in a apparent conflict with the fact that, on the original supersheet, we could consider
arbitrary SU(2)R fluctuations.
Another possible way to reconcile the two analyses above could be to consider multiple
superstrata and allow different superstrata (or even different parts of one superstratum) to
have different orientations so that the correlation with angular momentum might change
between superstrata. It is possible for the momentum modes on one of these superstrata
to be charged under SU(2)L and for the modes on the other to be charged under SU(2)R.
Thus, from a suitable distance, a generic collection of superstrata could appear to replicate
generic space-time shape modes. Moreover, it is possible to bring two superstrata close to
each other and to join them into a figure-eight configuration that looks like a deformation
of a superstratum with dipole charge two. One can similarly argue that a superstratum
with a very large dipole charge, of the type that is expected to describe the CFT states
that give the black-hole entropy, can be deformed into configurations that contain multiple
superstrata, which can in turn carry momentum modes with all angular momenta.
While these observations suggest that superstrata may have a much larger set of
spacetime configurations than the single, round superstratum considered in section 4, it does
14Recall that the perturbation taking the CFT away from the free orbifold point is a twist operator
insertion which mixes effective strings with different lengths.
and therefore reduces their influence on the geometry.
15This can appear paradoxical, but increasing the number of KKMs decreases the radius of the KKMs
not resolve the over-counting danger associated with having both the T 4 modes and the
full set of space-time shapes corresponding to states. However, one can argue that, in the
regime of parameters where the black hole exists, the modes that look like internal shape
modes in the perturbative analysis of section 4 will be suppressed and, in addition, it is
possible that they give rise to fluctuations in the transverse space.
Indeed, our analysis of section 4.3 indicates that in the fully back-reacted supergravity
regime where the classical black-hole solution exists, the modes that correspond to
fluctuations in the internal directions will have the wrong mass gap and will not be therefore
capable of describing the modes that give the black hole entropy. This will then suppress
such semi-classical states in the total entropy. A pessimist would then take the view that
only the perturbative space-time shapes have the correct energy gap and thus contribute
to the entropy, leading to the result (1.2).
However, based upon our experience with five-dimensional microstate geometries, we
know that details of internal sectors of the dual field theory corresponding to degrees of
freedom on the compactification directions can become visible within the space-time
geometry. The Coulomb-Higgs map [72, 73] is a classic example in which Higgs-branch fields
create composite operators that give rise to strong effects within the space-time geometry
that are more typically associated with the Coulomb branch of the field theory. Sometimes
this leakage of information onto the Coulomb branch can be complete in that it yields
complete information about the Higgs branch states and sometimes it can be very incomplete
in that it only captures a small fraction the data about the internal states of the system.
Thus one can take the optimistic view that the analysis of section 4.3 suppresses the shape
modes from exploring the T 4, thereby protecting us from over counting, but these modes
then leak into the floppier space-time directions for which the energy gap is much lower.
It is also possible that the missing 5/6ths will not be visible semi-classically within
supergravity and that we can only obtain the entropy (1.2). As we have already stressed,
this still represents major progress. On the other hand, we prefer to take the optimistic view
that the missing 5/6ths should still be visible within supergravity. One might therefore
hope that the internal shape modes of the single superstratum migrate to Coulomb branch
and become visible as space-time shape modes. It is interesting to ask whether these
modes will manifest themselves as superstratum modes, or as some other mode complicated
collective modes. The first possibility would reconcile the superstratum analysis in this
section with that of section 4. The second possibility would indicate there exists a
spacetime object more complicated than the single, isolated superstratum and such an object
will account for 5/6 of the modes that give the entropy of a black hole, while the single,
isolated superstratum accounts for the other 1/6. This more complicated object might be
some multi-superstrata state or even something new. Either way, finding and understanding
this more complicated object would clearly be a key priority.
We now make some first steps in suggesting the role of multi-superstrata states.
Independent of the bulk considerations of the previous section, we will argue that the
structure of the three-charge states in CFT suggests that bound states of multiple superstrata
on. The standard projection in the orbifold procedure imposes the condition P
have l1 quanta carrying k1 units of momentum, l2 quanta carrying k2 units of momentum, and so
k mlm/k Z.
are the most natural candidate for the holographic duals of the CFT states. To explain
this, we begin by unpacking more of the details of the states described in sections 3 and 4.
Structure of three-charge states in CFT
In the D1-D5 CFT, a two-charge BPS state, i.e. the RR ground state is made of multiple
effective strings of various length. Ignoring the SU(2)L SU(2)R charge, it is specified by
the numbers {nk}k1 satisfying (3.1) and is of the following form:
Y(|0ik)nk = (|0i1)n1 (|0i2)n2 (|0i3)n3 ,
k1
where |0ik is the ground state of the c = 6k CFT living on the effective string of length k.
See figure 1. The bulk dual of this is a D1-D5 supertube whose profile function f (w) has
Fourier coefficients ak given by
|a1|2 = n1,
|a2|2 = n2,
|a3|2 = n3,
Note that we are ignoring the SU(2)L SU(2)R charge for simplicity of presentation and
The three-charge states are obtained by exciting momentum-carrying modes on the
effective strings. In particular, on an effective string of length k lives the SU(2)L current
JL3 (z),16 whose modes we denote by J m , m Z. Note that the mode numbers are in units
k
of k1 because the length of the string is k. We can use these modes to obtain
momentumcarrying states on a single effective string as follows:
(J k1 )l1 (J k2 )l2 |0ik |l1, l2, . . .ik,
with the SN -orbifold constraint that the total momentum on the effective string is an
inte
m1 mlm/k Z. See figure 2 for a pictorial description of this state. Since
16Here, JL3 (z) is defined to be JL3 (z) = JL3(r)(z) with 2(r 1) arg(z) < 2r and is multi-valued, where
the modes J mk carry non-vanishing SU(2)L charge, they are visible in six-dimensional
space-time. If we excite the JL3 modes on all the effective strings in the two-charge
state (6.1), we obtain the general three-charge state that can be created by JL3
excitations.17 In doing so, we must remember that effective strings of identical length k are
indistinguishable if they are in the ground state but, once we excite JL3 modes, they become
distinguishable (unless they have identical excitation numbers {l1, l2, }). Thus, for each
k, the nk states will be broken into distinguishable and indistinguishable effective strings.
To be concrete, let us focus on effective strings with one particular value of length k,
n3 = 7 = 2 + 4 + 1 =
The seven strings are indistinguishable because they are all in the same ground state. So,
states are obtained by exciting momentum modes on these strings, as in (6.3). For example,
is an integer. The three-charge state thus obtained is
we have two strings, all in the state (J 13 )3|0i3 = |3, 0, 0, . . .i3. For four of the remaining
strings are in the state (J 13 )(J 23 )4|0i3 = |1, 4, 0, . . .i3. Finally, let the last string be in
the state (J 13 )6(J 33 )1|0i3 = |6, 0, 1, . . .i3. Note that the total momentum in each string
If n(3i) denotes the number of strings in the ith group, we have the splitting
The n3 = 7 indistinguishable strings in (6.4) have split into three distinguishable groups.
The n(i) strings in the ith group are all in the same excited state and indistinguishable.
Let n(3im), m 1 denote the momentum excitation numbers for the state of the ith group.
In the present example,
3rd group: (n(33) n30 ; n31 , n32 , n(333), . . . ) = (1; 6, 0, 1, . . . ),
(3) (3) (3)
between different groups with i 6= i0 means that {n3m}m1 6= {n3m}m1.
where we defined n(3i0) n(3i). More generally, it is clear that the general three-charge state
of length-3 strings is completely specified by the numbers {n3m}m0,i1. Distinguishability
17Of course, there are other momentum-carrying states that cannot be obtained by the action of J3 but,
for simplicity, we focus on the states that can be simply labeled as in (6.3).
The general three-charge state built on the general two-charge state (6.1) is obtained
by multiplying excited strings with different values of k together. Namely, for each k, we
index the distinguishable families of momentum excitations by (i) and let n(ki0) denote the
number of indistinguishable strings in each family (they are indistinguishable because they
have identical excitation numbers). Therefore, the two-charge constraint (3.1) is refined to:
X n(ki0) = nk ,
i1
X n(ki0) = N .
k1 i1
Let n(kim) (m 1) denote the momentum excitations, as in (6.7), of the ith set of effective
Distinguishability from the other strings of length k means that the momentum excitations
must be different: {n(kim)}m1 6= {n(kim0)}m1 if i 6= i0.
The three-charge states thus obtained are:
k1 i1
where the powers represent the fact that there are n(ki0) indistinguishable effective strings
in the same state. The three-charge states (6.10) are thus specified by the non-negative
integers, {n(kim)}. The index k 1 is associated with the Fourier mode in the w-direction
momentum Fourier modes in the v-direction. Note that we have identified n(ki0) introduced
4 of the original D1-D5 system) and the index m 0 is associated with the
shape modes as a function of two variables, as expected of a superstratum. However, there
remains an additional index (i) this means that the general three-charge states in the
D1-D5 CFT naturally parametrize multiple functions of two variables. What is the physical
interpretation of this fact?
Multi-superstrata interpretation
The index (i) labels distinguishable effective strings of the same length: sets of effective
strings that only became distinguishable by virtue of the momentum excitations on them.
It is therefore tempting to interpret (i) as labeling the multiple superstrata into which
the original D1-D5 supertube has split. The momentum excitations promote the original
profile function, f (w), into a function of two variables, f (v, w), but we conjecture that the
two-charge profile function actually gets promoted into multiple functions of two variables
where f (i)(w, v) describes the world-volume of the ith superstratum. The Fourier
coeffi|a(kim)|2 = n(kim).
superstratum interpretation. For each string length k, strings on which identical momentum modes
2nd superstratum is specified by {n(k2m) }, and so on. See the text for more detail.
are excited are grouped together. For fixed k, the n(k10) strings in group 1 are all in the same
state |n(k11), n(k12), . . .ik and are indistinguishable, the n(k20) strings in group 2 are all in the same state
|n(k21), n(k22), . . .ik and are indistinguishable, and so on. The shape of the 1st superstratum is specified
by the number of strings in group 1 for all possible values of k, namely by {n(k1m) }. The shape of the
See figure 3 for a pictorial description of the state (6.10) and the multi-superstrata
We hasten to note the important fact that the foregoing description of three-charge
states, such as (6.10), is valid only at the free orbifold point in the moduli space of the
D1-D5 CFT, whereas the actual supergravity sits at a very different point in the moduli
space. Deforming the CFT away from the orbifold point corresponds to turning on twist
operator perturbations (see [59] for a recent detailed account). Twist operators mix
different twist sectors and therefore the picture of each individual state gets modified. However,
it is the number of states that is important for our proposal, and it is not changed by such
deformations. Namely, the deformation does not change the crucial fact that more data
than can fit on a single superstratum is needed to account for general three-charge states.
Therefore, this does not invalidate our proposal that general three-charge states are
represented by multiple superstrata, although the precise dictionary between the superstrata
shape functions f (i)(w, v) and the CFT states may not be as simple as described above.
For example, it is quite conceivable a state that looks like a multi-strata state in CFT
corresponds to a single-stratum state in supergravity, and vice versa.18 This is analogous
to the fact that, in AdS5/CF T4, once interactions are turned on, the single/multi-trace
operator basis of the CFT Hilbert space is different from (and a unitary transformation
of) the single/multi-particle basis in of the supergravity Hilbert space.
Our multi-superstrata proposal raises several important issues. First, all the states we
are discussing in (6.10) are states within the same CFT and not states in distinct CFTs.
Arguing that some of these states correspond to different superstrata suggests that we are
factoring the CFT into different CFTs. At a more basic level, if one accepts that the
distinguishable families factor into different superstrata then why do we not accept that
the same must happen in the two-charge D1-D5 system: why arent effective strings of
different lengths simply different supertubes?
The resolution of all these issues comes from remembering that multiple supertubes
have no E B interactions, and therefore can be separated at arbitrary distances. If we
consider a solution that contains only two-charge supertubes placed at the bottom of a long
AdS throat, these supertubes are not trapped at the bottom of the throat and can move
freely out of the throat. They represent therefore unbound states dual to factorized CFTs.
On the other hand, two generic superstrata will always have non-trivial E B interactions,
and hence a solution that has multiple superstrata at the bottom of a long AdS throat will
represent a bound state of the CFT. Solutions with different numbers of superstrata will
have different topology, and hence will belong to different sectors of this CFT.
Another important consideration is the fact that the bubbling transition to create
microstate geometries with non-trivial cycles requires the three-charge system. The bubble
equations [7476], which relate the sizes of cycles to the fluxes through those cycles,
degenerate for two charges or if a flux through a cycle vanishes and so the corresponding bubble
collapses. Thus the possibility of separate superstrata forming a bound state in a CFT can
only occur if one excites the momentum modes in the D1-D5 system and only if one excites
momenta in distinct ways so that the fluxes on bubbles do not vanish. Conversely, if two
superstrata have exactly the same shape and charge distribution then they will coalesce
within a given AdS throat or, if they are not in an AdS throat, there will be no force
between them and they can be moved arbitrarily far away from each other, which is not
describable within one dual CFT [77].
It is worth noting that the moulting phase of the D1-D5 system [78] that appears in
the three-charge situation with large angular momentum has structures rather similar to
18This point is particularly clear for the three-charge states built on the two-charge state with n1 =
of (6.5)) says that we can build multiple-strata states on this state. On the other hand, in supergravity, the
Momentum-carrying excitations on it are small deformations of the S3, which do not seem to lead to
the ones proposed here. In [78], the following problem was studied: for given momentum
charge and angular momentum JL = O(N ), what is the ensemble of states that has the
largest entropy? In the CFT (at the orbifold point), the most entropic states were found to
be made of two sectors of effective strings, reminiscent of (6.11). The first sector is made
of a long string with length O(N ), which carries all the momentum charge as well as the
entropy, while the second sector consists of many (O(N )) short strings of length one, which
carry JL, JR = O(N ) but no entropy. On the other hand, in supergravity, the most entropic
configuration was found to be a two-center solution in an asymptotically AdS space. One
center is a BMPV black hole carrying all the momentum charge and entropy, while the
other center is a supertube carrying JL, JR = O(N ) but no entropy.19 (Because the BMPV
black hole can be thought of as shedding or moulting a supertube, it was dubbed the
moulting phase). The fact that the multi-sector states of the CFT correspond to a
multicenter solution in supergravity can be thought of as evidence in support of our conjecture
(even though these configurations are not microstates but phases with finite entropy).
Apart from the natural way in which the correspondence of distinguishable twisted
sectors and bound states of multiple superstrata appears to work, one can obtain further
evidence for the conjecture by re-examining the arguments of [7, 53, 55, 70] that obtain
the CFT gap from the supergravity solution. We first note that the longest effective string
nN = 1 ,
nk = 0 , 1 k < N ,
and so can only involve a single superstratum, no matter how we add momentum. This
sector of the theory is also the sector with Egap N11N5
and was obtained holographically
by considering an excitation of a bubbled geometry that has a wavelength equal to the
size of the AdS throat. Such a wavelength would be the natural fundamental oscillation
of a superstratum whose scale is that of the entire throat. In multiple, bound superstrata
the bubbles of geometry will be smaller than the throat and the scale of an individual
bubble will be roughly set by the scale of the throat divided by the some appropriate
power of the number of bubbles. Thus the fundamental modes of such individual bubbles
will have a shorter wavelength and a higher energy gap. Indeed, the energy gap of such
a configuration should be Egap N1pN5 , where p is the approximate number of bubbles
that span the diameter of the throat. This, at least qualitatively, fits very nicely with
the corresponding decreased lengths of the effective strings in the CFT. Obviously more
work is needed to fully substantiate our conjecture but we think it is promising enough to
warrant our description here.
In this paper we have argued that the BPS microstates of the D1-D5-P system will manifest
themselves in the regime in which the classical black hole exists as smooth horizonless
19Although the configurations in CFT and supergravity seem quite similar to each other, the entropy of
the CFT states and that of the bulk two-center solution do not quite agree (the CFT entropy is always larger
than the supergravity entropy), which is presumably caused by the partial lifting of states at strong coupling.
superstratum solutions. Despite the absence of an explicit solution describing the generic
superstratum, we have been able to account for their entropy using the intuition that adding
momentum modes to any system of branes will, upon back-reaction, emerge as shape modes
in supergravity, and, conversely, that the semi-classical quantization of such shape modes
will reconstruct the original Hilbert space of momentum states.
We first considered the construction of a superstratum in terms of fluctuations around
a maximally-spinning supertube and have argued, from the dual D1-D5 CFT, that the
number of supersymmetric momentum carriers of the superstratum is given by the product,
N1N5, of its D1 and D5 charges. This conservative estimate, which we believe can be
substantiated with a high level of confidence, gives the entropy:
and this is expected to come entirely from smooth supergravity solutions.
Then we went on to make a somewhat bolder proposal for counting the entropy of
superstrata using an approach similar to that of Maldacena, Strominger and Witten [56].
Specifically, we argued that the space of transverse fluctuations of two-charge supertubes
must have dimension 4N1N5. One can then view this as the moduli space of the
superstratum and, much as in the original construction of superstrata [10], all these moduli
could carry momentum. Assuming these moduli are independent and unobstructed, there
are thus 4N1N5 bosonic modes which, when combined with their fermionic superpartners,
would give an entropy:
This exactly matches the black-hole entropy. We have also discussed the possible ways to
reconcile this estimate to the more conservative estimate above, and have argued that, in
the regime of parameters where the black hole exists, all the modes in the internal directions
should somehow manifest themselves as fluctuations in the transverse space. We have also
argued that one cannot match all the states of the CFT by counting perturbatively around
a single superstratum solution, and that multiple superstrata bound states are a natural
candidate for matching these states.
Modulo the explicit construction of superstratum solutions that depend on arbitrary
functions, we have presented what we believe to be strong evidence that the so-called
fuzzball proposal is the correct description of extremal supersymmetric black holes within
string theory. Indeed, if one can obtain a macroscopic fraction of the black-hole entropy
from horizonless supergravity solutions, this implies that all the typical states that
contribute to the black-hole entropy will have a finite transverse size, and hence the entire
system will not be surrounded by horizon. This in turn would imply that the correct way
to think about the textbook black-hole solution is as a thermodynamic approximation of a
huge number of horizonless configurations, much as a continuous fluid is a thermodynamic
approximation of a huge number of molecule configurations.
The conservative and bolder views of superstrata lead to significant differences in the
structure of typical black-hole microstates. If all the black-hole microstates are visible
as transverse superstrata modes, then it is possible that upon full back-reaction these
modes will all give rise to low-curvature solutions that have a long black-hole-like throat
and end in a smooth cap. This would imply that the modes captured by six-dimensional
supergravity are enough to account for the black-hole entropy, which would establish the
fuzzball proposal in its strong form.
If, however, only 1/6 of the black-hole entropy comes from transverse modes, then
the typical black-hole microstates will still be horizonless, but will not be describable as
smooth solutions of six-dimensional supergravity: the typical microstates will necessarily
involve stringy or Kaluza-Klein modes. This would establish the weak version of the
fuzzball proposal, which is enough for solving the information paradox, but it may not
offer us a framework, at least within supergravity, for doing rigorous computations that
could help establish, for example, whether an incoming observer feels a firewall or falls
through the fuzzball states unharmed.
Clearly, there are two essential steps that should be done next. The first is the explicit
construction of the superstratum solutions that depend on functions of two variables. This
would represent major progress toward establishing the fuzzball proposal for extremal
black holes. The dramatic simplification of the BPS system of equations underlying these
solutions [47] means that it might be possible to construct the BPS supergravity excitations
at full non-linear order. The discussion at the beginning of section 4 showed that arbitrary
space-time shape modes break all the supersymmetry and that only the representations
(`, m; `, `) of SU(2)L SU(2)R can be excited in the 18 -BPS superstratum. This observation
also underlies the analysis in [37, 38] and it will provide invaluable insight into how to
address the construction of a fully back-reacted superstratum that depends upon a general
function of two variables.
The second, and most difficult, step is to extend this work to non-extremal black holes.
A very useful insight comes of our analysis here where we noted that certain momentum
carriers that are charged under SU(2)R may break supersymmetry.20 Hence, adding these
fluctuations to a typical BPS superstratum state may allow us to move away from
extremality and to argue that the supergravity structure of the black-hole microstates that we have
analyzed in this paper is robust when supersymmetry is broken. This, in turn, would imply
that near-extremal, and quite possibly generic, black holes are thermodynamic
approximations of horizonless solutions and that the pure states of a black hole would be represented
by horizonless configurations. This would solve the black-hole information paradox and
allow us to address, far more rigorously, the puzzles that the information-theory analysis
of black hole has revealed [1424, 79].
We would like to thank Stefano Giusto and Rodolfo Russo for extremely useful discussions
that strongly influenced the work presented here. MS thank the IPhT, CEA-Saclay for
hospitality where part of this work was done. NPW is grateful to the IPhT, CEA-Saclay,
the Institut des Hautes E tudes Scientifiques (IHE S), Bures-sur-Yvette and the Yukawa
20A similar phenomenon happens for supertubes, and there taking into account the
supersymmetrybreaking modes is crucial if one is to quantize correctly the supersymmetric modes [50].
Institute for hospitality while various parts of this work was done. NPW would also like to
thank the Simons Foundation for their support through a Simons Fellowship in Theoretical
We are all grateful to the Centro de Ciencias de Benasque for hospitality at
the Gravity New perspectives from strings and higher dimensions workshop, and
to the Yukawa Institute at Kyoto University for hospitality at the Exotic Structures of
Spacetime workshop (YITP-T-13-07). The work of IB was supported in part by the ERC
Starting Independent Researcher Grant 240210-String-QCD-BH, by the John Templeton
Foundation Grant 48222: String Theory and the Anthropic Universe and by a grant
from the Foundational Questions Institute (FQXi) Fund, a donor advised fund of the
Silicon Valley Community Foundation on the basis of proposal FQXi-RFP3-1321 to the
Foundational Questions Institute. The work of MS was supported in part by Grant-in-Aid
for Young Scientists (B) 24740159 from the Japan Society for the Promotion of Science
(JSPS). The work of NPW was supported in part by the DOE grant DE-FG03-84ER-40168.
This article is distributed under the terms of the Creative Commons
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any medium, provided the original author(s) and source are credited.
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