Mtheory superstrata and the MSW string
Accepted: June
Mtheory superstrata and the MSW string
Iosif Bena 0 1 2 5
Emil Martinec 0 1 2 3
David Turton 0 1 2 5
Nicholas P. Warner 0 1 2 4
0 University of Southern California , Los Angeles, CA 90089 , U.S.A
1 5640 S. Ellis Ave. , Chicago, IL 606371433 , U.S.A
2 CEA, CNRS, Orme des Merisiers , F91191 Gif sur Yvette , France
3 Enrico Fermi Inst. and Dept. of Physics, University of Chicago
4 Department of Physics and Astronomy, and Department of Mathematics
5 Institut de Physique Theorique, Universite Paris Saclay
The lowenergy description of wrapped M5 branes in compacti cations of Mtheory on a CalabiYau threefold times a circle is given by a conformal eld theory studied by Maldacena, Strominger and Witten and known as the MSW CFT. Taking the threefold to be T6 or K3 T2, we construct a map between a subsector of this CFT and a subsector of the D1D5 CFT. We demonstrate this map by considering a set of D1D5 CFT states that have smooth horizonless bulk duals, and explicitly constructing the supergravity solutions dual to the corresponding states of the MSW CFT. We thus obtain the largest known class of solutions dual to MSW CFT microstates, and demonstrate that vedimensional ungauged supergravity admits much larger families of smooth horizonless solutions than previously known.
AdSCFT Correspondence; Black Holes in String Theory

1 Introduction
2.1
2.2
2.3
2.4
3.1
3.2
3.3
4.1
4.2
5.1
5.2
2 D1D5P BPS solutions and spectral transformations
D1D5P BPS solutions
Canonical transformations
General spectral transformations
Fivedimensional solutions and spectral transformations
2.5 Spectral transformations for vindependent solutions
3 The multiwound supertube and mapping D1D5 to MSW
The multiwound D1D5 supertube con guration
Spectral transformations of the supertube
Mapping from D1D5KKM to Mtheory
4 General SL(2; Q) transformations in six dimensions
General spectral transformation of the metric functions
General spectral transformation of the gauge elds
5 Constructing Mtheory superstrata
D1D5P Superstrata
Transforming the superstrata
5.2.1
5.2.2
5.2.3
Transforming the metric quantities
Transforming the uxes
Mtheory superstrata
6 An explicit example
6.1
The D1D5P superstrata
6.2 The Mtheory superstrata
7 Comments on symmetric product orbifold CFTs
7.1
Review of the D1D5 CFT
7.2 The D1D5KKM CFT
8 Discussion
A Covariant form of BPS ansatz and equations
B Circular D1D5 supertube: parameterizations C Lattice of identi cations and fractional spectral ow
{ i {
D.1 Coordinate conventions D.2 MSW maximalcharge Ramond ground state solution
41
42
1
Introduction
For around twenty years there have been two wellknown routes to describe the entropy of
BPS black holes in some form of weakcoupling limit. The rst of these was done using
the original perturbative, weakcoupling microstate counting of momentum excitations of
the D1D5 system [1]. The corresponding threecharge black hole, in
ve dimensions, is
obtained by compactifying D1 and D5 branes of the IIB theory on M
either T
4 or K3, and adding momentum.
When the size of the S1 wrapped by N1 D1
S1, where M is
and N5 D5 branes is the largest scale in the problem, the lowenergy physics is given by
a superconformal eld theory (SCFT) with central charge 6N1N5. There is now strong
evidence that there is a locus in moduli space where this SCFT is a symmetric product
orbifold theory with target space M
N =SN , where N = N1N5 [2]. This orbifold point is
naturally thought of as the weakcoupling limit on the eld theory side, and it is far in
moduli space from the region in which supergravity is weakly coupled (see for example the
review [3]).
The D1D5 system has eight supersymmetries that, in terms of left and rightmoving
modes, are N = (4; 4). Adding NP units of leftmoving momentum breaks the
supersymmetry to (0; 4). The state counting emerges from the ways of partitioning this momentum
amongst the fundamental excitations of the SCFT, and the result matches the entropy of
the threecharge black hole in
ve dimensions.
The D1D5 black string in six dimensions has a nearhorizon limit that is AdS3
The holographic duality between the strongly coupled D1D5 CFT and supergravity on
AdS3
S3 has been widely studied and is, as far as these things go, relatively well
understood. In particular, one can often do \weakcoupling" calculations in the orbifold CFT
S3.
that can be mapped to \strongcoupling" physics described by the AdS3
S3 supergravity
dual of this theory. The computation of BPS black hole entropy is one such example.
However the holographic duality gives much more information than simple entropy counts,
enabling the study of strongcoupling physics of individual microstates of the CFT, and
their bulk descriptions (see, for example, [4{14]).
The second approach is to use the MaldacenaStromingerWitten (MSW) [15] \string."
Here one starts with a compacti cation of Mtheory on a CalabiYau threefold to
ve
dimensions. One then wraps an M5 brane around a suitably chosen divisor to obtain a
(1 + 1)dimensional string in ve dimensions. This breaks the supersymmetry to N = (0; 4)
right from the outset. There is a (1+1)dimensional SCFT on the worldvolume of this string
and its central charge is proportional to the number of moduli of the wrapped brane.1 This
1The central charge can be computed in terms of the intersection properties of the divisor through a
simple index theorem.
{ 1 {
MSW string is wrapped around another compacti cation circle to obtain a fourdimensional
compact bound state. One can add momentum excitations to the MSW string in a manner
that preserves the (0; 4) supersymmetry, and the entropy of these excitations matches (to
leading order) the entropy of the corresponding fourdimensional BPS black hole.
However, despite multiple attempts over the past twenty years, the holographic
description of the MSW black hole remains much more mysterious. The stronglycoupled
physics of this CFT is described by weaklycoupled supergravity in an asymptotically
AdS3
symmetricorbifold CFT anywhere in the moduli space [16, 17].
The purpose of this paper is to construct a map between a large subsector of the
MSW CFT and a large subsector of the D1D5 CFT. Our construction has two steps.
We rst construct a map between the Type IIB [global AdS3]
S
3
T4 solution dual to
the NS vacuum of the D1D5 CFT (and Z
orbifolds thereof) and the Mtheory [global
AdS3]
solutions are expanded into three sets of Fourier modes, labelled by (k; m; n). Our map
takes the modes with k = 2m to asymptotically AdS3
S2 solutions dual to
momentumcarrying microstates of the MSW CFT. Since, in principle, D1D5P superstrata with
generic (k; m; n) are described by functions of three variables,3 the restriction to enable
the map to MSW CFT reduces this to functions of two variables. In practice, in this paper
we have constructed solutions with k = 2, m = 1 and generic n and so our Mtheory
superstrata are parameterized by one integer n. Extrapolating to superpositions of two modes
will give, by the linearity of the BPS equations of sixdimensional supergravity, smooth
solutions parameterized by functions of one variable.4 Either way, we are able to build the
largest known class of smooth microstate geometries for the MSW black hole.
Precise dual CFT states for superstrata solutions have been identi ed [11{14], and our
map indicates that there is a onetoone map between the subset of these states that are
eigenstates of the Rcurrent JL3 corresponding to rotations around the Hopf ber of the
S3, and a certain class of states of the MSW CFT. Given that the MSW CFT does not
2More precisely, beginning with a set of D1D5 RR ground states that are related to (Z orbifolds of)
[global AdS3]
S
3
T4 [18, 19], we construct a map to the MSW maximallyspinning Ramond ground
state, which is related to [global AdS3]
S
2
T6 by spectral ow.
3Smooth solutions with generic Fourier modes have not yet been explicitly constructed, and it is
conceivable that interactions between such generic modes could introduce new, unanticipated singularities. This
is why we are using the phrasing \in principle."
4Another way to build Mtheory superstrata parameterized by a function of one variable is to impose
the k = 2m condition on the superstrata constructed in [11].
{ 2 {
seem to have a weaklycoupled symmetric orbifold description, the fact that one can map
a sector of this CFT into a sector of the D1D5 CFT may provide leverage in analyzing
some aspects of the MSW CFT, such as the set of protected threepoint functions where
two of the operators are heavy and one is light.
The [global AdS3]
S
2
T6 solution dual to the NS vacuum of the MSW CFT can be
obtained as an uplift of a Type IIA con guration with two
uxed D6 branes of opposite
charges (the uxes give rise to D4 charges which uplift to the M5 charges of the Mtheory
solution). The geometric transition that employs uxed D6 (and antiD6) branes to convert
black holes and black rings into smooth, horizonless geometries was rst described in [20,
21]. A particular example of this was studied in [22] in which a
uxed D6D6 bound state
the solution [22, 24, 25]. Uplifting this supergravity solution to Mtheory gives rise to a
singular supergravity PPwave solution, carrying angular momentum along both AdS3 and
S2; however, this naive extrapolation ignores the nonAbelian and nonlinear dynamics of
multiple D0branes. A second way to add momentum charge is to add a gas of supergravity
modes directly in Mtheory, in the smooth [global AdS3]
S
2
T6 solution. The entropy
of this \supergraviton gas" [26] scales in the same way as the added D0 branes described
above [25]. However the backreaction of this supergraviton gas was not constructed.
The solutions we nd are smooth Mtheory geometries carrying the same charges as
the foregoing ensembles of states, the D0's and the supergraviton gas, and so it is natural
to think of our solutions as examples of fully backreacted smooth geometries associated
to the supergraviton gas, or to correctlyuplifted D0 branes. Indeed, the linearized limit
of the superstratum modes is explicitly a supergraviton gas in [global AdS3]
Hence, at least outside the blackhole regime of parameters, it may well be that some of
S
2
our Mtheory solutions are fully backreacted supergraviton gas states.
In the blackhole regime of parameters, one desires more entropy than is provided
by the supergraviton gas. There are other methods of incorporating D0brane charge in
this regime, which often go beyond supergravity. For example, one can place branes in
the type IIA background that carry D0 charge as a worldvolume
ux. One possibility
is to add a D4D2D2D0 center, which uplifts in Mtheory to a M5M2M2P supertube
that rotates along the AdS3 [27]. Since supertubes can have arbitrary shapes [28], the
solutions corresponding to these con gurations can have a nontrivial dependence on the
Mtheory circle; however, like the pure D0brane sources, these are again naively singular
con gurations. Estimates of the entropy of backreacted solutions thus far yield results
subleading relative to the black hole entropy [27].
Another possibility is to add D0 branes via worldvolume ux on a dipolar, eggshaped
D2brane [22]. Counting the Landau levels of this twobrane has been argued to reproduce
the BPS black hole entropy. This \eggbrane" uplifts in M theory to an M2 brane wrapping
{ 3 {
the S2 and spinning in AdS3, and the corresponding solution is also singular [29, 30].
Moreover, simply wrapping branes around this S2 adds yet another charge to the system, which
either (a) introduces an uncanceled tadpole which changes the asymptotics of the
supergravity elds; or (b) breaks all of the supersymmetry [31]. Either way, such con gurations
cannot represent BPS microstates of the original black hole.
It is possible that smooth geometric oscillations of superstrata in deep scaling
geometries might contribute a nite fraction of black hole entropy [32], but this is by no means
proven. Such states lie well beyond the consideration of a supergraviton gas in the [global
AdS3] S2 background. Our construction gives new possibilities for deep superstrata in the
Mtheory frame, and thus represent another advance in the quest for a geometrical
understanding of black hole entropy. The elds that make smoothness of superstrata geometries
possible are exactly of the kind one expects to see when one considers the backreaction
of the momentumcarrying M5 brane source in the MSW system, or that are generated in
string emission calculations [33] in a Udual fourcharge con guration of D3 branes [34].5
We believe that this is not a coincidence but rather an indication that our construction is
closing in on a good holographic description of the microstates of this system.
Besides its interest for understanding the MSW CFT, our map is also a very powerful
solutiongenerating device. Indeed, we will use it to construct new smooth solutions of
vedimensional ungauged supergravity that are, in principle, parameterized by arbitrary
functions of at least one variable.6 There is a long history of constructing smooth solutions
in this theory [20, 21, 35]. However, while these solutions have nontrivial topology, they
also have much symmetry; the solution spaces depend on several continuous parameters
that describe the location of the topological bubbles. Until now it was not known how to
construct smooth solutions in these theories parameterized by arbitrary continuous
functions  such solutions were believed to exist only in supergravity theories in spacetime
dimensions greater than or equal to six, such as those in [4, 11, 13, 14, 36]. Our map
thus, in principle, yields the largest family, to date, of smooth solutions of vedimensional
ungauged supergravity. It also establishes that vedimensional supergravity can capture
smooth, horizonless solutions with blackhole charges, to a much greater extent than
previously thought.
The structure of the presentation is as follows. In section 2 we introduce the class
of sixdimensional BPS D1D5P geometries of interest, and the BPS equations that they
satisfy.
torus
We work with asymptotically AdS3
S3 geometries that can be written as a
bration, with the
ber coordinates (v; ) asymptotically identi ed with (roughly
speaking) the AdS3 angular coordinate and the Hopf ber coordinate of S3 respectively.
We introduce maps that involve an SL(2; Q), action on the torus ber and a rede nition of
the periodicities of these coordinates, and call these maps \spectral transformations". In
section 3 we illustrate the action of a particular spectral transformation on the example of
the round, wound multiwound supertube solution. This transformation introduces KK
5These elds are absent in the solutions of [24, 27, 29, 30].
6For our explicit example solutions we will restrict attention to a subclass of solutions parameterized by
one integer, however by the above discussion, the broader family of these solutions is in principle described
by arbitrary functions of at least one variable.
{ 4 {
monopole charge into the D1D5 system. We then recall the known Uduality that relates
the D1D5KKM system to the MSW system on T6 (or T2
K3). In section 4 we derive the
e ect of general SL(2; Q) transformations on the sixdimensional metric and gauge elds. In
section 5 we apply our particular transformation to the D1D5P superstrata of [11, 14, 37],
and we work out an explicit example in detail in section 6. In section 7 we investigate the
question of whether there is a weakly coupled symmetric orbifold CFT in the moduli space
of the MSW system, as there is for D1D5. When the compacti cation manifold, M, is T4,
the energetics of U(1) charged excitations can be inferred from a supergravity analysis [17],
and places strong constraints on the CFT, leading to a nogo theorem. In section 8 we
discuss our results, and the appendices contain various technical details.
2
2.1
D1D5P BPS solutions and spectral transformations
D1D5P BPS solutions
In the D1D5P frame, we work in type IIB string theory on M4;1
S
1
M is either T4 or K3. We shall take the size of M to be microscopic, and the S1 to be
macroscopic. The S1 is parameterized by the coordinate y which we take to have radius Ry,
M, where
y
y + 2 Ry :
We reduce on M and work in the lowenergy supergravity limit. That is, we work with
sixdimensional, N = 1 supergravity coupled to two (antiselfdual) tensor multiplets. This
theory contains all the elds expected from D1D5P string worldsheet calculations [33].
The system of equations describing all 18 BPS, D1D5P solutions of this theory was found
in [38]; it is a generalization of the system discussed in [39, 40] and greatly simpli ed in [41].
For supersymmetric solutions, the metric on M takes the local form:
coordinate, y, and a time coordinate t via
1
2
u = p (t
y) ;
v = p (t + y) :
1
2
However, there is some freedom in choosing such a relation, since the form of the metric
(and the ansatz in general) is invariant under the shift
u0
u
1
2 c0v ;
F
0
F + c0 ;
!0
!
1
2 c0 :
Using this freedom, we will shortly choose a di erent relation between u, v, t and y that is
more natural for spectral transformations and for reduction to ve dimensions.
While all the ansatz quantities may in principle depend upon v, throughout this paper
we shall require the metric, ds24(B), on the fourdimensional spatial base, and the bration
{ 5 {
(2.1)
(2.2)
(2.3)
(2.4)
vector, , to be independent of v. This greatly simpli es the BPS equations and, in
particular, requires that the base metric be hyperKahler and that d
be selfdual on B.
The metric and tensor gauge elds are determined as follows. We introduce an index
I = 1; : : : ; 4, and an index a that excludes I = 3 (which plays a preferred role): a = 1; 2; 4.
The ansatz then contains four functions Za and F , and four selfdual 2forms,
1; : : : ; 4. These can depend both upon the base, B, and upon the v
ber. The function, F ,
(I), I =
appears directly in (2.2) and the warp factor, P, in the metric is given by
The vector eld, , de nes
(3):
(3)
P = Z1 Z2
Z42 :
d ;
The individual functions, Za, and the remaining 2forms,
magnetic components of the tensor gauge elds.
(a), encode the electric and
Recall that the N = 1 supergravity
multiplet contains a selfdual tensor gauge eld, so that adding two antiselfdual tensor
multiplets means that the theory contains three tensor gauge elds.
Roughly speaking, the pairs (Z1; (2)) and (Z2; (1)) describe the elds sourced by
the D1 and D5 brane distributions. The function, F , and the vector eld, , encode the
details of the third momentum charge. In the IIB description, the addition of (Z4; (4))
allows for a nontrivial NSNS B eld as well as a linear combination of the RR axion
and fourform potential with all legs in the internal space M; these elds arise in D1D5P
string worldsheet calculations [33], so are expected to be generically present. For more
details, see [38].
The remaining simpli ed BPS equations come in two layers of linear equations. To
write them, we denote by d(4) the exterior derivative on the fourdimensional base, and
we de ne the operator, D, acting on a pform with legs on the fourdimensional base (and
possibly depending on v), by:
{ 6 {
D
d(4)
The rst layer of equations determines the Maxwell data. For notational convenience
throughout the paper, we work with a form of the BPS equations that is not explicitly
covariant in the indices a = 1; 2; 4. In particular, we will always label
index, while continuing to refer to these quantities collectively as
4 with a downstairs
(a); hopefully this will
not cause confusion. The covariant form of the BPS equations is given in appendix A.7 In
our conventions, the rst layer of the BPS equations takes the form
4DZ_1 =D (2) ;
4DZ_2 =D (1) ;
4DZ_4 =D 4 ;
D 4 DZ1 =
D 4 DZ2 =
D 4 DZ4 =
(2) ^ d ;
(1) ^ d ;
4 ^ d ;
(2) = 4 (2) ;
(1) = 4 (1) ;
4 = 4 4 :
7To pass to the covariant form, one rescales (Z4; 4; G4) ! (Z4; 4; G4)=p2; more details are given in
appendix A.
The second layer of equations determines the other parts of the metric in terms of the
Maxwell data:
2 4
Z2)
4
(1)
^
(Z_1Z_2
(Z_4)2)
(2)
4 ^
the following simple parametrization of solutions to (2.6):
=
K3
V
(d
ponents parallel and perpendicular to the  ber:
! =
(d
+ A) + $ :
We record here the ansatz for the threeform eld strengths in terms of the above data. A
discussion of how these eld strengths appear in the corresponding Type IIB ansatz may
be found in [38] and a simpli ed version without (Z4; 4) may be found in [41]. The BPS
ansatz for the uxes, where ds24 and
are vindependent, is given by:8
G(1) = d
G(2) = d
G4 = d
These elds satisfy a twisted selfduality condition; since this is most conveniently expressed
in covariant form [43] (see also [44]), we give it in appendix A.
2.2
Canonical transformations
As noted above around eq. (2.4), there is some freedom in relating the (u; v) coordinates to
the time and spatial coordinates, (t; y). As will shortly become clear, it will be convenient
for us to use the following relation throughout this paper:
u = t ;
v = t + y ;
y = y + 2 Ry :
(2.14)
8Note that, following [42], we have rescaled (1;2) ! 21 (1;2) relative to the conventions of [41]. See also
Footnote 7.
{ 7 {
(2.9)
(2.11)
(2.12)
2
One should note that the coordinates (t; y) are the same as those in eq. (2.3). To get from
eq. (2.3) to the above relation, one can make a shift (2.4) with c0 =
rescaling of u0 by p1 and v by p2, together with accompanying rescalings of the ansatz
2, followed by a
quantities, as follows. First, we take c0 =
2 in (2.4) and de ne:
u0
u + v ;
F
0
F
2 ;
!0
! +
One then arrives at (2.14) by dropping all the tildes.
Most importantly, with these rescalings, the ansatz for the metric, the ansatz for the
uxes and the BPS equations remain unchanged: the factors of p
2 cancel throughout.
Thus we are free to use either coordinate representation, (2.3) or (2.14), solve the BPS
equations and substitute into the ansatze: both will produce BPS solutions. The resulting
solutions will of course be related by (2.15) and (2.16). We illustrate this point with a
simple supertube solution in appendix B.
The u = t parameterization is much more convenient when comparing sixdimensional
solutions to vedimensional solutions. Assuming that F is everywhere negative, as it will
be in our solutions, one can write the metric as:
1
P F
ds62 = p
(du + !)2
F
p
P
dv +
+ F
1(du + !)
2 +
p
Since F is everywhere negative, the v coordinate is everywhere spacelike. Solutions with
an isometry along v can be reduced on v to obtain
vedimensional solutions. In this
reduction, u is the natural time coordinate in ve dimensions. This is the advantage of the
Finally, to be clear: in this paper we will use (2.14) and u = t will be kept xed in all
u = t parameterization for our purposes.
spectral transformations.
2.3
General spectral transformations
Note that for solutions with a GH base, the sixdimensional solution in the
form (2.17), (2.10) is written as a double circle
bration, de ned by (v; ), over the R
base of the GH metric. In this paper we will exploit a set of maps that involve coordinate
3
transformations of the (v; ) coordinates. We consider maps that act on (v; ) with
elements of SL(2; Q) and not just SL(2; Z), and so in general one must be careful to specify
how the map acts on the lattice of periodic identi cations of these coordinates. Our maps
consist of a composition of a coordinate transformation and an accompanying rede nition
of the lattice of coordinate identi cations.
The coordinate transformation component of our map is an SL(2; Q) map that
transforms a solution written in terms of (v; ) coordinates to a solution written in terms of
{ 8 {
new coordinates (v^; ^). We parameterize the SL(2; Q) action by rational numbers a; b; c
and d subject to ad
bc = 1, as follows:
v
R
= a
v^
R
For later convenience, in the above we have introduced the shorthand R for the ratio of the
periodicities of the v and
coordinates, so that the linear transformation acts on circles
with the same period of 4 . We emphasize again that the coordinate u is held
xed.
The lattice rede nition component of our map is as follows. We consider starting
con gurations for which the lattice of identi cations is9
We de ne the new lattice of identi cations of the new solution to be
v = v + 2 Ry ;
=
+ 4 :
that is, the new lattice is not the one that would follow from making the coordinate
transformation (2.18) on the original lattice (2.19), but is rede ned to be (2.20).
The fact that the lattice is rede ned means that when the parameters of the map are
noninteger, the maps are in general not di eomorphisms and can modify the presence or
absence of orbifold singularities in the spacetime, as has been observed in fractional spectral
ow transformations [10].
We illustrate the above procedure by reviewing the example
of fractional spectral ow transformations of multiwound circular D1D5 supertubes in
appendix C. We will refer to these maps as \spectral transformations".
Having made the above transformation, one can recast the metric and tensor gauge
elds back into their BPS form but in terms of the new coordinates, (v^; ^). For example,
one substitutes the coordinate change (2.18) into the metric (2.17) and (2.10), and then
rewrites the result as:
1
p
Pb Fb
+ p
This rearrangement of the background metric and tensor gauge elds in terms of the new
bers de nes new `hatted' functions and di erential forms in terms of the old functions and
forms. We will derive the explicit transformation rules for the individual ansatz quantities
in section 4, and use these rules to transform the family of superstrata solutions that
we consider.
After this transformation the local metric is still the same as the original one, so the
background is still locally supersymmetric, and so the hatted ansatz quantities solve the
BPS equations in the form (2.8), (2.9). More speci cally, if the original functions and forms
9For ease of exposition, here we suppress possible additional identi cations that involve
and an angle
in the threedimensional base; we will be more precise when we discuss explicit examples later.
{ 9 {
in the solution depend upon (v; ) then that dependence must, of course, be transformed
to (v^; ^) using (2.18), and the BPS equations satis ed by the hatted quantities will be
those of (2.8) and (2.9) but with (v; ) replaced by (v^; ^).
While the transformed solution is still locally supersymmetric, it is possible that the
rede ned lattice of identi cations may break some supersymmetry; indeed, we shall see
that the transformation that we employ in the current work will break half of the eight
real supersymmetries preserved by the D1D5 circular supertube solution.
Fivedimensional solutions and spectral transformations
Solutions that are independent of v can be dimensionally reduced from six to ve
dimensions. The BPS equations become those of N = 2 supergravity coupled to three vector
multiplets. In particular, the description of the four vector elds of this theory involves
totally symmetric structure constants, CIJK . Indeed, for the system we are considering
one has
with all other independent components equal to zero.10
written as follows [35]:11
The complete family of smooth solutions that are also
independent may then be
(2.22)
(2.23)
(2.24)
(2.25)
(2.26)
(I) = dBI ;
ZI = LI +
CIJK
1
2
=
M
2
+
KI LI +
We now reduce the metric, (2.17), on the v ber. Following (2.14), we set u = t, and we
relabel ! and F in terms of their more standard vedimensional analogs:
k
! ;
Z3
F :
This yields the standard vedimensional metric:
10One can convert this to the canonical normalization (in which C344 = 1) by the procedure described
in Footnote 7.
11Note that our convention for M di ers from that of [35] by a factor of 2.
Spectral transformations for vindependent solutions
The role of SL(2; Z) spectral transformations on (v; ) was studied in detail for
vindependent solutions in [45]. In particular, the spectral transformations could be reduced
to transformations on the harmonic functions V; KI ; LI and M . Moreover, from the
vedimensional perspective, any of the Maxwell elds can be promoted to the KaluzaKlein
eld of the sixdimensional formulation and so there as many di erent SL(2; Z) spectral
transformations as there are vector elds. Moreover, these SL(2; Z) actions do not commute
and, in fact, generate some even larger subgroup of the Uduality group.
The original study of spectral transformations was made for the system with two vector
0) but the results can be recast in a form that is valid for
vedimensional N = 2 supergravity coupled to (NV
1) vector multiplets, and so we will
give the relevant general results.
The spectral transformations considered in [45] included two important subclasses:
\gauge transformations" and \generalized spectral ows". A gauge transformation is
generated by choosing one of the Maxwell elds as the KK
eld, then leaving
xed and
shifting v by a multiple of . The choices of uplift lead to NV gauge parameters, gI , and
the gauge transformations reshu e the harmonic functions according to:
Vb = V ;
LbI = LI
Mb = M
b
KI = KI + gI V ;
CIJK gJ KK
gI LI +
1
2 CIJK gJ gK V ;
1
1
2 CIJK gI gJ KK +
3! CIJK gI gJ gK V :
While this is a highly nontrivial action on the harmonic functions, this transformation
leaves the physical elds, ZI ; (I),
transformations.
and $ invariant, and hence their designation as gauge
Spectral ows are induced by keeping v xed and shifting
by a multiple of v. Again
there are NV ways to do this with NV parameters, I , resulting in the full family of
generalized spectral ow transformations. In terms of CIJK de ned as
CIJK
II0 JJ0 KK0 CI0J0K0 ;
generalized spectral ow transformations act on the harmonic functions as follows:
Mb = M ;
b
KI = KI
Vb = V + I KI
LbI = LI
I M ;
CIJK J LK +
2
1 CIJK J K M ;
2
1 CIJK I J LK +
1
3!
In contrast to the gauge transformations, these transformations have a complicated and
very nontrivial action on the vedimensional physical elds (see [45]).
In our conventions, the polarization direction I = 3 in eq. (2.29) corresponds to the
(KaluzaKlein) vector eld in
ve dimensions that lifts to metric in six dimensions. We
reserve the term \spectral ow" for generalized spectral ows in this polarization direction.
(2.27)
(2.28)
(2.29)
Spectral ow transformations have the same e ect as the following large coordinate
transformation, where the new coordinates are denoted with a hat:
and where the other coordinates are invariant. In the D1D5 system CFT, the world
volume of the CFT lies along the v ber and translations along this ber are generated by
the Hamiltonian, L0. The  ber lies transverse to the D1 and D5 branes and so represents
an Rsymmetry transformation. The above transformation is thus a CFT spectral ow;
for a more detailed discussion, see [5, 6, 10].
Similarly, a gauge transformation in the polarization direction 3 with parameter g3 has
the same e ect as the following coordinate transformation, where again the new coordinates
are denoted with a hat:
v = v^ + g3 ^ ;
= ^ ;
(2.30)
(2.31)
and where the other coordinates are invariant. Note that, when the worldvolume of
the CFT lies along the v ber, this seemingly trivial (from a supergravity point of view)
transformation does not appear to have a simple interpretation in the dual CFT. The
transformation would appear to reorient the worldvolume of the CFT, and the question
of whether there is any sensible holographic interpretation of this gravity transformation
remains somewhat mysterious.
However, if one interchanges the roles of
and v such that the worldvolume of the CFT
lies along the  ber, and the Hopf ber of the S3 lies along v, then (in our conventions)
the above gauge transformation would correspond to spectral ow in the leftmoving sector
of the CFT.
3
The multiwound supertube and mapping D1D5 to MSW
Before proceeding to general spectral transformations, it is very instructive to see how
spectral transformations act on one of the most important vindependent BPS solutions:
the multiwound supertube [18, 19, 28, 46]. We start with its standard formulation as the
smooth geometry of the D1D5 supertube, and we map it to Mtheory with an SL(2; Q)
spectral transformation and a Uduality transformation. The SL(2; Q) spectral
transformation introduces a KKM charge along the Hopf ber of the S3, and the D1, D5 and KKM
charges transform under the Uduality into three independent M5brane charges
underlying an MSW string, where the resulting con guration is a particular form of that string,
with speci c dissolved M2brane charges and speci c angular momenta.
There is the following interesting conundrum: the D1D5 supertube is 14 BPS,
preserving eight supersymmetries, while the MSW string is 18 BPS and preserves only four
supersymmetries. We reconcile this di erence by carefully examining the lattice of
identications, and showing that our transformation accounts for the change in the number of
supersymmetries.
3.1
take F =
with (2.14) is given in appendix B.
The canonical starting point for the multiwound, circular D1D5 supertube is the
ansatz (2.2) with a time coordinate, t, and the asymptotic S1 coordinate, y, related to
the coordinates u; v via (2.3) and with F = 0. However, as we stipulated earlier, we are
going to use (2.14), and the transformations (2.15) and (2.16) then imply that one must
1. The precise relation between the supertube with (2.3) and the supertube
The wound supertube is a twocentered con guration de ned by the following
harmonic functions:
V =
L1 =
1
r+
;
Q1 ;
The base metric is at R4, which we write as
The remaining ansatz quantities for the supertube are then:
Z1 =
=
Q1
;
Rya2
The parameters are subject to the following regularity condition:
Q1Q5 =
2Ry2 a2 :
This solution may be written in GibbonsHawking form by de ning new coordinates,
( ; ; # ) via:
One then has
1
r
+ A)2 + V hdr+2 + r+2 (d#2+ + sin2 #+ d 2)i ;
V =
1
r+
=
4
;
A = cos #+ d
=
(r2 + a2) sin2
r2 cos2
(d'1
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
measure the distances in the at threedimensional base between two
centers, de ned by r
= 0; one can choose Cartesian coordinates in which we have
r
=
This solution describes a wound supertube, whose KKM dipole moment is . Given the
above choice of gauge for the oneform A, the lattice of identi cations for this solution is
HJEP06(217)3
generated by12
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(y; ; )
(y; ; )
(y; ; )
(y + 2 Ry; ; ) ;
(y;
(y;
+ 4 ; ) ;
+ 2 ;
+ 2 ) :
There is a Z orbifold singularity at the supertube locus, as we will now review.
Introducing the coordinates r=a sinh ; = 2 ; y~ =
y
Ry
;
t~ =
the metric can be written as
;
2
Under the further change of coordinates
we observe that the metric is locally AdS3 S3,
patches around the North and South Poles of the S2 respectively. The identi cations on 0 and
are then
simply 0 = 0 + 4 at xed , and
=
+ 2 at xed
y~; ~; ~
y~; ~; ~
y~; ~; ~
y~ +
2
; ~
2
; ~
2
;
y~; ~ + 4 ; ~ ;
y~; ~ + 2 ; ~ + 2
= 1, the full geometry is simply global AdS3 S3, and when
identi cation above is an orbifold identi cation that combines the AdS3 and the S3, and
that gives rise to a Z
orbifold singularity at the location of the supertube, (r = 0;
=
Spectral transformations of the supertube
We now perform an SL(2; Q) map on the above multiwound supertube con guration, that
maps the solution to a form in which it can be straightforwardly dualized to the MSW
frame. Our SL(2; Q) map can be decomposed into a product of gauge and spectral ow
transformations, and it will be instructive to go through these steps.
We rst perform a gauge transformation with parameters (g1; g2; g3) = (0; 0; 12 Ry).
The resulting harmonic functions are:
V =
L1 =
;
1
K1 = K2 = 0 ;
K3 =
L2 =
Q5 ;
4 r
Ry 1
2 r
;
We next perform a (fractional) spectral ow transformation with parameters ( 1; 2; 3) =
(0; 0; 2=( Ry)). The resulting harmonic functions are:
V =
L1 =
1
; K1 =
L2 =
Q5
1
2 Ry r
Q5 ;
4 r
; K2 =
L3 =
Q1
1
2 Ry r
Q1Q5
4 2Ry2 r
;
;
K3 =
M =
Ry 1
2 r
;
Ry +
2
=
( 4QR5y ; 4QR1y ; R4y ). The resulting harmonic functions are (here I = 1; 2; 3 and we employ
notation mod 3 for the I indices):
where we have de ned
V =
LI =
1
r+
;
k
2
KI =
M =
Q1 ;
2 Ry
1
r+
k
I
2
k k k
+
Ry
2
;
1
r
:
2c
: (3.18)
(3.19)
These harmonic functions are those that describe the MSW maximallycharged Ramond
ground state solution in
ve dimensions (related by rightmoving spectral ow to the NS
vacuum) [22, 47], as reviewed in appendix D. We will review the duality map momentarily.
The combination of these transformations corresponds to the following SL(2; Q) map
on the coordinates:
v
Ry
=
and where as discussed above, we rede ne the new lattice of identi cations. The new
lattice is generated by the appropriate smooth identi cations in the Mtheory frame;
combining (2.20) with the appropriate smooth identi cation on
as discussed around (D.10),
the new identi cations are:
where for each of the three generators in this equation, one holds the other two periodic
coordinates in the equation
xed.
becomes
Note that in the tilded coordinates, de ned in (3.13), the coordinate transformation
v
Ry
=
^
2
;
~ =
v^
k3
;
(u; ) xed ;
which can be described as a (fractional) \spectral interchange" transformation [42] between
rotating versions of the AdS3 circle coordinate and the Hopf ber coordinate of the S3.
Indeed, under this transformation, the metric (3.12) transforms to:
(3.21)
(3.22)
(3.23)
(3.24)
(3.25)
cosh2 dt~2 + d 2 + sinh2 d'2
where we de ne
+
1
4
dv^
k3 + cos d ~
2
It is interesting to reinterpret the coordinate transformation (3.22) in terms of the D1D5
and MSW CFTs. The relation between ~ and
in (3.13) corresponds to spectral ow
in the leftmoving sector of the D1D5 CFT; the solution in terms of
corresponds to a
particular Ramond ground state, while the solution in terms of ~ corresponds to the NS
vacuum of the leftmoving sector. (The analogous statement holds for ~ and
in terms
of spectral ow in the rightmoving sector of the CFT.) The coordinate transformation in
the tilded coordinates (3.22) is interesting, as we see from it that v^ is a rescaled version of
~, the leftmoving NS sector coordinate.
One can paraphrase these observations by describing the metric (3.23) as being written
in NSNS sector coordinates, corresponding to the NSNS vacuum of the dual CFT state,
which has
L0 = L0 = 0 :
Qe1 =
Q1
Qe5 =
Q5
:
Qe1 =
gs 03n1
V4
;
Qe5 = gs 0n5 ;
n1 =
N1
n5 =
N5
:
Recalling the usual relation between Qe1;5 and n1;5:
we see that the relation between the integer brane numbers on the two sides of the map is
If one rewrites ~ in terms of , one can describe the metric as being expressed in NSR
sector coordinates, and corresponding to the NSR ground state obtained from the NSNS
vacuum via rightmoving spectral ow with parameter 1/2, which has
ber, and these quantum numbers will correspond
respectively to the NS vacuum and the maximallycharged R ground state of the MSW
CFT (which we again emphasize are related by rightmoving spectral ow).
Let us analyze the lattice of identi cations (3.21) that has resulted from our
transformation. The AdS3 angle coordinate ' has period 2 , which is the correct periodicity for
a smooth global AdS3. The combination that appears in place of the Hopf ber of the S3
is v^=k3, which has period 4 = , corresponding to a smooth Z quotient of the Hopf ber,
appropriate for the decoupling limit of a D1D5KKM con guration with KKM charge .
Note now that the relation between the dimensionful parameters Q1, Q5 and the integer
number of D1 and D5 branes n1, n5 that correspond to this solution is changed, relative
to the solution without KKM charge. The Gaussian integral de ning the charges is now
done over a range of the Hopf ber coordinate that is smaller by a factor of , so that the
actual new supergravity charges are
;
;
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
We note that using (3.27), the relation (3.19) becomes
k1 =
Qe5 ;
2Ry
k2 =
Qe1 ;
2Ry
k3 =
Ry
2
:
so that, up to constant factors, the parameters kI correspond to the D1, D5 and KKM
charges, which map to the three di erent M5 charges in the M5M5M5 duality frame.
The other e ect of the rede ned lattice of identi cations is that, in the asymptotically
AdS3
S3 geometry, it breaks the SU(2)L
SU(2)R symmetry of the S3 down to U(1)L
SU(2)R. Since this is the Rsymmetry, it must break the N = 4 superalgebra of the
leftmoving sector down to an N = 2 superalgebra with this U(1)L Rsymmetry. Since
these remaining supercharges are charged under the translations along the Hopf ber, they
will not survive the dualization to Mtheory and thus the e ect of reassigning the lattice
identi cations and compactifying is to break all the leftmoving supersymmetries even in
the groundstate con guration we are studying here.
so we obtain
with
The duality map from D1D5KKM to Mtheory involves Tduality on the Hopf ber of the
S3 and two directions in the T4, followed by an Mtheory lift.13 This results in a solution
which asymptotically has a compact T6. The e ect of these dualities from the point of
view of the lowerdimensional theory is encoded in a dimensional reduction on the Hopf
ber of the S3, to ve dimensions (see e.g. [51, 52]).
The SL(2; Q) transformation we have performed means that the ansatz quantities have
already been rearranged to make this step straightforward: the coordinate v^ is precisely
the Hopf ber of the S3.
The metric (3.23) can be written (using an obvious shorthand) as
ds62 = pQ1Q5 ds2AdS3 +
pQ1Q5
4(k3)2
dv^ + k3 cos d ~ 2
+
4
pQ1Q5 ds2S2 :
The reduction ansatz for the sixdimensional metric takes the form
ds62
e 3A(dv^ + A^(3))2 + eA ds52 ;
Using the relation (3.19), and as reviewed in appendix D, this becomes exactly the
decoupled M5M5M5 metric in
ve dimensions that results from the set of harmonic
functions (3.18) [22] (see also [27, 47, 53]):
Note that the smooth Z quotient on the Hopf ber has migrated into an M5 charge, and
thus a parameter in the warp factors, and the vedimensional solution is smooth AdS3 S
with standard coordinate identi cations.
4
General SL(2; Q) transformations in six dimensions
Having understood how to map the groundstate of the D1D5 system onto that of the
MSW string, we now wish to extend our results to the transformation to families of
leftmoving excitations. This includes spectral transformations of superstrata [11, 13, 14].
We start more generally by considering a generic BPS background that can depend on
all of the coordinates, except, of course, u. We recall our parameterization of the general
spectral transformations on (v; ) from eq. (2.18):
13For early works on reduction and Tduality along Hopf bers, see [48{50].
v^
R
= a
where
The expression for ^(2) then simpli es signi cantly:
where in turn d^(4) is the exterior derivative on the transformed fourdimensional base. The
remaining uxes b
(1) and b 4 may be obtained similarly.
To reduce to ve dimensions we require solutions that are independent of the v^ ber. This
means restricting the modes to those with k = 2m, and hence with a phase, ^k;m;n, in (5.14)
given by:
^2m;m;n =
(n + m) ^
m :
(2) satisfy the rst layer of the equations:
One can then nd b
(2) using (4.19), whereupon one can explicitly verify that Zb1 and
Db b (2) = 0 ;
Db ^4Db Zb1 =
b
(2)
^ d b ;
b
(2) = ^4 b (2) ;
D
b
d^(4)
;
(5.28)
(5.29)
(5.30)
(5.32)
1
2
a2(m + n)
4 Ry
^(2) =
1
: (5.31)
The BPS equations also reduce to their vedimensional form and, in particular, (5.28)
implies that b
(2), is selfdual and closed:
d^b (2) = 0 ;
b
(2) = ^4 b (2) ;
and is thus \harmonic" on the GH base. One can explicitly verify this using (5.31)
and (4.19).
The harmonic forms on a standard Riemannian GH base are wellknown (see, for
example, [35]). For NC GH centers there are (NC
1) independent, smooth harmonic forms
given by the expressions in (2.23). In particular, these harmonic forms are independent of
the angles ( ; ). It may therefore seem surprising that there is, in fact, a doubly in nite
family of \harmonic forms" emerging from our solutions. However, this is because the
base is ambipolar and hence singular on the locus Vb = 0. This singular locus enables
the \harmonic forms" to have (singular) sources on this locus and thus the system admits
large families of solutions with oscillating magnetic
uxes. Again, as with everything else
in the ambipolar formulation, the physical eld strengths must be smooth. In this paper
smoothness is guaranteed because we derived the solution by a coordinate change of a
smooth sixdimensional solution.
Henceforth we will use the term pseudoharmonic forms to refer to the generalized
\harmonic forms" that are singular on the degeneration locus (Vb = 0) of an ambipolar
geometry, and yet give rise to smooth physical elds in the complete solution.
The rst analyses of vedimensional BPS solutions were done over a decade ago [20,
21, 35] and the pseudoharmonic forms were missed in that analysis. Given the ambipolar
structure of the base, many people were aware that the singular locus could allow the
presence of new sources that could generalize the usual known solutions. The problem was
that there was a vast range of singular sources available and no obvious systematic way
to nd precisely those sources that would lead to smooth physical eld strengths. That is,
the possibility, let alone the classi cation, of nontrivial pseudoharmonic forms remained
unclear. It is interesting to note that the possibility of ambipolar metrics was rst found
by Giusto and Mathur [55] by studying spectral ows of smooth supertube geometries.
In this paper we have used more general spectral transformations to discover precisely
how to go beyond the standard analogs of Riemannian harmonic forms in ve dimensions
to obtain (hopefully complete15) families of pseudoharmonic forms on our speci c
ambipolar geometry. It would be very interesting to see how pseudoharmonic forms might
be characterized, in terms of the di erential geometry, and then computed for generic
ambipolar hyperKahler metrics.
The bottom line is that we have obtained a huge class of pseudoharmonic forms and
these lead to new families of smooth
vedimensional solutions with
uctuating uxes. As
we have argued above, these solutions must be dual to microstates of the MSW string.
6
An explicit example
We now give a complete explicit example. It is one of the family of solutions discussed in
the previous section and has parameters (k; m; n) = (2; 1; n). Since k = 2m, this can be
dualized to a smooth vedimensional solution.
6.1
The D1D5P superstrata
The quantities
and ds24 are again as given in eqs. (3.4) and (2.10). The quantities of
the rst layer of the BPS equations are as given in (5.7) with (k; m; n) = (2; 1; n), for a
nonnegative integer, n, where (n + 1) is a multiple of . The phase dependence of this
solution is:
2;1;n = (n + 1)
v
Ry
)
^2;1;n =
(n + 1) ^
:
1
2
The relation between b and b4 required for regularity is
b2 =
b
2
4
Again generalizing the solution of [14] to
> 1, the solution to the second layer of BPS
equations is:
F =
1
! = !1 d'1 + !2 d'2 ;
b
2
2 a2 +
b
2
4
2 a2 sin2
4;2;2n
cos2
4 a2 +
1
;
(6.1)
(6.2)
(6.3)
Speci cally, while singular on the degeneration locus of ambipolar geometries, pseudoharmonic forms are
required to lead to smooth BPS solutions in ve dimensions.
!2 = b42 Ry
4;2;2n a2
1 +
b
4
1
4;2;2n
r2 + a2
1 +
r
2 (n + 1) a2 cos2
We record here the values of the other ansatz quantities that will be used when mapping
to the Mtheory frame:
P =
Q1Q5
2
Ry
2
1
a2 +
a2 +
2
2
2
2
Ry r2
4;2;2n ;
24 Ry
2
sin2 cos2
+ b24 Ry r2
4
4;2;2n
r2 + a2
a2
1 +
1
These quantities lead to a family of smooth, CTCfree solutions, due to the coi uring
ansatz and appropriate choices of homogeneous solutions to the BPS equations [14].
To transform to the Mtheory frame we convert the base metric to GH form and transform
the ansatz quantities recorded above. Using (4.11) and (4.2), the metric functions in the
Mtheory frame are
Zb1
Zb2 =
Pb =
Q1
Q5
Q1Q5
;
)2
+ Z(osc) =
1
Q1
Zb4 =
b
2
4
1
2 a2 + b2 4;2;2n
:
1 +
b
2
4
2 a2 + b2
b4 Ry
2;1;n cos ^2;1;n ;
4;2;2n cos ^4;2;2n
;
(6.5)
The oneforms, ^(I), are obtained from (4.18) with c =
1
,
^(1) =
1
R
D (Vb 1Z2) ;
1
R
(2)
D (Vb 1Z1) ;
^
4 =
4
where we have used that
(1) = 0 from (5.7). We nd
1
R
D (Vb 1Z4)
(6.6)
^(1) =
( + ) Zb2 ;
4 Ry
( + ) Zb1
4 Ry
( + ) Zb4
4 Ry
+
+
4 Ry
4 Ry
(n + 1) a2 h4 cot 2 Zb1(osc) d
(n + 1) a2 h2 cot 2 Zb4 d
+
i
;
(6.7)
+
i
where we recall our notation that d(3) is the exterior derivative on the R
3 base of the
GH space.
From the above expressions one obtains the b
(a) using (4.19). These are manifestly
selfdual, and it is straightforward to verify that they are indeed closed.
The transformations that yield the last layer of the BPS system are (4.6), (4.9)
and (4.10). Using these with c =
1 , we obtain:
Fb =
^ =
1
Ry
2(
2
2
2
2
2
4
4 4;2;2n
;
cos2
4;2;2n
a2 sin 2
2r2(r2 + a2)
1 + n
cot 2
a2(2r2 + a2) tan 2
;
(6.8)
One can then verify that these quantities, together with the hatted quantities and ansatz
given in (5.15){(5.20), do indeed satisfy the last layer of BPS equations for k = 2 and m = 1.
Smoothness in
ve dimensions requires that ^ and the ZbI are nite at the GH points
while the absence of CTC's requires that ^ vanishes at the GH points. The GH points lie
at (r; ) = (0; 0) and (r; ) = (0; =2) and if one sets r = 0 in (6.5) and (6.8), one has:
Zb1 =
Zb3 =
Q1
a2 cos 2
Fb =
;
1
a2 cos 2
Zb2 =
a2 +
a2 cos 2
Q5
b
2
2
; Zb4 = 0 ;
;
^ =
a2 +
tan2 2 :
Ry
4 a2
b
2
2
(6.9)
A complete analysis of the global absence of CTCs is in general a di cult problem, often
relying on numerical tests, and is beyond the scope of this paper. Here we content ourselves
with observing that the
vedimensional solution satis es the requisite local conditions,
providing evidence that the spectral transformation indeed maps the CTCfree D1D5P
superstratum onto a CTCfree solution in the Mtheory frame.
7
Comments on symmetric product orbifold CFTs
It is a tantalizing prospect that the D1D5KKM system might have a solvable CFT in its
moduli space, given that it is so similar to the D1D5 system  di ering only by a discrete
identi cation on the transverse angular S3. One might think that since, in the decoupling
limit, the introduction of KKM charge to the D1D5 geometry amounts to a Znk orbifold
of the Hopf ber of S3, that a similar quotient of the dual CFT by a chiral Rsymmetry
rotation would yield the corresponding dual CFT for the D1D5KKM system [56].16
The rst part of the construction in this paper maps a multiwound D1D5 supertube to
a D1D5KKM bound state. It is tempting to translate this into a map between states of the
D1D5 symmetric product orbifold CFT and the putative D1D5KKM symmetric product
16For related work on the microstates of the D1D5KKM system, see for example [57{63].
orbifold CFT. The multiwound D1D5 supertube con guration described in section 3
corresponds to a RR ground state of the D1D5 CFT with N1N5= strands each of winding
, with the same RR ground state on each strand.
If there existed a symmetric product D1D5KKM CFT, for n1 D1branes and n5
D5branes and KKM charge nk = , then this CFT should have total number of strands
n1n5 =
N1 N5
(7.1)
where we have used the relation between the brane numbers on the two sides of our map,
given in equation (3.29). The map appears to conserve the total number of strands, while
mapping strands of winding
in the D1D5 CFT to strands of winding 1 in the
D1D5HJEP06(217)3
KKM CFT.
However, when M = T4, strong constraints arise from the structure of U(1) currents
and the energetics of states carrying the corresponding charges [16, 17], as we now review.
Review of the D1D5 CFT
To begin, consider type IIB supergravity compacti ed on T5. The moduli space of this
theory is the 42dimensional
nE6(6)=USp(8), where the Uduality group
is E6(6)(Z).
Wrapped branes and momentum excitations transform as a 27 under this group; the
presence of the background D1D5 charge vector ~q reduces the moduli space to the
20dimensional Hq~nSO(5; 4)=(SO(5) SO(4)) through the attractor mechanism [64], and the
Uduality group reduces to the subgroup Hq~
SO(5; 4; Z)
vector decomposes as
that xes ~q. The charge
27 ! (1
9) 16
1
(7.2)
where the 1
9 represent the \heavy" charges (branes wrapping the y circle, including the
D1D5 background; the second singlet is the momentum charge along the y circle; and the
16 comprises branes and momentum along the T4 but not along the y circle.
Elements of SO(5; 4; Z) not in Hq~ do not preserve the charge vector ~q, instead they act
as nite motions on the moduli space Hq~nSO(5; 4)=(SO(5) SO(4)). Such transformations
are not symmetries of the CFT, any more than any other
nite motion on the moduli
space preserves the CFT. What such
nite motions do tell us is that, if there is a
weakcoupling cusp in the moduli space for a given pair of brane quanta (n1; n5), then there are
other cusps in the moduli space where the dual CFT becomes weakly coupled, one for each
factorization of N = N1N5 into any other pair of integers (N10 ; N50 ) with N = N10 N50 [16, 65].
Because these are motions on the moduli space and not symmetries, the existence of a locus
in the moduli space described by a symmetric product orbifold in one cusp does not imply
the existence of such a description in any other cusp.
The question then arises, in which cusp does the symmetric product orbifold (T4)N=SN
lie? The BPS mass formula for the 27 is on one hand protected by supersymmetry, and on
the other hand depends on the moduli and so determines the answer [16]. In the decoupling
limit of the D1D5 system, the energetics of the 16 is (for a rectangular torus with all the
antisymmetric tensor moduli switched o )
hR =
1
4
X
4N1 i=1
pg
pi r
i
1
4
X
4N5 i=1
wF 1
i r gs
:
(7.3)
There is no invariant notion of the \level" of a U(1) current algebra, as the normalization
of the currentcurrent twopoint function is moduli dependent. For instance, from the
previous considerations we know that SO(5; 4; Z) transformations can change the values
of n1 and n5 in the above formula. One can however compare the energetics of charged
states in the CFT with the above expression. The symmetric product orbifold has four
leftmoving translation currents (the diagonal sum of the translation currents in each copy
of T4), which realize the rst of the two terms in eq. (7.3), if we set N1 = N . The other
eight charges can be realized as the winding and momentum charges on a separate copy
of T4. The presence of this additional component of the CFT is necessary to realize all
the U(1) currents and the wrapped brane charges they couple to. Thus it is natural to
associate the symmetric product orbifold with the weakcoupling cusp of the moduli space
where the appropriate lowenergy description has N1 = N and N5 = 1.
The addition of KKmonopole charge compacti es one more dimension of the target space
 the
bered circle of the KKmonopole (the
circle), which is the Hopf
in the decoupling limit. One now has type IIB supergravity compacti ed on T6, whose
moduli space is E7(7)=SU(8). The charge vector ~q of wrapped branes and momentum on T6
transforms as a 56 of E7(7). The background D1D5KKM charges break the moduli space
down to the 28dimensional space Hq~nF4(4)=(SU(2) USp(6)), and the 56 decomposes as
ber of S3
56 ! (1
26) (1
26) 1
1
where once again the rst (1 26) is associated to the heavy background of branes wrapping
the y circle, and the second such factor is associated to wrapped branes and momentum
along the compacti cation S1
T4 transverse to the y circle; the remaining two charges
are KKmonopoles whose
bered circle is the y circle, and momentum along the y circle.
The (1
26) of wrapped branes/momentum charges along S1
T4 are again associated to
a set of U(1) currents in the CFT, and once again their energetics can be deduced from
the decoupling limit of the BPS mass formula [17]
(7.4)
(7.5)
2
:
hR =
pg
pi r
i
i
s + wDi1 pg
r
wD ne + wD~ 1
s
e
!2
2
1
i r gs
d1 n5 + p nk + d5 6789 n1
Here the third octet of charges related to U(1)'s of \level" nk are (f 1 ; n5 6789; d3 ij), and
the e are the corresponding volumes of the cycles they wrap, in appropriate units.
Once again there is a cusp of the moduli space for every factorization of N into a
triplet of background charges n1, n5, and nk; the supergravity description of the CFT
is thus merely a lowenergy e ective
eld theory approximation.
This fact also leads
to a minor puzzle. The only remnant of KK monopoles in the decoupling limit of the
background is a Znk quotient of the angular S3, which breaks the SU(2)L
symmetry down to U(1)L SU(2)R, and the supersymmetry from (4; 4) to (0; 4). But when
nk = 1, there is no quotient, and so it seems that there is an unbroken (4; 4) supersymmetry.
The resolution of this puzzle appears to be that indeed an accidental leftmoving N = 4
supersymmetry develops in the decoupling limit, on a codimension 8 sublocus of the cusp
SU(2)R
Rof the moduli space corresponding to nk = 1. The moduli space of the D1D5KKM system
get to any of the other supergravity limits with other values of nk, one must turn on the
additional eight moduli that break the accidental N = 4 supersymmetry of the leftmovers.
Again one can ask whether there is a symmetric product CFT W
N =SN somewhere in
the moduli space. It is again reasonable to suppose that the component CFT W has four
translation currents to generate the winding/momentum contributions in the rst term of
equation (7.5). The diagonal current that survives the orbifold projection yields a U(1)
of \level" N and so can only match the above energetics in the cusp where one of the
background charges is N , and again it is natural to take n5 = nk = 1 and n1 = N . The
second and third terms on the r.h.s. are then the contributions of eight more currents of
level one, and can be realized with a separate T
The last term in the wrapped brane energetics (7.5) is di cult to realize in a symmetric
product structure. With n1 = N , n5 = nk = 1, one seeks another current of level N . If
the building block is a c = 6 superconformal eld theory on T4 (with once again an extra
T
4
T
4 CFT to realize the \levelone" terms), the translation currents comprise c = 4,
and their superpartners are four free fermions comprising the remaining c = 2 (at least
for the rightmoving supersymmetric chirality). Bilinears in the free fermions form a
levelone SO(4) = SU(2) SU(2) current algebra, of which one SU(2) is the Rsymmetry. The
other, \auxiliary" SU(2) has energetics m2=4 for the individual component CFT W, where
m is the eigenvalue of J aux for this auxiliary (levelone) SU(2) current algebra.17
3
The
symmetric product structure then leads to an energetics m2=4N under the diagonal J3aux.
This energetics of SU(2) levelone current algebra is thus incompatible with the last term
in equation (7.5) by a factor of 3, and any attempt to engineer the requisite normalization
naively leads to a breaking of the (0,4) supersymmetry.
One can ask whether this lattice of auxiliary SU(2) charges with energies m2=4N is
a sublattice of some larger lattice of CFT zero modes, which also contains the values
present in (7.5). The possibilities are constrained by the full structure of U(1) charges in
supergravity. The (1 26) charges of wrapped branes/momentum on S1
T4 decompose as
(1; 1)
(2; 6)
(1; 14)
(7.6)
under the local SU(2)
USp(6) symmetry of the moduli space of the D1D5KKM
background. The thirteen rightmoving currents account for (1; 1)
(2; 6), with the singlet
17In the symmetric orbifold describing the D1D5 system, this auxiliary SU(2) is an accidental symmetry
of the orbifold locus, and does not survive perturbations away from this locus.
associated to the last term in (7.5) and the second factor associated to the translation
currents on the various copies of T4; the remaining (1; 14) are related to leftmoving currents,
for which there is less information due to the lack of supersymmetry in that chirality of the
CFT. A reasonable assumption is that twelve of the 14 are the leftmoving counterparts
of the rst three terms in (7.5) where one ips the relative sign of the \winding" and
\momentum" contributions. There are two more special currents whose energetics can then be
determined from the local SU(2) USp(6), leading to [17]
hL =
pg
pi r
i
s
1
4
X
1
2
The spectrum of the one rightmoving and two leftmoving \special" currents
associated to the charges p ; d1 ; d5 6789 in (7.5), (7.7) has also arisen in a related context,
in which spectral ows were used to generate a class of nonsupersymmetric solutions.18
Indeed, the charged states are all nonBPS, even though the starting point in the analysis
is a BPS mass formula; after the decoupling limit, none of the U(1) currents lie in the stress
tensor supermultiplet, even though before the decoupling limit, the rightmoving charges
did have that property. The U(1) charges in the CFT are thus no longer Rcharges, and
therefore there is no BPS condition involving them. Spectral ow remains a robust property
of the CFT that follows from symmetry, and leads to the same result as the combination of
the decoupling limit of the BPS formula for the rightmovers and the moduli space
considerations employed in [17] to obtain the charge spectrum. This gives us further con dence
in the applicability of these formulae, though with the caveat that the full energy of any
given state will typically not be saturated by the contributions of the U(1) charges.
A T4 symmetric product accounts for the rst term in (7.7) via the leftmoving T
4
translation currents, and similarly the second and third terms correlate with the
corresponding terms in (7.5). This leaves two additional leftmoving currents of level N . The
three \special" currents not associated to torus translations (two leftmoving and one
rightmoving), plus the rightmoving Rsymmetry current, thus all have level N and soak up all
the central charge of the rightmoving fermions in the symmetric product, and the
corresponding remaining central charge of the leftmovers. Bosonizing all four currents leads to
a (2,2) lattice of zero modes whose energetics must match (7.5), (7.7).
The energies of a general (2,2) lattice of zeromodes has the form
h
L# =
R# =
1
1
18In comparing equation (7.5) above to the spectrum equation 5.24 of [63], one notes a typo of a missing
factor of 1/2 in the rst term on the r.h.s. of the latter.
for complex
= 1 + i 2,
= 1 + i 2. Without loss of generality, we can write
1
2
m2 =
(mL + mR) ;
n2 =
(mL
mR)
1
2
(7.9)
(7.10)
(7.11)
(7.12)
(7.13)
HJEP06(217)3
and interpret mR as the eigenvalue of J R3 of the Rsymmetry. Demanding that mR appear
only in hR and only quadratically implies
reproduced for 1 = 1 = 1, 2 = 2 = p
3
. The rightmoving energetics (7.5) is
R# =
m2R +
4
(mL + 4m1
n1)2
12
mL + 4m1
n1 = d1 + p + d5 6789N :
Examining the contribution of the charges to the left and rightmoving energies, the closest
match comes if we identify
mL = p5
f5 ;
mL
n1 = p5 + f5 ;
4m1 = d5 6789N
which leads to a match between the lattice and supergravity expressions for the
rightmoving energy. The di erence between the supergravity and symmetric product formulae
then becomes
hL
L# =
(N d5 6789)2
4
(N d5 6789)p5
2
which is reminiscent of the structure of a spectral ow. The lowlying spectrum
(energies much less than order N ) is only compatible with d5 6789 = 0. The lattice of such
states, when chosen to match the results of the BPS mass formula, cannot simultaneously
accommodate the spectrum of free fermion superpartners of the torus translation currents.
To summarize, supersymmetry and a symmetric product of c = 6 building blocks
leads to a lattice of U(1) charges which is not compatible with the lattice inferred from
supergravity considerations. The rightmoving fermions which are the superpartners of
rightmoving translation currents have Rcharge 1/2 and dimension 1/2; on the other
hand, that lattice of charges inferred from supergravity does not have such a state in its
spectrum. This throws considerable doubt on the existence of a symmetric product orbifold
locus in the moduli space of the MSW CFT.
8
Discussion
Understanding the dynamics of multiple M5 branes has been one of the most challenging
and interesting issues in string theory for quite a number of years. There has been a huge
e ort in understanding how M5brane theories can describe stronglycoupled gauge theories
in four dimensions. Our purpose in this paper has been to study what should, perhaps,
be one of the simplest avatars of the M5brane eld theory: the (1+1)dimensional MSW
CFT that comes from wrappings of an M5 brane on a very ample divisor of a CalabiYau
manifold. This seemingly simple CFT remains enigmatic, almost twenty years after it was
rst shown to be able to encode microstate structure of fourdimensional black holes [15].
In this paper we have considered M5 branes wrapping 4cycles in T6 and T
2 K3, but our
resulting Mtheory solutions can be trivially extended to compacti cations with a more
general eld content.
As we have discussed, part of the di culty in analyzing this CFT is that it does not
seem to have any point in its moduli space with a canonical description in terms of
betterunderstood conformal eld theories, such as a symmetric orbifold theory. However, one
can use holographic methods to study this theory at strong coupling, and in this paper
we have made signi cant progress in that direction: we have obtained explicit families of
smooth, horizonless solutions of vedimensional supergravity that are dual to families of
BPS states of the MSW CFT.
We constructed these families of solutions to Mtheory by deriving a map between
them and a class of states of the D1D5 CFT, described as smooth, horizonless solutions to
sixdimensional supergravity. This was done by transforming asymptotically AdS3
T4, D1D5P superstratum solutions that are independent of the Hopf ber of the S3 to
asymptotically AdS3
S
2
T6 solutions19 dual to momentumcarrying microstates of the
MSW CFT. We therefore referred to our new families of solutions as Mtheory superstrata.
S
3
In principle, one should be able to obtain families of Mtheory superstrata that depend on
arbitrary functions of two variables (with arbitrary Fourier modes around the axis of the S2
and the spatial axis of the AdS3). In this paper we have constructed solutions which have
single Fourier mode excitations. However, based upon the success of the superstratum
program in six dimensions [11], we anticipate that one should be able to
nd smooth,
horizonless Mtheory superstrata with general families of Fourier modes excited.
It is important to emphasize that there are many more Mtheory superstrata solutions
constructed using our technology than those that we have directly mapped to smooth
D1D5P superstrata. As we have seen in section 5, when the KKM charge, , is greater than
one, the smooth D1D5P superstrata map to Mtheory superstrata with mode numbers
along the AdS3 circle that are multiples of . However, once in the Mtheory frame, nothing
prevents us from extrapolating these solutions to generic values of the mode numbers
compatible with smoothness and appropriate Mtheory periodicities. Under our map these
more generic Mtheory superstrata do not transform into geometric D1D5P states,20 and
yet they are perfectly good solutions.21
As we have noted in the Introduction, there have been several earlier approaches to the
construction of solutions dual to momentumcarrying BPS microstates of the MSW CFT.
The common goal of this paper and of previous work has been to examine the spacetime
structure of the microstates of black holes with a macroscopicallylarge horizon area. In
19Our solutions can trivially be extended by replacing T4 by K3 and T6 by T2 K3.
20A naive application of our map would give rise to solutions with multivalued
elds, and if one
extrapolates the candidate dual CFT states of [13, 14] to the appropriate values of the parameters, one
would not satisfy the condition of integer momentum per strand. Thus a straightforward application of
this holographic dictionary suggests that these con gurations should be discarded. For more discussion,
see [13].
Udualizing.
21Rather than using our map, one could also obtain these solutions by setting
= 1 in the D1D5
superstrata, restricting to k = 2m, introducing the smooth Z
quotient of the Hopf ber by hand and
this system, these black holes have a momentum charge along the AdS3 circle that, for
given M5 charges, must be larger than a certain threshold, which is of order the product
of the three M5 charges; once above this threshold, one is in the \black hole regime" of
parameters. In Type IIA, the M5 and momentum charges become D4 and D0 charges.
The microstate geometry corresponding to the maximallyspinning Ramond ground
state of the MSW CFT is obtained by blowing up the singlecenter D4D4D4 con guration
to a twocenter uxed D6D6 con guration, whose Mtheory uplift is [global AdS3] S
The addition of D0 charge via backreacted singular D0's was studied in [22, 47]. The
2
T6.
degeneracy of such \D0halo" solutions was counted in [24, 25], and found to give rise to
an entropy that matches that of an Mtheory supergraviton gas in [global AdS3]
for su ciently small D0 charge. The full backreaction of the supergraviton gas states has
never been computed, but since our Mtheory superstratum solutions represent smooth
waves in AdS3
S2, one may expect that at least some of them can be thought of as
coming from backreacted supergraviton gas states. Furthermore, if the full nonAbelian
and nonperturbative interactions of uplifted D0branes results in solutions that are
nonsingular and varying along the Mtheory circle, one expects these solutions to also resemble
our Mtheory superstrata. Hence, it may be that the smooth backreacted solutions we
construct are the missing link needed to connect the entropy counts in the
(nonbackS
2
T
HJEP06(217)3
reacted) supergraviton gas and (singular) D0halo approaches.
In the blackhole regime of parameters, the D0halo entropy exhibits a subleading
growth with the charges compared to the black hole entropy [47]. On the other hand, in
this regime the solutions have large deep AdS2 throats with high redshifts, and so are no
longer small perturbations of [global AdS3]
S
2
T6. A robust estimate of the number of
states comprised by superstrata remains to be carried out.
One can also add momentum by adding M2branes that wrap the twosphere of the
AdS3
S2, and that carry angular momentum on both the AdS3 and the S2 [22]. The
entropy of these con gurations comes from the high degeneracy of the Landau levels that
result from the dynamics of the M2branes on the compacti cation manifold in the presence
of M5 ux [22, 66, 67], and has been argued to scale in the same way as that of the black
hole. The backreaction of these \Wbrane" con gurations is fully worked out only in
some very simple examples [30]. However, on general grounds one expects uncancelled
tadpoles which give rise to asymptotics that are di erent from the asymptotics of bulk
duals of MSW CFT states. If, on the other hand, one cancels the tadpoles using additional
brane sources, there are no preserved supersymmetries whatsoever [31]. Furthermore, in
more generic multicenter solutions, the corresponding Wbrane con gurations also give
rise to tadpoles, which can only cancel when the Wbranes form a closed path among
the centers.22 Hence, when the multicenter solution has a throat of nite length, these
additional M2brane bound states break at least another half of supersymmetry (giving
116 BPS states), and typically all of the supersymmetry [31]. Thus these states cannot
correspond to microstates of the BPS MSW black hole.
22The counting of these closed paths gives an entropy that scales in the same way as the black hole
entropy as a function of the charges [67{69].
Given the large entropy of the Wbranes, one would like to somehow restore the
broken supersymmetry. This can only be achieved by going to a scaling limit, in which the
throat becomes in nitely deep. In the in nitethroat limit, the solitonic Wbranes become
massless, new dynamical elds emerge (corresponding to the Higgs branch of the
eld
theory for which the Wbranes are individual quanta) and the rich families of Wbranes
become re ections of the rich degeneracies of the vacua of these new dynamical elds. Thus
Wbranes should provide a semiclassical way to access the Higgs branch [67].
Another way to access the physics of the Higgs branch is via worldsheet disk
amplitudes. Using these techniques one can compute the supergravity backreaction of Dbrane
bound states upon an in nitesimal displacement on the Higgs branch. In the D1D5 system,
such calculations demonstrate that the additional tensor multiplet described by (Z4; G4) is
an integral part of the backreaction of generic Higgsbranch states [33]. Thus one expects
the con gurations that result from condensing the Wbranes to include such additional
species of supergravity
elds. For fourcharge black holes in four dimensions, in the
D3D3D3D3 system (which is Udual to the D1D5KKMP and the M5M5M5P systems),
a similar string emission calculation was recently performed [34, 70], con rming the
presence of this kind of additional species of supergravity
elds in the backreaction of these
bound states.
Remarkably, these new species of supergravity elds are exactly those needed to give
rise to smooth superstrata solutions, via the coi uring procedure we have used in sections 5
and 6. Furthermore, if one backreacts M5 branes of [15] wrapping smooth ample
divisors inside T6, one expects to source exactly these additional supergravity
elds. If one
combines these two features with the fact that our Mtheory superstrata solutions should
be parameterized by arbitrary continuous functions, and hence have a large entropy, it
appears very likely that these supergravity elds are a key component of the structure of
typical black hole microstates.
Our results raise some interesting questions about the formal mathematical
structures of vedimensional supergravity solutions. One should recall that the construction of
smooth microstate geometries in
ve dimensions was done via locally hyperKahler base
metrics whose signature changes from +4 to
4 on certain hypersurfaces. These singular
base metrics are referred to as ambipolar or pseudohyperKahler base metrics, and the
hypersurfaces where the signature changes are referred to as \degeneration loci". While
the fourdimensional spatial base metric is singular, all singularities cancel in the
vedimensional Lorentzian metric. There has been a growing mathematical interest in the
geometry of these ambipolar spaces [71], generalizing the notion of \folded" hyperKahler
metrics [72, 73]. Our results here indicate that harmonic analysis on such manifolds might
be extremely rich and interesting.
In particular, the rst step in solving the BPS equations is to nd smooth, harmonic
twoforms on the spatial base metric. In standard Riemannian geometry, this is a classical
exercise and the harmonic forms are dual to the homology cycles. The original work on
microstate geometries involving ambipolar bases [20, 21, 35] simply translated the
expressions for the standard harmonic forms of Riemannian geometry. The solutions constructed
in this paper have only one homology cycle, but we have exhibited in nitely many
\pseudoharmonic" twoforms. We de ned such twoforms to be those that are closed and coclosed
(\harmonic"), potentially singular on the degeneration loci of the base geometry, and yet
lead to completely regular,
vedimensional BPS solutions. This leads to several
interesting questions. Firstly, how does our result generalize to multicentered ambipolar GH
metrics? More generally, what is the classi cation of pseudoharmonic twoforms? This
paper shows that what seems to be a rigid topological problem actually has an in nite
amount of \wiggle room" on an ambipolar base.
Returning to our map between states of the MSW and D1D5 CFT's, the results
presented here suggest that this map should contribute more deeply to our understanding
of the physics of fourcharge black holes in four dimensions and to the question of how much
entropy of these black holes comes from smooth horizonless solutions. More broadly, we
believe that our map will also prove useful in gaining deeper understanding of the hitherto
mysterious MSW CFT. As we have seen, only a particular class of the MSW microstate
geometries are related to D1D5P ones, and hence only a subsector of the states of the
MSW CFT is mapped to a subsector of the D1D5 CFT. It would be extremely interesting
to explore and test possible extensions of this correspondence.
Indeed, several important questions remain about our map. First, the map is de ned
in terms of geometrical data, and it is interesting to see whether one can generalize it to
other CFT states that are not dual to smooth torusindependent horizonless supergravity
solutions, but may involve string or brane degrees of freedom, dependence on the internal
directions, or highcurvature corrections.23 A pessimistic possibility is that our construction
is merely an approximation that relates particular geometrical solutions in the supergravity
limit, but that does not map CFT physics beyond small perturbations around the particular
states that can be related to each other. On the other hand, it is tempting to speculate
that, if one accepts holography as a correct description of all physics in
asymptoticallyAdS backgrounds (including all 1=N corrections), a generalization of our map to degrees
of freedom beyond sixdimensional supergravity may exist, and it would be interesting to
investigate its properties.
A related question is whether our map is simply a useful device for counting and
classifying certain MSW states, or whether it is capable of capturing other CFT data such
as anomalous dimensions or threepoint functions. In the supergravity approximation,
these quantities can in principle be computed perturbatively around a given solution [74],
giving one hope that additional information about the MSW CFT could be gleaned.
One can reasonably expect at least some threepoint functions to be mapped from one
sector of one CFT to another sector of the other CFT, because of nonrenormalization
theorems [75]. However, if one considers the fourpoint functions of an MSW operator that
gets mapped to a D1D5 one, these fourpoint functions are computed by summing over all
operators in the intermediate channel, which may not belong to the relevant subsectors.
23For example, one can imagine constructing tendimensional supergravity solutions dual to D1D5
microstates that have a nontrivial dependence on the torus coordinates, and therefore cannot be described in a
sixdimensional truncation. Our map would take these solutions into holographic duals of MSW microstates
that contain an in nite tower of KaluzaKlein modes, and thus cannot be described in vedimensional
supergravity.
Furthermore, generic fourpoint functions are not protected when one deforms away from
the free orbifold point of the D1D5 CFT to the supergravity point, and hence there is no
reason to expect a map for this data. Nevertheless, one might hope to use our map to nd
a prescription that allows one to calculate at least certain conformal blocks of the MSW
CFT from D1D5 ones, which would already be remarkable progress.
What is clear is that, as a CFTtoCFT map, our construction is quite unusual. Indeed,
to go from the D1D5 NS vacuum to the MSW one, one needs to perform a combination of
spectral ow transformations and \gauge" transformations. While spectral ow
transformations have a clear CFT interpretation, as the rede ning of the CFT Hamiltonian by the
addition of a term proportional to the Rcharge, the gauge transformation would appear
to correspond to rede ning the Rcharge by the addition of a term proportional to the
Hamiltonian, which is much more mysterious. Hence, while spectral ow is an operation
that maps states to states within the CFT, the gauge transformation appears to change
the CFT itself. On the other hand, since the MSW CFT does not appear to have any point
in its moduli space with a symmetric product orbifold description, a map of the type we
have found may be the most one can hope for.
Acknowledgments
We thank Massimo Bianchi, Duiliu Emanuel Diaconescu, Stefano Giusto, Monica Guica,
Stefanos Katmadas, David Kutasov, Jose Francisco Morales, Rodolfo Russo, Masaki
Shigemori and Amitabh Virmani for valuable discussions. EM and NPW are very grateful to
the IPhT, CEASaclay for hospitality during the initial stages of this project. The work
of IB and DT was supported by the John Templeton Foundation Grant 48222 and by the
ANR grant BlackdSString. The work of NPW was supported in part by the DOE grant
DESC0011687; that of EJM was supported in part by DOE grant DESC0009924. The
work of DT was further supported by a CEA Enhanced Eurotalents Fellowship.
Covariant form of BPS ansatz and equations
To rewrite our ansatz in covariant form, we rescale (Z4; 4; G4) ! (Z4; 4; G4)=p2. Then
we have
C123 = 1 ;
C344 =
1 :
It should be understood that this rescaling holds throughout this appendix (and only in this
appendix). Then we de ne the (mostlyminus, lightcone) SO(1; 2) Minkowski metric via
ab = C3ab
12 = 21 = 1 ;
33 =
1 :
which can be used to raise and lower a; b indices, now that the above rescaling has been
done. After the rescaling we have
(A.1)
(A.2)
(A.3)
P =
2
1 abZaZb = Z1Z2
12 Z42 :
The rst layer of the BPS equations then takes the form
4DZ_a = abD (b) ;
D 4 DZa =
ab (b) ^ d ;
(a) = 4 (a) :
(A.4)
The second layer becomes
= Za (a) ;
4 ab 4
^
(b) :
Our ansatz for the tensor elds is
G(a) = d
P
2
1 abZb (du + !) ^ (dv + )
(dv + ) ^
In our conventions, the twisted selfduality condition for the eld strengths is
6 G(a) = M abG(b) ;
Mab =
ab :
ZaZb
P
B
Circular D1D5 supertube: parameterizations
In this appendix we recall the usual representation of a circular D1D5 supertube solution
within the sixdimensional ansatz (2.2), and the relation to the representation used in
this paper.
The usual representation (see, for example, [11, 13]) is given using the coordinate
transformation (2.3) and setting F = 0. This solution is then given by:
Z2 =
Q5
Z4 = 0 ;
(j) = 0 ; j = 1; 2; 4 ;
Rya2
p
2
!e =
! +
p
2
(A.5)
(A.7)
v~ = t + y ;
Z2 =
Q5
Fe =
1 ;
(j) = 0 ; j = 1; 2; 4 ;
Rya2
sin2 d'1 :
(B.2)
Z1 =
u~ = t ;
Z1 =
Rya2
This is the form of the solution used in the main text in (3.4).
asymptote to the form (B.2).
For superstratum solutions, the same rede nitions can be applied, and then the elds
C
Lattice of identi cations and fractional spectral ow
In this appendix we illustrate the step of rede ning the lattice of identi cations, with the
explicit example of fractional spectral ow of a multiwound circular D1D5 supertube
solution [10].
(sin2 d'1
(sin2 d'1 + cos2 d'2) :
(B.1)
Using the transformations (2.15) and (2.16), we obtain the following solution:
Equivalently, one can use the coordinate form of this fractional spectral ow transformation,
v^
v^ + 2 Ry ;
^ + 4 :
= ^
s
R
v^ ;
v = v^ :
and act with it on the explicit multiwound circular supertube metric (3.12), again
impos
Since ^ is the only coordinate that has transformed nontrivially, for ease of notation
we shall reuse the coordinates of the starting solution , , t~, y~ de ned in (3.11), as well
as , without writing hats explicitly. Then the transformed decouplinglimit solution is:
h
cosh2 dt~2 + d 2 + sinh2 dy~2i
4
d ^
(2s + 1)(dt~+ dy~) + cos d + (dt~
2
d + (dt~
dy~) 2 :
(C.1)
(C.2)
(C.3)
(C.4)
The starting con guration is the multiwound circular D1D5 supertube in the
decoupling limit, given in eqs. (3.1){(3.5). To this solution we apply a (fractional) spectral
ow transformation (2.29) with parameters ( 1; 2
; 3) = (0; 0; s=( R)), together with an
accompanying gauge transformation. The details and the resulting harmonic functions can
be found in appendix A of [10]. Recall that we have de ned R
Ry=2.
The point that we emphasize here is that to generate the transformed solution, one
inserts these new harmonic functions into a \hatted" version of the general ansatz, as
in (2.21), and importantly, one takes the lattice of identi cations to be the standard one
in the hatted coordinates:
namely
ing (C.1).
As in the starting solution (s = 0), there is a coordinate change to bring the metric to local
AdS3
S3 form. For the above transformed solution, it is of course
0 = ^
(2s + 1)(t~+ y~) ;
0 =
+ (t~
y~) :
The combination of (C.1) and (C.4) gives rise to an interesting variety of orbifold
singularities in the core of these solutions, depending on the common divisors of the integer
parameters s, s + 1 and , as noted in [76] and analyzed in detail in [10].
D
D.1
MSW
maximalcharge Ramond ground state solution
Coordinate conventions
We record here for convenience some of our coordinate conventions. We de ne the
threedimensional distances r from the centers, in Cartesian and cylindrical coordinates:
qy12 + y22 + (y3
p 2 + (z
c)2 ;
c =
1 2
a :
8
(D.1)
We de ne angular coordinates measured from the z = y3 axis at the two centers via
cos #
z
:
The relation between these coordinates and the (r; ) coordinates used throughout the
paper is
4r
4r+
(r2 + a2 sin2 ) ;
1
2
1
cos #
cos 2 #+ =
1=2
cos ;
1=2 sin ;
1=2
sin :
Prolate spheroidal coordinates centered on r
= 0 are useful for writing the metric as
z = c cosh 2 cos ;
= c sinh 2 sin ;
0 ; 0
:
= c (cosh 2
cos ) :
D.2
MSW maximalcharge Ramond ground state solution
The vedimensional MSW maximalcharge Ramond ground state solution is described by
the following harmonic functions. Using I = 1; 2; 3 and employing notation mod 3 for the
I indices, we have
V =
LI =
1
1
4
kI+1kI+2
1
1
KI =
M =
1
k k k
1 2 3
8
1
1
;
1
k k k
1 2 3
2c
:
The fourdimensional base metric can be written as
We write the oneform A as
ds42 = V 1 d
+ A)2 + V (d 2 + dz2 + 2
d 2) :
A = (cos #+
cos # )d :
Note that in this gauge, near the GH centers, A ' ( 1
identi cations that gives smoothness is (cf. Footnote 12)
cos # )d , so the lattice of
The oneform $ is given by
=
+ 4 ;
+ 2 :
$ =
k k k
1 2 3
4c
2 + (z
c + r+)(z + c
r )
r+r
d :
(D.2)
(D.4)
(D.6)
(D.8)
(D.9)
(D.10)
(D.11)
The metric of this solution is that of global AdS3
S2. To see this, we pass to the prolate
spheroidal coordinates ( ; ) de ned in (D.5), and de ne the coordinates
k1k2k3 t ;
1
2
+
in terms of which the vedimensional metric is manifestly global AdS3
S2,
(D.12)
(D.14)
(D.15)
(D.16)
HJEP06(217)3
with
Using the identity the above change of coordinates can be written as
R1 = 2R2 = 4(k1k2k3)1=3 :
c =
a
2
8
8Q12QR5y2 =
k k k
1 2 3
Ry
2
+ t~
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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