Integrability and blackhole microstate geometries
HJE
Integrability and blackhole microstate geometries
Iosif Bena 0 1 2 5
David Turton 0 1 2 5
Robert Walker 0 1 2 3
Nicholas P. Warner 0 1 2 3 4
0 Los Angeles , CA 90089 , U.S.A
1 Los Angeles , CA 900890484 , U.S.A
2 Orme des Merisiers , F91191 Gif sur Yvette , France
3 Department of Physics and Astronomy, University of Southern California , USA
4 Department of Mathematics, University of Southern California , USA
5 Institut de Physique Theorique, Universite Paris Saclay, CEA , CNRS
We examine some recentlyconstructed families of asymptoticallyAdS3 supergravity solutions that have the same charges and mass as supersymmetric D1D5P black holes, but that cap o smoothly with no horizon. These solutions, known as
Black Holes in String Theory; AdSCFT Correspondence

3
superstrata, are quite complicated, however we show that, for an in nite family of solutions,
the null geodesic problem is completely integrable, due to the existence of a nontrivial
conformal Killing tensor that provides a quadratic conservation law for null geodesics.
This implies that the massless scalar wave equation is separable. For another in nite
family of solutions, we nd that there is a nontrivial conformal Killing tensor only when
the leftmoving angular momentum of the massless scalar is zero. We also show that, for
both these families, the metric degrees of freedom have the form they would take if they
arose from a consistent truncation on S3 down to a (2 + 1)dimensional spacetime. We
discuss some of the broader consequences of these special properties for the physics of these
blackhole microstate geometries.
1 Introduction
2
3
4
5
1
3.1
3.2
4.1
The (1; 0; n) family of solutions 4.1.1 4.1.2 4.1.3
The special families of superstrata metrics
The last year has seen a signi cant breakthrough in the construction of microstate
geometries [1, 2]. In particular, microstate geometries corresponding to
vedimensional,
threecharge, supersymmetric black holes with arbitrarily small angular momenta have
been constructed. These solutions are horizonless and smooth and have an arbitrarilylong
BTZlike throat that interpolates between an AdS3
S3 asymptotic region and a long
verynearhorizon AdS2 region. Deep inside the AdS2 region, the throat caps o smoothly
just above where the blackhole horizon would be [1].
In the D1D5P frame these microstate geometries, known as \superstrata" [
3, 4
],
have been proposed as holographic duals of speci c families of pure states of the D1D5
CFT, involving particular leftmoving momentumcarrying excitations [1],1 with charges
in the regime of parameters in which a large BPS black hole exists. Hence these solutions
correspond to microstates of a black hole with a macroscopic horizon. The momentum
excitations may be thought of as creating the long AdS2 blackholelike throat; in the full
solution these momentum excitations are located deep inside that throat, and support its
macroscopic size. As one descends the AdS2 throat, to an excellent approximation it is
almost identical to a black hole throat until near the bottom, where one encounters the
momentum excitations, before the geometry caps o smoothly.
Given that we have a holographic understanding of these solutions, and that the
proposed dual CFT states live in the same ensemble as the states that give rise to the black
1A subset of these microstate geometries can be mapped to excitations of the MSW string that carry
momentum and angular momentum, via a sequence of solutiongenerating transformations and string
dualities [2].
{ 1 {
e.g. [5, 6]), and should also give rise to novel physics where the thermodynamic description
breaks down. Hence, we would like to understand how, for example, horizonless geometries
scatter and absorb incoming particles, and how this di ers from the classical black hole
result. The states we will study are still somewhat atypical, and will have interesting
differences from the corresponding classical black hole; we hope the present study will inform
future studies of progressively more typical microstates.
Superstratum solutions are parameterized by arbitrary functions of (at least) two
variables [4]. Generic superstratum solutions depend on all but one of the coordinates in six
dimensions, and hence may appear complicated and somewhat intimidating to the
uninitiated. However, in this paper we will show that two particular asymptoticallyAdS3
families, each parameterized by one positive integer, have much simpler physics than may
S
3
have been expected. One of these families, which we will call the (1; 0; n) family, has a
separable wave equation and a conformal Killing tensor.2 This implies that the equations
for null geodesics in these geometries are completely integrable: there is a complete set of
conserved quantities, that are linear or quadratic in velocities. Related work on geodesic
integrability in (twocharge) black hole microstate geometries and Dbrane metrics can be
found in [7, 8].
We will also show that the metrics of the (1; 0; n) family, and of another family that we
will call the (2; 1; n) family, can be rewritten in a form that they would take if they arose
from consistent truncations on S3 to (2 + 1)dimensions. By this, we mean the following:
we can write each metric in terms of a (deformed) S3
bration over a threedimensional
base, K, where the metric on K depends only on the coordinates on K. Moreover, the
bration and warp factors conform to the standard KK Ansatz for vector elds and Einstein
gravity on K.
In this paper we will restrict our attention to the metric degrees of freedom, and we
postpone a full analysis of the existence, or otherwise, of a complete consistent truncation
to future work. Such an investigation would require the inclusion of the sixdimensional
tensor gauge elds in the consistent truncation Ansatz, building upon the results of [9, 10]
to include more sixdimensional tensor multiplets. The purpose of the present work is to
elucidate the metric structure and how it takes the form it would take if it came from
a consistent truncation, since this is a remarkably strong constraint upon its structure.
The metrics of the (1; 0; n) and (2; 1; n) families also have several isometries, such that the
reduced metric on K, and the KK
elds of the
bration, depend only on one coordinate
(which asymptotically becomes the radial coordinate of the AdS3). Thus, the analysis
of these metrics can, in principle, be carried out entirely using threedimensional gravity
coupled to vector elds.
2This provides a quadratic conserved quantity for null geodesics.
{ 2 {
Unlike in the (1; 0; n) family, the wave equation is not generically separable in the
(2; 1; n) family, and the general geodesic problem does not appear to be completely
integrable. However, these microstate geometries come very close to having these properties:
for waves or geodesics that have zero SU(2)L angular momentum on the S3, one does have
separability of the wave equation and complete integrability of the null geodesic equations.
As we will discuss, complete integrability of geodesics may be both a blessing and
a curse. In particular, completely integrable systems have highly restricted spectra and
limited scattering behavior, and thus may not reveal some of the interesting physics of
generic microstate geometries. The complete set of conservation laws lead us to suspect that
the (1; 0; n) geometries will quickly eject infalling particles, and hence will not reproduce
the expected blackhole thermodynamic behavior.
On the other hand, the absence of
integrability in the (2; 1; n) family will allow for more complicated dynamics, and possibly
the trapping of incoming particles, which is a step closer to the behavior one expects from
typical microstates. In a sense, the (2; 1; n) family provides an ideal setting for investigating
more complicated dynamics: one can probe the behavior of the nonintegrable geodesics
by doing perturbation theory for waves and geodesics about the integrable (jL = 0) ones.
Thus one can probe quite nontrivial scattering properties in a controlled expansion.
There are three recent, seemingly unrelated, lines of investigation to which our results
should be relevant. The rst is an argument that supersymmetric microstate geometries
should exhibit a nonlinear instability [
11
] (see also [12{14]). This proposed nonlinear
instability is related to the existence of stably trapped null geodesics deep inside the core
of microstate geometries, however so far this has only been explicitly analyzed for very
symmetric geometries, none of which have charges and angular momenta corresponding to
a black hole with a large horizon area. Our results should help elucidate whether or not
this proposed nonlinear instability is an artifact of very symmetric microstate geometries.
The second line of investigation is the latetime behavior of correlation functions in
the D1D5 CFT, and its connection to quantum chaos [15]. One expects chaotic systems
to exhibit latetime
uctuations that come from their underlying microscopic description,
and that are not visible in the thermodynamic approximation. Hence, one expects typical
bulk microstates to give rise to latetime
uctuations that are not visible in the classical
blackhole solution. For a set of twocharge black hole microstates these
uctuations have
been computed in the dual CFT [16, 17], but the corresponding computations in the bulk
are beyond the capability of present technology. We hope that our results will open the
way for a new testing ground for these questions.
The third line of investigation is the computation of fourpoint functions of two heavy
and two light operators; for a few examples, see [18{24]. Such calculations can be done in
the CFT and can be matched to the lightlight twopoint function computed holographically
in the microstate geometry dual to the heavy state [22, 23]. So far, the bulk calculation
has only been done in microstate geometries that are dual to very special heavy states (a
particular set of RR ground states and spectral ows thereof [
25
]), essentially because of
the technical di culty of solving the wave equation. The integrability of geodesics and the
separability of wave operators in the in nite families of microstate geometries we consider
should simplify the calculation of the twopoint functions and vastly enhance the number
{ 3 {
of twoheavytwolight fourpoint functions that can be computed in the bulk. Moreover,
since there is an explicit proposal for the CFT states dual to the microstate geometries we
study [1], these fourpoint functions could be compared to those computed in the CFT.
Furthermore, since our geometries can have long blackholelike AdS2 throats and closely
resemble black holes far from the cap region, these fourpoint functions should shed light
on how unitarity is restored when replacing the black hole horizon with a fuzzball.
This paper is organized as follows. In section 2 we describe in more detail the special
properties of the metrics in the (1; 0; n) and (2; 1; n) families of microstate geometries. In
section 3 we give a brief description of these families of BPS solutions. In section 4 we
give the details of the metrics of the (1; 0; n) and (2; 1; n) families, separate variables in the
wave equation, and describe the conformal Killing tensor. Finally, in section 5 we discuss
the important features of the geometries described in this paper, and further discuss the
implications of our results.
2
Dimensional reduction, separability and Killing tensors
We work in Type IIB string theory on M4;1
S
1
M, where M is either T4 or K3.
The circle, S1, is taken to be macroscopic and is parameterized by the coordinate y, with
radius Ry:
y
y + 2 Ry :
We consider a bound state of D1branes wrapped on S1, D5branes wrapped on S1
and momentum P along S1. The internal manifold, M, is taken to be microscopic, and
we assume that all elds are independent of M. Upon dimensional reduction on M, one
obtains a theory whose lowenergy limit is sixdimensional N = 1 supergravity coupled
to two (antiselfdual) tensor multiplets. This theory contains all elds expected from the
study of D1D5P string worldsheet amplitudes [26]. The system of equations describing
all 18 BPS solutions of this theory was found in [27]; it is a generalization of the system
discussed in [28, 29]. Most importantly, this BPS system can be greatly simpli ed, and
largely linearized [30]. For supersymmetric solutions the sixdimensional metric is well
known to take the form [28]:
(2.1)
M,
(2.2)
(2.3)
3
ds62 =
2
p
P
where we take
1
2
u = p (t
y) ;
v = p (t + y) :
1
2
Supersymmetry requires that all elds be independent of u, but generic supersymmetric
solutions can depend upon all the other coordinates.
Upon taking the AdS/CFT decoupling limit [31], one obtains asymptotically AdS3 S
solutions. We will work exclusively in the decoupling limit throughout this paper. We shall
study solutions whose tensor elds have explicit dependence on v, as well as on the S3.
These solutions are known as \superstrata" [1{4, 32]. In the solutions we study, the metric,
(dv + ) du + ! +
1
2 F (dv + )
+
p
P ds42 (B)
gMN dzM dzN ;
{ 4 {
by (x1; x2; x3)
ds62
At in nity, the sets of coordinates (u; v; r) and ( ; '1; '2) parametrize AdS3 and S3
respectively. The superstratum solutions that we consider were constructed in [1], and they
have the property that the tensor elds depend explicitly on a single linear combination of
v, '1 and '2. We thus refer to them as singlemode superstrata. In these asymptotically
AdS3
S3 solutions, this phasedependence cancels in the energymomentum tensor, and
hence in the metric.3
Thus the metric has isometries not just along u (as required by
supersymmetry) but also along v, '1 and '2. In this paper we shall exploit these enhanced
symmetries and examine the remaining, highly nontrivial dependence on (r; ).
Our rst goal is to rewrite the general sixdimensional metric as a bration of the
compact threemanifold, S, described by (y1; y2; y3)
( ; '1; '2), over a base, K, parametrized
(u; v; r). Speci cally, we recast (2.2) in the following form:
gMN dzM dzN =
2 g^ dx dx + hij (dyi + Bi dx )(dyj + B
j dx ) ;
(2.6)
where g^
and hij are viewed as metrics on K and S respectively, and where
to be the volume of S divided by the volume of the S3 to which S limits at in nity:
is de ned
pdet(hij )
pdet(hij )jr!1
:
In a general BPS solution, g^ , hij , Bi and
It is convenient to de ne the metrics:
ds12;2
g^ dx dx ;
ds32
hij dyidyj :
At in nity, ds21;2 is asymptotic to the metric on AdS3, and ds23 is asymptotic to the metric
on S3. It is also useful to observe that one can invert the form of gMN in (2.6) explicitly:
can depend on all the coordinates, except u.
gMN =
gMN =
2
2 g^
+ hkm Bk Bm
hik Bk
Bk hkj
hij
g^
Bi g^
g^
Bj
2hij + g^ Bi Bj
!
;
!
:
ds24, on the fourdimensional base, B, is at and we write it in the standard bipolar form:
dr2
combination of the vector elds, Bi , and the metrics g^
and hij .
In particular, we note that the inverse metric on the internal space, S, is a nontrivial
The warp factor,
2, in front of ds21;2 in (2.6) is precisely the factor needed for the
dimensional reduction from six dimensions down to the threedimensional space time, K.
To be more speci c, this is the warp factor needed to reduce the sixdimensional Einstein
action down to the threedimensional Einstein action for g^
on K. In general, attempting
to perform such a dimensional reduction is of course not very useful, because g^
in (2.6)
will typically depend upon the yi.
3Upon completing these solutions to asymptotically R1;4 S1 solutions, the metric depends explicitly on
the linear combination of v, '1 and '2 [33].
{ 5 {
(2.4)
(2.5)
(2.7)
(2.8)
(2.9)
However, the (1; 0; n) and (2; 1; n) singlemode families of BPS geometries both have
the property that:
(i) The metric g^
is only a function of r.
Moreover, the (1; 0; n) family also has the following remarkable feature:
(ii) On ds26, the massless wave equation and the HamiltonJacobi equation for null
geodesics are separable. The \massive" wave equation on ds26 is also separable if
the mass term is induced from a mass as seen by the (2+1)dimensional metric ds21;2.
Finally, it is elementary to verify that the following is true for all metrics of the form (2.2):
(iii) If one computes p
along the base de ned by (r; ; '1; '2) are identical to the components of p
g~ g~ab,
g gMN for the sixdimensional metric, then the components of this
where g~ab is the metric de ned in (2.4).
Property (i), combined with equations (2.6) and (2.7), means that the complete
sixdimensional metric has the form it would take if it arose from a consistent truncation to
threedimensional physics on K. This threedimensional geometry is furthermore
determined entirely by functions of r alone.
Property (ii), the separability of the massless wave equation and of the massless
HamiltonJacobi equation for null geodesics, implies the existence of a \hidden
symmetry": there is another quadratic conserved quantity for the null geodesic equation [34, 35].4
That is, in addition to the usual conserved quantities along geodesics, there is a conformal
Killing tensor, MN , for which one has:
D
D
MN
dzM dzN
d
d
= F ( ) gMN
and two quadratic \energies," one involving vr
dd alone.
Properties (ii) and (iii) together mean that not only is the massless wave equation
separable, but its separability properties are precisely those of the atspace base metric
written in spherical bipolars (2.4). In particular, the angular modes on S are elementary:
they are simply the standard spherical harmonics on a round S3! Therefore the solutions of
the massless wave equation have an expansion in terms of functions of r alone, multiplied
by Jacobi polynomials in cos2 . Thus, most of the interesting physics is encoded in the
radial equation and in the functions of r alone that de ne g^ .
Finally, property (iii) suggests a rather interesting conjecture arising from the
general lore of consistent truncation of supergravity theories compacti ed on spheres.
Typically, purely internal, higherdimensional excitations reduce to scalar elds in the
lowerdimensional theory. Conversely, one of the most complicated aspects of obtaining \uplift"
4See also the recent review on geodesic integrability in blackhole backgrounds [36].
d
dr alone and the other involving v
{ 6 {
formulae for consistent truncations is the way in which lowerdimensional scalars encode
the details of the higherdimensional elds. For maximally supersymmetric theories on
spheres, there is now a large literature on this, but one of the earliest breakthroughs were
the metric uplift formulae [9, 10, 37, 38]. These formulae gave complete and explicit
expressions, in terms of the lowerdimensional scalar elds, for the inverse metric projected
onto the internal manifold. One of the simple consequences of this formula is that if the
inverse metric retains its original round form, then it means that the lowerdimensional
scalars are essentially trivial.
This piece of lore suggests that in the solutions we are considering, there are no
fundamental, lowerdimensional scalar excitations arising from the sixdimensional metric: all
the internal physics is encoded in the vector multiplets that descend from the Bi .
There are several caveats that come with this comment. First, we have not analyzed
the tensor gauge
elds to determine whether or not a complete, consistent truncation
containing the above solutions exists. Since these tensor elds have nontrivial dependence
on the S3 directions, they will descend to massive elds in three dimensions.
Moreover, while important results have been obtained in [10], the general consistent
truncation formulae have not been established for the S3 compacti cations considered here.
In addition, the formulae that have been established in other reductions are based on
massless vector elds that descend through Killing vectors on the sphere. For the tensor elds
and for some of the metric components in the (2; 1; n) family, we will need to allow more
general classes of elds, Bi , in which the internal components involve higher harmonics,
yielding massive vector elds on K. As we shall comment on below, it is not clear how
restricted an ansatz might be necessary in order to obtain a consistent truncation.
It is also important to note that, from the threedimensional perspective, Abelian
vector elds can trivially be rewritten as scalars. On the other hand, there are subtleties
in doing this for nonAbelian
elds and with o shell supermultiplet structure (see, for
example, [39]) and so the idea that there are only excitations of threedimensional vector
elds, descending from metric modes on S3, may be given some more precise formulation.
The bottom line is that the supergravity lore on consistent sphere truncations suggests
that property (iii) might imply that the only degrees of freedom that are being activated in
our solutions are the vector elds encoded in Bi and that there are no other independent
shape modes or lowerdimensional scalars coming from the sixdimensional metric.
3
Singlemode superstrata
In this section we review the construction of superstrata, before focusing on the set of such
solutions that involve a singlemode excitation. This will provide some background and
allow us to set up notation to be used when we present our main results in the next section.
3.1
D1D5P superstrata
The superstrata constructed to date have been obtained [1, 4, 32] by adding momentum
waves to the background of the circular supertube [40{44]. The starting point is therefore
{ 7 {
1=2
e1 =
(r2 + a2)1=2 dr ;
(1)
(2)
(3)
dr ^ d
(r2 + a2) cos
r
dr ^ d'1
r
One may then write:
and introduce a standard basis for the selfdual two forms:
+
r sin
d'1 ^ d'2 =
r2 + a2 dr ^ d'2 + tan d ^ d'1 =
1
1
(r2 + a2) 21 cos
12 (r2 + a2) 21 cos
(e1 ^ e2 + e3 ^ e4) ;
(e1 ^ e4 + e2 ^ e3) ; (3.3)
HJEP1(207)
1
1
2 r sin
cot d ^ d'2 =
(e1 ^ e3
e2 ^ e4) :
to take the vector eld
to be that of the standard magnetic ux of the supertube:
= p
We use the following frames on the four dimensional base, B, with metric (2.4):
= Z1 (1) + Z2 (2)
2 Z4 (4) ;
1
2
DF
(3.10)
(4)
^
(4) :
p
2 Ry a2
2
(3) = d
=
((r2 + a2) cos2
(2)
r2 sin2
(3)) :
In particular, (3) is selfdual.
The
rst part of the solution is de ned by three more potential functions, ZI , and
magnetic 2forms,
(I), I = 1; 2; 4, that are required to satisfy the \ rst layer" of the
linear system of equations governing all supersymmetric solutions of this theory:
The operator, D, acting on a pform with legs on the fourdimensional base (and possibly
depending on v), is de ned by:
where d(4) denotes the exterior derivative on B. The warp factor P in (2.2) is then
determined by a quadratic form in the electric potentials:
The remaining metric quantities are determined by the \second layer" of BPS
equations:
D
d(4)
P = Z1 Z2
Z42 :
1
2 4
Z2)
4
^
(1)
(2)
{ 8 {
(3.1)
(3.2)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
We will study solutions to the BPS equations with mode dependence of the form:
We also de ne:
p
2
Ry
k;m;n
(m + n) v + (k
m) '1
m '2 :
k;m;n
ak rn
(r2 + a2) k+2n sink m
cosm :
The smoothness of the solutions requires k to be a positive integer and m, n to be
nonnegative integers with m
k. This restriction has a clear holographic interpretation in the
description of the dual CFT states [1, 4, 32].
The most general solution to the rst layer is known for the singlebubble solutions
that are built upon the circular supertube [1, 4, 45{47]. This family of solutions can be
represented by a superposition of the following singlemode solutions for the pair (Z4; (4)):
HJEP1(207)
(3.11)
(3.12)
(3.13)
Z4 = b4
(4) =
Ry
p
k;m;n cos k;m;n ;
2 b4 k;m;n
h (m + n) r sin
+ n
+ cos k;m;n
m
m
k
n
k
1
+ 1
r sin
(2) + n
sin k;m;n
m
k
1
(1)
(3) i
;
with similar expressions for (Z1; (2)) and (Z2; (1)), with a priori independent coe cients.
The general families of regular solutions for the second layer have not been classi ed.
Classes of singlemode solutions are known [1, 2, 4, 32] and some multimode solutions
have been obtained [4]. Here we will focus entirely on the families of singlemode solutions
obtained in [1] and further studied in [2].
3.2
Coi ured singlemode solutions
In the maximallyrotating supertube solution, the data of the rst layer of BPS equations
takes the following simple form:
Z1 =
Q1
;
Z2 =
Q5
;
Z4 = 0 ;
(I) = 0 ;
I = 1; 2; 4 :
(3.14)
To this solution we add a single uctuating mode by taking Z4, 4 to be given by (3.13)
and by taking:
Z1 =
(2) =
Q1
b
2
2
b4 p
Ry
Observe that the Fourier frequencies appearing in (Z1; (2)) are twice those appearing in
(Z4; (4)) and that the Fourier coe cients of these modes have been tuned in terms of the
+cos 2k;2m;2n 2m
+1
{ 9 {
square of the Fourier coe cients of (Z4; (4)). This is an example of the procedure known
as \coi uring" [48, 49]. The problem is that generic
uctuations for the solutions to the
rst layer of BPS equations typically lead to singular solutions in the second layer. This
may be related to the nonlinear instabilities that have been suggested in [11{14]; this is
currently under investigation. Coi uring solves this problem by tuning other excitations
to remove the singularities in the solutions to the second layer of BPS equations. For a
single mode the result is particularly simple: all dependence on (v; '1; '2) cancels in the
sources for the second layer of BPS equations and in the warp factor, P. As a result, for a
single mode, the entire metric (2.2) is independent of (v; '1; '2). All that remains of the
uctuations is the \RMS values" proportional to b24.
The warp factor P now reduces to:
P =
Q1Q5
2
1
b
2
4
2a2 + b2 2k;2m;2n
It was shown in [1] that this is positive de nite for all r and .
Next, one must solve the second layer of BPS equations. Since the coi ured sources
are independent of (v; '1; '2), this means that we can use the following Ansatz:
(j++i1)N1
(J +1)m (L 1
J 3 1)n
m!
n!
j00ik
;
!1 d'1 + !2 d'2 = !0 + !^1(r; ) d'1 + !^2(r; ) d'2 ;
F = F (r; ) ;
where !0 the angular momentum vector of the round supertube:
!0
Ry a2
p
2
(sin2 d'1 + cos2 d'2) ;
and where !^1; !^2 and F are determined by solving (3.10), with the sources on the
righthand side given by the uctuating solution to the rst layer of BPS equations described
above.
The general family of (k; m; n) singlemode superstratum solution was obtained in [1].
Regularity at r = 0; =
=2 (the \supertube regularity" condition) imposes the constraint:
Ry2
Q1Q5 = a2 +
b
2
2
;
b
2
k
m
k + n
1
n
1
b42 :
The explicit solutions for (k; m; n) = (1; 0; n) and (k; m; n) = (2; 1; n) were studied in detail
in [1] and [2] respectively, and we will exhibit them momentarily.
In [1] it was shown that the complete (k; m; n) solution has the following values of the
conserved vedimensional angular momenta j, ~j, and ymomentum nP:
j = N
2
a2 +
k
m b2 ;
~j = N a2;
2
nP = N m + n 2
b ;
2
k
where N
n1n5Ry2=(Q1Q5), and n1; n5 are the numbers of D1 and D5 branes. It was
proposed in [1] that these solutions are holographic duals of coherent superpositions of
CFT states of the form:
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
for all values of N1 such that N1 + kNk;m;n = N . For an explanation of the above notation,
see [1]. The values of the conserved charges imply that, within the coherent superposition,
the average numbers of j++i1 and j00ik strands are given by N a2 and N b2=(2k) respectively.
4
The special families of superstrata metrics
We now examine the details of the solutions for which the parameters (k; m; n) take the
values (1; 0; n), (2; 1; n), and (2; 0; n). The solution for (k = 2; m = 0) is included simply to
illustrate that properties (i) and (ii) do not hold in general, since neither property holds for
this solution. We will therefore not discuss the details of this particular solution beyond
writing down the metric, and we will focus on the other two families.
For k = 1, m = 0 and general n > 0, the solution to the second layer of BPS equations
is [1]:
F =
b
2
a2
For k = 2, m = 1 and general n > 0, we have [2]:5
F =
!1 = p
!2 = p
b
2
(n + 1) a2
r2n sin2
;
r
2
2 (r2 + a2)n+1
2 (n + 1)
+ b24 r2(n+1) cos2
2 (r2 + a2)n+2
Finally, for k = 2, m = 0 the solution to the second layer of BPS equations is:
Regularity requires that b and b4 are related via the general relation (3.19), which
evalub2 = b42 for k = 1
and
b2 =
for k = 2 :
(4.4)
b
2
4
2(n + 1)
5To arrive at the following expression we have taken the results from section 6.1 of [2] and undone the
gauge transformations described in section 2.2 of that paper. We have thus ensured that (4.1) and (4.2)
are solutions in the conventions of this paper.
!2 = p
Ry
2
Ry
2
ates to:
Ry
Ry
2
2
b
2
+ a2 cos2
;
(4.2)
F =
(n+1)2 a4 na2 r
4
Note that killing o all the nontrivial modes by setting b4 = b = 0 in (4.1){(4.3) reduces
these expressions to !0 in (3.18).
The value of P in the starting supertube solution is Q1Q5= 2. It is convenient to factor
this o and introduce the quantity:
p
P
pQ1Q5
=
s
1
where the functions, Fi(r), are de ned by:
F0(r)
F2(r)
1
1
r2n
a2 b2
;
(2a2 + b2) (r2 + a2)n+1 :
F1(r)
r2n
a
6
b2 (2a2 + b2) r2 F0(r) ;
From (4.6) it is elementary to evaluate the determinant of the internal metric along S.
Recalling that (2.7) de nes the warp factor required by dimensional reduction, we nd:
In (4.6) we have extracted this warp factor from the rst part of the metric and so the
metric terms inside the rst set of square brackets yield the metric, g^ , on K de ned
in (2.6). The resulting threedimensional metric is, indeed, purely a function of r,
g^
dx dx = pQ1Q5
"
F2(r) dr2
+
We de ne the following oneforms on the threedimensional base, K:
(a2(du
dv)
b2 F0(r)dv) :
2 =
F2(r)
:
2 a2 r2 (r2 + a2) F2(r)
F1(r) Ry2
dv +
1
du2
F1(r)
p
2
(2a2 + b2) F2(r) Ry
(4.6)
2
;
(4.7)
(4.8)
(4.9)
(4.10)
We now observe that the o diagonal components, Bi of the bration form of the
metric (2.6) can be written as
Bi dx = K(1i) A(1) + K(2i) A(2) ;
(4.11)
only is the metric g^
components of the metric on S
where K(1M) = (0; 0; 0; 0; 1; 0) and K(2M) = (0; 0; 0; 0; 0; 1) are the components of the Killing
the vector elds, A(1) and A(2) are massless electromagnetic potentials on K. Thus, not
independent of the coordinates on S3, but so are the dynamical
. In light of Property (iii) and the comments made at the
end of section 2 about the absence of scalar excitations and shape modes, the dynamics of
the sixdimensional metric excitations in this solution reduces to dynamics of the metric
and massless vector
elds on K. Of course, one should recall that in the complete
sixdimensional solution, the threeform
elds depend upon v; '1 and '2, and so do not reduce
to massless elds in three dimensions. It is, however, still possible that such tensor gauge
modes give rise, in a consistent truncation, to a collection of massless and massive elds
on K. This possibility is currently under investigation.
4.1.2
Since the sixdimensional metric is independent of (u; v; '1; '2), this means that the
corresponding momenta are conserved:
L1 = K(1)M d
dzM
;
L2 = K(2)M d
dzM
;
P = K(3)M d
dzM
;
E = K(4)M d
dzM
;
(4.12)
In addition, there is the standard quadratic conserved quantity coming from the metric:
"
dzM dzN
d
d
:
These conservation laws determine all the velocities except vr
d
dr and v
In principle (4.13) allows exchange of energy between vr and v , and this could generate
interesting trapping of geodesics: as a particle falls in, some of its vr is traded for v and
thus the particle may lose radial momentum and be prevented from returning to where
it started.
However, at least for null geodesics in the (1; 0; n) family of metrics, there is a
hidden symmetry: there is an additional conserved quantity, that is quadratic in momenta.
The additional conserved quantity can be found by separating variables in the massless
HamiltonJacobi equation, and takes the form:
One can verify that for any geodesic one has which vanishes on null geodesics.
MN
d
d
dzM dzN
d
d
= Ry v
Q1Q5
2 v2 +
L2
1
sin2
+
L2
cos2
dzM dzN
d
d
;
(4.13)
dd .
(4.14)
(4.15)
F2(r)
n = 1 and, as (4.17) implies, the size of the dip decreases with n.
Were it not for the presence of 2 in (4.14), this conserved quantity would be the total
angular momentum on the round S3, and the motion on S would be essentially decoupled
from that on K. However, because of the factor of
On the other hand, the factor of 2 only exerts a minor in uence on geodesic motion. To
2, these motions are not decoupled.
see this, rst observe that
2jr=0 = 1 ;
2
! 1 as r ! 1 ;
2
j = 2
= F2(r) :
at
In fact 2 is very close to 1 for most values of (r; ), and its maximum deviation from 1 is
= 2 where it is given by F2(r). The function F2(r) is minimized at r = apn, and has
a minimum value of
b
2
nn
1
2a2 + b2 (n + 1)n+1 >
3
4
:
Moreover, as one can see in gure 1, the variation from 1 takes place in a short interval
around r = apn. The region around r = apn is also the region where the microstate
structure, in the form of momentumcarrying waves, is concentrated.
This means that v will increase brie y as it passes through r = apn and this can, in
turn, change the value of
and the value of vr. However, this e ect is quite localized and
makes little di erence to the asymptotic values or vr and r. A free particle falling in from
outside the throat will bounce o
the center and escape the throat. The only geodesics
that are a ected signi cantly by the microstate structure are the ones that are already
(4.16)
(4.17)
localized near r = apn.
4.1.3
The wave equation
Consider the sixdimensional scalar wave equation
1
p
= pQ1Q5
;
(4.18)
a2(! + p + q1)2
K = ( + M 2) K ;
S =
S ;
(4.21)
HJEP1(207)
for some eigenvalue .
Observe that the second equation (4.21) is the eigenvalue problem for the Laplace
operator on the round S3. The regular modes are therefore given in terms of Jacobi
S( ) = sinq1 cosq2 P (q1;q2)(cos 2 ) ;
j
P (q1;q2) (x) =
(q1 + 1)j 2F1
j!
1
2
j; 1 +
(` + q1 + q2) ; q1 + 1; (1
x) ;
1
2
m) is Pochhammer's symbol. The quantum numbers (`; j) are
= `(` + 2) ;
j =
(`
q1
q2) :
For the modes (4.19) to be singlevalued and regular, and for Pj(q1;q2) to be a polynomial,
one must have
where the factor of
1 has been included in the \mass term" on the righthand side for
reasons that will become apparent below.
Consider a generic mode for of the form
= K(r) S( ) ei Rpy2 ! u+ Ry2 pv+q1 '1+q2 '2 :
p
One then nds that the wave equation separates, yielding:
(4.19)
(4.20)
(4.22)
(4.23)
(4.24)
(4.25)
(4.26)
polynomials:
where
and (y)j = Qj
m=0(y
de ned by
q1; q2; `; j 2 Z ;
q1; q2
0 ;
`
q1 + q2 mod 2 :
The radial equation is considerably more involved, because of the presence of terms
proportional to b2, which encode the massless scattering from the detailed structure of the
superstratum. Note that setting r = a sinh
turns the di erential operator part of (4.20)
to a more canonical form:
1
sinh cosh
Note also that in the limit b ! 0, the background becomes the decoupling limit of the
circular supertube solution, and the radial equation simply becomes the hypergeometric
equation [50]. It has been known for some time that the massless scalar wave equation
is separable in the circular supertube solution [50], as well as in solutions obtained by
spectral ow thereof [
25, 51, 52
], and related black hole solutions [53]. The (1; 0; n) family
of singlemode superstrata is considerably more complicated than these solutions, and so
it is quite remarkable that separability is preserved.
Finally, note that the mass term in (4.18) has descended to a mass term in three
dimensions, as is evident in (4.20). Indeed the total threedimensional mass of the scalar
eld is the sum of the explicit mass, M 2, and the eigenvalue, .
We now analyze the (2; 1; n) family of solutions. The warp factor, , now takes the form:
=
s
1
Note that, compared to the corresponding warp factor in the (1; 0; n) solutions, (4.5), the
warp factor involves a higher harmonic mode with a stronger fallo at in nity. This means
that the nontrivial pro le in
is even smaller than the pro les depicted in gure 1. The
pro le is sharply peaked around
r =
r n
2
a :
= '1 + '2 ;
= '2
'1 ;
(4.28)
(4.29)
(4.30)
2
(4.31)
(4.32)
We introduce the coordinates
and the functions, Hi(r):
H0(r)
H2(r)
1
1
r2n
a4 (du + dv) + (2a2 + b2) r2 H1(r) dv
In terms of these, the sixdimensional metric, (2.2), can be rewritten as:
dv2
4
+ pQ1Q5
d 2 +
H2(r)
4
+
d
+ A^( ) 2
+
H2(r)
4
cos2(2 )H2(r) + sin2(2 )
cos 2 d
+ A^( ) d + A^( )
d + A^( ) 2 ;
where the vector elds A^( ) and A^( ) are given by:
A^( ) =
A^( ) =
p
p
Ry
2 2a2du
b2H0(r)dv
1
2
dv + cos 2
(2a2 + b2)
1
H2(r)
H2(r)
:
a
2
r
2
2a2 + b2 (du + dv) +
a2 H1(r)dv
;
The dimensionallyreduced metric on K, de ned in (2.6), is then:
dr2
+
As before, it is elementary to evaluate the determinant of the internal metric along S
and use (2.7) to obtain the warp factor required by dimensional reduction:
a4 (du+dv)+(2a2 +b2) r2 H1(r) dv
which is, again, purely a function of r.
Like the (1; 0; n) vector
elds in (4.10), the vector eld A^( ) is independent of the
coordinates on the S3 and, when incorporated in the metric, is multiplied by a Killing
vector. This means that it reduces to a massless KaluzaKlein vector eld on K. However,
A^( ), while also multiplying Killing vectors in the metric, has terms that are independent
of as well as terms proportional to cos 2 . The former are constant multiples of dv and are
thus pure gauge. The latter also depend on r and therefore represent nontrivial pro les for
massive vector elds on K. The metric of the (2; 1; n) family thus produces both massive
and massless KK vector elds on K.
4.2.1
Geodesics and separability
For the (2; 1; n) family of solutions, the massless wave equation and the HamiltonJacobi
equation for null geodesics are separable only for either vanishing frequency, or for a speci c
choice of angular modes on S3. Speci cally, if one seeks modes of the form (4.19) then
one
nds something very similar to (4.20){(4.21) except for a single problematic term.
One nds:
1 1
+
1
1
S sin cos
+ F (!; p; q1; q2; n; r)
a2b2 (n + 1) G(!; q1; q2; n; r; ) = 0
q
2
1
sin2
+
q
2
2
cos2
where F (!; p; q1; q2; n; r) is a complicated function of the coordinate r and the mode
numbers !; p; q1; q2 and n. The function G(!; q1; q2; n; r; ) is given by:
G(!; q1; q2; n; r; )
! (q1 + q2) (r2 + a2)n+2 cos 2
r2n
as r ! 0 and r ! 1.
equation:
and expresses the failure of separability. Note that this term vanishes if ! = 0 or q2 =
q1.
Moreover, the function G is strongly peaked at the dimple of , (4.28), and vanishes rapidly
One nds a similar result upon attempting to separate the massless HamiltonJacobi
= 0 :
(4.33)
(4.34)
2
;
S
(4.35)
(4.36)
(4.37)
Gb(E; `1; `2; n; r; )
E (`1 + `2) (r2 + a2)n+2 cos 2 :
r2n
This is manifestly the direct parallel of (4.36). Moreover, if either E vanishes or `1 + `2
vanishes, then Gb
0 and HamiltonJacobi theory tells us that
is a conserved quantity. Note that one has
2
P =
Ry2 (2a2 + b2) 1
=
Ry2 (2a2 + b2) 2
;
one obtains an equation of the form:
(S0( ))2 +
`
2
1
sin2
+
`
2
2
cos2
where
+
(r2 + a2) (K0(r))2 + Fb(E; `0; `1; `2; n; r)
+ (n + 1) a2b2 Gb(E; `1; `2; n; r; ) = 0 ;
where
is given in (4.27). Thus (4.41) is the analogue of the conserved quantity (4.14).
However, (4.41) is only conserved for E = 0 or `1 =
`2.
Recall that the momentum modes that underlie our solution depend upon the angles
according to (3.11), which now has the form:
p
Ry
2
2;1;n
(n + 1) v + ('1
'2) :
Thus the conservation and separation conditions, `1 =
`2 and q2 =
q1, mean that the
geodesic or wave must have the same angular dependence on the S3 as the underlying
momentum modes.
5
Discussion
In this paper we have found that two in nite families of superstratum solutions have quite
remarkable integrability properties for null geodesics. One of the families, the (1; 0; n)
family, has a separable massless KleinGordon equation and a complete set of conserved
quantities for null geodesics. The other family, the (2; 1; n) family, has a separable massless
KleinGordon equation and a complete set of conserved quantities only for a constrained
set of angular momenta on the S3. For the (2; 1; n) family (and for the (2; 0; n) family),
the failure of separation and failure of conservation is sharply localized in the region of the
solution where the momentum density is concentrated. We further found that the metrics
of these families of solutions can be reduced to interesting sets of degrees of freedom in
(2 + 1)dimensions.
The conservation laws and separability of the massless scalar wave equation for the
(1; 0; n) family means that this solution is readily amenable to detailed scattering
calculations. On the other hand, because of its integrability, this solution is likely to exhibit some
quite atypical behavior, particularly when it comes to the spectrum.
An interesting question to investigate is whether, and in what regime of parameters, a
given microstate geometry can capture or trap incoming particles. It was recently argued
that given any supersymmetric microstate geometry in six dimensions, there should exist
a stablytrapped null geodesic passing through every point of the spacetime [
11
]. These
null geodesics have tangent vector @=@u in our notation, so correspond to massless
particles moving purely in the y direction. In the geodesic approximation, massless particles
following such geodesics do not fall (deeper) into the throat. The main heuristic argument
of [
11
] considers such a particle that is coupled to gravitational radiation and other massless
elds, such that the probe gradually radiates some of its energy into these other elds, thus
evolving to follow geodesics of progressively lower energy. In this way a massless particle
can, slowly, descend the throat.
However, within the geodesic problem, one can ask whether an infalling particle with
nonzero radial momentum, falling from outside the throat, can be de ected nontrivially
from the region of the metric at the bottom of the throat and, through this de ection,
remain in the throat for arbitrarily long periods of time, as seen from in nity. If the
\radial kinetic energy," 12 vr2, is the only kinetic energy term that appears in a particular
conservation law, then, just as in any orbit problem, an infalling particle falling from
outside the throat will simply rebound and escape: it will not be captured and trapped
deep within the throat. For such capture to occur, particles and waves must be able
to scatter \radial kinetic energy," 12 vr2, into \angular kinetic energy." A complete set of
conserved quantities is thus largely antithetical to such behavior, although the conserved
quantity, (4.14), depends on both r and
via (4.5) and so, in principle, it is possible
that changing
along the trajectory can result in the loss of some radial kinetic energy.
However, in practice, in our solutions the
dependence dies out extremely rapidly for
large r, and so changing
will have only a minimal e ect upon the return of the particle
to large distances.
If a given solution does allow a signi cant de ection of radial momentum to angular
momentum, the angular motion can potentially prevent the particle from escaping the
throat for a long time. Our results imply that the (1; 0; n) family of solutions is likely
the wrong place to look for such behavior. However, the conservation laws present for the
(1; 0; n) family are not present for generic (k; m; n) superstrata, so such solutions should
allow the scattering of vr into angular motion, and it would be interesting to investigate
whether this can lead to geodesics that describe infalling particles with nonzero radial
momentum, falling from outside the throat, becoming trapped deep inside the throat for
long periods of time. One way to study this behavior analytically could be to use the (2; 1; n)
family (for which geodesics/waves with `1+`2 = 0 or q1+q2 = 0 are integrable) and examine
perturbatively how waves are scattered into angular directions for small jL = `1 + `2 or
small q1 + q2. It would be particularly interesting to investigate the timescale associated
with this trapping. This will depend on the depth of the throat, on how radial motion is
converted into angular motion, and on whether or not the trapping is chaotic. It might be
that the only way in which a particle can return to large distances after having scattered
o the microstate structure once is to scatter o it again in exactly the right manner as
to restore enough radial kinetic energy. This could lead to extremely large return times 
a desirable feature if one is to construct microstate geometries that describe typical black
hole states.
Finally, the results presented here underline the fact that the (1; 0; n) and (2; 1; n)
families have remarkably stringent constraints on their structure that suggests that the full
sixdimensional solutions might be written in a form that would come from a consistent
truncation to (2 + 1)dimensions. In particular, the Fourier expansions of the (2 +
1)dimensional elds may only have nontrivial dependence on one variable, r. If such a
structure indeed exists, it would be very interesting to investigate the existence or otherwise
of a consistent truncation containing these solutions. In doing so, an interesting question
will be to determine whether or not any consistent truncation ansatz will require some form
of coi uring to be built in. If a consistent truncation involving the modes of the tensor
gauge
elds exists, then this could provide a powerful new route for the construction
nonsupersymmetric solutions building on [54{56]: it would reduce the nonlinear,
nonBPS supergravity dynamics in the system of [54, 56] from functions of two variables in
six dimensions to the far more tractable problem of functions of one variable in (2 +
1)dimensions. These questions are currently under investigation.
Acknowledgments
NPW is very grateful to the IPhT of CEASaclay, and IB and DT to the Centro de Ciencias
de Benasque, for hospitality during this project. We would like to thank Stefano Giusto,
Mariana Gran~a, Monica Guica, Stefanos Katmadas, Emil Martinec, Rodolfo Russo, Masaki
Shigemori and Charles StricklandConstable for interesting discussions. The work of IB and
DT was supported by the ANR grant BlackdSString. The work of NPW was supported
in part by the DOE grant DESC0011687. The work of DT was also supported by a CEA
Enhanced Eurotalents Fellowship.
Open Access.
This article is distributed under the terms of the Creative Commons 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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