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Published for SISSA by Springer Received: August 18, 2015 Accepted: October 21, 2015 Published: November 9, 2015

Bidisha Chakrabarty,a David Turtonb and Amitabh Virmania;c

aInstitute of Physics, Sachivalaya Marg,

Bhubaneshwar, 751005 India

bInstitut de Physique Th eorique, CEA Saclay, CNRS URA 2306,

91191 Gif-sur-Yvette, France

cMax Planck Institute for Gravitational Physics (Albert Einstein Institute),

Am Muhlenberg 1, D-14476 Potsdam-Golm, Germany

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: Non-supersymmetric black hole microstates are of great interest in the context of the black hole information paradox. We identify the holographic description of the general class of non-supersymmetric orbifolded D1-D5-P supergravity solutions found by Jejjala, Madden, Ross and Titchener. This class includes both completely smooth solutions and solutions with conical defects, and in the near-decoupling limit these solutions describe degrees of freedom in the cap region. The CFT description involves a general class of states obtained by fractional spectral ow in both left-moving and right-moving sectors, generalizing previous work which studied special cases in this class. We compute the massless scalar emission spectrum and emission rates in both gravity and CFT and nd perfect agreement, thereby providing strong evidence for our proposed identi cation. We also investigate the physics of ergoregion emission as pair creation for these orbifolded solutions. Our results represent the largest class of non-supersymmetric black hole microstate geometries with identi ed CFT duals presently known.

Keywords: Black Holes in String Theory, AdS-CFT Correspondence, Black Holes

ArXiv ePrint: 1508.01231

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP11(2015)063

Web End =10.1007/JHEP11(2015)063

Holographic description of non-supersymmetric orbifolded D1-D5-P solutions

JHEP11(2015)063

Contents

1 Introduction 1

2 Orbifolded JMaRT solutions 42.1 Supergravity solutions 42.2 The near-decoupling limit 72.3 Smoothness analysis 82.4 Scalar wave equation 12

3 CFT description of orbifolded JMaRT solutions 143.1 The D1-D5 system on T4 and the orbifold CFT 143.2 Twisted Ramond sector ground states 153.3 Non-BPS states generated by general fractional spectral ow 163.4 Emission spectrum and emission rates from CFT 18

4 Ergoregion emission as pair creation 204.1 Solutions of the wave equation 214.2 Angular momenta of the perturbation 224.3 Energy and linear momentum of the perturbation 24

5 Discussion 26

A Solving the wave equation via matched asymptotic expansion 27

B Details of pair creation calculation 31B.1 Normalization of the asymptotic region wavefunction 31B.2 A hypergeometric function identity 32

C Conventions 32

1 Introduction

The black hole information paradox [1] is a profound and long-standing problem in quantum gravity. String theory has had many successes in black hole physics, including the microscopic derivation of the entropy of the large supersymmetric D1-D5-P black hole [2]. The evidence from constructions of black hole microstates in string theory points to a resolution of the information paradox whereby the true quantum bound state has a size of order the event horizon of the naive classical solution, and so the black hole event horizon and interior are replaced by quantum degrees of freedom. This is known as the fuzzball conjecture [3{10].

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JHEP11(2015)063

To probe the quantum degrees of freedom of the black hole, one often studies semi-classical microstates, which may be described within supergravity. There has been signi -cant progress in the study of three-charge BPS black hole microstates [11{23], culminating in the recent rst explicit construction of a superstratum [24].

Given a supergravity solution with the same charges as a black hole, it is important to establish whether the solution describes a bound state. In an AdS/CFT setup [25{27], one can do this by identifying a state in the holographically dual CFT. At present, there are many more such supergravity solutions than there are solutions with identi ed CFT duals. In the case of the superstratum, there is a proposal for the dual CFT states, evidence for which has recently been obtained [28] using precision holography techniques [29, 30].

Non-supersymmetric black hole microstate solutions are technically much more demanding; relatively few families have been explicitly constructed, and fewer still have known dual CFT states. The rst non-supersymmetric black hole microstate solutions to be discovered were the solutions found by Jejjala, Madden, Ross and Titchener (often abbreviated to JMaRT) [31]. For other studies of non-supersymmetric black hole microstate solutions, see [32{43].

The JMaRT family of solutions includes a positive integer parameter k; the solutions with k > 1 can be thought of as orbifolds of the k = 1 solutions, the orbifold acting on the asymptotic S1 coordinate y. It is important to note that the k = 1 solutions, while all smooth, do not exhaust the smooth solutions within this family, and that a signi cant parameter space of k > 1 solutions are also smooth [31]. In addition to the smooth solutions, there are also solutions with a rich possible structure of orbifold singularities; we will discuss this in detail in due course.

In this paper we identify the CFT duals of the general class of orbifolded JMaRT solutions. Physically, the k > 1 states are of particular interest; although the whole family of CFT states we study are atypical, states with larger k are closer to typical states than states with smaller k. This is because the typical three-charge state is in the maximally twisted sector, k = n1n5, so states with higher k are closer to typicality.

We work in the D1-D5 system on T4. In the holographically dual orbifold CFT [2, 25, 44], it has been proposed that semiclassical states obtained by the action of the superconformal algebra generators on Ramond-Ramond (R-R) ground states are dual to bulk solutions involving di eomorphisms that do not vanish at the boundary of the AdS throat [45{47]. Similarly, solutions involving non-trivial deformations (with respect to a reference R-R ground state) in the region deep inside the AdS throat known as the cap should be dual to CFT states that cannot be expressed in terms of superconformal algebra generators acting on R-R ground states; examples of three-charge BPS states which support this proposal were found in [48].

The CFT states studied in [48] involve fractional spectral ow in the left-moving sector, starting from the twisted R-R ground states studied in [49, 50]. The dual geometries are the BPS orbifold solutions found in [13, 31]. It was anticipated in [48] that applying fractional spectral ow in both left- and right-moving sectors of the CFT, one should obtain states dual to the general orbifolded JMaRT geometries. In this paper we con rm this expectation, make precise the map between gravity and CFT, and provide strong

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evidence for the identi cation by studying the emission spectrum and emission rates of the states in both gravity and CFT. Although our main interest is in R-R states, the general class of CFT states we study also contains NS-NS states.

Since non-BPS, non-extremal states may be expected to be generically unstable, it is far from clear how many states might be described by stationary supergravity solutions. However the decay of such states is an opportunity to gain insight into the unitary mechanism that should replace Hawking radiation for generic states. In the case of the k = 1 JMaRT solutions, soon after the discovery of these solutions it was shown that these geometries decay via a classical ergoregion instability [51].

A microscopic dual CFT explanation of the instability was proposed in [52]: the unitary CFT process of Hawking radiation is enhanced for the atypical CFT states dual to the JMaRT solutions, such that it manifests in the bulk as the ergoregion instability. Certain aspects of the CFT arguments were somewhat heuristic at the time, but were later made more precise in a series of papers [53{56]. The spectrum and emission rate of minimal scalars from the microscopic considerations were found to be in exact agreement with the instability found on the gravity side. In this paper we extend these studies to the general k > 1 case.

Finally, we explore the physical picture of ergoregion emission as pair creation [57, 58]. This picture was investigated in reference [53] for the two-charge k = 1 JMaRT solutions. It was shown that to a good approximation, radiation from these solutions can be split into two distinct parts. One part escapes to in nity, and the other remains deep inside the AdS region, at the cap. In the present work, we generalize this picture to include all three charges and the orbifolding parameter k, and consider the most general form of the probe scalar wavefunction. We con rm that also in this more elaborate set-up, the radiation splits into two distinct parts: one part escapes to in nity and the other part remains deep inside in the AdS region.

Our results generalize various previous studies (already mentioned above) of both BPS and non-BPS states arising from spectral ow of R-R ground states. We comment in detail on the relation of our work to these previous works after we have introduced the CFT states in full detail in section 3.

There has been a resurgence of interest in the black hole information paradox in recent years, in particular with regard to the experience of an infalling observer; see [59{66] and references within. Our results develop further the AdS/CFT dictionary for non-BPS black hole microstates, and such technical progress may ultimately shed light on these questions.

The remainder of this paper is organized as follows. In section 2 we study the general family of orbifolded JMaRT solutions and solve the wave equation on these backgrounds. In section 3 we identify the CFT description of these geometries. The emission spectrum and rates obtained from the CFT are shown to be in perfect agreement with the gravity computation. In section 4 we analyze the pair creation picture of ergoregion emission for these orbifolds. We close with a brief discussion in section 5.

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2 Orbifolded JMaRT solutions

After a brief review of the supergravity solutions in section 2.1, we study the near-decoupling limit in section 2.2 in which the geometries have a large AdS inner region, weakly coupled to at asymptotics. In section 2.3 we analyze the smoothness properties and categorize the possible orbifold singularities of the solutions. In section 2.4 we study the scalar wave equation on these orbifolds. We obtain the real and imaginary parts of the instability eigen-frequencies in the near-decoupling limit.

2.1 Supergravity solutions

The JMaRT solutions [31] are special cases of the non-extremal rotating three-charge Cveti c-Youm [67] solutions. In general, Cveti c-Youm geometries can have singularities, horizons, and closed timelike curves. Reference [31] derived the conditions that need to be imposed on the parameter space of the Cveti c-Youm geometries so that we get smooth solitonic solutions, possibly with orbifold singularities.

We consider type IIB string theory compacti ed on

M4;1 [notdef] S1 [notdef] T4 : (2.1)

We consider the S1 to be macroscopic, and consider the T4 to be string-scale. We consider n1 D1-branes wrapped on S1, n5 D5-branes wrapped on S1[notdef]T4, and np units of momentum

P along the S1. We parameterize the S1 with coordinate y and the T4 with coordinates zi.

Our supergravity analysis begins with the general non-extremal three-charge Cveti c-Youm metric, lifted to type IIB supergravity [67, 68]. The 10D string frame metric is [31]

ds2 =

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(dt2 dy2) +

(spdy cpdt)2

r2dr2(r2 + a21)(r2 + a22) Mr2

+ d 2

15 (a22 a21)

(1 +5 f) cos2

!cos2 d 2

15 + (a22 a21)

(1 +5 f) sin2

!sin2 d2

+ M

(a1 cos2 d + a2 sin2 d)2

+ 2M cos2

[(a1c1c5cp a2s1s5sp)dt + (a2s1s5cp a1c1c5sp)dy]d

+ 2M sin2

[(a2c1c5cp a1s1s5sp)dt + (a1s1s5cp a2c1c5sp)dy]d

Xi=1dz2i ; (2.2)

{ 4 {

where we use the shorthand notation ci = cosh i, si = sinh i, for i = 1; 5; p, and where

f = r2+a21 sin2 +a22 cos2 ;1 = f +M sinh2 1 ;5 = f +M sinh2 5 : (2.3)

Explicit expressions for the six-dimensional dilaton and Ramond-Ramond two-form eld can be found, e.g., in [31]; we will not need those details in our discussion below. Upon compacti cation to ve dimensions, one obtains asymptotically at con gurations carrying three U(1) charges, corresponding to D1, D5, and P. These charges are given by Qi = Msici:

The y circle will play a key role in the following; we take it to have radius R at spacelike in nity, y y + 2R. In addition, we take the volume of T4 to be (2)4V at spacelike

in nity. The integer quantization of the three charges is then given by

Q1 = gs [prime]3V n1; Q5 = gs [prime]n5; Qp =

g2s [prime]4

V R2 np: (2.4)

The ADM mass and angular momenta of the ve-dimensional asymptotically at con- gurations are

MADM = M 4G5

s21 + s25 + s2p + 32 ; (2.5)

J =

M4G5 (a1c1c5cp a2s1s5sp); (2.6)

J =

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M4G5 (a2c1c5cp a1s1s5sp); (2.7)

where G5 is the ve-dimensional Newton constant. The ten-dimensional Newton constant is as usual G10 = 86g2s [prime]4, so we have G5 = g

2s [prime]44V R : To have positive ADM mass we take

M 0, and without loss of generality we take 1; 5; p 0 and a1 a2 0.

The singularities H1 = 0 and H5 = 0 in metric (2.2) are curvature singularities, and there are also singularities where the function

g(r) (r2 + a21)(r2 + a22) Mr2 (2.8)

has roots, i.e. at

(M a21 a22) [notdef] q(M a21 a22)2 4a21a22 : (2.9)

Smooth geometries without horizons are obtained by demanding that at r = r+ an S1 should shrink smoothly, with the singularity at r = r+ being that of polar coordinates at the origin of a two-dimensional factor of the metric [31]. The parameter analysis is slightly di erent for the two-charge (Qp = 0) and three-charge cases; in this paper we focus on the general case of three non-vanishing charges. In this case four conditions on the parameters must be satis ed for the geometries to be smooth (up to possible orbifold singularities). We now present a brief summary of the analysis of [31]; for further details we refer the reader to that reference.

The function g(r) has real roots if and only if M > (a1 + a2)2 or M < (a1 a2)2.

For r = r+ to be an origin, rather than a horizon, the determinant of the metric in the

{ 5 {

r2[notdef] =

constant t and r subspace must vanish at r = r+. This rules out the case M > (a1 + a2)2 and gives the rst condition on the parameters,

M = a21 + a22 a1a2

c21c25c2p + s21s25s2p

s1c1s5c5spcp : (2.10)

In order that r = r+ be an origin, a spacelike Killing vector with closed orbits must smoothly degenerate there. The most general Killing vector with closed orbits is

Killing = @y @ @; (2.11)

and the one that degenerates at r = r+, given the condition (2.10), is given by

spcp

(a1c1c5cp a2s1s5sp)

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; =

spcp

(a2c1c5cp a1s1s5sp)

: (2.12)

We introduce new coordinates appropriate to the neighborhood of r = r+,

spcp

(a1c1c5cp a2s1s5sp)

spcp

(a2c1c5cp a1s1s5sp)

y: (2.13)

Then the coordinate which shrinks at r = r+ is y at constant ,

In order to nd the most general smooth solutions and to allow for the possibility of orbifold singularities at r = r+, we introduce a positive integer k and impose that y ! y + 2kR at constant

, is a closed orbit. This gives two further conditions,

spcp

(a1c1c5cp a2s1s5sp)

(kR) = m 2 Z : (2.14)

Furthermore, demanding regularity at the origin r = r+ under y ! y + 2kR xes the size

of the y-circle at in nity,

R = 1

Ms1c1s5c5(s1c1s5c5spcp)1=2

pa1a2(c21c25c2p s21s25s2p)

(kR) = n 2 Z ;

spcp

(a2c1c5cp a1s1s5sp)

: (2.15)

To summarize, the full regularity conditions for the three-charge orbifolded case are

(a) a1a2 = Q1Q5 k2R2

s21c21s25c25spcp (c21c25c2p s21s25s2p)2

; (2.16)

(b) M = a21 + a22 a1a2

c21c25c2p + s21s25s2p

s1c1s5c5spcp ; (2.17)

(a1c1c5cp a2s1s5sp)

(kR) = n 2 Z; (2.18)

(d)

spcp

(a2c1c5cp a1s1s5sp)

(kR) = m 2 Z: (2.19)

Using these conditions we record here some relations between the parameters which will be useful in what follows,

r2+ = a1a2

s1s5sp

c1c5cp ; r2 = a1a2

c1c5cp

s1s5sp ; M = a1a2nm

c1c5cp

s1s5sp

c1c5cp

2: (2.20)

The ADM angular momenta, in terms of the parameters introduced above, take the following simple form,

J =

mk n1n5; J =

nk n1n5: (2.21)

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2.2 The near-decoupling limit

In order to study AdS/CFT in this system one must isolate low-energy excitations of the D1-D5 bound state, which is achieved by taking the large R limit. In the gravity description, this corresponds to taking the near-decoupling limit in which one obtains a large inner region involving an AdS3[notdef]S3[notdef]T4 throat, weakly coupled to the at asymptotics.

The large R limit is de ned by keeping Q1; Q5 xed and taking R (Q1Q5)

14 , which

makes R the largest scale in the problem. We then have the small dimensionless parameter

= (Q1Q5)

1 4

R 1: (2.22)

In the Cvetic-Youm metric, the near-decoupling limit is obtained by taking

a21; a22; M Q1; Q5 ) s1 [similarequal] c1 1 ; s5 [similarequal] c5 1: (2.23)

We refer to the limit (2.22), (2.23) as the large R limit or the near-decoupling limit.In this limit we can identify the region r2 Q1; Q5 as an asymptotically AdS region.

This amounts to taking1 Q1 and

5 Q5 and using approximations (2.23) in the

metric (2.2). We thus obtain an asymptotically AdS3 [notdef] S3 metric,

ds2 =

2 M3 +

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J23 42

d2 +

2 M3 +J23 42

1 d2 + 2

J3 22 d

pQ1Q5

(d 2+sin2

d + RpQ1Q5 (a1cpa2sp)d+RpQ1Q5 (a2cpa1sp)d 2

+ cos2

d + RpQ1Q5 (a2cp a1sp)d +RpQ1Q5 (a1cp a2sp)d 2)(2.24)

where we have de ned new coordinates

= y

R; =

tR; (2.25)

2 = R2

Q1Q5 [r2 + (M a21 a22) sinh2 p + a1a2 sinh 2 p]: (2.26)

The AdS length and the size of the S3 is (Q1Q5)

14 . In writing the above expressions we

have also de ned

M3 = R2

Q1Q5 [(M a21 a22) cosh 2 p + 2a1a2 sinh 2 p]; (2.27)

J3 = R2

Q1Q5 [(M a21 a22) sinh 2 p + 2a1a2 cosh 2 p]: (2.28)

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The regularity conditions (2.16){(2.19) simplify in the large R limit as follows,

(a[prime]) a1a2 [similarequal]

Q1Q5

k2R2 spcp; (2.29)

(b[prime]) M [similarequal] a21 + a22 a1a2

c2p + s2p

spcp ; (2.30)

(c[prime]) spcp c1c5(a1cp a2sp)

(kR) [similarequal] n 2 Z; (2.31)

(d[prime])

spcp

c1c5(a2cp a1sp)

(kR) [similarequal] m 2 Z: (2.32)

A useful form of condition (2.30) via (2.20) is

M [similarequal]

Q1Q5

(kR)2

nmspcp ; (2.33)

and another expression that will be useful later is

r2+ r2 [similarequal]

Q1Q5

k2R2 : (2.34)

Substituting conditions (2.29){(2.32) into (2.24) we nd that the geometry is an orbifold of AdS3 [notdef] S3,

ds2 =

pQ1Q5"

1k2 + 2

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d2 + d2

1k2 + 2 1

+ 2d2 (2.35)

+d 2 + sin2

d + mk d nk d

2+ cos2

d nk d +mk d

We will analyze the smoothness properties of these orbifold geometries in the next subsection.

Let us now look at how various physical quantities behave in the large R limit. It is only in this limit that we expect physical parameters in the gravity description to be reproduced by a dual CFT analysis. In this limit the mass above the mass of the D1 and D5 branes is

MADM [similarequal]

M 4G5

s2p + 12 [similarequal]n1n5Rm2 + n2 12k2 ; (2.36)

and the 6D ADM linear momentum PADM is

PADM = np

R =

M4G5 spcp [similarequal]

mnk2 : (2.37)

In section 3 we will observe agreement between the gravity quantities (2.21), (2.36), and (2.37) from the CFT description.

2.3 Smoothness analysis

As observed above, the decoupling limit of the general JMaRT orbifolded solution is an orbifold of AdS3[notdef] S3, with metric (2.36). These orbifolds were brie y discussed in [31]; here

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n1n5

we present a detailed smoothness analysis following [48]. For convenience let us introduce the coordinates

nk +

mk ;

nk : (2.38)

Working in the covering space for coordinates (; ~

; ~

), the periodicities of (; ; ) trans-

late into the following identi cations:

A : (; ~

; ~

) ! (;

~ ) + 2

1; nk ;mk ; (2.39)

B : (; ~

; ~

) ! (;

JHEP11(2015)063

~ ) + 2(0; 1; 0); (2.40)

C : (; ~

; ~

) ! (;

~ ) + 2(0; 0; 1): (2.41)

Note that the coordinate in the metric (2.36) is ill-de ned at = 0, where a conical singularity can occur; the periodicity required for smoothness is ! +2k at xed

; ~

Conical singularities only occur at points that remain invariant under the operation

AmABmBCmC (2.42)

for some mI 2 Z. The conical singularities all arise at = 0 and may be localized at = 0

and/or = 2 , or may occur everywhere in . We will continue to focus on the case of three non-zero charges; the two-charge case is discussed in [31].

Case 1: gcd(k, m) = gcd(k, n) = 1: if there are no common divisors between the pairs (m; k) and (n; k), then there are no conical singularities and the spacetime is completely smooth.

To see this, we rst examine the possibility of having a xed point where ~

has a

non-zero size, i.e., at = 0; [negationslash]= 0. For a xed point to occur here,

must remain invariant

under (2.42). This implies that m

k mA + mC = 0. Since mC is an integer this requires

mA k

to be an integer; we write mA = km[prime]A. The periodic identi cations of and ~ are then

! + 2km[prime]A ;

~ + 2 mB nm[prime]A

: (2.43)

In the range 0 < <

2 , ~ also has a nite size. So for a xed point to occur there, ~ must also remain invariant. This xes mB = nm[prime]A, and as a result the above identi cation becomes

! + 2km[prime]A ; (2.44) which, being an integer multiple of 2k, is the correct identi cation for smoothness.

At = 2 , ~ has zero size. Thus, under the di eomorphism (2.43) the point = 0; = 2 is a xed point. The relevant identi cation is simply (2.44) and we again have smoothness.

It remains to examine = 0; = 0. Here ~

has zero size but ~ has non-zero size.

Requiring ~ to be invariant xes nkmA + mB = 0. Since mB is an integer, this implies

mA should be an integer multiple of k; we again write it as mA = km[prime]A. The relevant

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identi cation is again (2.44). This shows that the spacetime is free of conical singularity here also.

In summary, there are no conical singularities anywhere, and so the spacetime is completely smooth. From the point of view of the k = 1 JMaRT solitons, one can say that the

Zk quotient is freely acting in this case [31].

Case 2: gcd(k, m) > 1, gcd(k, n) = 1: if gcd(k; m) l1 > 1 and gcd(k; n) = 1, there

is a Zl1 orbifold singularity at = 0; = 2 and the spacetime is otherwise smooth.

To see this, we rst note that at = 0; = 0, the analysis is the same as in Case 1, and there is no orbifold singularity at these points.

Next, let us write k = l1^

k , m = l1 ^

m. For a xed point at = 0; [negationslash]= 0,

~ also has a non-zero

nl1 m[prime]A.

Since mB is an integer, m[prime]A must be an integer multiple of l1, i.e., m[prime]A = l1m[prime][prime]A. Then the identi cation ! + 2mA becomes ! + (2k)m[prime][prime]A since we have

mA = ^

km[prime]A = ^

kl1m[prime][prime]A = km[prime][prime]A: (2.45)

Hence we have smoothness at = 0; 0 < < 2 .

For = 0; =

2 , ~ has zero size. Invariance of ~

gives mA = ^

m[prime]Al1 : (2.46)

Since m[prime]A is a general integer, there is a Zl1 orbifold singularity at = 0; = 2 .

Case 3: gcd(k, m) = 1, gcd(k, n) > 1: if gcd(k; m) = 1 and gcd(k; n) l2 > 1, there

is a Zl2 orbifold singularity at = 0; = 0 and the spacetime is otherwise smooth.

To see this, rstly an analysis similar to Case 2 shows that there are no conical singularities at = 0; = 2 or at = 0; 0 < < 2 .

Next, let us write k = l2^

k, n = l2^

n. For = 0; = 0 to be a xed point ~ must remain invariant. This xes mA = ^

km[prime]A; mB = ^

nm[prime]A. The identi cation ! + 2mA then

becomes ! + 2

km[prime]A, which is

m[prime]Al2 : (2.47)

This results in a Zl2 orbifold singularity at = 0; = 0.

Case 4: gcd(k, m) > 1, gcd(k, n) > 1, gcd(k, m, n) = 1: when both gcd(k; m)

l1 > 1 and gcd(k; n) l2 > 1, the spacetime has both a Zl1 orbifold singularity at

= 0; =

2 and a Zl2 orbifold singularity at = 0; = 0. Away from these points the metric is smooth. The analysis is similar to the previous two cases.

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must remain

invariant. This xes mA = ^

km[prime]A , mC = ^

mm[prime]A. At points [negationslash]= 2 ,

size, so it must also remain invariant. This xes nk(

km[prime]A) + mB = 0, i.e, mB =

km[prime]A. So the

identi cation ! + 2mA becomes ! + 2(

km[prime]A), i.e.,

! + (2k)

Case 5: gcd(k, m) > 1, gcd(k, n) > 1 gcd(k, m, n) > 1: when gcd(k; m) l1 > 1, gcd(k; n) l2 > 1, and gcd(k; m; n) l3 > 1, then the orbifold has a rich singularity

structure with

Zl1 orbifold singularity at = 0; = 2 ,

Zl2 orbifold singularity at = 0; = 0, and

Zl3 orbifold singularity at = 0; 0 < < 2 .

Thus at = 0 there is at least a Zl3 orbifold singularity all over the three-sphere,1 which may be enhanced to a singularity of higher degree at the poles if l1 and/or l2 are greater than l3.

To see this, we rst observe that an analysis similar to Cases 2 and 3 shows that there is a Zl1 orbifold singularity at = 0; = 2 and a Zl2 orbifold singularity at = 0; = 0.

To see the orbifold singularity at = 0; 0 < < 2 we introduce the following notation,

m = l1 ^

m; n = l2^

n; l1 = l3^l1; l2 = l3^l2; k = l3^l1^l2^

k: (2.48)

must remain invariant. From the

^ k mA + mC = 0 : (2.49)

Since mC is an integer, mA must be a multiple of ^l2^

k, so we write mA = ^l2^

km[prime]A: This gives

mC = ^

mm[prime]A:

Similarly, from the invariance of ~ we get

^l1^l2^

Hence there is a Zl3 orbifold at = 0 and [negationslash]= 0; [negationslash]= 2 .

We nally note that although we presented the above analysis for the decoupled asymptotically AdS3 [notdef] S3 geometries, it applies equally well to the asymptotically at geometries

before taking the decoupling limit using (2.13).

1Note that the orbifold singularity at = 0, 0 < < 2 only arises in the non-BPS case, since in the BPS limit we have m = n + 1 and so m and n have no common divisors.

{ 11 {

JHEP11(2015)063

To have a xed point at = 0; 0 < < 2 , both ~ and ~

invariance of ~

we get

mk mA + mC = 0 )

nk mA + mB = 0 )

^ m[prime]A + mB = 0: (2.50)

Since mB is an integer, m[prime]A must be a multiple of ^l1, i.e., m[prime]A = ^l1m[prime][prime]A: This implies mA = ^l1^l2^

km[prime][prime]A; and mB = ^

n^l2m[prime][prime]A: The identi cation now becomes

! + 2

km[prime][prime]A = + (2k)m[prime][prime]A

l3 : (2.51)

2.4 Scalar wave equation

We next study a minimally coupled scalar in six dimensions on the general orbifolded JMaRT solutions. For the k = 1 solutions, such a computation showed that these geometries su er from a classical ergoregion instability [51]. We extend this study to the case of general k; m; n, obtaining the real and imaginary parts of the instability eigen-frequencies in the large R limit. We will later reproduce these results from the CFT.

Let us consider a minimally coupled complex scalar in six dimensions, on the background of the dimensionally reduced 6D Einstein frame metric. If one takes the 10D string frame metric written in (2.2) and discards the T4 directions, one obtains exactly the 6D Einstein frame metric, and so we will not rewrite it here. Such a minimal scalar arises for example from the dimensional reduction of the ten-dimensional IIB graviton having both its indices along the four-torus. We can separate variables using the ansatz,

= exp i!t + im + im + i

Ry ~( )h(r); (2.52)

which gives equation for the angular part

1 sin 2

d d

sin 2 dd ~ +

2 R2

(a21 sin2 + a22 cos2 ) m2cos2 m2 sin2

~ = ~:

JHEP11(2015)063

(2.53)

We are looking for wave functions with frequency !

1R . In the large R limit, in terms of

de ned in (2.22), we observe that

!2 2 R2

a2i 4 (2.54)

and so we nd

= l(l + 2) + O( 4) : (2.55)

The radial equation takes the form

1 r

d dr

g(r)rddr h h +

2 R2

(r2 + Ms21 + Ms25) +

!cp

Rsp

k2 r2+ r2r2 r2+

nk m +

mk m

2h + k2 r2+ r2r2 r2 !%R #nk m+mk m

2h = 0;

(2.56)

where g(r) = (r2 r2+)(r2 r2). Introducing the dimensionless variable x for the radial

coordinate via

x = 1

r2 r2+ r2+ r2

; (2.57)

we can write the radial equation in the form

x + 1 k2

@xh + 14 2x + 1 2 +2x + k2 2x h = 0; (2.58)

{ 12 {

with

2 =

!2 2 R2

(r2+ r2)k2; (2.59)

= !%R # m

nk + m

mk ; (2.60)

= m

nk + m

mk ; (2.61)

% = c21c25c2p s21s25s2ps1c1s5c5 ; (2.62)

# = c21c25 s21s25s1c1s5c5 spcp; (2.63)

2 = 1 +

2 R2

(r2+ + Ms21 + Ms25) !cp Rsp

2M: (2.64)

For later use, we note from (2.64) that the correction to is O( 2),

= l + 1 + O( 2) : (2.65)

The radial di erential equation (2.58) cannot be solved exactly. It can however be solved via matched asymptotic expansion. This is done in detail in appendix A. The instability frequencies are given by solutions to the transcendental equation (A.16). To the leading order in the large R expansion, we let one of the functions in the denominator of (A.16) develop a pole,

12(1 + + k[notdef] [notdef] + k) [similarequal] N; (2.66) with N a non-negative integer. From equations (2.62) and (2.63) we see that in the large R limit % ! 1 and # 2. Hence to leading order one obtains

[similarequal] !R m

nk + m

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mk : (2.67)

Substituting this relation along with (2.65) into equation (2.66), we get the real part !R of the instability frequencies to leading order, which are given by

!R [similarequal]

1 kR

: (2.68)

For certain values of the parameters, !R can become negative or zero. For those cases there is no emission.

One obtains the imaginary part !I of the instability frequencies to leading order by iterating the above approximation to the next order, setting N ! N + N. This computation

is discussed in detail in appendix A. The result is

!I [similarequal]

1 kR

l m m + mn [notdef]k m n + mm[notdef] 2(N + 1)

22l+1(l!)2

2 R2

Q1Q5

k2R2

l+1

N + l + 1l + 1 N+ k[notdef] [notdef] + l + 1 l + 1

: (2.69)

Since !I > 0, we have an instability: i.e., an exponentially growing perturbation. In the following section we reproduce (2.68) and (2.69) from the dual CFT.

{ 13 {

3 CFT description of orbifolded JMaRT solutions

3.1 The D1-D5 system on T4 and the orbifold CFT

In order to discuss the CFT interpretation of the general orbifolded JMaRT solutions, we next review some properties of the D1-D5 system on T4 and the corresponding orbifold CFT. We follow in places the presentations of [69] and [48].

As mentioned in the previous section, we work in type IIB string theory compacti ed on M4;1[notdef]S1[notdef]T4, with n1 D1-branes wrapped on S1 and n5 D5-branes wrapped on S1[notdef]T4.

We work in the limit of large R, which corresponds to the low-energy limit of the gauge theory on the D-brane bound state.

At low energies, the gauge theory on the bound state ows to a (4; 4) SCFT. It is conjectured that there is a point in moduli space where this SCFT is a symmetric product orbifold theory, consisting of n1n5 symmetrized copies of a free (4; 4) SCFT with target space T4 [2, 44].

Each copy of T4 gives 4 bosonic elds X1; X2; X3; X4, along with 4 left-moving fermionic excitations 1; 2; 3; 4 and the corresponding right-moving excitations, which we denote with a bar ( 1, etc.). The total central charge of the CFT is c = 6n1n5.

The CFT has a (small) N = 4 superconformal symmetry in both the left and right-

moving sectors. Each superconformal algebra contains an R-symmetry SU(2). Therefore we have the global symmetry SU(2)L [notdef] SU(2)R, whose quantum numbers we denote as

SU(2)L : (jL; mL); SU(2)R : (jR; mR): (3.1)

In addition there is a broken SO(4) [similarequal] SU(2)[notdef]SU(2) symmetry, corresponding to rotations

in the four directions of the T4. We label this symmetry by

SU(2)1 [notdef] SU(2)2: (3.2)

We use indices ; _

for SU(2)L and SU(2)R respectively, and indices A; _

A for SU(2)1 and

SU(2)2 respectively. The 4 real fermion elds of the left sector are grouped into complex fermions A. The right fermions are grouped into fermions _

A. The boson elds Xi are a vector in T4 and have no charge under SU(2)L or SU(2)R, so are grouped as XA _

Di erent copies of the c = 6 CFT are denoted with a copy label in brackets, e.g.,

X(1) ; X(2) ; [notdef] [notdef] [notdef] ; X(n1n5) : (3.3)

It will be convenient to describe the states of interest in terms of spectral ow [70]. Under a spectral ow transformation on the left-moving sector with parameter , the dimensions and charges of states change as follows:

h[prime] = h + mL + 2 c

24 ; m[prime]L = mL +

c12 : (3.4)

An independent spectral ow operation exists in the right-moving sector, with parameter

{ 14 {

JHEP11(2015)063

3.2 Twisted Ramond sector ground states

We next brie y review the construction of twist operators and twisted Ramond sector ground states by mapping to a local covering space [69, 71]. Let us consider the permutation (123 : : : k). The bare twist operator k corresponding to this permutation imposes the following periodicity conditions on the cylinder:

X(1) ! X(2) ! [notdef] [notdef] [notdef] ! X(k) ! X(1) (3.5)

(1) ! (2) ! [notdef] [notdef] [notdef] ! (k) ! (1): (3.6)

Note that the last sign in the second line above is minus, and is the only physically meaningful sign, as the intermediate signs can be absorbed by eld rede nitions.2 This state is then in the NS sector in the covering space, which we will sometimes refer to simply as the NS sector. A similar expression holds for the right-moving fermions; for ease of presentation we will write only the left-moving expressions in various places in the following. It is convenient to describe these k twisted copies of the CFT as a component string of length k.

One de nes the bare twist operator k by mapping rst to the plane with coordinate z = ew and then to a local covering plane with coordinate t via a map of the local form

z z b (t t )k ; (3.7)

where z and t are the respective images of w in the z plane and the t plane. The k bosonic

elds in (3.5) map to one single-valued bosonic eld X(t) in the t plane, and similarly for the fermions. In the t plane, one inserts the identity operator at the point t , obtaining the

lowest-dimension operator in the k-twisted sector. If we take t = 0, we obtain the NS-NS

vacuum in the covering space. We thus refer to it as the \k-twisted NS-NS vacuum", and denote it by [notdef]0k[angbracketright](r)NS, where r is an index labelling the di erent component strings. The

quantum numbers of this state are

h =

h = 14

JHEP11(2015)063

1 k

; mL = mR = 0 : (3.8)

We next de ne an excited (spin-)twist operator k as follows. Follow the procedure used to de ne the bare twist k, but in the covering t plane, insert a spin eld3 S at t .

If we take t = 0, we obtain the (left-moving) R vacuum [notdef]0[notdef]R[angbracketright]t of the t plane. We write

k = S kk ; = +; : (3.9)

Back on the original cylinder, with coordinate w, as the elds circle the operator [notdef]k, they transform as

X(1) ! X(2) ! [notdef] [notdef] [notdef] ! X(k) ! X(1) (3.10)

(1) ! (2) ! [notdef] [notdef] [notdef] ! (k) ! + (1): (3.11)

2One of the authors (DT) thanks Oleg Lunin for a discussion on this point.

3If b [negationslash]= 1 in (3.7), one must also include an appropriate normalization factor [71].

{ 15 {

The elds are thus in the Ramond sector in the covering space; as before, we will sometimes refer to this simply as the Ramond sector. We write the corresponding state on the original cylinder (with coordinate w) as [notdef]0[notdef]k[angbracketright](r)R.

Adding in the right-moving sector, we obtain the full spin-twist eld

_ k = S k

S _ kk (3.12)

and we denote the corresponding twisted R-R ground state by [notdef]0 _ k[angbracketright](r)R.

3.3 Non-BPS states generated by general fractional spectral ow

We now consider spectral ow operations in the k-fold covering space. Spectral ow by c units in the k-fold covering space corresponds to an e ective spectral ow in the base space by an amount [48]

= c

k : (3.13)

On the base space, this may then be described as fractional spectral ow; for previous discussions of fractional spectral ow, see [56, 72, 73].

Using this operation we now describe the general AdS/CFT dictionary for the k > 1 JMaRT solutions. All of the states we consider consist of nc = N1N5=k component strings of length k, with each component string in the same state;4 spectral ow acts simultaneously on all component strings. On a component string of length k, excitations are spaced in units of 1=k. Fractional spectral ow generates states with lled Fermi seas with this fractional moding, as we will see explicitly shortly.

Let us rst de ne the reference state from which we will perform the fractional spectral ows. This state has all of its component strings in the k-twisted NS-NS vacuum:

|0k[angbracketright]NS = [notdef]0k[angbracketright](1)NS [notdef]0k[angbracketright](2)NS [notdef] [notdef] [notdef] [notdef]0k[angbracketright](nc)NS : (3.14)

The quantum numbers of this state are (here c = 6n1n5 for the full CFT)

h =

h = c

1 1k2 ; mL = mR = 0 : (3.15)

The AdS dual of this state is the decoupled orbifolded AdS solution (2.36) with m = n = 0. The full asymptotically at JMaRT solitonic solutions exist only when [notdef]m[notdef] [negationslash]= [notdef]n[notdef] (if one

works with a1 a2 0, this becomes m > n 0), so this solution does not directly come

from the decoupling limit of an asymptotically at JMaRT solution.5The states we are interested in are obtained by general fractional spectral ow from

|0k[angbracketright]NS. The map to the JMaRT solutions is that the spectral ow parameters are given by

= m + n

k ;

= m n

4In this paper we consider parameters such that nc is an integer.

5It is however related to the other decoupled JMaRT solutions by (fractional) spectral ow coordinate transformations, which do not go to zero at the boundary of AdS.

{ 16 {

JHEP11(2015)063

k : (3.16)

Using (3.4), the quantum numbers of the spectral owed states are

h = c

1 + (m + n)2 1k2 ; mL = c12m + nk ; (3.17)

h = c

1 + (m n)2 1k2 ; mR = c12m nk : (3.18)

Therefore the CFT energy above the R-R ground state and momentum are

E = h +

hR =

n1n5

m2 + n2 12k2 ; P =

mnk2 : (3.19)

Note that in the orbifold CFT, the momentum on each component string must be an integer (see e.g. [74]), so in the orbifold CFT one has mn=k 2 Z.

Using the map between CFT and gravity SU(2) quantum numbers,

m = (mL + mR) ; m = (mL mR) ; (3.20)

these parameters exactly match those computed on the gravity side in (2.21), (2.36) and (2.37), providing a rst check on our proposed identi cation.

The above states are R-R in the covering space when m + n is odd, and NS-NS in the covering space when m + n is even. Our main interest is in the R-R states; in order to connect with the discussion in [48], let us present the free fermion description of these states, focussing on the states with positive mL and mR.

Let us rst consider a single component string. Recall that on a component string of length k, excitations are spaced in units of 1=k. The state on the component string involves Fermi seas lled to a general fractional level s=k in both species of fermions, +1 and +2, and similarly to a level

s=k for the right-movers:

| s; s;k[angbracketright](r) = +1sk +2 s k

: : :

hR =

n1n5

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+12k +22k +11k +21k

[notdef]

0++k[angbracketright](r)R : (3.21)

Then as before the state of the full CFT is obtained by taking all nc = N1N5=k component strings to be in the same state:

| s; s;k[angbracketright] = [notdef] s; s;k[angbracketright](1) [notdef] s; s;k[angbracketright](2) [notdef] [notdef] [notdef] [notdef] s; s;k[angbracketright](nc) : (3.22)

The twisted R-R ground state [notdef]0++k[angbracketright](r)R may be obtained from the twisted NS-NS

vacuum [notdef]0k[angbracketright]NS by performing fractional spectral ow with parameters = 1=k,

: : :

[notdef]

= 1=k.

The above state [notdef] s; s;k[angbracketright] is generated by a further fractional spectral ow with parameters

= 2s=k,

= 2s=k. So in total, [notdef] s; s;k[angbracketright] is generated by starting with the state [notdef]0k[angbracketright]NS and

performing fractional spectral ow with parameters

= 2s + 1

k ;

= 2 s + 1

k : (3.23)

{ 17 {

We then have the relations

m + n = 2s + 1 ; m n = 2 s + 1 : (3.24) The NS-NS states obtained for even m + n have analogous Fermi sea representations, built on the twisted NS-NS vacuum [notdef]0k[angbracketright]NS.

We now return to our main discussion. For general k; m; n, we have observed the agreement of conserved charges above. As was noted in [48] in the BPS case however, generically these states are degenerate and so further evidence is required to support the identi cation. In principle, one could compute the one-point functions of operators following [29, 30], however the states we are considering are R-charge eigenstates, and therefore all one-point functions of R-charged operators vanish [29]. Instead, we provide further evidence for our proposed identi cation by matching the scalar excitation spectrum between gravity and CFT.

3.4 Emission spectrum and emission rates from CFT

The vertex operator for emission (or absorption) of a minimal scalar of angular momentum l has the following form [55]. It involves a chiral primary in the twisted sector of degree (l+1), ~

l+1, dressed with fermion and supercurrent excitations G+ _A12

which add the

T4 polarization indices, and further dressed with powers of SU(2) current zero modes J0,

J0 which ll out the SU(2) representation. There is also a non-trivial normalization factor; the explicit form can be found in [55].

Since the vertex operator involves a twisted chiral primary ~

l+1, when it acts on a state it introduces new fractionated degrees of freedom. It is thus capable of lowering the energy of the state, with the remainder energy being carried away by the emitted particle.

Our initial state (3.21) is composed of component strings which are all of length k. In the limit of a large number of component strings, nc = n1n5=k 1, the process which

dominates is that in which ~

l+1 acts on l + 1 distinct component strings, combining them into a component string of length k(l + 1). There is a family of resulting nal states labelled by left- and right-moving excitation numbers NL, NR, which correspond to acting with the Virasoro generators L1,

L1 in the form LNL1

LNR1 on the nal state of lowest

s are multiples of k. This was rst done for k = 1 in [55] and then for k > 1 in [56]. For the k > 1 case, the calculation involved mapping the amplitude to a covering space, and using the method of [71, 75].6 In each case one observes a Bose enhancement e ect: the probability for emission of the Nth quantum is N

times the probability for emission of the rst quantum [52]. Since the CFT is a symmetric product orbifold, one must also take care of various combinatorial factors in computing the amplitude.

6For a recent application of this method in a di erent context, see [69, 76].

{ 18 {

_ 12

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possible energy.

This CFT amplitude, corresponding to emission of a minimal scalar, can be mapped to a technically simpler amplitude by spectral ow and hermitian conjugation. This technique has been employed in the special case of excited R-R states arising from integer spectral ow of the state [notdef]0++k[angbracketright]R, i.e., when s and

Having proposed the identi cation of the general orbifolded JMaRT solutions with the general fractional spectral owed CFT states, we can now make a straightforward generalization of the results of [56] to fractional spectral owed CFT states. We do this by simply taking the emission spectrum, expressed in terms of ,

, and substituting the values appropriate for the general fractional spectral owed states that we study. This technique works because all of the states under consideration are fractional spectral ows of the twisted NS-NS vacuum [notdef]0k[angbracketright]NS.

The emission spectrum computed in [56] for the integer spectral owed k > 1 JMaRT states, translated into our conventions,7 is

! = 1

12 k(m m ) 12 k(m + m ) (l + 2 + NL + NR) ;

= 1

JHEP11(2015)063

12 k(m m ) +12 k(m + m ) + NR NL : (3.25)

We now generalize this by substituting the parameters appropriate to fractional spectral ow from the twisted NS vacuum [notdef]0k[angbracketright]NS,

= m + n

k ;

= m n

k ; (3.26)

which yields the spectrum

! = 1

kR [m m + mn (l + 2 + NL + NR)]

= 1

k [mm m n + NR NL] : (3.27)

Now, generalizing the discussion in [52], note that = (NL NR)=k. If > 0, ! may be

written as

! = 1

kR [l m m + mn 2 k 2NR] (3.28)

and if < 0, ! may be written as

! = 1

kR [l m m + mn 2 + k 2NL] : (3.29)

In either case, ! has the form

! = 1

kR (l m m + mn [notdef]k m n + mm[notdef] 2(N + 1)) (3.30) for some N 0, which exactly matches the real part of the instability frequencies computed

from the gravity side, given in eq. (2.68).

The CFT emission rate computed in [56] for the Nth particle, writing (!; ) as a schematic delta function which imposes that ! and must take their speci c allowed values, in our conventions takes the form

dd! = N

1 kR

2 22l+1(l!)2

2 R2

Q1Q5

k2R2

l+1

NL + l + 1l + 1 NR + l + 1 l + 1

(!; ) : (3.31)

7The map between conventions is given in appendix C.

{ 19 {

In this form, the expression for the emission rate immediately generalizes to the present situation of fractional spectral owed states, with the allowed frequencies and wavelengths given in (3.27).

Treating separately the cases for > 0 and < 0 as above, one nds

NL + l + 1l + 1 NR + l + 1l + 1 =

N + l + 1l + 1 N+ k[notdef] [notdef] + l + 1 l + 1

: (3.32)

The imaginary part of the frequency !I is given by 1/2 the value of the emission rate for the rst quantum, as discussed in [52]. Thus we have

!I [similarequal]

1 kR

JHEP11(2015)063

22l+1(l!)2

2 R2

Q1Q5

k2R2

l+1

N + l + 1l + 1 N+ k[notdef] [notdef] + l + 1 l + 1

; (3.33)

in exact agreement with the value (2.69) obtained from the gravity calculation.

Relation to previous work. We pause here to comment on the relation of our results to previous literature.

The class of states generated by (3.16) is the general set of R-R and NS-NS states obtained by fractional spectral ow from the twisted NS-NS vacuum [notdef]0k[angbracketright]NS. Various special

cases of this class of CFT states have been studied previously in the literature, as we now describe.

For BPS states, the two-charge states (k 2 Z+; m = 1; n = 0) were studied in [49, 50].

The three-charge family (k = 1; s 2 Z;

s = 0) was studied in [11, 12]. The family (k 2

Z+; s = nk; n 2 Z;

s = 0) was studied in [13]. Such values of s correspond to integer spectral ow of the states [notdef]0 _ k[angbracketright]R. The general BPS family obtained from fractional spectral

ow, (k 2 Z+ s 2 Z;

s = 0) was studied in [48].

For non-BPS states, the CFT states obtained by setting k = 1 in (3.14){(3.16) were proposed to be the dual CFT states of the k = 1 JMaRT solutions in the original paper [31] and the CFT emission was studied in [52, 55]. The two-charge family (k 2 Z+; s =

s =

nk; ^

^ n 2 Z) was studied in [54]. The family (k 2 Z+; s = ^

nk; ^

n 2 Z;

s =

nk;

n 2 Z) was

studied in [56]. Again, such values of s;

s correspond to integer spectral ow of the states

|0 _ k[angbracketright]R. The general non-BPS family of R-R and NS-NS states arising from fractional

spectral ow of [notdef]0k[angbracketright]NS (or [notdef]0 _ k[angbracketright]R) is the subject of the present work.

Regarding the wave equation calculation on the gravity side, for k = 1 the instability was rst derived in [51], and was revisited in slightly di erent forms in [52, 53]. The two-charge case with k > 1 was studied in [54]. In the present work we have analyzed the general three-charge case with arbitrary k; m; n.

4 Ergoregion emission as pair creation

Having demonstrated that the general class of orbifolded JMaRT solutions decay via an ergoregion instability, with emission spectrum and emission rate in agreement with the dual CFT, we now examine more explicitly some features of the produced radiation. In particular we investigate the physical picture of ergoregion emission as pair creation [57, 58].

{ 20 {

The ergoregion contains negative energy excitations as measured by the Killing vector that generates time translations at spatial in nity. The pair creation picture involves a positive energy excitation that escapes to in nity and a negative energy excitation that remains in the ergoregion. The two excitations also carry equal and opposite values of other conserved charges.

For two-charge, k = 1 JMaRT solutions, this picture was investigated in [53] for the simplest form of the probe scalar wavefunction. It was shown that to a good approximation, the radiation from these solutions can be split into two distinct parts. One part escapes to in nity and the other remains deep inside in the AdS region. The two parts carry equal and opposite energy and angular momentum. For large angular momenta, when the wavefunctions can be thought of as approximately localized, it was argued that the inner region part has its main support in the ergoregion.

In this section we generalize this discussion to include three non-zero charges, two non-zero angular momenta, the orbifolding parameter k, and the most general form of the wavefunction. We start with a summary of the solutions of the scalar wave equation in section 4.1. We then compute the contributions to angular momenta (section 4.2) and energy (section 4.3) from the inner and asymptotic regions due to the scalar perturbation.

4.1 Solutions of the wave equation

In order to calculate the contributions to conserved charges from the inner and asymptotic regions (which are de ned in appendix A), we need the explicit form of the wavefunctions in these regions. We work exclusively in the large R limit, as only in this limit is there a clear separation between the inner and asymptotic regions.

Let us start by relating the di erent radial coordinates so that we can easily change from one to the other. The coordinate transformation (2.26) upon using (2.20) and (2.30) is simply

2 = R2Q1Q5 (r2 r2+) : (4.1)

In terms of the dimensionless radial variable x used in section 2.4, this relation is x = 2:

The metric in the inner region is (2.36), and from (A.6) the wavefunction in the inner region is

in = exp i!t + i

Ry + im + im ~( )

where

2(1 + k[notdef] [notdef] + k); c = 1 + k[notdef] [notdef]; (4.3)

with , , de ned in (2.60), (2.61), (2.64). Recall also from (2.65), (2.66) that to leading order in we have

[similarequal] l + 1; 1 + + k[notdef] [notdef] + k [similarequal] 2N: (4.4)

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2 + 1 k2

k2k[notdef] [notdef]

2F1(a; b; c; k22) ;(4.2)

a = 12(1 + + k[notdef] [notdef] + k); b =

For small , we have in k[notdef] [notdef], and for large , using (A.8) and (4.4) we have

in (l+2). The norm of the wavefunction is

( )in = 2k[notdef] [notdef]

|~( )[notdef]2e2!It 2F1(N; N l 1; 1 + k[notdef] [notdef]; k22) 2; (4.5)

where as before !I is the imaginary part of the frequency !.

In the asymptotic region the metric is at spacetime to leading order. Using the asymptotic region wavefunction (A.12) together with the requirement of only outgoing waves (A.15), in terms of a normalization constant C2 and the quantity de ned in (2.59), the wavefunction is

out = C2 exp i!t + im + im + i

Ry ~( ) 1p2 132ei ei 4

ei 2 ei3 2 :

(4.6)

Therefore the norm of the wavefunction is

( )out = [notdef]C2[notdef]2[notdef]~( )[notdef]2e2!It

2 k4N+2 (kR)!I

Q1Q5 r2; and as a result the exponent in (4.7) can be written as

2!R!I

T dS ; L =

2 + 1 k2

JHEP11(2015)063

2 [notdef] [notdef]

13 ei( ) sin2( ) : (4.7)

We next x the normalization of in the asymptotic region given its form in the inner region.This is done in detail in appendix B. For our purposes we do not need an expression for C2 itself, but only its norm. From (B.7) we have

|C2[notdef]2 sin2( ) =

(1 + k[notdef] [notdef])2 (N + 1) (N + + 1)

(N + + 1 + k[notdef] [notdef]) (N + 1 + k[notdef] [notdef])

; (4.8)

where !I takes the value given in (2.69). Finally we note that in the asymptotic region 2 [similarequal]

q!2R 2R2

r ; (4.9)

where as before !R is the real part of the frequency ! and takes the value given in (2.68). In the neck region, the exponent (4.9) is very small, [notdef]( )[notdef] !I(Q1Q5)1=4 4l+5.

4.2 Angular momenta of the perturbation

The general JMaRT solution has four Killing vectors, namely @t; @y; @; @ . In general the

geometries carry angular momentum in both and directions and momentum in the y direction. As in the previous sections, we consider scalar perturbations that also carry all these charges. The conserved quantities for the scalar perturbation associated to the two angular momenta are

L =

i( ) [similarequal]

T dS ; (4.10)

where T is the energy momentum tensor of the (complex) scalar eld,

T = @ @ + @ @ g @ @ : (4.11)

{ 22 {

The integrals in (4.10) extend over a spacelike hypersurface in the spacetime. We choose the surface to be simply given by t = constant. It was shown in reference [31] that gtt < 0

everywhere, therefore the t = constant surface is everywhere spacelike.

Substituting the separation ansatz (2.52) in (4.11), we nd the following expressions for angular momenta of the scalar perturbation,

L = 2m

pgdrdA

gtt!R + gt m + gtm + gty R

; (4.12)

L = 2m

pgdrdA

gtt!R + gt m + gtm + gty R

; (4.13)

where dA = d d ddy. Note that the integrals involved in computing L and L are the

same. For this reason we focus on L ; the discussion for L is entirely analogous.In the asymptotic region the metric is at spacetime to leading order. We have pg =

r3 cos sin . There are no cross terms in the metric, so the integral (4.12) simply becomes

(L )out = 2m !R

Zout drdA(r3 cos sin )( )out;

= 4m !RRCe2!ItZout dr(r3h(r)h(r) )out; (4.14)

where C =

RS3 d dd cos sin [notdef]~( )[notdef]2: Using relations (4.7) and (4.9) expression (4.14) becomes

(L )out = 8Q1Q5m !RC R

q!2R 2R2e2!It[notdef]C2[notdef]2 sin2( ) Z

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1 14 dr exp

2!R!I

q!2R 2R2r

A: (4.15)

To leading order in the large R limit this integral gives

(L )out [similarequal] 2m Q1Q5Ce2!Itk4N+2l+3

(1 + k[notdef] [notdef])2 (N + 1) (N + l + 2)

(N + k[notdef] [notdef] + l + 2) (N + k[notdef] [notdef] + 1)

; (4.16)

where we have used the normalization (4.8). This is our nal expression for the angular momentum L of the scalar perturbation that ows o to in nity. For N = 0, k = 1, and

| [notdef] = 0 this expression reduces to the corresponding expression of reference [53].

Exactly the same expression but with opposite sign is obtained from the inner region. Using the coordinate de nitions (4.1) and (2.25), and the metric in the inner region (2.36), the integral (4.12) in the inner region becomes

(L )in = 2m Q1Q5

1 ( )in:

Substituting the norm (4.5) of the inner region wavefunction, we observe that the integrand falls o in the large limit as 2l5. Thus to leading order in we can set the upper limit of the integration to in nity. Thus we obtain

(L )in [similarequal] 4m Q1Q5Ce2!It

!RR + mk m

!RR+ mk m nk m Z1= 0 dZdA cos sin

2 + 1 k2

nk m

2F1(N; N l 1; 1 + k[notdef] [notdef]; k22) 2 :

[notdef] 0 d2k[notdef] [notdef]+1

2 + 1 k2

{ 23 {

Making the substitution ~

= k and using the integer relations (4.4), this expression can be converted to the form

(L )in [similarequal] 4m Q1Q5Ce2!It

!RR + mk m

nk m

k4N+2l+4

1 2k[notdef] [notdef]+1 ~

2 + 1

2Nlk[notdef] [notdef]3

2F1(N; N l 1; 1 + k[notdef] [notdef]; ~ 2)

2 :

This integral can be calculated using the hypergeometric function identity (B.8). We get

(L )in [similarequal] 4m Q1Q5Ce2!It

!RR + mk m

nk m

k4N+2l+4

[notdef] 2(2N + k[notdef] [notdef] + l + 2)

(1 + k[notdef] [notdef])2 (N + 1) (N + l + 2)

(N + k[notdef] [notdef] + l + 2) (N + k[notdef] [notdef] + 1)

: (4.17)

Using the de nition of from (2.60) and the integer relations (4.4), we obtain a contribution that is exactly the opposite of (4.16),

(L )in [similarequal] 2m Q1Q5Ce2!Itk4N+2l+3

(1 + k[notdef] [notdef])2 (N + 1) (N + l + 2)

(N + k[notdef] [notdef] + l + 2) (N + k[notdef] [notdef] + 1)

: (4.18)

At a technical level the analysis presented above is signi cantly more involved compared to that of [53], however various technical pieces precisely t together to give exactly equal and opposite contributions to L (and hence L) from the inner and asymptotic regions.

4.3 Energy and linear momentum of the perturbation

A similar set of considerations applies to energy and linear momentum along y. Let us start with linear momentum along y. The conserved linear momentum associated to the scalar perturbation is

Py =

Zt=constpgdrdA T ty: (4.19)

Using the separation ansatz (2.52) in the scalar stress tensor (4.11), the linear momentum expression reduces to

Py = 2

pgdrdA

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: (4.20)

Since the integral involved is exactly what we discussed above, it follows that the inner and asymptotic region wavefunctions give equal and opposite contributions to the linear momentum.

The conserved energy of the scalar eld is

H =

Zt=constpgdrdA T tt : (4.21)

It is convenient to write this expression as a part which involves the integral already computed for the angular momentum, plus a remainder which is a total derivative [53]. We denote these as the bulk and boundary terms respectively,

H = Hbulk + Hbdy ; (4.22)

{ 24 {

gtt!R + gt m + gtm + gty

where

Hbulk =

pgdrdAh

gtt@t @t

drdA

h @i pggij@j

+ @i pggij@j

;

hpggij@j ( )i: (4.23)

Using the equation of motion for the scalar @ (pg@ ) = 0 and the ansatz (2.52), the

bulk term simpli es to

Hbulk = 2!R Z

pgdrdA

Hbdy = 12 Z

drdA @i

: (4.24)

We now apply the decomposition into Hbulk and Hbdy separately in the inner and asymptotic regions. For this purpose, we approximate both the outer boundary of the inner region and the inner boundary of the outer region by the surface r = (Q1Q5)

14 . Since

gtt!R gtm gt m gty

the integral involved in Hbulk is exactly the one discussed above, it follows that the inner and asymptotic region wavefunctions give equal and opposite contributions to Hbulk. We

need only be concerned with the boundary terms. The only non-zero boundary terms arise from the terms with radial derivatives. We have

Hinbdy = 12 Z

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pggrr@r( )

r=(Q1Q5)

= Hin;neckbdy + Hin;r=r+bdy ; (4.25)

Houtbdy = 12 Z

r=r+

r=1

r=(Q1Q5)

pggrr@r( )

14 = Hout;r=1bdy + Hout;neckbdy: (4.26)

We observe that

Hin;neckbdy = Hout;neckbdy: (4.27)

Let us now estimate the various boundary terms. Firstly, for Hin;r=r+bdy, counting powers of we see that as ! 0,

pggrr@r( ) = pg

@( ) 2@( ) 2k[notdef] [notdef]+1; (4.28)

which vanishes at = 0 (i.e. at r = r+) and so we have Hin;r=r+bdy = 0.

Next, for Hout;r=1bdy we observe that falls o exponentially with exponent (4.9) in the r ! 1 limit. Therefore, in the limit r ! 1 it also vanishes.

Since we have observed in (4.27) that the neck terms are equal and opposite, it is not necessary to evaluate them to conclude that the contribution to the energy from the asymptotic region and the inner region are equal and opposite. Nevertheless, out of interest we now observe that these terms are parametrically subleading with respect to the contributions from Hbulk.

{ 25 {

dr d

At the neck, we have 1, so for Hin;neckbdy we nd the parametric dependence

Hin;neckbdy R pg

dr d

@( ) =1=

R (Q1Q5)

Q1Q5

[notdef]

2 (Q1Q5)

[notdef]

(Q1Q5)

1 4

R2 [notdef] @2l4

=1=

(Q1Q5)

4 2l+3 ; (4.29)

and therefore Hin;neckbdy is subleading with respect to Hinbulk.

Again it is not necessary to separately estimate Hout;neckbdy, however it is straightforward to observe that as a result of the matching of solutions at the neck, the asymptotic region wavefunction also behaves as l2 in the neck, and with the same coe cient as the inner solution, giving precisely (4.29).

To summarize, we have seen explicitly that the inner and asymptotic region wave-functions give equal and opposite contributions to the conserved angular momenta, linear momentum along y and energy of the scalar eld. Since the inner part of the wavefunction carries negative energy with respect to the Killing vector @t, it has its main support in the ergoregion. One can also see this fact explicitly by plotting a selection of examples. Thus we see explicitly in this setup the physical picture of ergoregion emission as pair creation.

5 Discussion

In this paper we have proposed the holographic description of the general family of orbifolded JMaRT solutions, with orbifolding parameter k. The k > 1 states are of signi cant physical interest since states with larger k are closer to typical states than states with smaller k. We have proposed that the dual CFT states are the general set of R-R and NS-NS states obtained by fractional spectral ow in both left- and right-moving sectors from the twisted NS-NS vacuum [notdef]0k[angbracketright]NS. We reviewed the fact that the orbifolded JMaRT

solutions are completely smooth when the integer parameters m, n, and k have no common divisors, and presented a full analysis of the orbifold singularities which arise depending on the common divisors between these parameters.

To support our proposed identi cation, we matched the minimal scalar emission spectrum and emission rate between gravity and CFT. On the gravity side, this involved solving the wave equation on the general orbifolded solution, generalizing previous studies [51, 54]. On the CFT side, our results were obtained via a straightforward generalization of the results of [56].

We also investigated the physical picture of ergoregion emission as pair creation, generalizing the results of [53] to include three non-zero charges, two non-zero angular momenta, the orbifolding parameter k, and the most general form of the probe scalar wavefunction. We showed that radiation from the general orbifolded JMaRT solutions can be split into two distinct parts, one escaping to in nity and the other remaining deep inside the AdS region. Since the inner part of the wavefunction carries negative energy with respect to

{ 26 {

JHEP11(2015)063

the Killing vector @t which generates time translations at spatial in nity, it has its main support in the ergoregion.

The states we have studied are non-BPS, and there is no known non-renormalization theorem protecting the quantities we have studied. Thus the fact that the orbifold CFT and gravity calculations agree exactly is quite non-trivial and better than might have been naively expected of the orbifold CFT. Naturally, the agreement observed in the k = 1 solutions was reason for optimism on this point. The fact that our proposed dual states are related to BPS states by fractional spectral ow may perhaps be the feature which enables this non-trivial agreement.

It would be interesting to study string theory in the subset of these backgrounds that have orbifold singularities. String theory on orbifolds of AdS3 [notdef] S3 has previously been

studied in [72, 73]. In the presence of orbifold singularities, twisted sectors of closed strings typically give rise to light (or tachyonic) degrees of freedom that are not taken into account by supergravity [77{79]. Furthermore, non-supersymmetric orbifolds are expected to decay to a region of smooth spacetime together with an expanding pulse of excitations [80, 81] (see also [82, 83]).8 If such a mechanism is present here, one can ask whether it interacts with the pair creation mechanism; for example one might imagine that the pair creation excitation that remains deep in the cap might interact with the orbifold and/or its decay products.

Indeed one might wonder whether such additional modes could a ect the matching of emission frequencies and rates between the supergravity and CFT for the solutions with orbifold singularities. We did not nd any such discrepancy; the calculations agree exactly between gravity and CFT regardless of the presence or absence of orbifold singularities. This strongly suggests that the ergoregion emission spectrum and rates are una ected by such light degrees of freedom. It would be interesting to investigate this physics in more detail in the future.

Acknowledgments

We thank Guillaume Bossard, Borun Chowdhury, Oleg Lunin, Emil Martinec, Samir Mathur, Kostas Skenderis, Marika Taylor and Nick Warner for fruitful discussions. The work of DT was supported by the CEA Eurotalents program and by John Templeton Foundation Grant 48222: \String Theory and the Anthropic Universe". BC thanks TIFR for hospitality where part of this work was done. DT thanks AEI Potsdam for hospitality where part of this work was carried out. AV thanks IPhT Saclay and IISER Bhopal for hospitality where part of this work was completed.

A Solving the wave equation via matched asymptotic expansion

In this appendix we solve the wave equation in a matched asymptotic expansion analysis. We obtain the instability frequencies and also x the normalization of the wavefunction in the asymptotic region given its form in the inner region.

8One of the authors (DT) thanks Emil Martinec for a discussion on this point.

{ 27 {

JHEP11(2015)063

We de ne the following regions of the geometry, in which we set up the matched asymptotic expansion. In section 2.2 we speci ed that when studying AdS/CFT on the JMaRT solutions, one works in the regime of parameters

= (Q1Q5)

1 4

R 1: (A.1)

In terms of the dimensionless radial variable x de ned in eq. (2.57), we de ne the inner region to be the range 0 x 2; to be more speci c, let us introduce another parameter

1 and de ne the inner region to be given by9

0 x [lessorsimilar]

1 2 : (A.2)

We then de ne the asymptotic region to be given by the range

x [greaterorsimilar] 1

1 2 : (A.3)

The inner and asymptotic regions do not overlap. We will match solutions in the neck region x

1 2 , or more speci cally

1 2 (A.4)

where solutions to the radial wave equation are power law in x [68]. Solutions from the inner and asymptotic regions match on to these power law solutions from the two sides.10

Inner region. In the inner region one can neglect 2x relative to the other terms, and so the radial wave equation (2.58) simpli es to

4@x

JHEP11(2015)063

2 [lessorsimilar] x [lessorsimilar]

x + 1 k2

@xh +

1 2 +2x + k2 2 x

h = 0: (A.5)

Demanding regularity at the origin we get the solution for this equation

h =

x + 1 k2

k2x k[notdef] [notdef] 2

2F1(a; b; c; k2x) ; (A.6)

where

a = 12(1 + + k[notdef] [notdef] + k); b =

2(1 + k[notdef] [notdef] + k); c = 1 + k[notdef] [notdef]: (A.7)

In writing this solution we have chosen to normalize the wavefunction (A.6) by setting its overall normalization constant to unity. The behaviour of the inner solution near x ! 0

9While one must consider [epsilon1] to be exponentially small in order to get a large AdS inner region (see for example the discussion in [45]), here is simply a bookkeeping device. The important point is that the inner and asymptotic regions do not overlap, and must be matched onto the neck region.

10Note that the regions involved in the present matched asymptotic expansion analysis are di erent to those employed, e.g., in [45].

{ 28 {

is simply h kkx

k[notdef] [notdef]

2 ; and its expansion for large x is

h [similarequal] (1 + k[notdef] [notdef])

k1 k[notdef] [notdef]k ( )

2 (1 + k[notdef] [notdef] + k)

2 (1 + k[notdef] [notdef] k))

+1 2

+ k1+ k[notdef] [notdef]k ( )

2 (1 + + k[notdef] [notdef] + k)

2 (1 + + k[notdef] [notdef] k)

2 #: (A.8)

We will match this onto the power law behaviour in the neck region below.

Asymptotic region. In the asymptotic region, one can neglect 2

x+k2

x relative to

the other terms, and so the radial wave equation simpli es to

@2x(xh) +

2 4x +

1 2

4x2

(xh) = 0: (A.9)

The most general solution to this equation is a linear combination of Bessel functions

h = 1

C1J ( px) + C2J ( px) : (A.10)

For px 1, its behaviour is h

(1 + )

x 12 + C2 (1 )

x +12 ; (A.11)

and its large px behaviour is

1 x

JHEP11(2015)063

1 p2 h

ei pxei

4 (C1ei

2 + C2ei

2 ) + ei pxei

4 (C1ei

2 + C2ei

: (A.12)

x can be neglected, and the

Neck region. In the neck region, both 2x and 2

x+k2

wave equation approximates to

@2x(xh) +

1 24x2 (xh) = 0: (A.13)

The general solution is

h = A x

2 + B x

2 : (A.14)

Matching the solutions. We can now match the solutions at each end of the neck region, and thereby patch together the three matching regions.

We are interested in instability of the geometry where there are no incoming waves, yet we have outgoing waves carrying energy and other charges to in nity. The requirement of no incoming waves gives the relation

C1 + C2ei = 0: (A.15)

Matching the two asymptotic expansions (A.8) and (A.11) to the solutions in the neck region, we obtain

(1 )

(1 + )

2 = ( ) ( ) 12 (1 + k[notdef] [notdef] + k)

2 (1 + k[notdef] [notdef] k)

:(A.16)

The emission frequencies are given by the solutions to this transcendental equation.

{ 29 {

12(1 + + k[notdef] [notdef] + k)

2 (1 + + k[notdef] [notdef] k)

Instability frequencies. Let us now analyze equation (A.16). Recall that we work in the large R limit, 1. In this limit, taking ! 1=R and 1, one nds 2 4, as can be seen from eqs. (2.59), (2.34), and (2.22). Therefore the l.h.s. of equation (A.16) is parametrically small. The r.h.s. is parametrically small when one of the functions in the denominator is parametrically close to developing a pole. To leading order, the values of the parameters will be those which give poles. Let us set11

2(1 + + k[notdef] [notdef] + k) [similarequal] N; (A.17)

with N a non-negative integer. From equations (2.62) and (2.63) we see that in the large R limit % ! 1 and # 2. Hence to leading order one obtains

[similarequal] !R m

nk + m

: (A.20)

At next-to-leading order, we will obtain the leading imaginary part of !. To do this, we replace N ! N + N and eliminate in favour of N. From equation (2.67) we have

= R !, which upon using (A.17) gives the change in ! due to shifting N to be

! =

2kRIm( N): (A.22)

The small deformation N controls the pole of the divergent function. We assume that N , so that to leading order in it can be neglected in the argument of all the

other functions. In what follows we shall verify the consistency of this assumption. The residue at the pole of the function is given by

(N N) =

Using the relations

(n + 1 + x) = xn![x]n (x); (n x) =

{ 30 {

mk ; (A.18)

[similarequal] l + 1: (A.19) Replacing these relations in equation (A.17) gives the leading order instability frequencies.

To leading order, the instability frequencies are real; we de ne !R to be the real part of !, thus obtaining

!R [similarequal]

1 kR

JHEP11(2015)063

l m m + mn [notdef]k m n + mm[notdef] 2(N + 1)

2kR N : (A.21)

There are also contributions to the subleading part of ! from corrections to and at order 2, however these a ect only the real part of !. Therefore, denoting by !I the imaginary part of !, to leading order in we have

!I [similarequal]

(1)N+1

1 N : (A.23)

; (A.24)

11Taking parameters for which the other Gamma function in the denominator of (A.16) develops a pole leads to an exponentially decaying mode, rather than an exponentially growing mode.

(x)

(1)nn![x]n

where [x]n =

Qi=1 1 + xi

; we obtain

N = ei

( )

2 [ ]N[ ]N+k[notdef] [notdef]: (A.25)

For p and q integers, we have [p]q = p+qCp = p+q

The identity

( ) ( ) =

sin( ) (A.26)

allows us to extract Im( N). From (A.25) we obtain

!I [similarequal]

2 kR

JHEP11(2015)063

( )2

2 [ ]N[ ]N+k[notdef] [notdef] : (A.27)

Recalling that = l+1+O( 2), we observe that Im( N) 4l+4, and that Re( N) 4l+2,

which demonstrates the consistency of our approach. Then to leading order the imaginary part of the frequency is

!I [similarequal]

1 kR

22l+1(l!)2

2 R2

Q1Q5

k2R2

l+1

N + l + 1l + 1 N+ k[notdef] [notdef] + l + 1 l + 1

: (A.28)

B Details of pair creation calculation

B.1 Normalization of the asymptotic region wavefunction

In this appendix we x the normalization of the asymptotic region wavefunction, for use in section 4.

Using the asymptotic region wavefunction (A.12) together with the requirement of only outgoing waves (A.15), we obtain

hout(x) = C2 1 p2

1 x

ei pxei

ei 2 ei3 2 ; (B.1)

and thus

hout(x)h out(x) = [notdef]C2[notdef]2

2 [notdef] [notdef]

ei( )px sin2( ): (B.2)

Matching the two asymptotic expansions | (A.11) and (A.8) | say by comparing coe -cients of x

2 , we get an equation that determines C2,

k1+ k[notdef] [notdef]k (1 + k[notdef] [notdef]) ( )

2 (1 + + k[notdef] [notdef] + k)

1 x

3 2

= (C2ei )

(1 + )

: (B.3)

2 (1 + + k[notdef] [notdef] k)

To nd the real and imaginary frequencies we matched the solution using

12(1 + + k[notdef] [notdef] + k)

= (N N) =

(1)N+1

N! N : (B.4)

Replacing this expression in (B.3) we get

k2N+2 (1 + k[notdef] [notdef]) ( )

(N + + 1 + k[notdef] [notdef])

(1)N+1N! N = (C2ei )

(1 + )

: (B.5)

{ 31 {

Taking modulus of the above relation allows us to extract [notdef]C2[notdef]2. We get,

k4N+4 (1 + k[notdef] [notdef])2 ( )2 (N + 1)2

(N + + 1 + k[notdef] [notdef])2

( N)( N) = [notdef]C2[notdef]2

(1 + )2

[notdef] [notdef] 2

2 : (B.6)

We now use (A.25), and working to leading order in , we approximate [notdef] [notdef]2 [similarequal] 2. For use in the main text, it is convenient to extract one power of !I using (A.27). We thus obtain

|C2[notdef]2 sin2( ) =

2 k4N+2 (kR)!I

(1 + k[notdef] [notdef])2 (N + 1) (N + + 1)

(N + + 1 + k[notdef] [notdef]) (N + 1 + k[notdef] [notdef])

: (B.7)

B.2 A hypergeometric function identity

Identity: for positive and for arbitrary positive integers N and l,

0 d2 +1(1 + 2)2Nl3 (2F1(N; N l 1; 1 + ; 2))2

= 1

2(2N + + l + 2)

(1 + )2 (N + 1) (N + l + 2)

(N + + l + 2) (N + + 1) : (B.8)

Proof: a proof of the above identity can be given by relating hypergeometric functions in the integral to Jacobi polynomials. From identity 8.962.1 (third line) of Gradshteyn and Ryzhik [84], page 999, we have

P ( ;l+1)N(y) =

(N + 1 + )

(N + 1) (1 + )

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1 + y 2

N 2F1

N; N l 1; 1 + ;y 1 y + 1

: (B.9)

De ning

y + 1 = 2; (B.10) the integral can be converted into

2 +l+3

(N + 1) (1 + ) (N + 1 + )

y 1

2 1 dy(1 y) (1 + y)l+1

P ( ;l+1)N(y)

2; (B.11)

which simply gives the right hand side of (B.8) upon using identity 7.391.1 (second line) on page 806 of [84].

C Conventions

In this appendix we record our conventions and their relation to those of ref. [56], which we use to obtain eq. (3.25) of the main text.

Our conventions are that

left moving $ holomorphic $ positive Py; (C.1)

where Py is momentum along y. So the holomorphic coordinate in the CFT is related to the null coordinate v = (t y) in the spacetime.

{ 32 {

Our map between CFT and gravity SU(2) quantum numbers is given in (3.20),

m = (m +

m) ; m = (m

m) : (C.2)

Let us compare our conventions to those of Avery-Chowdhury [56], whose quantities we denote with a superscript AC. In that paper, the anti-holomorphic coordinate corresponds to positive y. Therefore we interchange L and R in mapping between the two papers, so the spectral ow parameters are

AC ;

= AC : (C.3)

Next, the parameter controlling the twist is

AC = k : (C.4)

In the conventions of [56], the emission of a scalar with gravity quantum numbers (l; mAC ; mAC) corresponds to the CFT vertex

Vl;mAC ;mAC (C.5)

where

mAC = l kAC

In addition, similarly to our conventions we have the relation

mAC = (mAC +

mAC) ; mAC = mAC

Since L and R are interchanged between the two papers, we have

mL =

mAC ; mR = mAC ) m = mAC ; m = mAC : (C.8)

Therefore we obtain

kAC = 12 (l + m + m) ;

{ 33 {

JHEP11(2015)063

kAC ; mAC = kAC

kAC : (C.6)

mAC : (C.7)

kAC = 12 (l + m m) : (C.9)

Using these relations in eq. (10.3) of [56], we arrive at (3.25).

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

[1] S.W. Hawking, Breakdown of predictability in gravitational collapse, http://dx.doi.org/10.1103/PhysRevD.14.2460

Web End =Phys. Rev. D 14 (1976)

http://dx.doi.org/10.1103/PhysRevD.14.2460

Web End =2460 [http://inspirehep.net/search?p=find+J+"Phys.Rev.,D14,2460"

Web End =INSPIRE ].[2] A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, http://dx.doi.org/10.1016/0370-2693(96)00345-0

Web End =Phys.

http://dx.doi.org/10.1016/0370-2693(96)00345-0

Web End =Lett. B 379 (1996) 99 [http://arxiv.org/abs/hep-th/9601029

Web End =hep-th/9601029 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9601029

Web End =INSPIRE ].

[3] O. Lunin and S.D. Mathur, AdS/CFT duality and the black hole information paradox, http://dx.doi.org/10.1016/S0550-3213(01)00620-4

Web End =Nucl.

http://dx.doi.org/10.1016/S0550-3213(01)00620-4

Web End =Phys. B 623 (2002) 342 [http://arxiv.org/abs/hep-th/0109154

Web End =hep-th/0109154 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0109154

Web End =INSPIRE ].[4] S.D. Mathur, The fuzzball proposal for black holes: an elementary review, http://dx.doi.org/10.1002/prop.200410203

Web End =Fortsch. Phys. 53

http://dx.doi.org/10.1002/prop.200410203

Web End =(2005) 793 [http://arxiv.org/abs/hep-th/0502050

Web End =hep-th/0502050 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0502050

Web End =INSPIRE ].[5] I. Bena and N.P. Warner, Black holes, black rings and their microstates, http://dx.doi.org/10.1007/978-3-540-79523-0_1

Web End =Lect. Notes Phys.

http://dx.doi.org/10.1007/978-3-540-79523-0_1

Web End =755 (2008) 1 [http://arxiv.org/abs/hep-th/0701216

Web End =hep-th/0701216 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0701216

Web End =INSPIRE ].[6] K. Skenderis and M. Taylor, The fuzzball proposal for black holes, http://dx.doi.org/10.1016/j.physrep.2008.08.001

Web End =Phys. Rept. 467 (2008)

http://dx.doi.org/10.1016/j.physrep.2008.08.001

Web End =117 [http://arxiv.org/abs/0804.0552

Web End =arXiv:0804.0552 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0804.0552

Web End =INSPIRE ].[7] V. Balasubramanian, J. de Boer, S. El-Showk and I. Messamah, Black holes as e ective

geometries, http://dx.doi.org/10.1088/0264-9381/25/21/214004

Web End =Class. Quant. Grav. 25 (2008) 214004 [http://arxiv.org/abs/0811.0263

Web End =arXiv:0811.0263 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0811.0263

Web End =INSPIRE ].[8] B.D. Chowdhury and A. Virmani, Modave lectures on fuzzballs and emission from the D1-D5

system, http://arxiv.org/abs/1001.1444

Web End =arXiv:1001.1444 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1001.1444

Web End =INSPIRE ].[9] S.D. Mathur, Black holes and beyond, http://dx.doi.org/10.1016/j.aop.2012.05.001

Web End =Annals Phys. 327 (2012) 2760 [http://arxiv.org/abs/1205.0776

Web End =arXiv:1205.0776 ]

[http://inspirehep.net/search?p=find+EPRINT+arXiv:1205.0776

Web End =INSPIRE ].

[10] I. Bena and N.P. Warner, Resolving the structure of black holes: philosophizing with a

hammer, http://arxiv.org/abs/1311.4538

Web End =arXiv:1311.4538 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1311.4538

Web End =INSPIRE ].[11] S. Giusto, S.D. Mathur and A. Saxena, Dual geometries for a set of 3-charge microstates,

http://dx.doi.org/10.1016/j.nuclphysb.2004.09.001

Web End =Nucl. Phys. B 701 (2004) 357 [http://arxiv.org/abs/hep-th/0405017

Web End =hep-th/0405017 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0405017

Web End =INSPIRE ].[12] O. Lunin, Adding momentum to D1-D5 system, http://dx.doi.org/10.1088/1126-6708/2004/04/054

Web End =JHEP 04 (2004) 054 [http://arxiv.org/abs/hep-th/0404006

Web End =hep-th/0404006 ]

[http://inspirehep.net/search?p=find+EPRINT+hep-th/0404006

Web End =INSPIRE ].

[13] S. Giusto, S.D. Mathur and A. Saxena, 3-charge geometries and their CFT duals, http://dx.doi.org/10.1016/j.nuclphysb.2005.01.009

Web End =Nucl.

http://dx.doi.org/10.1016/j.nuclphysb.2005.01.009

Web End =Phys. B 710 (2005) 425 [http://arxiv.org/abs/hep-th/0406103

Web End =hep-th/0406103 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0406103

Web End =INSPIRE ].[14] I. Bena and N.P. Warner, One ring to rule them all. . . and in the darkness bind them?, http://dx.doi.org/10.4310/ATMP.2005.v9.n5.a1

Web End =Adv.

http://dx.doi.org/10.4310/ATMP.2005.v9.n5.a1

Web End =Theor. Math. Phys. 9 (2005) 667 [http://arxiv.org/abs/hep-th/0408106

Web End =hep-th/0408106 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0408106

Web End =INSPIRE ].[15] I. Bena and N.P. Warner, Bubbling supertubes and foaming black holes, http://dx.doi.org/10.1103/PhysRevD.74.066001

Web End =Phys. Rev. D 74

http://dx.doi.org/10.1103/PhysRevD.74.066001

Web End =(2006) 066001 [http://arxiv.org/abs/hep-th/0505166

Web End =hep-th/0505166 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0505166

Web End =INSPIRE ].[16] P. Berglund, E.G. Gimon and T.S. Levi, Supergravity microstates for BPS black holes and

black rings, http://dx.doi.org/10.1088/1126-6708/2006/06/007

Web End =JHEP 06 (2006) 007 [http://arxiv.org/abs/hep-th/0505167

Web End =hep-th/0505167 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0505167

Web End =INSPIRE ].[17] I. Bena, C.-W. Wang and N.P. Warner, Mergers and typical black hole microstates, http://dx.doi.org/10.1088/1126-6708/2006/11/042

Web End =JHEP 11

http://dx.doi.org/10.1088/1126-6708/2006/11/042

Web End =(2006) 042 [http://arxiv.org/abs/hep-th/0608217

Web End =hep-th/0608217 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0608217

Web End =INSPIRE ].[18] S. Giusto, R. Russo and D. Turton, New D1-D5-P geometries from string amplitudes, http://dx.doi.org/10.1007/JHEP11(2011)062

Web End =JHEP

http://dx.doi.org/10.1007/JHEP11(2011)062

Web End =11 (2011) 062 [http://arxiv.org/abs/1108.6331

Web End =arXiv:1108.6331 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1108.6331

Web End =INSPIRE ].[19] S. Giusto and R. Russo, Perturbative superstrata, http://dx.doi.org/10.1016/j.nuclphysb.2012.12.012

Web End =Nucl. Phys. B 869 (2013) 164

[http://arxiv.org/abs/1211.1957

Web End =arXiv:1211.1957 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1211.1957

Web End =INSPIRE ].[20] S. Giusto, L. Martucci, M. Petrini and R. Russo, 6D microstate geometries from 10D

structures, http://dx.doi.org/10.1016/j.nuclphysb.2013.08.018

Web End =Nucl. Phys. B 876 (2013) 509 [http://arxiv.org/abs/1306.1745

Web End =arXiv:1306.1745 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1306.1745

Web End =INSPIRE ].[21] I. Bena, J. de Boer, M. Shigemori and N.P. Warner, Double, double supertube bubble, http://dx.doi.org/10.1007/JHEP10(2011)116

Web End =JHEP

http://dx.doi.org/10.1007/JHEP10(2011)116

Web End =10 (2011) 116 [http://arxiv.org/abs/1107.2650

Web End =arXiv:1107.2650 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1107.2650

Web End =INSPIRE ].[22] B.E. Nieho , O. Vasilakis and N.P. Warner, Multi-superthreads and supersheets, http://dx.doi.org/10.1007/JHEP04(2013)046

Web End =JHEP 04

http://dx.doi.org/10.1007/JHEP04(2013)046

Web End =(2013) 046 [http://arxiv.org/abs/1203.1348

Web End =arXiv:1203.1348 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1203.1348

Web End =INSPIRE ].[23] B.E. Nieho and N.P. Warner, Doubly- uctuating BPS solutions in six dimensions, http://dx.doi.org/10.1007/JHEP10(2013)137

Web End =JHEP

http://dx.doi.org/10.1007/JHEP10(2013)137

Web End =10 (2013) 137 [http://arxiv.org/abs/1303.5449

Web End =arXiv:1303.5449 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1303.5449

Web End =INSPIRE ].

{ 34 {

JHEP11(2015)063

[24] I. Bena, S. Giusto, R. Russo, M. Shigemori and N.P. Warner, Habemus superstratum! A

constructive proof of the existence of superstrata, http://dx.doi.org/10.1007/JHEP05(2015)110

Web End =JHEP 05 (2015) 110 [http://arxiv.org/abs/1503.01463

Web End =arXiv:1503.01463 ]

[http://inspirehep.net/search?p=find+EPRINT+arXiv:1503.01463

Web End =INSPIRE ].

[25] J.M. Maldacena, The large-N limit of superconformal eld theories and supergravity, http://dx.doi.org/10.1023/A:1026654312961

Web End =Int. J.

http://dx.doi.org/10.1023/A:1026654312961

Web End =Theor. Phys. 38 (1999) 1113 [http://arxiv.org/abs/hep-th/9711200

Web End =hep-th/9711200 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9711200

Web End =INSPIRE ].[26] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253

[http://arxiv.org/abs/hep-th/9802150

Web End =hep-th/9802150 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9802150

Web End =INSPIRE ].[27] S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical

string theory, http://dx.doi.org/10.1016/S0370-2693(98)00377-3

Web End =Phys. Lett. B 428 (1998) 105 [http://arxiv.org/abs/hep-th/9802109

Web End =hep-th/9802109 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9802109

Web End =INSPIRE ].[28] S. Giusto, E. Moscato and R. Russo, AdS3 Holography for 1=4 and 1=8 BPS geometries,

http://arxiv.org/abs/1507.00945

Web End =arXiv:1507.00945 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1507.00945

Web End =INSPIRE ].[29] K. Skenderis and M. Taylor, Fuzzball solutions and D1-D5 microstates, http://dx.doi.org/10.1103/PhysRevLett.98.071601

Web End =Phys. Rev. Lett. 98

http://dx.doi.org/10.1103/PhysRevLett.98.071601

Web End =(2007) 071601 [http://arxiv.org/abs/hep-th/0609154

Web End =hep-th/0609154 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0609154

Web End =INSPIRE ].[30] I. Kanitscheider, K. Skenderis and M. Taylor, Holographic anatomy of fuzzballs, http://dx.doi.org/10.1088/1126-6708/2007/04/023

Web End =JHEP 04

http://dx.doi.org/10.1088/1126-6708/2007/04/023

Web End =(2007) 023 [http://arxiv.org/abs/hep-th/0611171

Web End =hep-th/0611171 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0611171

Web End =INSPIRE ].[31] V. Jejjala, O. Madden, S.F. Ross and G. Titchener, Non-supersymmetric smooth geometries

and D1-D5-P bound states, http://dx.doi.org/10.1103/PhysRevD.71.124030

Web End =Phys. Rev. D 71 (2005) 124030 [http://arxiv.org/abs/hep-th/0504181

Web End =hep-th/0504181 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0504181

Web End =INSPIRE ].[32] E.G. Gimon, T.S. Levi and S.F. Ross, Geometry of non-supersymmetric three-charge bound

states, http://dx.doi.org/10.1088/1126-6708/2007/08/055

Web End =JHEP 08 (2007) 055 [http://arxiv.org/abs/0705.1238

Web End =arXiv:0705.1238 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0705.1238

Web End =INSPIRE ].[33] S. Giusto, S.F. Ross and A. Saxena, Non-supersymmetric microstates of the D1-D5-KK

system, http://dx.doi.org/10.1088/1126-6708/2007/12/065

Web End =JHEP 12 (2007) 065 [http://arxiv.org/abs/0708.3845

Web End =arXiv:0708.3845 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0708.3845

Web End =INSPIRE ].[34] J.H. Al-Alawi and S.F. Ross, Spectral ow of the non-supersymmetric microstates of the

D1-D5-KK system, http://dx.doi.org/10.1088/1126-6708/2009/10/082

Web End =JHEP 10 (2009) 082 [http://arxiv.org/abs/0908.0417

Web End =arXiv:0908.0417 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0908.0417

Web End =INSPIRE ].[35] I. Bena, A. Puhm and B. Vercnocke, Metastable supertubes and non-extremal black hole

microstates, http://dx.doi.org/10.1007/JHEP04(2012)100

Web End =JHEP 04 (2012) 100 [http://arxiv.org/abs/1109.5180

Web End =arXiv:1109.5180 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1109.5180

Web End =INSPIRE ].[36] I. Bena, A. Puhm and B. Vercnocke, Non-extremal black hole microstates: fuzzballs of re or

fuzzballs of fuzz?, http://dx.doi.org/10.1007/JHEP12(2012)014

Web End =JHEP 12 (2012) 014 [http://arxiv.org/abs/1208.3468

Web End =arXiv:1208.3468 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1208.3468

Web End =INSPIRE ].[37] S.D. Mathur and D. Turton, Oscillating supertubes and neutral rotating black hole

microstates, http://dx.doi.org/10.1007/JHEP04(2014)072

Web End =JHEP 04 (2014) 072 [http://arxiv.org/abs/1310.1354

Web End =arXiv:1310.1354 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1310.1354

Web End =INSPIRE ].[38] S. Banerjee, B.D. Chowdhury, B. Vercnocke and A. Virmani, Non-supersymmetric

microstates of the MSW system, http://dx.doi.org/10.1007/JHEP05(2014)011

Web End =JHEP 05 (2014) 011 [http://arxiv.org/abs/1402.4212

Web End =arXiv:1402.4212 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1402.4212

Web End =INSPIRE ].[39] D. Katsimpouri, A. Kleinschmidt and A. Virmani, An inverse scattering construction of the

JMaRT fuzzball, http://dx.doi.org/10.1007/JHEP12(2014)070

Web End =JHEP 12 (2014) 070 [http://arxiv.org/abs/1409.6471

Web End =arXiv:1409.6471 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1409.6471

Web End =INSPIRE ].[40] G. Bossard and S. Katmadas, A bubbling bolt, http://dx.doi.org/10.1007/JHEP07(2014)118

Web End =JHEP 07 (2014) 118 [http://arxiv.org/abs/1405.4325

Web End =arXiv:1405.4325 ]

[http://inspirehep.net/search?p=find+EPRINT+arXiv:1405.4325

Web End =INSPIRE ].

[41] G. Bossard and S. Katmadas, Floating JMaRT, http://dx.doi.org/10.1007/JHEP04(2015)067

Web End =JHEP 04 (2015) 067 [http://arxiv.org/abs/1412.5217

Web End =arXiv:1412.5217 ]

[http://inspirehep.net/search?p=find+EPRINT+arXiv:1412.5217

Web End =INSPIRE ].

[42] G.W. Gibbons and N.P. Warner, Global structure of ve-dimensional fuzzballs, http://dx.doi.org/10.1088/0264-9381/31/2/025016

Web End =Class. Quant.

http://dx.doi.org/10.1088/0264-9381/31/2/025016

Web End =Grav. 31 (2014) 025016 [http://arxiv.org/abs/1305.0957

Web End =arXiv:1305.0957 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1305.0957

Web End =INSPIRE ].[43] P. de Lange, D.R. Mayerson and B. Vercnocke, Structure of six-dimensional microstate

geometries, http://dx.doi.org/10.1007/JHEP09(2015)075

Web End =JHEP 09 (2015) 075 [http://arxiv.org/abs/1504.07987

Web End =arXiv:1504.07987 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1504.07987

Web End =INSPIRE ].

{ 35 {

JHEP11(2015)063

[44] C. Vafa, Gas of D-branes and Hagedorn density of BPS states, http://dx.doi.org/10.1016/0550-3213(96)00025-9

Web End =Nucl. Phys. B 463 (1996) 415

[http://arxiv.org/abs/hep-th/9511088

Web End =hep-th/9511088 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9511088

Web End =INSPIRE ].[45] S.D. Mathur and D. Turton, Microstates at the boundary of AdS, http://dx.doi.org/10.1007/JHEP05(2012)014

Web End =JHEP 05 (2012) 014

[http://arxiv.org/abs/1112.6413

Web End =arXiv:1112.6413 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1112.6413

Web End =INSPIRE ].[46] S.D. Mathur and D. Turton, Momentum-carrying waves on D1-D5 microstate geometries,

http://dx.doi.org/10.1016/j.nuclphysb.2012.05.014

Web End =Nucl. Phys. B 862 (2012) 764 [http://arxiv.org/abs/1202.6421

Web End =arXiv:1202.6421 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1202.6421

Web End =INSPIRE ].[47] O. Lunin, S.D. Mathur and D. Turton, Adding momentum to supersymmetric geometries,

http://dx.doi.org/10.1016/j.nuclphysb.2012.11.017

Web End =Nucl. Phys. B 868 (2013) 383 [http://arxiv.org/abs/1208.1770

Web End =arXiv:1208.1770 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1208.1770

Web End =INSPIRE ].[48] S. Giusto, O. Lunin, S.D. Mathur and D. Turton, D1-D5-P microstates at the cap, http://dx.doi.org/10.1007/JHEP02(2013)050

Web End =JHEP 02

http://dx.doi.org/10.1007/JHEP02(2013)050

Web End =(2013) 050 [http://arxiv.org/abs/1211.0306

Web End =arXiv:1211.0306 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1211.0306

Web End =INSPIRE ].[49] V. Balasubramanian, J. de Boer, E. Keski-Vakkuri and S.F. Ross, Supersymmetric conical

defects: Towards a string theoretic description of black hole formation, http://dx.doi.org/10.1103/PhysRevD.64.064011

Web End =Phys. Rev. D 64

http://dx.doi.org/10.1103/PhysRevD.64.064011

Web End =(2001) 064011 [http://arxiv.org/abs/hep-th/0011217

Web End =hep-th/0011217 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0011217

Web End =INSPIRE ].[50] J.M. Maldacena and L. Maoz, Desingularization by rotation, http://dx.doi.org/10.1088/1126-6708/2002/12/055

Web End =JHEP 12 (2002) 055

[http://arxiv.org/abs/hep-th/0012025

Web End =hep-th/0012025 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0012025

Web End =INSPIRE ].[51] V. Cardoso, O.J.C. Dias, J.L. Hovdebo and R.C. Myers, Instability of non-supersymmetric

smooth geometries, http://dx.doi.org/10.1103/PhysRevD.73.064031

Web End =Phys. Rev. D 73 (2006) 064031 [http://arxiv.org/abs/hep-th/0512277

Web End =hep-th/0512277 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0512277

Web End =INSPIRE ].[52] B.D. Chowdhury and S.D. Mathur, Radiation from the non-extremal fuzzball, http://dx.doi.org/10.1088/0264-9381/25/13/135005

Web End =Class. Quant.

http://dx.doi.org/10.1088/0264-9381/25/13/135005

Web End =Grav. 25 (2008) 135005 [http://arxiv.org/abs/0711.4817

Web End =arXiv:0711.4817 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0711.4817

Web End =INSPIRE ].[53] B.D. Chowdhury and S.D. Mathur, Pair creation in non-extremal fuzzball geometries, http://dx.doi.org/10.1088/0264-9381/25/22/225021

Web End =Class.

http://dx.doi.org/10.1088/0264-9381/25/22/225021

Web End =Quant. Grav. 25 (2008) 225021 [http://arxiv.org/abs/0806.2309

Web End =arXiv:0806.2309 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0806.2309

Web End =INSPIRE ].[54] B.D. Chowdhury and S.D. Mathur, Non-extremal fuzzballs and ergoregion emission, http://dx.doi.org/10.1088/0264-9381/26/3/035006

Web End =Class.

http://dx.doi.org/10.1088/0264-9381/26/3/035006

Web End =Quant. Grav. 26 (2009) 035006 [http://arxiv.org/abs/0810.2951

Web End =arXiv:0810.2951 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0810.2951

Web End =INSPIRE ].[55] S.G. Avery, B.D. Chowdhury and S.D. Mathur, Emission from the D1-D5 CFT, http://dx.doi.org/10.1088/1126-6708/2009/10/065

Web End =JHEP 10

http://dx.doi.org/10.1088/1126-6708/2009/10/065

Web End =(2009) 065 [http://arxiv.org/abs/0906.2015

Web End =arXiv:0906.2015 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0906.2015

Web End =INSPIRE ].[56] S.G. Avery and B.D. Chowdhury, Emission from the D1-D5 CFT: higher twists, http://dx.doi.org/10.1007/JHEP01(2010)087

Web End =JHEP 01

http://dx.doi.org/10.1007/JHEP01(2010)087

Web End =(2010) 087 [http://arxiv.org/abs/0907.1663

Web End =arXiv:0907.1663 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0907.1663

Web End =INSPIRE ].[57] J.L. Friedman, Ergosphere instability, Comm. Math. Phys. 63 (1978) 243.[58] N. Comins and B.F. Schutz, On the ergoregion instability, Proc. Roy. Soc. London A 364

(1978) 211.[59] S.D. Mathur, The information paradox: a pedagogical introduction, http://dx.doi.org/10.1088/0264-9381/26/22/224001

Web End =Class. Quant. Grav. 26

http://dx.doi.org/10.1088/0264-9381/26/22/224001

Web End =(2009) 224001 [http://arxiv.org/abs/0909.1038

Web End =arXiv:0909.1038 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0909.1038

Web End =INSPIRE ].[60] S.L. Braunstein, S. Pirandola and K. _

Zyczkowski, Better late than never: information

retrieval from black holes, http://dx.doi.org/10.1103/PhysRevLett.110.101301

Web End =Phys. Rev. Lett. 110 (2013) 101301 [http://arxiv.org/abs/0907.1190

Web End =arXiv:0907.1190 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0907.1190

Web End =INSPIRE ].[61] A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or

rewalls?, http://dx.doi.org/10.1007/JHEP02(2013)062

Web End =JHEP 02 (2013) 062 [http://arxiv.org/abs/1207.3123

Web End =arXiv:1207.3123 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1207.3123

Web End =INSPIRE ].[62] A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An apologia for rewalls,

http://dx.doi.org/10.1007/JHEP09(2013)018

Web End =JHEP 09 (2013) 018 [http://arxiv.org/abs/1304.6483

Web End =arXiv:1304.6483 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1304.6483

Web End =INSPIRE ].[63] S.D. Mathur and D. Turton, Comments on black holes I: the possibility of complementarity,

http://dx.doi.org/10.1007/JHEP01(2014)034

Web End =JHEP 01 (2014) 034 [http://arxiv.org/abs/1208.2005

Web End =arXiv:1208.2005 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1208.2005

Web End =INSPIRE ].[64] S.D. Mathur and D. Turton, The aw in the rewall argument, http://dx.doi.org/10.1016/j.nuclphysb.2014.05.012

Web End =Nucl. Phys. B 884 (2014)

http://dx.doi.org/10.1016/j.nuclphysb.2014.05.012

Web End =566 [http://arxiv.org/abs/1306.5488

Web End =arXiv:1306.5488 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1306.5488

Web End =INSPIRE ].

{ 36 {

JHEP11(2015)063

[65] E.J. Martinec, The Cheshire cap, http://dx.doi.org/10.1007/JHEP03(2015)112

Web End =JHEP 03 (2015) 112 [http://arxiv.org/abs/1409.6017

Web End =arXiv:1409.6017 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1409.6017

Web End =INSPIRE ].[66] S.D. Mathur, A model with no rewall, http://arxiv.org/abs/1506.04342

Web End =arXiv:1506.04342 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1506.04342

Web End =INSPIRE ].[67] M. Cveti c and D. Youm, General rotating ve-dimensional black holes of toroidally

compacti ed heterotic string, http://dx.doi.org/10.1016/0550-3213(96)00355-0

Web End =Nucl. Phys. B 476 (1996) 118 [http://arxiv.org/abs/hep-th/9603100

Web End =hep-th/9603100 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9603100

Web End =INSPIRE ].[68] M. Cveti c and F. Larsen, General rotating black holes in string theory: Grey body factors and

event horizons, http://dx.doi.org/10.1103/PhysRevD.56.4994

Web End =Phys. Rev. D 56 (1997) 4994 [http://arxiv.org/abs/hep-th/9705192

Web End =hep-th/9705192 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9705192

Web End =INSPIRE ].[69] Z. Carson, S. Hampton, S.D. Mathur and D. Turton, E ect of the deformation operator in

the D1-D5 CFT, http://dx.doi.org/10.1007/JHEP01(2015)071

Web End =JHEP 01 (2015) 071 [http://arxiv.org/abs/1410.4543

Web End =arXiv:1410.4543 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1410.4543

Web End =INSPIRE ].[70] A. Schwimmer and N. Seiberg, Comments on the N = 2, N = 3, N = 4 superconformal

algebras in two-dimensions, http://dx.doi.org/10.1016/0370-2693(87)90566-1

Web End =Phys. Lett. B 184 (1987) 191 [http://inspirehep.net/search?p=find+J+"Phys.Lett.,B184,191"

Web End =INSPIRE ].[71] O. Lunin and S.D. Mathur, Three point functions for M(N)/S(N) orbifolds with N = 4

supersymmetry, http://dx.doi.org/10.1007/s002200200638

Web End =Commun. Math. Phys. 227 (2002) 385 [http://arxiv.org/abs/hep-th/0103169

Web End =hep-th/0103169 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0103169

Web End =INSPIRE ].[72] E.J. Martinec and W. McElgin, String theory on AdS orbifolds, http://dx.doi.org/10.1088/1126-6708/2002/04/029

Web End =JHEP 04 (2002) 029

[http://arxiv.org/abs/hep-th/0106171

Web End =hep-th/0106171 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0106171

Web End =INSPIRE ].[73] E.J. Martinec and W. McElgin, Exciting AdS orbifolds, http://dx.doi.org/10.1088/1126-6708/2002/10/050

Web End =JHEP 10 (2002) 050

[http://arxiv.org/abs/hep-th/0206175

Web End =hep-th/0206175 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0206175

Web End =INSPIRE ].[74] R. Dijkgraaf, G.W. Moore, E.P. Verlinde and H.L. Verlinde, Elliptic genera of symmetric

products and second quantized strings, http://dx.doi.org/10.1007/s002200050087

Web End =Commun. Math. Phys. 185 (1997) 197

[http://arxiv.org/abs/hep-th/9608096

Web End =hep-th/9608096 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9608096

Web End =INSPIRE ].[75] O. Lunin and S.D. Mathur, Correlation functions for M(N)=S(N) orbifolds, http://dx.doi.org/10.1007/s002200100431

Web End =Commun.

http://dx.doi.org/10.1007/s002200100431

Web End =Math. Phys. 219 (2001) 399 [http://arxiv.org/abs/hep-th/0006196

Web End =hep-th/0006196 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0006196

Web End =INSPIRE ].[76] Z. Carson, S. Hampton, S.D. Mathur and D. Turton, E ect of the twist operator in the D1D5

CFT, http://dx.doi.org/10.1007/JHEP08(2014)064

Web End =JHEP 08 (2014) 064 [http://arxiv.org/abs/1405.0259

Web End =arXiv:1405.0259 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1405.0259

Web End =INSPIRE ].[77] L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on orbifolds, http://dx.doi.org/10.1016/0550-3213(85)90593-0

Web End =Nucl. Phys. B 261

http://dx.doi.org/10.1016/0550-3213(85)90593-0

Web End =(1985) 678 [http://inspirehep.net/search?p=find+J+"Nucl.Phys.,B261,678"

Web End =INSPIRE ].[78] L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on orbifolds. 2, http://dx.doi.org/10.1016/0550-3213(86)90287-7

Web End =Nucl. Phys. B 274

http://dx.doi.org/10.1016/0550-3213(86)90287-7

Web End =(1986) 285 [http://inspirehep.net/search?p=find+J+"Nucl.Phys.,B274,285"

Web End =INSPIRE ].[79] L.J. Dixon, D. Friedan, E.J. Martinec and S.H. Shenker, The conformal eld theory of

orbifolds, http://dx.doi.org/10.1016/0550-3213(87)90676-6

Web End =Nucl. Phys. B 282 (1987) 13 [http://inspirehep.net/search?p=find+J+"Nucl.Phys.,B282,13"

Web End =INSPIRE ].[80] A. Adams, J. Polchinski and E. Silverstein, Dont panic! Closed string tachyons in ALE

space-times, http://dx.doi.org/10.1088/1126-6708/2001/10/029

Web End =JHEP 10 (2001) 029 [http://arxiv.org/abs/hep-th/0108075

Web End =hep-th/0108075 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0108075

Web End =INSPIRE ].[81] M. Gutperle, M. Headrick, S. Minwalla and V. Schomerus, Space-time energy decreases

under world sheet RG ow, http://dx.doi.org/10.1088/1126-6708/2003/01/073

Web End =JHEP 01 (2003) 073 [http://arxiv.org/abs/hep-th/0211063

Web End =hep-th/0211063 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0211063

Web End =INSPIRE ].[82] C. Vafa, Mirror symmetry and closed string tachyon condensation, http://arxiv.org/abs/hep-th/0111051

Web End =hep-th/0111051

[http://inspirehep.net/search?p=find+EPRINT+hep-th/0111051

Web End =INSPIRE ].

[83] J.A. Harvey, D. Kutasov, E.J. Martinec and G.W. Moore, Localized tachyons and RG ows,

http://arxiv.org/abs/hep-th/0111154

Web End =hep-th/0111154 [http://inspirehep.net/search?p=find+EPRINT+hep-th/0111154

Web End =INSPIRE ].[84] I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products, 7th edition,

Elsevier/Academic Press, Amsterdam, The Netherlands (2007).

{ 37 {

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Abstract

Non-supersymmetric black hole microstates are of great interest in the context of the black hole information paradox. We identify the holographic description of the general class of non-supersymmetric orbifolded D1-D5-P supergravity solutions found by Jejjala, Madden, Ross and Titchener. This class includes both completely smooth solutions and solutions with conical defects, and in the near-decoupling limit these solutions describe degrees of freedom in the cap region. The CFT description involves a general class of states obtained by fractional spectral flow in both left-moving and right-moving sectors, generalizing previous work which studied special cases in this class. We compute the massless scalar emission spectrum and emission rates in both gravity and CFT and find perfect agreement, thereby providing strong evidence for our proposed identification. We also investigate the physics of ergoregion emission as pair creation for these orbifolded solutions. Our results represent the largest class of non-supersymmetric black hole microstate geometries with identified CFT duals presently known.

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Title

Holographic description of non-supersymmetric orbifolded D1-D5-P solutions

Author

Chakrabarty, Bidisha; Turton, David; Virmani, Amitabh

Pages

1-38

Publication year

2015

Publication date

Nov 2015

Publisher

Springer Nature B.V.

e-ISSN

10298479

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/JHEP11(2015)063

ProQuest document ID

1749589321

SISSA, Trieste, Italy 2015

Holographic description of non-supersymmetric orbifolded D1-D5-P solutions

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