xxx EFI-21-8

Holography and Irrelevant Operators
Chih-Kai Chang1, Christian Ferko2 and Savdeep Sethi1
1 Enrico Fermi Institute & Kadanoff Center for Theoretical Physics
University of Chicago, Chicago, IL 60637, USA

2 Center for Quantum Mathematics and Physics (QMAP)
Department of Physics & Astronomy, University of California, Davis, CA 95616,
USA

We explore the holographic proposal involving spacetimes with linear dilaton
asymptotics in three dimensions from a gravity perspective. The holographic
dual shares some properties with a symmetric product conformal field theory
deformed by a single-trace analogue of the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation. We
present solutions of ten-dimensional supergravity which interpolate from BTZ
black holes in the interior to either a linear dilaton spacetime near
infinity, or to flat space. This allows a precise identification of field
theory parameters with gravity parameters. The solutions manifestly exhibit
the square root structure that is characteristic of ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed conformal
field theories. We compute the mass of the spacetimes using the covariant
phase space formalism and find agreement with the square root formula for the
case of black holes without spin. We also discuss whether closed string
tachyons might play a role when the deformation parameter becomes too large
and the vacuum becomes unstable.

###### Contents

  1. 1 Introduction
  2. 2 Type IIB Supergravity Brane Solutions
    1. 2.1 Constraints from Flux Quantization
    2. 2.2 Review of NS5-Brane and F-String Bound State
  3. 3 Generalized NS5-Brane and F1-String Solutions
    1. 3.1 Characterizing the General Solution
    2. 3.2 Solutions with ALD Asymptotics
    3. 3.3 Solutions with Flat Space Asymptotics
  4. 4 Mass Calculation
    1. 4.1 Surface Charges in Gravitational Theories
    2. 4.2 Evaluating the Charge
    3. 4.3 Winding Tachyon
  5. A Derivation of General Solutions
    1. A.1 The partially decoupled case $k_{5}=0$
    2. A.2 The asymptotically flat case where $k_{5}>0$

##  1 Introduction

Although holography for asymptotically $\mathrm{AdS}$ spacetimes has been
extensively studied and tested in a variety of contexts, holography for
spacetimes with non-$\mathrm{AdS}$ asymptotics is poorly understood. This
certainly includes the cases of asymptotically flat and asymptotically de
Sitter spacetimes. There are good reasons to suspect that new structures in
quantum field theory are needed to define quantum gravity in such spacetimes,
should such gravity theories exist. If one assumes a correspondence similar in
spirit to the AdS/CFT correspondence then any holographic description is
forced to have a high-energy density of states that grows faster than a local
quantum field theory in order to match the spectrum of large black holes. If
one is in low enough dimension, this obstruction might be avoided but it seems
hard to avoid in theories with propagating gravitons.

It seems reasonable then to suspect that a field theory dual to a
non-$\mathrm{AdS}$ spacetime, with a potentially rich spectrum of black holes,
might be controlled by some structure other than a local quantum field theory
at high energies. On the other hand, we have recently learned that some
special irrelevant deformations of local quantum field theories possess
exactly the property of dramatically modifying the ultraviolet behavior of the
theory. It is an exciting prospect that the new structures seen in quantum
field theory by turning on controlled irrelevant deformations might play a
role in defining quantum gravity on non-$\mathrm{AdS}$ spacetimes.

The most prominent family of controlled irrelevant deformations, which has
generated considerable recent excitement, involves operators constructed from
bilinears of conserved currents in two-dimensional quantum field theories. The
first example in this family is the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator of [1]. This
operator, which can be used to deform any $2d$ QFT, is given by the
combination $\det(T)$,

$\displaystyle{T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}(x)=\lim_{y\to
x}\left(T^{\mu\nu}(x)T_{\mu\nu}(y)-\tensor{T}{{}^{\mu}_{\mu}}(x)\tensor{T}{{}^{\nu}_{\nu}}(y)\right)\,$

where $T_{\mu\nu}$ is the stress tensor of the theory. Although this
definition involves a coincident-point limit of local operators, all OPE
divergences which arise in this limit are proportional to total derivatives.
These total derivatives do not contribute to one-point functions in
translationally invariant states. Therefore, up to fairly harmless total
derivative ambiguities, one can always define a local irrelevant operator from
the stress tensor using (1). Since any $2d$ QFT with translation invariance
admits a stress-energy tensor, this operator is universal. We will call this
the double-trace ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ deformation to distinguish it from a different deformation of interest
in this work.

We define a flow in the space of theories with tangent vector $\int{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}(x)$. The flow parameter
$\lambda$ has length dimension $2$. The stress tensor must be recomputed at
each step along the flow, and the composite operator (1) is constructed using
this $\lambda$-dependent stress tensor. Quantizing this theory on a cylinder
of radius $L$ gives an energy spectrum which satisfies the inviscid Burgers’
equation:

$\displaystyle\frac{\partial}{\partial\lambda}E_{n}(L,\lambda)=\frac{1}{2}\frac{\partial}{\partial
L}\left(E_{n}(L,\lambda)^{2}\right)+\frac{P_{n}(L)^{2}}{L}\,.$ (1.1)

Here $E_{n}$ are the energies and $P_{n}$ are the quantized momenta [1, 2, 3].

There is considerable evidence that this deformation defines a theory at the
quantum level for flat and even ${\rm AdS}_{2}$ spacetimes [4], and leads to a
new structure beyond local quantum field theory. For example, the high-energy
density of states exhibits a Hagedorn growth which can be inferred from the
explicit solution of $(\ref{burgers})$ for a ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed
$\mathrm{CFT}_{2}$ on a cylinder of radius $L$,

$\displaystyle E_{n}(\lambda)=\frac{L}{2\lambda}\left(\sqrt{1+\frac{4\lambda
E_{n}}{L}+\frac{4\lambda^{2}P_{n}^{2}}{L^{2}}}-1\right)\,,$ (1.2)

where $E_{n}(\lambda)$ is the deformed energy, $E_{n}$ is the undeformed
$\mathrm{CFT}_{2}$ energy and $P_{n}$ is the momentum of the $n^{\text{th}}$
eigenstate.111All of our discussion is for the “good” sign of the deformation,
which corresponds to $\lambda>0$. For this sign, all of the energies are real
for sufficiently small $\lambda$. For the “bad” sign $\lambda<0$, most of the
energies are complex. If one considers sequential flows by irrelevant
operators then the bad sign behavior can be cured [5]. Many of the basic
questions that one might ask about this new structure, like the nature of
physical observables, are still unsettled.

If one instead considers the symmetric product orbifold ${\rm
Sym}^{N}(\mathrm{CFT}_{2})=(\mathrm{CFT}_{2})^{N}/S_{N}$ then we could deform
by the single-trace operator

$D(x)=\lim_{y\to
x}\sum_{i=1}^{N}\left[\left(T_{i}(x)\right)^{\mu\nu}\left(T_{i}(y)\right)_{\mu\nu}-\tensor{\left(T_{i}(x)\right)}{{}^{\mu}_{\mu}}\tensor{\left(T_{i}(y)\right)}{{}^{\nu}_{\nu}}\right]\,,$
(1.3)

which exists at the orbifold point [6], instead of the usual double-trace
operator

$\displaystyle{T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}(x)=\lim_{y\to
x}\left[\Big{(}\sum_{i}T_{i}(x)\Big{)}^{\mu\nu}\Big{(}\sum_{i}T_{i}(y)\Big{)}_{\mu\nu}-\tensor{\Big{(}\sum_{i}T_{i}(x)\Big{)}}{{}^{\mu}_{\mu}}\tensor{\Big{(}\sum_{i}T_{i}(y)\Big{)}}{{}^{\nu}_{\nu}}\right]\,.$
(1.4)

Here $T_{i}$ denotes the stress tensor for the $i^{\text{th}}$ copy of
$\mathrm{CFT}_{2}$ in the symmetric product.

There is a very intriguing holographic proposal for defining quantum gravity
in spacetimes with linear dilaton asymptotics in three dimensions [7, 6, 8].
Such backgrounds include solutions that interpolate between ${\rm AdS}_{3}$
spacetimes and linear dilaton spacetimes [9, 10]. The proposal is that
deforming by an operator like $(\ref{singletrace_ttbar})$, although not
precisely this operator,222The holographic definition of $\mathrm{AdS}_{3}$
string theory with NS-flux cannot be the symmetric product orbifold, aside
possibly from the stringy case of a single NS5-brane [11]. Rather the
holographic theory is suspected to correspond to a marginal deformation of the
orbifold theory. The existence of a well-defined irrelevant operator analogous
to $(\ref{singletrace_ttbar})$ away from the orbifold point is in no sense
obvious. For our purpose, what is important is that an energy formula like
$(\ref{energyformula})$ exists for at least some states in the holographic
description. provides the holographic description of this system [7, 6, 8].
The specific backgrounds studied arise as limits of the gravitational solution
describing a collection of type II NS5-branes and fundamental strings. Such a
gravitational solution has a holographic description controlled in the
ultraviolet by the conjectured little string theory supported on the
NS5-branes; for reviews, see [12, 13].

This example is especially interesting because an asymptotically linear
dilaton (ALD) spacetime is structurally quite different from an asymptotically
$\mathrm{AdS}$ spacetime. For instance, $\mathrm{AdS}$ has a timelike boundary
but an ALD spacetime has a null component in the boundary. We might therefore
expect holography in the ALD setting to differ significantly from the familiar
$\mathrm{AdS}/\mathrm{CFT}$ correspondence. Much of the progress in realizing
that deformations in the spirit of $D(x)$ play a role in holography has come
from worldsheet considerations. In the specific case of the $M=0$ BTZ
background with pure NS-flux, there is a marginal worldsheet deformation that
generates the ALD spacetime [9, 10]. The effect of this worldsheet deformation
on the long string excitation spectrum has been argued to match the way $D(x)$
deforms the energies of the symmetric product orbifold [6]. This is compelling
evidence that something like a single-trace ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator, with a
similar effect on the energy spectrum, should exist in the actual holographic
$\mathrm{CFT}_{2}$ which defines $\mathrm{AdS}_{3}$ with pure NS-flux.

Our goal in the present work is to provide a complementary view of this
holographic proposal: rather than using a worldsheet construction, we will
work primarily in the target spacetime and perform an analysis using only
classical general relativity. We find families of solutions which look like
${\rm AdS}_{3}$ spacetimes in the interior but with either ALD asymptotics
$(\ref{spinning_case})$ or flat space asymptotics
$(\ref{final_undecoupled_soln_J_bulk})$. What is striking about these
solutions, which are parametrized by the mass and spin
$(\widetilde{M},\widetilde{J})$ of the interior BTZ black hole, is that the
dilaton solution exhibits exactly the square root form seen in
$(\ref{energyformula})$:

$\displaystyle
e^{2\Phi}=\frac{r_{5}^{2}\sqrt{1+8k_{1}\frac{\widetilde{M}r_{5}^{2}}{R^{2}}+16\left(k_{1}\right)^{2}\frac{\widetilde{J}^{2}r_{5}^{4}}{R^{4}}}}{\hat{r}_{1}^{2}+k_{1}r^{2}}\,.$
(1.5)

The radial coordinate in $(\ref{dilaton_form_intro})$ is $r$ while the
parameter $k_{1}$ corresponds to $\lambda$ of $(\ref{energyformula})$ via the
relation:

$\displaystyle\lambda=\frac{(\alpha^{\prime})^{2}|m_{5}|}{2R^{2}}k_{1}\,.$
(1.6)

The constants $r_{5}$ and $\hat{r}_{1}$ in (1.5) are fixed length scales
determined by the number of NS5-branes, $m_{5}$, and the number of fundamental
strings, $m_{1}$. The string length is $\ell_{s}=\sqrt{\alpha^{\prime}}$ and
$R$ is the size of an asymptotic circle which is finite in string-frame and
characteristic of the ALD spacetime.

It is worth noting that the effect of the $k_{1}$ deformation is detectable
even in the deep interior where $r$ is very small. For example, the ${\rm
AdS}_{3}$ length scale is $k_{1}$-deformed in a mass and spin-dependent way.
Alternately, the value of the dilaton at $r=0$ in $(\ref{dilaton_form_intro})$
is now mass and spin-dependent while it is simply constant in the case of
${\rm AdS}_{3}$. Said differently: the gravity solution ‘knows’ about the
irrelevant deformation of the holographic CFT even in regions of spacetime
normally associated to the deep infrared of the CFT.

Our conventions can be found in section 2 along with a review of the
$\widetilde{M}=0$ brane solutions. In section 3 we summarize the final form of
the various families of solutions with mass and spin. A more detailed analysis
leading to these solutions can be found in Appendix A. In section 4, we use
the covariant phase space formalism to compute the mass of the ALD solutions.
For the case without spin, we find a square root form for the mass of the
spacetime $(\ref{sqrt_charge_soln})$ in accord with our expectations for a
${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like
deformation of the holographic description. The case with spin is more subtle
because of issues with charge integrability which we discuss in section 4.2.
Lastly, in section 4.3 we briefly explore whether a closed string tachyon
might play a role in understanding the fate of the vacuum state when the
deformation parameter becomes too large.

Some generalizations of the holography discussed here for the pure NS
background to cases with $(p,q)$ $5$-branes, which really require the kind of
gravity approach we have employed, will appear elsewhere.

Note Added: The results in this paper have been in a state of partial
completion for quite a long time. During that period two interesting papers
appeared with related results from a different approach based on TsT
transformations [14, 15]. The long delay can, in part, be attributed to the
arrival of a new baby during the COVID lockdown.

##  2 Type IIB Supergravity Brane Solutions

We first want to find general solutions of type II supergravity with NS5-brane
and fundamental string charge that asymptote to flat space. There will be two
decoupling limits that play a role in our discussion and we will want to write
the solutions in convenient coordinates for exhibiting these limits.

As a starting point, in this section we review brane solutions of type IIB
supergravity, which fully decouple to the Poincaré patch of the $M=0$ BTZ
background. In section 3, we will find the general class of solutions with
non-zero mass and spin. Because the solutions involve a torus, we can dualize
between type IIA or type IIB string theory; for concreteness, we will discuss
solutions of the type IIB equations of motion.

###  2.1 Constraints from Flux Quantization

To fix conventions, let us begin with the bosonic action for ten-dimensional
type IIB supergravity in string-frame:

$\displaystyle S_{{\rm IIB}}={1\over(2\pi)^{7}{\alpha^{\prime}}^{4}}\int
d^{10}x\,\sqrt{-\hat{g}}\Bigg{\\{}$ $\displaystyle
e^{-2\Phi}\left(\hat{R}+4(\partial\Phi)^{2}-{1\over
12}|H_{3}|^{2}\right)-{1\over 2}|F_{1}|^{2}$ (2.1) $\displaystyle-{1\over
12}|F_{3}-C_{0}H_{3}|^{2}-\frac{1}{4\cdot
5!}|\widetilde{F}_{5}|^{2}\Bigg{\\}},$ (2.2)

where $\boldsymbol{\widetilde{F}}_{5}=d\boldsymbol{C}_{4}-{1\over
2}\boldsymbol{C}_{2}\wedge\boldsymbol{H}_{3}+{1\over
2}\boldsymbol{B}_{2}\wedge\boldsymbol{F}_{3}$.333We use boldface to denote
differential forms to conform with the notation often found in the gravity
literature and used in section 4.  We define the gravitational constant

$\displaystyle\kappa_{10}^{2}:={1\over
2}(2\pi)^{7}{\alpha^{\prime}}^{4}e^{2\Phi_{0}}\,,$ (2.3)

in terms of the string length, $\ell_{s}=\sqrt{\alpha^{\prime}}$, and the
asymptotic value of the string coupling $g_{s}=e^{\Phi_{0}}$. The hatted
variables $\hat{g}$ and $\hat{R}$ refer to string-frame quantities.

The analysis of ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$-like deformations and holography in general flux backgrounds – when
the Ramond fluxes $C_{0},\boldsymbol{C}_{2}$, $\boldsymbol{C}_{4}$ may be
nonzero – will be explored in future work. For our present purposes, we will
restrict to the case of pure NS fluxes. In this case, it is consistent to set
$C_{0}=\boldsymbol{C}_{2}=\boldsymbol{C}_{4}=0$.

We will consider solutions of the form $\mathcal{M}_{3}\times S^{3}\times
T^{4}$. The $T^{4}$ factor is a spectator which could also be replaced by K3
without changing the analysis. Although not visible in supergravity, in string
theory flux quantization through the $S^{3}$ requires that

$\displaystyle{1\over
4\pi^{2}\alpha^{\prime}}\int_{S^{3}}\boldsymbol{H}_{3}=m_{5}\,,$ (2.4)

where $m_{5}$ is an integer corresponding to the NS$5$-brane charge.

There is another quantization condition on the dual field strength. If we
define

$\displaystyle\boldsymbol{H}_{7}=e^{-2\Phi}\ast\boldsymbol{H}_{3}\,,$ (2.5)

then this form satisfies the following quantization condition for any
$7$-cycle $\Sigma_{7}$,

$\displaystyle{1\over(2\pi)^{6}(\alpha^{\prime})^{3}}\int_{\Sigma_{7}}\boldsymbol{H}_{7}=m_{1}\in\mathbb{Z}\,.$
(2.6)

###  2.2 Review of NS5-Brane and F-String Bound State

There is a well-known class of solutions in type IIB supergravity which
satisfy the flux quantization conditions outlined in the previous subsection
and which form the prototype for the generalized solutions that we will
consider in section 3. These much-studied solutions are interpreted as bound
states of $m_{1}$ fundamental strings and $m_{5}$ NS5 branes, where
$m_{1},m_{5}$ are the integers appearing in the quantization conditions (2.6)
and (2.4), respectively.

One way of presenting these solutions [16], which is nicely reviewed in [17],
is

$\displaystyle e^{-2\Phi}=\frac{1}{g_{s}^{2}}\frac{f_{1}}{f_{5}}\,,\qquad
B_{05}=\frac{1}{f_{1}}-1\,,\qquad
H_{mnp}=\tensor{\epsilon}{{}_{m}{}_{n}{}_{p}^{q}}\,\partial_{q}(\log f_{5})$
$\displaystyle\widehat{ds}^{2}=\frac{-dt^{2}+dx_{5}^{2}}{f_{1}}+f_{5}\left(dx_{1}^{2}+\cdots+dx_{4}^{2}\right)+\left(dx_{6}^{2}+\cdots
dx_{9}^{2}\right)\,,$ (2.7)

where $m,n,p,q=1,\ldots,4$, $\epsilon_{mnpq}$ is the volume form for
$f_{5}\sum_{i=1}^{4}dx_{i}^{2}$ and

$\displaystyle
f_{1}=1+\frac{r_{1}^{2}}{r^{2}}=1+\frac{16\pi^{4}g_{s}^{2}\alpha^{\prime
3}m_{1}}{V_{4}r^{2}}\,,\qquad
f_{5}=1+\frac{r_{5}^{2}}{r^{2}}=1+\frac{\alpha^{\prime}m_{5}}{r^{2}}\,.$ (2.8)

In these expressions, $r$ represents a radial coordinate in the four-
dimensional space transverse to the NS5-branes parameterized by
$x_{1},x_{2},x_{3},x_{4}$, while the coordinates $x^{6},x^{7},x^{8},x^{9}$
parametrize a torus with volume $V_{4}$. The asymptotic region corresponds to
large $r$. The constant $g_{s}$ specifies the asymptotic value of the string
coupling. We have a choice about how to treat $x_{5}$. To patch to ${\rm
AdS}_{3}$ in the interior, we will periodically identify

$\displaystyle x_{5}\sim x_{5}+2\pi R\,.$ (2.9)

Note that the flux $H_{3}$ has two separate contributions, one arising from
the explicit $H_{mnp}$ components and one arising from
$\boldsymbol{H}=d\boldsymbol{B}$ with $\boldsymbol{B}=B_{05}\,dx^{0}\wedge
dx^{5}$ and $B_{05}$ as indicated in (2.2).

The solution (2.2) contains at least two qualitatively different asymptotic
regions. For $r\gg r_{1},r_{5}$, the solution approaches flat $10$-dimensional
Minkowski space. On the other hand, in the near-horizon limit $r\ll
r_{1},r_{5}$, the sphere and torus become spectators and we recover an
effective three-dimensional gravity solution which is diffeomorphic to a
Poincaré patch of $\mathrm{AdS}_{3}$.

We will refer to the full spacetime with the asymptotically flat region
retained as the undecoupled or totally undecoupled solution, and the near-
horizon solution in the deep bulk as the decoupled or totally decoupled
solution. The process of taking the full decoupling limit may be thought of as
simultaneously sending the asymptotic string coupling $g_{s}\rightarrow 0$ and
sending $\alpha^{\prime}\to 0$, along with a re-scaling of coordinates.
Alternatively, one may think of this decoupling limit as dropping the $1$’s in
the harmonic functions $f_{1}$ and $f_{5}$ so that
$f_{i}=1+\frac{r_{i}^{2}}{r^{2}}\,\to\,f_{i}=\frac{r_{i}^{2}}{r^{2}}$.

#### Partial Decoupling

There is also an intermediate regime in this spacetime which looks neither
like $\mathrm{AdS}_{3}$ in the Poincaré patch nor like flat space. To take
this limit, define $r=g_{s}\hat{r}$ and take $g_{s}\to 0$ giving,

$\displaystyle\begin{split}&e^{-2\Phi}=\frac{\hat{r}^{2}}{r_{5}^{2}}\left(1+\frac{\hat{r}_{1}^{2}}{\hat{r}^{2}}\right)\,,\quad\qquad\widehat{ds}^{2}=\frac{-dt^{2}+dx_{5}^{2}}{f_{1}}+\frac{r_{5}^{2}}{\hat{r}^{2}}d\hat{r}^{2}+r_{5}^{2}d\Omega_{3}^{2}+ds^{2}_{T^{4}}\,,\\\
&f_{1}=1+\frac{\hat{r}_{1}^{2}}{\hat{r}^{2}}=1+\frac{16\pi^{4}\alpha^{\prime
3}m_{1}}{V_{4}\,\hat{r}^{2}},\qquad r_{1}=g_{s}\hat{r}_{1}\,.\end{split}$
(2.10)

This amounts to replacing $f_{5}=1+\frac{r_{5}^{2}}{r^{2}}$ by
$f_{5}=\frac{r_{5}^{2}}{r^{2}}$, but retaining the full function $f_{1}$
without dropping the $1$ there, since $\frac{r_{1}^{2}}{r^{2}}$ is not
necessarily big.

#### Full Decoupling

The fully decoupled solution is found by making $r$ very small and dropping
the $1$ in $f_{1}$ [18]. To put the resulting metric in a conventional ${\rm
AdS}_{3}$ form, we define dimensionless variables

$\displaystyle\tilde{r}^{2}=\hat{r}^{2}\frac{R^{2}}{r_{5}^{2}\,\hat{r}_{1}^{2}},\qquad\tilde{\varphi}=\frac{x_{5}}{R}\sim\tilde{\varphi}+2\pi,\qquad{\tilde{t}}=\frac{t}{R}.$
(2.11)

In these variables the metric takes the form

$\displaystyle\begin{split}\widehat{ds}^{2}&=\ell^{2}\left(\tilde{r}^{2}\left(-d{\tilde{t}}^{2}+d\tilde{\varphi}^{2}\right)+\frac{d\tilde{r}^{2}}{\tilde{r}^{2}}+d\Omega_{3}^{2}\right)+ds^{2}_{T^{4}}\,,\\\
e^{-2\Phi}&=\frac{\tilde{r}^{2}r_{1}^{2}}{R^{2}}+\frac{\hat{r}_{1}^{2}}{\ell^{2}}\,\xrightarrow{\text{decoupling}}\,\frac{\hat{r}_{1}^{2}}{\ell^{2}}\,,\end{split}$
(2.12)

where $\ell=r_{5}$ is the ${\rm AdS}_{3}$ length scale. We note that the size,
$R$, of the $x_{5}$ circle at infinity is absorbed into a redefinition of time
and therefore energy. This will play a role later when we compare energies in
the partially decoupled theory to energies in the ${\rm AdS}_{3}$ region. The
last comments on the fully decoupled case concern the value of the
gravitational coupling in the ${\rm AdS}_{3}$ theory which is given by,

$\displaystyle\left(\kappa_{3}\right)^{2}=8\pi
G_{3}=\frac{2\pi\ell_{s}}{|m_{1}|\sqrt{|m_{5}|}},$ (2.13)

and the central charge of the dual holographic CFT which is given by

$\displaystyle c=\frac{3\ell}{2G_{3}}=6|m_{1}m_{5}|\,.$ (2.14)

We can identify the radius $L$ of the holographic CFT by noting that the mass
of global ${\rm AdS}_{3}$ is $-\frac{1}{8G_{3}}$. This corresponds to the
ground state of the CFT with energy $-\frac{c}{12L}$ so we identify:

$\displaystyle L=\ell=\sqrt{|m_{5}|}\ell_{s}\,.$ (2.15)

This is independent of the radius $R$ of the $x_{5}$ circle which appears in
$(\ref{periodicity_x5})$.

As we discussed in the introduction, gravity solutions of the partially-
decoupled form have been argued to have field theory duals which are obtained
by deforming a conformal field theory by an irrelevant operator related to
${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ [7, 6, 8].
These solutions have also been shown to arise from TsT transformations [19,
14, 15]. In the following section we will generalize these well-known
solutions for the $M=0$ BTZ case without spin to metrics which interpolate
from a general spinning BTZ black hole in the interior to either an
asymptotically linear dilaton spacetime or to an asymptotically flat
spacetime.

##  3 Generalized NS5-Brane and F1-String Solutions

In this section, we will present a class of supergravity solutions with the
desired quantized charges, symmetries and asymptotic behavior. These solutions
can be interpreted as bound states of a system of F1 strings and NS5-branes
that interpolate between an $\mathrm{AdS}_{3}$ black hole in the deep interior
with some mass and spin to either a linear dilaton solution in the asymptotic
region, or to asymptotically flat six dimensions at large distances. We will
separately present those two cases because the latter is a more complicated
class of solutions.

The full spacetime equations of motion are quite difficult to solve in
generality without some physically reasonable metric ansatz as an input. Our
ansatz is spelled out in Appendix A along with the calculations of the metric
and dilaton functions, and the details of the fairly involved asymptotic
matching needed to identify the physical parameters of the solution. The aim
of this section is to present clearly the assumptions going into the analysis
and the final solutions. There are other classes of solutions in Appendix A,
which are even more surprising: they appear to interpolate from an ${\rm
AdS}_{3}$ spacetime in the interior to an asymptotic $3$-dimensional spacetime
with positive curvature. Whether those solutions are fully sensible will be
explored elsewhere.

###  3.1 Characterizing the General Solution

We will look for solutions to the equations of motion of type IIB
supergravity, written using the same coordinates as $(\ref{kluson_soln})$,
which satisfy the following assumptions:

  1. 1.

Asymptotics. We assume that near infinity the family of solutions approach
either the linear dilaton asymptotic form seen in
$(\ref{partialasymptotics})$, or the flat space asymptotic form seen in
$(\ref{kluson_soln})$. We will insist that the isometries of $T^{4}$, $S^{3}$
and $S^{1}$ are preserved by the solutions. This collapses the unknown
functions in the metric, fluxes and dilaton to functions of $r$ and possibly
time $t$. We assume these coordinates can be extended to global coordinates
for our spacetime soution. Lastly, we assume the metric, dilaton, and NS-flux
admit an expansion in powers of $\frac{1}{r}$ near $r\to\infty$.

Near $r=0$, we insist the solution takes the form $\mathcal{M}_{3}\times
S^{3}\times T^{4}$ with the dilaton $\Phi$ tending to a constant. We need this
form if we are to interpret the holographic theory as some kind of deformed
CFT.

  2. 2.

Symmetries. We assume that the solution is stationary and therefore only
depends on the radial coordinate $r$. Because the spacetime is assumed to be
stationary but not static, the timelike Killing vector $\partial_{t}$ need not
be irrotational. However, we will constrain the allowed rotation to occur only
in the $x_{5}$ direction. Operationally, this means that a non-vanishing
metric component $g_{tx_{5}}$ is allowed – corresponding to a spinning BTZ
black hole at small $r$ – but no other off-diagonal metric components are
permitted.

  3. 3.

Fluxes. We assume that the Ramond potentials $\boldsymbol{C}_{p}$ are all
vanishing and that the NS flux $\boldsymbol{H}_{3}=d\boldsymbol{B}_{2}$ only
threads the three-dimensional submanifolds $S^{3}$ and $\mathcal{M}_{3}$. So
$\boldsymbol{H}_{3}$ is a sum of the volume forms
$\boldsymbol{\epsilon}_{S^{3}}$ and $\boldsymbol{\epsilon}_{\mathcal{M}_{3}}$
on $S^{3}$ and $\mathcal{M}_{3}$, respectively, with coefficients that can in
principle depend on $r$. Finally, we impose the usual flux quantization
conditions $(\ref{H3_quantization})$ and $(\ref{dualquantization})$ on $H_{3}$
required by string theory.

###  3.2 Solutions with ALD Asymptotics

#### The $\widetilde{J}=0$ case

From the analysis presented in Appendix A, we find the following solutions in
string-frame for the case where the black hole has no spin:

$\displaystyle
ds^{2}=-\frac{1-\frac{\alpha_{e}^{2}}{r^{2}}}{k_{1}+\frac{\hat{r}_{1}^{2}}{r^{2}}}\,dt^{2}+\frac{1}{k_{1}+\frac{\hat{r}_{1}^{2}}{r^{2}}}\,dx_{5}^{2}+\frac{r_{5}^{2}}{r^{2}-\alpha_{e}^{2}}\,dr^{2}+r_{5}^{2}\,d\Omega_{3}^{2}+ds^{2}_{T^{4}}\,,$
$\displaystyle
e^{2\Phi}=\frac{r_{5}^{2}\sqrt{1+8k_{1}\frac{\widetilde{M}r_{5}^{2}}{R^{2}}}}{\hat{r}_{1}^{2}+k_{1}r^{2}}\,,\qquad\alpha_{e}^{2}=8\widetilde{M}\frac{\hat{r}_{1}^{2}r_{5}^{2}}{R^{2}}\,,\quad
r_{5}^{2}=\alpha^{\prime}m_{5}\,,\quad\hat{r}_{1}^{2}=\frac{16\pi^{4}\alpha^{\prime
3}m_{1}}{V_{4}}\,.$ (3.1)

These solutions depend on a dimensionless parameter $k_{1}$. The fully
decoupled pure ${\rm AdS}_{3}$ solutions correspond to $k_{1}=0$. We will
momentarily identify $k_{1}$ with the deformation parameter $\lambda$ of a
${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed
theory. It is important to point out that the $\widetilde{M}=0$ solutions,
which match $(\ref{partialasymptotics})$, are sensitive to whether $k_{1}$ is
zero or non-zero. However, the precise value of $k_{1}$ is not important and
can be absorbed in a rescaling of coordinates. This reflects the fact that
$E=0$ states remain zero energy under a ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation: namely,
zero energy does not flow. Any case with $\widetilde{M}\neq 0$, however, is
sensitive to the value of $k_{1}$.

Note that the $x_{5}$ circle has periodicity $\frac{2\pi R}{\sqrt{k_{1}}}$ at
$r=\infty$ in string-frame. In Einstein frame, the $x_{5}$ circle has a proper
size that goes to infinity as $r\rightarrow\infty$. The dimensionless mass
$\widetilde{M}$ is related to the dimensionful mass of the BTZ black hole near
$r=0$ via $\widetilde{M}=MG_{3}$, where $G_{3}$ is specified in
$(\ref{kappa3_defn})$.

On first glance, it might appear that the solutions
$(\ref{final_string_frame_soln_bulk})$ have the wrong large $r$ asymptotics in
Einstein-frame because the dilaton is mass-dependent. However a careful
redefinition of the radial coordinate in the large $r$ region, discussed
around $(\ref{defrho})$, shows that these solutions have the correct
asymptotic behavior. Also notice that the value of the dilaton at $r=0$ is now
mass-dependent and differs from the pure ${\rm AdS}_{3}$ case where $k_{1}=0$.
As a consequence, the ${\rm AdS}_{3}$ length scale is also mass-dependent in
Einstein frame.

Global ${\rm AdS}_{3}$ has mass $\widetilde{M}=-\frac{1}{8}$. This corresponds
to the ground state of the holographic CFT which should have dimensionless
energy $\widetilde{E}=EL=\widetilde{M}\frac{L}{G_{3}}$. Notice that the
dilaton solution is only real if the condition,

$\displaystyle k_{1}<\frac{R^{2}}{r_{5}^{2}}\,,$ (3.2)

is satisfied. It is important to note that this condition is independent of
$m_{1}$ but depends on the value of $m_{5}$ via $r_{5}^{2}$. As we increase
$m_{5}$, the maximum value of $k_{1}$ decreases. It is this asymmetry between
$m_{1}$ and $m_{5}$ that singles out the single-trace ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation rather
than the conventional double-trace deformation. In the latter case, we should
have seen the maximum value of $k_{1}$ reduce as we increase the total central
charge $(\ref{central_charge})$ which depends on both $m_{1}$ and $m_{5}$. We
can then compare this bound with (1.2) which states that

$\displaystyle\lambda<\frac{3L^{2}}{c_{\rm block}}=\frac{3L^{2}}{6|m_{5}|}\,,$
(3.3)

where $c_{\rm block}=6|m_{5}|$ is the central charge of a block that would
appear if the holographic dual were actually a symmetric product [6]. Via this
comparison we can identify,

$\displaystyle\lambda=\frac{(\alpha^{\prime})^{2}|m_{5}|}{2R^{2}}k_{1}\,,\quad\qquad{\tilde{\lambda}}=\frac{\lambda}{L^{2}}=\frac{\alpha^{\prime}}{2R^{2}}k_{1}\,,$
(3.4)

where we have also defined the natural dimensionless ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformation parameter
$\tilde{\lambda}$. This makes clear that the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation is
controlled by the asymptotic size of the $x_{5}$ circle in string frame. We
will see further evidence for this identification in the spinning case. It is
striking that the square root solution of the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flow equation for a
CFT emerges directly from gravity in the structure of the dilaton solution.

#### The $\widetilde{J}\neq 0$ case

The solutions for spinning black holes are similar,

$\displaystyle
ds^{2}=-\frac{1-\frac{\alpha_{e}^{2}}{r^{2}}+\frac{\alpha_{j}^{4}}{r^{4}}}{k_{1}+\frac{\hat{r}_{1}^{2}}{r^{2}}}\,dt^{2}+\frac{1}{k_{1}+\frac{\hat{r}_{1}^{2}}{r^{2}}}\,\left(dx_{5}-\frac{\alpha_{j}^{2}}{r^{2}}dt\right)^{2}+\frac{r_{5}^{2}}{r^{2}-\alpha_{e}^{2}+\frac{\alpha_{j}^{4}}{r^{2}}}\,dr^{2}+r_{5}^{2}\,d\Omega_{3}^{2}+ds^{2}_{T^{4}}\,,$
$\displaystyle
e^{2\Phi}=\frac{r_{5}^{2}\sqrt{1+8k_{1}\frac{\widetilde{M}r_{5}^{2}}{R^{2}}+16\left(k_{1}\right)^{2}\frac{\widetilde{J}^{2}r_{5}^{4}}{R^{4}}}}{\hat{r}_{1}^{2}+k_{1}r^{2}}\,,\qquad\alpha_{e}^{2}=8\widetilde{M}\frac{\hat{r}_{1}^{2}r_{5}^{2}}{R^{2}}\,,\qquad\alpha_{j}^{2}=4\widetilde{J}\,\frac{\hat{r}_{1}^{2}r_{5}^{2}}{R^{2}}.$
(3.5)

Here $\widetilde{J}=\frac{JG_{3}}{\ell}$ is a dimensionless spin, but this is
not the natural quantity to study from the perspective of the dual CFT. The
$\mathrm{AdS}/\mathrm{CFT}$ correspondence maps the dimensionless parameter
$J=\widetilde{J}\frac{\ell}{G_{3}}$ to the dimensionless momentum quantum
number $\widetilde{P}=PL$ of the holographic CFT. With this identification,
the form of the dilaton solution again beautifully matches the structure of
the ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed
energy formula $(\ref{energyformula})$,

$\displaystyle\sqrt{1+8k_{1}\frac{\widetilde{M}r_{5}^{2}}{R^{2}}+16\left(k_{1}\right)^{2}\frac{\widetilde{J}^{2}r_{5}^{4}}{R^{4}}}\quad\longrightarrow\quad\sqrt{1+4\widetilde{\lambda}\widetilde{E}+4\widetilde{\lambda}^{2}\widetilde{P}^{2}}\,,$
(3.6)

where we identify

$\displaystyle\widetilde{P}=\frac{J}{m_{1}}\,,\qquad\widetilde{E}=4m_{5}\widetilde{M}=\frac{1}{m_{1}}\left(\widetilde{M}\cdot\frac{\ell}{G_{3}}\right)\,.$
(3.7)

Here $\widetilde{E}$ and $\widetilde{P}$ are the energy and momentum,
respectively, of a single block of the symmetric product.

###  3.3 Solutions with Flat Space Asymptotics

Lastly we turn to the fully undecoupled solutions that asymptote to a flat
$6$-dimensional spacetime. In string-frame, the metric and dilaton take the
form:

$\displaystyle
ds^{2}=-\frac{1-\frac{\alpha_{e}^{2}}{r^{2}}+\frac{\alpha_{j}^{4}}{r^{4}}}{k_{1}+\frac{\hat{r}_{1}^{2}}{r^{2}}}\,dt^{2}+\frac{1}{k_{1}+\frac{\hat{r}_{1}^{2}}{r^{2}}}\,\left(dx_{5}-\frac{\alpha_{j}^{2}}{r^{2}}dt\right)^{2}$
$\displaystyle\quad+\frac{k_{5}r^{2}-\frac{k_{5}\alpha_{e}^{2}}{2}+\sqrt{r_{5}^{4}-k_{5}^{2}\alpha_{j}^{4}+\frac{1}{4}k_{5}^{2}\alpha_{e}^{4}}}{r^{4}-r^{2}\alpha_{e}^{2}+\alpha_{j}^{4}}\,r^{2}\,dr^{2}+r^{2}\left(k_{5}+\frac{\alpha_{5}^{2}}{r^{2}}\right)\,d\Omega_{3}^{2}+ds^{2}_{T^{4}}\,,$
$\displaystyle
e^{2\Phi}=\frac{g_{s}^{2}+\frac{\gamma\alpha_{5}^{2}}{k_{1}r^{2}}}{1+\frac{\hat{r}_{1}^{2}}{k_{1}r^{2}}}\,,\qquad\alpha_{e}^{2}=8\widetilde{M}\frac{\hat{r}_{1}^{2}r_{5}^{2}}{R^{2}}\,,\qquad\alpha_{j}^{2}=4\widetilde{J}\,\frac{\hat{r}_{1}^{2}r_{5}^{2}}{R^{2}}\,,\qquad
k_{5}=\frac{k_{1}g_{s}^{2}}{\gamma}\,,$
$\displaystyle\alpha_{5}^{2}=\frac{1}{2}\left(\sqrt{k_{5}^{2}\alpha_{e}^{4}+4\left(r_{5}^{4}-k_{5}^{2}\alpha_{j}^{4}\right)}-k_{5}\alpha_{e}^{2}\right)\,,\qquad\gamma=\sqrt{1+\frac{k_{1}\alpha_{e}^{2}}{\hat{r}_{1}^{2}}+\frac{k_{1}^{2}\alpha_{j}^{4}}{\hat{r}_{1}^{4}}}\,.$
(3.8)

This final form for the metric depends on the dimensionless parameters
$(g_{s},k_{1})$ where $g_{s}$ is the $r=\infty$ value of the string coupling.
The solution also depends on the flux quantum numbers via
$(\hat{r}_{1},r_{5})$ defined in $(\ref{final_string_frame_soln_bulk})$ and
the black hole mass and spin $(\widetilde{M},\widetilde{J})$. We note that the
$3$-sphere now grows as $r$ becomes large which is why the asymptotic theory
is $6$-dimensional.

The square root structure of the solution to the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flow equation is
encoded in $\gamma$ which now appears both in the metric via $k_{5}$ and the
dilaton. Notice that the size of the $S^{3}$ at $r=0$ is also deformed from
the pure ${\rm AdS}_{3}$ case by an amount that depends on the mass and spin
of the interior black hole. Lastly we note that the parameter $k_{1}$ still
controls the size of the $x_{5}$ circle near infinity.

##  4 Mass Calculation

The supergravity solutions derived in the previous section appear to have
something to do with a ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed theory. We
made a tentative identification of the deformation parameter with $k_{1}$ of
the spacetime solution based on the appearance of the square root in the
dilaton solution of $(\ref{spinning_case})$. This identification assumes that
the holographic CFT is of symmetric product type and that the deformation is
purely a single-trace ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation. The only
cases where we have formulae for the deformed energies are the single and
double-trace ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformations, and sequential flows by such irrelevant operators [5]. However,
the undeformed holographic CFT is not a symmetric product though it is
believed to be connected to a symmetric product by a marginal deformation.
Regardless, we can provide further evidence for a correspondence with some
kind of single-trace ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$-like deformation of that undeformed theory by directly computing the
mass of the solutions $(\ref{final_string_frame_soln_bulk})$.

The definition of mass in general relativity is not unique. There are a
variety of proposals that depend on the asymptotics of the background. A way
to uniformly define the mass is by using the covariant phase space formalism
[20]. This will provide an independent way of confirming the parameter
identification we made in section 3.

###  4.1 Surface Charges in Gravitational Theories

The covariant phase space formalism allows us to define conserved surface
charges in generally covariant theories. In this section we will review this
formalism before using it to compute the charges associated with the solutions
in section 3. Our discussion of this formalism will be very brief and many
facts will be stated without justification; for a more comprehensive
treatment, we refer the reader to a review such as [21].

First consider a theory described by a general Lagrangian density
$\boldsymbol{L}(\Phi^{i})$ in a theory with $n$ spacetime dimensions, where
$\Phi^{i}$ represents an arbitrary collection of fields. Following common
conventions in the gravity literature we denote differential forms in
spacetime by boldface symbols, so $\boldsymbol{L}=L\,d^{n}x$ is the top form
associated with the usual Lagrangian density $L$. In pure Einstein gravity,
for instance, the only field $\Phi^{i}$ is the metric $g_{\mu\nu}$ and
$L=\frac{1}{16\pi G}\sqrt{-g}R$. Upon variation of the fields $\Phi^{i}$, one
sees

$\displaystyle\delta\boldsymbol{L}=\frac{\delta\boldsymbol{L}}{\delta\Phi^{i}}\,\delta\Phi^{i}-d\boldsymbol{\Theta}[\delta\Phi^{i};\Phi^{i}]\,.$
(4.1)

The form $\boldsymbol{\Theta}[\delta\Phi^{i};\Phi^{i}]$ is called the
presymplectic potential and depends both on the fields $\Phi^{i}$ and their
variations $\delta\Phi^{i}$. One can think of $\boldsymbol{\Theta}$ as
simultaneously being an $(n-1)$-form in spacetime and a $1$-form in the space
of field variations. In what follows, we will suppress the index $i$ when
writing the functional dependence on fields and simply write
$\boldsymbol{\Theta}[\delta\Phi;\Phi]$. To define conserved charges, it is
convenient to introduce a related object $\boldsymbol{\omega}$ called the
presymplectic form via

$\displaystyle\boldsymbol{\omega}[\delta_{2}\Phi,\delta_{1}\Phi;\Phi]=\delta_{2}\boldsymbol{\Theta}[\delta_{1}\Phi;\Phi]-\delta_{1}\boldsymbol{\Theta}[\delta_{2}\Phi;\Phi]\,.$
(4.2)

Here we write the dependence of $\boldsymbol{\omega}$ on both variations
$\delta_{1,2}\Phi$ to emphasize that $\boldsymbol{\Theta}$ itself depends on
one variation $\delta_{1}\Phi$ of the fields, and $\boldsymbol{\omega}$ is
defined by again performing a second variation $\delta_{2}$ of the fields and
finding how $\boldsymbol{\Theta}$ varies (with appropriate
antisymmetrization). From this perspective, the symbol $\delta$ acts as the
exterior derivative in field space so that $\boldsymbol{\omega}$ is an
$(n-1)$-form in spacetime and a $2$-form in the space of field variations.

We will be especially interested in field variations $\delta_{\xi}\Phi$ that
are associated with an infinitesimal diffeomorphism generated by $x^{\mu}\to
x^{\mu}+\xi^{\mu}$. For instance, under such a transformation any tensor field
$T_{\mu_{1}\cdots\mu_{p}}$ transforms via the Lie derivative
$\mathcal{L}_{\xi}$ as

$\displaystyle\delta_{\xi}T_{\mu_{1}\cdots\mu_{p}}=\mathcal{L}_{\xi}T_{\mu_{1}\cdots\mu_{p}}\,.$
(4.3)

We will consider the case where the field configuration $\Phi$ satisfies the
equations of motion for the theory, the first variation
$\delta_{1}\Phi=\delta\Phi$ solves the linearized equations of motion about
the solution $\Phi$, and the second variation
$\delta_{2}\Phi=\delta_{\xi}\Phi$ is generated by a diffeomorphism of this
form. In this case, one can show that
$\boldsymbol{\omega}[\delta_{\xi}\Phi,\delta\Phi;\Phi]$ is exact, so that

$\displaystyle\boldsymbol{\omega}[\delta_{\xi}\Phi,\delta\Phi;\Phi]=d{\boldsymbol{k}}_{\xi}[\delta\Phi;\Phi]\,.$
(4.4)

This differential form ${\boldsymbol{k}}_{\xi}$ is an $(n-2)$-form in
spacetime and a $1$-form in the space of field variations. Note that we will
also identify the $(n-2)$-form ${\boldsymbol{k}}_{\xi}$ with its Hodge dual,
an anti-symmetric $2$-tensor with components $k^{\mu\nu}$, which we write
without boldface type. This ${\boldsymbol{k}}_{\xi}$ can also be expressed as

$\displaystyle{\boldsymbol{k}}_{\xi}[\delta\Phi;\Phi]=-\delta\boldsymbol{Q}_{\xi}[\delta\Phi;\Phi]+i_{\xi}\boldsymbol{\Theta}[\delta\Phi;\Phi]\,,$
(4.5)

up to the ambiguity of adding a total (spacetime) derivative term to
${\boldsymbol{k}}_{\xi}$. This ambiguity will not be relevant for us, since we
are primarily interested in the integral

$\displaystyle\delta
Q_{\xi}=\oint_{S}{\boldsymbol{k}}_{\xi}[\delta\Phi;\Phi]\,,$ (4.6)

where $S$ is a closed codimension-$2$ surface. Typically we will think of a
spacetime with a radial direction $r$ and let the surface $S$ be a sphere at a
fixed (large) value of $r$ and at a fixed time $t$. For this case, the total
derivative ambiguity of ${\boldsymbol{k}}_{\xi}$ will not contribute to the
integral in (4.6).

Finally, we will specialize to the case where $\xi$ is an exact Killing vector
of the solution. In this case, the expression $\delta Q_{\xi}$ in (4.6) can
always be integrated in order to define a conserved charge $Q_{\xi}$.
Operationally, this procedure always involves first computing the _change_ in
the charge $Q_{\xi}$ associated with a variation $\delta\Phi$ around a
particular solution $\Phi$. In examples of interest, we usually think of a
one-parameter family of solutions $\Phi(\alpha)$ which solve the equations of
motion for a continuous range of values of $\alpha$. In this case, we can
consider the variation

$\displaystyle\delta_{\alpha}\Phi=\frac{\partial\Phi(\alpha)}{\partial\alpha}\,.$
(4.7)

Since $\Phi(\alpha)$ is a solution for any $\alpha$, the variation (4.7)
always satisfies the linearized equations of motion. We can then compute

$\displaystyle\frac{dQ_{\xi}}{d\alpha}=\oint_{S}{\boldsymbol{k}}_{\xi}[\delta_{\alpha}\Phi;\Phi]\,,$
(4.8)

and integrate the resulting expression with respect to $\alpha$ (using a
suitable initial condition) to find $Q_{\xi}(\alpha)$. For example, the BTZ
black hole with mass $M$ is a solution to the three-dimensional equations of
motion for pure gravity (with negative cosmological constant) for any value of
$M$. We can compute the change in the charge associated with the exact Killing
vector $\xi=\partial_{t}$ in these backgrounds using the known expression for
${\boldsymbol{k}}_{\xi}$ in Einstein gravity. The resulting integral gives

$\displaystyle\frac{dQ_{\xi}}{dM}$
$\displaystyle=\oint_{S}{\boldsymbol{k}}_{\xi}[\delta_{M}g;g]\,,$
$\displaystyle=1\,.$ (4.9)

Using the initial condition $Q_{\xi}(M=0)=0$, we can then trivially integrate
to find

$\displaystyle Q_{\xi}=M\,,$ (4.10)

which recovers the fact that the mass of the black hole is the conserved
charge associated with the global timelike Killing vector $\partial_{t}$.

#### Charge Integrability

Given a family of spacetimes labeled by some parameter $\alpha$, and
possessing some exact Killing vector $\xi$, we have seen that the covariant
phase space formalism allows us to compute the quantity $\frac{\partial
Q_{\xi}}{\partial\alpha}$ which controls how the conserved charge associated
with $\xi$ varies with $\alpha$. If $\alpha$ is the only parameter in our
family of solutions, it is then trivial to integrate $\frac{\partial
Q_{\xi}}{\partial\alpha}$ given an initial condition and recover the
expression $Q_{\xi}(\alpha)$ for the conserved charge.

Now consider a family of solutions which depend on several parameters
$\alpha_{1},\ldots,\alpha_{n}$. The formalism permits us to compute a
collection of partial derivatives $\frac{\partial
Q_{\xi}}{\partial\alpha_{i}}$ for each $i$. We are not guaranteed that
integrating any two of these quantities will yield equivalent charges. For
instance, in the simple case of a solution for a black hole which depends on a
mass parameter $M$ and a spin parameter $J$, we could obtain two separate
derivatives

$\displaystyle\begin{split}\frac{\partial Q_{\xi}}{\partial
M}&=\oint_{S}{\boldsymbol{k}}_{\xi}[\partial_{M}g;g]\,,\\\ \frac{\partial
Q_{\xi}}{\partial
J}&=\oint_{S}{\boldsymbol{k}}_{\xi}[\partial_{J}g;g]\,.\end{split}$ (4.11)

If computing $\partial_{J}\partial_{M}Q_{\xi}$ using the first line of (4.11)
yields the same result as computing $\partial_{M}\partial_{J}Q_{\xi}$ using
the second line of (4.11), then we can unambiguously define the charge
$Q_{\xi}(M,J)$ in a way which does not involve any choice of how to perform
the integration. In this case, the charge $Q_{\xi}(M,J)$ is said to be
integrable. More generally, the integrability condition will hold so long as

$\displaystyle\delta_{1}\oint_{S}{\boldsymbol{k}}_{\xi}[\delta_{2}\Phi;\Phi]=\delta_{2}\oint_{S}{\boldsymbol{k}}_{\xi}[\delta_{1}\Phi;\Phi]\,,$
(4.12)

for any pair of variations $\delta_{1}\Phi$, $\delta_{2}\Phi$.

Charge integrability holds in many cases of physical interest, such as the
Kerr black hole in $D>3$ dimensions or the spinning BTZ black hole in $D=3$,
both of which possess charges that are integrable in the space of $M$ and $J$
parameters. However, it was pointed out in [22] that solutions to the
equations of motion for type IIB supergravity with multiple parameters
$\alpha_{i}$ do not, in general, possess manifestly integrable charges. In
such cases, if we are given only the information about the gravity solution,
the definitions of conserved charges like mass and spin can be ambiguous; one
obtains different answers depending on which quantity $\frac{\partial
Q_{\xi}}{\partial\alpha_{i}}$ one integrates.

To resolve this ambiguity, one requires additional input which identifies a
preferred notion of the surface charges. For instance, in the cases analyzed
in [22], the additional input comes from holography: if a type IIB
supergravity solution is dual to a conformal field theory, one can use the
scale invariance of the field theory to identify particular choices of the
gravitational charges that are natural from the perspective of the boundary
theory.

However, for more general solutions that are not dual to CFTs, there does not
appear to be a general principle for constructing unambiguous charges in the
case of non-integrable solutions. We will see shortly that the supergravity
solutions constructed in section 3 possess exactly this ambiguity when
$\widetilde{J}\neq 0$ . Because these bulk geometries are believed to be dual
to field theories which are obtained through deforming a CFT by an irrelevant
operator – namely, something akin to a single-trace ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator – we cannot
rely on the scale invariance of the dual theory to construct a preferred set
of gravitational charges.

#### Surfaces Charges in Type IIB-Like Theories

The formalism reviewed above applies to any generally covariant Lagrangian
$\boldsymbol{L}$. Next we will restrict attention to a class of Lagrangians
which are relevant for type IIB supergravity theories in Einstein frame and
with pure NS flux. Consider an action of the form

$\displaystyle S=\frac{1}{16\pi G}\int
d^{D}x\,\sqrt{-g}\left(R-\frac{1}{2}\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{1}{2}f(\Phi)|H|^{2}\right)\,.$
(4.13)

Here $\boldsymbol{H}=d\boldsymbol{B}$ is the field strength associated with a
now general $p$-form potential
$\boldsymbol{B}=B_{\mu_{1}\ldots\mu_{p}}\,dx^{\mu_{1}}\ldots dx^{\mu_{p}}$,
$|H|^{2}=\boldsymbol{H}\wedge\ast\boldsymbol{H}$, and $f(\Phi)$ is an
arbitrary function which controls the scalar coupling to the $p$-form kinetic
term. When $D=10$, $p=2$, and $f(\Phi)=e^{-\Phi}$, this gives the Einstein-
frame action for type IIB supergravity. After performing the conformal mapping
from string-frame to Einstein-frame, the general solutions which we derived in
section 3 satisfy the equations of motion associated with this action.

The surface charges associated with the Lagrangian (4.13) were worked out in
[23]; here we will recall the results which are relevant for our
analysis.444These formulas have also been implemented in Mathematica by the
author of [23] in a convenient package which may be found here.

The contribution from the scalar is

$\displaystyle{\boldsymbol{k}}^{\Phi}_{\xi}[\delta
g,\delta\Phi;g;\Phi]=i_{\xi}\boldsymbol{\Theta}_{\Phi}\,,\qquad\boldsymbol{\Theta}_{\Phi}=\ast(d\Phi\,\delta\Phi)\,.$
(4.14)

In components, (4.14) is

$\displaystyle k^{\mu\nu,\Phi}_{\xi}[\delta
g,\delta\Phi;g;\Phi]=2\left(\delta\Phi\right)\cdot\xi^{[\nu}\partial^{\mu]}\Phi,$
(4.15)

where $T^{[ab]}=\frac{1}{2}\left(T^{ab}-T^{ba}\right)$ denotes the usual
antisymmetrization. Recall that the non-bolface $k^{\mu\nu}_{\xi}$ is the
antisymmetric $2$-tensor whose Hodge dual is the $(n-2)$-form
${\boldsymbol{k}}_{\xi}$.

The contribution from the $p$-form $\boldsymbol{B}$ is more complicated and
can be expressed as

$\displaystyle{\boldsymbol{k}}^{B}_{\xi}[\delta g,\delta
B;g,B,\Phi]=-\delta\boldsymbol{Q}^{B}_{\xi}+i_{\xi}\boldsymbol{\Theta}_{B}-\boldsymbol{E}_{\mathcal{L}}^{B}[\mathcal{L}_{\xi}B,\delta
B]\,,$ (4.16)

where

$\displaystyle\boldsymbol{Q}^{B}_{\xi}$
$\displaystyle=f(\Phi)\left(i_{\xi}B\right)\wedge\ast H\,,$
$\displaystyle\boldsymbol{\Theta}^{B}$ $\displaystyle=f(\Phi)\left(\delta
B\right)\wedge\ast H\,,$
$\displaystyle\boldsymbol{E}_{\mathcal{L}}^{B}[\mathcal{L}_{\xi}B,\delta B]$
$\displaystyle=f(\Phi)\ast\left(\frac{1}{2(p-1)!}\delta
B_{\mu\alpha_{1}\cdots\alpha_{p-1}}\tensor{\left(\mathcal{L}_{\xi}B\right)}{{}_{\nu}^{\alpha_{1}}{}^{\ldots}{}^{\alpha_{p-1}}}\,dx^{\mu}\wedge
dx^{\nu}\right)\,.$ (4.17)

It is convenient to separate the component expression for $k_{\xi}^{B,\mu\nu}$
into its contributions from the tensor field variation $\delta\boldsymbol{B}$
(and its field strength variation
$\delta\boldsymbol{H}=d\left(\delta\boldsymbol{B}\right)$), the metric
variation $\delta g_{\mu\nu}$, and the dilaton variation $\delta\Phi$:

$\displaystyle k_{\xi}^{B,\mu\nu}$ $\displaystyle=2K_{\delta
B}^{[\mu\nu]}+2K_{\delta g}^{[\mu\nu]}+2K_{\delta\Phi}^{[\mu\nu]}\,,$
$\displaystyle K_{\delta B}^{\mu\nu}$
$\displaystyle=-f(\Phi)\Bigg{[}\frac{2}{p}\xi^{\mu}H^{\nu\alpha_{1}\cdots\alpha_{p}}\left(\delta
B\right)_{\alpha_{1}\cdots\alpha_{p}}-\left(\mathcal{L}_{\xi}B\right)^{\mu\alpha_{1}\cdots\alpha_{p-1}}\tensor{\left(\delta
B\right)}{{}^{\nu}_{\alpha_{1}}{}_{\cdots}{}_{\alpha_{p-1}}}$
$\displaystyle\qquad\qquad\quad+H^{\mu\nu\alpha_{1}\cdots\alpha_{p-1}}\xi^{\rho}\left(\delta
B\right)_{\rho\alpha_{1}\cdots\alpha_{p-1}}+\left(\delta
H\right)^{\mu\nu\alpha_{1}\cdots\alpha_{p-1}}\xi^{\rho}B_{\rho\alpha_{1}\cdots\alpha_{p-1}}\Bigg{]}\,,$
$\displaystyle K_{\delta g}^{\mu\nu}$
$\displaystyle=f(\Phi)\Bigg{[}-\frac{1}{2}\tensor{\left(\delta
g\right)}{{}^{\alpha}_{\alpha}}H^{\mu\nu\alpha_{1}\cdots\alpha_{p-1}}\xi^{\rho}B_{\rho\alpha_{1}\cdots\alpha_{p-1}}+(p-1)\left(\delta
g\right)^{\mu\rho}\tensor{H}{{}_{\rho}^{\nu}{}^{\alpha_{1}}{}^{\cdots}{}^{\alpha_{p-1}}}\xi^{\sigma}B_{\sigma\alpha_{1}\cdots\alpha_{p-1}}$
$\displaystyle\qquad\qquad\quad+2\tensor{\left(\delta
g\right)}{{}^{\mu}_{\rho}}H^{\rho\nu\alpha_{1}\cdots\alpha_{p-1}}\xi^{\sigma}B_{\sigma\alpha_{1}\cdots\alpha_{p-1}}\Bigg{]}\,,$
$\displaystyle K_{\delta\Phi}^{\mu\nu}$ $\displaystyle=-\frac{\partial
f}{\partial\Phi}\left(\delta\Phi\right)H^{\mu\nu\alpha_{1}\cdots\alpha_{p-1}}\xi^{\rho}B_{\rho\alpha_{1}\cdots\alpha_{p-1}}\,.$
(4.18)

Here we write $\delta g=g^{\mu\nu}\delta g_{\mu\nu}$ for the trace of the
metric fluctuation.555This differs from the usage in expressions like
${\boldsymbol{k}}_{\xi}^{g}[\delta g;g]$, where $\delta g$ refers to the full
metric fluctuation rather than its trace, but where we have suppressed indices
for ease of notation.

Finally, the contribution from the Einstein-Hilbert term $\sqrt{-g}R$ is

$\displaystyle{\boldsymbol{k}}_{\xi}^{g}[\delta
g;g]=-\delta\boldsymbol{Q}_{\xi}[g]-i_{\xi}\boldsymbol{\Theta}[\delta g;g]\,,$
(4.19)

where

$\displaystyle\Theta^{\mu}$ $\displaystyle=\frac{\sqrt{-g}}{16\pi
G}\left(\nabla_{\nu}\,\delta g^{\mu\nu}-\nabla^{\mu}\delta g\right)\,,$
$\displaystyle\boldsymbol{Q}_{\xi}$ $\displaystyle=\frac{\sqrt{-g}}{8\pi
G}\nabla^{\mu}\xi^{\nu}\left(d^{D-2}x\right)_{\mu\nu}\,.$ (4.20)

Here we follow the conventions of [21] for differential forms; for instance,
$\left(d^{n-p}x\right)_{\mu_{1}\ldots\mu_{p}}$ is defined by
$\frac{1}{p!(n-p)!}\varepsilon_{\mu_{1}\ldots\mu_{p}\nu_{p+1}\ldots\nu_{n}}dx^{\nu_{p+1}}\wedge\ldots\wedge
dx^{\nu_{n}}$ where $\varepsilon$ is the Levi-Civita symbol with entries
$-1,0$, or $1$ (without the factor of $\sqrt{-g}$). In components, (4.19) can
be written as

$\displaystyle k^{g,\mu\nu}_{\xi}=$
$\displaystyle\xi^{[\nu}\nabla^{\mu]}\delta g-\xi^{[\nu}\nabla_{\alpha}\delta
g^{\mu]\alpha}+\xi_{\alpha}\nabla^{[\nu}\delta
g^{\mu]\alpha}+\frac{1}{2}\delta g\nabla^{[\nu}\xi^{\mu]}$ (4.21)
$\displaystyle-\frac{1}{2}\delta
g^{\alpha[\nu}\nabla_{\alpha}\xi^{\mu]}+\frac{1}{2}\delta
g^{\alpha[\nu}\nabla^{\mu]}\xi_{\alpha}\,.$ (4.22)

The total contribution to the change in a charge $Q_{\xi}$ as a parameter
$\alpha$ of the solution is varied, therefore, is given by

$\displaystyle\frac{\partial Q_{\xi}}{\partial\alpha}$
$\displaystyle=\oint_{S}{\boldsymbol{k}}_{\xi}[\partial_{\alpha}g;g]\,,$
$\displaystyle{\boldsymbol{k}}_{\xi}$
$\displaystyle={\boldsymbol{k}}_{\xi}^{\Phi}+{\boldsymbol{k}}_{\xi}^{B}+{\boldsymbol{k}}_{\xi}^{g}\,.$
(4.23)

We will take the surface $S$ to lie at a fixed time and fixed large value $R$
of the radial coordinate $r$ in our solutions, and then take the limit as
$R\to\infty$. The integral (4.1) then extracts the component $k_{\xi}^{tr}$ of
the antisymmetric $2$-tensor which is Hodge dual to ${\boldsymbol{k}}_{\xi}$.
For the purpose of computing the integral (4.1), the compact directions
$S^{3}\times T^{4}$ are merely spectators which are integrated over to yield
an appropriate volume factor.

###  4.2 Evaluating the Charge

Our goal is to compute the Noether-Wald surface charge $Q_{\xi}$ associated
with the exact Killing vector $\xi=\partial_{t}$ of the solutions with ALD
asymptotics $(k_{5}=0)$ derived in section 3. We will restrict to the case
given in $(\ref{final_string_frame_soln_bulk})$ of a non-rotating black hole
with $\widetilde{J}=0$. We will comment on the case with both mass and spin at
the close of this section. That case involves interesting subtleties related
to the non-integrability discussed in section 4.1.

As is typical in the covariant phase space formalism, rather than computing
the charge $Q_{\xi}$ directly, we first find $\frac{\partial
Q_{\xi}}{\partial\alpha}$ where $\alpha$ is one of the parameters in these
solutions. At this stage we have a choice, since our spacetimes depend on the
parameters $(k_{1},\widetilde{M})$ with quantized fixed charges determined by
$(m_{1},m_{5})$. If we vary both $(k_{1},\widetilde{M})$ then we find that the
charge $Q_{\xi}$ is _not_ integrable in the space of these parameters; that
is, we obtain different expressions for the mass if we, for instance, (1)
compute $\frac{\partial Q_{\xi}}{\partial k_{1}}$ and integrate with respect
to $k_{1}$, or (2) compute $\frac{\partial Q_{\xi}}{\partial\widetilde{M}}$
and integrate with respect to $\widetilde{M}$. In this case, we view the
failure of integrability as reflecting the definition of an asymptotically
linear dilaton spacetime. Namely, the parameter $k_{1}$ is a part of the
definition of the asymptotic behavior and should not be varied.

For this reason, we will consider a one-parameter family of solutions where
$\alpha_{e}$, which controls $\widetilde{M}$, is varied but all other
parameters are held fixed. Therefore we will use the variations of the fields

$\displaystyle\delta_{\alpha_{e}}g_{\mu\nu}=\frac{\partial
g_{\mu\nu}}{\partial\alpha_{e}}\,,\qquad\delta_{\alpha_{e}}B_{\mu\nu}=\frac{\partial
B_{\mu\nu}}{\partial\alpha_{e}}\,,\qquad\delta_{\alpha_{e}}\Phi=\frac{\partial\Phi}{\partial\alpha_{e}}\,.$
(4.24)

Under this linearized variation, the combined contribution from $k_{\xi}^{g}$,
$k_{\xi}^{B}$, and $k_{\xi}^{\Phi}$ to the change in the charge is

$\displaystyle\delta_{\alpha_{e}}Q_{\xi}=\oint_{S}\left({\boldsymbol{k}}_{\xi}^{g}+{\boldsymbol{k}}_{\xi}^{B}+{\boldsymbol{k}}_{\xi}^{\Phi}\right)=\frac{\hat{r}_{1}\alpha_{e}}{4\sqrt{\hat{r}_{1}^{2}+k_{1}\alpha_{e}^{2}}}\,.$
(4.25)

Here $S$ is a fixed time slice at large $r$ in $\mathcal{M}_{3}$ and includes
an integral over all the compact directions. Because we have used the
variations (4.24) associated with changing $\alpha_{e}$, we have
$\delta_{\alpha_{e}}Q_{\xi}=\frac{\partial Q_{\xi}}{\partial\alpha_{e}}$ so
that

$\displaystyle\frac{\partial
Q_{\xi}}{\partial\alpha_{e}}=\frac{\hat{r}_{1}\alpha_{e}}{4\sqrt{\hat{r}_{1}^{2}+k_{1}\alpha_{e}^{2}}}\,.$
(4.26)

This equation can be trivially integrated to find

$\displaystyle
Q_{\xi}=\frac{\hat{r}_{1}}{4k_{1}}\sqrt{\hat{r}_{1}^{2}+k_{1}\alpha_{e}^{2}}+C\,,$
(4.27)

where $C$ is an integration constant which may be a function of $k_{1}$ and
$\alpha_{1}$ but not of $\alpha_{e}$. We fix the integration constant by
requiring that the conserved charge $Q_{\xi}$ have a finite limit as $k_{1}\to
0$. This is only possible if the constant $C$ is chosen so that the charge
takes the form

$\displaystyle
Q_{\xi}=\frac{\hat{r}_{1}^{2}}{4k_{1}}\left(\sqrt{1+k_{1}\frac{\alpha_{e}^{2}}{\hat{r}_{1}^{2}}}-1\right)\,.$
(4.28)

To recover the undeformed black hole, we take $k_{1}\to 0$. This is an easy
way to fix the overall normalization. We will choose to normalize the charge,
denoted $\widetilde{Q}_{\xi}$, so that $k_{1}\rightarrow 0$ gives
$\widetilde{M}$:

$\displaystyle\widetilde{Q}_{\xi}=\frac{R^{2}}{4r_{5}^{2}k_{1}}\left(\sqrt{1+k_{1}\frac{8\widetilde{M}r_{5}^{2}}{R^{2}}}-1\right)\,.$
(4.29)

This square root is identical to the expression that appears in the dilaton
solution found in $(\ref{final_string_frame_soln_bulk})$. It is strong
evidence for a holographic correspondence in the spirit proposed in [6].

A straightforward extension of this analysis to the case with mass and spin
encounters the issue of non-integrability in the space of
$(\widetilde{M},\widetilde{J})$ solutions. Similar issues have been seen in
past work [22, 24, 25]. It would be interesting to resolve the non-
integrability and see if one can reproduce the mass and charge for these
asymptotically linear dilaton spacetimes expected from the form of the dilaton
solution seen in $(\ref{spinning_case})$.

###  4.3 Winding Tachyon

We have seen that the solution which interpolates from global ${\rm AdS}_{3}$
to ALD asymptotics, presented in section 3 with $\widetilde{M}=-\frac{1}{8}$,
has a spacetime mass $(\ref{sqrt_charge_soln})$ which becomes complex when
$k_{1}$ is made too large.

What might be happening in string theory when this ground state approaches
this instability? Note that the string-frame metric sees an approximately
constant circle for large $r$ of size $\frac{R}{\sqrt{k_{1}}}$ and a very weak
string coupling. Global ${\rm AdS}$ in the interior corresponds to anti-
periodic boundary conditions for spacetime fermions on the $x_{5}$ circle.
Boundary conditions of this type tend to produce tachyons for strings wrapping
the circle if the circle becomes too small [26]. Qualitatively, this is the
kind of phenomenon we might hope would happen prior to the spacetime becoming
unstable, although the endpoint of condensing such a bulk closed string
tachyon is often non-geometric [27]. If the holographic theory exists for
$k_{1}$ beyond the value where the ground state energy becomes complex, we
will see that the more likely resolution lies in the low-energy gravity
theory.

To estimate where a tachyon might emerge, we note that a fundamental string on
a circle of radius $\frac{R}{\sqrt{k_{1}}}$ with winding $w$ and a twisted
boundary condition breaking supersymmetry satisfies a mass-shell condition,

$\displaystyle
m^{2}=\frac{w^{2}R^{2}}{k_{1}(\alpha^{\prime})^{2}}+\frac{2}{\alpha^{\prime}}\left(N_{L}+N_{R}\right)\,,$
(4.30)

for bosonic excitations with no momentum on the circle. In the $(NS-,NS-)$
sector with odd winding and no oscillators, the zero point energy is given by
$N_{L}=N_{R}=-\frac{1}{2}$. The first state to become tachyonic has $w=1$
which happens for $k_{1}>\frac{R^{2}}{2\alpha^{\prime}}$. This is not quite
right in our case because we have not taken the linear dilaton background into
account. The linear dilaton slope is often denoted $Q$ in the literature on
worldsheet descriptions of linear dilaton backgrounds, where in this case
$Q=\frac{2}{\sqrt{m_{5}\alpha^{\prime}}}$; see, for example [13]. For a string
with no radial momentum, the mass-shell condition for a singly wound string is
modified slightly from the simple circle of $(\ref{simplecircle})$,

$m^{2}=\frac{R^{2}}{k_{1}(\alpha^{\prime})^{2}}+\frac{Q^{2}}{4}-\frac{2}{\alpha^{\prime}}\,.$
(4.31)

Now the condition to avoid a tachyon becomes

$\displaystyle
k_{1}<\frac{R^{2}}{\alpha^{\prime}\left(2-\frac{1}{m_{5}}\right)}\,.$ (4.32)

This should be contrasted with the condition $(\ref{gravity_condition})$
needed to avoid a complex dilaton solution, which we repeat here for
convenience:

$\displaystyle k_{1}<\frac{R^{2}}{m_{5}\alpha^{\prime}}\,.$ (4.33)

These conditions only agree for $m_{5}=1$. For large $m_{5}$ the gravity
condition is violated first, long before one expects a closed string tachyon,
suggesting a possible resolution in the low-energy theory without stringy
ingredients. There is one final observation which is curious: if there were a
‘long’ string with tension reduced by a factor of $m_{5}$ so the tension
$T_{\rm long}=\frac{1}{2\pi\alpha^{\prime}m_{5}}$ then a tachyon for this
string would emerge before the gravity solution becomes complex as $k_{1}$ is
increased.666Such long strings are expected from the low-energy physics of the
D1-D5 system but are not expected to be visible in the weakly coupled
perturbative string spectrum of the F1-NS5 system.

## Acknowledgements

It is our pleasure to thank D. Kutasov and E. Martinec for helpful
discussions. C. C., C. F. and S. S. are supported in part by NSF Grant No.
PHY1720480. C. F. is also supported by U.S. Department of Energy grant DE-
SC0009999 and by funds from the University of California.

##  Appendix A Derivation of General Solutions

In this Appendix we derive a general class of type IIB supergravity solutions
which satisfy the assumptions laid out in section 3.1. We will work in string
frame with the action (2.1), for which the associated equations of motion for
the metric, dilaton, and flux are

$\displaystyle
R_{\mu\nu}+2\nabla_{\mu}\nabla_{\nu}\Phi-\frac{1}{4}|H|^{2}_{\mu\nu}=0\ ,$
(A.1) $\displaystyle
R+4\nabla^{2}\Phi-4|\nabla\Phi|^{2}-\frac{1}{12}|H|^{2}=0\,,$ (A.2)
$\displaystyle\nabla^{\mu}\left(e^{-2\Phi}H_{\mu\nu\rho}\right)=0\,,$ (A.3)

respectively. A combination of the first two equations gives a nice equation
for the dilaton. See, for example [28]:

$\displaystyle\nabla^{2}e^{-2\Phi}=e^{-2\Phi}|H|^{2}.$ (A.4)

We will begin with the following general ansatz consistent with the symmetry
and flux assumptions stated in section 3.1:

$\displaystyle ds^{2}$
$\displaystyle=-\frac{f_{e}(r)}{f_{1}(r)}\,dt^{2}+\frac{1}{f_{x}(r)}\,\left\\{dx_{5}+f_{j}(r)\,dt\right\\}^{2}+f_{r}(r)\,dr^{2}+r^{2}f_{5}(r)\,d\Omega_{3}^{2}+f_{T^{4}}(r)ds^{2}_{T^{4}}\,,$
$\displaystyle\hskip 70.0pt\Phi=\Phi(r)\,,\qquad
H_{3}=\frac{2m_{5}}{\ell_{s}}\epsilon_{S^{3}}+f_{\mathcal{M}_{3}}(r)e^{2\Phi}\epsilon_{3}^{\mathcal{M}_{3}}\,.$
(A.5)

Here $\epsilon_{S^{3}}$ is the volume form for $S^{3}$ with
$\int\epsilon_{S^{3}}=2\pi^{2}\ell_{s}^{3}$. Flux quantization through $S^{3}$
is automatically satisfied. Flux quantization through ${\mathcal{M}}_{3}$
requires that

$\displaystyle f_{\mathcal{M}_{3}}(r)=\frac{32m_{1}\pi^{4}\alpha^{\prime
3}g_{s}^{2}}{V_{4}r^{3}f_{5}(r)^{3/2}}.$ (A.6)

That fixes one unknown function. This form for $H_{3}$ also solves
$(\ref{flux_eom})$. One way to make $R_{\mu\nu}$ vanish in the $T^{4}$
directions is by setting $f_{T^{4}}(r)$ to a constant. It is not clear that
any other solution exists so will set $f_{T^{4}}(r)=1$. This is in accord with
the torus playing no significant role in the physics.

The choice of $f_{r}$ is the freedom to parametrize the radial direction in a
convenient way. At this point, we need to understand what asymptotic
conditions to impose on our spacetime at small $r$ and large $r$.

###  A.1 The partially decoupled case $k_{5}=0$

#### Interpolating ansatz

It will simplify our analysis to take the following ansatz for the ten-
dimensional metric,

$\displaystyle\begin{split}&f_{1}(r)=k_{1}+\frac{\alpha_{1}^{2}}{r^{2}},\qquad
f_{e}(r)=1-\frac{\alpha_{e}^{2}}{r^{2}},\qquad
f_{5}(r)=\frac{\alpha_{5}^{2}}{r^{2}},\\\ &f_{x}(r)=k_{x}+{\alpha_{x}^{2}\over
r^{2}}\,,\qquad f_{j}(r)=0\,,\end{split}$ (A.7)

with $f_{j}$ set to $0$ because $J=0$. We could have considered the more
general $f_{5}(r)=k_{5}+\frac{\alpha_{5}^{2}}{r^{2}}$ but for this partially
decoupled case, we set $k_{5}=0$. We also demand that $k_{1}>0$ and $k_{x}>0$
for a sensible metric at large $r$. We have chosen to absorb the constant
$k_{e}$ that might have appeared in $f_{e}$ into a redefinition of the other
parameters. Because this is the string-frame metric rather than the Einstein-
frame metric, we will leave $k_{1}$ in the ansatz until later when we
determine how to normalize the time direction at large $r$.

#### Constraints from equations of motion

With this assumption, plugging our ansatz into the Einstein and dilaton
equations gives constraints,

$\displaystyle\alpha_{x}^{2}=\frac{k_{x}}{k_{1}}\alpha_{1}^{2}\,,\qquad\alpha_{5}=r_{5}\,,$
(A.8)

and the following solution for $e^{2\Phi}$,

$\displaystyle
e^{2\Phi}=c_{2}^{2}\left(\frac{1+k_{1}c_{1}\left(\alpha_{1}^{2}+k_{1}r^{2}\right)}{\alpha_{1}^{2}+k_{1}r^{2}}\right)\,,$
(A.9)

where $c_{1}$ and $c_{2}$ are unknown constants. Imposing the dilaton equation
of motion relates $c_{1}$ and $c_{2}$ as follows,

$\displaystyle\frac{r_{5}^{4}}{\hat{r}_{1}^{4}}-\frac{(k_{1}c_{1}\alpha_{1}^{2}+1)(k_{1}c_{1}(\alpha_{1}^{2}+k_{1}\alpha_{e}^{2})+1)}{\alpha_{1}^{4}+k_{1}\alpha_{1}^{2}\alpha_{e}^{2}}c_{2}^{4}=0\,.$
(A.10)

Note that $c_{2}$ cannot be set to zero. At this juncture, we have not imposed
either small $r$ or large $r$ asymptotics on the dilaton. The constants
$(c_{1},c_{2})$ determine the dilaton and $f_{r}$. Everything else is fixed by
both matching onto BTZ at small $r$ and imposing large $r$ asymptotics, which
we have yet to examine.

If we now demand that $g_{s}\rightarrow 0$ as $r\rightarrow\infty$ then we
need to set $c_{1}=0$ in $(\ref{dilatonsolution})$ to get a dilaton:

$\displaystyle
e^{2\Phi}=\left(\frac{c_{2}^{2}}{\alpha_{1}^{2}+k_{1}r^{2}}\right)\,.$ (A.11)

We also require $\frac{c_{2}^{2}}{k_{1}}\geq 0$ since it is the coefficient of
the large $r$ string coupling of $(\ref{dilatonsolution})$ so $c_{2}^{2}\geq
0$. For a sensible value of $e^{2\Phi}$ in a neighborhood of $r=0$ we must
also then require $\alpha_{1}^{2}>0$. This also fixes $c_{2}$ in terms of the
other parameters of the ansatz using $(\ref{relatec1c2})$:

$\displaystyle
c_{2}^{4}=\frac{r_{5}^{4}}{\hat{r}_{1}^{4}}(\alpha_{1}^{4}+k_{1}\alpha_{1}^{2}\alpha_{e}^{2})\,.$
(A.12)

With the dilaton determined, we can now solve for $f_{r}$:

$\displaystyle f_{r}(r)=\frac{r_{5}^{2}}{r^{2}-\alpha_{e}^{2}}\,.$ (A.13)

#### Small $r$ asymptotics

At very small $r$ we demand that the solution in Einstein-frame look like a
three-dimensional BTZ black hole. This expectation is at least reasonable for
low mass black holes whose structure should not care very much about the large
$r$ asymptotic behavior of the metric. For simplicity let us first consider a
black hole with mass but no spin. In the notation of the full decoupling limit
presented in equation (2.12), the metric of a three-dimensional BTZ black hole
of mass $\widetilde{M}$ and spin $J=0$ is

$\displaystyle\widehat{ds}^{2}=\ell^{2}\left(-\left(-8\widetilde{M}+\tilde{r}^{2}\right)d\tilde{t}^{2}+\tilde{r}^{2}d\tilde{\varphi}^{2}+\frac{d\tilde{r}^{2}}{-8\widetilde{M}+\tilde{r}^{2}}\right)\,,$
(A.14)

where the mass parameter $\widetilde{M}$ has been made dimensionless using the
effective three-dimensional Newton constant.

We will initially be noncommittal about the relation between these
dimensionless variables and the dimensionful variables of our ansatz
$(\ref{metricansatzk5=0})$. Because $\alpha_{5}=r_{5}$ by the equations of
motion, the reduction on $S^{3}$ and $T^{4}$ to a three-dimensional theory is
unchanged from the discussion around (2.12). There are no mass-dependent
factors from integrating over those volumes in string-frame.

The three-dimensional Einstein-frame metric is given by,

$\displaystyle ds^{2}_{E}=e^{-4\Phi}ds^{2}_{\rm string}\,.$ (A.15)

The most commonly studied backgrounds involve a constant dilaton so moving
between string-frame and Einstein-frame is straightforward. This is no longer
immediately the case in backgrounds where the dilaton varies so we will want
to keep careful track of dilaton factors. For this $c_{1}=0$ case we can
expand $e^{-4\Phi}$ at small $r$ to find,

$\displaystyle
e^{-4\Phi}=\frac{\alpha_{1}^{4}}{c_{2}^{4}}\left(1+\frac{2k_{1}r^{2}}{\alpha_{1}^{2}}\right)+O(r^{4})\,,$
(A.16)

where $c_{2}^{4}$ is given in $(\ref{constraintonc2})$.

How shall we fix the small $r$ asymptotics? Let us first note that there is no
singularity in $e^{2\Phi}$ which must be positive for all $r$ so the conformal
factor in $(\ref{einsteinframe})$ does not introduce any potential
singularities. On the other hand, $f_{r}$ appearing in $(\ref{frc1=0})$ does
have a singularity at $r=\alpha_{e}$. Comparing with $(\ref{BTZM=0})$ tells us
that

$\displaystyle\alpha_{e}^{2}\sim\widetilde{M}\,.$ (A.17)

We still need to relate the coordinates of our ansatz to the dimensionless
coordinates of the BTZ solution. There is a natural place to begin such an
identification with,

$\displaystyle e^{-4\Phi}{1\over
f_{x}(r)}dx_{5}^{2}\,\rightarrow\,\ell^{2}\tilde{r}^{2}d\tilde{\varphi}^{2}\,,$
(A.18)

for small $r$. There is no freedom to redefine the angular coordinate which
allows us to unambiguously identify:

$\displaystyle
e^{-4\Phi}r^{2}=\ell^{2}\frac{\alpha_{x}^{2}}{R^{2}}\,\tilde{r}^{2}=\ell^{2}\frac{k_{x}}{k_{1}}\frac{\alpha_{1}^{2}}{R^{2}}\,\tilde{r}^{2}\quad\Rightarrow\quad
r^{2}=\ell^{2}\,\frac{k_{x}}{k_{1}}\frac{c_{2}^{4}}{\alpha_{1}^{2}R^{2}}\,\tilde{r}^{2}+O(\tilde{r}^{4})\,.$
(A.19)

Once we have this relation between the dimensionless $\tilde{r}$ and $r$, we
can determine a small $\tilde{r}$ regime where the dilaton $(\ref{dilc1=0})$
is approximately constant. This requires that $\frac{d\Phi}{d\tilde{r}}$ be
small:

$\displaystyle{k_{x}}\frac{\ell^{2}c_{2}^{4}}{\alpha_{1}^{4}R^{2}}\tilde{r}=\sqrt{k_{1}k_{x}}\frac{\ell
c_{2}^{2}}{\alpha_{1}^{3}R}r\ll 1.$ (A.20)

In this regime we demand that the solution looks like a BTZ black hole in
approximately ${\rm AdS}$ space.

Now we can examine the metric coefficient of $dr^{2}$ in
$(\ref{black_hole_ansatz})$,

$\displaystyle e^{-4\Phi}f_{r}(r)dr^{2}$
$\displaystyle=\left(\frac{\alpha_{1}^{4}}{c_{2}^{4}}\left(1+\frac{2k_{1}r^{2}}{\alpha_{1}^{2}}\right)+\ldots\right)\left(\frac{r_{5}^{2}}{r^{2}-\alpha_{e}^{2}}\right)dr^{2}\,.$
(A.21)

Comparison with $(\ref{BTZM=0})$ teaches us that

$\displaystyle\ell^{2}=\frac{\alpha_{1}^{4}}{c_{2}^{4}}r_{5}^{2},\qquad\alpha_{e}^{2}=8\widetilde{M}\frac{k_{x}}{k_{1}}\frac{\alpha_{1}^{2}r_{5}^{2}}{R^{2}}\,.$
(A.22)

As expected, $\alpha_{e}^{2}$ is related to the mass of the black hole.
Interestingly the definition of $\ell$ looks mass-dependent when $k_{1}\neq 0$
unlike the case of constant dilaton. We can also simplify $(\ref{relatingr})$,

$\displaystyle\frac{r^{2}}{\tilde{r}^{2}}=\frac{k_{x}}{k_{1}}\frac{\alpha_{1}^{2}r_{5}^{2}}{R^{2}},$
(A.23)

and the regime where the solution will look approximately like ${\rm AdS}$:

$\displaystyle\sqrt{k_{1}k_{x}}\frac{r_{5}r}{\alpha_{1}R}\ll 1.$ (A.24)

The remaining task is to relate $t$ and $\tilde{t}$, which follows by
requiring the singularity at $r^{2}=\alpha_{e}^{2}$ be a coordinate
singularity. Let us recall that $e^{-4\Phi}\frac{1}{f_{1}(r)}$ is positive and
introduces neither singularities nor changes to the signature of the time
coordinate. On the other hand, $f_{r}(r)$ does change sign at $r=\alpha_{e}$
and this change must be matched by the behavior of the $dt^{2}$ metric
coefficient. This constrains the $dt^{2}$ metric coefficient to match BTZ,

$\displaystyle\frac{\alpha_{1}^{2}}{c_{2}^{4}}\left({r^{2}-\alpha_{e}^{2}}\right)\left\\{1+O(r^{4})\right\\}dt^{2}=\ell^{2}\left(\tilde{r}^{2}-8\widetilde{M}\right)d\tilde{t}^{2}\,.$
(A.25)

From the $O(r^{2})$ term we conclude:

$\displaystyle
d\tilde{t}^{2}=\frac{r^{2}}{\tilde{r}^{2}}\frac{\alpha_{1}^{2}}{c_{2}^{4}\ell^{2}}\,dt^{2}\quad\implies\quad\tilde{t}=\frac{t}{R}\sqrt{\frac{k_{x}}{k_{1}}}\,.$
(A.26)

The other $r$-independent condition from $(\ref{timecomponentofmetric})$ is
then automatically satisfied.

#### Matching large and small $r$ asymptotics

To go further we need to examine the large $r$ asymptotics. In essence, the
issue is what large $r$ criteria define metrics which are asymptotically
linear dilaton? To understand the large $r$ asymptotics, we need an
appropriate large $r$ coordinate. From the $e^{-4\Phi}f_{r}(r)dr^{2}$ term
with the explicit form for $f_{r}$ given in $(\ref{frc1=0})$, we see that the
natural radial coordinate for large $r$ takes the form,

$\displaystyle\rho=\frac{r_{5}k_{1}}{2c_{2}^{2}}r^{2}=\frac{r^{2}}{\alpha_{\rho}}\,,\qquad\alpha_{\rho}=\frac{2c_{2}^{2}}{r_{5}k_{1}}\,.$
(A.27)

In terms of $\rho$, the three-dimensional asymptotic metric takes the form:

$\displaystyle ds^{2}$
$\displaystyle=-\frac{4\rho^{2}}{r_{5}^{2}}\left(\frac{1}{k_{1}}+O\left(\frac{1}{\rho}\right)\right)\,dt^{2}+\frac{4\rho^{2}}{r_{5}^{2}}\left(\frac{1}{k_{x}}+O\left(\frac{1}{\rho}\right)\right)\,dx_{5}^{2}+d\rho^{2}\,.$
(A.28)

What is important is that the leading term at large $\rho$ is completely
independent of the mass $\widetilde{M}$. This defines the vacuum to which each
solution should asymptote with mass-dependent subleading corrections.

Our remaining task is to identify $\alpha_{1}^{2}$. We have one example of an
asymptotically linear dilaton metric in $(\ref{partialasymptotics})$ which
corresponds to $\widetilde{M}=0$ BTZ. This case turns out to be somewhat
special when compared with black holes with $\widetilde{M}\neq 0$. For
example, any non-zero value of $k_{1}$ can be set to $1$ by a redefinition of
the radial coordinate.777There is a similar special feature for $E=0$ states
under a ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformation; namely the energy does not flow. This is readily visible from
$(\ref{energyformula})$.  This is not the case when $\widetilde{M}\neq 0$.
Regardless, we can still try to use this case to normalize $\alpha_{1}^{2}$ in
our metric ansatz $(\ref{metricansatzk5=0})$. Let us examine the large $\rho$
behavior of our dilaton:

$\displaystyle\Phi$
$\displaystyle=-\frac{1}{2}\log\rho+\frac{1}{2}\log\frac{r_{5}}{2}-\frac{1}{2}\log\left(1+\frac{\alpha_{1}^{2}r_{5}}{2c_{2}^{2}}\frac{1}{\rho}\right)\,$
$\displaystyle=-\frac{1}{2}\log\rho+\frac{1}{2}\log\frac{r_{5}}{2}-\frac{\alpha_{1}^{2}r_{5}}{4c_{2}^{2}}\frac{1}{\rho}+O\left(\frac{1}{\rho^{2}}\right)\,.$
(A.29)

All the potential mass-dependence, which is hidden in $c_{2}^{2}$, is
$\rho$-suppressed. If we set $\alpha_{e}=0$ and compare these expressions for
the dilaton and metric with the expressions found in
$(\ref{partialasymptotics})$, we can identify $\alpha_{1}^{2}$:

$\displaystyle\alpha_{1}^{2}=\hat{r}_{1}^{2}\,.$ (A.30)

At this point we could redefine time to absorb the $\frac{1}{k_{1}}$ factor
with the replacement $\alpha_{1}^{2}\rightarrow\frac{\alpha_{1}^{2}}{k_{1}}$.
However the ${\rm AdS}$ limit where $k_{1}\rightarrow 0$ is more transparent
with the explicit $k_{1}$. On the other hand, redefining the periodicity of
$x_{5}$ so that $k_{x}\rightarrow k_{1}$ does look natural. With this
redefinition, $\alpha_{x}\rightarrow\alpha_{1}$ using $(\ref{solveax})$. The
final string-frame metric and dilaton take the form,

$\displaystyle
ds^{2}=-\frac{1-\frac{\alpha_{e}^{2}}{r^{2}}}{k_{1}+\frac{\hat{r}_{1}^{2}}{r^{2}}}\,dt^{2}+\frac{1}{k_{1}+\frac{\hat{r}_{1}^{2}}{r^{2}}}\,dx_{5}^{2}+\frac{r_{5}^{2}}{r^{2}-\alpha_{e}^{2}}\,dr^{2}+r_{5}^{2}\,d\Omega_{3}^{2}+ds^{2}_{T^{4}}\,,$
$\displaystyle
e^{2\Phi}=\frac{r_{5}^{2}\sqrt{1+8k_{1}\frac{\widetilde{M}r_{5}^{2}}{R^{2}}}}{\hat{r}_{1}^{2}+k_{1}r^{2}}\,,\qquad\alpha_{e}^{2}=8\widetilde{M}\frac{\hat{r}_{1}^{2}r_{5}^{2}}{R^{2}}\,.$
(A.31)

Note that the $x_{5}$ circle has periodicity $\frac{2\pi R}{\sqrt{k_{1}}}$ at
$r=\infty$ in string-frame.

#### The case of $c_{1}\neq 0$

In solving for the dilaton, we set $c_{1}=0$ in $(\ref{dilatonsolution})$. In
this subsection we will examine whether there are any sensible solutions with
$c_{1}>0$. The case of $c_{1}<0$ is ruled out because there would be a
singularity in the dilaton at some finite $r$. If a solution exists with
$c_{1}>0$, it would be quite curious because it interpolates from one
approximately constant value of the dilaton near $r=0$ to a different
approximately constant value near $r=\infty$.

The first step is to solve for $c_{2}^{4}$ using $(\ref{relatec1c2})$. This is
just algebraic and gives:

$\displaystyle
c_{2}^{4}=\frac{r_{5}^{4}}{\hat{r}_{1}^{4}}\left\\{\frac{\alpha_{1}^{4}+k_{1}\alpha_{1}^{2}\alpha_{e}^{2}}{(k_{1}c_{1}\alpha_{1}^{2}+1)(k_{1}c_{1}(\alpha_{1}^{2}+k_{1}\alpha_{e}^{2})+1)}\right\\}\,.$
(A.32)

The $r\rightarrow\infty$ UV value of the dilaton is determined by
$e^{2\Phi}=k_{1}c_{1}c_{2}^{2}$ while the $r\rightarrow 0$ IR value,

$\displaystyle
e^{2\Phi}=\left(\frac{1+k_{1}c_{1}\alpha_{1}^{2}}{\alpha_{1}^{2}}\right)c_{2}^{2}>k_{1}c_{1}c_{2}^{2}\,,$
(A.33)

is strictly larger. If the asymptotic solutions are both approximately ${\rm
AdS}$ solutions with the same string-frame length scale $\ell$ then this
relation would be in accord with the two-dimensional $c$-theorem. This
suggests a potential holographic interpretation of this class of solutions as
describing a flow from a UV two-dimensional CFT to an IR CFT. What is
particularly interesting about this possibility is that by taking $c_{1}$ very
small, this background would have a long intermediate region that looks like a
linear dilaton but which eventually becomes ${\rm AdS}_{3}$ again.

When $c_{1}>0$, $f_{r}$ is constrained to be

$\displaystyle
f_{r}(r)=\frac{r_{5}^{2}(1+k_{1}\alpha_{1}^{2}c_{1})(1+k_{1}c_{1}(\alpha_{1}^{2}+k_{1}\alpha_{e}^{2}))}{(r-\alpha_{e})(r+\alpha_{e})(1+k_{1}c_{1}(\alpha_{1}^{2}+k_{1}r^{2}))^{2}}\,,$
(A.34)

which diverges at $r=\alpha_{e}$ just like the $c_{1}=0$ solution. Again it
appears that this is only a coordinate singularity and not a genuine curvature
singularity. The full Ricci scalar associated with this ten-dimensional
solution is given by

$\displaystyle R=\frac{2k_{1}}{r_{5}^{2}}\times$
$\displaystyle\frac{\left(3k_{1}r^{4}+10r^{2}\alpha_{1}^{2}+2k_{1}r^{2}\alpha_{e}^{2}-5\alpha_{1}^{2}\alpha_{e}^{2}+5k_{1}c_{1}(k_{1}r^{2}+\alpha_{1}^{2})(r^{2}(2\alpha_{1}^{2}+k_{1}\alpha_{e}^{2})-\alpha_{1}^{2}\alpha_{e}^{2})\right)}{(k_{1}r^{2}+\alpha_{1}^{2})^{2}(1+k_{1}c_{1}\alpha_{1}^{2})\left(1+k_{1}c_{1}(\alpha_{1}^{2}+k_{1}\alpha_{e}^{2})\right)}\,.$
(A.35)

If $k_{1}>0$ and $\alpha_{1}^{2}>0$, the Ricci scalar is finite for all values
of $r$. It appears to interpolate from a constant negative value at small $r$
to a constant positive value at large $r$. One finds

$\displaystyle\lim_{r\to
0}R=-\frac{10k_{1}\alpha_{e}^{2}}{r_{5}^{2}\alpha_{1}^{2}(1+k_{1}\alpha_{1}^{2}c_{1}+k_{1}^{2}\alpha_{e}^{2}c_{1})}\,,$
(A.36)

while

$\displaystyle\lim_{r\to\infty}R=\frac{2\left(3+10k_{1}\alpha_{1}^{2}c_{1}+5c_{1}k_{1}^{2}\alpha_{e}^{2}\right)}{r_{5}^{2}(1+k_{1}c_{1}\alpha_{1}^{2})(1+k_{1}\alpha_{1}^{2}c_{1}+k_{1}^{2}\alpha_{e}^{2}c_{1})}\,.$
(A.37)

We can also compute the Ricci scalar $R^{(3)}$ associated with the three-
dimensional spacetime $\mathcal{M}_{3}$ parameterized by $t,r,x_{5}$. This is
given by

$\displaystyle R^{(3)}=\,$ $\displaystyle
2\left(1+k_{1}c_{1}(k_{1}r^{2}+\alpha_{1}^{2})\right)\times$ (A.38)
$\displaystyle\frac{\left(4k_{1}r^{2}\alpha_{1}^{2}-3\alpha_{1}^{4}+2k_{1}^{2}r^{2}\alpha_{e}^{2}-5k_{1}\alpha_{1}^{2}\alpha_{e}^{2}-3k_{1}c_{1}\alpha_{1}^{2}(k_{1}r^{2}+\alpha_{1}^{2})(\alpha_{1}^{2}+k_{1}\alpha_{e}^{2})\right)}{r_{5}^{2}(k_{1}r^{2}+\alpha_{1}^{2})(1+k_{1}c_{1}\alpha_{1}^{2})(1+k_{1}c_{1}(\alpha_{1}^{2}+k_{1}\alpha_{e}^{2}))}\,.$
(A.39)

Again we can look at the small $r$ and large $r$ limits,

$\displaystyle\lim_{r\to 0}R^{(3)}$
$\displaystyle=-\frac{2\left(3\alpha_{1}^{4}+5k_{1}\alpha_{1}^{2}\alpha_{e}^{2}+3k_{1}c_{1}\alpha_{1}^{4}(\alpha_{1}^{2}+k_{1}\alpha_{e}^{2})\right)}{r_{5}^{2}(1+k_{1}c_{1}\alpha_{1}^{2})(1+k_{1}c_{1}(\alpha_{1}^{2}+k_{1}\alpha_{e}^{2}))}\,,$
$\displaystyle\lim_{r\to\infty}R^{(3)}$
$\displaystyle=\frac{2c_{1}\left(4k_{1}\alpha_{1}^{2}+2k_{1}^{2}\alpha_{e}^{2}-3k_{1}^{2}c_{1}\alpha_{1}^{2}(\alpha_{1}^{2}+k_{1}\alpha_{e}^{2})\right)}{r_{5}^{2}(1+k_{1}c_{1}\alpha_{1}^{2})(1+k_{1}c_{1}(\alpha_{1}^{2}+k_{1}\alpha_{e}^{2}))}\,.$
(A.40)

We see that the small $r$ limit of $R^{(3)}$ is negative-definite,
corresponding to an ${\rm AdS}_{3}$ in the deep interior, but that the large
$r$ value of $R^{(3)}$ has competing contributions with opposite signs. In
order to have negative curvature at large $r$, we require

$\displaystyle
c_{1}>\frac{2}{3k_{1}}\left(\frac{1}{\alpha_{1}^{2}}+\frac{1}{\alpha_{1}^{2}+k_{1}\alpha_{e}^{2}}\right)\,.$
(A.41)

Notice that this condition is easy to violate by taking $k_{1}$ small. The
surprising result is that we seem to be able to construct a three-dimensional
spacetime ${\mathcal{M}}_{3}$ with positive scalar curvature! This conclusion
also seems to hold in Einstein frame. Whether this class of solutions is
sensible definitely merits a deeper investigation, which will appear
elsewhere.

#### Adding spin $\widetilde{J}$

Now let us return to the case $c_{1}=0$ and extend our solutions
$(\ref{final_string_frame_soln})$ to include spin. This will also help us
identify the relation between the deformation parameter $k_{1}$ and $\lambda$
of $(\ref{energyformula})$. The BTZ solution with spin takes the form,

$\displaystyle\widehat{ds}^{2}=$
$\displaystyle\ell^{2}\left(-\left(-8\widetilde{M}+\tilde{r}^{2}+\frac{16\widetilde{J}^{2}}{\tilde{r}^{2}}\right)d\tilde{t}^{2}+\tilde{r}^{2}\left(d\tilde{\varphi}-\frac{4\widetilde{J}}{\tilde{r}^{2}}d{\tilde{t}}\right)^{2}\right.$
(A.42)
$\displaystyle\left.+\frac{d\tilde{r}^{2}}{-8\widetilde{M}+\tilde{r}^{2}+\frac{16\widetilde{J}^{2}}{\tilde{r}^{2}}}\right)\,.$
(A.43)

In parallel with the mass $\widetilde{M}=MG_{3}$, we have absorbed factors of
the gravitational coupling into the angular momentum via
$\widetilde{J}=\frac{JG_{3}}{\ell}$ with $G_{3}$ defined in
$(\ref{kappa3_defn})$. This redefinition is quite important because we have
already seen that $\ell$ is mass-dependent in (A.22). Soon we will see that it
also becomes spin-dependent when $k_{1}\neq 0$. We will use the following
ansatz for $f_{j}$ and $f_{e}$ which have the expected behavior at small $r$,

$\displaystyle f_{j}(r)=-\frac{\alpha_{j}^{2}}{r^{2}},\qquad
f_{e}(r)=1-\frac{\alpha_{e}^{2}}{r^{2}}+\frac{\tilde{\alpha}_{e}^{4}}{r^{4}}.$
(A.44)

We can then try solving the equations of motion for the dilaton, which again
gives

$\displaystyle
e^{2\Phi}=c_{2}^{2}\left(\frac{1}{\alpha_{1}^{2}+k_{1}r^{2}}\right)\,,$ (A.45)

where $c_{2}$ is an unknown constant, as in the solution (A.9) with
$\widetilde{J}=0$. We also find the algebraic constraints

$\displaystyle\alpha_{x}=\sqrt{\frac{k_{x}}{k_{1}}}\alpha_{1}\,,\qquad\tilde{\alpha}_{e}=\left(\frac{k_{1}}{k_{x}}\right)^{1/4}\alpha_{j}\,,\qquad\alpha_{5}=r_{5}\,.$
(A.46)

As in the discussion around equation $(\ref{final_string_frame_soln})$, we can
rescale $x_{5}$ to set $k_{x}=k_{1}$ which significantly simplifies the number
of parameters. In the spinning case, the relationship between $c_{2}$ and the
other parameters, imposed by the dilaton equation of motion, is modified. In
this case we find

$\displaystyle
c_{2}^{4}=\frac{r_{5}^{4}\left(\alpha_{1}^{4}+k_{1}\left(\alpha_{1}^{2}\alpha_{e}^{2}+k_{1}\alpha_{j}^{4}\right)\right)}{\hat{r}_{1}^{4}}\,.$
(A.47)

Likewise, the equations of motion determine $f_{r}(r)$ to be

$\displaystyle
f_{r}(r)=\frac{r_{5}^{2}}{r^{2}-\alpha_{e}^{2}+\frac{\alpha_{j}^{4}}{r^{2}}}\,.$
(A.48)

In determining the parameter map, we see that the relations,

$\displaystyle\frac{r^{2}}{\tilde{r}^{2}}=\frac{\alpha_{1}^{2}r_{5}^{2}}{R^{2}}\,,\qquad\ell^{2}=\frac{\alpha_{1}^{4}}{c_{2}^{4}}r_{5}^{2}\,,$
(A.49)

remain unchanged from the case without spin. With this identification, we can
determine $\alpha_{j}$ in terms of $\widetilde{J}$ appearing in the metric
(A.42). The final string-frame metric and dilaton take the form,

$\displaystyle
ds^{2}=-\frac{1-\frac{\alpha_{e}^{2}}{r^{2}}+\frac{\alpha_{j}^{4}}{r^{4}}}{k_{1}+\frac{\hat{r}_{1}^{2}}{r^{2}}}\,dt^{2}+\frac{1}{k_{1}+\frac{\hat{r}_{1}^{2}}{r^{2}}}\,\left(dx_{5}-\frac{\alpha_{j}^{2}}{r^{2}}dt\right)^{2}+\frac{r_{5}^{2}}{r^{2}-\alpha_{e}^{2}+\frac{\alpha_{j}^{4}}{r^{2}}}\,dr^{2}+r_{5}^{2}\,d\Omega_{3}^{2}+ds^{2}_{T^{4}}\,,$
$\displaystyle
e^{2\Phi}=\frac{r_{5}^{2}\sqrt{1+8k_{1}\frac{\widetilde{M}r_{5}^{2}}{R^{2}}+16\left(k_{1}\right)^{2}\frac{\widetilde{J}^{2}r_{5}^{4}}{R^{4}}}}{\hat{r}_{1}^{2}+k_{1}r^{2}}\,,\qquad\alpha_{e}^{2}=8\widetilde{M}\frac{\hat{r}_{1}^{2}r_{5}^{2}}{R^{2}}\,,\qquad\alpha_{j}^{2}=4\widetilde{J}\,\frac{\hat{r}_{1}^{2}r_{5}^{2}}{R^{2}}.$
(A.50)

The form of the dilaton solution is eerily reminiscent of the solution of the
${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed
energy flow equation $(\ref{energyformula})$.

###  A.2 The asymptotically flat case where $k_{5}>0$

We will now attempt to recouple the asymptotic region by taking $k_{5}>0$ for
the general class of metrics with mass and spin. This means that asymptotic
theory will be $6$-dimensional rather than $3$-dimensional because the
$3$-sphere will decompactify as $r\rightarrow\infty$, while the $T^{4}$
remains compact.

#### Interpolating Ansatz

To simplify our analysis, we take the following ansatz for the ten-dimensional
metric which is a slight generalization of the prior cases,

$\displaystyle\begin{split}&f_{1}(r)=k_{1}+\frac{\alpha_{1}^{2}}{r^{2}},\qquad
f_{e}(r)=1-\frac{\alpha_{e}^{2}}{r^{2}}+\frac{\tilde{\alpha}_{e}^{4}}{r^{4}},\qquad
f_{5}(r)=k_{5}+\frac{\alpha_{5}^{2}}{r^{2}},\\\
&f_{x}(r)=k_{x}+{\alpha_{x}^{2}\over r^{2}}\,,\qquad
f_{j}(r)=-\frac{\alpha_{j}^{2}}{r^{2}}\,.\end{split}$ (A.51)

We also demand that $k_{1}>0,k_{x}>0$ and $k_{5}>0$ for a sensible metric at
large $r$. We can repeat the steps of the preceeding analysis and see what
constraints emerge from trying to solve the spacetime equations of motion.
Firstly, there are algebraic constraints generalizing what we found in the
$k_{5}=0$ case:

$\displaystyle\tilde{\alpha}_{e}=\left(\frac{k_{1}}{k_{x}}\right)^{1/4}\alpha_{j}\,,\qquad\alpha_{x}=\sqrt{\frac{k_{x}}{k_{1}}}\alpha_{1}\,,\qquad\alpha_{e}=\frac{\sqrt{k_{x}(r_{5}^{4}-\alpha_{5}^{4})-k_{1}k_{5}^{2}\alpha_{j}^{4}}}{\alpha_{5}\sqrt{k_{5}k_{x}}}\,.$
(A.52)

Once again it is natural to rescale the $x_{5}$ circle to set $k_{x}=k_{1}$
and replace both $\tilde{\alpha}_{e}$ by $\alpha_{j}$ and $\alpha_{x}$ by
$\alpha_{1}$. With these replacements the solution for $f_{r}(r)$ is given by,

$\displaystyle
f_{r}(r)=\frac{k_{5}\alpha_{5}^{2}r^{2}(k_{5}r^{2}+\alpha_{5}^{2})}{r^{2}\left(\alpha_{5}^{4}+k_{5}\alpha_{5}^{2}r^{2}-r_{5}^{4}\right)+k_{5}\alpha_{j}^{4}\left(k_{5}r^{2}+\alpha_{5}^{2}\right)}\,,$
(A.53)

while the dilaton takes the form:

$\displaystyle
e^{2\Phi}=\frac{k_{5}r^{2}+\alpha_{5}^{2}}{\hat{r}_{1}^{2}\left(k_{1}r^{2}+\alpha_{1}^{2}\right)}\cdot\sqrt{\alpha_{1}^{4}+k_{1}^{2}\alpha_{j}^{4}-\frac{k_{1}\alpha_{1}^{2}\left(\alpha_{5}^{4}+k_{5}\alpha_{j}^{4}-r_{5}^{4}\right)}{k_{5}\alpha_{5}^{2}}}\,.$
(A.54)

It is more convenient to invert the relation between $\alpha_{5}$ and
$\alpha_{e}$,

$\displaystyle\alpha_{5}^{2}=\frac{1}{2}\left(\sqrt{k_{5}^{2}\alpha_{e}^{4}+4\left(r_{5}^{4}-k_{5}^{2}\alpha_{j}^{4}\right)}-k_{5}\alpha_{e}^{2}\right)$
(A.55)

As a check, notice that $\alpha_{5}\rightarrow r_{5}$ when we take
$k_{5}\rightarrow 0$ which is the solution we found earlier in
$(\ref{solveax})$. In this more general case, $\alpha_{5}$ is mass-dependent.
In a small $k_{5}$ expansion, we see that

$\displaystyle\alpha_{5}^{2}=r_{5}^{2}-\frac{k_{5}\alpha_{e}^{2}}{2}+O(k_{5}^{2})\,.$
(A.56)

Substituting (A.55) into (A.54) gives the following nicer form for the
dilaton:

$\displaystyle
e^{2\Phi}=\frac{k_{5}r^{2}+\alpha_{5}^{2}}{k_{1}r^{2}+\alpha_{1}^{2}}\cdot\sqrt{\frac{\alpha_{1}^{4}+k_{1}\alpha_{1}^{2}\alpha_{e}^{2}+k_{1}^{2}\alpha_{j}^{4}}{\hat{r}_{1}^{4}}}\,.$
(A.57)

Let us count parameters versus expectations. First $\alpha_{e}^{2}$ and
$\alpha_{j}^{2}$ are determined by the mass and spin of the deep interior BTZ
black hole via $(\ref{final_string_frame_soln_J})$. Together with $k_{5}$,
these parameters determine $\alpha_{5}^{2}$ using $(\ref{solalpha5})$. That
leaves $\alpha_{1},k_{1}$ and $k_{5}$ to be identified with physical
parameters of the asymptotic solution. Now the existence of a non-zero $k_{5}$
is the statement that we have not taken the partial decoupling limit.
Therefore it should be possible to identify $k_{5}$ with the asymptotic value
of the string coupling, called $g_{s}$ in $(\ref{kluson_soln})$.

To determine the precise identifications, we can start by examining the
asymptotic value of the dilaton. This cannot be mass or spin-dependent because
it is part of the data defining the quantum gravity Hilbert space. Unlike the
prior discussion, we now need to make a conformal transformation in a
$6$-dimensional gravity theory rather than a $3$-dimensional theory. The
Einstein frame metric is now given by,

$\displaystyle ds^{2}_{\rm Einstein}=e^{-(\Phi-\Phi_{0})}ds^{2}_{\rm
string}\,,$ (A.58)

where $\Phi_{0}$ is the now finite $r\rightarrow\infty$ value of the dilaton.
We will require that $\Phi_{0}$ be independent of mass and spin since it
determines the gravitational constant. To determine how to implement this
condition, it is useful to examine the nicer form of the dilaton appearing in
$(\ref{nicerdilsolution})$. It is not possible to make the square root
independent of mass and spin by modifying $\alpha_{1}$ in a way proportional
to $k_{5}$ so we expect that the relation

$\displaystyle\alpha_{1}=\hat{r}_{1}\,,$ (A.59)

is unchanged. This means we must absorb the mass and spin-dependence in
$k_{5}$. So we define

$\displaystyle\hat{k}_{5}=k_{5}\cdot\gamma,\qquad\gamma=\sqrt{1+\frac{k_{1}\alpha_{e}^{2}}{\alpha_{1}^{2}}+\frac{k_{1}^{2}\alpha_{j}^{4}}{\alpha_{1}^{4}}}\,.$
(A.60)

In terms of $\hat{k}_{5}$, the dilaton now takes the form

$\displaystyle
e^{2\Phi}=\frac{\hat{k}_{5}r^{2}+\gamma\alpha_{5}^{2}}{k_{1}r^{2}+\alpha_{1}^{2}}\,.$
(A.61)

As $r\rightarrow\infty$ the Einstein frame metric now contains the terms,

$\displaystyle e^{-(\Phi-\Phi_{0})}ds^{2}_{\rm
string}=\frac{\hat{k}_{5}}{\gamma}\left(dr^{2}+r^{2}d\Omega_{3}^{2}\right)+\ldots\,.$
(A.62)

The natural radial coordinate at large distances is now

$\displaystyle\rho^{2}=\frac{\hat{k}_{5}}{\gamma}r^{2}\,,$ (A.63)

and the dilaton takes the form

$\displaystyle e^{2\Phi}$
$\displaystyle=\frac{\hat{k}_{5}}{k_{1}}+\frac{k_{1}\alpha_{5}^{2}\gamma-\hat{k}_{5}\alpha_{1}^{2}}{k_{1}^{2}r^{2}}+O\left(\frac{1}{r^{4}}\right)\,,$
$\displaystyle=\frac{\hat{k}_{5}}{k_{1}}+\frac{\hat{k}_{5}}{\gamma\rho^{2}}\frac{k_{1}\alpha_{5}^{2}\gamma-\hat{k}_{5}\alpha_{1}^{2}}{k_{1}^{2}}+O\left(\frac{1}{\rho^{4}}\right)\,.$
(A.64)

The mass-dependence is now encoded in the subleading terms of the $\rho$
expansion. Now we can identify $g_{s}^{2}=\frac{\hat{k}_{5}}{k_{1}}$, or
equivalently $\hat{k}_{5}=k_{1}g_{s}^{2}$. The periodicity of the $x_{5}$
circle is $\frac{2\pi R}{\sqrt{k_{1}}}$ at $r=\infty$.

The final string-frame metric and dilaton take the form,

$\displaystyle
ds^{2}=-\frac{1-\frac{\alpha_{e}^{2}}{r^{2}}+\frac{\alpha_{j}^{4}}{r^{4}}}{k_{1}+\frac{\hat{r}_{1}^{2}}{r^{2}}}\,dt^{2}+\frac{1}{k_{1}+\frac{\hat{r}_{1}^{2}}{r^{2}}}\,\left(dx_{5}-\frac{\alpha_{j}^{2}}{r^{2}}dt\right)^{2}$
$\displaystyle\quad+\frac{k_{5}r^{2}-\frac{k_{5}\alpha_{e}^{2}}{2}+\sqrt{r_{5}^{4}-k_{5}^{2}\alpha_{j}^{4}+\frac{1}{4}k_{5}^{2}\alpha_{e}^{4}}}{r^{4}-r^{2}\alpha_{e}^{2}+\alpha_{j}^{4}}\,r^{2}\,dr^{2}+r^{2}\left(k_{5}+\frac{\alpha_{5}^{2}}{r^{2}}\right)\,d\Omega_{3}^{2}+ds^{2}_{T^{4}}\,,$
$\displaystyle
e^{2\Phi}=\frac{g_{s}^{2}+\frac{\gamma\alpha_{5}^{2}}{k_{1}r^{2}}}{1+\frac{\hat{r}_{1}^{2}}{k_{1}r^{2}}}\,,\qquad\alpha_{e}^{2}=8\widetilde{M}\frac{\hat{r}_{1}^{2}r_{5}^{2}}{R^{2}}\,,\qquad\alpha_{j}^{2}=4\widetilde{J}\,\frac{\hat{r}_{1}^{2}r_{5}^{2}}{R^{2}}\,,\qquad
k_{5}=\frac{k_{1}g_{s}^{2}}{\gamma}\,,$
$\displaystyle\alpha_{5}^{2}=\frac{1}{2}\left(\sqrt{k_{5}^{2}\alpha_{e}^{4}+4\left(r_{5}^{4}-k_{5}^{2}\alpha_{j}^{4}\right)}-k_{5}\alpha_{e}^{2}\right)\,,\qquad\gamma=\sqrt{1+\frac{k_{1}\alpha_{e}^{2}}{\hat{r}_{1}^{2}}+\frac{k_{1}^{2}\alpha_{j}^{4}}{\hat{r}_{1}^{4}}}\,.$
(A.65)

This final form for the metric depends on $(g_{s},k_{1},\hat{r}_{1},r_{5})$
along with the mass and spin $(\widetilde{M},\widetilde{J})$ of the interior
BTZ black hole.

## References

  * [1] A. B. Zamolodchikov, “Expectation value of composite field T anti-T in two-dimensional quantum field theory,” hep-th/0401146.
  * [2] F. A. Smirnov and A. B. Zamolodchikov, “On space of integrable quantum field theories,” Nucl. Phys. B915 (2017) 363–383, 1608.05499.
  * [3] A. Cavaglià, S. Negro, I. M. Szécsényi, and R. Tateo, “$T\bar{T}$-deformed 2D Quantum Field Theories,” JHEP 10 (2016) 112, 1608.05534.
  * [4] T. D. Brennan, C. Ferko, E. Martinec, and S. Sethi, “Defining the $T\overline{T}$ Deformation on $\mathrm{AdS}_{2}$,” 2005.00431.
  * [5] C. Ferko and S. Sethi, “Sequential Flows by Irrelevant Operators,” 2206.04787.
  * [6] A. Giveon, N. Itzhaki, and D. Kutasov, “$\mathrm{T}\overline{\mathrm{T}}$ and LST,” JHEP 07 (2017) 122, 1701.05576.
  * [7] A. Giveon, N. Itzhaki, and D. Kutasov, “A solvable irrelevant deformation of AdS3/CFT2,” JHEP 12 (2017) 155, 1707.05800.
  * [8] M. Asrat, A. Giveon, N. Itzhaki, and D. Kutasov, “Holography Beyond AdS,” Nucl. Phys. B 932 (2018) 241–253, 1711.02690.
  * [9] S. Forste, “A Truly marginal deformation of SL(2, R) in a null direction,” Phys. Lett. B 338 (1994) 36–39, hep-th/9407198.
  * [10] D. Israel, C. Kounnas, and M. P. Petropoulos, “Superstrings on NS5 backgrounds, deformed AdS(3) and holography,” JHEP 10 (2003) 028, hep-th/0306053.
  * [11] L. Eberhardt, M. R. Gaberdiel, and R. Gopakumar, “The Worldsheet Dual of the Symmetric Product CFT,” JHEP 04 (2019) 103, 1812.01007.
  * [12] O. Aharony, “A Brief review of ’little string theories’,” Class. Quant. Grav. 17 (2000) 929–938, hep-th/9911147.
  * [13] D. Kutasov, “Introduction to little string theory,” ICTP Lect. Notes Ser. 7 (2002) 165–209.
  * [14] L. Apolo, S. Detournay, and W. Song, “TsT, $T\bar{T}$ and black strings,” JHEP 06 (2020) 109, 1911.12359.
  * [15] L. Apolo and W. Song, “TsT, black holes, and $T\bar{T}+J\bar{T}+T\bar{J}$,” 2111.02243.
  * [16] A. A. Tseytlin, “Extreme dyonic black holes in string theory,” Mod. Phys. Lett. A 11 (1996) 689–714, hep-th/9601177.
  * [17] J. Kluson, “(p, q)-five brane and (p, q)-string solutions, their bound state and its near horizon limit,” JHEP 06 (2016) 002, 1603.05196.
  * [18] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2 (1998) 231–252, hep-th/9711200.
  * [19] T. Araujo, E. O. Colgáin, Y. Sakatani, M. M. Sheikh-Jabbari, and H. Yavartanoo, “Holographic integration of $T\bar{T}$ \& $J\bar{T}$ via $O(d,d)$,” JHEP 03 (2019) 168, 1811.03050.
  * [20] V. Iyer and R. M. Wald, “Some properties of Noether charge and a proposal for dynamical black hole entropy,” Phys. Rev. D 50 (1994) 846–864, gr-qc/9403028.
  * [21] G. Compère and A. Fiorucci, “Advanced Lectures on General Relativity,” 1801.07064.
  * [22] O. J. C. Dias, G. S. Hartnett, and J. E. Santos, “Covariant Noether charges for type IIB and 11-dimensional supergravities,” Class. Quant. Grav. 38 (2021), no. 1, 015003, 1912.01030.
  * [23] G. Compere, “Note on the First Law with p-form potentials,” Phys. Rev. D 75 (2007) 124020, hep-th/0703004.
  * [24] R. Ruzziconi and C. Zwikel, “Conservation and Integrability in Lower-Dimensional Gravity,” JHEP 04 (2021) 034, 2012.03961.
  * [25] A. Fiorucci, Leaky covariant phase spaces: Theory and application to $\Lambda$-BMS symmetry.  PhD thesis, Brussels U., Intl. Solvay Inst., Brussels, 2021.  2112.07666.
  * [26] R. Rohm, “Spontaneous supersymmetry breaking in supersymmetric string theories,” Nuclear Physics B 237 (1984), no. 3, 553–572.
  * [27] E. J. Martinec, D. Robbins, and S. Sethi, “Non-Supersymmetric String Theory,” JHEP 10 (2011) 078, 0904.3498.
  * [28] T. Maxfield and S. Sethi, “Domain Walls, Triples and Acceleration,” JHEP 08 (2014) 066, 1404.2564.