# Infinite $\mathrm{T\bar{T}}$-like symmetries of compactified LST

Silvia Georgescu§,† and Monica Guica§

§ _Université Paris-Saclay, CNRS, CEA,_ _Institut de Physique Théorique, 91191
Gif-sur-Yvette, France_
† _CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120
Palaiseau, France_

###### Abstract

We show that the three-dimensional asymptotically linear dilaton background
that arises in the near-horizon decoupling region of NS5-branes compactified
on $T^{4}$ admits boundary conditions that lead to an infinite set of
symmetries. The associated conserved charges, which implement field-dependent
coordinate transformations, are found to be identical to the corresponding
generators in a symmetric product orbifold of $T\bar{T}$ \- deformed CFTs.
Their algebra is a non-linear modification of the $\mathrm{Virasoro\times
Virasoro}$ algebra, which precisely coincides with the algebra of the
“unrescaled” symmetry generators in $T\bar{T}$-deformed CFTs. This further
strengthens a previously proposed link between the single-trace $T\bar{T}$
deformation and compactified little string theory.

###### Contents

  1. 1 Introduction
  2. 2 Setup
    1. 2.1 The backgrounds
    2. 2.2 Consistent truncation to three dimensions
    3. 2.3 Relationship to $T\bar{T}$ \- deformed CFTs
  3. 3 Asymptotic analysis of the black hole backgrounds
    1. 3.1 The linearized solutions
    2. 3.2 Choice of boundary conditions
    3. 3.3 Conserved charges
  4. 4 The charge algebra
    1. 4.1 Perturbative calculation setup
    2. 4.2 Allowed diffeomorphisms upon the perturbed backgrounds
    3. 4.3 Algebra of the conserved charges and match to $T\bar{T}$
  5. 5 Discussion
  6. A Conserved charges in double-trace $T\bar{T}$-deformed CFTs
    1. A.1 Asymptotic symmetries of the dual gravitational background
    2. A.2 Conserved charges from the covariant phase space formalism
    3. A.3 The charge algebra
  7. B Perturbative analysis around constant backgrounds
    1. B.1 Perturbative expansion of the symmetry generators
    2. B.2 The perturbative charge algebra
    3. B.3 Comments on the representation theorem

##  1 Introduction

One of the most important open problems in quantum gravity is to generalize
the highly successful AdS/CFT correspondence [1] to other spacetimes, such as
the more realistic asymptotically flat ones. While a fascinating link [2, 3]
has been recently uncovered between the rich asymptotic structure of Minkowski
spacetimes [4, 5] and properties of flat space scattering amplitudes [6] \-
culminating in the celestial holography program [7, 8, 9, 10] \- the
underlying quantum-field-theoretical structure of the holographic dual to flat
space has yet to be understood. Basic considerations based on e.g., the
behaviour of black hole and entanglement entropy [11], as well as the
structure of correlation functions [12], suggest the corresponding QFT is non-
local.

Part of the difficulty lies in the lack of tractable string-theoretical
examples of flat holography. In this article, we will concentrate on the
asymptotically linear dilaton spacetime in string theory [13], where the
metric is asymptotically flat, but the dilaton has a non-trivial asymptotic
profile. This spacetime is obtained from a decoupling limit of the geometry
created by NS5-branes and can be viewed as intermediary between the well-
understood anti-de Sitter case and the challenging asymptotically flat one;
for example, the growth of the black hole entropy with energy is Hagedorn,
which is intermediary between the sub-Hagedorn growth in AdS and the super-
Hagedorn one in flat space. The conjectured holographic dual to string theory
in this background is known as little string theory (LST) [14, 15, 16, 17]: a
non-local, non-gravitational theory containing strings, which arises in the
low-energy decoupling limit of the NS5-branes.

The behaviour of observables in LST has traditionally been inferrred from its
holographically dual spacetime. However, more recently [18] put forth a very
interesting alternate QFT definition of little string theory, at least when
compactified on $T^{4}$, in terms of a solvable irrelevant deformation of a
two-dimensional CFT known as the $T\bar{T}$ deformation.

The $T\bar{T}$ deformation [19, 20] is a universal deformation of two-
dimensional QFTs by an operator that is bilinear in the stress tensor
components. Remarkably, this deformation appears well-defined - despite its
irrelevant nature - and leads to a theory that is UV complete [21], albeit
non-local at the scale set by the dimensionful deformation parameter.
Moreover, many observables in the deformed theory, such as the finite-size
spectrum and the deformed S-matrix, are entirely determined by their
counterparts in the original QFT [19, 20, 22, 23]. For the case of a
$T\bar{T}$ \- deformed _CFT_ , the deformed spectrum is known in closed form
and can be used to determine the density of states, $e^{S}$, as a function of
the energy

$S(E)=2\pi\sqrt{\frac{cER}{3}+\frac{\mu c}{6\pi}E^{2}}$ (1.1)

where $c$ is the central charge of the undeformed CFT and $\mu$ is the
$T\bar{T}$ parameter. This expression interpolates between a Cardy regime at
low energies and Hagedorn behaviour at high energies, which is suggestive of a
connection to strings.

The original $T\bar{T}$ deformation put forth by Smirnov and Zamolodchikov is
“double-trace” from the perspective of AdS/CFT. As a result, its holographic
interpretation is to simply change the boundary conditions on the dual AdS3
metric from Dirichlet to mixed [24], without affecting the local curvature. In
order for the irrelevant deformation to locally modify the geometry of the
dual spacetime, one needs to consider instead the so-called “single-trace”
$T\bar{T}$ deformation [18], which is possible when the undeformed CFT is a
symmetric product orbifold (SPO) of some seed CFT. This deformation takes the
form

$\sum_{i=1}^{p}T_{i}\bar{T}_{i}$ (1.2)

where $p$ is the number of copies in the SPO, and produces a symmetric product
orbifold of $T\bar{T}$\- deformed CFTs, many of whose properties are
universally determined by the original seed [25].

The connection between single-trace $T\bar{T}$ and LST arises from considering
the NS5-F1 system instead of just NS5 [26]. In this case, the decoupled
background geometry interpolates between AdS3 in the IR and a three-
dimensional linear dilaton spacetime in the UV. Worldsheet string theory is
well understood in this entire background [27]; in particular, the IR AdS3
region is described by an $SL(2,\mathbb{R})$ WZW model [28]. Holographically,
it is expected [29] that at least the long string sector of this theory should
be described by a symmetric product orbifold with respect to the number of F1
strings, $p$, of a seed CFT of central charge $c=6k$, where $k$ is the number
of NS5 branes. In [18], it has been argued that turning on the single-trace
$T\bar{T}$ deformation (1.2) in this theory corresponds precisely to the
deformation of the dual background from AdS3 to asymptotically linear dilaton.
The $T\bar{T}$ coupling is proportional to the inverse string tension,
$\alpha^{\prime}$.

If correct, this proposal provides an alternate QFT definition of compactified
LST in terms of an essentially solvable theory. This proposal has passed
several non-trivial checks, such as the fact that black hole entropy precisely
follows the $T\bar{T}$ relation (1.1) between entropy and energy [18, 30], and
that the spectrum of long strings in the deformed massless BTZ background
precisely matches the single-trace $T\bar{T}$ \- deformed spectrum [31]. It
has also made some interesting predictions [32, 33] for the behaviour of
entanglement entropy in $T\bar{T}$ \- deformed CFTs. Its ultimate status is,
however, somewhat unclear, due to the fact that the full CFT dual to the IR
AdS3 is _not_ a symmetric product orbifold (see [34] for a recently proposed
dual). More work is thus needed before the full picture emerges.

In this article, we further strengthen the link between the single-trace
$T\bar{T}$ deformation and compactified LST by showing that the asymptotically
linear dilaton background dual to the latter possesses an infinite set of
symmetries, whose form and algebra _precisely match_ those of a single-trace
$T\bar{T}$ \- deformed CFT [25]. These symmetries are a straightforward
generalization of the infinite symmetries of standard $T\bar{T}$ \- deformed
CFTs, which have been first predicted by the holographic analysis of [24],
then studied at classical level in [35], proven to exist at full quantum level
in [36], and are further explored in [37].

More specifically, [24] showed that the asymptotic symmetry algebra of the
spacetime dual to $T\bar{T}$-deformed CFTs - namely, AdS3 with mixed boundary
conditions - is Virasoro $\times$ Virasoro with the same central extension as
that of the undeformed CFT. The asymptotic symmetry generators are
parametrised by two arbitrary functions of certain “field-dependent
coordinates”, $u,v$, whose radii depend on the left- and right-moving energies
of the background as $R_{u,v}=R+\mu H_{R,L}/\pi$, where $R$ is the radius of
the circle on which the theory is defined. The Virasoro generators
$L_{m},\bar{L}_{m}$ correspond to the natural Fourier basis of functions
_multiplied_ by the field-dependent radii. Writing $L_{m}=R_{u}Q_{m}$,
$\bar{L}_{m}=R_{v}\bar{Q}_{m}$, the algebra of the “unrescaled” generators
$Q_{m},\bar{Q}_{m}$ is a non-linear modification of the Virasoro $\times$
Virasoro algebra

$i\\{Q_{m},Q_{n}\\}=\frac{1}{R_{u}}(m-n)Q_{m+n}+\frac{\mu^{2}H_{R}}{\pi^{2}RR_{H}R_{u}}(m-n)Q_{m}Q_{n}+\frac{c}{12}\,\frac{m^{3}}{R_{u}^{2}}\delta_{m+n}$
$i\\{Q_{m},\bar{Q}_{n}\\}=-\frac{\mu(m-n)}{\pi RR_{H}}Q_{m}\bar{Q}_{n}$ (1.3)

where $R_{H}=R+\mu H/\pi$; a similar bracket holds for
$\\{\bar{Q}_{m},\bar{Q}_{n}\\}$. Even though this algebra is more complicated
and likely receives quantum corrections to all orders in $\hbar$, the
“unrescaled” basis of generators: i) is very natural to consider, being just
the Fourier basis; in particular, the left/right-moving energies simply
correspond to $Q_{0},\bar{Q}_{0}$ and ii) may be preferred for defining
pseudo-local observables in the theory, in analogy to the case of $J^{1}\wedge
J^{2}$ and $J\bar{T}$ \- deformed CFTs111 The Virasoro generators of
$T\bar{T}$ \- deformed CFTs are obtained at full quantum level by transporting
the ones of the seed CFT along the $T\bar{T}$ flow [36]. The same procedure
yields flowed generators in $J^{1}\wedge J^{2}$ and $J\bar{T}$ \- deformed
CFTs, whose algebra is Virasoro by construction, but which are explicitly
_different_ from the Virasoro generators of conformal symmetries in these
theories. The interplay between physical and flowed generators is essential
for defining analogues of primary operators in the non-local
$J\bar{T}$-deformed CFTs and for computing their correlation functions [38]. .
It is not hard to argue [25] that for single-trace $T\bar{T}$ \- deformed
CFTs, the algebra of the corresponding “unrescaled” generators is still given
by the above, but with the replacement $\mu\rightarrow\mu/p$.

The main technical result of this article is to show that the asymptotic
symmetry algebra of asymptotically linear dilaton spacetimes is precisely
(1.3) with $\mu\rightarrow\pi\alpha^{\prime}/p$. To capture the field-
dependence of the asymptotic symmetry generators, we perform the asymptotic
analysis around non-trivial black hole solutions in these spacetimes. Due to
the strong background dependence of the asymptotic diffeomorphisms, we can
only compute the charge algebra _perturbatively_ around the black hole
backgrounds, where the perturbation corresponds to adding boundary gravitons.
The terms we compute are however sufficient to see the entire structure of
(1.3) emerge.

This article is organised as follows. In section 2, we review the black hole
backgrounds we study, present a consistent truncation to three dimensions that
captures them, and recall the relationship between their thermodynamics and
that of single-trace $T\bar{T}$. In section 3, we classify the general
linearized perturbations of these backgrounds and use their symplectic product
to motivate a set of boundary conditions that lead to an infinite set of
asymptotic symmetries. We also show that the associated conserved charges take
an identical form to those in $T\bar{T}$. In section 4, we present the
perturbative computation of the charge algebra around the black hole
backgrounds. Finally, in appendix A we revisit the calculation of the
asymptotic symmetry algebra for the spacetimes dual to double-trace $T\bar{T}$
\- deformed CFTs, explaining how to reproduce both the algebra (1.3) and the
originally found Virasoro using covariant phase space methods.

##  2 Setup

In this section, we review the construction of the asymptotically linear
dilaton back hole backgrounds from a decoupling limit of string theory, and
present a consistent truncation to three dimensions that captures their
essential properties. We then review the link between the thermodynamics of
these black holes and that of $T\bar{T}$ \- deformed CFTs.

###  2.1 The backgrounds

We start by considering the non-extremal NS5 - F1 string frame solution of
type IIB string theory, with $n$ units of momentum along the common $S^{1}$
[39, 30]

$ds^{2}=\frac{1}{f_{1}}\left[d\sigma^{2}-dt^{2}+\frac{r_{0}^{2}}{r^{2}}(\cosh\alpha_{n}dt+\sinh\alpha_{n}d\sigma)^{2}\right]+f_{5}\left(\frac{dr^{2}}{f}+r^{2}d\Omega_{3}^{2}\right)+\sum_{i=1}^{4}dx_{i}^{2}$
$H=\frac{g_{s}^{2}\alpha^{\prime}p}{v}\,d\sigma\wedge dt\wedge
d\left(\frac{1}{f_{1}r^{2}}\right)+2\alpha^{\prime}k\,\omega_{S^{3}}\;,\;\;\;\;\;e^{2\phi}=g_{s}^{2}\,\frac{f_{5}}{f_{1}}$
(2.1)

where $d\Omega_{3}^{2}$ and $\omega_{S^{3}}$ are the metric and, respectively,
the volume form of the unit $S^{3}$, and the various harmonic functions are
given by

$f=1-\frac{r_{0}^{2}}{r^{2}}\;,\;\;\;\;\;f_{1,5,n}=1+\frac{r_{0}^{2}\sinh^{2}\alpha_{1,5,n}}{r^{2}}$
(2.2)

with

$\sinh
2\alpha_{1}=\frac{2g_{s}^{2}\alpha^{\prime}p}{vr_{0}^{2}}\;,\;\;\;\;\;\sinh
2\alpha_{5}=\frac{2\alpha^{\prime}k}{r_{0}^{2}}\;,\;\;\;\;\;\sinh
2\alpha_{n}=\frac{2g_{s}^{2}\alpha^{\prime 2}n}{R^{2}vr_{0}^{2}}$ (2.3)

Here, $p$ and $k$ are the number of F1 strings and, respectively, NS5 branes
supporting the solution, $g_{s}$ is the ten-dimensional string coupling, $R$
is the radius of the $S^{1}$ parametrized by $\sigma$ and $v$ is the volume of
the $T^{4}$ in string units, $V_{T^{4}}=(2\pi)^{4}\alpha^{\prime 2}v$. This
way of writing the metric makes manifest the boost that generated the momentum
of the solution.

We now take the standard NS5-brane decoupling limit, where the asymptotic
string coupling $g_{s}\rightarrow 0$, with

$\hat{r}_{0}\equiv\frac{r_{0}}{g_{s}}\;,\;\;\;\;\;\hat{r}\equiv\frac{r}{g_{s}}$
(2.4)

held fixed. We also introduce the null coordinates $U,V=\sigma\pm t$, which
are identified mod $2\pi R$. In this limit, the six dimensional string frame
decoupled geometry becomes

$ds^{2}=\frac{1}{f_{1}}\left(dUdV+\frac{\hat{r}_{0}^{2}}{4\hat{r}^{2}}(e^{\alpha_{n}}dU-e^{-\alpha_{n}}dV)^{2}\right)+\alpha^{\prime}k\left(\frac{d\hat{r}^{2}}{\hat{r}^{2}-\hat{r}_{0}^{2}}+d\Omega_{3}^{2}\right)\;,\;\;\;\;e^{2\phi}=\frac{k\alpha^{\prime}}{f_{1}\hat{r}^{2}}$
(2.5)

and

$H=\frac{2p\,e^{2\phi}}{v\sqrt{\alpha^{\prime}k^{3}}}\,\omega_{3}+2k\alpha^{\prime}\omega_{S^{3}}$
(2.6)

where $\omega_{3}$ is the volume form on the non-compact part of the space-
time. This background interpolates between AdS3 as $\hat{r}\rightarrow 0$ and
an asymptotically flat spacetime with a linear dilaton as
$\hat{r}\rightarrow\infty$. The AdS3 geometry in the IR can also be obtained
from the standard $\alpha^{\prime}\rightarrow 0$ decoupling limit. As such, it
is useful to work in coordinates in which this limit is smooth, by letting

$\hat{r}=\alpha^{\prime}\rho\;,\;\;\;\;\;\;\hat{r}_{0}=\alpha^{\prime}\rho_{0}$
(2.7)

Since the metric only depends on the ratio
$\hat{r}_{0}/\hat{r}=\rho_{0}/\rho$, it will take exactly the same form in
terms of the new variables, and so will $\omega_{3}$; only $e^{2\phi}$ will
pick up a factor of $(\alpha^{\prime})^{-2}$, as will the first term in $H$.
Finally, it is useful to introduce a new coordinate $r$ (different from the
previous one) and notation

$\rho=r+\frac{\rho_{0}^{2}}{4r}\;,\;\;\;\;\;\;L_{u}\equiv\frac{\rho_{0}^{2}\,e^{2\alpha_{n}}}{4}\;,\;\;\;\;L_{v}\equiv\frac{\rho_{0}^{2}\,e^{-2\alpha_{n}}}{4}$
(2.8)

in terms of which the decoupled metric and dilaton become

$ds^{2}=\frac{r^{2}}{r^{4}+\beta
r^{2}/(\alpha^{\prime}v)+L_{u}L_{v}}\left(r^{2}dUdV+L_{u}dU^{2}+L_{v}dV^{2}+\frac{L_{u}L_{v}}{r^{2}}dUdV\right)+\alpha^{\prime}k\left(\frac{dr^{2}}{r^{2}}+d\Omega_{3}^{2}\right)$
$e^{2\phi}=\frac{kr^{2}}{\alpha^{\prime}(r^{4}+\beta
r^{2}/(v\alpha^{\prime})+L_{u}L_{v})}$ (2.9)

where the quantity $\beta$ is defined as

$\frac{\beta}{\alpha^{\prime}v}\equiv\rho_{0}^{2}\left(\sinh^{2}\alpha_{1}+\frac{1}{2}\right)=\frac{\rho_{0}^{2}}{2}\sqrt{\left(\frac{2p}{v\rho_{0}^{2}\alpha^{\prime}}\right)^{2}+1}=\frac{1}{\alpha^{\prime}v}\sqrt{p^{2}+4\alpha^{\prime
2}v^{2}L_{u}L_{v}}$ (2.10)

The reason for pulling out a factor of $\alpha^{\prime}v$ is that we would
like $\beta$ to have a finite limit as $\alpha^{\prime}\rightarrow 0$. In the
second step, we used the definition of $\alpha_{1}$, $\sinh
2\alpha_{1}=\frac{2p}{v\rho_{0}^{2}\alpha^{\prime}}$, to solve for
$\alpha_{1}$ in terms of $\rho_{0}$ and the relation
$\rho_{0}^{4}=16L_{u}L_{v}$ to write $\beta$ in terms of these variables.

The energy and momentum of these solutions are given by [30]

$E=\frac{Rv\rho_{0}^{2}}{2}(e^{-2\alpha_{1}}+\cosh
2\alpha_{n})=Rv\left(L_{u}+L_{v}+\sqrt{\left(\frac{p}{v\alpha^{\prime}}\right)^{2}+4L_{u}L_{v}}-\frac{p}{v\alpha^{\prime}}\right)$

$P=\frac{n}{R}=Rv(L_{u}-L_{v})$ (2.11)

where we measure the energy with respect to deformed massless BTZ. One last
useful manipulation is to absorb the factor of $v$ above into a redefinition
of the radial coordinate and of $L_{u},L_{v}$. More precisely, we let

$r^{2}\rightarrow\frac{r^{\prime
2}}{v}\;,\;\;\;\;\;\;L_{u}\rightarrow\frac{L_{u}^{\prime}}{v}\;,\;\;\;\;\;L_{v}\rightarrow\frac{L_{v}^{\prime}}{v}$
(2.12)

and then drop the primes. This will simply amount to dropping the factors of
$v$ from the metric and from the formulae for the energy and momentum. To
remove this factor also from the dilaton, one can simply define

$e^{2\phi^{\prime}}=\frac{1}{v}\,e^{2\phi}$ (2.13)

The decoupling limit $\alpha^{\prime}\rightarrow 0$ can now be
straightforwardly taken, taking into account the fact that the metric scales
with an overall factor of $\alpha^{\prime}$, as usual. The dilaton reduces to
a constant, $k/p$, which can be made as small as desired by taking the number
of F1 strings to be large. To further study these solutions, it is convenient
to introduce a consistent truncation to three dimensions that captures all of
their essential properties.

###  2.2 Consistent truncation to three dimensions

The six-dimensional string frame action is

$S=\frac{1}{2\kappa_{6}^{2}}\int
d^{6}x\,\sqrt{g}\,e^{-2\phi}\left(R+4\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{12}H^{2}\right)\;,\;\;\;\;\;\frac{1}{2\kappa_{6}^{2}}=\frac{(2\pi)^{4}v\alpha^{\prime
2}}{2\kappa_{10}^{2}}=\frac{v}{(2\pi)^{3}\alpha^{\prime 2}}$ (2.14)

and the associated equations of motion read

$R_{MN}+2\nabla_{M}\nabla_{N}\phi-\frac{1}{4}H_{MAB}H_{N}{}^{AB}=0\;,\;\;\;\;\;\;\Box\phi-2(\partial\phi)^{2}=-\frac{1}{12}H^{2}\;,\;\;\;\;\;d(e^{-2\phi}\star
H)=0$ (2.15)

We would like to perform a consistent truncation to three dimensions on
$S^{3}$ using the following simple Ansatz, inspired by the family of solutions
we described

$ds^{2}=ds_{3}^{2}+\ell^{2}ds_{S_{3}}^{2}\;,\;\;\;\;\;\;H=2\ell^{2}\,\omega_{S^{3}}+b\,e^{2\phi}\,\omega_{3}$
(2.16)

where $\ell,b$ are constants and $\phi$ is assumed to be a function of only
the three-dimensional non-compact coordinates, $x^{\mu}$. Comparing with the
background $H$-field (2.6), we find

$\ell=\sqrt{\alpha^{\prime}k}\;,\;\;\;\;\;\;b=\frac{2p}{\sqrt{\alpha^{\prime}k^{3}}}=\frac{2p}{k\ell}$
(2.17)

Note there should be an additional factor of $v$ in the denominator of $b$; we
have removed it by rescaling $e^{\phi}$ as in (2.13). This rescaling also
removes the factor of $v$ from the six-dimensional Newton’s constant. However,
it does not affect the string equations of motion (2.15), since $\phi$ itself
only shifts by a constant.

Thus, $\ell$ and $b$ are fixed parameters, depending only on the number of
branes and $\alpha^{\prime}$, but not on the state-dependent properties of the
black hole solutions. Performing the truncation is trivial: the Ansatz for the
$H$-field automatically solves the last equation in (2.15), while the middle
equation reads

$\Box_{3}\phi-2(\partial\phi)^{2}=-\frac{1}{2}(a^{2}-b^{2}e^{4\phi})$ (2.18)

where we have introduced the shorthand $a\equiv 2/\ell$. The sphere components
of the Einstein equations set ${}^{(3)}R_{ab}=\frac{2}{\ell^{2}}g_{ab}$, and

${}^{(3)}R_{\mu\nu}+2\nabla_{\mu}\nabla_{\nu}\phi+\frac{1}{2}b^{2}e^{4\phi}g_{\mu\nu}=0$
(2.19)

One can easily check that the black hole solutions satisfy these equations. To
derive them from an action, we can introduce
$\tilde{g}_{\mu\nu}=e^{-4\phi}g_{\mu\nu}$, which obeys

$\tilde{R}_{\mu\nu}+\left(a^{2}e^{4\phi}-\frac{b^{2}}{2}e^{8\phi}\right)\tilde{g}_{\mu\nu}-4\partial_{\mu}\phi\partial_{\nu}\phi=0\;,\;\;\;\;\;\tilde{\Box}\phi=-\frac{1}{2}e^{4\phi}(a^{2}-b^{2}e^{4\phi})$
(2.20)

These equations can be derived from the action

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi G_{3}}\int
d^{3}x\sqrt{\tilde{g}}\left[\tilde{R}-4(\tilde{\partial}\phi)^{2}+\left(a^{2}-\frac{b^{2}}{2}e^{4\phi}\right)e^{4\phi}\right]$
(2.21) $\displaystyle=$ $\displaystyle\frac{1}{16\pi G_{3}}\int
d^{3}x\sqrt{g}\,e^{-2\phi}\left(R+4(\partial\phi)^{2}+a^{2}-\frac{b^{2}}{2}e^{4\phi}\right)$

where $16\pi G_{3}=4\pi{\sqrt{\alpha^{\prime}k^{-3}}}$. One may drop the
factors of $\sqrt{\alpha^{\prime}}$ from $a$ (or $\ell),b$ and $G_{3}$ by
rescaling the metric by $\alpha^{\prime}$, as we will do in the next section.

###  2.3 Relationship to $T\bar{T}$ \- deformed CFTs

As mentioned in the introduction, the finite-size spectrum of a $T\bar{T}$ \-
deformed CFT is entirely determined by that of the original CFT. The
relationship between the undeformed and deformed energies takes the simple
form

$E_{L}^{(0)}=E_{L}\left(1+\frac{\mu E_{R}}{\pi
R}\right)\;,\;\;\;\;\;\;E_{R}^{(0)}=E_{R}\left(1+\frac{\mu E_{L}}{\pi
R}\right)$ (2.22)

where $E_{L,R}=(E\pm P)/2$ and $E_{L,R}^{(0)}$ are their undeformed
counterparts. This relationship can be used to compute the entropy in
$T\bar{T}$ \- deformed CFTs, using the fact that the number of energy levels
does not change with the deformation. Consequently, the entropy is still given
by Cardy’s formula

$S_{Cardy}=2\pi\sqrt{\frac{c\,E_{L}^{(0)}R}{6}}+2\pi\sqrt{\frac{c\,E_{R}^{(0)}R}{6}}$
(2.23)

when written in terms of $E_{L,R}^{(0)}$. To obtain the entropy as a function
of the physical energies $E_{L,R}$, one simply needs to plug in the expression
(2.22) into the formula above, obtaining

$S_{T\bar{T}}=2\pi\sqrt{\frac{cE_{L}(R+\mu
E_{R}/\pi)}{6}}+2\pi\sqrt{\frac{cE_{R}(R+\mu E_{L}/\pi)}{6}}$ (2.24)

This analysis holds for the standard, double-trace $T\bar{T}$ deformation. The
“single-trace” $T\bar{T}$ deformation relevant to our case corresponds to a
symmetric product orbifold, $S_{p}$, of $T\bar{T}$ \- deformed CFTs, and its
(untwisted sector) entropy is obtained by equipartitioning the energy between
the $p$ copies [18]

$S_{T\bar{T}/S_{p}}=2\pi\sqrt{\frac{cE_{L}(R\,p+\mu
E_{R}/\pi)}{6}}+2\pi\sqrt{\frac{cE_{R}(R\,p+\mu E_{L}/\pi)}{6}}$ (2.25)

where $c$ is the central charge of the seed CFT.

One of the interesting observations of [18] was that the relationship between
the entropy and the energy of the asymptotically linear dilaton black holes
reviewed in the previous section takes precisely this form, where $p$ is given
by the number of F1 strings supporting the background and the central charge
of the seed CFT is $c=6k$. Concretely, the entropy of the decoupled black
holes is

$S=2\pi Rv\ell_{s}\sqrt{k}\,\rho_{0}^{2}\cosh\alpha_{1}\cosh\alpha_{n}$ (2.26)

To compare with the $T\bar{T}$ formula, we first compute the single-trace
analogue of the relation (2.22) between the undeformed and deformed
energies222If the $T\bar{T}$ interpretation of these backgrounds holds, then
the quantity $E_{L,R}^{(0)}$ should be fixed as the $T\bar{T}$ coupling,
$\alpha^{\prime}$, is turned on. Note this is different from turning on the
$\alpha_{1}$ parameter of the solution via a TsT transformation, as then
$\rho_{0}$ would be fixed instead. This can be easily seen by writing
$E_{L,R}^{(0)}$ as
$E_{L,R}^{(0)}=\frac{Rv\rho_{0}^{2}}{2}\frac{\cosh^{2}(\alpha_{1}\pm\alpha_{n})}{\sinh
2\alpha_{1}}\;,\;\;\;\;\mbox{with}\;\;\;\sinh
2\alpha_{1}=\frac{2p}{v\alpha^{\prime}\rho_{0}^{2}}$ , using
$\mu=\pi\alpha^{\prime}$

$E_{L,R}^{(0)}\equiv
E_{L,R}+\frac{\alpha^{\prime}}{R\,p}E_{L}E_{R}=\frac{Rp\cosh^{2}(\alpha_{1}\pm\alpha_{n})}{\alpha^{\prime}\sinh^{2}2\alpha_{1}}$
(2.27)

and the explicit expression (2.11) for the deformed energies. Thus, the
entropy can be written as

$S=2\pi\sqrt{pkE_{L}R+\alpha^{\prime}kE_{L}E_{R}}+2\pi\sqrt{pkE_{R}R+\alpha^{\prime}kE_{L}E_{R}}$
(2.28)

exactly matching that of a symmetric product orbifold333Note this does not
mean the dual is necessarily a symmetric product orbifold - indeed, the seed
CFT we started from is not, so neither should be its deformation.  of $p$
$T\bar{T}$ \- deformed CFTs, with $\mu=\pi\alpha^{\prime}$ and $c_{seed}=6k$.
This entropy interpolates between a Cardy regime at low energies and a
Hagedorn one at high energies. Note the number of F1 strings drops out from
the high-energy entropy, as expected from the fact that that this regime (for
which $E_{L,R}\alpha^{\prime}>>pR$) is controlled by the physics of the
NS5-branes alone.

Given this interesting connection to $T\bar{T}$, it is useful to translate the
parameters we have used to describe the black hole solutions to a notation
that may be more natural from the $T\bar{T}$ perspective. This
parametrization, which has been extensively used and explored in [24, 35],
relies on the observation that the stress tensor of a $T\bar{T}$ \- deformed
CFT only has two independent components off-shell, which can be conveniently
written in terms of two functions of the coordinates, $\mathcal{L}$ and
$\bar{\mathcal{L}}$, as in (A.4). For the black hole backgrounds, these
functions are simply constant. The energy and momentum of the solutions are
given in terms of them by [24]

$E=\pi
R\,p\,\frac{\mathcal{L}+\bar{\mathcal{L}}+2\mu\mathcal{L}\bar{\mathcal{L}}}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}\;,\;\;\;\;\;P=\pi
R\,p\,\frac{\mathcal{L}-\bar{\mathcal{L}}}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}$
(2.29)

where, as explained in the appendix, $\mathcal{L},\bar{\mathcal{L}}$ and $\mu$
are rescaled versions of those in [24] and the factor of $p$ is related to the
symmetric orbifold. Comparing these expressions with (2.11) written in terms
of the rescaled $L_{u,v}$, namely

$E=R(L_{u}+L_{v})+\frac{R}{\alpha^{\prime}}\left(\sqrt{p^{2}+4\alpha^{\prime
2}L_{u}L_{v}}-p\right)\;,\;\;\;\;\;\;P=R(L_{u}-L_{v})$ (2.30)

we find the relation between the parameters to be

$L_{u}=\frac{\pi
p\,\mathcal{L}}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}\;,\;\;\;\;\;\;L_{v}=\frac{\pi
p\,\bar{\mathcal{L}}}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}\;,\;\;\;\;\;\;\alpha^{\prime}=\frac{\mu}{\pi}$
(2.31)

The inverse relations read

$\mathcal{L}=\frac{2L_{u}}{\pi(\beta+p)}\;,\;\;\;\;\;\bar{\mathcal{L}}=\frac{2L_{v}}{\pi(\beta+p)}\;,\;\;\;\;\;\;\mu^{2}\mathcal{L}\bar{\mathcal{L}}=\frac{\beta-p}{\beta+p}\;,\;\;\;\;\;\beta=\sqrt{p^{2}+4\alpha^{\prime
2}L_{u}L_{v}}$ (2.32)

In the following, we will continue to use the $L_{u,v},\alpha^{\prime}$
notation for the analysis of the perturbations of the black hole backgrounds.
Then, we will switch to the more natural $T\bar{T}$ notation
$\mathcal{L},\bar{\mathcal{L}},\mu$ when computing the conserved charges and
making the link to single-trace $T\bar{T}$.

##  3 Asymptotic analysis of the black hole backgrounds

In the previous section, we have shown that the asymptotically linear dilaton
backgrounds of interest can be captured by a consistent truncation of the type
IIB action to three dimensions, which we now quickly summarize. The truncated
theory consists of three-dimensional gravity non-trivially coupled to the
dilaton, for a total of one propagating degree of freedom. The equations of
motion read

${}^{(3)}R_{\mu\nu}+2\nabla_{\mu}\nabla_{\nu}\phi+\frac{2p^{2}}{k^{3}}e^{4\phi}g_{\mu\nu}=0\;,\;\;\;\;\;\;\Box_{3}\phi-2(\partial\phi)^{2}=-\frac{2}{k}\left(1-\frac{p^{2}}{k^{2}}e^{4\phi}\right)$
(3.1)

where, as noted at the end of subsection (2.2), we have pulled out an explicit
factor of $\alpha^{\prime}$ from the metric. As a consequence, now
$G_{3}=(4k^{3/2})^{-1}$. The black hole solutions are given by

$d\bar{s}^{2}=\frac{r^{2}}{\alpha^{\prime}r^{4}+\beta
r^{2}+\alpha^{\prime}L_{u}L_{v}}\left(r^{2}dUdV+L_{u}dU^{2}+L_{v}dV^{2}+\frac{L_{u}L_{v}}{r^{2}}dUdV\right)+k\frac{dr^{2}}{r^{2}}$
$e^{2\bar{\phi}}=\frac{kr^{2}}{\alpha^{\prime}r^{4}+\beta
r^{2}+\alpha^{\prime}L_{u}L_{v}}\;,\;\;\;\;\;\;\;\;\beta=\sqrt{p^{2}+4\alpha^{\prime
2}L_{u}L_{v}}$ (3.2)

where the overbars denote the fact that these are the background values of the
fields. The goal of this section is to understand the boundary conditions that
linearized perturbations of these backgrounds satisfy. Of particular interest
are the perturbations generated by large diffeomorphisms, as these correspond
to the asymptotic symmetries of the system. Our choice of boundary conditions
will be guided by the asymptotic vanishing of the symplectic form evaluated on
these solutions, which in turn ensures the conservation and finiteness of the
corresponding charges.

We start with a thorough characterization of the linearized solutions above
the black hole backgrounds (3.2). We then compute the symplectic form in the
covariant phase space formalism and use it to select a set of boundary
conditions. We subsequently compute the asymptotic conserved charges and
compare them them to the conserved charges in $T\bar{T}$ \- deformed CFTs.

###  3.1 The linearized solutions

We consider linearized perturbations of the backgrounds (3.2)

$g_{\mu\nu}=\bar{g}_{\mu\nu}+\epsilon\,h_{\mu\nu}\;,\;\;\;\;\;\;\phi=\bar{\phi}+\epsilon\,\delta\phi$
(3.3)

where $\epsilon$ is a small parameter. We expand these perturbations in
Fourier modes, fixing radial gauge for the string frame metric. In components,
we have

$h_{\mu\nu}=e^{-i\omega U+i\kappa
V}\left(\begin{array}[]{ccc}f_{1}(r)&f_{2}(r)&0\\\ f_{2}(r)&f_{3}(r)&0\\\
0&0&0\end{array}\right)\;,\;\;\;\;\;\;\delta\phi=e^{-i\omega U+i\kappa V}f(r)$
(3.4)

The linearized equations of motion couple these four functions in a non-
trivial way. Below, we briefly explain their structure and how the solutions
are obtained. We assume that $L_{u,v}\neq 0$; a separate analysis would be
needed otherwise. We then classify the solutions into modes generated by
diffeomorphisms and propagating ones; the mode counting turns out to be
analogous to that in AdS3 and is summarized in table 1.

#### Solving the linearized equations of motion

To disentangle the equations of motion, we first note that the scalar equation
only mixes $\delta\phi$ with the trace of the metric perturbation, which is
proportional to

$h(r)\equiv(r^{4}+L_{u}L_{v})f_{2}-L_{u}r^{2}f_{3}-L_{v}r^{2}f_{1}$ (3.5)

The scalar equation then only contains $f,h$ and their derivatives up to
$f^{\prime\prime}$ and, respectively, $h^{\prime}$. Another simplifying
observation is that a linear combination of the Einstein equations (more
precisely, $L_{v}E_{uu}-L_{u}E_{vv}$) only depends on the following linear
combination of the functions

$g(r)\equiv L_{u}f_{3}-L_{v}f_{1}$ (3.6)

and its derivatives up to $g^{\prime\prime}$ and444The coefficient of $f$ in
this equation is proportional to $(L_{u}\kappa^{2}-L_{v}\omega^{2})$, so $g$
decouples from $f$ at zero frequency. $f$. Thus, $h$ and $g$ are in principle
determined by $f$. Trading $f_{2,3}$ for $h$, $g$, one can then solve the
remaining Einstein equations555For completeness, we list herein the steps
leading to the solution, which were carefully chosen so as to not have to
divide by the frequency at any point. After replacing $f_{2,3}$ by $g,h$, we
solve the $rr$ Einstein equation for $f_{1}^{\prime}$ (as a function of
$f_{1},h$ and up to third derivatives of $f$ and up to the first derivative of
$g$). Plugging the result back into Einstein’s equations one finds that a
particular linear combination of the $uu$ and $uv$ components is purely a
fourth order differential equation for $f$. Subtracting this equation from the
$uv$ component so that the $f^{\prime\prime\prime\prime}$ term cancels yields
an expression that involves $f$ and its derivatives up to
$f^{\prime\prime\prime}$ , plus some terms linear in $f_{1},h$ (and $g$), with
coefficients that vanish in the zero frequency limit. It turns out this is
exactly the form of the $ur$ and $vr$ Einstein equations (which additionally
have factors of $g^{\prime}$), so we can add them to the previous equation in
such a way that the $f_{1},h$ terms cancel. We are left with a third order
differential equation for $f$ only, the $g$ and $g^{\prime}$ terms having
cancelled out by themselves. One can explicitly check that the previous fourth
order equation for $f$ is a linear combination of the third order one and its
derivative, so we can just concentrate on the latter. for $f_{1}$ and $f$. One
finds that $f$ obeys a third order differential equation, which is decoupled
from all the other functions. The asymptotic large $r$ expansion of the
coefficients of this equation (multiplied just by $-4$) reads

$\displaystyle\left(\frac{\omega\kappa
r^{7}}{4L_{u}L_{v}}+\frac{(\kappa^{2}L_{u}+\omega^{2}L_{v}+\beta\omega\kappa/\alpha^{\prime})r^{5}}{4L_{u}L_{v}}+\frac{1}{\alpha^{\prime}k}\left(1+\frac{k\beta\kappa^{2}}{4L_{v}}+\frac{k\beta\omega^{2}}{4L_{u}}+k\alpha^{\prime}\omega\kappa\right)r^{3}+\mathcal{O}(r)\right)f^{\prime\prime\prime}$
$\displaystyle+$ $\displaystyle\left(\frac{5\omega\kappa
r^{6}}{4L_{u}L_{v}}+\frac{(7\kappa^{2}L_{u}+7\omega^{2}L_{v}+5\beta\omega\kappa/\alpha^{\prime})r^{4}}{4L_{u}L_{v}}+\frac{1}{\alpha^{\prime}k}\left(9+\frac{7k\beta\kappa^{2}}{4L_{v}}+\frac{7k\beta\omega^{2}}{4L_{u}}+8k\alpha^{\prime}\omega\kappa\right)r^{2}+\mathcal{O}(r^{0})\right)f^{\prime\prime}$
$\displaystyle+$
$\displaystyle\left(\frac{\omega\kappa(3+4k\alpha^{\prime}\omega\kappa)r^{5}}{4L_{u}L_{v}}+\frac{(5\kappa^{2}L_{u}+5\omega^{2}L_{v}+\omega\kappa\cdot\alpha_{1}(\omega,\kappa))r^{3}}{4L_{u}L_{v}}+\frac{1}{\alpha^{\prime}k}\bigl{(}15+\alpha_{2}(\omega,\kappa)\bigr{)}r+\mathcal{O}(r^{-1})\right)f^{\prime}$
$\displaystyle+$
$\displaystyle\left(-\frac{2k\beta\omega^{2}\kappa^{2}r^{2}}{L_{u}L_{v}}+\frac{2(\beta\kappa^{2}L_{u}+\beta\omega^{2}L_{v}+\omega\kappa\cdot\alpha_{3}(\omega,\kappa))}{\alpha^{\prime}L_{u}L_{v}}+\mathcal{O}(r^{-2})\right)f=0\;\;\hskip
34.14322pt\;$ (3.7)

where $\alpha_{i}(\omega,\kappa)$ are known polynomials in the given arguments
and the background parameters, whose particular form is not very illuminating
for the general structure of this equation.

The three independent solutions of this equation behave asymptotically as
$r^{-s}$, where the power $s$ is determined by the asymptotic behaviour of the
coefficients of $f$ and of its derivatives. Interestingly, the asymptotic
behaviour of the solutions changes _discontinuously_ as one or both of the
(null) frequencies is set to zero. Concretely, we find

$\displaystyle\omega,\kappa\neq 0\;\;$ $\displaystyle:$
$\displaystyle\;\;\;\;\;\;s=1\pm\sqrt{1-4k\alpha^{\prime}\omega\kappa}\;,\;\;0$
$\displaystyle\omega=0\;\;or\;\;\kappa=0\;\;$ $\displaystyle:$
$\displaystyle\;\;\;\;\;\;s=0,\;4,\;0$ $\displaystyle\omega=\kappa=0\;\;$
$\displaystyle:$ $\displaystyle\;\;\;\;\;\;s=2,\;4,\;0$ (3.8)

Thus, except for one solution with $s=0$ that we will discuss shortly, the
$\omega,\kappa\rightarrow 0$ limit of the remaining weights is _not smooth_ ,
which is a rather puzzling behaviour from the point of view of the dual
holographic interpretation. The form of the asymptotic expansion above makes
it clear why this happens at a technical level: the $r\rightarrow\infty$ and
$\omega,\kappa\rightarrow 0$ limits of the equation do not commute. One can
easily check this does not represent the generic behaviour of the wave
equation upon these backgrounds; if we studied a probe scalar, for example, or
simply ignored the coupling between the dilaton and the background metric, we
would find the equation of motion has a standard expansion near infinity, with
a smooth zero frequency limit. We conclude that it is the particular couplings
of the dilaton to the background - as dictated by string theory - and the
inclusion of backreaction that conspire to produce a linearized equation of
motion with this curious property.

Another interesting observation is that the black hole energies parametrized
by $L_{u,v}$ do not enter the asymptotic formula for the weights, but only
appear at subleading order in the asymptotic solution666Asymptotically, the
equation for $f$ (normalized so that the coefficient of
$f^{\prime\prime\prime}$ is $r^{3}$) takes the form  $\displaystyle
r^{3}f^{\prime\prime\prime}+\left[5r^{2}+\frac{2}{\omega\kappa}(L_{u}\kappa^{2}+L_{v}\omega^{2})+\mathcal{O}(r^{-2})\right]f^{\prime\prime}+\left[(3+4k\alpha^{\prime}\omega\kappa)r+\frac{4\beta(1+k\alpha^{\prime}\omega\kappa)}{\alpha^{\prime}r}+\right.$
$\displaystyle\hskip
79.6678pt\left.+\frac{2(1+2\alpha^{\prime}k\omega\kappa)}{\omega\kappa
r}(L_{u}\kappa^{2}+L_{v}\omega^{2})+\mathcal{O}(r^{-3})\right]f^{\prime}-\left[\frac{8k\beta\omega\kappa}{r^{2}}+\mathcal{O}(r^{-4})\right]f=0$
We can thus write the (propagating) solution asymptotically as  $f(r)\sim
r^{-s}\left(1-\frac{2\left[\alpha^{\prime}s(2k\alpha^{\prime}\omega\kappa-s)(L_{u}\kappa^{2}+L_{v}\omega^{2})+2\omega\kappa\beta\left(s+(s+2)k\alpha^{\prime}\omega\kappa\right)\right]}{r^{2}\omega\kappa\alpha^{\prime}(s+2)\left(4\kappa
k\alpha^{\prime}\omega+s^{2}+2s\right)}+\mathcal{O}(r^{-4})\right)\;,\;\;\;s\in\\{s_{+},s_{-}\\}$
. This property is not _a priori_ guaranteed; indeed, previous analyses of the
$SL(2,\mathbb{R})$ spectrum of warped AdS3 backgrounds in string theory have
found that the weights did generically depend on the black hole temperature
[40].

Given the solution for $f$ (parametrized by three arbitrary integration
constants), one can easily solve for the remaining functions (namely $g,h$ and
$f_{1}$), for which $f$ and its various derivatives act as an inhomogenous
source term. As mentioned above, the equation for $g$ is second order, which
will lead to two more integration constants. The equations that one obtains
for $h,f_{1}$ depend on whether one works at zero or non-zero frequency. In
the former case, $f_{1},h$ satisfy first-order differential equations, which
lead to two more integration constants; in the latter case, they are entirely
determined777The difference comes from solving the remaining $ur$ and $vr$
components of the Einstein equations, which were not discussed in the previous
footnote. If the frequencies are zero, then these equations vanish
identically, so the previously obtained equations for $h^{\prime}$,
$f_{1}^{\prime}$ are the ones that determine them, yielding two additional
integration constants. If the frequencies are non-zero, $\omega,\kappa\neq 0$,
then the $ur$ and $vr$ Einstein equations yield a solution for $f_{1}$ and
$h$, which can be shown to be compatible with the previously discussed first
order differential equation. Thus, in this case one no new integration
constants are generated.  by the solutions for $f$ and $g$.

#### Analysis of the solutions

As is well known, three-dimensional gravity does not have any propagating
degrees of freedom; the only propagating field in the consistent truncation is
the dilaton. We thus expect to have two asymptotically independent propagating
solutions to the equations of motion (one normalizable and one non-
normalizable) and a number of pure large diffeomorphism modes, corresponding
to the asymptotic data of the boundary gravitons.

It is easy to check that the two propagating modes we expect correspond to two
of the three solutions to the equation for the scalar perturbation, $f$. These
solutions are most easily identified for generic $\omega,\kappa$, where they
correspond to the first two values of $s$ in (3.8). Since the weights, denoted
$s_{\pm}$, sum to $2=d$, their most natural holographic interpretation is as
the source and the expectation value for a scalar operator of a momentum-
dependent dimension

$\Delta=1+\sqrt{1-4k\alpha^{\prime}\omega\kappa}$ (3.9)

Such a behaviour is characteristic of irrelevant deformations. The combination
$\omega\kappa$ is Lorentz invariant, and an anomalous dimension that is a
function of $\alpha^{\prime}\omega\kappa$ is precisely what one expects from
the broken scale invariance of the system [41]. Note, however, that the
$\alpha^{\prime}\rightarrow 0$ limit of (3.9) is $2$, so the dual operator
does not appear to correspond - as one would have naïvely expected - to the
continuation of the “single-trace $T\bar{T}$” operator defined in the IR
theory, which has dimension $4$. This behaviour continues to hold for
perturbations around the asymptotically linear dilaton vacuum, to which the
arguments of [41] truly apply; at a technical level, it is due to the fact
that the $r\rightarrow\infty$ and $\alpha^{\prime}\rightarrow 0$ limits of the
dilaton equation do not commute. This does not happen for probe fields [42,
43, 44] or if one ignores the backreaction of the metric on the dilaton. A
more thorough analysis of holography in this spacetime is clearly needed to
shed light on this issue, as well as on the discontinuous behaviour of the
weights (3.8) in a black hole background.

Note that the conformal dimension (3.9) becomes imaginary at large
frequencies, and the associated oscillatory behaviour at infinity is precisely
the one expected for fields in asymptotically flat spacetimes. This regime
would again deserve a more in-depth study. However, in this article we
restrict to $\omega\kappa<1/(4k\alpha^{\prime})$ so that the weights are real
(in particular, $1\leq s_{+}<2$) and thus we only need to consider Dirichlet
boundary conditions at infinity for these modes.

The solution for $f$ that corresponds to the $s=0$ asymptotic behaviour can be
induced by a diffeomorphism $\xi=rF_{r}\partial_{r}$, which acts as
$\delta\phi=\xi^{r}\partial_{r}\bar{\phi}$. This solution can be worked out
explicitly

$f^{(s=0)}(r)=-\frac{\alpha^{\prime}\,(r^{4}-L_{u}L_{v})}{\alpha^{\prime}r^{4}+\alpha^{\prime}L_{u}L_{v}+r^{2}\beta}\cdot
F_{r}$ (3.10)

and this knowledge can be used to reduce the third order differential equation
for $f$ to a second order one for the propagating modes only888For this, one
writes $f(r)=f^{(s=0)}(r)\int^{r}dr^{\prime}\hat{f}(r^{\prime})$ and finds
that $\hat{f}(r)$ satisfies a second order differential equation, whose
leading coefficients are the same as those given above for
$f^{\prime\prime\prime},f^{\prime\prime}$ and $f^{\prime}$ (the subleading
coefficients are different).. It can be explicitly checked that the additional
homogenous solutions to the remaining equations can be induced by a (string
frame) radial gauge-preserving diffeomorphism, which takes the form

$\displaystyle\xi_{rad}^{string}$ $\displaystyle=$
$\displaystyle\left[F_{U}(U,V)+k\,\frac{(r^{2}\beta+2\alpha^{\prime}L_{u}L_{v})\partial_{V}\\!F_{r}-L_{v}(\beta+2r^{2}\alpha^{\prime})\partial_{U}\\!F_{r}}{r^{4}-L_{u}L_{v}}-2k\alpha^{\prime}\,\partial_{V}\\!F_{r}\ln
r\right]\partial_{U}+\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (3.11)
$\displaystyle+\;\left[F_{V}(U,V)+k\,\frac{(r^{2}\beta+2\alpha^{\prime}L_{u}L_{v})\partial_{U}\\!F_{r}-L_{u}(\beta+2r^{2}\alpha^{\prime})\partial_{V}\\!F_{r}}{r^{4}-L_{u}L_{v}}-2k\alpha^{\prime}\,\partial_{U}\\!F_{r}\ln
r\right]\partial_{V}+rF_{r}(U,V)\partial_{r}$

The action of this diffeomorphism on the asymptotic metric yields

$h_{ij}=_{r\rightarrow\infty}\left[\frac{1}{\alpha^{\prime}}\left(\begin{array}[]{cc}\partial_{{U}}F_{V}&\frac{\partial_{U}F_{U}+\partial_{V}F_{V}}{2}\\\
\frac{\partial_{U}F_{U}+\partial_{V}F_{V}}{2}&\partial_{V}F_{U}\end{array}\right)-2k\ln
r\left(\begin{array}[]{cc}\partial_{U}^{2}F_{r}&\partial_{U}\partial_{V}F_{r}\\\
\partial_{U}\partial_{V}F_{r}&\partial_{V}^{2}F_{r}\end{array}\right)\right]$
(3.12)

where the log term comes from solving the non-homogenous differential equation
for $h$. Both parantheses receive corrections at $\mathcal{O}(r^{-2})$ and we
have ommitted the radial components of the metric perturbation, which are zero
in this gauge.

From a holographic point of view, it is more natural to parametrize the
solutions in terms of the components of the asymptotic metric that they
induce, namely $\partial_{U}F_{V}$ and $\partial_{V}F_{U}$. Note that, unlike
in AdS3, here the $UV$ component of the boundary metric is not independent of
the other two for generic functions $F_{U,V}$. Indeed, if we act with the
$\alpha^{\prime}\rightarrow 0$ limit of (3.11) on the BTZ black hole (as they
are are just the radial-gauge-preserving diffeomorphisms of that background),
we find that their action on the diagonal component of the boundary metric is
instead

$\delta
g_{\text{\tiny{$UV$}}}^{(0)}=F_{r}+\frac{\partial_{U}F_{U}+\partial_{V}F_{V}}{2}$
(3.13)

in the standard Fefferman-Graham notation. Thus, the $UV$ component of the
boundary metric can be considered as an independent piece of boundary data,
which is not the case for asymptotically flat spacetimes999To obtain a
behaviour more similar to AdS, one may consider the Einstein frame metric
$\tilde{g}_{\mu\nu}=e^{-4\phi}g_{\mu\nu}$, or $e^{-2\phi}g_{\mu\nu}$.. This
additional factor of $F_{r}$ is important in off-setting the effect of
boundary conformal transformations in AdS. Due to the different structure of
(3.12), this does not happen in the asymptotically flat case where, apart from
the log term, $F_{r}$ only appears at subleading order in the metric.

#### Zero-frequency perturbations

We have so far analysed the general solutions to the equations of motion. Let
us now concentrate on the case of zero-frequency perturbations
($\omega=\kappa=0$) which, as we already noted, cannot be treated as a subcase
of the general case.

Let us start with the $f$ equation, whose solution can now be found explicitly
also for the propagating modes, and reads

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle-\frac{\alpha^{\prime}\,C_{r}\left(r^{4}-L_{u}L_{v}\right)}{\alpha^{\prime}L_{u}L_{v}+\alpha^{\prime}r^{4}+\beta
r^{2}}+\frac{\phi_{(2)}\left(\beta+2\alpha^{\prime}r^{2}\right)}{2(\alpha^{\prime}L_{u}L_{v}+\alpha^{\prime}r^{4}+\beta
r^{2})}+$ (3.14) $\displaystyle\hskip
11.38092pt+\;\frac{\phi_{(4)}\left[\beta+2r^{2}\alpha^{\prime}+\left(\frac{r^{4}\beta}{L_{u}L_{v}}+\beta+4\alpha^{\prime}r^{2}\right)\left(\frac{r^{2}}{2\sqrt{L_{u}L_{v}}}\ln\frac{r^{2}+\sqrt{L_{u}L_{v}}}{r^{2}-\sqrt{L_{u}L_{v}}}-1\right)\right]}{2r^{2}(\alpha^{\prime}L_{u}L_{v}+\alpha^{\prime}r^{4}+\beta
r^{2})}$

There are three integration constants: $C_{r}$ \- the constant mode of $F_{r}$
\- and the propagating modes $\phi_{(2)}$ and $\phi_{(4)}$, which we have
labeled according to their leading large $r$ behaviour. Note this behaviour
does not match the zero frequency limit of the general one (3.8), which would
have predicted modes behaving as $r^{0}$ and $r^{-2}$ at infinity. Since their
asymptotic weights do not satisfy the holographic pairing relation (nor are
they paired inside the symplectic form, as we will soon show) the modes
$\phi_{(2)}$ and $\phi_{(4)}$ cannot be interpreted holographically as a
source and, respectively, expectation value for some operator, despite being
the only propagating modes at zero frequency.

Having solved for $f$, we can find the solutions for the remaining functions.
Whenever possible, we label the integration constants by the components of the
asymptotic metric that they induce

$\displaystyle f_{1}$ $\displaystyle=$
$\displaystyle\frac{C_{uu}(r^{4}+L_{u}L_{v}+\frac{r^{2}\beta}{2\alpha^{\prime}})+2C_{uv}r^{2}L_{u}+(L_{u}L_{v}C_{b}-\frac{\beta
C_{vv}L_{u}}{\alpha^{\prime}})\frac{r^{2}}{2L_{v}}}{\alpha^{\prime}L_{u}L_{v}+\alpha^{\prime}r^{4}+\beta
r^{2}}-\frac{2\alpha^{\prime}C_{r}L_{u}r^{2}(r^{4}-L_{u}L_{v})}{(\alpha^{\prime}L_{u}L_{v}+\alpha^{\prime}r^{4}+\beta
r^{2})^{2}}-$ $\displaystyle\hskip
8.53581pt-\;\frac{r^{4}p^{2}-\alpha^{\prime}r^{2}\beta
L_{u}L_{v}+r^{6}\beta\alpha^{\prime}}{2\alpha^{\prime}L_{v}(\alpha^{\prime}L_{u}L_{v}+\alpha^{\prime}r^{4}+\beta
r^{2})^{2}}\,\phi_{(2)}+\left(\frac{2}{\alpha^{\prime}L_{v}r^{2}}+\mathcal{O}(r^{-4})\right)\phi_{(4)}$
$\displaystyle f_{2}$ $\displaystyle=$
$\displaystyle\frac{C_{uv}(r^{4}+L_{u}L_{v})+C_{uu}L_{v}r^{2}+C_{vv}L_{u}r^{2}}{\alpha^{\prime}L_{u}L_{v}+\alpha^{\prime}r^{4}+\beta
r^{2}}+\frac{r^{2}\left[\alpha^{\prime}(r^{4}+L_{u}L_{v})-\frac{\beta^{2}}{2\alpha^{\prime}}\right]\phi_{(2)}+C_{r}r^{2}\beta(r^{4}-L_{u}L_{v})}{(\alpha^{\prime}L_{u}L_{v}+\alpha^{\prime}r^{4}+\beta
r^{2})^{2}}+$ $\displaystyle\hskip 8.53581pt+\;\mathcal{O}(r^{-4})\phi_{(4)}$
$\displaystyle f_{3}$ $\displaystyle=$
$\displaystyle\frac{C_{vv}(r^{4}+L_{u}L_{v}+\frac{r^{2}\beta}{2\alpha^{\prime}})+2C_{uv}r^{2}L_{v}-(L_{u}L_{v}C_{b}+\frac{\beta
C_{uu}L_{v}}{\alpha^{\prime}})\frac{r^{2}}{2L_{u}}}{\alpha^{\prime}L_{u}L_{v}+\alpha^{\prime}r^{4}+\beta
r^{2}}-\frac{2\alpha^{\prime}C_{r}L_{v}r^{2}(r^{4}-L_{u}L_{v})}{(\alpha^{\prime}L_{u}L_{v}+\alpha^{\prime}r^{4}+\beta
r^{2})^{2}}-$ (3.15) $\displaystyle\hskip
8.53581pt-\;\frac{r^{4}p^{2}-\alpha^{\prime}r^{2}\beta
L_{u}L_{v}+r^{6}\beta\alpha^{\prime}}{2\alpha^{\prime}L_{u}(\alpha^{\prime}L_{u}L_{v}+\alpha^{\prime}r^{4}+\beta
r^{2})^{2}}\,\phi_{(2)}+\left(\frac{2}{\alpha^{\prime}L_{u}r^{2}}+\mathcal{O}(r^{-4})\right)\phi_{(4)}$

Note that since $C_{r}=const$, no log terms are induced. It is easy to see
that the $C_{ij},C_{r}$ and $C_{b}$ modes can be generated by acting on the
black hole background with the diffeomorphism

$\xi_{rad}^{string,ct}=\big{[}C_{vv}V+(C_{uv}+\gamma)\,U\big{]}\partial_{U}+\big{[}C_{uu}U+(C_{uv}-\gamma)V\big{]}\partial_{V}+C_{r}r\partial_{r}$

$\gamma=\frac{C_{b}}{4}+\frac{\beta}{\alpha^{\prime}}\left(\frac{C_{uu}}{4L_{u}}-\frac{C_{vv}}{4L_{v}}\right)$
(3.16)

From here, it is easy to identify the significance of each diffeomorphism-
induced mode: $C_{r}$ is a simple rescaling of the radius, $C_{b}$ is a boost,
$C_{uv}$ is a rescaling of space and time, etc101010By comparison, the
diffeomorphism used to generate constant perturbations of the BTZ black hole
is
$\xi_{rad}^{BTZ}=\left(C^{AdS}_{vv}V+\frac{c_{uu}^{AdS}U}{2L_{u}}\right)\partial_{U}+\left(C_{uu}^{AdS}U+\frac{c_{vv}^{AdS}V}{2L_{v}}\right)\partial_{V}+\left(C_{uv}^{AdS}-\frac{c_{uu}^{AdS}}{4L_{u}}-\frac{c_{vv}^{AdS}}{4L_{v}}\right)r\partial_{r}$
where $C_{ij}^{AdS}$ parametrize the AdS boundary metric and $c_{ij}^{AdS}$,
the change in the stress tensor expectation value. . Note that this form of
the diffeomorphism breaks down in the vacuum.

We are ultimately interested in classifying these diffeomorphisms as allowed
(normalizable), disallowed (non-normalizable) and trivial (no effect on
physical observables), where normalizability is measured by the symplectic
norm of the induced metric perturbations. This is the goal of the next
section. In the mean time, a simple diagnostic that may shed light on this
question is to study how their coefficients respond to varying the parameters
of the background. Since the perturbations are in string frame radial gauge,
the identification is straightforward

$\displaystyle\delta\alpha^{\prime}\;\;\;$ $\displaystyle:$
$\displaystyle\;C_{r}=-C_{uv}=\frac{\delta\alpha^{\prime}}{2\alpha^{\prime}}\;,\;\;\;\;\phi_{(2)}=-\frac{2L_{u}L_{v}\delta\alpha^{\prime}}{\beta}\;,\;\;\;\;C_{uu}=C_{vv}=C_{b}=\phi_{(4)}=0~{}~{}~{}~{}~{}~{}$
$\displaystyle\delta L_{u}\;\;\;$ $\displaystyle:$
$\displaystyle\;\phi_{(2)}=-\frac{\alpha^{\prime}L_{v}\delta
L_{u}}{\beta}\;,\;\;\;\;\;\;C_{b}=\;\frac{\delta
L_{u}}{L_{u}}\;,\;\;\;\;C_{uu}=C_{vv}=C_{uv}=C_{r}=\phi_{(4)}=0$
$\displaystyle\delta L_{v}\;\;\;$ $\displaystyle:$
$\displaystyle\;\phi_{(2)}=-\frac{\alpha^{\prime}L_{u}\delta
L_{v}}{\beta}\;,\;\;\;\;\;C_{b}=-\frac{\delta
L_{v}}{L_{v}}\;,\;\;\;\;C_{uu}=C_{vv}=C_{uv}=C_{r}=\phi_{(4)}=0$ (3.17)

Since variations of $L_{u,v}$ should definitely be part of the phase space, we
conclude that the modes $\phi_{(2)}$ and $C_{b}$ should be allowed by the
boundary conditions. It is interesting to note that, contrary to the $AdS_{3}$
case, changes in the energy and momentum are not generated by diffeomorphisms
only, but one also needs the propagating mode $\phi_{(2)}$. Since the modes
$C_{r}$ and $C_{uv}$ vary as the (irrelevant) coupling constant of the theory
is changed, we would expect them to be fixed. Note that
$\alpha^{\prime}\phi_{(2)}$ corresponds precisely to the variations of
$-\beta/2$.

Another useful exercise is to compute the left and right-moving energies
(i.e., the charges associated to the isometries $\partial_{U},\partial_{-V}$)
carried by these modes

$\displaystyle E_{L}$ $\displaystyle=$ $\displaystyle
2R\left[\frac{\beta}{4\alpha^{\prime}}C_{uu}+L_{u}C_{vv}\left(1-\frac{\beta}{4\alpha^{\prime}L_{v}}\right)+L_{u}C_{uv}+\frac{L_{u}C_{b}}{4}-\frac{1}{4}\left(2+\frac{\beta}{\alpha^{\prime}L_{v}}\right)\phi_{(2)}+\frac{\phi_{(4)}}{L_{v}}\right]~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle E_{R}$ $\displaystyle=$ $\displaystyle
2R\left[\frac{\beta}{4\alpha^{\prime}}C_{vv}+L_{v}C_{uu}\left(1-\frac{\beta}{4\alpha^{\prime}L_{u}}\right)+L_{v}C_{uv}-\frac{L_{v}C_{b}}{4}-\frac{1}{4}\left(2+\frac{\beta}{\alpha^{\prime}L_{u}}\right)\phi_{(2)}+\frac{\phi_{(4)}}{L_{u}}\right]$
(3.18)

The charges are finite for all the modes, as the only potential quadratic
divergence (proportional to $C_{r}$) cancels between the Einstein (3.49) and
the scalar (3.50) contribution to the conserved charges. Thus, the energies do
not provide information regarding normalizability of the modes. Moreover, one
notes that all modes but $C_{r}$ carry non-trivial energy and momentum,
indicating that they are all physical, with the possible exception of $C_{r}$.

The table below summarizes the most general solution to the linearized
equations of motion and its parametrization

frequency | $\omega=\kappa=0$ |  $\omega,\kappa\neq 0$
---|---|---
propagating | $\phi_{(2)},\phi_{(4)}$ | $\phi_{(s_{+})},\phi_{(s_{-})}$
pure diffeomorphisms | $C_{r},C_{uu},C_{uv},C_{vv},C_{b}$ | $F_{U},F_{V},F_{r}$
Table 1:  Classification of the solutions to the linearized equations of
motion.

Note that for the non-zero frequency solutions, the propagating modes must be
described in momentum space, whereas the ones generated by diffeomorphisms are
more naturally analysed in position space.

###  3.2 Choice of boundary conditions

Having characterized all the linearized perturbations of the geometries, the
next step is to understand the boundary conditions that these perturbations
obey. As we saw, a simple analysis of the asymptotic behaviour of the
solutions is insufficient to answer this question, and a more systematic
approach is needed.

In general, in order for a theory to have a well-defined phase space, one
requires that the symplectic $d-1$ form associated with any two allowed
perturbations vanish asymptotically. More precisely, in three dimensions one
imposes that

$\boldsymbol{\omega}_{ab}[\Phi,\delta_{1}\Phi,\delta_{2}\Phi]=o(r^{0}),\qquad\boldsymbol{\omega}_{ra}[\Phi,\delta_{1}\Phi,\delta_{2}\Phi]=o(r^{-1})$
(3.19)

where $a$ denotes the tangent indices to the boundary located at
$r\rightarrow\infty$, $\Phi$ stands generically for the fields in the theory,
$\delta\Phi$ are arbitrary variations thereof and the notation $o(r^{-c})$
signifies that the fall-off must be faster than the indicated power of $r$.
The first condition above ensures that the symplectic flux is conserved,
whereas the second leads to normalizability of the symplectic form at
infinity. The boundary conditions are chosen such that these requirements are
satisfied. Since $\boldsymbol{\omega}$ is only defined up to the addition of
an exact form,
$\boldsymbol{\omega}\rightarrow\boldsymbol{\omega}+d\boldsymbol{\omega}_{Y}$,
where $\boldsymbol{\omega}_{Y}$ is antisymmetric in the variations, the
boundary conditions only need to be satisfied for some choice of this boundary
term. Note that the analysis of the symplectic form will not necessarily
indicate all the modes that need to be fixed, but rather which modes shoud not
be allowed simultaneously.

When one of the perturbations is generated by a diffeomorphism,
$\delta\phi=\mathcal{L}_{\xi}\phi$, the symplectic form can be related on-
shell to a $d-2$ form $\boldsymbol{k}_{\xi}$

$\boldsymbol{\omega}[\Phi,\delta_{\xi}\Phi,\delta\Phi]=d\boldsymbol{k}_{\xi}[\Phi,\delta\Phi]$
(3.20)

whose spatial integral at infinity yields the conserved charge difference
(3.47) associated to the corresponding diffeomorphism. The boundary conditions
(3.19) then ensure that the charges are conserved and, respectively, finite.
The above-mentioned ambiguities in the construction of $\boldsymbol{\omega}$
may contribute to the conserved charges (see e.g. [40]) and would, in
principle, be related to counterterms in the action.

The covariant phase space formalism constructs the symplectic form directly
from the action, in our case (2.21). The result for Einstein gravity coupled
to a scalar with kinetic term $-\frac{\varkappa}{2}(\partial\phi)^{2}$ and an
arbitrary potential is

$\boldsymbol{\omega}=\boldsymbol{\omega}_{g}+\boldsymbol{\omega}_{scal}\equiv\frac{1}{16\pi
G_{3}}\tilde{\epsilon}_{\mu\nu\rho}(\omega_{g}^{\mu}+\omega_{scal}^{\mu})\,dx^{\nu}\wedge
dx^{\rho}$ (3.21)

with

$\omega^{\mu}_{g}[\tilde{g},\delta_{1}\tilde{g},\delta_{2}\tilde{g}]=\frac{1}{2}\left[(2\tilde{\nabla}_{\lambda}h_{1}^{\mu}{}_{\rho}-\tilde{\nabla}^{\mu}h^{1}_{\lambda\rho})h_{2}^{\lambda\rho}-\tilde{\nabla}_{\rho}h_{1}\,h_{2}^{\rho\mu}+h_{1}(\tilde{\nabla}_{\rho}h^{\rho\mu}_{2}-\tilde{\nabla}^{\mu}h_{2})-(1\leftrightarrow
2)\right]$ (3.22)

where $\tilde{g}$ is the Einstein frame metric and
$h_{1\mu\nu}=\delta_{1}\tilde{g}_{\mu\nu}$,
$h_{2\mu\nu}=\delta_{2}\tilde{g}_{\mu\nu}$. The scalar contribution to the
symplectic form is

$\omega^{\mu}_{scal}=-\delta_{1}(\varkappa\tilde{\nabla}^{\mu}\phi)\,\delta_{2}\phi-\frac{\varkappa}{2}\,h_{1}\tilde{\nabla}^{\mu}\phi\,\delta_{2}\phi-(1\leftrightarrow
2)$ (3.23)

For the action (2.21), $\varkappa=8$. Note $\boldsymbol{\omega}$ is closed on-
shell, $d\boldsymbol{\omega}=0$. In the following, we compute the symplectic
form for all the linearized perturbations of the black hole backgrounds that
we have presented. We will perform a separate treatment of the zero and non-
zero frequency modes.

#### Symplectic pairing of the zero-frequency modes

We start by considering two general zero-frequency perturbations, parametrized
by two sets of constants $C_{ij},C_{r},C_{b},\phi_{(2)}$ and $\phi_{(4)}$, and
compute their symplectic form. We find that all but the $UV$ component of the
two contributions to $\boldsymbol{\omega}$ vanish. Since $\boldsymbol{\omega}$
is closed on-shell, its $UV$ component must be $r$ \- independent. We indeed
notice that a potential quadratic divergence in $r$ (proportional to
$C_{uv}\wedge C_{r}$) cancels between the gravitational and scalar
contributions. The symplectic form reads, up to an overall factor of twice the
radius

$\displaystyle\omega_{\text{\tiny{$UV$}}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\,(C_{uu}L_{v}-C_{vv}L_{u})\wedge
C_{b}+C_{uv}\wedge(C_{uu}L_{v}+C_{vv}L_{u})+$ (3.24) $\displaystyle+$
$\displaystyle\phi_{(2)}\wedge\left(C_{uv}-\frac{\beta}{4\alpha^{\prime}L_{u}L_{v}}(C_{uu}L_{v}+C_{vv}L_{u})\right)+\phi_{(4)}\wedge\,\frac{C_{uu}L_{v}+C_{vv}L_{u}}{L_{u}L_{v}}$

where the wedge notation stands for the antisymmetric product of the two modes
$a\wedge b\equiv a^{(1)}b^{(2)}-a^{(2)}b^{(1)}$, with the superscript
indicating which of the two perturbations we are referring to. From a
holographic perspective, this formula should provide us with the symplectic
pairing between sources and expectation values for the operators of the dual
theory that correspond to the fields of the consistent truncation [45].

As discussed previously, the modes $C_{b}$ and $\phi_{(2)}$ must be allowed on
the phase space, since they correspond to variations of the energy and
momentum of the black hole solutions. We conclude that their coefficient
inside the symplectic form must be fixed. This yields the boundary condition

$C_{uu}=\frac{2\alpha^{\prime}L_{u}}{\beta}C_{uv}\;,\;\;\;\;\;\;C_{vv}=\frac{2\alpha^{\prime}L_{v}}{\beta}C_{uv}$
(3.25)

which determines the two off-diagonal components of the asymptotic metric
fluctuation in terms of the diagonal piece. Upon making this choice, the
second term on the first line of (3.24) cancels due to the antisymmetry of the
wedge product. We are nevertheless left with a coupling

$\phi_{(4)}\wedge\hat{C}_{uv}$ (3.26)

where the hat on the $C_{uv}$ mode indicates it must be accompanied by
$C_{uu}$ and $C_{vv}$ perturbations, whose coefficients are fixed by the mixed
boundary conditions (3.25). The diffeomorphism that would generate the
$\hat{C}_{uv}$ mode is

$\xi\mbox{\footnotesize{$[\hat{C}_{uv}]$}}=\hat{C}_{uv}\left[\left(U+\frac{2\alpha^{\prime}L_{v}}{\beta}V\right)\partial_{U}+\left(V+\frac{2\alpha^{\prime}L_{u}}{\beta}U\right)\partial_{V}\right]$
(3.27)

Note this diffeomorphism changes the radius of the $\sigma$ circle.

It is not entirely clear which one of the modes in (3.26) should be fixed.
Both of them are zero on the black hole backgrounds, which is consistent with
either a source interpretation, or with an expectation value that happens to
vanish in thermal states. It would seem natural to classify $\phi_{(4)}$ as an
allowed mode because it is asymptotically subleading to $\phi_{(2)}$; however,
we note from the explicit solution (3.14) that the full solution is divergent
on the black hole horizon $r=\sqrt[4]{L_{u}L_{v}}$. The fact that it is not
the zero-frequency limit of any of the propagating modes also does not aid in
understanding its holographic interpretation. The $\hat{C}_{uv}$ mode could in
principle be allowed, since it carries finite energy; however, the fact that
the energy of the black hole backgrounds is entirely carried by the
$\phi_{(2)}$ mode, and not $\hat{C}_{uv}$, is somewhat puzzling111111It is
interesting to note that upon imposing the boundary conditions (3.25), the
energies (3.18) only depend on the combination
$\phi_{(2)}-4\alpha^{\prime}L_{u}L_{v}\hat{C}_{uv}/\beta$.. As we show in the
next section, the presence of the latter mode is in fact required when
perturbing the black hole backgrounds, in order to offset the change in the
radius induced by an asymptotic symmetry generator.

As far as the asymptotic charges are concerned, the term (3.26) would result
in the non-conservation of the charge associated with the diffeomorphism
(3.27) on a background where $\phi_{(4)}$ is turned on. However,
$\xi\mbox{\tiny{$[\hat{C}_{uv}]$}}$ is not an allowed diffeomorphism upon the
constant backgrounds - as it would change the asymptotic radius of the
$\sigma$ circle - and thus we are not led to any inconsistency from the point
of view of the asymptotic symmetry group analysis. The holographic
interpretation of these modes as sources or expectation values of dual
operators deserves, nevertheless, a more in-depth analysis, which we leave to
future work.

In conclusion, of the seven zero-frequency modes listed in table 1, we find
that $C_{uu}$ and $C_{vv}$ must be fixed in terms of $C_{uv}$ as in (3.25),
$C_{b}$ and $\phi_{(2)}$ must be allowed, whereas the interpretation of the
remaining modes $\hat{C}_{uv},\phi_{(4)}$ and $C_{r}$ remains unclear. The
fact that $C_{r}$ does not yet appear in the symplectic form, nor in the
energies, suggests it could be a trivial gauge mode; our subsequent analysis
will nevertheless indicate that it is in fact a source mode, which should be
fixed. This interpretation is corroborated by the fact that from the point of
view of the IR AdS3, the $C_{r}$ mode corresponds to the source for the
dilaton.

#### Symplectic pairing of the zero and non-zero frequency modes

Let us now study the symplectic pairing between zero-frequency
$(\omega=\kappa=0)$ and non-zero frequency ($\omega,\kappa\neq 0$) modes. The
latter come in two types: i) those generated by a diffeomorphism of the form
(3.11) and ii) those which are propagating, with asymptotic falloffs given in
(3.8). We first discuss the symplectic pairing of the zero frequency modes
with the non-zero frequency ones generated by a diffeomorphism.

Before we start, let us make some remarks about the choice of gauge. The zero-
frequency perturbations are naturally described in string frame radial gauge,
since then their identification with the change in the black hole parameters
is very clear121212This is due to the fact that in string frame, the black
holes’ metric has $g_{rr}=\frac{1}{r^{2}}$. In Einstein frame, the radial
component depends on the black hole parameters, and thus their variation will
not preserve $\tilde{g}_{rr}$. While it is in principle possible to redefine
the radial coordinate so that the new radial metric component is independent
of the black hole parameters, in practice the resulting metric is very
difficult to work with.. On the other hand, the mode generated by the non-zero
frequency diffemorphism can be in any gauge that is compatible with the
boundary conditions, i.e. one that does not exclude _physical_ large
diffeomorphism modes. As it turns out131313If we work with a diffeomorphism in
string frame, the symplectic form has log divergences that constrain $F_{r}$
to be at most linear in $U,V$. We also find $r^{2}$ divergences that depend on
the second derivatives of $F_{r}$, though they vanish if $F_{r}$ takes the
form (3.37), $C_{ij}$ satisfies (3.25) and $C_{r}=0$. While fixing $F_{r}=0$
in string frame would resolve these issues, it would also impair our ability
to find a central extension to the asymptotic symmetry algebra we discuss in
the next section. , it is better to consider diffeomorphisms that preserve the
radial components of the metric in Einstein frame, rather than string frame.
To reach this frame, one simply adds in a “correcting” diffeomorphism to
(3.11) so that $\delta\tilde{g}_{rr}=0$ in the final result, where
$\tilde{g}_{\mu\nu}=e^{-4\phi}g_{\mu\nu}$ is the Einstein frame metric.
Interestingly, this requirement completely changes the asymptotic behaviour of
$\xi^{r}$ and cancels away the log terms present in (3.11), which leads to
important simplifications in the calculation of the symplectic form.

The most general diffeomorphism that preserves the radial metric in Einstein
frame reads

$\displaystyle\xi_{rad}^{Einst.}$ $\displaystyle=$
$\displaystyle\left(F_{U}(U,V)+\frac{k(r^{2}\,\partial_{V}F_{r}-L_{v}\,\partial_{U}F_{r})}{r^{4}-L_{u}L_{v}}\right)\partial_{U}+\left(F_{V}(U,V)+\frac{k(r^{2}\,\partial_{U}F_{r}-L_{u}\,\partial_{V}F_{r})}{r^{4}-L_{u}L_{v}}\right)\partial_{V}+$
(3.28) $\displaystyle+$
$\displaystyle\frac{r^{3}F_{r}(U,V)}{\alpha^{\prime}r^{4}+r^{2}\beta+\alpha^{\prime}L_{u}L_{v}}\partial_{r}$

where $F_{U},F_{V}$ and $F_{r}$ are arbitrary at this stage. Note that the
radial component of this diffeomorphism, as well as the associated
perturbation of the dilaton it induces, is now more subleading at large $r$
than it was for the string frame radial gauge perturbations. This allows for
the possibility that it be an allowed mode while its string frame counterpart
is not, as we will soon argue is the case. From now on, $F_{r}$ will
exclusively denote the function labeling the radial component of the
diffeomorphism in Einstein frame (3.28), and not (3.11).

We thus compute the symplectic product between perturbations generated by the
diffeomorphisms above and arbitrary zero-frequency modes. Unlike when the
diffeomorphism was in string frame radial gauge, now all the log terms in the
symplectic form vanish. Moreover, the potential quadratic divergence ($\propto
C_{r}$) in the $\omega_{\text{\tiny{$UV$}}}$ component now completely cancels
between the metric and the scalar contribution. The finite part of
$\omega_{\text{\tiny{$UV$}}}$ has terms proportional to all the parameters of
the zero-frequency solution. Since $C_{b}$ and $\phi_{(2)}$ parametrize the
black hole backgrounds, we must require that their coefficients vanish
independently, which constrains the functions appearing in the diffeomorphism
(3.28) as follows

$L_{u}\,\partial_{V}F_{U}=L_{v}\,\partial_{U}F_{V}\;,\;\;\;\;\;\partial_{U}F_{U}+\partial_{V}F_{V}=\frac{\beta}{2\alpha^{\prime}L_{u}L_{v}}(L_{v}\partial_{U}F_{V}+L_{u}\partial_{V}F_{U})$
(3.29)

The most general solution to these equations is

$F_{U}=f\left(u\right)+\frac{2\alpha^{\prime}L_{v}}{p+\beta}\,\bar{f}\left(v\right)+c_{U}\;,\;\;\;\;\;F_{V}=\bar{f}\left(v\right)+\frac{2\alpha^{\prime}L_{u}}{p+\beta}\,f\left(u\right)+c_{V}$
(3.30)

where $c_{U,V}$ are integration constants and the “field-dependent
coordinates” $u,v$ are defined as

$u\equiv\frac{(p+\beta)\,U+2\alpha^{\prime}L_{v}V}{2p}\;,\;\;\;\;\;\;v\equiv\frac{(p+\beta)\,V+2\alpha^{\prime}L_{u}U}{2p}$
(3.31)

where their overall normalization has been chosen for later convenience. Note
these coordinates $u,v$ have identifications

$u\sim u+2\pi R\,r_{u}\;,\;\;\;\;\;\;v\sim v+2\pi R\,r_{v}$ (3.32)

where

$r_{u}=\frac{\beta+p+2\alpha^{\prime}L_{v}}{2p}=1+\frac{\alpha^{\prime}H_{R}}{pR}\;,\;\;\;\;\;r_{v}=\frac{\beta+p+2\alpha^{\prime}L_{u}}{2p}=1+\frac{\alpha^{\prime}H_{L}}{pR}$
(3.33)

where we used (2.30). The reason for using the notation $H_{L,R}$ for the
left/right-moving energies, rather than the previously used $E_{L,R}$, is to
emphasize the operatorial origin of the field-dependent radius of these
coordinates; the two quantities are identical as far as the classical analysis
of this article is concerned. Acting with these diffeomorphisms produces a
change in the components of the asymptotic metric that precisely obeys (3.25),
as one can easily check, where now $C_{ij}$ is space-time dependent.

Note that for $F_{r}=0$, these diffeomorphisms _precisely coincide_ with the
asymptotic symmetry generators (A.7) of the AdS3 background with mixed
boundary conditions that is holographically dual to double-trace $T\bar{T}$ \-
deformed CFTs, particularized to constant parameters and with the radial
component removed. This background and its associated asymptotic symmetries is
reviewed in detail in appendix A. A perturbative expansion of these results,
which is more appropriate for comparison with the analysis of the
asymptotically linear dilaton backgrounds, is presented in appendix B.

Evaluated on this solution, the radial symplectic form contains a term

$\phi_{(4)}\frac{(L_{u}\partial_{V}F_{U}+L_{v}\partial_{U}F_{V})}{L_{u}L_{v}}$
(3.34)

that is directly analogous to the term (3.26) encountered in the analysis of
the constant modes. Note that since $\phi_{(4)}$ is constant, this term is a
total $\sigma$ derivative, so it will not immediately affect the conservation
of the charges associated with the allowed diffemorphisms (3.45) upon a
background with $\phi_{(4)}$ turned on, as one might worry. In any case, these
charges vanish except when $f$ is a constant.

Next, we also find a coupling

$2\left(k\partial_{U}\partial_{V}F_{r}-\frac{2}{\alpha^{\prime}}F_{r}\right)C_{r}$
(3.35)

If $C_{r}\neq 0$, then the vanishing of this term would constrain $F_{r}$ to
be a Bessel function with argument $UV$, which is not the structure we expect.
We thus set $C_{r}=0$. Note that $C_{r}=0$ does not imply that $F_{r}=0$,
because the former is the radial mode of a diffeomorphism in string frame,
whereas the latter is the radial mode of a diffeomorphism in Einstein frame,
which has a different radial dependence. In particular, its asymptotic
behaviour is subleading by a factor of $r^{-2}$ with respect to that of
$C_{r}$. Fixing the string frame constant radial mode $C_{r}$ is also
consistent with keeping $\alpha^{\prime}$ \- here interpreted as the
coefficient of the irrelevant deformation - fixed.

Finally, writing $C_{uu},C_{vv}$ in terms of $C_{uv}$ using the boundary
conditions (3.25) and plugging in the solution for $F_{U,V}$, we find the term
proportional to $\hat{C}_{uv}$ is

$\frac{k\left(\alpha^{\prime}L_{u}F_{r}^{(0,2)}(U,V)+\alpha^{\prime}L_{v}F_{r}^{(2,0)}(U,V)-\beta
F_{r}^{(1,1)}(U,V)\right)}{\beta}\hat{C}_{uv}$ (3.36)

If $\hat{C}_{uv}\neq 0$, then the most general solution for $F_{r}$ is a
linear combination of two arbitrary functions of the field-dependent
coordinates

$F_{r}=f_{r}(u)+\bar{f}_{r}(v)$ (3.37)

As we shall see, this constraint also follows from the symplectic pairing of
two diffeomorphisms.

Let us now analyse the tangent components, ${\boldsymbol{\omega}}_{ra}$, of
the symplectic form, which should fall off faster than $1/r$ in order to
ensure finiteness of the conserved charges. We find that both components
diverge linearly with $r$, with divergent terms that are proportional with
$C_{r}$ and, respectively, $C_{ij}$ multiplied by various derivatives of
$F_{U}$ and $F_{V}$, and that they are both total $\partial_{a}$ derivatives
of the same scalar quantity. If we plug in the boundary condition (3.25) for
$C_{uu},C_{vv}$, the term proportional to $\hat{C}_{uv}$ vanishes on the
solution (3.30) for $F_{U},F_{V}$. As for the other term, we must set
$C_{r}=0$ in order to have it vanish.

Moving on to the $1/r$ term, we find that it is not zero. However, it can be
written as

$\omega_{ra}=\frac{1}{r}\partial_{a}Y^{(1)}+\ldots$ (3.38)

before imposing any constraint on the form of the diffeomorphism, and thus the
finiteness of the integrated charge is not affected. Such a term can be
absorbed into a redefinition of the symplectic form
$\boldsymbol{\omega}\rightarrow\boldsymbol{\omega}+d\boldsymbol{\omega}_{Y}$,
where the one-form $\boldsymbol{\omega}_{Y}=r^{-1}Y^{(1)}dr$ and $Y^{(1)}$ is
local. Such a redefinition does not affect the leading behaviour of
$\boldsymbol{\omega}_{ab}$, nor does it affect the conserved
charges141414Writing
$\boldsymbol{\omega}_{Y}(\Phi,\delta_{1}\Phi,\delta_{2}\Phi)=\delta_{1}\boldsymbol{Y}(\Phi,\delta_{2}\Phi)-\delta_{2}\boldsymbol{Y}(\Phi,\delta_{1}\Phi)$,
the shift in the charge integrand is
$\boldsymbol{k}_{\xi}\rightarrow\boldsymbol{k}_{\xi}+\delta\boldsymbol{Y}-\xi\cdot
d\boldsymbol{Y}$..

The components of the symplectic form for one zero-frequency mode and one non-
zero frequency propagating mode with asymptotic weight $s\in\\{s_{+},s_{-}\\}$
scale as

$\boldsymbol{\omega}_{ab}\sim
r^{1-s}\;,\;\;\;\;\;\;\boldsymbol{\omega}_{ra}\sim r^{-s}$ (3.39)

For the range of frequencies we are considering, the asymptotic weights
$s_{\pm}=1\pm\sqrt{1-4k\alpha^{\prime}\omega\kappa}$ satisfy $s_{+}\in[1,2)$
and $s_{-}\in(0,1]$ . We immediately conclude that $s_{+}$ should be an
allowed mode, and $s_{-}$ should be disallowed. As previously mentioned, we do
not consider the case where $s_{\pm}$ become imaginary, where a different
analysis is necessary.

#### Symplectic pairing of non-zero frequency modes

Finally, we consider the symplectic pairing for two non-zero frequency modes,
which can be either generated by diffeomorphisms or be propagating modes.

We start with two pure diffeomorphism modes and evaluate the symplectic form
on the solution (3.30) for $F_{U},F_{V}$, though leaving $F_{r}$ arbitrary.
The boundary condition on $\boldsymbol{\omega}_{ab}$ leads to the following
constraint

$\displaystyle\partial_{u}\partial_{v}F_{r}=0$ (3.40)

Hence, the radial function must take the form (3.37). Note this is the same
constraint as that which emerged from (3.36), but without having to assume
that $\hat{C}_{uv}\neq 0$. The $\boldsymbol{\omega}_{ra}$ components of the
symplectic behave as $\frac{1}{r}$ asymptotically, but they again take the
form (3.38) and thus can be absorbed into a shift of $\boldsymbol{\omega}$ by
a total derivative.

Next, we compute the symplectic form for two propagating modes, denoted by
their asymptotic behaviour $s_{1,2}$, which can be either normalizable or non-
normalizable. We find

$\boldsymbol{\omega}_{ab}(s_{1},s_{2})\sim
r^{2-s_{1}-s_{2}}\;,\;\;\;\;\;\;\;\boldsymbol{\omega}_{ra}(s_{1},s_{2})\sim
r^{1-s_{1}-s_{2}}$ (3.41)

which confirms our identification of $s_{+}$ as a normalizable mode and
$s_{-}$ as a non-normalizable one. The symplectic pairing between one
normalizable and one non-normalizable mode of equal but opposite frequency (so
that $s_{1}=s_{+}$ and $s_{2}=s_{-}$) is

$\boldsymbol{\omega}_{ab}(s_{-},s_{+})=2\sqrt{1-4k\alpha^{\prime}\omega\kappa}\;,\;\;\;\;\;\boldsymbol{\omega}_{ra}=0$
(3.42)

This precisely agrees with the radial symplectic form for a scalar field in
AdSd+1 [45]

$\Omega_{rad}=(d-2\Delta)\int\delta\phi_{vev}\wedge\delta\phi_{source}$ (3.43)

with $\Delta$ given in (3.9). Lastly, the symplectic form for one
(normalizable) propagating mode and one pure diffeomorphism reads

$\boldsymbol{\omega}_{ab}(s_{+},\xi)\sim
r^{2-s_{+}}\;,\;\;\;\;\;\;\;\boldsymbol{\omega}_{ra}(s_{+},\xi)\sim
r^{1-s_{+}}$ (3.44)

These falloffs violate the boundary conditions (3.19) for the $s_{+}\in[1,2)$
range we have been considering. However, one can explicitly construct a 1-form
$\boldsymbol{\omega}_{Y}$ such that $\omega=d\boldsymbol{\omega}_{Y}$, when
evaluated on the modes above. Hence, these possibly problematic terms can be
absorbed in an exact form. It would of course be intersting to provide
covariant expressions for all the counterterms that we have added to the
symplectic form.

To summarize, the symplectic form analysis fixes the components $F_{U},F_{V}$
of the Einstein frame radial-gauge-preserving diffeomorphisms in terms of two
arbitrary functions of the field-dependent coordinates (3.36), which precisely
match those in $T\bar{T}$ \- deformed CFTs. The radial function $F_{r}$ is
also constrained to depend on these coordinates via (3.37). It is an
interesting question whether $F_{r}$ should be _completely_ determined by
$F_{U,V}$, as it is in AdS. While we do suspect this to be the case, our
analysis - which is solely based on the symplectic form - is unable to see
such a constraint. The same situation occurs in AdS3, where the symplectic
form analysis does not fix the radial component of the allowed
diffeomorphisms, which is instead determined by a boundary condition on the
asymptotic metric.

We additionally find that, generically, one of the propagating modes,
$\phi_{(s_{-})}$, should be fixed, whereas the other, $\phi_{(s_{+})}$, should
be allowed to fluctuate, as is standard in holography. The constant mode
$C_{r}$ should be fixed, whereas the boundary conditions on the remaining
constant modes $\hat{C}_{uv}$ and $\phi_{(4)}$ are still unclear. These
results are summarized in the following table

frequency | $\omega=\kappa=0$ | $\omega,\kappa\neq 0$
---|---|---
mode | $\phi_{(2)},C_{b}$ | $C_{uu},C_{vv}$ | $C_{r}$ | $\phi_{(4)},\hat{C}_{uv}$ | $F_{U},F_{V}$ | $F_{r}$ | $\phi_{(s_{+})}$ | $\phi_{(s_{-})}$
status | allowed | fixed in terms | fixed | unclear | constrained | constrained | allowed | fixed
of $C_{uv}$ |  (3.30) |  (3.37)
Table 2:  Classification of the linearized perturbations into allowed modes
and fixed ones.

###  3.3 Conserved charges

In the previous section, we have found that the symplectic form vanishes for
perturbations generated by Einstein frame radial-gauge-preserving
diffeomorphisms of the form

$\displaystyle\xi_{f,\bar{f}}$ $\displaystyle=$
$\displaystyle\left(f\left(u\right)+\frac{2\alpha^{\prime}L_{v}}{p+\beta}\,\bar{f}\left(v\right)+\frac{k(r^{2}\,\partial_{V}F_{r}-L_{v}\,\partial_{U}F_{r})}{r^{4}-L_{u}L_{v}}+c_{U}\right)\partial_{U}+\left(\bar{f}\left(v\right)+\frac{2\alpha^{\prime}L_{u}}{p+\beta}\,f\left(u\right)+\right.$
(3.45)
$\displaystyle+\;\left.\frac{k(r^{2}\,\partial_{U}F_{r}-L_{u}\,\partial_{V}F_{r})}{r^{4}-L_{u}L_{v}}+c_{V}\right)\partial_{V}+\frac{r^{3}F_{r}}{\alpha^{\prime}r^{4}+r^{2}\beta+\alpha^{\prime}L_{u}L_{v}}\partial_{r}$

where $f(u),\bar{f}(v)$ are arbitrary functions of the field-dependent
coordinates (3.36) and

$\displaystyle F_{r}=f_{r}(u)+\bar{f}_{r}(v)$ (3.46)

with $f_{r},\bar{f}_{r}$ arbitrary so far. This indicates that the asymptotic
charges associated with the above diffeomorphisms should be conserved.

We would now like to compute these charges using the covariant phase space
formalism, in which the charge difference between two nearby on-shell
backgrounds is given by

$\not{\\!\delta}\boldsymbol{Q}_{\xi}=\oint\boldsymbol{k}_{\xi}[\delta\Phi,\Phi]$
(3.47)

Here, $\delta\Phi$ stands for an arbitrary variation of all the background
fields and $\boldsymbol{k}_{\xi}$ is a $d-2$ form constructed algorithmically
from the action, which depends on the background fields, their variations and
the diffeomorphism $\xi$. For the problem at hand, we have

$\boldsymbol{k}_{\xi}=\frac{1}{2}\tilde{\epsilon}_{\mu\nu\rho}K^{\nu\rho}_{\xi}\;,\;\;\;\;\;\;\;\;K^{\nu\rho}_{\xi}=\frac{1}{8\pi
G}(K^{\nu\rho}_{\xi,g}+K^{\nu\rho}_{\xi,scal})$ (3.48)

with

$K^{\mu\nu}_{g}=\left(\xi^{\nu}\tilde{\nabla}^{\mu}h-\xi^{\nu}\tilde{\nabla}_{\sigma}h^{\mu\sigma}+\xi_{\sigma}\tilde{\nabla}^{\nu}h^{\mu\sigma}+\frac{1}{2}h\tilde{\nabla}^{\nu}\xi^{\mu}-h^{\rho\nu}\tilde{\nabla}_{\rho}\xi^{\mu}\right)$
(3.49)

$K^{\mu\nu}_{scal}=\varkappa\,\xi^{\nu}\tilde{\nabla}^{\mu}\phi\,\delta\phi$
(3.50)

where $\varkappa=8$ and the $\tilde{}$ indicates that we are working in the
Einstein frame. This charge difference needs to be integrable in order for the
charges to be well-defined, a property that is not automatically guaranteed by
the definition (3.47), but needs to be checked separately. Assuming this is
the case, the total conserved charge is obtained by integrating
$\cancel{\delta}\boldsymbol{Q}_{\xi}$ along a path in configuration space
between the background of interest and a reference background. We will
alternatively be using the notation $\cancel{\delta}Q_{f,\bar{f}}$ for the
charge difference associated with a diffeomorphism of the form (3.45)
parametrized by the functions $f(u),\bar{f}(v)$.

In our case, if we take $\delta\Phi$ to correspond to the difference between
two nearby black hole solutions, we find that all the charges vanish except
for the energy and momentum. These charge differences are given by

$\not{\\!\delta}Q_{f_{0},\bar{f}_{0}}=(f_{0}+\mu\bar{\mathcal{L}}\bar{f}_{0}+c_{U})\delta
H_{L}-(\bar{f}_{0}+\mu\mathcal{L}f_{0}+c_{V})\delta H_{R}$ (3.51)

where $f_{0},\bar{f}_{0}$ are the Fourier zero modes of the respective
functions and we used (2.32) to write the answer in terms of the natural
$T\bar{T}$ parametrization. Given that

$\mathcal{L}=\frac{H_{L}}{\pi Rp+\mu
H_{R}}\;,\;\;\;\;\;\bar{\mathcal{L}}=\frac{H_{R}}{\pi Rp+\mu H_{L}}$ (3.52)

we immediately note that the charges are not integrable for generic (field-
independent) $f_{0}$ and $\bar{f}_{0}$. A natural way to remedy this problem,
which recovers the standard energies of the background when $f_{0}=1$ and
$\bar{f}_{0}=-1$, is to choose the constants $c_{U,V}$ as

$c_{U}=-\mu\bar{\mathcal{L}}\bar{f}_{0}\;,\;\;\;\;\;c_{V}=-\mu\mathcal{L}f_{0}$
(3.53)

case in which the charges become trivially integrable, and read

$Q_{f}^{(0)}=f_{0}H_{L}=\frac{1}{2}\oint d\sigma
r_{u}\,p\,\mathcal{L}f(u)\;,\;\;\;\;\;\;\;\bar{Q}_{\bar{f}}^{(0)}=-\bar{f}_{0}H_{R}=-\frac{1}{2}\oint
d\sigma r_{v}\,p\,\bar{\mathcal{L}}\bar{f}(v)$ (3.54)

where we have used (2.29) to write the charges in a notation that resembles
the $T\bar{T}$ one (A.11), and we note that for constant perturbations
$\partial_{\sigma}u=r_{u}$, defined in (3.33). The (0) superscript indicates
that this is the only contribution to the conserved charges upon the constant
backgrounds.

To obtain conserved charges that are generically non-zero, we first need to
perturb the black hole background by an arbitrary allowed diffeomorphism
$\eta$, which takes the form (3.45) for some other functions
$h(u),\bar{h}(v),h_{r}(u),\bar{h}_{r}(v)$. We obtain the following expression
for the difference in the charge associated with a diffeomorphism $\xi$ on the
background generated by $\eta$

$\displaystyle\delta_{\eta}Q_{\xi}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\oint
d\sigma\left[\frac{4pL_{u}r_{u}}{p+\beta}f(u)h^{\prime}(u)-\frac{4pL_{v}r_{v}}{p+\beta}\bar{f}(v)\bar{h}^{\prime}(v)+kr_{u}h_{r}^{\prime\prime}(u)f(u)-kr_{v}\bar{h}_{r}^{\prime\prime}(v)\bar{f}(v)\right.+$
(3.55)
$\displaystyle+\;\left.\frac{k}{2}(r_{u}f_{r}^{\prime}(u)-r_{v}\bar{f}_{r}^{\prime}(v))\left(h^{\prime}(u)+\bar{h}^{\prime}(v)\right)-\frac{k}{2}(f_{r}(u)+\bar{f}_{r}(v))\left(r_{u}h^{\prime\prime}(u)-r_{v}\bar{h}^{\prime\prime}(v)\right)\right]$

Integrating by parts, the cross left-right terms cancel, and we are left with

$\displaystyle\delta_{\eta}Q_{\xi}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\oint d\sigma
r_{u}\left(\frac{4pL_{u}}{p+\beta}f(u)h^{\prime}(u)+kh_{r}^{\prime\prime}(u)f(u)-kf_{r}^{\prime\prime}(u)h(u)\right)+RM$
(3.56) $\displaystyle=$ $\displaystyle\frac{1}{2}\oint d\sigma
r_{u}\left(2p\mathcal{L}fh^{\prime}+\frac{k}{\pi}h_{r}^{\prime\prime}f-\frac{k}{\pi}f_{r}^{\prime\prime}h\right)+RM$

where in the second line we have translated the result to the $T\bar{T}$
parametrization (2.31).

Let us now compare this with the corresponding expression (A.47) for double-
trace $T\bar{T}$ \- deformed CFTs, specialized to $\mathcal{L}=const.$

$\delta_{\eta}Q_{\xi}^{\,double-trace}=\frac{1}{2}\oint
d\sigma\partial_{\sigma}u\left(2\mathcal{L}fh^{\prime}-\frac{c}{12\pi}fh^{\prime\prime\prime}\right)$
(3.57)

where we have used the fact that $\ell/8\pi G=c/(12\pi)$ in (A.47), where $c$
is the central charge of the undeformed CFT, and we dropped the winding term,
which vanishes on the constant $\mathcal{L},\bar{\mathcal{L}}$ black hole
backgrounds. The expressions (3.56) and (3.57) are almost the same, up to a
factor of $p$ in the first term - which is related to the symmetric product
orbifold - and the fact that the $\mathcal{L}$ \- independent terms in (3.56)
depend on more functions than their $T\bar{T}$ counterpart.

As discussed at the end of the previous subsection, we would expect that the
full set of boundary conditions on the asymptotically linear dilaton
backgrounds includes one that determines the radial function $F_{r}$ in terms
of $f,\bar{f}$; however, the symplectic form analysis that we have performed
is insensitive to this additional boundary condition. Requiring that the
central contribution to the charge in (3.56) match that in symmetric product
orbifolds of $T\bar{T}$ \- deformed CFTs, which is $6kp$ in our case, allows
us to fix the radial function in terms of the other ones as

$f_{r}(u)=-\frac{p}{4}\,f^{\prime}(u)\;,\;\;\;\;\;\bar{f}_{r}(v)=-\frac{p}{4}\,\bar{f}^{\prime}(v)$
(3.58)

It would be very interesting if we could independently justify this form of
$F_{r}$ from an asymptotic boundary condition on the fields.

Thus, we find that not only do the asymptotic symmetries of the asymptotically
linear dilaton background _precisely match_ (for $F_{r}=0$) those of the AdS3
backgrounds with mixed boundary conditions that are dual to double-trace
$T\bar{T}$ \- deformed CFTs, but also the conserved charges match _exactly_
for a particular choice (3.58) of the radial function $F_{r}$ that is
consistent with the results of the symplectic form analysis, but could not be
singled out by it. This coincidence is quite remarkable, given that the
background metrics (3.2) and (A.2) are entirely different, as are the boundary
conditions that their fluctuations obey. Even more remarkably, this
identification continues to hold when the two backgrounds are perturbed, as we
show in the next section.

##  4 The charge algebra

In this section, we use the covariant phase space formalism to compute the
algebra of the conserved charges _perturbatively_ above the asymptotically
linear dilaton black hole backgrounds and show the result is identical to the
nonlinear $T\bar{T}$ charge algebra (1.3) to the order we checked. We start by
reviewing a few generalities.

The standard definition of the charge algebra in the covariant phase space
formalism is

$\\{Q_{\xi},Q_{\chi}\\}\equiv\delta_{\chi}Q_{\xi}=\oint\boldsymbol{k}_{\xi}(\mathcal{L}_{\chi}\Phi,\Phi)$
(4.1)

where the last term corresponds to the difference in the charge associated
with some diffeomorphism, $\xi$, between two backgrounds that differ by some
other diffeomorphism, $\chi$. This definition relies on the representation
theorem [46, 47], which relates the right-hand-side of (4.1) to the charge
associated to the Lie bracket $[\xi,\chi]$ of the two diffeomorphisms

$\delta_{\chi}Q_{\xi}=Q_{[\xi,\chi]}$ (4.2)

assuming the charges are integrable. This conclusion continues to hold when
the diffeomorphisms are field-dependent, provided one replaces the Lie bracket
with a modified Lie bracket [48, 49]

$[\xi,\chi]_{*}\equiv[\xi,\chi]_{L.B.}-\delta_{\xi}\chi+\delta_{\chi}\xi$
(4.3)

in the representation theorem. It is important to note that in this formalism
the integrability [50] of the conserved charges - which is not guaranteed by
the definition (3.47), but needs to be checked separately - plays a crucial
role in obtaining the correct algebra, especially when the diffeomorphisms are
field-dependent. This fact is neatly exemplified in our analysis, presented in
appendix A, of the asymptotic symmetries of the spacetime dual to the double-
trace $T\bar{T}$ deformation, where integrability is essential for reproducing
the correct charge algebra when the generators are rescaled by a field-
dependent factor.

###  4.1 Perturbative calculation setup

The aim of this section is to set up the computation of the charge algebra for
the asymptotically linear dilaton black hole backgrounds. This is
significantly more complicated than for the AdS3 spacetimes with mixed
boundary conditions dual to double-trace $T\bar{T}$-deformed CFTs because the
allowed modes are only known _perturbatively_ around these constant
backgrounds, whereas in the double-trace case the general, non-linear solution
(A.2) for the allowed metrics is known.

A quick glance at (4.2) suffices to understand that the allowed
diffeomorphisms we have found so far are _insufficient_ to determine the full
algebra of the conserved charges. The limitation stems from the fact that all
the non-trivial charges (i.e., except the energy and momentum) evaluate to
zero on the constant black hole backgrounds, and thus $Q_{[\xi,\chi]}$ can
only receive non-vanishing contributions from the zero mode of the Lie bracket
of the two diffeomorphisms. Working in a Fourier basis for the functions $f,g$
that parametrize the two diffeomorphisms, with $f=e^{imu/R_{u}}$ and
$g=e^{inu/R_{u}}$, the computation of the algebra upon the constant
backgrounds will only be able to recover the contributions to the
$\\{Q_{m},Q_{n}\\}$ commutator that have $m+n=0$.

Let us briefly discuss this zeroth-order result before moving on to the more
general contributions. The computation of the charge variation
$\delta_{\eta}Q_{\xi}$ associated to two diffeomorphisms $\xi_{f}$ and
$\eta_{h}$ upon the constant backgrounds has already been performed in (3.56),
and now we simply need to interpret it as the commutator
$\\{Q_{\xi},Q_{\eta}\\}$ of the two conserved charges. The result can be made
more transparent by an integration by parts, which replaces the $2fh^{\prime}$
factor in the first term of (3.56) by $fh^{\prime}-hf^{\prime}$. We thus
obtain

$\\{Q_{f},Q_{h}\\}=Q^{(0)}_{fh^{\prime}-f^{\prime}h}-\frac{k}{\pi}\oint
d\sigma r_{u}f_{r}^{\prime\prime}h$ (4.4)

where the superscript (0) indicates that the charge associated with the given
diffeomorphism should be evaluated on the constant black hole backgrounds,
where only the zero mode of the function contributes. This algebra is similar
to a centrally-extended Witt algebra. Further making the choice
$f_{r}=-pf^{\prime}/4$ that was discussed previously and working in the
Fourier basis for the generators, the charge algebra we obtain to this order
is

$\bigg{.}\\{Q_{m},Q_{n}\\}\bigg{|}_{m+n=0}=\;\frac{2m}{R_{u}}Q_{0}+\frac{6kp}{12}\cdot\frac{m^{3}}{R_{u}^{2}}\,$
(4.5)

where $Q_{0}=H_{L}$ and $R_{u}=R\,r_{u}$, with $r_{u}$ given in (3.33). The
factors of $R_{u}$ are due to the fact that the periodicity of the coordinate
$u$ that enters the diffeomorphisms is field-dependent; note that they
perfectly match the structure (1.3) expected from $T\bar{T}$ \- deformed CFTs.
A similar result holds for the right-movers. The central term has been
_chosen_ to match the prediction from a symmetric product orbifold of
$T\bar{T}$ \- deformed CFTs with $c=6kp$. While it would have been far more
valuable to have _derived_ this central extension by finding a natural
boundary condition that leads to (3.58), the fact that were were able to find
some form of the radial function that yields precisely this value is still a
non-trivial result; for example, such a choice would not have been available
had we been working in string frame radial gauge.

Let us now discuss the computation of the charge algebra for generic $m,n$.
For this, we need to consider a perturbed background that generically has non-
zero charges with respect to arbitrary diffeomorphisms belonging to the
asymptotic symmetry group. As in the previous section, we can generate such a
perturbed background by acting with an allowed diffeomorphism, $\eta$, on the
constant background, $\bar{g}$

$g=\bar{g}+\epsilon\,\mathcal{L}_{\eta}\bar{g}$ (4.6)

According to our discussion, the charges associated to allowed diffeomorphisms
$\xi_{f,\bar{f}}$ of such a background satisfy

$Q_{\xi_{f,\bar{f}}}[g]=\left\\{\begin{array}[]{ccc}\mathcal{O}(1)&\mbox{for}&f,\bar{f}=const.\vspace{2mm}\\\
\mathcal{O}(\epsilon)&\mbox{for}&f,\bar{f}\neq const.\end{array}\right.$ (4.7)

To compute the charge algebra, we will subsequently act with another allowed
diffeomorphism, $\chi$, on this deformed background and then, according to the
recipe (4.1), we will compute the charge difference $\delta_{\chi}Q_{\xi}$
associated to a third allowed diffeomorphism, $\xi$. The terms in the algebra
with $m+n\neq 0$ are visible at $\mathcal{O}(\epsilon)$. Note, however, that
the diffeomorphisms $\xi,\chi$ themselves can receive corrections at
$\mathcal{O}(\epsilon)$, which will, in principle, contribute to the charge
algebra at the same order. Consequently, in order to find the full answer for
the charge algebra at $\mathcal{O}(\epsilon)$, we do need to work out the
corresponding correction to the diffeomorphisms. From the point of view of the
expected algebra (1.3), the $\mathcal{O}(\epsilon)$ analysis reveals the
generic linear terms, as well as the non-linear terms with either $m$ or $n$
zero, whereas the generic non-linear terms appear only at
$\mathcal{O}(\epsilon^{2})$.

The $\mathcal{O}(\epsilon)$ correction to the diffeomorphisms (3.45) can again
be determined using the symplectic form, now evaluated on the perturbed
backgrounds (4.6). This analysis does not, however, completely fix the
correction, and complementary arguments need to be invoked. One such argument
is based on the representation theorem, which relates the
$\mathcal{O}(\epsilon)$ modification to the vectors (which appears in the
modified Lie bracket (4.3), but without the factor of $\epsilon$) to the
charge algebra at $\mathcal{O}(1)$. Thus, knowing the latter, we can obtain
further constraints on the diffeomorphisms themselves. This argument still
does not completely fix the vectors, and a “minimal continuation” assumption
on the functions that parametrize them is also needed.

The final solution for the asymptotic diffeomorphisms up to
$\mathcal{O}(\epsilon)$ _precisely agrees_ with the corresponding expression
in double-trace $T\bar{T}$ \- deformed CFTs, particularized to perturbations
around a constant background, that we work out in appendix B. Finally equipped
with the solution for the vectors, we derive the charge algebra to
$\mathcal{O}(\epsilon)$ and find a _perfect match_ to the charge algebra in
double-trace $T\bar{T}$ \- deformed CFTs, computed to the same order.
Throughout this section, we use the natural $T\bar{T}$ parametrization (2.31)
of the black hole backgrounds, in order to facilitate the comparison with the
double-trace results.

###  4.2 Allowed diffeomorphisms upon the perturbed backgrounds

As explained, we consider again the perturbed background obtained by acting
with an allowed diffeomorphism $\eta_{h,\bar{h}}$, of the form (3.45) with
$c_{U,V}$ given by (3.53), on the black hole background $\bar{g}$. For
simplicity, we set the radial component of $\eta$ to zero, since the
symplectic form analysis is not expected to constrain it very much, and
keeping it would significantly complicate the calculations. Thus, $\eta$ is
parametrized by two periodic functions, $h(u)$ and $\bar{h}(v)$, of the field-
dependent coordinates (3.36) only. In terms of the $T\bar{T}$ notation, it
takes the form

$\displaystyle\eta$
$\displaystyle=\left[h(u)-\mu\bar{\mathcal{L}}(\bar{h}(v)-\bar{h}_{0})\right]\partial_{U}+\left[\bar{h}(v)-\mu\mathcal{L}(h(u)-h_{0})\right]\partial_{V}$
(4.8)

The goal of this subsection is to determine the first-order correction to the
allowed diffeomorphisms of the perturbed background (4.6).

As for the case of the black hole backgrounds, we expect the answer to be
largely fixed by requiring that the symplectic product of the diffeomorphism
we search for with the perturbation of the background that is generated by
varying the black hole parameters $L_{u,v}$ vanish asymptotically. The result
takes the form

$\xi=\xi^{(0)}+\epsilon\,\xi^{(1)}$ (4.9)

where $\xi^{(0)}$ was previously determined in (3.45), and $\xi^{(1)}$ is to
be computed by imposing the requirements (3.19). More concretely, this
perturbative computation takes the form

$\displaystyle\omega_{\bar{g}+\epsilon\mathcal{L}_{\eta}\bar{g}}\left(\frac{\delta}{\delta
L_{i}}(\bar{g}+\epsilon\mathcal{L}_{\eta}\bar{g}),\mathcal{L}_{\xi^{(0)}+\epsilon\xi^{(1)}}(\bar{g}+\epsilon\mathcal{L}_{\eta}\bar{g})\right)=\omega_{\bar{g}+\epsilon\mathcal{L}_{\eta}\bar{g}}\left(\frac{\delta\bar{g}}{\delta
L_{i}},\mathcal{L}_{\xi^{(0)}}\bar{g}\right)+\epsilon\,\omega_{\bar{g}}\left(\frac{\delta\mathcal{L}_{\eta}\bar{g}}{\delta
L_{i}},\mathcal{L}_{\xi^{(0)}}\bar{g}\right)+$ $\displaystyle\hskip
34.14322pt+\;\epsilon\,\omega_{\bar{g}}\left(\frac{\delta\bar{g}}{\delta
L_{i}},\mathcal{L}_{\xi^{(1)}}\bar{g}+\mathcal{L}_{\xi^{(0)}}(\mathcal{L}_{\eta}\bar{g})\right)+\mathcal{O}(\epsilon^{2})$
(4.10)

where for simplicity we have suppressed the factor of the dilaton from the
background notation and $L_{i}=L_{u,v}$. One important remark is that our
analysis determines the metric perturbation only up to terms that have zero
symplectic product with $\delta_{L_{i}}\bar{g}$, such as the constant modes
$C_{b}$ and $\hat{C}_{uv}$ discussed in the previous section. In principle, we
have the freedom to add such contributions to $\mathcal{L}_{\xi^{(1)}}\bar{g}$
without violating the requirements on $\boldsymbol{\omega}$, as long as we do
not generate winding for the fixed coordinates $U,V$.

While we perform the computations on the string backgrounds, it is useful to
transcribe the results in terms of the natural $T\bar{T}$ notation (2.32). The
vanishing of the symplectic flux through the boundary at
$\mathcal{O}(\epsilon)$ imposes the following constraints on the components of
the correction $\xi^{(1)}$ to the allowed diffeomorphisms

$\partial_{u}\left(\xi^{(1)V}-\mu\mathcal{L}\,\xi^{(1)U}\right)=\;2\mu\mathcal{L}\,h^{\prime}(f^{\prime}(u)+\bar{f}^{\prime}(v))$

$\partial_{v}\left(\xi^{(1)U}-\mu\bar{\mathcal{L}}\,\xi^{(1)V}\right)=\;2\mu\bar{\mathcal{L}}\,\bar{h}^{\prime}(f^{\prime}(u)+\bar{f}^{\prime}(v))$
(4.11)

where $u,v$ are the field-dependent coordinates (3.36) associated with the
unperturbed background. The solution is given by

$\displaystyle\xi^{(1)U}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}\left[F_{h}(u)+\mu\bar{\mathcal{L}}\bar{F}_{\bar{h}}(v)+2\mu^{2}\mathcal{L}\bar{\mathcal{L}}\left(\int^{u}h^{\prime}f^{\prime}+(h-h_{0})\bar{f}^{\prime}\right)+2\mu\bar{\mathcal{L}}\left(\int^{v}\bar{h}^{\prime}\bar{f}^{\prime}+(\bar{h}-\bar{h}_{0})f^{\prime}\right)\right]~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\xi^{(1)V}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}\left[\bar{F}_{\bar{h}}(v)+\mu\mathcal{L}F_{h}(u)+2\mu^{2}\mathcal{L}\bar{\mathcal{L}}\left(\int^{v}\bar{h}^{\prime}\bar{f}^{\prime}+(\bar{h}-\bar{h}_{0})f^{\prime}\right)+2\mu\mathcal{L}\left(\int^{u}h^{\prime}f^{\prime}+(h-h_{0})\bar{f}^{\prime}\right)\right]$
(4.12)

where $F_{h},\bar{F}_{\bar{h}}$ are so far unfixed, background-dependent
functions of the field-dependent coordinates, which represent the continuation
of the $\mathcal{O}(1)$ functions $f,\bar{f}$ to the perturbed background
generated by $h,\bar{h}$, and the primitives of $h^{\prime}f^{\prime}$ and
$\bar{h}^{\prime}\bar{f}^{\prime}$ are integrated over a corresponding field-
dependent variable $\tilde{u}\,$/ $\tilde{v}$ that we have ommitted in order
to simplify the notation. These primitives are defined to not contain any
constant Fourier mode, and thus all the possible integration constants that
arise are included in the functions $F_{h},\bar{F}_{\bar{h}}$; the zero modes
of $h,\bar{h}$ were subtracted for further convenience. The most general
solution for the vector field should also include contributions from the zero-
frequency modes $C_{b}$ and $\hat{C}_{uv}$ with arbitrary coefficients, since
including them does not affect the symplectic form. While these terms will
turn out to be essential for obtaining the correct algebra, we can simply add
them at the end, without affecting the result of the discussion below.

It is interesting to compare the corrections (4.12) with the expansion of the
allowed diffeomorphisms for the spacetime dual to double-trace $T\bar{T}$ \-
deformed CFTs, a computation we perform in appendix B. We find that the choice

$F_{h}=2\mu^{2}\mathcal{L}\bar{\mathcal{L}}\bigg{(}\int^{u}(h-h_{0})f^{\prime\prime}+(hf^{\prime})_{zm}\bigg{)}\;,\;\;\;\;\;\bar{F}_{\bar{h}}=2\mu^{2}\mathcal{L}\bar{\mathcal{L}}\bigg{(}\int^{v}(\bar{h}-\bar{h}_{0})\bar{f}^{\prime\prime}+(\bar{h}\bar{f}^{\prime})_{zm}\bigg{)}$
(4.13)

\- where the subscript stands for the “zero mode” of the corresponding
expression - precisely reproduces the expansion (B.7) of the allowed
diffeomorphisms in that theory up to $\mathcal{O}(\epsilon)$ about the
background of constant parameters, barring winding terms. However, from the
point of view of the asymptotic analysis of the charge algebra in
asymptotically linear dilaton backgrounds, this choice needs to be justified
from first principles.

For this purpose, we use the representation theorem, which should be satisfied
order by order in the perturbation, provided we use the modified Lie bracket
(4.3) that takes into account the field-dependence of vector fields151515As
noted in appendix A, the modified Lie bracket (4.3) needs to be further
modified in order for the representation theorem to hold for the spacetimes
dual to double-trace $T\bar{T}$ \- deformed CFTs. These subtleties are however
linked to winding terms, and do not appear at the order in perturbation theory
that we are discussing. The representation theorem then (almost) takes its
standard modified form, as we show in appendix (B.3). . To ascertain the
necessity of this modified bracket, let us consider the representation theorem
at $\mathcal{O}(1)$. The correction to the usual Lie bracket is given by the
change in a vector field under the action of another, which precisely
corresponds to the $\mathcal{O}(\epsilon)$ term in (4.9). For simplicity, we
will focus on the “left” vector fields, obtained by setting all the functions
of $v$ to zero in (4.8). The standard Lie bracket of two such $\mathcal{O}(1)$
vector fields is

$[\xi,\chi]_{L.B}=\frac{1+\mu^{2}\mathcal{L}\bar{\mathcal{L}}}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}(fg^{\prime}-gf^{\prime})(\partial_{U}+\mu\mathcal{L}\partial_{V})+\frac{\mu^{2}\mathcal{L}\bar{\mathcal{L}}}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}(g_{0}f^{\prime}-f_{0}g^{\prime})(\partial_{U}+\mu\mathcal{L}\partial_{V})$
(4.14)

Its associated conserved charge on the constant backgrounds only receives
contributions from the first term, and one can easily check that it does not
agree with the $\mathcal{O}(1)$ charge algebra that we have already computed
in (4.4). Thus, in order for the representation theorem to be obeyed, the
modification (4.3) to the Lie bracket is required to be

$\displaystyle\delta_{\xi}\chi-\delta_{\chi}\xi=\frac{2\mu^{2}\mathcal{L}\mathcal{\bar{L}}}{1-\mu^{2}\mathcal{L}\mathcal{\bar{L}}}(fg^{\prime}-gf^{\prime})(\partial_{U}+\mu\mathcal{L}\partial_{V})+\ldots$
(4.15)

where the $\ldots$ stand for terms that integrate to zero upon the constant
backgrounds. It seems natural that these terms should take the form
$\sum_{n>0}c_{n}(f_{0}g^{(n)}-g_{0}f^{(n)})(\partial_{U}+\mu\mathcal{L}\partial_{V})$
for some constants $c_{n}$, as the correction needs to be bilinear in $f,g$,
antisymmetric, and generically not posess a zero mode.

At the same time, $\delta_{\xi}\chi=\chi^{(1)}$ is the correction to $\chi$ on
the background generated by $\xi$, which is given in (4.12), and vice-versa.
We therefore have

$\delta_{\xi}\chi-\delta_{\chi}\xi=\frac{1}{1-\mu^{2}\mathcal{L}\mathcal{\bar{L}}}\big{(}G_{f}-F_{g})(\partial_{U}+\mu\mathcal{L}\partial_{V})$
(4.16)

Equating the last two expressions immediately yields a constraint on the
difference $G_{f}-F_{g}$. To further narrow it, note that when, say,
$f=const$, both $G_{f}$ and $F_{g}$ need to vanish, irrespectively of what
$g(u)$ is: $G_{f}$ vanishes because a constant coordinate shift is an isometry
of the black hole backgrounds, and thus the asymptotic symmetries cannot
depend on it, while $F_{g}$ vanishes because $\xi_{f}$ is a background-
independent diffeomorphism in this case. Taking this into account, the
$\mathcal{O}(\epsilon)$ information from the Lie bracket fixes the difference
to

$\displaystyle
G_{f}-F_{g}=2\mu^{2}\mathcal{L}\bar{\mathcal{L}}(fg^{\prime}-gf^{\prime}+g_{0}f^{\prime}-f_{0}g^{\prime})$
(4.17)

Applying the same reasoning for to the right-moving vector fields, we obtain

$\displaystyle\bar{G}_{\bar{f}}-\bar{F}_{\bar{g}}=2\mu^{2}\mathcal{L}\bar{\mathcal{L}}(\bar{f}\bar{g}^{\prime}-\bar{g}\bar{f}^{\prime}+\bar{g}_{0}\bar{f}^{\prime}-\bar{f}_{0}\bar{g}^{\prime})$
(4.18)

Thus, the zeroth order modified Lie bracket can partly fix the
$\mathcal{O}(\epsilon)$ correction to the allowed diffeomorphisms. The fact
that the representation theorem does not fully fix these functions is not
surprising. A comparison with the analogous computation in standard AdS3 with
Brown-Henneaux boundary conditions shows that an assumption that $f$ be
minimally continued to higher orders is also necessary. The difference is that
for the asymptotically linear dilaton backgrounds, the minimal continuation
does require $F_{h}$ to be non-zero.

Guided by our $T\bar{T}$ intuition, and since $u$ depends on the background
parameters, the minimal continuation at $\mathcal{O}(\epsilon)$ of $f(u)$
should be due to a change in the argument of the function, which implies that
$\xi^{(1)U}=f^{\prime}\delta_{h}u$. Comparing this with the general solution
(4.12) with all the functions of $v$ set to zero, we identify

$\displaystyle\frac{1}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}\bigg{(}F_{g}+2\mu^{2}\mathcal{L}\bar{\mathcal{L}}\int^{u}g^{\prime}f^{\prime}\bigg{)}=f^{\prime}\delta_{g}u$
(4.19)

Using (4.17), it follows that

$\displaystyle
g^{\prime}\delta_{f}u-f^{\prime}\delta_{g}u=\frac{1}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}(G_{f}-F_{g})=\frac{2\mu^{2}\mathcal{L}\bar{\mathcal{L}}}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}\big{(}fg^{\prime}-gf^{\prime}+g_{0}f^{\prime}-f_{0}g^{\prime}\big{)}$
(4.20)

From here we conclude that
$\delta_{f}u=\frac{2\mu^{2}\mathcal{L}\bar{\mathcal{L}}(f-f_{0})}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}+\gamma
f^{\prime}$, where $\gamma$ can be any constant, reflecting the possibility of
adding terms that will drop out from the expression above because of
antisymmetry. The minimal choice, which sets such terms to zero161616In the
next subsection, we show such terms do not contribute to the charge algebra.,
plugged into (4.19), yields

$\displaystyle F_{g}$
$\displaystyle=2\mu^{2}\mathcal{L}\bar{\mathcal{L}}\big{(}f^{\prime}(g-g_{0})-\int^{u}f^{\prime}g^{\prime}\big{)}=2\mu^{2}\mathcal{L}\bar{\mathcal{L}}\bigg{(}\int^{u}(f^{\prime}g)^{\prime}+(f^{\prime}g)_{zm}-\int^{u}f^{\prime}g^{\prime}-g_{0}f^{\prime}\bigg{)}=$
$\displaystyle=2\mu^{2}\mathcal{L}\bar{\mathcal{L}}\big{(}\int^{u}(g-g_{0})f^{\prime\prime}+(f^{\prime}g)_{zm}\big{)}$
(4.21)

where we are still using the convention that our primitives do not contain a
zero mode. The result precisely coincides with the value of this function in
$T\bar{T}$ \- deformed CFTs, which is rather remarkable, given that our
analysis only used the $\mathcal{O}(1)$ representation theorem in the
asymptotically linear dilaton backgrounds and the assumption that the
$\mathcal{O}(1)$ vector fields should be minimally continued at next order.
The right-movers work in an analogous way.

The full solution for the correction to the vector field is171717The leading
behaviour of the $\boldsymbol{\omega}_{ra}$ components of the symplectic form
evaluated on this solution is given by $1/r$ terms that can be written as
(3.38), and will thus be ignored.

$\displaystyle\xi^{(1)U}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{2\mu\bar{\mathcal{L}}(\mu\mathcal{L}(h-h_{0})+\bar{h}-\bar{h}_{0})}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}f^{\prime}+\mu\bar{\mathcal{L}}\frac{2\mu\mathcal{L}(h-h_{0}+\mu\bar{\mathcal{L}}(\bar{h}-\bar{h}_{0}))}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}\bar{f}^{\prime}+2\mu\bar{\mathcal{L}}\int^{v}\bar{f}^{\prime}\bar{h}^{\prime}+\xi^{U}[\hat{C}_{uv},C_{b}]~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\xi^{(1)V}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{2\mu\mathcal{L}(\mu\bar{\mathcal{L}}(\bar{h}-\bar{h}_{0})+h-h_{0})}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}\bar{f}^{\prime}+\mu\mathcal{L}\frac{2\mu\bar{\mathcal{L}}(\bar{h}-\bar{h}_{0}+\mu\mathcal{L}(h-h_{0}))}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}f^{\prime}+2\mu\mathcal{L}\int^{u}f^{\prime}h^{\prime}+\xi^{V}[\hat{C}_{uv},C_{b}]$
(4.22)

where, as promised, we have included back the terms proportional to
$\hat{C}_{uv}$ and $C_{b}$ that can be freely added to the solution and read

$\xi^{U}[\hat{C}_{uv},C_{b}]=\bigg{(}\hat{C}_{uv}+\frac{C_{b}}{4}\bigg{)}U+\frac{2\mu\bar{\mathcal{L}}\,\hat{C}_{uv}}{1+\mu^{2}\mathcal{L}\bar{\mathcal{L}}}V\;,\;\;\;\;\xi^{V}[\hat{C}_{uv},C_{b}]=\frac{2\mu\mathcal{L}\,\hat{C}_{uv}}{1+\mu^{2}\mathcal{L}\bar{\mathcal{L}}}U+\bigg{(}\hat{C}_{uv}-\frac{C_{b}}{4}\bigg{)}V$
(4.23)

Remember that upon the constant backgrounds, diffeomorphisms of this form were
disallowed, as they would change the identification of the compact coordinate,
$\sigma$. Here, however, the part of the correction to the allowed
diffeomorphisms that _is_ fixed by the symplectic form analysis already has
winding (due to the terms under the integral in (4.22)). This winding can be
off-set by choosing the constants $\hat{C}_{uv},C_{b}$ that parametrize (4.23)
in such a way that the periodicity of the coordinates $U,V$ is overall
unaffected. This choice corresponds to

$\hat{C}_{uv}=-\frac{\mu(\bar{\mathcal{L}}\oint
dv\bar{f}^{\prime}\bar{h}^{\prime}+\mathcal{L}\oint
duf^{\prime}h^{\prime})(1+\mu^{2}\mathcal{L}\bar{\mathcal{L}})}{2\pi
R(1+\mu\mathcal{L})(1+\mu\bar{\mathcal{L}})}$

$C_{b}=\frac{4\mu\mathcal{L}(1+\mu^{2}\mathcal{L}\bar{\mathcal{L}}+2\mu\bar{\mathcal{L}})\oint
duf^{\prime}h^{\prime}-4\mu\bar{\mathcal{L}}(1+\mu^{2}\mathcal{L}\bar{\mathcal{L}}+2\mu\mathcal{L})\oint
dv\bar{f}^{\prime}\bar{h}^{\prime}}{2\pi
R(1+\mu\mathcal{L})(1+\mu\bar{\mathcal{L}})}$ (4.24)

With this choice, the diffeomorphisms (4.22) are _identical_ to their double-
trace counterparts (B.8).

We will henceforth find it convenient to write the full allowed
diffeomorphisms up to $\mathcal{O}(\epsilon)$ in the form

$\xi=\xi_{(p)}+\xi_{(w)}$ (4.25)

where $\xi_{(p)}$ corresponds to the part of the diffeomorphism that is fixed
by the symplectic form analysis and the representation theorem, while
$\xi_{(w)}$ \- which is given by (4.23) with the choice (4.24) for the
parameters - will be referred to as a “compensating” diffeomorphism, in the
sense that its presence is required to off-set the unphysical winding
introduced by $\xi_{(p)}$. As is clear from the solution above, $\xi_{(p)}$
receives contributions at both zeroth and first order in the perturbation,
whereas the expansion of $\xi_{(w)}$ starts at $\mathcal{O}(\epsilon)$. It is
interesting to rewrite the compensating diffeomorphisms in terms of the field-
dependent coordinates $u,v$, with the result

$\xi_{w}^{U}=w_{f}u+\mu\bar{\mathcal{L}}w_{\bar{f}}\;,\;\;\;\;\;\xi^{V}_{w}=w_{\bar{f}}v+\mu\mathcal{L}w_{f}u$
(4.26)

where

$w_{f}=\frac{2\mu^{2}\mathcal{L}\bar{\mathcal{L}}\oint
duf^{\prime}h^{\prime}-2\mu\bar{\mathcal{L}}\oint
dv\bar{h}^{\prime}\bar{f}^{\prime}}{2\pi
R(1+\mu\bar{\mathcal{L}})}\,\epsilon\;,\;\;\;\;\;w_{\bar{f}}=\frac{2\mu^{2}\mathcal{L}\bar{\mathcal{L}}\oint
dv\bar{f}^{\prime}\bar{h}^{\prime}-2\mu\mathcal{L}\oint
duh^{\prime}f^{\prime}}{2\pi R(1+\mu\mathcal{L})}\,\epsilon$ (4.27)

These expressions for the winding precisely coincide with the corresponding
perturbative expansion (B.12) of the winding in double-trace $T\bar{T}$ \-
deformed CFTs. They can alternatively be rewritten in the form (A.39) by
noting that $\epsilon\,\mathcal{L}\oint duf^{\prime}h^{\prime}$ and
$\epsilon\,\bar{\mathcal{L}}\oint dv\bar{f}^{\prime}\bar{h}^{\prime}$ are
nothing but the conserved charges $Q_{f^{\prime}}/p$ and, respectively,
$-\bar{Q}_{\bar{f}^{\prime}}/p$ (3.56) upon the perturbed background.

Finally, we need to show, as promised, that adding these winding terms in the
final step of our computation does not affect the previous steps. Had they
been included from the beginning, they could have affected the expression
(4.16) for $\delta_{\xi}\chi-\delta_{\chi}\xi$. However, from the explicit
form of the solution above, we see that $C_{b},\hat{C}_{uv}$ are symmetric in
$f\leftrightarrow g,\bar{f}\leftrightarrow\bar{g}$. Hence, (4.16) is not
affected by adding the compensating diffeomorphisms, and thus the arguments
presented in this section are unaffected.

###  4.3 Algebra of the conserved charges and match to $T\bar{T}$

Armed with the knowledge of the allowed diffeomorphisms up to
$\mathcal{O}(\epsilon)$, we are now finally ready to compute the algebra of
conserved charges to the same order. For this, it is useful to split the
allowed diffeomorphisms as in (4.25) and to organise the computation into four
separate pieces that correspond, schematically, to

$\displaystyle\delta_{\chi_{(p)}+\chi_{(w)}}Q_{\xi_{(p)}+\xi_{(w)}}$
$\displaystyle=\delta_{\chi_{(p)}}Q_{\xi_{(p)}}+\delta_{\chi_{(w)}}Q_{\xi_{(p)}}+\delta_{\chi_{(p)}}Q_{\xi_{(w)}}+\delta_{\chi_{(w)}}Q_{\xi_{(w)}}$
(4.28)

Since the compensating diffeomorphisms are at least $\mathcal{O}(\epsilon)$,
the last term in (4.28) does not contribute to the order we are interested in.

We first concentrate on the algebra of the “left-movers”, obtained by setting
all the periodic functions of $v$ to zero. The diffeomorphism $\xi$ is
parametrized by the periodic function $f$, and $\chi$ by181818In principle,
one may worry that the variation $\delta_{\chi}\xi$ of the diffeomorphism
between the two backgrounds could contribute to the charge difference. One can
nevertheless show (see e.g. [Compere:2015knw]) that this is not the case.
$g$. After various integrations by parts, the $\mathcal{O}(\epsilon)$
contribution to the periodic-periodic charge difference is

$\displaystyle\bigg{.}\delta_{\chi_{(p)}}Q_{\xi_{(p)}}\bigg{|}_{\mathcal{O}(\epsilon)}\\!\\!\\!\\!$
$\displaystyle=p\,\mathcal{L}\\!\oint\\!d\sigma
r_{u}(fg^{\prime}-gf^{\prime})h^{\prime}+\frac{\mu^{2}\mathcal{L}\bar{\mathcal{L}}}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}\,p\,\mathcal{L}\\!\oint\\!d\sigma
r_{u}(f_{0}g^{\prime}h^{\prime}+g_{0}f^{\prime}h^{\prime})-2\pi
R_{u}p\mathcal{L}w_{f}g_{0}-$ $\displaystyle-\frac{p\mathcal{L}}{2\pi
R}\frac{2\mu^{2}\mathcal{L}\bar{\mathcal{L}}}{1-\mu^{2}\mathcal{L}\bar{\mathcal{L}}}\oint
d\sigma r_{u}f^{\prime}h^{\prime}\oint d\sigma ug^{\prime}$ (4.29)

where we have introduced the field-dependent radius of the $u$ coordinate,
$R_{u}=Rr_{u}$, and similarly $R_{v}=Rr_{v}$. The first term above is
precisely the $\mathcal{O}(\epsilon)$ contribution to
$Q_{fg^{\prime}-gf^{\prime}}$, which agrees with the zeroth order result
(4.4), but is now generically non-zero. Using (3.54), the zero modes $f_{0}$,
$g_{0}$ of the functions can be traded, at this order, for the zeroth order
conserved charges. The expression can be further simplified by writing it in
terms of the the winding (4.27), particularized to $\bar{f}=0$. The final
result is

$\displaystyle\delta_{\chi_{(p)}}Q_{\xi_{(p)}}$
$\displaystyle=Q_{fg^{\prime}-gf^{\prime}}+w_{g}Q_{f}^{(0)}-w_{f}Q_{g}^{(0)}-p\,\mathcal{L}w_{f}\oint
d\sigma r_{u}ug^{\prime}$ (4.30)

This precisely matches the analogous charge difference in the perturbative
double-trace $T\bar{T}$ case, which is computed in appendix (B.2).

The second term in (4.28) vanishes for the particular choice (3.53) of
integration constants in the $\mathcal{O}(1)$ vector fields that was required
by integrability. The details are given in appendix (B.2) for the double-trace
$T\bar{T}$ case, which works identically. Finally, the third term in (4.28)
cancels the integral of the non-periodic function in (4.30) with no finite
contribution, leading to the following final result for the algebra of the
left-movers

$\displaystyle\\{Q_{f},Q_{g}\\}\equiv\delta_{g}Q_{f}$
$\displaystyle=Q_{fg^{\prime}-gf^{\prime}}+w_{g}Q_{f}^{(0)}-w_{f}Q_{g}^{(0)}$
(4.31)

which holds up to $\mathcal{O}(\epsilon)$. This can be rewritten in terms of
the conserved charges as

$\boxed{\\{Q_{f},Q_{g}\\}=Q_{fg^{\prime}-gf^{\prime}}+\frac{\mu^{2}H_{R}}{p^{2}\pi^{2}RR_{H}}\left(Q_{g^{\prime}}Q_{f}^{(0)}-Q_{f^{\prime}}Q_{g}^{(0)}\right)}$
(4.32)

where we defined
$R_{H}=R(r_{u}+r_{v}-1)=R(1+\mu\mathcal{L})(1+\mu\bar{\mathcal{L}})/(1-\mu^{2}\mathcal{L}\bar{\mathcal{L}})$.
Using a Fourier basis for $f,g$ and taking into account the fact that
$f^{\prime}=imf/R_{u}$, we find precisely the algebra (1.3) with
$\mu\rightarrow\mu/p$ that was advertised in the introduction. A similar
result holds for the right-movers.

The commutation relations of the charges associated with one left-moving and
one right-moving diffeomorphism can be obtained by setting $\bar{f}(v)=0$ in
$\xi$ and $g(u)=0$ in $\chi$. The steps of the computation are explained in
appendix (B.2) for the case of the double-trace $T\bar{T}$ deformation, which
turn out to be identical to this order in perturbation theory to our
computation in the asymptotically linear dilaton background. The result is

$\boxed{\\{Q_{f},\bar{Q}_{\bar{g}}\\}=w_{g}Q_{f}^{(0)}-w_{\bar{f}}\bar{Q}_{\bar{g}}^{(0)}=\frac{\mu}{\pi
pRR_{H}}\left(R_{v}\bar{Q}_{\bar{g}^{\prime}}Q_{f}^{(0)}+R_{u}Q_{f^{\prime}}\bar{Q}_{\bar{g}}^{(0)}\right)}$
(4.33)

where the windings are again given by (4.27), particularized to $\bar{f}=g=0$.

Hence, the charge algebra that we obtain at $\mathcal{O}(\epsilon)$ is _non-
linear_ in the conserved charges $Q_{f},\bar{Q}_{\bar{f}}$. The source of non-
linearity is the presence of charge-dependent parameters191919A similar
mechanism for obtaining a non-linear charge algebra has recently appeared in
[54, 55] for the case of five-dimensional asymptotically flat spacetimes.,
namely the windings which, as discussed in the previous subsection, are
required for consistency of the asymptotic coordinate transformations. This
algebra _precisely matches_ the algebra of the “unrescaled” generators in
single-trace $T\bar{T}$ \- deformed CFTs, where the factors of $p$ are exactly
those required by the symmetric product orbifold [25]. We have thus succeeded
to establish an additional important link between asymptotically linear
dilaton backgrounds and the single-trace $T\bar{T}$ deformation, by showing
that the full doubly-infinite set of extended symmetries of the two theories
are _identical_ to the order we checked.

We end this section with a few technical comments. First, we checked that the
redefinition $\delta_{f}u\mapsto\delta_{f}u+\gamma f^{\prime}$ that adds a
term $\gamma f^{\prime}h^{\prime}/(1-\mu^{2}\mathcal{L}\bar{\mathcal{L}})$ to
$F_{h}$ and could not be fixed by the arguments of the previous section
contributes to the charge difference (4.28) only a total derivative term,
which vanishes upon integration. We were therefore justified to neglect this
term.

Second, it would be interesting to show that we obtain the charge algebra
(A.49) for _arbitrary_ functions $f,g$, rather than just when one of them is a
constant. This requires computing the $\mathcal{O}(\epsilon^{2})$ corrections
to the charge algebra. While, in principle, this computation can be performed
along the same lines of the $\mathcal{O}(\epsilon)$ one, it nevertheless
becomes extremely cumbersome. One may however wonder whether the interesting
non-linear terms in the algebra, which are proportional to the windings, could
not be recovered just from the $\mathcal{O}(\epsilon)$ solution for the vector
fields, given that they are always multiplied by the winding, which starts at
$\mathcal{O}(\epsilon)$. This expectation turns out to be too naïve, for two
reasons: first, the non-linear contributions include terms where the charges
are at $\mathcal{O}(1)$ and therefore the windings are at
$\mathcal{O}(\epsilon^{2})$; these “compensating” windings cannot be computed
without knowing the full $O(\epsilon^{2})$ correction to the $\xi_{(p)}$ part
of the vector fields that is determined from the symplectic form analysis.
Second, as can be seen (B.8) for the doube-trace $T\bar{T}$ computation
expanded up to $\mathcal{O}(\epsilon^{2})$ around the constant backgrounds,
the $\xi_{(p)}$ piece of the vector fields receives
$\mathcal{O}(\epsilon^{2})$ contributions that depend on the windings at
$\mathcal{O}(\epsilon)$ and enter the result of the charge algebra. Thus, the
computation cannot be performed without explicit knowledge of the
$\mathcal{O}(\epsilon^{2})$ corrections to the allowed diffeomorphisms.

Finally, it is natural to ask whether the algebra (4.32), (4.33) can be
linearized by a redefinition of the generators. Guided by the $T\bar{T}$
results, we expect that the rescaled generators $R_{u}Q_{f}$ and
$R_{v}\bar{Q}_{\bar{f}}$ will simply satisfy a Virasoro $\times$ Virasoro
algebra. As explained at length in appendix A, the effects of a field-
dependent rescaling of the diffeomorphisms on the associated charge algebra
are somewhat non-trivially implemented in the covariant phase space formalism
via a different solution to the charge integrability constraints. Concretely,
starting from the $\mathcal{O}(1)$ expression (3.51) and temporarily setting
$\bar{f}_{0}=0$ for simplicity, we can easily note that the charges are again
integrable for the field-dependent choice

$f_{0}=R_{u}\tilde{f}_{0}=\left(R+\frac{\mu H_{R}}{\pi
p}\right)\tilde{f}_{0}\;,\;\;\;\;\;c_{V}=-\mu\mathcal{L}f_{0}-\frac{\mu
H_{L}}{\pi p}\tilde{f}_{0}=-\frac{2\mu H_{L}}{\pi p}\tilde{f}_{0}$ (4.34)

where $\tilde{f}_{0}$ is now field-independent, and we have used (3.33) and
the fact that $H_{L}=\pi p\mathcal{L}R_{u}$ to manipulate the expressions. A
similar choice holds for the right movers. The charges (3.51) associated to
the diffeomorphisms $\tilde{f}_{0}R_{u}\partial_{U}$ and, respectively,
$-\tilde{\bar{f}}_{0}R_{v}\partial_{V}$ become

$\displaystyle\cancel{\delta}Q_{\tilde{f}_{0}R_{u}\partial_{U}}$
$\displaystyle=$ $\displaystyle\;\tilde{f}_{0}R_{u}\delta H_{L}+\frac{\mu
H_{L}}{\pi p}\tilde{f}_{0}\delta H_{R}=\delta(\tilde{f}_{0}R_{u}H_{L})$
$\displaystyle\cancel{\delta}Q_{-\tilde{\bar{f}}_{0}R_{v}\partial_{V}}$
$\displaystyle=$ $\displaystyle-\tilde{\bar{f}}_{0}R_{v}\delta H_{R}-\frac{\mu
H_{R}}{\pi p}\tilde{\bar{f}}_{0}\delta
H_{L}=-\delta(\tilde{\bar{f}}_{0}R_{v}H_{R})$ (4.35)

The set of allowed vector fields that are associated with this new
integrability constraint are, to leading order

$\displaystyle\xi_{L}$
$\displaystyle=R_{u}\left(\tilde{f}(u)\partial_{U}+\mu\mathcal{L}(\tilde{f}-2\tilde{f}_{0})\partial_{V}\right)\;,\;\;\;\;\;\;\xi_{R}=R_{v}(\tilde{\bar{f}}(v)\partial_{V}+\mu\bar{\mathcal{L}}(\tilde{\bar{f}}-2\tilde{\bar{f}}_{0})\partial_{U})$
(4.36)

One can straightforwardly repeat the steps of the previous section and compute
the $\mathcal{O}(\epsilon)$ correction to these vector fields. The constraints
(4.11) are not affected and the correction takes exactly the same form as
before, depending on two functions analogous to $F_{h},\bar{F}_{\bar{h}}$. The
arguments that determine these functions still hold. However, fixing the
constant field-dependent piece of the $\mathcal{O}(\epsilon)$ correction to
$\xi_{(p)}$ (i.e., the $\mathcal{O}(\epsilon)$ correction to $c_{V}$ in (4.34)
above) is somewhat more challenging in this case, as the full non-linear
$T\bar{T}$ answer (A.41) shows it is only fixed by integrability - a property
that we did not check beyond leading order in the asymptotically linear
dilaton background. Fortunately, this extra constant does not contribute to
the charge algebra at $\mathcal{O}(\epsilon)$ \- as can also be seen from
(A.36), and the final result for the left-moving algebra corresonding to this
rescaled set of vector fields is

$\displaystyle\delta_{\chi}(R_{u}Q_{\xi})$
$\displaystyle=R_{u}Q_{fg^{\prime}-gf^{\prime}}$ (4.37)

in perfect agreement with the single-trace $T\bar{T}$ prediction [25]. A
similar result holds for the right-movers, with $R_{u}$ replaced by $R_{v}$.

##  5 Discussion

The main result of this article was to show that the asymptotic symmetry
algebra of asymptotically linear dilaton backgrounds in string theory
_precisely matches_ that of single-trace $T\bar{T}$ \- deformed CFTs, at least
to the order in perturbation theory to which we have checked. This further
strengthens the connection proposed in [18] between compactified little string
theory and a symmetric product orbifold of $T\bar{T}$ \- deformed CFTs.

This asymptotic symmetry group analysis of the linear dilaton backgrounds was
performed in parallel to that of the spacetime dual to double-trace $T\bar{T}$
\- deformed CFTs - namely, AdS3 with mixed boundary conditions [24] \- using
the same formalism. Despite the entirely different asymptotics of these two
space-times and the different boundary conditions for the allowed
fluctuations, the asymptotic symmetry generators and conserved charges were
found to be virtually identical at each step of the calculation, up to a
constant rescaling related to the symmetric product orbifold. The only
quantity that our study of the asymptotically linear dilaton background could
not fix was the central extension of the asymptotic symmetry algebra, as the