# Boundary Conditions for Warped AdS3 in Quadratic Ensemble

Ankit Aggarwal,a,b Luca Ciambelli,b Stéphane Detournayb and Antoine
Somerhausenb

_ a Institute for Theoretical Physics Amsterdam and Delta Institute for
Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH
Amsterdam, The Netherlands
_

_ b Physique Mathématique des Interactions Fondamentales & International
Solvay Institutes, Université Libre de Bruxelles, Campus Plaine - CP 231, 1050
Bruxelles, Belgium<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
_

###### Abstract

In the context of warped conformal field theories (WCFT), the derivation of
the warped Cardy formula relies on the zero mode spectrum being bounded from
below. Generically, this is not true for holographic WCFTs in “canonical”
ensemble, whereas this condition is satisfied in the “quadratic” ensemble,
making it more natural in holographic setups. In this work, we find a new set
of boundary conditions in three-dimensional Topologically Massive Gravity
(TMG) such that the putative dual theory is a WCFT in quadratic ensemble.
Surprisingly, imposing the equations of motion, we obtain a closed form metric
spanned by two arbitrary chiral functions, analogous to the Bañados metrics in
Einstein gravity. Surface charges for these boundary conditions are not a
priori integrable and we discuss two choices of boundary conditions to make
them so. We obtain the bulk thermodynamic entropy of warped BTZ black holes by
making use of the warped Cardy formula, in its regime of validity. We also
discuss the issue of identifying the vacuum in our solution space: demanding
the enhancement of global symmetries selects only a family of solutions, out
of which the unique vacuum must be carefully singled out.

###### Contents

  1. 1 Introduction
  2. 2 From WAdS3 to Warped BTZ and warped Cardy formula
  3. 3 New boundary conditions
    1. 3.1 Setup
    2. 3.2 Limits
  4. 4 Symmetries and charges
    1. 4.1 Residual symmetries and variation of solution space
    2. 4.2 Surface charges and algebra
      1. 4.2.1 The case $\delta j_{++}=0$ ($\Delta$ fixed)
      2. 4.2.2 Varying $\Delta$ and generators redefinition
  5. 5 Entropy matching
    1. 5.1 Bulk thermodynamics
    2. 5.2 WCFT Cardy formula and matching
  6. 6 Conclusions and outlook
  7. A New boundary conditions including pp waves

##  1 Introduction

In the search for toy models able to capture the main features of the most
salient black hole conundrums, lower-dimensional theories [1, 2, 3, 4] have
always stood out.111The literature, especially for $3$-dimensional gravity, is
vast. Some standard useful books and reviews are [5, 6, 7]. The seminal work
of Brown and Henneaux [8] has for the first time pointed at a connection
between AdS gravity and a CFT, and BTZ black holes [9, 10] have successfully
been used in a variety of contexts to explore properties of gravity beyond the
classical regime. In recent years, the principles of AdS/CFT have been applied
to Kerr black holes with potential groundbreaking applications in astronomy
[11]. The status of the correspondence is however not settled as several
consistent and possibly related proposals exist [12, 13, 14, 15, 16, 17].

A useful lower-dimensional toy model for Kerr black holes has emerged over the
years in the form of a deformed BTZ spacetime, the so-called Warped AdS3
(WAdS3) black holes, [18, 19, 20, 21, 22, 23, 24, 25, 26]. WAdS3 black holes
display features departing from AdS3 black holes and closer quantitatively to
Kerr. For instance, the near-horizon geometry of extremal black holes has
$SL(2,R)\times U(1)$ isometry like the near horizon region of extremal Kerr
black holes (NHEK) [27]; actually, it is a section of constant polar angle of
NHEK. Also, their Penrose diagram resembles that of an asymptotically flat
black hole [28]. The study of the asymptotic symmetry group of WAdS3 black
holes in the spirit of Brown and Henneaux has revealed that they belong to a
phase space with symmetries consisting in the semi-direct product of a chiral
Virasoro algebra with an affine u(1) current [24, 29, 30, 31]. This suggested
the existence of a new class of 2d field theories, Warped Conformal Field
Theories (WCFTs) [14], invariant under translations and chiral scaling but not
rotations [32], to which WAdS3 would be dual to. The symmetries of WCFT are,
to some extend, almost as powerful as a $2$-dimensional CFT. For this reason,
many of the questions that can be explored and answered in a $2$-dimensional
CFT have also become a subject of interest for WCFTs in the last few years:
bootstrap techniques [33], the study of partition functions [14, 34], and
their entanglement entropy [35, 36, 37]. Interestingly, examples of WCFTs have
been found in chiral Liouville gravity [38], free Weyl fermions [39], and free
scalars [40].

States in a WCFT can be labelled by the zero modes of the algebra (2.4-2.6):
$L_{0}$ and $P_{0}$. However, we are free to use another combination of
generators to refer to them, for instance, the zero modes of the algebra
(2.28-2.30), $\tilde{L}_{0}$ and $\tilde{P}_{0}$. From these generators, one
can build two different density matrices, $\rho_{c}=e^{-\beta P_{0}+i\theta
L_{0}}$ corresponding to what we refer to as canonical ensemble and
$\rho_{q}=e^{-\beta_{L}\tilde{P_{0}}-\beta_{R}\tilde{L_{0}}}$ corresponding to
what we will call quadratic ensemble. The entropy of a thermal state is
independent of the ensemble. Now, for $\rho_{c}$, which is the ensemble
usually considered for WAdS3 black holes, we cannot use a Cardy like formula
to derive the black hole entropy, because the conditions to derive this Cardy-
like formula are not satisfied by the dual bulk spacetime, because the
spectrum of $L_{0}$ is not bounded from below. For $\rho_{q}$ instead, we do
know how to proceed, and we can compute the entropy. The charges
$(L_{n},P_{n})$ (called “canonical”) and $(\tilde{L}_{n},\tilde{P}_{n}$)
(dubbed “quadratic”) are related to one another, and yield slightly different
algebras (referred to as canonical (2.4)-(2.6) and quadratic algebras
(2.28)-(2.30)). This operation at the boundary has a counterpart in the bulk:
it is a non-local change of coordinates, which amounts to two different
choices of boundary conditions. Performing it on the regular WAdS3 black hole
solution (dual to $\rho_{c}$), one gets the black hole solution dual to
$\rho_{q}$ (Warped BTZ). The goal of this paper is to write boundary
conditions including these solutions, and to have a bulk realization of the
boundary relation between canonical and quadratic generators. For the zero
mode charges, and solutions, this is trivial and spelled out in [14]. We here
extended this to the whole phase space and asymptotic Killing vectors. The
phase space obtained is shown to have a symmetry algebra of WCFT expressed in
quadratic charges, as expected. The boundary conditions we obtain are a
natural generalization of the Compère, Song, and Strominger (CSS) boundary
conditions [41] for AdS3 spaces (which in our terminology are in quadratic
ensemble, satisfying (2.28-2.30)), to allow for more leading metric components
as one approaches the boundary.222In that sense, the boundary conditions of
[29] for $\mu\ell=\pm 3$, i.e. in the AdS3 limit, are the CSS boundary
conditions in canonical ensemble described in App. B of [41].

The paper is organized as follows. In Sec. 2, we review WAdS3 black holes and
the bulk change of coordinates to go to the WBTZ line element. We discuss how
WAdS3 black holes are dual to a WCFT in canonical ensemble while the WBTZ line
element is dual to the quadratic ensemble. We further show that the warped
Cardy formula is well-defined only for WCFTs dual to WBTZ coordinates. This
serves as a motivation for introducing in Sec. 3 new boundary conditions
naturally encompassing WBTZ black holes. We further show in this section two
relevant limits of our new solution space: the null warped and CSS limit. A
discussion of the various regimes as a function of the TMG Chern-Simons
coupling constant is also presented. In Sec. 4, we compute the surface charges
in TMG, and find that they are not integrable in general. We propose two ways
to render them integrable: a restriction of the solution space, giving rise to
a solution dual to a thermal state in a WCFT in quadratic ensemble; and a
redefinition of the symmetry generators, giving rise to a solution dual to a
thermal state in a WCFT in canonical ensemble. We subsequently perform the
entropy analysis in Sec. 5. We compute the bulk thermodynamic entropy and
compare it with the WCFT Cardy formula, showing that they match once the
vacuum is correctly identify. This is a delicate procedure because the
standard enhancement of isometries is not enough to single out the vacuum.
This fact opens the door to interesting future investigations, that we discuss
in the conclusions, after summarizing the main achievements of the paper. We
exile to Appendix A the study of new boundary conditions that include pp
waves.

##  2 From WAdS3 to Warped BTZ and warped Cardy formula

We present in this section a self-contained review of the WAdS3 analysis, the
bulk change of coordinates to WBTZ black holes, and the warped Cardy formula
which is counting the degeneracy of states in the dual WCFT. We detail the
conditions on the spectrum to ensure the well-definiteness of this formula,
and show that these conditions are met in holographic setup in the quadratic
ensemble, but not in the canonical ensemble.

The line element for Warped black holes is (see e.g. [21, 22, 23, 24])

$\displaystyle\text{d}s^{2}=\frac{\text{d$\rho$}^{2}}{\frac{12j}{\mu}+\frac{1}{9}{\rho^{2}\over\ell^{2}}\left(\mu^{2}\ell^{2}+27\right)-12m\rho}+\ell^{2}\text{dT}^{2}-\frac{4}{3}\mu\ell\rho\
\text{dT}\ \text{d$\psi$}+\left({\rho^{2}\over
3\ell^{2}}\big{(}\mu^{2}\ell^{2}-3\big{)}+12m\rho-\frac{12j}{\mu}\right)\text{d$\psi$}^{2}$
(2.1)

with $\rho\in[0,\infty)$, $T\in(-\infty,\infty)$, and $\psi\sim\psi+2\pi$.
This solves the $3$-dimensional Topologically Massive Gravity (TMG) [42]
equations of motion

$R_{\mu\nu}-{R\over 2}g_{\mu\nu}+\Lambda g_{\mu\nu}+{1\over\mu}C_{\mu\nu}=0.$
(2.2)

This theory depends on two parameters: the cosmological constant $\Lambda$ and
the Chern-Simons coupling $\mu$. While the equations of motion (2.2) have
solutions for all values of $\mu$, we can, without loss of generality, assume
$\mu>0$ in the following. This comes about because each negative-$\mu$
solution can be rewritten with $\mu>0$ sending $\mu\to-\mu$, $\psi\to-\psi$,
and $j\to-j$. Indeed, due to the conventions in the Cotton Hodge dual, defined
below, eq. (2.2) are invariant under this symmetry. The Cotton Hodge dual is
defined as

$C_{\mu\nu}=\eta_{\mu}{}^{\rho\sigma}\nabla_{\rho}(R_{\sigma\nu}-{1\over
4}Rg_{\sigma\nu})\qquad\text{with}\qquad\eta_{\mu\nu\rho}=\sqrt{|g|}\varepsilon_{\mu\nu\rho},$
(2.3)

where $\eta_{\mu\nu\rho}$ and $\varepsilon_{\mu\nu\rho}$ are the Levi-Civita
tensor and symbol, respectively, and $\Lambda=-{1\over\ell^{2}}$. It is worth
noting that the metric (2.1) becomes locally AdS3 at $\mu={3\over\ell}$ (it is
the BTZ black hole line element [9, 10], albeit in unusual coordinates).

The metric (2.1) has been extensively studied in the literature, and boundary
conditions have been outlined in previous accounts on the topic, [25, 29, 30,
26, 31]. It belongs to a phase space whose asymptotic symmetry algebra is
given by that of a WCFT in _canonical_ ensemble

$\displaystyle\left[L_{n},L_{m}\right]$ $\displaystyle=$
$\displaystyle(n-m)L_{n+m}+\frac{c}{12}n(n-1)(n+1)\delta_{n+m},$ (2.4)
$\displaystyle\left[L_{n},P_{m}\right]$ $\displaystyle=$ $\displaystyle-
mP_{m+n},$ (2.5) $\displaystyle\left[P_{n},P_{m}\right]$ $\displaystyle=$
$\displaystyle\frac{k}{2}n\delta_{n+m},$ (2.6)

where the central extensions are given by [24, 29, 30, 31]333In comparing with
these references, we have different factors of $\ell$ because our time
coordinate is dimensionless.

$\displaystyle c=\frac{15\ell^{2}\mu^{2}+81}{G\ell^{2}\mu^{3}+27G\mu},\qquad
k=-\frac{\ell^{2}\mu^{2}+27}{18G\mu}.$ (2.7)

For black hole solutions, the zero modes of the charges are given by

$\displaystyle L_{0}$ $\displaystyle=$ $\displaystyle
Q_{\partial_{\psi}}=-{j\over
6G\mu^{2}\ell^{2}}\left(27+5\ell^{2}\mu^{2}\right)+\frac{9m^{2}}{2G\mu},$
(2.8) $\displaystyle P_{0}$ $\displaystyle=$ $\displaystyle
Q_{\partial_{T}}={\ell m\over G}.$ (2.9)

Requiring the absence of naked singularities we find, in terms of the
parameters $m$ and $j$, that

$m^{2}\geq\frac{j}{\ell^{2}\mu}+\frac{j\mu}{27}=-{2Gkj\over 3\ell^{2}}.$
(2.10)

This can be translated in terms of the zero mode charges as

$L_{0}\geq{P_{0}^{2}\over k}.$ (2.11)

For these boundary conditions, we get a WCFT in canonical ensemble. The warped
Cardy formula [14] is

$\displaystyle S=-{4\pi iP_{0}P_{0}^{vac}\over
k}+4\pi\sqrt{-\Big{(}L_{0}^{vac}-{(P_{0}^{vac})^{2}\over
k}\Big{)}\Big{(}L_{0}-{P_{0}^{2}\over k}\Big{)}}.$ (2.12)

The vacuum values of the charges were previously identified as

$\displaystyle\quad
L_{0}^{vac}=-\frac{c}{24}+\frac{(P_{0}^{vac})^{2}}{k},\qquad(P_{0}^{vac})^{2}=-\frac{\ell^{2}}{36}.$
(2.13)

Notice that we have $c>0$ and thus

$\displaystyle L_{0}^{vac}-{(P_{0}^{vac})^{2}\over k}=-\frac{c}{24}<0.$ (2.14)

Consequently, the first condition for the well-definiteness of (2.12) is

$L_{0}-{P_{0}^{2}\over k}>0,$ (2.15)

which is automatically satisfied thanks to (2.11). The result (2.12) was shown
to match the Bekenstein-Hawking entropy of the corresponding dual black hole
metrics in TMG. However, another condition on the spectrum necessary to derive
(2.12) is that $L_{0}$ is bounded from below, see [14]. Given (2.11), and the
fact that $j$ can attain any value, we are in a setup where $L_{0}$ is
unbounded, which means that we are not in the regime of validity of the warped
Cardy formula. This makes the use of the canonical ensemble in holographic
instances questionable.

In [14], WAdS3 black holes were reinterpreted as a deformation of the BTZ
metrics – of the form (2.23). The resulting spaces, already mentioned in the
introduction and dubbed “Warped BTZ” (WBTZ) are related to (2.1) through a
charge-dependent change of coordinates. This change of coordinates induces a
boundary change of coordinates between the vacuum metric on the plane (the
warped version of the Poincare metric) and the black holes of Rindler type
[14, 43], allowing an interpretation of the black holes as being dual to a
thermal density matrix in the spirit of [44]. This change of coordinates has
the effect of redefining the zero mode charges and suggests that the symmetry
algebra differs from that of a WCFT in canonical ensemble. From the bulk
gravitational field space viewpoint, this field-dependent change of
coordinates leads to the same metric falloffs at the boundary but different
varying quantities, that is, different boundary conditions.444An equivalent
way to formulate the problem is to say that we are looking for the field-
dependent behavior of the (ADM) lapse and shift for which the charges are
integrable. In particular, the level becomes charge-dependent (see also [33]
for a discussion), and the new zero-mode charges are related to the former
ones by a quadratic redefinition (see Sect.4, eq.(55) of [14]). A WCFT
partition function involving these new charges is therefore said to be in
_quadratic ensemble_. In the latter, the conditions on the spectrum to ensure
the validity of the warped Cardy formula are satisfied, as we review below.
Nonetheless, intrinsic boundary conditions for the quadratic ensemble,
containing these black hole solutions, have not been derived so far. As stated
in the introduction, the main goal of this paper is to derive the
aforementioned boundary conditions, such that the asymptotic symmetry algebra
is that of a WCFT in quadratic ensemble and the spectrum is in the regime of
validity of the warped Cardy formula, as just explained.

The change of coordinates aforementioned brings (2.1) to the so-called WBTZ
metric. Calling the new coordinates $(t,r,\phi)$, it is explicitly given by

$\displaystyle T(t,r,\phi)$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{2G\left(1-2H^{2}\right)\left(2H^{2}+3\right)(LM-J)}}{\sqrt{3L}L}\
t,$ (2.16) $\displaystyle\rho(t,r,\phi)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3L}L}{4\sqrt{2G\left(2H^{2}+3\right)(LM-J)}}r^{2}-\frac{\sqrt{3GL}JL^{2}}{\sqrt{2\left(2H^{2}+3\right)(LM-J)}},$
(2.17) $\displaystyle\psi(t,r,\phi)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2H^{2}+3}(L\phi+t)}{\sqrt{3}L^{2}}.$ (2.18)

The parameters $M,J,H,$ and $L$ just introduced are related to the WAdS3
parameters as

$\displaystyle\ell=\frac{\sqrt{3}L}{\sqrt{3+2H^{2}}},\quad\mu=\frac{3\sqrt{1-2H^{2}}}{L},\quad
m=\sqrt{2}\sqrt{\frac{GL(LM-J)}{3(2H^{2}+3)}},\quad
j=-\frac{3G\sqrt{1-2H^{2}}JL^{2}}{2H^{2}+3}.$ (2.19)

The change of coordinates (2.16-2.18) is a non-local/charge dependent
transformation. Indeed, defining $x^{\pm}={t\over L}\pm\phi$, we observe in
particular that

$\displaystyle
T=\frac{2\sqrt{2}\sqrt{G}\sqrt{\sqrt{1-2H^{2}}\left(2H^{2}+3\right)}}{L^{3/2}}\sqrt{\tilde{{{P}}}_{0}}\
t,$ (2.20)

where we introduced

$Q_{\partial_{-}}\equiv\tilde{{P}}_{0}=\frac{1}{3}\sqrt{1-2H^{2}}(LM-J).$
(2.21)

The time coordinate is therefore rescaled by the charge. As we will shortly
see, this affects the asymptotic symmetry algebra in a non-trivial way.

The resulting metric after the change of coordinates (2.16-2.18) is

$\displaystyle\text{d}s^{2}_{WBTZ}$ $\displaystyle=$
$\displaystyle\frac{L^{2}r^{2}}{16G^{2}J^{2}L^{2}-8GL^{2}Mr^{2}+r^{4}}\text{d}r^{2}-2(H^{2}-{1\over
2})(4GL^{2}M-r^{2})\text{d}x^{+}\text{d}x^{-}$
$\displaystyle-{8G^{2}L^{2}(J^{2}+(2H^{2}-1)L^{2}M^{2})-8GH^{2}L^{2}Mr^{2}+H^{2}r^{4}\over
4GL(LM-J)}\text{d}x^{+2}-2GL(2H^{2}-1)(LM-J)\text{d}x^{-2}.$

This is the so-called WBTZ line element, which as advertised earlier can be
obtained as a deformation of the classic BTZ black hole spacetime [9, 10] of
the form

$\text{d}s^{2}_{WBTZ}=\text{d}s^{2}_{BTZ}-2H^{2}\xi\otimes\xi,$ (2.23)

with

$\displaystyle\text{d}s^{2}_{BTZ}=\Big{(}-{r^{2}\over
L^{2}}+8MG\Big{)}\text{d}t^{2}+{r^{2}L^{2}\text{d}r^{2}\over
r^{4}+16L^{2}J^{2}G^{2}-8MGL^{2}r^{2}}+8JG\text{d}t\text{d}\phi+r^{2}\text{d}\phi^{2},$
(2.24)

the BTZ metric solving Einstein equations with $\Lambda=-{1\over L^{2}}$. The
deformation $\xi$ is the one form dual under $\text{d}s^{2}_{BTZ}$ to the
vector field

$\underline{\xi}={1\over\sqrt{8GL(LM-J)}}(-L\partial_{t}+\partial_{\phi})=-{1\over\sqrt{2GL(LM-J)}}\partial_{-},$
(2.25)

where the prefactor is obtained requiring the norm of this vector being $1$.

The metric (2.23) solves the TMG equations of motion (2.2) for555The signature
here is $(-1,1,1)$ and we choose $\varepsilon^{tr\phi}={1\over\sqrt{-g}}$. If
one changes the convention on $\varepsilon$, i.e., changes the order of the
coordinates, then one has to change the sign of $\mu$ accordingly.

$\Lambda=-{1\over 27L^{2}}\big{(}36-\mu^{2}L^{2}\big{)},\quad
H^{2}={9-\mu^{2}L^{2}\over 18},\quad\mu>0.$ (2.26)

The metric has two globally defined Killing vectors $\partial_{-}$ and
$\partial_{+}$ with corresponding conserved charges given by (2.21) and

$Q_{\partial_{+}}\equiv\tilde{{L}}_{0}={2(1-H^{2})\over
3\sqrt{1-2H^{2}}}{(LM+J)}.$ (2.27)

The family of solutions (2) was conjectured in [14] to be dual to thermal
states in a WCFT in quadratic ensemble, that is a $2$-dimensional field theory
with symmetry algebra

$\displaystyle\left[\tilde{L}_{n},\tilde{L}_{m}\right]$ $\displaystyle=$
$\displaystyle(n-m)\tilde{L}_{n+m}+\frac{c}{12}\left(n^{3}-n\right)\delta_{n+m},$
(2.28) $\displaystyle\left[\tilde{L}_{n},\tilde{P}_{m}\right]$
$\displaystyle=$ $\displaystyle-m\tilde{P}_{m+n},$ (2.29)
$\displaystyle\left[\tilde{P}_{n},\tilde{P}_{m}\right]$ $\displaystyle=$
$\displaystyle-2n\tilde{P}_{0}\delta_{m+n}.$ (2.30)

The asymptotic degeneracy of states of field theories with algebras
(2.28-2.30) has been determined in [14] in a Cardy-like regime assuming
certain conditions on the spectrum

$\displaystyle
S=4\pi\sqrt{-\tilde{P}_{0}^{vac}\tilde{P}_{0}}+4\pi\sqrt{-\tilde{L}_{0}^{vac}\tilde{L}_{0}}.$
(2.31)

Here, $\tilde{P}_{0}^{vac}$ and $\tilde{L}_{0}^{vac}$ are the zero modes of
the vacuum of the theory in the quadratic ensemble. From the change of
coordinates, the new zero modes are

$\displaystyle\tilde{L}_{0}={L}_{0}-{{P}_{0}^{2}\over
k},\qquad\tilde{P}_{0}=-{{P}_{0}^{2}\over k}.$ (2.32)

The condition on the spectrum is that $\tilde{L}_{0}$ must be bounded from
below which, thanks to (2.15), is now satisfied in this ensemble, because
$\tilde{L}_{0}\geq 0$. Hence, we are in the domain of validity, as claimed
above.

The goal of this manuscript is to determine a phase space in TMG with metrics
falling off at infinity like the WBTZ metrics, which exhibit a different fall-
off from BTZ due to the warping. Indeed, comparing (2) with the unwarped BTZ
metric,

$\text{d}s^{2}_{BTZ}=\frac{L^{2}r^{2}}{16G^{2}J^{2}L^{2}-8GL^{2}Mr^{2}+r^{4}}\text{d}r^{2}+(4GL^{2}M-r^{2})\text{d}x^{+}\text{d}x^{-}+2GL(LM+J)\text{d}x^{+2}+2GL(LM-J)\text{d}x^{-2},$
(2.33)

we see that the warping is over-leading in the $\text{d}x^{+}$ component.
Eventually, this means that we are looking for relaxed boundary conditions, in
order to capture WBTZ metrics, of the form

$g_{++}=O(r^{4}),\quad g_{+-}=O(r^{2}),\quad g_{--}=O(1),\quad
g_{rr}={L^{2}\over r^{2}}+O(r^{-4}).$ (2.34)

and show that its symmetries are given by (2.28-2.30).

##  3 New boundary conditions

Guided by the motivations of the previous section, we derive here new boundary
conditions encompassing WBTZ black holes. We first outline the boundary
conditions, then show how the solution space is restricted to give WBTZ black
holes, and finally study two limits in which we retrieve familiar line
elements.

###  3.1 Setup

A solution of (2.2) is given, trading $\Lambda$ for $L$ as

$\Lambda=-{1\over 27L^{2}}\big{(}36-\mu^{2}L^{2}\big{)},$ (3.1)

and using the Fefferman-Graham (FG) gauge, with coordinates
$x^{a}=(x^{+},x^{-})$ such that $x^{\pm}={t\over L}\pm\phi$,666One could have
chosen any other length $\ell$ to normalize time. As long as we pick a
quantity with dimension length it is the same. Note that $L$ becomes $\ell$,
and so becomes the usual definition of $x^{\pm}$, only at $\mu L=\pm 3$. The
conventions on $x^{\pm}$ are here the same as in [45] (see [109] there).

$\text{d}s^{2}=g_{\mu\nu}\text{d}x^{\mu}\text{d}x^{\nu}={L^{2}\over
r^{2}}\text{d}r^{2}+g_{ab}(r,x)\text{d}x^{a}\text{d}x^{b},$ (3.2)

by the following metric components

$\displaystyle g_{++}$ $\displaystyle=$ $\displaystyle
r^{4}j_{++}+r^{2}h(x^{+})+f_{++}(x^{+})+\frac{h(x^{+})\left(\mu^{2}L^{2}-9\right)\left(4j_{++}\mu^{2}L^{2}f_{++}(x^{+})-h(x^{+})^{2}\left(\mu^{2}L^{2}-9\right)\right)}{8r^{2}j_{++}^{2}\mu^{2}L^{2}\left(\mu^{2}L^{2}+9\right)}$
(3.3)
$\displaystyle+\frac{\left(\mu^{2}L^{2}-9\right)^{2}\left(4j_{++}\mu^{2}L^{2}f_{++}(x^{+})+h(x^{+})^{2}\left(9-\mu^{2}L^{2}\right)\right)^{2}}{64r^{4}j_{++}^{3}\mu^{4}L^{4}\left(\mu^{2}L^{2}+9\right)^{2}}$
$\displaystyle g_{+-}$ $\displaystyle=$
$\displaystyle-\frac{1}{18}\mu^{2}L^{2}r^{2}-\frac{h(x^{+})\left(\mu^{2}L^{2}-9\right)}{36j_{++}}-\frac{\left(\mu^{2}L^{2}-9\right)\left(4j_{++}\mu^{2}L^{2}f_{++}(x^{+})+h(x^{+})^{2}\left(9-\mu^{2}L^{2}\right)\right)}{144j_{++}^{2}r^{2}\left(\mu^{2}L^{2}+9\right)}$
(3.4) $\displaystyle g_{--}$ $\displaystyle=$
$\displaystyle\frac{\mu^{2}L^{2}\left(\mu^{2}L^{2}-9\right)}{324j_{++}}.$
(3.5)

This metric solves (2.2), with the convention $\varepsilon_{r+-}=-1$, which is
consistent with $\varepsilon^{tr\phi}=1$. Note that the first term in (3.4) is
the would-be boundary metric, which is thus flat. One could generalize this
ansatz including a Weyl factor, and reproduce in this context what has been
done in [46]. We will not do this enhancement here, leaving it for future
work, and this will consequently freeze the dilatation symmetry in the
asymptotic vectors below.

The parameter $L$ is from now on assumed to be positive, $L>0$, which implies
that we have a solution of the equations of motion only in the range $\mu>0$.
The solution space is characterized by three quantities: a constant $j_{++}$
and two chiral functions $h(x^{+})$ and $f_{++}(x^{+})$. Various comments are
in order here. The first is that the parameter $L$ becomes the usual AdS
radius $\ell$ only for $\mu L=\pm{3}$. At these points indeed
$\Lambda=-{1\over L^{2}}$. These values of $\mu$ are critical, and we will
study shortly what happens there. In general, the $L$ appearing in the FG
gauge is not the usual one, in contrast with what happens e.g. in [26, 31].
The second remark is that the line element has been directly defined on-shell,
whereas it would have been more rigorous to find first appropriate boundary
conditions off-shell. It can be viewed as a counterpart of the Bañados metrics
for AdS3.

It is important to note that the ansatz we made to solve the TMG equations of
motion involved only integer powers of $r$. This in turn leads to a family of
metrics that is locally equivalent to the Warped AdS3 black holes, as can be
checked by inspecting curvature invariants built out of the Ricci and Cotton
tensors. Therefore, our analysis does not include the massive graviton around
global WAdS3. Including more general, non-diffeomorphic, solutions capturing
the massive mode would require to introduce $mu$-dependent fall-offs, in the
spirit of [47, 26] for AdS3 in TMG, and of [31] for WAdS3. We leave this for
future work.

Our boundary conditions are interesting because they allow to navigate through
the various space-like, time-like, and null warped solutions. The WBTZ black
holes in Fefferman-Graham gauge are given by

$\displaystyle\text{d}s^{2}$ $\displaystyle=$
$\displaystyle\frac{\text{d}r^{2}L^{2}}{r^{2}}+\frac{2}{9}\text{d}x^{-2}G\mu^{2}L^{3}(LM-J)-\frac{\text{d}x^{-}\text{d}x^{+}\mu^{2}L^{2}\left(4G^{2}L^{2}(L^{2}M^{2}-J^{2})+r^{4}\right)}{9r^{2}}$
(3.7)
$\displaystyle+\frac{\text{d}x^{+2}\left(\mu^{2}L^{2}\left(4G^{2}L^{2}(L^{2}M^{2}-J^{2})+r^{4}\right)^{2}-9\left(r^{4}-4G^{2}L^{2}(L^{2}M^{2}-J^{2})\right)^{2}\right)}{72GLr^{4}(LM-J)}.$

These solutions are included in our boundary conditions (3.3-3.5) for

$\displaystyle j_{++}$ $\displaystyle=$
$\displaystyle\frac{\mu^{2}L^{2}-9}{72GL(LM-J)},$ (3.8) $\displaystyle h$
$\displaystyle=$ $\displaystyle 0,$ (3.9) $\displaystyle f_{++}$
$\displaystyle=$
$\displaystyle\frac{1}{9}GL\left(\mu^{2}L^{2}+9\right)(J+LM),$ (3.10)
$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle{\mu^{2}L^{2}-9\over
144GLj_{++}}={(LM-J)\over 2}.$ (3.11)

Here we introduced also the quantity $\Delta$, which plays an important role
in what follows.

Note that, given (3.1), if we want to keep the cosmological constant negative
we have to assume $-6<\mu L<6$, which is further restricted to $0<\mu L<6$,
since we assumed both $\mu$ and $L$ positive. The scalar curvature of (3.2) is
given by

$R={2(\mu L+6)(\mu L-6)\over 9L^{2}}=6\Lambda$ (3.12)

Consequently, it is negative as long as $\Lambda$ is. Although extensions to
asymptotically flat and WdS spacetimes are part of our agenda (see e.g. [48]),
from now on we will focus on these values of the Chern-Simons coupling.

###  3.2 Limits

Our new line element has two interesting limits. The first one is the null
warped (NW) limit obtained setting $\mu L=3$ while keeping $j_{++}$ arbitrary.
The cosmological constant becomes $\Lambda=-{1\over L^{2}}=-{1\over\ell^{2}}$
and the line element collapses to

$\displaystyle\text{d}s^{2}={\ell^{2}\over
r^{2}}\text{d}r^{2}+(r^{4}j_{++}+r^{2}h(x^{+})+f_{++}(x^{+}))(\text{d}x^{+})^{2}-r^{2}\text{d}x^{+}\text{d}x^{-}.$
(3.13)

This metric is not an Einstein space, for the Cotton tensor has a non-
vanishing component $C_{++}={12j_{++}r^{4}\over\ell^{3}}$. The line element
(3.13) falls inside a bigger class of boundary conditions, where the constant
$j_{++}$ is allowed to be chiral $j_{++}(x^{+})$ [49]. See Appendix A for
further details.

The second limit is the CSS limit. This limit involves both $\mu$ and
$j_{++}$. It is performed sending $j_{++}$ to zero and $\mu L\to 3$ keeping
their ratio constant, i.e. keeping $g_{--}$ finite in the limit. The quantity
$\Delta={\mu^{2}L^{2}-9\over 144GLj_{++}}$ introduced in (3.11) is
particularly relevant for this limit. Note that its sign depends on the sign
of $j_{++}$. Using $\Lambda\to-{1\over L^{2}}=-{1\over\ell^{2}}$, we obtain

$\displaystyle\text{d}s^{2}$ $\displaystyle=$ $\displaystyle{\ell^{2}\over
r^{2}}\text{d}r^{2}+\Big{(}r^{2}h(x^{+})+f_{++}(x^{+})-\frac{4\Delta G\ell
h(x^{+})(4\Delta G\ell
h(x^{+})^{2}-f_{++}(x^{+}))}{r^{2}}\Big{)}(\text{d}x^{+})^{2}+4\Delta
G\ell(dx^{-})^{2}$ (3.14) $\displaystyle-\Big{(}r^{2}+8\Delta G\ell
h(x^{+})-\frac{4\Delta G\ell(4\Delta G\ell
h(x^{+})^{2}-f_{++}(x^{+}))}{r^{2}}\Big{)}\text{d}x^{+}\text{d}x^{-}.$

We call this the CSS limit because the metric falls into CSS boundary
conditions [41], embedded in TMG as in [45]. This metric has identically
vanishing Cotton tensor, so it is a solution of Einstein equations. The exact
matching between (3.14) and the CSS line element (eq. (1) in [45]) is made via
the identification

$\displaystyle h(x^{+})=\partial_{+}P(x^{+}),\qquad
f_{++}(x^{+})=4G\ell(\bar{L}(x^{+})+\Delta(\partial_{+}P(x^{+}))^{2}).$ (3.15)

We have therefore introduced new boundary conditions that encompass previous
results. Furthermore, they are useful because the asymptotic symmetry algebra
turns out to be that of a WCFT in quadratic ensemble, as we now demonstrate.

We may also observe that the metric (3.7) becomes ill-defined when $J=ML$.
However, as we will see in the forthcoming sections and can already be seen on
(2.21) and (2.27), the charges themselves are well-defined at these values of
the parameters, indicating a mere coordinate singularity. The metric can be
made regular and non-degenerate by a rescaling $r\rightarrow(-J+LM)^{1/4}r$
and $x^{-}\rightarrow(-J+LM)^{-1/2}x^{-}$, and still belonging to our family
of metrics.

##  4 Symmetries and charges

We begin this section with the analysis of residual symmetries and the most
general variation of solution space. We then show that surface charges are in
general not integrable, and discuss two possible consistent sets of boundary
conditions: a restriction of the solution space, or a field-dependent
redefinition of the generators. In both cases, we study the charge algebra and
consider WBTZ black holes.

###  4.1 Residual symmetries and variation of solution space

We denote by $\chi$ the solution space,
$\chi=\\{j_{++},h(x^{+}),f_{++}(x^{+})\\}$. We impose to preserve the FG gauge

$\displaystyle\pounds_{\underline{\xi}}g_{rr}=0=\pounds_{\underline{\xi}}g_{ra},$
(4.1)

where $\pounds$ denotes the Lie derivative. As reviewed in [50, 51], the most
general solution is simply

$\displaystyle\xi^{r}=r\eta(x^{+},x^{-}),\qquad\xi^{b}=\xi^{b}_{0}(x^{+},x^{-})-L^{2}\partial_{a}\eta(x^{+},x^{-})\int_{\infty}^{r}\text{d}r^{\prime}{g^{ab}(x^{+},x^{-})\over
r^{\prime}},$ (4.2)

and depends on three arbitrary functions $\eta(x^{+},x^{-})$ and
$\xi^{b}_{0}(x^{+},x^{-})$. Given the discussion above, it is natural to
furthermore impose that residual symmetries leave the metric chiral:

$\displaystyle\xi^{r}=r\eta(x^{+}),\qquad\xi^{b}=\xi^{b}_{0}(x^{+})-L^{2}\partial_{a}\eta(x^{+})\int_{\infty}^{r}\text{d}r^{\prime}{g^{ab}(x^{+})\over
r^{\prime}}.$ (4.3)

To avoid confusion, and because everything is chiral, we call
$\xi^{+}_{0}(x^{+})=\epsilon$ and $\xi^{-}_{0}(x^{+})=\sigma$ in the
following. Given our line element, we obtain :

$\displaystyle\underline{\xi}$ $\displaystyle=$
$\displaystyle\xi^{r}\partial_{r}+\xi^{+}\partial_{+}+\xi^{-}\partial_{-}$
(4.5) $\displaystyle=$ $\displaystyle
r\eta\partial_{r}+\Big{(}\epsilon-\frac{2j_{++}\mu^{2}L^{4}(\mu^{4}L^{4}-81)\eta^{\prime}}{9(-4j_{++}\mu^{2}L^{2}f_{++}(\mu^{2}L^{2}-9)+h^{2}(\mu^{2}L^{2}-9)^{2}+8j_{++}^{2}\mu^{2}L^{2}r^{4}(\mu^{2}L^{2}+9))}\Big{)}\partial_{+}$
$\displaystyle+\Big{(}\sigma-\frac{2j_{++}L^{2}(\mu^{2}L^{2}+9)\eta^{\prime}(h(\mu^{2}L^{2}-9)+4j_{++}\mu^{2}L^{2}r^{2})}{-4j_{++}\mu^{2}L^{2}f_{++}(\mu^{2}L^{2}-9)+h^{2}(\mu^{2}L^{2}-9)^{2}+8j_{++}^{2}\mu^{2}L^{2}r^{4}(\mu^{2}L^{2}+9)}\Big{)}\partial_{-}.$

We remark that, since the line element has a finite expansion in the
coordinate $r$, these vector fields are written in closed form in $r$.

We now proceed to compute the variation of the solution under residual
symmetries $\delta_{\underline{\xi}}\chi$ using as usual
$\delta_{\underline{\xi}}g_{\mu\nu}=\pounds_{\underline{\xi}}g_{\mu\nu}$,
expanded in powers of $r$. We first focus on the leading term in the $++$
component:

$\delta_{\underline{\xi}}j_{++}=2j_{++}(2\eta(x^{+})+\epsilon^{\prime}(x^{+})).$
(4.6)

We then require $j_{++}$ to remain constant, which implies

$\eta=-{1\over 2}\epsilon^{\prime}+\eta_{0},$ (4.7)

with $\eta_{0}$ a constant transforming $j_{++}$:

$\delta_{\underline{\xi}}j_{++}=4j_{++}\eta_{0}.$ (4.8)

Although this is admissible for $j_{++}$, if we compute the $+-$ component we
obtain

$\pounds_{\underline{\xi}}g_{+-}=-r^{2}{\mu^{2}L^{2}\eta_{0}\over
9}+O(r^{0}),$ (4.9)

which is not allowed, and thus we must impose $\eta_{0}=0$. It would be
interesting to enhance this construction including an arbitrary boundary
conformal factor as done in [46] for Einstein gravity, allowing us to make
contact with the recent work [52]. Keeping $\eta_{0}=0$ means that $j_{++}$ is
fixed along the residual orbits. We proceed and find the full residual
variation of $\chi$, that we summarize here:

$\displaystyle\delta_{\underline{\xi}}j_{++}$ $\displaystyle=$ $\displaystyle
0,$ (4.10) $\displaystyle\delta_{\underline{\xi}}h$ $\displaystyle=$
$\displaystyle\epsilon
h^{\prime}+h\epsilon^{\prime}-\frac{1}{9}\mu^{2}L^{2}\sigma^{\prime},$ (4.11)
$\displaystyle\delta_{\underline{\xi}}f_{++}$ $\displaystyle=$
$\displaystyle\epsilon
f_{++}^{\prime}+2f_{++}\epsilon^{\prime}-{(\mu^{2}L^{2}-9)h\sigma^{\prime}\over
18j_{++}}-{(\mu^{2}L^{2}+9)L^{2}\epsilon^{\prime\prime\prime}\over 36}.$
(4.12)

The second expression suggests that $h$ is a $u(1)$ current with level related
to the last term. The last expression indicates that $f_{++}$ is a Virasoro
current, where the last term is the one related to the central extension.

The most general residual symmetries are thus generated by the on-shell
vectors:

$\displaystyle\underline{\xi}$ $\displaystyle=$
$\displaystyle\xi^{r}\partial_{r}+\xi^{+}\partial_{+}+\xi^{-}\partial_{-},$
(4.14) $\displaystyle=$ $\displaystyle-{r\over
2}\epsilon^{\prime}\partial_{r}+\Big{(}\epsilon+\frac{j_{++}\mu^{2}L^{4}(\mu^{4}L^{4}-81)\epsilon^{\prime\prime}}{9(-4j_{++}\mu^{2}L^{2}f_{++}(\mu^{2}L^{2}-9)+h^{2}(\mu^{2}L^{2}-9)^{2}+8j_{++}^{2}\mu^{2}L^{2}r^{4}(\mu^{2}L^{2}+9))}\Big{)}\partial_{+}$
$\displaystyle+\Big{(}\sigma+\frac{j_{++}L^{2}(\mu^{2}L^{2}+9)\epsilon^{\prime\prime}(h(\mu^{2}L^{2}-9)+4j_{++}\mu^{2}L^{2}r^{2})}{-4j_{++}\mu^{2}L^{2}f_{++}(\mu^{2}L^{2}-9)+h^{2}(\mu^{2}L^{2}-9)^{2}+8j_{++}^{2}\mu^{2}L^{2}r^{4}(\mu^{2}L^{2}+9)}\Big{)}\partial_{-},$

which depend on two arbitrary chiral functions $\epsilon(x^{+})$ and
$\sigma(x^{+})$. For the asymptotic symmetry algebra, $\sigma$ generates an
abelian algebra

$[\underline{\sigma}^{1},\underline{\sigma}^{2}]=0,\quad\text{with}\quad\underline{\sigma}^{i}=\sigma_{i}(x^{+})\partial_{-},$
(4.15)

while $\epsilon$ generates the usual Witt algebra, under the modified Lie
brackets, see [53], due to the field dependence of the vectors. The total
asymptotic symmetry algebra is therefore a semi-direct sum of a Witt and a
$u(1)$ algebra.

The notation we will adopt is

$\displaystyle\underline{\sigma}$ $\displaystyle=$
$\displaystyle\sigma(x^{+})\partial_{-},$ (4.16)
$\displaystyle\underline{\epsilon}$ $\displaystyle=$ $\displaystyle-{r\over
2}\epsilon^{\prime}\partial_{r}+\Big{(}\epsilon+\frac{j_{++}\mu^{2}L^{4}(\mu^{4}L^{4}-81)\epsilon^{\prime\prime}}{9(-4j_{++}\mu^{2}L^{2}f_{++}(\mu^{2}L^{2}-9)+h^{2}(\mu^{2}L^{2}-9)^{2}+8j_{++}^{2}\mu^{2}L^{2}r^{4}(\mu^{2}L^{2}+9))}\Big{)}\partial_{+}$
(4.17)
$\displaystyle+\frac{j_{++}L^{2}(\mu^{2}L^{2}+9)\epsilon^{\prime\prime}(h(\mu^{2}L^{2}-9)+4j_{++}\mu^{2}L^{2}r^{2})}{-4j_{++}\mu^{2}L^{2}f_{++}(\mu^{2}L^{2}-9)+h^{2}(\mu^{2}L^{2}-9)^{2}+8j_{++}^{2}\mu^{2}L^{2}r^{4}(\mu^{2}L^{2}+9)}\partial_{-}.$

At this stage, we have found the variation of solution space and the residual
symmetries, so we have all the ingredients to compute surface charges.

###  4.2 Surface charges and algebra

We start with the $u(1)$ sector and compute TMG surfaces charges. We obtain
that the Iyer-Wald [54, 55, 22, 25] charges read777In this paper, charges are
computed using the package [56].

$\cancel{\delta}Q_{\underline{\sigma}}[g,h]=-\frac{\mu^{2}L^{2}-9}{1296\pi
G\mu L^{2}j_{++}^{2}}\int_{0}^{2\pi}\text{d}\phi\ \sigma(x^{+})\Big{(}\delta
j_{++}(\mu^{2}L^{2}+9h(x^{+}))-18j_{++}\delta h(x^{+})\Big{)}.$ (4.18)

Here we have computed the surface charges first at $(r,x^{-})$ fixed, then at
$(r,x^{+})$ fixed, added them and sent $r\to\infty$. These charges are finite,
conserved but integrable only if $\delta j_{++}=0$. However, as we will see
one can find combination of residual vectors such that these charges become
integrable even when $\delta j_{++}\neq 0$. Although there they are
integrable, we observe some similarities with [45] and [46], where one needs
to find specific orbits of the residual symmetry vectors to obtain a direct
sum algebra. Indeed, in cases where the charges are not integrable, such
redefinition makes them so, as discussed in [57, 58, 59, 60].

Before proceeding, we report here also the Virasoro charges

$\displaystyle\cancel{\delta}Q_{\underline{\epsilon}}[g,h]$ $\displaystyle=$
$\displaystyle{1\over 144\pi
GL^{4}\mu^{3}j_{++}^{2}}\int^{2\pi}_{0}\text{d}\phi\Big{(}27(\mu^{2}L^{2}-9)h^{2}\epsilon\delta
j_{++}+(\mu^{2}L^{2}-9)h\epsilon(\mu^{2}L^{2}\delta j_{++}-54j_{++}\delta h)$
(4.19) $\displaystyle+\mu^{2}L^{2}j_{++}\Big{(}L^{2}(\mu^{2}L^{2}-9)\delta
j_{++}\epsilon^{\prime\prime}+72j_{++}\epsilon\delta f_{++}\Big{)}\Big{)}.$

where we used

$\displaystyle\underline{\epsilon}$ $\displaystyle=$ $\displaystyle-{r\over
2}\epsilon^{\prime}\partial_{r}+\Big{(}\epsilon+\frac{j_{++}\mu^{2}L^{4}(\mu^{4}L^{4}-81)\epsilon^{\prime\prime}}{9(-4j_{++}\mu^{2}L^{2}f_{++}(\mu^{2}L^{2}-9)+h^{2}(\mu^{2}L^{2}-9)^{2}+8j_{++}^{2}\mu^{2}L^{2}r^{4}(\mu^{2}L^{2}+9))}\Big{)}\partial_{+}$
(4.20)
$\displaystyle+\Big{(}\frac{j_{++}L^{2}(\mu^{2}L^{2}+9)\epsilon^{\prime\prime}(h(\mu^{2}L^{2}-9)+4j_{++}\mu^{2}L^{2}r^{2})}{-4j_{++}\mu^{2}L^{2}f_{++}(\mu^{2}L^{2}-9)+h^{2}(\mu^{2}L^{2}-9)^{2}+8j_{++}^{2}\mu^{2}L^{2}r^{4}(\mu^{2}L^{2}+9)}\Big{)}\partial_{-}.$

We would then then to compute the charge algebra. To do so, we need to find a
set of integrable charges. As we discussed, there exist two possibilities that
do not further constraint the residual vectors

  1. 1.

Set $\delta j_{++}=0$ on the solution space.

  2. 2.

Perform a field dependent redefinition of the generators of residual
symmetries $\sigma$ and $\epsilon$.

Both possibilities have interesting consequences, so we study them in detail
in the following subsections.

####  4.2.1 The case $\delta j_{++}=0$ ($\Delta$ fixed)

A simple situation is attained imposing $\delta j_{++}=0$, which does not
further reduce the residual symmetries. The charges then read

$\displaystyle\cancel{\delta}Q_{\underline{\sigma}}[g,h]$ $\displaystyle=$
$\displaystyle\frac{\mu^{2}L^{2}-9}{72\pi G\mu
L^{2}j_{++}}\int_{0}^{2\pi}\text{d}\phi\ \sigma(x^{+})\delta h(x^{+}),$ (4.21)
$\displaystyle\cancel{\delta}Q_{\underline{\epsilon}}[g,h]$ $\displaystyle=$
$\displaystyle{1\over 8\pi GL^{4}\mu^{3}j_{++}}\int^{2\pi}_{0}\text{d}\phi\
\epsilon(x^{+})\Big{(}4\mu^{2}L^{2}j_{++}\delta
f_{++}-3(\mu^{2}L^{2}-9)h\delta h\Big{)}.$ (4.22)

These can now be integrated up to an arbitrary background value (integration
constant). We fix this by demanding that the zero mode of the charge
$Q_{\underline{\sigma}}[g]$ matches its zero mode when $j_{++}$ is allowed to
vary, as also done in [41]. This also ensures that we reproduce the charges
for the exact isometries of WBTZ black holes. We thus get

$\displaystyle Q_{\underline{\sigma}}[g]$ $\displaystyle=$
$\displaystyle\frac{\mu^{2}L^{2}-9}{72\pi G\mu
L^{2}j_{++}}\int_{0}^{2\pi}\text{d}\phi\ \sigma(h+{\mu^{2}L^{2}\over 18}),$
(4.23) $\displaystyle Q_{\underline{\epsilon}}[g]$ $\displaystyle=$
$\displaystyle{1\over 16\pi GL^{4}\mu^{3}j_{++}}\int^{2\pi}_{0}\text{d}\phi\
\epsilon\Big{(}8\mu^{2}L^{2}j_{++}f_{++}-3(\mu^{2}L^{2}-9)h^{2}\Big{)}.$
(4.24)

We can therefore directly compute the charge algebra here. We start with the
$u(1)$ sector and get

$\displaystyle\\{Q_{\underline{\sigma}^{1}}[g],Q_{\underline{\sigma}^{2}}[g]\\}=\delta_{\underline{\sigma}^{2}}Q_{\underline{\sigma}^{1}}[g]=\frac{\mu^{2}L^{2}-9}{72\pi
G\mu L^{2}j_{++}}\int_{0}^{2\pi}\text{d}\phi\
\sigma_{1}\delta_{\underline{\sigma}^{2}}h=-\frac{\mu(\mu^{2}L^{2}-9)}{648\pi
Gj_{++}}\int_{0}^{2\pi}\text{d}\phi\ \sigma_{1}\sigma_{2}^{\prime}.$ (4.25)

Since $Q_{[\underline{\sigma}^{1},\underline{\sigma}^{2}]_{M}}[g]=0$, (4.25)
is the central extension for the $u(1)$ sector. Using the mode decomposition
representation $\sigma_{1}=e^{imx^{+}}$ and $\sigma_{2}=e^{inx^{+}}$, and
calling $Q_{\underline{\sigma}^{1}}[g]=\tilde{{P}}_{m}$ and
$Q_{\underline{\sigma}^{2}}[g]=\tilde{{P}}_{n}$:

$\displaystyle\tilde{{P}}_{m}=\frac{\mu^{2}L^{2}-9}{72\pi G\mu
L^{2}j_{++}}\int_{0}^{2\pi}\text{d}\phi\ e^{imx^{+}}(h+{\mu^{2}L^{2}\over
18}){=\frac{2\Delta}{\mu L\pi}\int_{0}^{2\pi}\text{d}\phi\
e^{imx^{+}}(h+{\mu^{2}L^{2}\over 18})},$ (4.26)

we obtain888Conventions: $\int_{0}^{2\pi}\text{d}\phi
e^{i(m+n)\phi}=2\pi\delta_{m+n,0}$.

$i\\{\tilde{{P}}_{m},\tilde{{P}}_{n}\\}=m{\tilde{k}\over
2}\delta_{m+n,0},\qquad\tilde{k}=-\frac{\mu(\mu^{2}L^{2}-9)}{162Gj_{++}}=-\frac{8\mu
L}{9}\Delta.$ (4.27)

This is a centrally extended $u(1)$ algebra with central extension $\tilde{k}$
– colloquially called Kac-Moody level. We want to make connection with the
quadratic algebra (2.28) and that the zero modes of our metric represent WBTZ
black holes. This requires that $h(x^{+})$ doesn’t have a zero mode (see
(3.9)) which can be achieved by demanding that $h(x+)=\partial_{+}H(x^{+})$
for some arbitrary periodic function $H(x^{+})$. This is analogous to the
discussion in [41], where one of the arbitrary chiral function that appears in
the metric is $\partial_{+}P(x^{+})$, ensuring that the zero modes of the CSS
metric coincides with those of the BTZ black hole (see (3.15))999Also, note
that (2.4)-(2.6) of [33] is correct only if $H(x^{+})$ is periodic.. Thus, the
condition $h(x^{+})=\partial_{+}H(x^{+})$ with $H(x^{+})$ periodic implies
that

$\tilde{k}=-4\tilde{{P}}_{0}.$ (4.28)

For the Virasoro sector we have:

$\displaystyle\\{Q_{\underline{\epsilon}^{1}}[g],Q_{\underline{\epsilon}^{2}}[g]\\}=\delta_{\underline{\epsilon}^{2}}Q_{\underline{\epsilon}^{1}}[g]$
$\displaystyle=$ $\displaystyle{1\over 16\pi
GL^{4}\mu^{3}j_{++}}\int^{2\pi}_{0}\text{d}\phi\Big{(}\Big{(}\epsilon_{1}\epsilon^{\prime}_{2}-\epsilon_{2}\epsilon^{\prime}_{1}\Big{)}\Big{(}8\mu^{2}L^{2}j_{++}f_{++}-3(\mu^{2}L^{2}-9)h^{2}\Big{)}$
(4.29) $\displaystyle-{(\mu^{2}L^{2}+9)\over 72\pi
G\mu}\int^{2\pi}_{0}\text{d}\phi\epsilon_{2}^{\prime\prime\prime}\epsilon_{1}.$

The first line in this expression is clearly
$Q_{[\underline{\epsilon}^{1},\underline{\epsilon}^{2}]_{M}}[g]$, where one
should use the modified Lie bracket

$\displaystyle\big{[}\underline{\epsilon}^{1},\underline{\epsilon}^{2}\big{]}_{M}:=\big{[}\underline{\epsilon}^{1},\underline{\epsilon}^{2}\big{]}-\delta_{\underline{\epsilon}^{1}}\underline{\epsilon}^{2}+\delta_{\underline{\epsilon}^{2}}\underline{\epsilon}^{1},$
(4.30)

because the vectors are field dependent, while the second line is the central
extension. Using the mode decomposition representation
$\epsilon_{1}=e^{imx^{+}}$ and $\epsilon_{2}=e^{inx^{+}}$, and calling
$Q_{\underline{\epsilon}^{1}}[g]=\tilde{{L}}_{m}$ and
$Q_{\underline{\epsilon}^{2}}[g]=\tilde{{L}}_{n}$:

$\tilde{{L}}_{m}={1\over 16\pi
GL^{4}\mu^{3}j_{++}}\int^{2\pi}_{0}\text{d}\phi\
e^{imx^{+}}\Big{(}8\mu^{2}L^{2}j_{++}f_{++}-3(\mu^{2}L^{2}-9)h^{2}\Big{)},$
(4.31)

we obtain

$i\\{\tilde{{L}}_{m},\tilde{{L}}_{n}\\}=(m-n)\tilde{{L}}_{m+n}+{c\over
12}m^{3}\delta_{m+n,0},\qquad c=\frac{\mu^{2}L^{2}+9}{3G\mu}.$ (4.32)

Note that both central charges are sensitive to the sign of $\mu$.

Finally the mixed sector, with the conventions established, reads

$i\\{\tilde{{L}}_{m},\tilde{{P}}_{n}\\}=-n\tilde{{P}}_{m+n},$ (4.33)

which is the expected semi-direct action of the Virasoro sector on the Abelian
one.

To summarize, we have obtained the algebra

$\displaystyle i\\{\tilde{{L}}_{m},\tilde{{L}}_{n}\\}$ $\displaystyle=$
$\displaystyle(m-n)\tilde{{L}}_{m+n}+{c\over 12}m^{3}\delta_{m+n,0},$ (4.34)
$\displaystyle i\\{\tilde{{L}}_{m},\tilde{{P}}_{n}\\}$ $\displaystyle=$
$\displaystyle-n\tilde{{P}}_{m+n},$ (4.35) $\displaystyle
i\\{\tilde{{P}}_{m},\tilde{{P}}_{n}\\}$ $\displaystyle=$ $\displaystyle
m{\tilde{k}\over 2}\delta_{m+n,0},$ (4.36)

with central extensions

$\displaystyle
c=\frac{\mu^{2}L^{2}+9}{3G\mu},\qquad\tilde{k}=-\frac{\mu(\mu^{2}L^{2}-9)}{162Gj_{++}}=-\frac{8\mu
L}{9}\Delta,$ (4.37)

where we recall $\Delta={\mu^{2}L^{2}-9\over 144GLj_{++}}$. This algebra is
the one of the centrally extended group

$Vir\ltimes U(1).$ (4.38)

While $c$ is field independent, the $U(1)$ central extension $\tilde{k}$
depends on $j_{++}$. With (4.28), the algebra precisely becomes (2.28-2.30),
confirming the expectation that our phase space has the same symmetries as
that of a WCFT in quadratic ensemble. This is why we used the tilde notation
for the generators.

Let us study the Null Warped limit, reached simply taking $\mu L\to 3$. In
this case the $u(1)$ level and charges vanish identically, and we are left
with a Virasoro symmetry algebra with central extension

$c=\frac{2\ell}{G}\quad\text{at}\quad\mu L\to 3,$ (4.39)

where again $L\to\ell$, the AdS radius, in this limit. The disappearance of
the $u(1)$ is the reason why we typically think of null warped solutions as
extremal. This is also suggested by the fact that the chiral function $h$
disappears from the residual vectors in (4.5) and the charges (4.21-4.22) in
the limit, where therefore it is a pure gauge datum.

As we have seen, the CSS limit is reached setting $\mu L\to 3$ and $j_{++}\to
0$ while keeping ${\mu^{2}L^{2}-9\over 144GLj_{++}}=\Delta$ constant. The
charges read

$\displaystyle
Q_{\underline{\sigma}}[g]=\frac{2\Delta}{3\pi}\int_{0}^{2\pi}\text{d}\phi\
\sigma(h+{1\over 2}),\qquad Q_{\underline{\epsilon}}[g]={1\over 6\pi
G\ell}\int^{2\pi}_{0}\text{d}\phi\ \epsilon\Big{(}f_{++}-6\Delta\ell
Gh^{2}\Big{)},$ (4.40)

while the central extensions become

$\tilde{k}=-\frac{8\Delta}{3},\qquad c=\frac{2\ell}{G}.$ (4.41)

Thus, this limit coincides with our findings in [45], where, for generic
$\mu$, the central extensions were found to be

$\displaystyle k_{\text{CSS}}=-\big{(}1-{1\over\mu\ell}\big{)}4\Delta,\quad
c_{\text{CSS}}=\big{(}1+{1\over\mu\ell}\big{)}\frac{3\ell}{2G}.$ (4.42)

We remark that the CSS metrics, even when embedded in TMG, have vanishing
Cotton, so the equations of motion are invariant under chiral parity
transformations. On the other hand, our boundary conditions admit solutions
with non-vanishing Cotton so they are sensitive to chirality. This ultimately
constraints the sign of $\mu$ (here assumed positive) as we previously saw.

We now turn our attention to WBTZ black holes. These are reached restricting
the solution space to

$\displaystyle j_{++}=\frac{\mu^{2}L^{2}-9}{72GL(LM-J)},\quad h=0,\quad
f_{++}=\frac{1}{9}GL\left(\mu^{2}L^{2}+9\right)(J+LM),\quad\Delta={\mu^{2}L^{2}-9\over
144GLj_{++}}={(LM-J)\over 2}.$ (4.43)

Therefore, their charges in the quadratic ensemble take the form

$\displaystyle\tilde{{P}}_{m}=\frac{(LM-J)\mu
L}{9}\delta_{m,0},\qquad\tilde{{L}}_{m}=(\mu^{2}L^{2}+9){(LM+J)\over
9L\mu}\delta_{m,0}.$ (4.44)

with $M$ and $J$ the Einstein charges. The TMG mass and angular momentum of
these solutions are defined as

$\displaystyle\tilde{M}=Q_{\partial_{t}}={1\over
L}\Big{(}Q_{\partial_{+}}+Q_{\partial_{-}}\Big{)},\qquad\tilde{J}=Q_{\partial_{\phi}}=Q_{\partial_{+}}-Q_{\partial_{-}},$
(4.45)

and we also have

$\displaystyle Q_{\partial_{-}}=\tilde{{P}}_{0},\qquad
Q_{\partial_{+}}=\tilde{{L}}_{0}.$ (4.46)

Subsequently, we obtain the relationship between the TMG mass and angular
momentum and the zero modes of the charges

$\displaystyle\tilde{M}={1\over
L}\Big{(}\tilde{{L}}_{0}+\tilde{{P}}_{0}\Big{)}={2\mu LM\over 9}+{LM+J\over\mu
L^{2}},\qquad\tilde{J}=\tilde{{L}}_{0}-\tilde{{P}}_{0}={2\mu LJ\over
9}+{LM+J\over\mu L}.$ (4.47)

For BTZ, it has been found in [45] that101010In comparing, we should change
the sign of $J$ and send the charge for $\partial_{\phi}$ to the charge for
$-\partial_{\phi}$.

$\displaystyle
Q_{\underline{\sigma}}[g]=\frac{1}{2\pi\mu\ell}\int_{0}^{2\pi}\text{d}\phi\
\sigma(\mu\ell-1)\Delta,\qquad
Q_{\underline{\epsilon}}[g]=\frac{1}{2\pi\mu\ell}\int_{0}^{2\pi}\text{d}\phi\
\epsilon(\mu\ell+1)\bar{\Delta},$ (4.48)

with $\Delta={1\over 2}(LM-J)$, $\bar{\Delta}={1\over 2}(LM+J)$, and
$f_{++}=4GL\bar{\Delta}$. At $\mu\ell=3$, this consistently coincides with the
results above. In particular, the mass and angular momentum (4.47) in the CSS
limit at $\mu L=3$ read

$\displaystyle\tilde{M}=M+{J\over 3\ell},\qquad\tilde{J}=J+{\ell M\over 3},$
(4.49)

which coincide exactly with $(21)$ and $(22)$ of [45] for $\mu\ell=3$, if we
consistently change the sign of $J$ and $\tilde{J}$, because there we used
$\tilde{J}=Q_{-\partial_{\phi}}$.

To summarize, we showed in this subsection that the bulk solution space has a
symmetry algebra identified with that of a WCFT in quadratic ensemble.

####  4.2.2 Varying $\Delta$ and generators redefinition

Another set of boundary conditions that makes our charges integrable is
achieved performing a redefinition of the generators of residual symmetries
following CSS [41]

$\displaystyle\underline{\sigma}\to\underline{\boldsymbol{\sigma}}={3\underline{\sigma}\over\mu
L\sqrt{|\Delta|}}\quad\Rightarrow\quad\boldsymbol{\sigma}={3\sigma\over\mu
L\sqrt{|\Delta|}},\qquad\underline{\epsilon}\to\underline{\boldsymbol{\epsilon}}=\underline{\epsilon}-\epsilon(x^{+})\partial_{-}\quad\Rightarrow\quad\boldsymbol{\epsilon}={\epsilon}.$
(4.50)

Here we took the absolute value of $\Delta$ because this constant can assume
both positive and negative values. So the two new chiral vector fields are
$\underline{\boldsymbol{\epsilon}}$ and $\underline{\boldsymbol{\sigma}}$, and
depend on two arbitrary chiral functions $\epsilon(x^{+})$ and
$\sigma(x^{+})$. We are allowed to do so because all the generators are
functions of $x^{+}$, so the Fourier decomposition is the same. Furthermore,
the redefinition of $\boldsymbol{\epsilon}$ uses the fact that we subtract to
the old one the $u(1)$ generator, which eventually keeps the algebra
invariant.

The $u(1)$ charges then become

$\displaystyle\cancel{\delta}Q_{\underline{\boldsymbol{\sigma}}}[g,h]$
$\displaystyle=$ $\displaystyle-\frac{\mu^{2}L^{2}-9}{1296\pi G\mu
L^{2}j_{++}^{2}}\int_{0}^{2\pi}\text{d}\phi\
\boldsymbol{\sigma}(x^{+})\Big{(}\delta
j_{++}(\mu^{2}L^{2}+9h(x^{+}))-18j_{++}\delta h(x^{+})\Big{)}$
$\displaystyle=$
$\displaystyle-\frac{(\mu^{2}L^{2}-9)}{36\pi\sqrt{GL^{5}}\mu^{2}|j_{++}|\sqrt{|j_{++}|}\sqrt{|\mu^{2}L^{2}-9|}}\int_{0}^{2\pi}\text{d}\phi\
\sigma(x^{+})\Big{(}\delta j_{++}(\mu^{2}L^{2}+9h(x^{+}))-18j_{++}\delta
h(x^{+})\Big{)},$

which are easily integrated in solution space.

$\displaystyle Q_{\underline{\boldsymbol{\sigma}}}[g]$ $\displaystyle=$
$\displaystyle
sg(j_{++})\frac{(\mu^{2}L^{2}-9)}{18\pi\sqrt{GL^{5}}\mu^{2}\sqrt{|\mu^{2}L^{2}-9|}\sqrt{|j_{++}|}}\int_{0}^{2\pi}\text{d}\phi\
\sigma(x^{+})\Big{(}\mu^{2}L^{2}+9h(x^{+})\Big{)}$ (4.52) $\displaystyle=$
$\displaystyle sg(\Delta)\frac{2\sqrt{|\Delta|}}{3\pi
L^{2}\mu^{2}}\int_{0}^{2\pi}\text{d}\phi\
\sigma(x^{+})\Big{(}\mu^{2}L^{2}+9h(x^{+})\Big{)},$

where we use that $sg(\Delta)=sg(j_{++})sg(\mu^{2}L^{2}-9)$. Similarly, for
the Virasoro sector the charges become

$\displaystyle\cancel{\delta}Q_{\underline{\boldsymbol{\epsilon}}}[g,h]$
$\displaystyle=$ $\displaystyle{1\over 1296\pi
GL^{4}\mu^{3}j_{++}^{2}}\int^{2\pi}_{0}\text{d}\phi\
\epsilon(x^{+})\Big{(}(\mu^{2}L^{2}-9)\delta
j_{++}(\mu^{4}L^{4}+18\mu^{2}L^{2}h(x^{+})+243h(x^{+})^{2})$ (4.53)
$\displaystyle+18j_{++}\big{(}36\mu^{2}L^{2}j_{++}\delta
f_{++}(x^{+})-(\mu^{2}L^{2}-9)(\mu^{2}L^{2}+27h(x^{+}))\delta
h(x^{+})\big{)}\Big{)},$

where we have thrown away total derivatives. Also this can be integrated in
solution space. Trading $j_{++}$ for $\Delta$, which organizes the expressions
better, the result is

$\displaystyle Q_{\underline{\boldsymbol{\epsilon}}}[g]=\frac{1}{18\pi
G\mu^{3}L^{4}}\int_{0}^{2\pi}\text{d}\phi\
\epsilon(x^{+})\Big{(}-2GL\Delta(\mu^{4}L^{4}+18\mu^{2}L^{2}h(x^{+})+243h(x^{+})^{2})+9\mu^{2}L^{2}f_{++}(x^{+})\Big{)}.$
(4.54)

The charge algebra, calling
$\underline{\boldsymbol{\xi}}^{i}=(\underline{\boldsymbol{\epsilon}}^{i},\underline{\boldsymbol{\sigma}}^{i})$
and given the new gauge orbits, is given by

$\\{Q_{\underline{\boldsymbol{\xi}}^{1}}[g],Q_{\underline{\boldsymbol{\xi}}^{2}}[g]\\}=\delta_{\underline{\boldsymbol{\xi}}^{2}}Q_{\underline{\boldsymbol{\xi}}^{1}}[g]=Q_{[\underline{\boldsymbol{\xi}}^{1},\underline{\boldsymbol{\xi}}^{2}]_{M}}[g]+{\cal
K}_{\underline{\boldsymbol{\xi}}^{1},\underline{\boldsymbol{\xi}}^{2}},$
(4.55)

where ${\cal
K}_{\underline{\boldsymbol{\xi}}^{1},\underline{\boldsymbol{\xi}}^{2}}$ is the
central extension. In this expression, since the generators are now field
dependent, one should use the modified bracket (4.30). However, since
$\delta_{\underline{\xi}}j_{++}=0$, in this specific instance the modified
bracket coincides with the ordinary Lie bracket. For the $u(1)$ sector we
gather

$\\{Q_{\underline{\boldsymbol{\sigma}}^{1}}[g],Q_{\underline{\boldsymbol{\sigma}}^{2}}[g]\\}=\delta_{\underline{\boldsymbol{\sigma}}^{2}}Q_{\underline{\boldsymbol{\sigma}}^{1}}[g]=-sg(\Delta)\frac{2}{\pi\mu
L}\int_{0}^{2\pi}\text{d}\phi\ \sigma_{1}\sigma_{2}^{\prime}.$ (4.56)

Since
$Q_{[\underline{\boldsymbol{\sigma}}^{1},\underline{\boldsymbol{\sigma}}^{2}]_{M}}[g]=0$,
(4.56) is the central extension for the $\hat{u}(1)$ sector. Using the mode
decomposition representation $\sigma_{1}=e^{imx^{+}}$ and
$\sigma_{2}=e^{inx^{+}}$, and calling
$Q_{\underline{\boldsymbol{\sigma}}^{1}}[g]={{P}}_{m}$ and
$Q_{\underline{\boldsymbol{\sigma}}^{2}}[g]={{P}}_{n}$:111111Note that the new
vector fields are written with respect to $\boldsymbol{\sigma}$, but the field
independent quantity is $\sigma$, which can thus be expanded in modes.

${{P}}_{m}=sg(\Delta)\frac{2\sqrt{|\Delta|}}{3\pi
L^{2}\mu^{2}}\int_{0}^{2\pi}\text{d}\phi\
e^{imx^{+}}\Big{(}\mu^{2}L^{2}+9h(x^{+})\Big{)},$ (4.57)

we obtain121212At $\mu L=3$, this result is the analogue of $(13)$ of [45] but
for the canonical ensemble, namely, the level is equal to
$\Big{(}(1-{1\over\mu\ell})\text{CSS}\Big{)}|_{\mu\ell\to 3}$, where CSS
stands for the CSS level in Einstein gravity in canonical ensemble (as in App.
B of [41]).

$i\\{{{P}}_{m},{{P}}_{n}\\}=m{k\over 2}\delta_{m+n,0},\qquad
k=-sg(\Delta)\frac{8}{\mu L}.$ (4.58)

This is a centrally extended $\hat{u}(1)$ algebra with central extension
(level) $k$. As anticipated, this level coincides exactly with (2.7), except
for the $sg(\Delta)$ term, determining the sector of the theory. We remark
that such extension depends on the sign of $j_{++}$, which is the warped
equivalent of the result in [41] for TMG. For the other sector, generated by
$\underline{\boldsymbol{\epsilon}}$, using the mode decomposition
representation $\boldsymbol{\epsilon}_{1}=e^{imx^{+}}$ and
$\boldsymbol{\epsilon}_{2}=e^{inx^{+}}$, and calling
$Q_{\underline{\boldsymbol{\epsilon}}^{1}}[g]={{L}}_{m}$ and
$Q_{\underline{\boldsymbol{\epsilon}}^{2}}[g]={{L}}_{n}$:

${{L}}_{m}=\frac{1}{18\pi G\mu^{3}L^{4}}\int_{0}^{2\pi}\text{d}\phi\
e^{imx^{+}}\Big{(}-2GL\Delta(\mu^{4}L^{4}+18\mu^{2}L^{2}h(x^{+})+243h(x^{+})^{2})+9\mu^{2}L^{2}f_{++}(x^{+})\Big{)},$
(4.59)

we obtain

$i\\{{{L}}_{m},{{L}}_{n}\\}=(m-n){{L}}_{m+n}+{c\over
12}m^{3}\delta_{m+n,0},\qquad c=\frac{\mu^{2}L^{2}+9}{3G\mu}.$ (4.60)

Note that both central charges are sensitive to the sign of $\mu$. And finally
the mixed sector, given our conventions, gives the usual semi-direct action of
Virasoro on the $\hat{u}(1)$ algebra

$i\\{{{L}}_{m},{{P}}_{n}\\}=-n{{P}}_{m+n}.$ (4.61)

So, to summarize, the total symmetry group, also when $j_{++}$ varies, is

$\text{Vir}\ltimes U(1).$ (4.62)

The total algebra is

$\displaystyle i\\{{{L}}_{m},{{L}}_{n}\\}=(m-n){{L}}_{m+n}+{c\over
12}m^{3}\delta_{m+n,0},\quad i\\{{{L}}_{m},{{P}}_{n}\\}=-n{{P}}_{m+n},\quad
i\\{{{P}}_{m},{{P}}_{n}\\}=m{k\over 2}\delta_{m+n,0},$ (4.63)

with central extensions

$\displaystyle c=\frac{\mu^{2}L^{2}+9}{3G\mu},\qquad k=-sg(\Delta)\frac{8}{\mu
L}.$ (4.64)

Notice that, once the sign of $j_{++}$ given, both central extensions are
field independent, in contrast with the previous case. Therefore, this bulk in
these coordinates is dual to thermal states of a WCFT in the canonical
ensemble. This is the reason why we denote the charges mode decomposition
without the tilde, in line with (2.4-2.6).

Finally, we can see how WBTZ black holes are described in this setup. Setting

$\displaystyle j_{++}=\frac{\mu^{2}L^{2}-9}{72GL(LM-J)},\quad h=0,\quad
f_{++}=\frac{1}{9}GL\left(\mu^{2}L^{2}+9\right)(J+LM),\quad\Delta={\mu^{2}L^{2}-9\over
144GLj_{++}}={LM-J\over 2},$ (4.65)

we observe that, since the absence of naked singularities requires
$LM\geq|J|$, the quantity $\Delta$ is always positive, so that $sg(\Delta)=1$
and $|\Delta|=\Delta$. Therefore the charges in the canonical ensemble take
the form

$\displaystyle{{P}}_{m}=\frac{2\sqrt{2(LM-J)}}{3}\delta_{m,0},\qquad{{L}}_{m}=\Big{(}\frac{2J\mu
L}{9}+\frac{J}{\mu L}+\frac{M}{\mu}\Big{)}\delta_{m,0}.$ (4.66)

The TMG mass and angular momentum of these solutions are defined as

$\displaystyle\boldsymbol{M}=Q_{\partial_{t}}={1\over
L}\Big{(}Q_{\partial_{+}}+Q_{\partial_{-}}\Big{)},\qquad\boldsymbol{J}=Q_{\partial_{\phi}}=Q_{\partial_{+}}-Q_{\partial_{-}}.$
(4.67)

As a consequence of the boundary conditions used here, the relationship
between ${{P}}_{0}$ and $Q_{\partial_{-}}$ is non-trivial. In particular,
since we have performed a field-dependent redefinition involving a
multiplication rather than a linear shift, the charges are non-linearly
related. Since $\partial_{-}$ is an exact Killing, we can compute separately
this charge for WBTZ backgrounds, and then compare it with the results here.
We find131313This relationship has an important extra factor of ${1\over 2}$
with respect to the naive linear rescaling. This comes about because of the
field dependent redefinition. Note that it is only for WBTZ black holes that
we can separately compute $Q_{\partial_{-}}$, which is otherwise non
integrable.

$\displaystyle Q_{\partial_{-}}={{P}}_{0}{\mu L\sqrt{|\Delta|}\over 6},\qquad
Q_{\partial_{+}}={{L}}_{0}+{{P}}_{0}{\mu L\sqrt{|\Delta|}\over 6},$ (4.68)

which implies

$\displaystyle\boldsymbol{M}={1\over L}\Big{(}{{L}}_{0}+{{P}}_{0}{2\mu
L\sqrt{|\Delta|}\over 3}\Big{)}={2\mu LM\over 9}+{LM+J\over\mu
L^{2}},\qquad\boldsymbol{J}={{L}}_{0}={2\mu LJ\over 9}+{LM+J\over\mu L}.$
(4.69)

While the expression for the TMG mass and angular momentum in terms of the
zero modes depends on the explicit realization of the algebra, their intrinsic
value as charges is invariant. Indeed, as non-trivial consistency check, we
observe that $\boldsymbol{M}=\tilde{M}$ and $\boldsymbol{J}=\tilde{J}$, with
$\tilde{M}$ and $\tilde{J}$ given in (4.47).

Recalling that for WBTZ black holes we have that $sg(\Delta)=1$, we obtain a
relationship between the zero modes here and those in the quadratic ensemble
of the previous section given by

$\displaystyle\tilde{{L}}_{0}={{L}}_{0}+\frac{\mu
L}{8}{{P}}_{0}^{2}={{L}}_{0}-{{{P}}_{0}^{2}\over k},\qquad\tilde{{P}}_{0}={\mu
L\over 8}{{P}}_{0}^{2}=-{{{P}}_{0}^{2}\over k},$ (4.70)

which exactly reproduces (2.32).

To summarize, we have seen how two different restriction on the solution space
leading to different boundary conditions can be used to make the charges
integrable, yielding different realizations of the asymptotic symmetry
algebra. With our new metric falloffs, the two methods give a bulk that is
dual to a WCFT in either canonical or quadratic ensemble. We will focus in
particular on the latter, whose modes are reported in (4.44), because, as
explained previously, the warped Cardy formula is well defined in this case.
The novelty is that we do not reach this ensemble from a bulk change of
coordinates, instead we found enhanced boundary conditions to automatically
accommodate it.

##  5 Entropy matching

We turn in this section to the matching between thermodynamic bulk entropy in
TMG and boundary warped Cardy formula in quadratic ensemble. As is clear from
(2.31), this will require identifying the vacuum state on which the partition
function projects in the Cardy regime. Although we will only indirectly
identify the vacuum via intuition gathered with limiting procedures, we
discuss at the end of this section possible techniques to intrinsically
identify it. These techniques require a deep understanding of the global
structure of these solutions, which is currently under investigation and on
which we will report later.

###  5.1 Bulk thermodynamics

To study the thermodynamics of WBTZ solutions, following [61, 62, 63, 22, 64],
we bring the line element to the ADM form:

$\text{d}s^{2}=-N(r)^{2}\text{d}t^{2}+{\text{d}r^{2}\over
f(r)^{2}}+R(r)^{2}(N^{\phi}(r)\text{d}t+\text{d}\phi)^{2}.$ (5.1)

In the coordinates $(t,r,\phi)$, with $\varepsilon^{tr\phi}=1$, using
$x^{\pm}={t\over L}\pm\phi$, the WBTZ metric is given by141414In comparing
with [43], one should set $G=1$, $L=1$.

$\displaystyle g_{rr}$ $\displaystyle=$
$\displaystyle\frac{L^{2}r^{2}}{16G^{2}J^{2}L^{2}-8GL^{2}Mr^{2}+r^{4}},$ (5.2)
$\displaystyle g_{tt}$ $\displaystyle=$
$\displaystyle-\frac{H^{2}\left(4GL(J-2LM)+r^{2}\right)^{2}}{4GL^{3}(LM-J)}+8GM-\frac{r^{2}}{L^{2}},$
(5.3) $\displaystyle g_{t\phi}$ $\displaystyle=$ $\displaystyle
4GJ-\frac{H^{2}\left(r^{2}-4GJL\right)\left(4GL(J-2LM)+r^{2}\right)}{4GL^{2}(LM-J)},$
(5.4) $\displaystyle g_{\phi\phi}$ $\displaystyle=$ $\displaystyle
r^{2}-\frac{H^{2}\left(r^{2}-4GJL\right)^{2}}{4GL(LM-J)},$ (5.5)

which is exactly (2.23). Therefore, the ADM data read

$\displaystyle N(r)^{2}$ $\displaystyle=$
$\displaystyle-\frac{4G\left(2H^{2}-1\right)(J-LM)\left(16G^{2}J^{2}L^{2}-8GL^{2}Mr^{2}+r^{4}\right)}{L\left(16G^{2}H^{2}J^{2}L^{2}-4GLr^{2}\left(\left(2H^{2}-1\right)J+LM\right)+H^{2}r^{4}\right)},$
(5.6) $\displaystyle f(r)^{2}$ $\displaystyle=$
$\displaystyle\frac{16G^{2}J^{2}}{r^{2}}-8GM+\frac{r^{2}}{L^{2}},$ (5.7)
$\displaystyle R(r)^{2}$ $\displaystyle=$
$\displaystyle-\frac{16G^{2}H^{2}J^{2}L^{2}-4GLr^{2}\left(2H^{2}J-J+LM\right)+H^{2}r^{4}}{4GL(-J+LM)},$
(5.8) $\displaystyle N^{\phi}(r)$ $\displaystyle=$
$\displaystyle\frac{-16G^{2}JL^{2}\left(\left(H^{2}-1\right)J+\left(1-2H^{2}\right)LM\right)-8GH^{2}L^{2}Mr^{2}+H^{2}r^{4}}{L\left(16G^{2}H^{2}J^{2}L^{2}-4GLr^{2}\left(\left(2H^{2}-1\right)J+LM\right)+H^{2}r^{4}\right)}.$
(5.9)

Applying in particular the results of [22], the black hole entropy in TMG,
given (5.1), is

$S^{TMG}_{\pm}={\pi\over 2G}R(r_{\pm})-{\pi\over 2\mu
G}{R(r)^{2}f(r)N^{\phi}(r)^{\prime}\over 2N(r)}\Big{|}_{r=r_{\pm}},$ (5.10)

where $\mu$ is the Chern-Simons coupling and $r_{\pm}$ are the two positive
roots, solutions of the equation $f(r)=0$ and are given by

$\displaystyle r_{\pm}=2\sqrt{GL}\sqrt{LM\pm\sqrt{L^{2}M^{2}-J^{2}}},$ (5.11)

for all values of $J$. For WBTZ we thus gather the outer entropy

$\displaystyle S^{TMG}_{+}$ $\displaystyle=$ $\displaystyle\frac{\pi
L\left(\left(3-4H^{2}\right)\sqrt{\sqrt{L^{2}M^{2}-J^{2}}+LM}+\text{sg}(J)\sqrt{LM-\sqrt{L^{2}M^{2}-J^{2}}}\right)}{3\sqrt{G\left(1-2H^{2}\right)L}},$
(5.12)

where $sg$ is again the sign function. The presence of $sg(J)$ is due to the
fact that we are dealing with a theory sensitive to parity, due to the
presence of the Hodge-dual Cotton tensor in the equations of motion.

The WBTZ mass
$\tilde{M}=Q_{\partial_{t}}=\frac{1}{L}\left({\tilde{{L}}}_{0}+{\tilde{{P}}}_{0}\right)$
and angular momentum
$\tilde{J}=Q_{\partial_{\phi}}={\tilde{{L}}}_{0}-{\tilde{{P}}}_{0}$ are given
by (4.47), i.e.

$\displaystyle\tilde{M}$ $\displaystyle=$ $\displaystyle{2\mu LM\over
9}+{LM+J\over\mu L^{2}}={\left(3-4H^{2}\right)LM+J\over 3\sqrt{1-2H^{2}}L},$
(5.13) $\displaystyle\tilde{J}$ $\displaystyle=$ $\displaystyle{2\mu LJ\over
9}+{LM+J\over\mu L}={\left(3-4H^{2}\right)J+LM\over 3\sqrt{1-2H^{2}}}.$ (5.14)

As expected, the entropy (5.12) satisfies the first law

$\text{d}\tilde{M}=T\text{d}S^{TMG}_{+}+\Omega\text{d}\tilde{J},$ (5.15)

with

$T=\frac{r_{+}^{2}-r_{-}^{2}}{2\pi
r_{+}L^{2}}=\frac{2}{\pi}\sqrt{\frac{G\left(L^{2}M^{2}-J^{2}\right)}{L^{3}\left(\sqrt{L^{2}M^{2}-J^{2}}+LM\right)}},$
(5.16)

and

$\Omega=sg(J)\frac{r_{-}}{Lr_{+}}=\frac{1}{LJ}\sqrt{2LM\left(LM-\sqrt{L^{2}M^{2}-J^{2}}\right)-J^{2}}.$
(5.17)

Finally, we note that in the CSS limit we find the BTZ entropy in TMG, as
found e.g. in [45]. This is the standard derivation of the black hole
thermodynamic entropy, we now turn our attention to the dual viewpoint.

###  5.2 WCFT Cardy formula and matching

In this section we would like to obtain the entropy from the counting of
degeneracy of states in the boundary field theory. The first important step is
to define the vacuum of the theory. The usual approach is to use as a
guideline the enhancement of isometries. We now compute the local Killing
vectors preserving the WBTZ metric in the quadratic ensemble (5.2-5.5). These
are $\partial_{t}$, $\partial_{\phi}$ and $\underline{\chi}_{\pm}$ given by

$\displaystyle\chi_{\pm}^{r}$ $\displaystyle=$ $\displaystyle e^{\pm
2\sqrt{2}\sqrt{\frac{G(J+LM)}{L}}\left(\frac{t}{L}+\phi\right)}\frac{\sqrt{16G^{2}J^{2}L^{2}-8GL^{2}Mr^{2}+r^{4}}}{2r},$
(5.18) $\displaystyle\chi_{\pm}^{t}$ $\displaystyle=$ $\displaystyle\mp e^{\pm
2\sqrt{2}\sqrt{\frac{G(J+LM)}{L}}\left(\frac{t}{L}+\phi\right)}\frac{L\left(4GJL+r^{2}\right)}{4\sqrt{2}\sqrt{\frac{G(J+LM)\left(16G^{2}J^{2}L^{2}-8GL^{2}Mr^{2}+r^{4}\right)}{L}}},$
(5.19) $\displaystyle\chi_{\pm}^{\phi}$ $\displaystyle=$ $\displaystyle\pm
e^{\pm
2\sqrt{2}\sqrt{\frac{G(J+LM)}{L}}\left(\frac{t}{L}+\phi\right)}\frac{\left(4GL(J+2LM)-r^{2}\right)}{4\sqrt{2}\sqrt{\frac{G(J+LM)\left(16G^{2}J^{2}L^{2}-8GL^{2}Mr^{2}+r^{4}\right)}{L}}}.$
(5.20)

These vectors, combined as
$\\{\frac{L}{2\sqrt{2}\sqrt{GL(J+LM)}}(L\partial_{t}+\partial_{\phi}),\underline{\chi}_{\pm}\\}$,
satisfy an $SL(2,\mathbb{R})$ algebra.

We observe that these local isometries are extended to hold globally if the
exponential is $2\pi$ periodic in $\phi$. Thus, the WBTZ solution admits
global isometries if and only if

$2\sqrt{2}\sqrt{\frac{G(J+LM)}{L}}=\pm i.$ (5.21)

This equation therefore has solution $M=-\frac{1}{8G}-\frac{J}{L}$. The metric
given by (5.2-5.5) with $M$ taking this value therefore represents a family of
solutions with enhanced global symmetries. A crucial observation is that the
enhancement of symmetries does not single out the vacuum, contrary to what
happens for BTZ black holes. Indeed, for the latter, we impose to enhance the
symmetries to global $SL(2,\mathbb{R})_{L}\times SL(2,\mathbb{R})_{R}$, which
imposes more restrictions on the metric parameters, allowing to immediately
single out the unique vacuum.

We notice that inside the family of solutions with enhanced symmetry, one
particular solution deserves special attention, the one with J = 0. In this
case we obtain $M=-\frac{1}{8G}$, and the line element is explicitly written

$\text{d}s^{2}_{\text{vac}}=(L^{2}+r^{2})(2H^{2}r^{2}+L^{2}(-1+2H^{2})){\text{d}t^{2}\over
L^{4}}+{L^{2}\text{d}r^{2}\over(L^{2}+r^{2})}+4H^{2}r^{2}(L^{2}+r^{2}){\text{d}t\text{d}\phi\over
L^{3}}+(r^{2}+{2H^{2}r^{4}\over L^{2}})\text{d}\phi^{2}.$ (5.22)

This metric differs from the AdS3 global vacuum, but reduces to it for $H\to
0$ (i.e. $\mu\ell=3$ with $L=\ell$). We take this external input on the
analysis to be the feature singling out the true vacuum of the theory. As
already pointed out, we stress that this is not an intrinsic definition of the
vacuum, but rather comes from a limiting procedure. We will return to this at
the end of the section, where we present a more detailed discussion. For this
specific solution, the values of the TMG charges are

$\displaystyle\tilde{M}=-{\left(3-4H^{2}\right)\over
24\sqrt{1-2H^{2}}G},\qquad\tilde{J}=-{L\over 24\sqrt{1-2H^{2}}G}.$ (5.23)

We remark that the TMG angular moment $\tilde{J}$ is non-vanishing on the
vacuum solution, even though the metric parameter $J$, which corresponds to
the Einstein angular momentum in the Einstein limit, is vanishing.

We want now to evaluate the WBTZ entropy (5.12), which is reproduced by
counting the degeneracy of states in the dual WCFT. In the quadratic ensemble,
the Warped Cardy formula takes the form (2.31):

$S_{\mathrm{WCFT}}=4\pi\sqrt{-{\tilde{{P}}}_{0}^{vac}{\tilde{{P}}}_{0}}+4\pi\sqrt{-{\tilde{{L}}}_{0}^{vac}{\tilde{{L}}}_{0}}.$
(5.24)

For the specific solution identified above, the zero modes (4.44) become

$\displaystyle\tilde{{{P}}}_{0}^{\text{vac}}=-\frac{\sqrt{1-2H^{2}}L}{24G},\qquad\tilde{{{L}}}_{0}^{\text{vac}}=\frac{\left(H^{2}-1\right)L}{12G\sqrt{1-2H^{2}}}.$
(5.25)

Plugging this and (4.44) in (5.24), and using $\mu L=3\sqrt{(1-2H^{2})}$, we
find

$S_{\mathrm{WCFT}}=\frac{1}{3}\sqrt{2}\pi\left(\sqrt{\frac{\left(2H^{2}-1\right)L(J-LM)}{G}}+2\sqrt{-\frac{\left(H^{2}-1\right)^{2}L(J+LM)}{G\left(2H^{2}-1\right)}}\right).$
(5.26)

After some non-trivial manipulations, this expression exactly matches the bulk
thermodynamic WBTZ entropy (5.12).

In conclusion, we return here to the discussion on the vacuum. To match the
entropy, we used the specific solution whose charges coincide with the AdS3
global vacuum in the limit $H\to 0$. We argue that this argument for singling
out this vacuum must be improved with a more intrinsic and fundamental
argument, lacking at present. There are two possible resolutions that we are
currently exploring. First, it could be useful to introduce supersymmetry, and
require that the true vacuum inside the family of solutions found is the one
for which also all the supersymmetry generators are globally well-defined,
which could potentially give further constraints. Indeed, it is known that
certain BTZ and WAdS3 black holes in canonical ensemble exhibit supersymmetry
[65, 66], and one could expect the vacuum state to display maximal
supersymmetry. Second, a thorough analysis of the global structure of these
solutions could give us a better understanding of the properties of this
family of solutions, such as geodesic completeness, chronological and/or
conical singularities, and the issue of closed time-like curves.

##  6 Conclusions and outlook

We have introduced a consistent set of metric falloffs allowing to extract the
asymptotic symmetry algebra directly in the quadratic ensemble. In this setup,
asymptotic charges are generically non integrable. Two possible boundary
conditions arose, fixing part of the solution space or performing a field
redefinition of the symmetry generators. These two procedures give an
asymptotic symmetry algebra in quadratic and canonical ensemble, respectively.
Using the former, we tested that we are in the regime of validity of the Cardy
warped formula and show that the boundary counting of degeneracy of states
correctly reproduces the bulk thermodynamic entropy for WBTZ black holes. To
do so, we had to identify the vacuum of the theory. We showed that the
enhancement of Killing isometries to globally well-defined symmetries is not
enough, and propose the identification of the vacuum via a limiting procedure.

This project opens the door to both short-term and long-term investigations.
The most pressing and natural continuation of this project is to find an
intrinsic definition of the vacuum for the family of solutions with enhanced
symmetry. As we discussed, we plan to introduce supersymmetry, to see if it
imposes further constraints on the solutions. Another direction is to study in
detail the global structure of these solutions. This might shed light on their
topological properties. We would like to find a complete Penrose-Carter
diagram of these solutions, and study their inextensibility. Another direction
to pursue is the question of integrability of charges. We confirmed here that
it seems always possible to render charges integrable by redefining the
symmetry generators by introducing field/ state dependence, as has also been
carried out in [67, 68, 69, 60]. Another mechanism to make charges integrable,
recently proposed in [70] (see also [71]), is to carefully treat embeddings.
It would be interesting to apply this new mechanism to our specific
construction here. Other questions stemming from the covariant phase-space
formalism, when applied to TMG, are also worth investigating, such as the
classification of ambiguities. On the long term, there are some fascinating
directions to pursue. First, we here restricted our analysis to $0<\mu L<6$.
At $\mu L=6$, the cosmological constant vanishes, whereas for $\mu L>6$ the
metric has positive cosmological constant. Our analysis of surface charges is
not expected to break down, allowing us to sail from negative to positive
cosmological constants, an idea already explored in [48]. Secondly, we saw
that in our analysis we restricted the would-be boundary metric to have non-
varying conformal factor. This restriction could be lifted, in the spirit of
[46]. A similar construction appeared recently in [52]. Eventually, we expect
to have some more general boundary conditions arising at specific values of
the Chern-Simons coupling. We briefly touch upon this for $\mu L=3$ in
Appendix A. In this setup, we expect to be able to make contact with [26]. A
far-reaching consequence could be to shed light on the still elusive
properties of logarithmic CFTs [47, 72, 73, 74], and study the possibility of
constructing their warped version.

##### Acknowledgments

We thank Marc Henneaux for valuable discussions. AA is a Research Fellow of
the Fonds de la Recherche Scientifique F.R.S.-FNRS (Belgium). The research of
LC was partially supported by a Marina Solvay Fellowship, by the ERC Advanced
Grant “High-Spin-Grav” and by FNRS-Belgium (convention FRFC PDR T.1025.14 and
convention IISN 4.4503.15). SD is a Research Associate of the Fonds de la
Recherche Scientifique F.R.S.-FNRS (Belgium). SD was supported in part by IISN
– Belgium (convention 4.4503.15) and benefited from the support of the Solvay
Family. SD acknowledges support of the Fonds de la Recherche Scientifique
F.R.S.-FNRS (Belgium) through the CDR project C 60/5 - CDR/OL “Horizon
holography : black holes and field theories” (2020-2022). This research was
supported in part by the National Science Foundation under Grant No. NSF
PHY-1748958. The work of AS is supported by the “Fonds pour la Formation et la
Recherche dans l’Industrie et dans l’Agriculture”, FRIA (Belgium).

##  Appendix A New boundary conditions including pp waves

At $\mu={3\over L}={3\over\ell}$, (where $\ell={1\over\sqrt{|\Lambda|}}$), the
solutions (3.3-3.5) can be generalized to allow $j_{++}$ to be an arbitrary
function $j_{++}(x^{+})$. The resulting class of metric reads

$\displaystyle\text{d}s^{2}$ $\displaystyle=$
$\displaystyle(\text{d}x^{+})^{2}\left(f_{++}(x^{+})+r^{2}h(x^{+})+j_{++}(x^{+})r^{4}\right)+\text{d}x^{-}\text{d}x^{+}(2j_{+-}(x^{+})+1)r^{2}+\text{d}r^{2}\frac{\ell^{2}}{r^{2}}.$
(A.1)

In fact, these solutions can be generalized for arbitrary values of positive
TMG coupling, $\mu>0$, as follows

$\displaystyle\text{d}s^{2}$ $\displaystyle=$
$\displaystyle(\text{d}x^{+})^{2}\left(f_{++}(x^{+})+r^{2}h(x^{+})+j_{++}(x^{+})r^{\mu\ell+1}\right)+\text{d}x^{-}\text{d}x^{+}(2j_{+-}(x^{+})+1)r^{2}+\text{d}r^{2}\frac{\ell^{2}}{r^{2}},$
(A.2)

where $j_{+-}(x^{+})\leq-{1\over 2}$. This class of solutions includes the pp-
waves. The asymptotic symmetry group is again Vir$\ltimes$ U(1).

For $\mu\ell<1$, these boundary conditions generalize CSS boundary conditions
[41] in TMG in the same way [26] generalizes Brown-Hennaux boundary conditions
[8] in TMG. For $\mu\ell\geq 3$, these generalize the null warped and null
z-warped boundary conditions of [49]. The details about the charges and other
features of these solutions will be presented elsewhere. We also plan to
explore their relationship with the boundary conditions involving non-integer
($\mu$-dependent) powers of $r$ and $\log r$ introduced in [26] (see also [47,
72, 73, 74]).

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