aaaffiliationtext: Centro de Estudios Científicos (CECs), Av. Arturo Prat 514,
Valdivia, Chile.bbaffiliationtext: Departamento de Física, Universidad de
Santiago de Chile, Av. Víctor Jara 3493, Santiago, Chile.ccaffiliationtext:
Center for Interdisciplinary Research in Astrophysics and Space Exploration
(CIRAS), Universidad de Santiago de Chile, Av. Libertador Bernardo O’Higgins
3363, Santiago, Chile.

#  1/c deformations of AdS3 boundary conditions and the Dym hierarchy

Kristiansen Lara<EMAIL_ADDRESS>Miguel<EMAIL_ADDRESS>Francisco<EMAIL_ADDRESS>
###### Abstract

This work introduces a novel family of boundary conditions for AdS3 General
Relativity, constructed through a polynomial expansion in negative integer
powers of the Brown-Henneaux central charge. The associated dynamics is
governed by the Dym hierarchy of integrable equations. It is shown that the
infinite set of Dym conserved charges generates an abelian asymptotic symmetry
group. Additionally, these boundary conditions encompass black hole solutions,
whose thermodynamic properties are examined.

###### Contents

  1. 1 Introduction
  2. 2 Constructing the family of boundary conditions
  3. 3 Consistency of the boundary conditions
    1. 3.1 Action principle
    2. 3.2 Asymptotic symmetries
    3. 3.3 Stationary black hole solution and its thermodynamics
  4. 4 Final comments
  5. A Chern-Simons formulation of AdS3 gravity
  6. B List of Dym conserved charges $H^{\pm}_{n}$
  7. C Euclidean analysis of black hole solution

##  1 Introduction

The holographic principle [1, 2, 3] is a valuable tool for exploring General
Relativity and gauge theories. For any realization of holography, it is
crucial to define well-posed boundary conditions for the dynamical fields,
since the corresponding asymptotic symmetries match the global symmetries of
the boundary theory. The precursory example by Brown and Henneaux in three-
dimensional General Relativity with negative cosmological constant [4],
highlights the relevance of boundary conditions and the identification of
asymptotic symmetries. Although the best known realization of holography, the
AdS/CFT correspondence [5, 6, 7] was proposed in a string theory framework,
the Brown-Henneaux example provides valuable insights into holography without
relying on string theory.

The consequences of choosing a wide variety of boundary conditions for the
gravitational field on several different scenarios has been extensively
explored in the literature. The seminal example concerns four-dimensional
asymptotically flat spacetimes, for which the infinite dimensional BMS4 group
[8, 9] arises as the asymptotic symmetries at null infinity. This classic
result has been scrutinized and extended for several decades [10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46].

The choice of boundary conditions have been used to explore dualities in a
broader context. A nonexhaustive list includes the Kerr/CFT duality [47] (for
a review, see [48]), Schrödinger-invariant spacetimes (see [49] and references
thereof), dualities with condensed matter physics (for a review, see [50]),
among many others. It also plays a rol in the context of the soft hair
proposal [51, 52]: it amounts to choose the behavior of the gravitational
field in the near-horizon region of rotating black holes [53, 54, 55, 56]. The
same ideas has been applied to cosmological horizons [57, 58, 59, 60] and
plays a key role in the recent celestial holography program (for reviews, see
e.g., [61, 62, 63] and references therein.)

Three-dimensional gravity is a successful toy model to explore the holographic
consequences of a variety of boundary conditions. Because it is devoid of bulk
degrees of freedom, the complete dynamical content is captured by boundary
excitations. Out of simplicity, throughout the paper we will be focusing on
pure AdS3 General Relativity. The aforementioned Brown-Henneaux boundary
conditions is the precursor example, on which the asymptotic symmetry algebra
corresponds to two copies of Virasoro algebra with central charge

$c_{\mathrm{BH}}=\frac{3\ell}{2G},$ (1.1)

where $\ell$ stands for the AdS radius, while $G$ represents Newton’s
constant. However, many others possibilities for asymptotic behavior have been
explored in the literature [64, 65, 66, 67, 68, 69, 70, 53, 71, 72, 73, 74,
75]

Following [76, 77], this paper concerns the possibility of exploiting the near
boundary behavior of the lapse and shift functions, while leaving the spatial
part of the metric in standard AdS3 asymptotic form. To make the following
discussion more precise, consider a manifold with coordinates $(t,\rho,\phi)$,
where $0\leq t<\infty$, $0\leq\phi\leq 2\pi$ and $0\leq\rho<\infty$. The
boundary is located at $\rho\to\infty$. The formulation of gravitational
boundary conditions can be most simply achieved within the Chern-Simons
formulation of AdS3 General Relativity [78, 79] (See A for notation and
conventions). The asymptotic structure of the gravitational field is encoded
in the behavior of the two $sl(2,\mathbb{R})$ connections $A^{\pm}$ near the
boundary. As explained in [80], asymptotically AdS spacetimes are described by
connections of the form
$A^{\pm}=b^{-1}_{\pm}a^{\pm}b_{\pm}+b^{-1}_{\pm}db_{\pm}$, where the group
element $b_{\pm}(\rho)$ gauges away the radial dependence. The spatial part of
the auxiliary connection is given by

$\displaystyle a_{\phi}^{\pm}$ $\displaystyle=L_{\pm
1}-\frac{12\pi}{c_{\mathrm{BH}}}\mathcal{L}^{\pm}L_{\mp 1}\,,$ (1.2a)
$\displaystyle a_{\rho}^{\pm}$ $\displaystyle=0\,,$ (1.2b)

where $\mathcal{L}^{\pm}$ are functions of $t$ and $\phi$.

As shown in [76], the most general gauge transformation preserving the form of
(1.2) has a gauge parameter of the form

$\Lambda^{\pm}[\eta^{\pm}]=\pm\left(\eta^{\pm}a_{\phi}^{\pm}\mp\partial_{\phi}\eta^{\pm}L_{0}+\frac{1}{2}\partial_{\phi}^{2}\eta^{\pm}L_{\mp
1}\right)\,.$ (1.3)

Regarding the temporal component $a_{t}^{\pm}$, its choice should ensure that
the spatial boundary conditions (1.2) are preserved under temporal evolution.
Therefore, it should also follow the form given by equation (1.3),

$a^{\pm}_{t}=\frac{1}{\ell}\Lambda^{\pm}[\mu^{\pm}]\,.$ (1.4)

According to the map among Chern-Simons and metric fields, the functions
$\mu^{\pm}$ encode the asymptotic behavior of the lapse and shift functions.

It is important to highlight that $\mu^{\pm}$ should be chosen consistently in
order to describe well defined boundary conditions for the gravitational
field. For example, Brown-Henneaux boundary conditions [4] are obtained with
the choice $\mu^{\pm}=1$, yielding
$a_{t}^{\pm}=\pm\frac{1}{\ell}a_{\phi}^{\pm}$. In this case, the gauge
parameters $\eta^{\pm}$ fulfill a chiral boson equation, rendering the
asymptotic symmetry group infinite-dimensional and spanning two copies of
Virasoro algebra with central charge $c_{\mathrm{BH}}$. Another noteworthy
example was given in [77], where $\mu^{\pm}$ are chosen as functions of
$\mathcal{L}^{\pm}$ and its derivatives; by fixing $\mu^{\pm}$ as the $n$-th
Gelfand-Dikii polynomial [81] of $\mathcal{L}^{\pm}$, Einstein’s equation
reduces to the $n$-th member of the Korteweg-de Vries (KdV) hierarchy of
integrable equations, unveiling a connection between integrable systems and 3D
gravity. The integrability of the KdV hierarchy translates into an infinite
dimensional and abelian asymptotic symmetry algebra generated by the KdV
conserved charges. This connection with KdV hierarchy has been further
explored [82, 83, 84, 85, 86, 87, 88]. The relation between 3D gravity and
integrable systems was extended to include the Ablowitz-Kaup-Newell-Segur
(AKNS) system [89].

This paper aims to enlarge the aforementioned landscape of boundary conditions
for AdS3 gravity. We will construct a new class of functions $\mu^{\pm}$,
dependent on $\mathcal{L}^{\pm}$ and its derivatives, that yield an infinite
family of well-defined boundary conditions labeled by a positive integer $N$.
The corresponding dynamical equations are those of the Harry Dym hierarchy of
nonlinear integrable equations [90]. The asymptotic symmetries are generated
by an infinite tower of conserved charges, forming an abelian algebra under
the Poisson bracket, as expected from the integrability properties of the
dynamical equations.

Inspired by the ansatz of AKNS [91], the key step in the construction is the
assumption that the functions $\mu^{\pm}$ can be expanded as a polynomial in
negative powers of the Brown-Henneaux central charge $c_{\mathrm{BH}}$. As
explained below, the coefficients of such polynomials are constructed
following a recursion relation, much in the spirit of integrable systems (see
e.g., [92]).

The paper is structured as follows: Section 2 details the construction of
boundary conditions, while Section 3 shows their well-defined nature. Some
final comments and appendixes are found at the end of the paper.

##  2 Constructing the family of boundary conditions

The aim of this section is to construct an infinite family of possible choices
for the functions $\mu^{\pm}$ in (1.4) as local functions of
$\mathcal{L}^{\pm}$ and its derivatives. The starting point is the spatial
behavior of the connection (1.2), supplemented with the temporal component
given by (1.4). The first-order Einstein equations translates in the Chern-
Simons formulation as two independent copies of the zero-curvature condition
$\partial_{t}a^{\pm}_{\phi}-\partial_{\phi}a^{\pm}_{t}+\left[a^{\pm}_{t},a^{\pm}_{\phi}\right]~{}=~{}0$,
yielding the evolution equations

$\pm\ell\partial_{t}\mathcal{L}^{\pm}=2\mathcal{L}^{\pm}\partial_{\phi}\mu^{\pm}+\partial_{\phi}\mathcal{L}^{\pm}\mu^{\pm}-\frac{c_{\mathrm{BH}}}{24\pi}\partial^{3}_{\phi}\mu^{\pm}\,.$
(2.1)

These dynamical equations will serve as a guide for constructing the sought
functions $\mu^{\pm}$. In this regard, the above is conveniently expressed as
follows

$\pm\ell\partial_{t}\mathcal{L}^{\pm}=\left(\mathcal{D}^{\pm}-c\mathcal{E}\right)\mu^{\pm},$
(2.2)

where the operators $\mathcal{D}^{\pm}$ and $\mathcal{E}$ are defined as

$\mathcal{D}^{\pm}\equiv
2\mathcal{L}^{\pm}\partial_{\phi}+\partial_{\phi}\mathcal{L}^{\pm},\quad\quad\quad\mathcal{E}\equiv\partial_{\phi}^{3}.$
(2.3)

These two differential operators are Hamiltonian, in the sense that they are
linear, anti-symmetric and fulfill Jacobi identity [93]. As shown below, these
form the bi-Hamiltonian structure of the Harry Dym hierarchy. The central
charge in (2.2) is rescaled $c\equiv c_{\mathrm{BH}}/24\pi$ to avoid $24\pi$
terms in the forthcoming expressions.

As mentioned in the introduction, the main assumption is that the functions
$\mu^{\pm}$ can be expanded in negative powers of the central charge $c$ up to
some fixed integer power $N\geq 1$

$\mu^{\pm}=\mu^{\pm}_{0}+\sum^{N}_{n=1}\mu^{\pm}_{n}\frac{1}{c^{N-n+1}}\,.$
(2.4)

The positive integer $N$ is arbitrary and can be chosen freely. It plays the
role of labeling the family of possible choices for $\mu^{\pm}$; different
values for $N$ corresponds to inequivalent boundary conditions. Although the
integer $N$ can have different values on both sectors $\pm$, here are chosen
the same for simplicity.

When substituted in equation of motion (2.2), the polynomial ansatz (2.4)
allows for grouping coefficients with the same power of $c$, resulting in
relations among the coefficients $\mu^{\pm}_{n}$. Firstly, the terms linear in
$c$ renders the condition

$\mathcal{E}\mu_{0}^{\pm}=0\,,$ (2.5)

which fixes the first coefficient $\mu_{0}^{\pm}$. The solution to (2.5),
respecting periodic boundary conditions, is given by a constant
$\mu_{0}^{\pm}=\alpha_{0}^{\pm}$.

The $c^{0}$ terms, involving the time derivative of $\mathcal{L}^{\pm}$, yield
the evolution equation

$\pm\ell\partial_{t}\mathcal{L}^{\pm}=\left(\mathcal{D}^{\pm}\mu_{0}^{\pm}-\mathcal{E}\mu_{N}^{\pm}\right)\,.$
(2.6)

Since $\mu^{\pm}_{0}=\alpha_{0}^{\pm}$, the first term on the right-hand side
of the equation becomes $\alpha_{0}^{\pm}\partial_{\phi}\mathcal{L}^{\pm}$.
Therefore, the dynamical equation corresponds to the chiral boson deformed by
the second term, which depends on $\mu_{N}^{\pm}$. To determine the value of
$\mu_{N}^{\pm}$, we can use the remaining $c^{-1}$ to $c^{-N}$ terms, as they
imply the following relations

$\displaystyle\mathcal{D}^{\pm}\mu^{\pm}_{1}$ $\displaystyle=0\,,$ (2.7a)
$\displaystyle\mathcal{D}^{\pm}\mu^{\pm}_{n+1}$
$\displaystyle=\mathcal{E}\mu^{\pm}_{n}\,,\quad n\geq 1\,.$ (2.7b)

The former differential equation governs $\mu_{1}^{\pm}$, while the latter
establishes a recursive relation for deriving $\mu^{\pm}_{n+1}$ from its
predecessor. These relations allows to iteratively construct all coefficients
$\mu^{\pm}_{n}$ as functions of ${\mathcal{L}^{\pm}}$ and its spatial
derivatives. Once the coefficients $\mu_{n}^{\pm}$ are determined, the
function $\mu^{\pm}$ is fully specified by the summation in equation (2.4),
thereby completely defining the boundary conditions.

It is important to remark that solving the recursion relation leads to
integration constants. These arise due to the linearity of the differential
equation (2.7b); the solution for for $\mu_{n+1}^{\pm}$ consists of a
homogeneous component with an integration constant, as well as a particular
solution that depends on the previous coefficient $\mu_{n}^{\pm}$. To shed
light on this observation, lets examine the role of the differential operators
in the recursion relation. Consider the inverse of the operator
$\mathcal{D}^{\pm}$, given by

$\displaystyle{\mathcal{D}^{\pm}}^{-1}[X(\phi)]=\frac{1}{2\sqrt{\mathcal{L}^{\pm}}}\partial_{\phi}^{-1}\left(\frac{X(\phi)}{\sqrt{\mathcal{L}^{\pm}}}\right)\,.$
(2.8)

It allows to define the recursion operator
$\mathcal{R}^{\pm}\equiv{\mathcal{D}^{\pm}}^{-1}\circ\mathcal{E}$, yielding

$\displaystyle\mathcal{R}^{\pm}[X(\phi)]=\frac{1}{2\sqrt{\mathcal{L}^{\pm}}}\partial_{\phi}^{-1}\left(\frac{1}{\sqrt{{\mathcal{L}^{\pm}}}}\partial_{\phi}^{3}X(\phi)\right)\,.$
(2.9)

Accordingly, the recursion relation (2.7b) can then be expressed as

$\displaystyle\mu_{n+1}^{\pm}=\mathcal{R}^{\pm}[\mu_{n}^{\pm}]+\mathcal{R}^{\pm}[0]\,.$
(2.10)

The first term on the right-hand side produces the particular solution,
whereas the subsequent term corresponds to the homogeneous one, i.e.,
$\mathcal{R}^{\pm}[0]=\frac{\alpha}{2\sqrt{\mathcal{L}^{\pm}}}$, where
$\alpha$ is an integration constant. Then, the complete solution to the
differential equation (2.7b) is given by

$\displaystyle\mu_{n+1}^{\pm}=\mathcal{R}^{\pm}[\mu_{n}^{\pm}]+\frac{\alpha^{\pm}_{n+1}}{2\sqrt{\mathcal{L}^{\pm}}}\,.$
(2.11)

This result can be successively used to derive an explicit expression for any
of the coefficients $\mu_{n}^{\pm}$,

$\displaystyle\mu_{n}^{\pm}=\sum_{i=1}^{n}\alpha_{i}^{\pm}{\mathcal{R}^{\pm}}^{n-i}\left[\frac{1}{2\sqrt{\mathcal{L}^{\pm}}}\right]\,,\quad
n\geq 1\,.$ (2.12)

At this point, it is important to highlight a remarkable property of the
recursion operator $\mathcal{R}^{\pm}$; the successive action of
$\mathcal{R}^{\pm}$ on the expression $\frac{1}{2\sqrt{\mathcal{L}^{\pm}}}$
consistently yields functional derivatives of quantities $H_{n}^{\pm}$,
according to

$\displaystyle{\mathcal{R}^{\pm}}^{n}\left[\frac{1}{2\sqrt{\mathcal{L}^{\pm}}}\right]=\frac{\delta
H_{n+1}^{\pm}}{\delta\mathcal{L}}\,.$ (2.13)

The reason behind this lies in the properties of the operators
$\mathcal{D}^{\pm}$ and $\mathcal{E}$; these are not just Hamiltonian
operators but are _compatible_ among each other, meaning that any linear
combination of them also renders a Hamiltonian operator. This compatibility
ensures (2.13), as demonstrated in [93]. Such a property is common in bi-
Hamiltonian integrable systems, proving useful as it allows the construction
of an infinite tower of conserved quantities of the system. As illustrated
below, the quantities $H_{n}^{\pm}$ found in our construction are conserved
charges and serve as generators for the asymptotic symmetry algebra. An
explicit list of the first $H_{n}^{\pm}$ along with its functional derivatives
can be found in Appendix B.

According to the previous discussion, the coefficients $\mu_{n}^{\pm}$ take
the form

$\displaystyle\mu_{n}^{\pm}=\sum_{i=1}^{n}\alpha_{n-i+1}^{\pm}\functionalderivative{H_{i}^{\pm}}{{\mathcal{L}^{\pm}}}\,,\quad
n\geq 1\,,$ (2.14)

rendering the functions $\mu^{\pm}$ as

$\displaystyle\mu^{\pm}=\alpha_{0}^{\pm}+\sum_{n=1}^{N}\sum_{i=1}^{n}\frac{1}{c^{N-n+1}}\alpha_{n-i+1}^{\pm}\functionalderivative{H_{i}^{\pm}}{{\mathcal{L}^{\pm}}}\,.$
(2.15)

While the above expression is sufficient to specify the boundary conditions,
for the sake of simplicity, we aim to make specific choices for the arbitrary
constants $\alpha_{n}^{\pm}$. As mentioned earlier, opting for
$\alpha^{\pm}_{i}=\delta_{i,0},\,i\geq 0,$ yields the Brown-Henneaux boundary
conditions, since all the $1/c$ terms in the $\mu^{\pm}$ functions disappear.
Another viable option is $\alpha_{0}^{\pm}=0$ and
$\alpha_{i}^{\pm}=\delta_{i,1},\,i\geq 1$. This selection has the virtue of
extending the asymptotic symmetries to include Lifshitz scaling. However, this
possibility will be explored in a separate study.

For the current analysis, we set $\alpha_{0}^{\pm}=1$ and
$\alpha_{i}^{\pm}=\delta_{i,1},\,i\geq 1$. With these choices, the
coefficients $\mu_{n}^{\pm}$ take on the following form

$\displaystyle\mu^{\pm}_{n}=\functionalderivative{H^{\pm}_{n}}{{\mathcal{L}^{\pm}}}\,.$
(2.16)

(Note that within this choice, the same expression also holds for
$\mu_{0}^{\pm}$ due to the relation $\mu_{0}^{\pm}=\frac{\delta
H_{0}^{\pm}}{\delta\mathcal{L}^{\pm}}$, where
$H_{0}^{\pm}\equiv\int\mathcal{L}^{\pm}\,d\phi$). The functions $\mu^{\pm}$
result in

$\displaystyle\mu^{\pm}=1+\sum_{n=1}^{N}\frac{1}{c^{N-n+1}}\functionalderivative{H^{\pm}_{n}}{{\mathcal{L}^{\pm}}}\,.$
(2.17)

The latter expression reinforces the title of this work: these boundary
conditions can be interpreted as the standard Brown-Henneaux case, denoted by
the “1” in the latter expression, deformed by suitable terms in negative
powers of the central charge, symbolized by the sum.

To conclude this section, let us briefly delve into the dynamical equation
(2.6). Given the choices outlined above for the arbitrary constants
$\alpha_{n}^{\pm}$, the simplest scenario arises for $N=1$, reducing (2.6) to
the well-known Dym’s equation

$\displaystyle\pm\ell\partial_{t}{\mathcal{L}^{\pm}}$
$\displaystyle=\partial_{\phi}{\mathcal{L}^{\pm}}-\partial_{\phi}^{3}\left(\frac{1}{2\sqrt{{\mathcal{L}^{\pm}}}}\right).$
(2.18)

This non-linear integrable evolution equation was initially discovered in an
unpublished manuscript by Harry Dym and later revisited in [90]. It manifests
in various physical scenarios, including the classical string [94, 95] and
problems in fluid mechanics, specifically governing the dynamics at the
interface between two fluids with different viscosities, known as the Saffman-
Taylor problem [96]. Dym’s equation admits a diverse set of traveling solitons
[97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107]. An extension to higher
dimensions was later uncovered in [108].

The resulting higher-order differential equations obtained for $N>1$ are
collectively referred to as the Dym hierarchy.

Concerning the integrability of Dym’s hierarchy, it is important to highlight
that each equation within the hierarchy shares an identical infinite tower of
conserved quantities $H_{n}^{\pm}$ [109, 110]. To demonstrate this, consider
equations (2.7b) and (2.14), which give rise to the recursion relation

$\displaystyle\mathcal{D}^{\pm}\left(\functionalderivative{H^{\pm}_{n+1}}{{\mathcal{L}^{\pm}}}\right)=\mathcal{E}\left(\functionalderivative{H^{\pm}_{n}}{{\mathcal{L}^{\pm}}}\right),\quad
n\geq 0.$ (2.19)

By virtue of the latter, we can express the $N$-th member of the hierarchy
(2.6) in two distinct Hamiltonian forms

$\displaystyle\pm\ell\partial_{t}{\mathcal{L}^{\pm}}=\mathcal{D}^{\pm}\left(\functionalderivative{H_{0}^{\pm}}{{\mathcal{L}^{\pm}}}-\functionalderivative{H^{\pm}_{N+1}}{{\mathcal{L}^{\pm}}}\right)=\mathcal{E}\left(\functionalderivative{H_{-1}^{\pm}}{{\mathcal{L}^{\pm}}}-\functionalderivative{H_{N}^{\pm}}{{\mathcal{L}^{\pm}}}\right),$
(2.20)

(where $H^{\pm}_{-1}=-\frac{1}{2}\int
d\phi\,\left(\partial_{\phi}^{-1}{\mathcal{L}^{\pm}}\right)^{2}$), thereby
establishing each element of Dym’s hierarchy as a bi-Hamiltonian system.
Furthermore, the differential operators $\mathcal{D}^{\pm}$ and $\mathcal{E}$
enable the construction of two nonequivalent Poisson structures

$\displaystyle\poissonbracket{F}{G}_{\mathcal{D}}=\int
d\phi\,\functionalderivative{F}{{\mathcal{L}^{\pm}}}\mathcal{D}^{\pm}\left(\functionalderivative{G}{{\mathcal{L}^{\pm}}}\right),\quad\poissonbracket{F}{G}_{\mathcal{E}}=\int
d\phi\,\functionalderivative{F}{{\mathcal{L}^{\pm}}}\mathcal{E}\left(\functionalderivative{G}{{\mathcal{L}^{\pm}}}\right).$
(2.21)

A well-established result follows from the recursion relation (2.19),
asserting that all quantities $H_{n}^{\pm}$ Poisson commute with each other,
utilizing either of the two defined Poisson brackets (see [93]),

$\displaystyle\poissonbracket{H_{n}^{\pm}}{H_{m}^{\pm}}_{\mathcal{D},\mathcal{E}}=0,\quad\forall\,n,m\geq
0.$ (2.22)

The involution of the quantities $H_{n}^{\pm}$ demonstrates that each of them
serves as a conserved charge for the evolution equation, conclusively
establishing the integrability of each equation within the hierarchy.

##  3 Consistency of the boundary conditions

Well-defined boundary conditions must meet three criteria [111]. First, they
must yield a differentiable action principle. Second, they should permit an
asymptotic symmetry group with finite generators. Lastly, it is desirable for
them to encompass appealing gravitational configurations, such as black holes.

The objective of this section is to demonstrate that the chosen boundary
conditions (1.2)(1.4) with (2.17) fulfill all three aforementioned criteria.

###  3.1 Action principle

The Hamiltonian Chern-Simons action is given by

$\displaystyle
I_{H}^{\pm}=-\frac{k}{4\pi}\int\,dt\,d^{2}x\,\epsilon^{ij}\left\langle{A}_{i}^{\pm}\dot{A}_{j}^{\pm}-A_{t}^{\pm}F_{ij}^{\pm}\right\rangle+B^{\pm}\,,$
(3.1)

where the boundary term $B^{\pm}$ fulfills

$\displaystyle\delta
B^{\pm}=-\frac{k}{2\pi}\int_{\rho\to\infty}dt\,d\phi\,\left\langle
A_{t}^{\pm}\delta A_{\phi}^{\pm}\right\rangle\,,$ (3.2)

By using the asymptotic behavior of the fields (1.2) and (1.4), the variation
of this surface integral reads

$\displaystyle\delta B^{\pm}=\mp\frac{k}{4\pi
c\ell}\int_{\rho\to\infty}dt\,d\phi\,\mu^{\pm}\delta\mathcal{L}^{\pm}\,.$
(3.3)

By means of equation (2.17), the boundary term becomes integrable, rendering

$\displaystyle
B^{\pm}_{N}=\mp\int\frac{dt}{\ell}\left(H^{\pm}_{0}+\sum_{n=1}^{N}\frac{1}{c^{N-n+1}}H^{\pm}_{n}\right)\,.$
(3.4)

The subscript $N$ is included as a reminder that the boundary term is
different for distinct choices of $N$.

Since the full gravitational action is the difference of two Chern-Simons
functionals $I=I_{H}^{+}-I_{H}^{-}$, then the full boundary term
$B_{N}=B_{N}^{+}-B_{N}^{-}$ becomes

$\displaystyle
B_{N}=-\int\frac{dt}{\ell}\left[H_{0}^{+}+H_{0}^{-}+\sum_{n=1}^{N}\frac{1}{c^{N-n+1}}\left(H_{n}^{+}+H_{n}^{-}\right)\right]\,.$
(3.5)

Therefore, the action principle is differentiable.

###  3.2 Asymptotic symmetries

Here, we show that the previously identified conserved quantities
$H_{n}^{\pm}$ serve as canonical generators for the asymptotic symmetry group
associated with the given boundary conditions.

To begin, let us examine the asymptotic behavior of the angular component of
the connection $a_{\phi}$, shown in (1.2a). As mentioned in the introduction,
the most general gauge transformation that preserves such behavior has a
parameter of the form (1.3) [76]. The corresponding transformation of the
dynamical fields $\mathcal{L}^{\pm}$ can be expressed as

$\displaystyle\pm\delta\mathcal{L}^{\pm}=\left(\mathcal{D}^{\pm}-c\mathcal{E}\right)\eta^{\pm}\,.$
(3.6)

At this point, the typical approach involves exploring the preservation of the
temporal component of the connection $a_{t}^{\pm}$, often leading to
additional conditions on the functions $\eta^{\pm}$. However, we opt for an
alternative approach. First, by following the methodology outlined in the
previous section, we construct a family of possible choices for $\eta^{\pm}$.
Subsequently, we determine the canonical charges and their algebra. Towards
the end of the section, we demonstrate that these transformations also respect
the form of $a_{t}^{\pm}$.

The construction employs the same methodology as in section 2, where
$\eta^{\pm}$ is treated as an expansion in negative powers of the central
charge up to some arbitrary integer $M$. It is then inserted into (3.6), and
the coefficients of the expansion are determined by solving order by order.
Following the selection of arbitrary constants in a similar fashion to the
previous section, this construction yields the following result

$\displaystyle\eta^{\pm}=\epsilon^{\pm}_{0}+\epsilon^{\pm}_{M}\sum_{m=1}^{M}\frac{1}{c^{M-m+1}}\functionalderivative{H^{\pm}_{m}}{{\mathcal{L}^{\pm}}}\,,$
(3.7)

where $\epsilon_{0}$ and $\epsilon_{M}$ are arbitrary infinitesimal constants.

From a canonical perspective, the variation of the generators for these
transformations can be expressed as

$\displaystyle\delta Q^{\pm}[\eta^{\pm}]=\mp\frac{1}{\ell}\int
d\phi\,\eta^{\pm}\delta{\mathcal{L}^{\pm}}\,.$ (3.8)

This expression is integrable due to (3.7). Consequently, we designate the
generators as $Q_{0}^{\pm}\equiv
Q^{\pm}[\epsilon^{\pm}_{0}=1,\epsilon^{\pm}_{M}=0]$ and $Q_{M}^{\pm}\equiv
Q^{\pm}[\epsilon^{\pm}_{0}=0,\epsilon^{\pm}_{M}=1]$, leading to the following
identifications

$\displaystyle Q_{0}^{\pm}$ $\displaystyle=\mp\frac{1}{\ell}H_{0}^{\pm}\,,$
(3.9a) $\displaystyle Q_{M}^{\pm}$
$\displaystyle=\mp\frac{1}{\ell}\sum_{m=1}^{M}\frac{1}{c^{M-m+1}}H^{\pm}_{m}\,.$
(3.9b)

It is important to emphasize that $M$ can take any positive integer value.
Therefore, the generator $Q_{M}^{\pm}$ corresponds to an infinite family of
generators, each labeled by a distinct value of $M$. According to (3.6), the
generators induce the following transformations for the dynamical fields

$\displaystyle\pm\delta_{0}\mathcal{L}^{\pm}$
$\displaystyle=\partial_{\phi}\mathcal{L}^{\pm}\,,$ (3.10a)
$\displaystyle\pm\delta_{M}\mathcal{L}^{\pm}$
$\displaystyle=-\mathcal{E}\left(\frac{\delta
H_{M}^{\pm}}{\delta\mathcal{L}^{\pm}}\right)\,.$ (3.10b)

With the generators now identified, we can proceed to establish their algebra.
For this purpose, the following relation proves to be useful [4]

$\displaystyle\left\\{Q^{\pm}\left[\eta_{1}^{\pm}\right],Q^{\pm}\left[\eta_{2}^{\pm}\right]\right\\}=\delta_{1}Q^{\pm}_{2}\,.$
(3.11)

It is important to distinguish the above Poisson bracket from those defined in
(2.21). The former corresponds to the gravitational phase space, while the
latter relates to the bi-Hamiltonian structure of the Dym hierarchy. These are
not necessarily equivalent. However, through the utilization of relation
(3.11) and considering the transformations in (3.10), it is straightforward to
show that

$\displaystyle\left\\{Q^{\pm}_{M},Q^{\pm}_{\bar{M}}\right\\}=\sum_{m=1}^{\bar{M}}\frac{1}{c^{\bar{M}-m+1}}\int
d\phi\,\frac{\delta
H^{\pm}_{m}}{\delta\mathcal{L}^{\pm}}\mathcal{E}\left(\frac{\delta
H_{M}^{\pm}}{\delta\mathcal{L}^{\pm}}\right)\,.$ (3.12)

The right-hand side of the latter equation corresponds to
$\poissonbracket{H_{m}^{\pm}}{H_{M}^{\pm}}_{\mathcal{E}}$, which vanishes due
to (2.22). Consequently, the generators satisfy an abelian algebra, as
expected from the integrability of the dynamical equations. A similar
procedure can be executed with $Q_{0}^{\pm}$, leading to the following
generator algebra

$\displaystyle\left\\{Q_{M}^{\pm},Q_{\bar{M}}^{\pm}\right\\}=0\quad\forall\,M,\bar{M}\geq
0.$ (3.13)

To finalize this section, we show that the aforementioned asymptotic gauge
transformations also preserve the form of the temporal component
$a_{t}^{\pm}$. The corresponding gauge transformation is given by

$\displaystyle\delta
a_{t}^{\pm}=\partial_{t}\Lambda^{\pm}+\commutator{a_{t}^{\pm}}{\Lambda^{\pm}}\,.$
(3.14)

As mentioned previously, due to the temporal derivative acting on the gauge
parameter, it is expected that the latter equation will further impose
constraints on the form of $\eta^{\pm}$. For instance, in the case of Brown-
Henneaux boundary conditions, a consecuence of (3.14) is that the functions
obey a chiral boson equation. In the case at hand, it turns out that the
equation (3.14) implies no further conditions on the functions $\eta^{\pm}$
but rather reduces to a combination of the equation of motion (2.6) and the
transformation of the field (3.10), thereby becoming an identity. To prove
this statement, we follow [89] to establish a general relation applicable to
any gauge transformation of the form (1.3) along with (3.7).

First, consider two gauge transformations of parameters $\Lambda^{\pm}$ and
$\bar{\Lambda}^{\pm}$, whose action on an auxiliary connection $a$ is given by

$\displaystyle\delta a=$ $\displaystyle d\Lambda+\left[a,\Lambda\right]\,,$
(3.15) $\displaystyle\bar{\delta}a=$ $\displaystyle
d\bar{\Lambda}+\left[a,\bar{\Lambda}\right]\,.$ (3.16)

A straightforward calculation reveals that the commutator of two such gauge
transformations results in another gauge transformation of parameter
$\bar{\bar{\Lambda}}^{\pm}$,

$\displaystyle\commutator{\delta}{\bar{\delta}}a^{\pm}=\bar{\bar{\delta}}a^{\pm}\,,$
(3.17)

where $\bar{\bar{\Lambda}}$ is given by

$\displaystyle\bar{\bar{\Lambda}}^{\pm}=\delta\bar{\Lambda}^{\pm}-\bar{\delta}\Lambda^{\pm}+\commutator{\Lambda^{\pm}}{\bar{\Lambda}^{\pm}}\,.$
(3.18)

Now, let us assume that $\Lambda$ and $\bar{\Lambda}$ belong to the class of
transformations generated by (3.9). Each transformation (3.15) is canonically
generated by $\delta a=\\{Q[\Lambda],a\\}$ and
$\bar{\delta}a=\\{Q[\bar{\Lambda}],a\\}$, where $Q[\Lambda]$ and
$Q[\bar{\Lambda}]$ correspond to some linear combinations of (3.9). It follows
from the Jacobi identity that the commutator
$\commutator{\delta}{\bar{\delta}}a^{\pm}$ is generated by

$\displaystyle\commutator{\delta}{\bar{\delta}}a^{\pm}=\left\\{\left\\{Q^{\pm}[\Lambda],{Q[\bar{\Lambda}]}^{\pm}\right\\},a^{\pm}\right\\}\,.$
(3.19)

As the generators satisfy the involution condition (3.13), the right-hand side
of the latter equation vanishes, indicating that $\bar{\bar{\delta}}a=0$, or
equivalently

$\displaystyle
d\bar{\bar{\Lambda}}^{\pm}+\commutator{a^{\pm}}{\bar{\bar{\Lambda}}^{\pm}}=0\,.$
(3.20)

Since this relation holds for arbitrary connections $a^{\pm}$, we conclude
that

$\displaystyle\delta\bar{\Lambda}^{\pm}-\bar{\delta}\Lambda^{\pm}+\commutator{\Lambda^{\pm}}{\bar{\Lambda}^{\pm}}=0\,.$
(3.21)

In summary, considering the transformations (3.15), the preceding identity
remains valid for any pair of gauge transformations whose generators Poisson-
commute. Since the temporal component of the connection, $a_{t}$, belongs to
the class of allowable gauge parameters (1.4), the above identity encompasses
(3.14) as a specific case, as anticipated.

###  3.3 Stationary black hole solution and its thermodynamics

The third requirement for a well-defined boundary condition is to encompass
physically sensible gravitational configurations, such as black holes.

Solutions to the equations of the Dym hierarchy (2.6) have been extensively
explored in the literature [97, 98, 99, 100, 101, 102, 103, 104, 105, 106,
107]. A prerequisite for these solutions to represent stationary black holes
is that they must be independent of time and exhibit periodicity in the
angular coordinate. For instance, solitons typically do not meet such
criteria. Therefore, our focus lies in the periodic solutions to the equation

$0=\partial_{\phi}\mathcal{L}^{\pm}-\partial^{3}_{\phi}\left(\frac{\delta
H_{N}^{\pm}}{\delta\mathcal{L}^{\pm}}\right)\,.$ (3.22)

To solve the latter, we need to delve into the toolkit provided by the
literature on integrable systems, work which will be pursued elsewhere. For
our current objectives, we will concentrate on a much simpler family of
solutions, given by constant functions

$\displaystyle{\mathcal{L}^{\pm}}(t,\phi)=\mathcal{L}^{\pm}\,.$ (3.23)

Note that this solution satisfies the stationary equation for any choice of
$N$. These solutions correspond to black hole configurations, as shown in this
section using the metric formulation.

Before proceeding, a few comments are in order. Firstly, note that
$\mathcal{L}^{\pm}=0$ is a singularity of the dynamical equation for any
$N\geq 1$ (see B); hence, it is not encompassed in the set of allowed
solutions. As shown below, this implies that extremal black holes are not part
of this class of solutions. Secondly, due to the presence of square roots in
the functions $\mu^{\pm}$, negative values for $\mathcal{L}^{\pm}$ must be
avoided to prevent complex configurations. Finally, it is worth noting that in
this class of solutions, the only non-vanishing conserved quantities are
$H_{0}^{\pm}$ and $H_{1}^{\pm}$. As a result of this, the functions
$\mu^{\pm}$ become constants

$\displaystyle\mu^{\pm}=1+\frac{1}{2c^{N}\sqrt{\mathcal{L}^{\pm}}}\,.$ (3.24)

We initiate the analysis of the geometry by constructing the metric, in
accordance to the relation

$\displaystyle
g_{\mu\nu}=\frac{\ell^{2}}{2}\left\langle\left(A_{\mu}^{+}-A_{\nu}^{-}\right)\left(A_{\mu}^{+}-A_{\nu}^{-}\right)\right\rangle\,.$
(3.25)

where $A^{\pm}=b^{-1}_{\pm}\left(d+a^{\pm}\right)b_{\pm}$ and the radial group
element is chosen as
$b_{\pm}(\rho)=\exp\left[\pm\log\left(\frac{\rho}{\ell}\right)L_{0}\right]$.
Thus, the stationary line element reads

$\displaystyle\begin{split}ds^{2}&=-\left(\frac{\ell}{2c\rho}\mu^{-}\mathcal{L}^{-}-\frac{\rho}{\ell}\mu^{+}\right)\left(\frac{\ell}{2c\rho}\mu^{+}\mathcal{L}^{+}-\frac{\rho}{\ell}\mu^{-}\right)dt^{2}+\frac{\ell^{2}d\rho^{2}}{\rho^{2}}\\\
&\hskip
28.45274pt+\left\\{\bigg{[}\frac{\rho^{2}}{\ell}+\frac{\ell}{c}\left(1+\frac{\ell^{2}}{4c\rho^{2}}\mathcal{L}^{-}\right)\mathcal{L}^{+}\bigg{]}\mu^{+}-\bigg{[}\frac{\rho^{2}}{\ell}+\frac{\ell}{c}\left(1+\frac{\ell^{2}}{4c\rho^{2}}\mathcal{L}^{+}\right)\mathcal{L}^{-}\bigg{]}\mu^{-}\right\\}dtd\phi\\\
&\hskip
199.16928pt+\left(\rho+\frac{\ell^{2}}{2c\rho}\mathcal{L}^{+}\right)\left(\rho+\frac{\ell^{2}}{2c\rho}\mathcal{L}^{-}\right)d\phi^{2}\,.\end{split}$
(3.26)

It is convenient to rewrite the latter metric using Schwarzschild-like
coordinates. For this purpose, it is helpful to redefine the radial coordinate
according to

$\displaystyle
r^{2}=\left(\rho+\frac{\ell^{2}}{2c\rho}\mathcal{L}^{+}\right)\left(\rho+\frac{\ell^{2}}{2c\rho}\mathcal{L}^{-}\right)\,,$
(3.27)

through which it can be expressed as

$\displaystyle
ds^{2}=-N^{2}(r)dt^{2}+\frac{dr^{2}}{f^{2}(r)}+r^{2}\left(N^{\phi}(r)dt+d\phi\right)^{2}\,.$
(3.28)

The lapse and shift functions $N(r)$ and $N^{\phi}(r)$ are given by

$\displaystyle N(r)=\left(\frac{{\mu^{+}}+{\mu^{-}}}{2}\right)f(r)\,,\quad
N^{\phi}(r)=\frac{{\mu^{+}}-{\mu^{-}}}{2\ell}+\frac{\ell\left({\mu^{+}}+{\mu^{-}}\right)\left({\mathcal{L}}^{+}-{\mathcal{L}}^{-}\right)}{4cr^{2}}\,,$
(3.29)

while the function $f(r)$ corresponds to

$\displaystyle
f^{2}(r)=\frac{r^{2}}{\ell^{2}}-\frac{{\mathcal{L}}^{+}+{\mathcal{L}}^{-}}{c}+\frac{\ell^{2}\left({\mathcal{L}}^{+}-{\mathcal{L}}^{-}\right)^{2}}{4c^{2}r^{2}}\,.$
(3.30)

The horizon is determined by the condition $N(r)=0$, indicating the positions
of both the inner and outer horizons

$\displaystyle
r_{\pm}^{2}=\frac{\ell^{2}}{2c}\left(\mathcal{L}^{+}+\mathcal{L}^{-}\pm
2\sqrt{{\mathcal{L}}^{+}{\mathcal{L}}^{-}}\right)\,.$ (3.31)

Extremality occurs when either $\mathcal{L}^{+}=0$ or $\mathcal{L}^{-}=0$, a
condition not supported in the solution spectrum. Consequently, this family of
solutions does not include extremal black holes.

The black hole’s mass and angular momentum are conserved quantities associated
with the Killing vectors $\partial_{t}$ and $-\partial_{\phi}$, respectively.
In the Chern-Simons formulation, these are expressed as

$\displaystyle\delta
Q^{\pm}\left[\xi\right]=\frac{k}{2\pi}\oint_{\rho\to\infty}d\phi\,{\xi}^{\mu}\bigg{\langle}a^{\pm}_{\mu}\delta
a^{\pm}_{\phi}\bigg{\rangle}\,,$ (3.32)

where $\xi$ represents the Killing vector. Considering both $\pm$ sectors, the
total charge is $Q=Q^{+}-Q^{-}$. Consequently, the mass and angular momentum
correspond to

$\displaystyle
M=\frac{2\pi}{\ell}\left[\left(\mathcal{L}^{+}+\mathcal{L}^{-}\right)+\frac{1}{c^{N}}\left(\sqrt{\mathcal{L}^{+}}+\sqrt{\mathcal{L}^{-}}\right)\right]\,,\quad
J=-2\pi\left(\mathcal{L}^{+}-\mathcal{L}^{-}\right)\,.$ (3.33)

Figure 1 illustrates the black hole spectrum in terms of $M$ and $J$.

$M$$J$$\mathcal{L}^{-}=0$$\mathcal{L}^{+}=0$ Figure 1: Black hole solution
spectrum in terms of the mass $M$ and angular momentum $J$. The hatched area
corresponds to black holes, while the dashed lines represent extremal
configurations, which are not permissible within this solution family.

The thermal properties of the black hole can be examined within the Euclidean
counterpart of the metric (3.28). The Hawking temperature $\beta$ and horizon
angular velocity $\Omega_{h}$ are selected to regularize the Euclidean version
of metric (3.28) in the near-horizon sector (see appendix C for details). This
yields

$\displaystyle\beta$
$\displaystyle=\frac{4\pi\ell^{2}r_{+}}{\left({\mu^{+}}+{\mu^{-}}\right)\left(r_{+}^{2}-r_{-}^{2}\right)}=\frac{\sqrt{2c}\ell\pi(\sqrt{\mathcal{L}^{+}}+\sqrt{\mathcal{L}^{-}})}{\left({\mu^{+}}+{\mu^{-}}\right)\sqrt{\mathcal{L}^{-}}\sqrt{\mathcal{L}^{+}}}\,,$
(3.34a) $\displaystyle\Omega_{h}$
$\displaystyle=-N^{\phi}(r_{+})=\frac{{\mu^{+}}-{\mu^{-}}}{2\ell}+\frac{\left({\mu^{+}}+{\mu^{-}}\right)\left({\mathcal{L}}^{+}-{\mathcal{L}}^{-}\right)}{2\ell(\sqrt{\mathcal{L}^{+}}+\sqrt{\mathcal{L}^{-}})^{2}}\,.$
(3.34b)

Finally, the black hole entropy can be deduced from the boundary term $B$
needed to regularize the Euclidean action. Through a straightforward
calculation detailed in Appendix C, the variation $\delta B$ takes the
following form

$\displaystyle\delta B=i\beta\left(\delta M-\frac{1}{\beta}\delta
S-\Omega_{h}\delta J\right)\,,$ (3.35)

where the entropy $S$ is expressed as

$\displaystyle S=\frac{2\pi
r_{+}}{4G}=\frac{\pi\ell\left(\sqrt{{\mathcal{L}}^{+}}+\sqrt{{\mathcal{L}}^{-}}\right)}{2G\sqrt{2c}}\,,$
(3.36)

as expected by the universality of the Bekenstein-Hawking result.

##  4 Final comments

We have shown the construction of Dym boundary conditions for AdS3 pure
gravity and established their well-defined nature. Notably, these conditions
give rise to an infinite and abelian asymptotic symmetric group, where the
generators correspond to the Dym conserved charges, and encompass black hole
solutions within their spectrum.

We conclude this work with a discussion of avenues worth exploring.

It is important to note that the choice of the integration constants
$\alpha_{n}^{\pm}$ is arbitrary and was inspired by the goal of constructing
deformations of Brown-Henneaux boundary conditions. However, as mentioned
earlier, alternative possibilities exist. In particular, selecting
$\alpha_{0}^{\pm}=0$ leads to the dynamical equation

$\pm\ell\partial_{t}\mathcal{L}^{\pm}=-\partial^{3}_{\phi}\left(\frac{\delta
H_{N}^{\pm}}{\delta\mathcal{L}^{\pm}}\right)\,,$ (4.1)

which enhances symmetry to include Lifshitz scaling. This might allow to
explore Lifshitz holography within this context [112, 113, 114, 115].

It would also be desirable to find solutions describing black holes with non-
vanishing Dym charges. It is worth mentioning that when applying large gauge
transformations in the form of (3.10) to a gravitational configuration, the
values of the global charges $H_{n}^{\pm}$ remain unchanged due to the
involution of the generators (3.13). Therefore, to find solutions, and in
particular black holes, with non-trivial Dym charges, it is not sufficient to
apply large gauge transformations to known solutions; rather, it is necessary
to draw on techniques from integrable systems, which should prove useful for
this purpose.

Regarding the holographic implications of this family of boundary conditions,
these will be explored elsewhere. Noting that the expansion in (2.4) utilizes
negative integer powers of the Brown-Henneaux central charge, it is tempting
to conjecture that the bulk theory described by this choice of boundary
conditions is somehow related to a boundary conformal theory at large central
charge. These has been extensively explored in the context of holography [116,
117, 118, 119, 120, 121, 122, 123].

Finally, an extension of this work can be pursued for higher-order algebras,
such as $sl(N,R)$, potentially establishing connections between higher spin
gravity and a broader hierarchy of integrable equations that includes the Dym
hierarchy as a particular case.

#### Acknowledgments

The authors express their gratitude to Marcela Cárdenas, Francisco Correa,
José Figueroa, Hernán A. González, Luis Guajardo, and Ricardo Troncoso for
their insightful comments. This work has received support from FONDECYT grants
1231810, 3230618, ANID Beca Doctorado Nacional 21232318, and DICYT 042231PR PS
530. K.L. would like to extend thanks to the Theoretical and Mathematical
Physics group at Université libre de Bruxelles for their warm hospitality,
where part of this work was carried out. Furthermore, M.P. and F.R. also
acknowledge Centro de Estudios Científicos (CECs) at Universidad San Sebastián
for their hospitality during part of this work.

##  Appendix A Chern-Simons formulation of AdS3 gravity

We provide a concise overview of the Chern-Simons formulation of AdS3 gravity,
focusing solely on essential details to ensure the self-contained nature of
the preceding presentation.

$(2+1)$ dimensional General Relativity with a negative cosmological constant
can be written as the difference of two Chern-Simons action [78, 79]

$\displaystyle I=I_{H}[A^{+}]-I_{H}[A^{-}]\,,\quad
I_{H}[A]=-\frac{k}{4\pi}\int dt\,d^{2}x\,\epsilon^{ij}\left\langle
A_{i}\dot{A}_{j}-A_{t}F_{ij}\right\rangle+B\,,$ (A.1)

where
$F_{ij}^{\pm}=\partial_{i}A^{\pm}_{j}-\partial_{j}A_{i}^{\pm}+\commutator{A_{i}^{\pm}}{A_{j}^{\pm}}$.
The Chern-Simons level is given by $k=\ell/4G$, where $\ell$ stands for the
AdS3 radius while $G$ is the three-dimensional Newton constant. The fields
$A^{\pm}$ are $sl(2,\mathbb{R})$ connections. The three algebra generators
$L_{n}$, where $n\in\left\\{0,\pm 1\right\\}$, fulfill the algebra

$\displaystyle\commutator{L_{m}}{L_{n}}=(m-n)L_{m+n}\,.$ (A.2)

The nonvanishing components of the bilinear product
$\left\langle\,,\right\rangle$ are

$\displaystyle\left\langle
L_{0},L_{0}\right\rangle=\frac{1}{2}\,,\quad\left\langle
L_{1},L_{-1}\right\rangle=-1\,.$ (A.3)

The boundary term $B$ in (A.1) is constructed to assure the differentiability
of the action, yielding

$\displaystyle\delta
B=-\frac{k}{2\pi}\int_{\rho\to\infty}dt\,d\phi\,\left\langle A_{t}\delta
A_{\phi}\right\rangle\,.$ (A.4)

Therefore, adequate boundary conditions must be inforced in order for $\delta
B$ to be integrable.

The infinitesimal gauge transformation

$\displaystyle\delta A=d\Lambda+\commutator{A}{\Lambda}\,,$ (A.5)

is generated by the smeared integral

$\displaystyle G\left[\Lambda\right]=\frac{k}{4\pi}\int
d^{2}x\,\epsilon^{ij}\left\langle\Lambda F_{ij}\right\rangle+Q\,,$ (A.6)

where $Q$ is a surface term assuring the differentiability of the generator,
implying

$\displaystyle\delta
Q\left[\Lambda\right]=-\frac{k}{2\pi}\int_{\rho\to\infty}d\phi\,\left\langle\Lambda\delta
A_{\phi}\right\rangle\,.$ (A.7)

If the gauge parameter $\Lambda$ decay fast enough towards the boundary such
that $Q$ vanishes, then the transformation is a proper gauge transform. On the
other hand, if the boundary behavior of the gauge parameter is such that $Q$
is finite, then the transformation is a large gauge transformation [124]. More
details and discussion on this subject can be found in the review [125] and
references therein.

Finally, the link between Chern-Simons and the metric fields is established
through $A^{\pm}=\omega\pm\frac{e}{\ell}$. Consequently, the metric can be
constructed as

$\displaystyle
g_{\mu\nu}=\frac{\ell^{2}}{2}\left\langle\left(A_{\mu}^{+}-A_{\mu}^{-}\right),\left(A_{\nu}^{+}-A_{\nu}^{-}\right)\right\rangle\,.$
(A.8)

##  Appendix B List of Dym conserved charges $H^{\pm}_{n}$

Following the recursive procedure outlined in Section 2, we present the
explicit form of the first five functionals $H^{\pm}_{n}$ along with their
respective functional derivatives. In what follows, the prime denotes
derivatives with respect to $\phi$. Introducing the notation $H^{\pm}_{n}=\int
d\phi\,\mathcal{H}^{\pm}_{n}$, the first five charge densities are as follows:

$\displaystyle\begin{split}\mathcal{H}^{\pm}_{1}&=\sqrt{{\mathcal{L}^{\pm}}}\,,\\\
\mathcal{H}^{\pm}_{2}&=-\frac{1}{2\sqrt{{\mathcal{L}^{\pm}}}}\left(\frac{5}{8}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{2}}{{\mathcal{L}^{\pm}}^{2}}-\frac{1}{2}\frac{{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}}\right)\,,\\\
\mathcal{H}^{\pm}_{3}&=-\frac{1}{2\sqrt{{\mathcal{L}^{\pm}}}}\left(\frac{385}{256}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{4}}{{\mathcal{L}^{\pm}}^{5}}-\frac{77}{32}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{2}{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{4}}+\frac{7}{16}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}^{2}}{{\mathcal{L}^{\pm}}^{3}}+\frac{7}{12}\frac{{\mathcal{L}^{\pm}}^{\prime}{\mathcal{L}^{\pm}}^{\prime\prime\prime}}{{\mathcal{L}^{\pm}}^{3}}-\dfrac{1}{12}\frac{{\mathcal{L}^{\pm}}^{(4)}}{{\mathcal{L}^{\pm}}^{2}}\right)\,,\\\
\mathcal{H}^{\pm}_{4}&=-\frac{1}{2\sqrt{{\mathcal{L}^{\pm}}}}\left(\frac{85085}{4096}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{6}}{{\mathcal{L}^{\pm}}^{8}}-\frac{51051}{1024}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{4}{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{7}}+\frac{35607}{1280}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{2}{{\mathcal{L}^{\pm}}^{\prime\prime}}^{2}}{{\mathcal{L}^{\pm}}^{6}}-\frac{671}{320}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}^{3}}{{\mathcal{L}^{\pm}}^{5}}\right.\\\
&\hskip
56.9055pt+\frac{429}{32}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{3}{\mathcal{L}^{\pm}}^{\prime\prime\prime}}{{\mathcal{L}^{\pm}}^{6}}-\frac{1419}{160}\frac{{\mathcal{L}^{\pm}}^{\prime}{\mathcal{L}^{\pm}}^{\prime\prime}{\mathcal{L}^{\pm}}^{\prime\prime\prime}}{{\mathcal{L}^{\pm}}^{5}}+\frac{69}{160}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}^{2}}{{\mathcal{L}^{\pm}}^{4}}-\frac{165}{64}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{2}{\mathcal{L}^{\pm}}^{(4)}}{{\mathcal{L}^{\pm}}^{5}}\\\
&\hskip
170.71652pt\left.+\frac{57}{80}\frac{{\mathcal{L}^{\pm}}^{\prime\prime}{\mathcal{L}^{\pm}}^{(4)}}{{\mathcal{L}^{\pm}}^{4}}+\frac{27}{80}\frac{{\mathcal{L}^{\pm}}^{\prime}{\mathcal{L}^{\pm}}^{(5)}}{{\mathcal{L}^{\pm}}^{4}}-\frac{1}{40}\frac{{\mathcal{L}^{\pm}}^{(6)}}{{\mathcal{L}^{\pm}}^{3}}\right)\,,\\\
\mathcal{H}^{\pm}_{5}&=-\frac{1}{2\sqrt{\mathcal{L}^{\pm}}}\left(\frac{185910725}{262144}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{8}}{{\mathcal{L}^{\pm}}^{11}}-\frac{37182145}{16384}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{\prime}}^{6}}{{\mathcal{L}^{\pm}}^{10}}+\frac{9006855}{14336}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}{{\mathcal{L}^{\pm}}^{\prime}}^{5}}{{\mathcal{L}^{\pm}}^{9}}\right.\\\
&\hskip
28.45274pt+\frac{123740331}{57344}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}^{2}{{\mathcal{L}^{\pm}}^{\prime}}^{4}}{{\mathcal{L}^{\pm}}^{9}}-\frac{1859715}{14336}\frac{{{\mathcal{L}^{\pm}}^{(4)}}{{\mathcal{L}^{\pm}}^{\prime}}^{4}}{{\mathcal{L}^{\pm}}^{8}}+\frac{36465}{1792}\frac{{{\mathcal{L}^{\pm}}^{(5)}}{{\mathcal{L}^{\pm}}^{\prime}}^{3}}{{\mathcal{L}^{\pm}}^{7}}\\\
&\hskip
14.22636pt-\frac{3201627}{3584}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}{{\mathcal{L}^{\pm}}^{\prime}}^{3}}{{\mathcal{L}^{\pm}}^{8}}+\frac{282711}{3584}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}^{2}{{\mathcal{L}^{\pm}}^{\prime}}^{2}}{{\mathcal{L}^{\pm}}^{7}}+\frac{226083}{1792}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{(4)}}{{\mathcal{L}^{\pm}}^{\prime}}^{2}}{{\mathcal{L}^{\pm}}^{7}}\\\
&-\frac{2145}{896}\frac{{{\mathcal{L}^{\pm}}^{(6)}}{{\mathcal{L}^{\pm}}^{\prime}}^{2}}{{\mathcal{L}^{\pm}}^{6}}-\frac{629629}{1024}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}^{3}{{\mathcal{L}^{\pm}}^{\prime}}^{2}}{{\mathcal{L}^{\pm}}^{8}}+\frac{97383}{448}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}^{2}{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}{{\mathcal{L}^{\pm}}^{\prime}}}{{\mathcal{L}^{\pm}}^{7}}+\frac{11}{56}\frac{{{\mathcal{L}^{\pm}}^{(7)}}{{\mathcal{L}^{\pm}}^{\prime}}}{{\mathcal{L}^{\pm}}^{5}}\\\
&-\frac{1287}{112}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{(5)}}{{\mathcal{L}^{\pm}}^{\prime}}}{{\mathcal{L}^{\pm}}^{6}}-\frac{3861}{224}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}{{\mathcal{L}^{\pm}}^{(4)}}{{\mathcal{L}^{\pm}}^{\prime}}}{{\mathcal{L}^{\pm}}^{6}}+\frac{180323}{7168}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}^{4}}{{\mathcal{L}^{\pm}}^{7}}+\frac{253}{448}\frac{{{\mathcal{L}^{\pm}}^{(4)}}^{2}}{{\mathcal{L}^{\pm}}^{5}}\\\
&\hskip
28.45274pt+\frac{209}{224}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}{{\mathcal{L}^{\pm}}^{(5)}}}{{\mathcal{L}^{\pm}}^{5}}+\frac{121}{224}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{(6)}}}{{\mathcal{L}^{\pm}}^{5}}-\frac{1}{112}\frac{{{\mathcal{L}^{\pm}}^{(8)}}}{{\mathcal{L}^{\pm}}^{4}}-\frac{13299}{896}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}^{2}}{{\mathcal{L}^{\pm}}^{6}}\\\
&\hskip
227.62204pt\left.-\frac{10725}{896}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}^{2}{{\mathcal{L}^{\pm}}^{(4)}}}{{\mathcal{L}^{\pm}}^{6}}\right)\,.\end{split}$
(B.1)

The corresponding functionals derivatives are:

$\displaystyle\begin{split}\frac{\delta
H_{1}^{\pm}}{\delta\mathcal{L}^{\pm}}&=\frac{1}{2\sqrt{{\mathcal{L}^{\pm}}}}\,,\\\
\frac{\delta
H_{2}^{\pm}}{\delta\mathcal{L}^{\pm}}&=\frac{1}{2\sqrt{{\mathcal{L}^{\pm}}}}\left(\frac{5}{16}\frac{{{\mathcal{L}^{\pm}}}^{\prime
2}}{{\mathcal{L}^{\pm}}^{3}}-\frac{1}{4}\frac{{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{2}}\right)\,,\\\
\frac{\delta
H_{3}^{\pm}}{\delta\mathcal{L}^{\pm}}&=\frac{1}{2\sqrt{\mathcal{L}^{\pm}}}\left(\frac{1155}{512}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{4}}{{\mathcal{L}^{\pm}}^{6}}-\frac{231}{64}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{2}{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{5}}+\frac{21}{32}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}^{2}}{{\mathcal{L}^{\pm}}^{4}}+\frac{7}{8}\frac{{\mathcal{L}^{\pm}}^{\prime}{\mathcal{L}^{\pm}}^{\prime\prime\prime}}{{\mathcal{L}^{\pm}}^{4}}-\frac{1}{8}\frac{{\mathcal{L}^{\pm}}^{(4)}}{{\mathcal{L}^{\pm}}^{3}}\right)\,,\\\
\hfil\displaystyle\begin{split}\frac{\delta
H_{4}^{\pm}}{\delta\mathcal{L}^{\pm}}&=\frac{1}{2\sqrt{{\mathcal{L}^{\pm}}}}\left(\frac{425425}{8192}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{6}}{{\mathcal{L}^{\pm}}^{9}}-\frac{255255}{2048}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{4}{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{8}}+\frac{35607}{512}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{2}{{\mathcal{L}^{\pm}}^{\prime\prime}}^{2}}{{\mathcal{L}^{\pm}}^{7}}-\frac{671}{128}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}^{3}}{{\mathcal{L}^{\pm}}^{6}}\right.\\\
&\hskip
42.67912pt+\frac{2145}{64}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{3}{\mathcal{L}^{\pm}}^{\prime\prime\prime}}{{\mathcal{L}^{\pm}}^{7}}-\frac{1419}{64}\frac{{{\mathcal{L}^{\pm}}^{\prime}}{{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}}{{\mathcal{L}^{\pm}}^{6}}+\frac{69}{64}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}^{2}}{{\mathcal{L}^{\pm}}^{5}}-\frac{825}{128}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{2}{\mathcal{L}^{\pm}}^{(4)}}{{\mathcal{L}^{\pm}}^{6}}\\\
&\hskip
142.26378pt\left.+\frac{57}{32}\frac{{\mathcal{L}^{\pm}}^{\prime\prime}{\mathcal{L}^{\pm}}^{(4)}}{{\mathcal{L}^{\pm}}^{5}}+\frac{27}{32}\frac{{\mathcal{L}^{\pm}}^{\prime}{\mathcal{L}^{\pm}}^{(5)}}{{\mathcal{L}^{\pm}}^{5}}-\frac{1}{16}\frac{{\mathcal{L}^{\pm}}^{(6)}}{{\mathcal{L}^{\pm}}^{4}}\right)\,,\\\
\functionalderivative{H^{\pm}_{5}}{{\mathcal{L}^{\pm}}}&=\frac{1}{2\sqrt{{\mathcal{L}^{\pm}}}}\left(\frac{1301375075}{524288}\frac{{{\mathcal{L}^{\pm}}^{\prime}}^{8}}{{\mathcal{L}^{\pm}}^{12}}-\frac{260275015}{32768}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{\prime}}^{6}}{{\mathcal{L}^{\pm}}^{11}}+\frac{9006855}{4096}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}{{\mathcal{L}^{\pm}}^{\prime}}^{5}}{{\mathcal{L}^{\pm}}^{10}}\right.\\\
&\hskip
56.9055pt+\frac{123740331}{16384}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}^{2}{{\mathcal{L}^{\pm}}^{\prime}}^{4}}{{\mathcal{L}^{\pm}}^{10}}-\frac{1859715}{4096}\frac{{{\mathcal{L}^{\pm}}^{(4)}}{{\mathcal{L}^{\pm}}^{\prime}}^{4}}{{\mathcal{L}^{\pm}}^{9}}+\frac{36465}{512}\frac{{{\mathcal{L}^{\pm}}^{(5)}}{{\mathcal{L}^{\pm}}^{\prime}}^{3}}{{\mathcal{L}^{\pm}}^{8}}\\\
&\hskip
28.45274pt-\frac{3201627}{1024}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}{{\mathcal{L}^{\pm}}^{\prime}}^{3}}{{\mathcal{L}^{\pm}}^{9}}+\frac{282711}{1024}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}^{2}{{\mathcal{L}^{\pm}}^{\prime}}^{2}}{{\mathcal{L}^{\pm}}^{8}}+\frac{226083}{512}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{(4)}}{{\mathcal{L}^{\pm}}^{\prime}}^{2}}{{\mathcal{L}^{\pm}}^{8}}\\\
&\hskip
56.9055pt-\frac{2145}{256}\frac{{{\mathcal{L}^{\pm}}^{(6)}}{{\mathcal{L}^{\pm}}^{\prime}}^{2}}{{\mathcal{L}^{\pm}}^{7}}-\frac{4407403}{2048}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}^{3}{{\mathcal{L}^{\pm}}^{\prime}}^{2}}{{\mathcal{L}^{\pm}}^{9}}+\frac{97383}{128}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}^{2}{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}{{\mathcal{L}^{\pm}}^{\prime}}}{{\mathcal{L}^{\pm}}^{8}}\\\
&\hskip
56.9055pt+\frac{11}{16}\frac{{{\mathcal{L}^{\pm}}^{(7)}}{{\mathcal{L}^{\pm}}^{\prime}}}{{\mathcal{L}^{\pm}}^{6}}-\frac{1287}{32}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{(5)}}{{\mathcal{L}^{\pm}}^{\prime}}}{{\mathcal{L}^{\pm}}^{7}}-\frac{3861}{64}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}{{\mathcal{L}^{\pm}}^{(4)}}{{\mathcal{L}^{\pm}}^{\prime}}}{{\mathcal{L}^{\pm}}^{7}}\\\
&\hskip
56.9055pt+\frac{180323}{2048}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}^{4}}{{\mathcal{L}^{\pm}}^{8}}+\frac{253}{128}\frac{{{\mathcal{L}^{\pm}}^{(4)}}^{2}}{{\mathcal{L}^{\pm}}^{6}}+\frac{209}{64}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}{{\mathcal{L}^{\pm}}^{(5)}}}{{\mathcal{L}^{\pm}}^{6}}+\frac{121}{64}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{(6)}}}{{\mathcal{L}^{\pm}}^{6}}\\\
&\hskip
99.58464pt\left.-\frac{1}{32}\frac{{{\mathcal{L}^{\pm}}^{(8)}}}{{\mathcal{L}^{\pm}}^{5}}-\frac{13299}{256}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}{{\mathcal{L}^{\pm}}^{\prime\prime\prime}}^{2}}{{\mathcal{L}^{\pm}}^{7}}-\frac{10725}{256}\frac{{{\mathcal{L}^{\pm}}^{\prime\prime}}^{2}{{\mathcal{L}^{\pm}}^{(4)}}}{{\mathcal{L}^{\pm}}^{7}}\right)\,.\end{split}\end{split}$
(B.2)

Note that these satisfy the following identity [126]

$\displaystyle H^{\pm}_{n}=\frac{2}{3-2n}\int
d\phi\,\frac{\delta\mathcal{H}_{n}^{\pm}}{\delta\mathcal{L}^{\pm}}{\mathcal{L}^{\pm}}\,.$
(B.3)

##  Appendix C Euclidean analysis of black hole solution

In this analysis, the Euclidean version of the metric (3.28) is examined to
extract the thermodynamic properties of the black hole solution.

Consider first the Wick rotation

$\displaystyle t\to i\tau\,.$ (C.1)

Upon implementing, the metric takes on the Euclidean form

$\displaystyle
ds^{2}_{\mathrm{E}}=N^{2}(r)d\tau^{2}+\frac{dr^{2}}{f^{2}(r)}+r^{2}\left(N^{\phi}_{\mathrm{E}}(r)d\tau+d\phi\right)^{2}\,,$
(C.2)

where $N^{j}_{\mathrm{E}}=iN^{j}$.

To determine the temperature and velocity of the horizon, the Euclidean metric
near the surface $r=r_{+}$ is addressed. It results in

$\displaystyle
ds^{2}_{\mathrm{E}}\simeq\frac{1}{4}\left(\frac{{\mu^{+}}+{\mu^{-}}}{2}\right)^{2}\left(\left.{f^{2}(r)}^{\prime}\right|_{r_{+}}\right)^{2}R^{2}d\tau^{2}+dR^{2}+r_{+}^{2}\left(iN^{\phi}(r_{+})d\tau+d\phi\right)^{2}\,,$
(C.3)

Here, the radial coordinate has been redefined as
$R=2\sqrt{\frac{(r-r_{+})}{\left.{f^{2}(r)}^{\prime}\right|_{r_{+}}}}$. In
order for the latter to represent a cylinder of the form
$ds^{2}_{\mathrm{E}}=R^{2}d\theta^{2}+dR^{2}+dz^{2}$, the following
identification of coordinates is performed

$\displaystyle\theta=\left(\frac{{\mu^{+}}+{\mu^{-}}}{4}\right)\left(\left.{f^{2}(r)}^{\prime}\right|_{r_{+}}\right)\tau\,,\quad
dz=r_{+}\left(iN^{\phi}(r_{+})d\tau+d\phi\right)\,.$ (C.4)

The angular periodicity $\theta\sim\theta+2\pi$ implies that the temporal
coordinate is identified as $\tau\sim\tau+\beta$, while the absence of
periodicity in $z$ (the transverse section) results in
$\phi\sim\phi+i\beta\Omega_{h}$. Consequently, the inverse of the Hawking
temperature $\beta$ and the velocity of the horizon $\Omega_{h}$ are in
agreement with (3.34a) and (3.34b), respectively.

Now, details about the derivation of (3.35) are presented. To determine the
variation of the boundary term $\delta B$, the Hamiltonian formulation is
employed. The Euclidean Hamiltonian action of General Relativity is expressed
as

$\displaystyle I_{E}=I_{H}+B\,.$ (C.5)

Focusing our attention solely on the boundary term, its variation is given by
[127]

$\displaystyle\delta B=-\frac{1}{16\pi G}\int
dS^{2}_{l}\,\left[G^{ijkl}\left(N_{|k}\delta\gamma_{ij}-N\delta\gamma_{ij|k}\right)-2N_{\mathrm{E},k}\delta\pi^{kl}_{\mathrm{E}}-\left(2\pi^{il}_{\mathrm{E}}N^{j}_{\mathrm{E}}-\pi^{ij}_{\mathrm{E}}N^{l}_{\mathrm{E}}\right)\delta\gamma_{ij}\right]\,.$
(C.6)

Here, $\gamma_{ij}$ represents the spatial metric, and $\pi^{ij}$ is its
conjugate momentum. Additionally, $N$ and $N_{E}^{i}$ denote the lapse and
shift vectors, respectively.

Considering the stationary and axisymmetric symmetry of the solution (3.28),
the variation of $B$ simplifies to

$\displaystyle\delta B$ $\displaystyle=\frac{i}{16\pi
G}\int_{0}^{\beta}d\tau\int_{0}^{2\pi}d\phi\left.\left(G^{ijkr}N\delta\gamma_{ij|k}+2N_{\mathrm{E},j}\delta\pi^{rj}_{\mathrm{E}}\right)\right\rvert_{r=r_{+}}^{r=\infty}\,.$
(C.7)

Expressed in terms of Dym’s fields, the above can be written as

$\displaystyle\begin{split}\delta B&=\frac{i}{16\pi
G}\beta\left(2\pi\right)\left[\frac{1}{c}\left({\mu^{+}}\delta{\mathcal{L}}^{+}+{\mu^{-}}\delta{\mathcal{L}}^{-}\right)-\frac{4\pi}{\beta}\delta
r_{+}\right.\\\ &\hskip
142.26378pt\left.-\frac{1}{c}\frac{\left({\mu^{+}}\sqrt{{\mathcal{L}}^{+}}-{\mu^{-}}\sqrt{{\mathcal{L}}^{-}}\right)}{\sqrt{{\mathcal{L}}^{+}}+\sqrt{{\mathcal{L}}^{-}}}\left(\delta{\mathcal{L}}^{+}-\delta{\mathcal{L}}^{-}\right)\right]\,.\end{split}$
(C.8)

Recognizing the mass and angular momentum (3.33), we arrive at (3.35), as
promised.

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