# Exploration of Near-Horizon CFT Duality and $AdS_{2}/CFT_{1}$ in Conformal
Weyl Gravity

Daksh Aggarwal1, Dominic Chang2,3,4, Quentin Dancewicz Helmers1, Nesibe
Sivrioglu1, L. R. Ram-Mohan5, Leo Rodriguez1,5, Shanshan Rodriguez1 and Raid
Suleiman4,5 1 Department of Physics, Grinnell College, Grinnell IA, 50112, USA
2 The Event Horizon Telescope Collaboration  3 Black Hole Initiative, Harvard
University, Cambridge, MA 02138, USA  4 Center for Astrophysics, Harvard &
Smithsonian, Cambridge MA, 02138, USA  5 Department of Physics, Worcester
Polytechnic Institute, Worcester MA, 01609, USA<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compute near horizon black hole entropy via the Nöther current method
within the conformal Weyl gravity paradigm for vacuum and non-vacuum
spacetimes. We do this, in the vacuum case, for the near horizon near extremal
Kerr metric and for the non-vacuum case we couple the conformal Weyl gravity
field equations to a near horizon (linear) $U(1)$ gauge potential and analyze
the respective found solutions. We highlight the non-universality of black
hole entropy between black hole solutions of varying symmetries, yet their
congruence with Wald’s entropy formula for the respective gravity theory.
Finally, we implement an $AdS_{2}/CFT_{1}$ construction to compute the full
asymptotic symmetry group of one of the non-vacuum conformal Weyl black holes.
We do this by performing a Robinson-Wilczek two dimensional reduction, thus
enabling the construction of an effective quantum theory of the remaining
field content. The effective stress energy tensor generates an asymptotic
Virasoro algebra, to $s$-wave approximation, whose center in conjunction with
their proper regularized lowest Virasoro eigen-mode is implemented to compute
black hole entropy via the statistical Cardy formula. We additionally
implement quantum holomorphic fluxes (of the dual CFT) in the near horizon to
compute the Hawking temperature of the respective black hole spacetime. We
conclude with a discussion and outlook for future work.

  *

##  1 Introduction

Conformal Weyl gravity (CWG) and fourth order gravity theories relating to
string theory and in general [3, 4] have recently reemerged as helpful tools
in gravitational/cosmological physics [5, 6], black hole physics [7], their
thermodynamics [8, 9] and role in quantum gravity [6, 10]. CWG is given by:

$\displaystyle S_{CWG}=\alpha_{c}\int
d^{4}x\sqrt{-g}W^{\alpha\mu\beta\nu}W_{\alpha\mu\beta\nu},$ (1)

which is a purely covariant and conformally invariant Weyl tensor squared
action [1, 2] for unit-less coupling $\alpha_{c}$. In addition, CWG includes
all of the Einstein-Hilbert (EH) vacuum solutions as part of its solution
space [11] and cosmological dynamics appear naturally within this formalism
[12]. Additionally, several Einstein-Hilbert non-vacuum solutions have
analogue counterparts within CWG [13].

However, the fourth order nature and unitarity issues endow it with some
difficulties. It is nonetheless an attractive theory and, since its vacuum
solution space includes Einstein-Hilbert vacua, CWG includes all the standard
solar-system tests of general relativity and warrants investigation on its own
right. One aspect and line of questioning that has not been extensively
explored yet is the implementation of the near horizon black hole conformal
field theory duality paradigm (with the $Kerr/CFT$ correspondence [14, 15] as
a notable example) within CWG. Since the seminal work of Brown and Henneaux
[16] and the $Kerr/CFT$ correspondence, it has become universally accepted
that black holes are holographically dual to lower dimensional conformal field
theories ($CFT$) at their boundaries. This is reminiscent of the more general
anti-de Sitter($AdS$)/$CFT$ correspondence of string theory [17]. This
paradigm has surged a tremendous program led by Strominger [18, 14], Carlip
[19, 20, 21, 22], Park [23, 24, 25] and others, in applying $CFT$ techniques
to compute Bekenstein-Hawking entropy of various black holes, in particular
(near-)extremal ones. This is based upon the feature that (near-)extremal
black holes admit a local $AdS_{2}\times S^{2}$ topology in their near
horizons. This fact allows for the implementation of $AdS/CFT$ techniques to
construct lower dimensional black hole $CFT$ duals, effectively providing a
statistical quantum mechanical description of black hole horizon microstates.
The results of these dualities include the standard black hole thermodynamic
quantities [26, 27, 28, 29, 30],

$\displaystyle\begin{cases}T_{H}=\frac{\hbar\kappa}{2\pi}&\text{Hawking
Temperature}\\\ S_{BH}=\frac{A}{4\hbar G}&\text{Bekenstein-Hawking
Entropy}\end{cases},$ (2)

which are the usual tests for any candidate theory of quantum gravity.

In general, the argument is that the asymptotic symmetry group (ASG), which
preserves specific metric fall off conditions, is given by a Virasoro algebra
with a calculable center:

$\displaystyle\left[\mathcal{Q}_{m},\mathcal{Q}_{n}\right]=(m-n)\mathcal{Q}_{m+n}+\frac{c}{12}m\left(m^{2}-1\right)\delta_{m+n,0},$
(3)

where $m,n\in\mathbb{Z}$. Then using Cardy’s formula [31, 32]:

$\displaystyle S=2\pi\sqrt{\frac{c\cdot\mathcal{Q}_{0}}{6}}.$ (4)

to count the horizon microstates, in terms of the dual $CFT$’s central charge
$c$ and normalized lowest eigenmode $\mathcal{Q}_{0}$, provides a holographic
computation of black hole entropy. Traditionally, a finite time regulator in
terms of surface gravity is employed to regulate the quantum charges of (3).
This leads to difficulty in using (4) for extremal black holes, where surface
gravity, and thus Hawking temperature, vanishes identically. To circumvent
this, a thermal Cardy formula may be used:

$\displaystyle S\sim\left(cT\right)$ (5)

where $T$ is the inverse finite time regulator which for extremal black holes
corresponds to the sum of left and right Frolov-Thorne vacuum temperatures for
chiral $CFT$s. Additionally, $c=c_{L}=c_{R}$ assuming diffeomorphism
invariance of the two chiral halves [33, 15, 34, 35].

There has been recent and semi-recent interest in black hole holography
applied within CWG and other fourth order gravity theories in general [36, 37,
38, 39, 40], with several interesting results, some of which that do not align
with expectations from general relativity. One such misalignment is the loss
of universality, i.e., black hole entropy still exhibits an area law in CWG,
but the coefficient is not always $1/4$ and in fact differs depending on what
spacetime is chosen. This loss of universality can immediately be deduced by
inspection of the the Wald entropy for CWG [41, 42, 43, 38]:

$\displaystyle S=-\frac{\alpha_{c}}{8}\int
W_{\mu\nu\alpha\beta}\epsilon^{\mu\nu}\epsilon^{\alpha\beta}d\Sigma,$ (6)

where $d\Sigma$ is the orthogonal hypersurface area element and
$\epsilon^{\mu\nu}$ is its unit normal bivector. From the above we see that
the Wald entropy in CWG depends on the Weyl tensor. Thus black holes that are
conformally flat, with universal area entropy laws in general relativity, will
exhibit zero entropy in CWG. Specific examples include the $RN/CFT$
correspondence [44, 45, 46, 47] with spacetime element:

$\displaystyle
ds^{2}=Q^{2}\left(-r^{2}dt^{2}+\frac{dr^{2}}{r^{2}}+d\Omega^{2}\right),$ (7)

where $Q$ is the black hole charge parameter and $d\Omega^{2}$ is the unit two
sphere. The above line element is conformally flat for the spacetime
diffeomorphism and conformal factor ($e^{2\omega}$) given by:

$\displaystyle
T=t,~{}x=-\frac{\sin\theta\cos\phi}{r},~{}y=-\frac{\sin\theta\sin\phi}{r},~{}z=-\frac{\cos\theta}{r},~{}\text{and}~{}e^{2\omega}=Q^{2}r^{2},$
(8)

and thus has vanishing Weyl tensor. Another example is the Weyl rescaled near
horizon Schwarzschild spacetime in finite mass/temperature gauge [37, 48]:

$\displaystyle
ds^{2}=-\frac{r^{2}-2GMr}{\ell^{2}}dt^{2}+\frac{\ell^{2}}{r^{2}-2GMr}dr^{2}+{\ell^{2}}d\theta^{2}+{\ell^{2}}\sin^{2}\theta
d\phi^{2},$ (9)

which is conformally flat via the diffeomorphism and conformal factor:

$\displaystyle\begin{split}T=&\frac{\exp{\left\\{\frac{t}{4GM}\right\\}}}{GM}\cosh
u;~{}x=\frac{\exp{\left\\{\frac{t}{4GM}\right\\}}}{GM}\sinh
u\cos\phi\sin\theta;\\\
y=&\frac{\exp{\left\\{\frac{t}{4GM}\right\\}}}{GM}\sinh
u\sin\phi\sin\theta;~{}z=\frac{\exp{\left\\{\frac{t}{4GM}\right\\}}}{GM}\sinh
u\cos\theta;\\\
e^{2\omega}=&\exp{\left\\{\frac{-t}{2GM}\right\\}}\ell^{2}r(r-2GM);\text{and}~{}u=\ln\sqrt{1-\frac{2GM}{r}}.\end{split}$
(10)

The last black hole example we discuss here was first introduced in [37] by
directly forging for a local $AdS_{2}\times S^{2}$ CWG vacuum solution with
the Kaluza-Klein ansatz:

$\displaystyle ds^{2}=$ $\displaystyle
K_{1}(\theta)\left[-f(r)dt^{2}+f(r)^{-1}dr^{2}+\ell^{2}d\theta^{2}\right]+\frac{\ell^{2}sin^{2}{\theta}}{K_{2}(\theta)}\left(d\phi-
A_{\mu}(r)dx^{\mu}\right)^{2}$ (11)

and yielding:

$\displaystyle\begin{split}ds^{2}=&\sin{\theta}\left[-\frac{r^{2}-2GMr}{\ell^{2}}dt^{2}+\frac{\ell^{2}}{r^{2}-2GMr}dr^{2}+\ell^{2}d\theta^{2}\right]\\\
&+\ell^{2}\sin{\theta}\left(d\phi+\frac{r-2GM}{\ell^{2}}dt\right)^{2}.\end{split}$
(12)

This spacetime is very interesting since it is a vacuum solution with local
axisymmetry and $AdS_{2}\times S^{2}$ topology with respect to CWG, however
from a general relativistic interpretation it is globally of _Petrov Type O_
and _Segre Type_ $[(111),1]$111We employ the standard spacetime classification
notation of [49].. In other words (and not previously discussed in [37]), from
an Einstein perspective it is conformally flat and sourced by a perfect fluid
of energy momentum tensor:

$\displaystyle\begin{split}T_{\mu\nu}=&(\rho+p)U_{\mu}U_{\nu}+pg_{\mu\nu},~{}\text{where}\\\
\rho=&p=-\frac{3\csc^{3}\theta}{32\pi
G\ell^{2}}~{}\text{and}~{}U^{\mu}=\left(0,0,\frac{i\sqrt{\csc\theta}}{\ell},0\right).\end{split}$
(13)

The diffeomorphism and conformal factor which maps (12) to conformal flatness
is given by:

$\displaystyle\begin{split}T=&2e^{\alpha}\sqrt{\frac{r}{r-2GM}}\cosh\frac{\phi}{2},~{}x=2e^{\alpha}\sqrt{\frac{r}{r-2GM}}\sinh\frac{\phi}{2},\\\
y=&2e^{\alpha}\sqrt{\frac{2GM}{r-2GM}}\sin\frac{\theta}{2},~{}z=2e^{\alpha}\sqrt{\frac{2GM}{r-2GM}}\cos\frac{\theta}{2},\\\
e^{2\omega}=&(r-2GM)\exp{\left\\{-\frac{t}{\ell}+\frac{\ell}{2GM}\phi\right\\}}\sin\theta,\text{and}~{}\alpha=\frac{t}{2\ell}-\frac{\ell}{4GM}\phi.\end{split}$
(14)

As mentioned earlier, there has been recent interest in black hole holography
and CWG. Specifically, and for a non-zero result, the entropy for spherical
black holes:

$\displaystyle ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}+r^{2}d\Omega^{2}$ (15)

that are CWG vacuum solutions and not conformally flat were studied from the
holographic perspective via the Nöther current method [50, 51, 52, 53] in [38]
and shown to agree with the Wald entropy (6). In particular, for this case the
result obtained was:

$\displaystyle
S=\frac{\alpha_{c}}{8}\left(\frac{4}{3r^{2}_{+}}+\frac{4f^{\prime}\left(r_{+}\right)}{3r_{+}}-\frac{2f^{\prime\prime}\left(r_{+}\right)}{3}\right)A,$
(16)

where $r_{+}$ is the horizon radius of (15) and $A$ its horizon area. Our goal
will be to add to the above examples by studying the holographic $CFT$ duals
of some additional black holes that arise within the solution space of CWG. We
will do this by first computing black hole entropy, employing and extending
the Nöther current method to previously unexplored spacetimes within this
paradigm, and comparing our results to the Wald entropy (6). Finally we
comment on the possible construction of suitable two dimensional $CFT$s that
would be holographically dual to our chosen black hole examples of CWG within
an $AdS_{2}/CFT_{1}$ correspondence. We end with a discussion of our results
and outlook for future directions.

##  2 Quick Review of the Boundary Nöther Current and Its Charge

In this part we will briefly review the boundary Nöther current method
according to [50, 51, 52, 53]. We will begin with the boundary term of a
general Lagrangian density $\mathcal{L}$, written as a total divergence, such
that:

$\displaystyle\sqrt{g}\mathcal{L}=\sqrt{g}\nabla_{\mu}B^{\mu}.$ (17)

Next, considering an infinitesimal diffeomorphism $x^{\mu}\to
x^{\mu}+\xi^{\mu}$ applied to the left hand side of (17) we have:

$\displaystyle\delta_{\xi}\left(\sqrt{g}\mathcal{L}\right)=\delta_{\xi}\sqrt{g}\mathcal{L}+\sqrt{g}\delta_{\xi}\mathcal{L}=\sqrt{g}\nabla_{\mu}\left(\mathcal{L}\xi^{\mu}\right),$
(18)

after using the fact that
$\delta_{\xi}\sqrt{g}=\frac{1}{2}\sqrt{g}g^{\alpha\beta}\delta_{\xi}g_{\alpha\beta}$.
For the right hand side of (17) we obtain:

$\displaystyle\begin{split}\delta_{\xi}\left(\sqrt{g}\nabla_{\mu}B^{\mu}\right)=&\delta_{\xi}\left(\sqrt{g}\frac{1}{\sqrt{g}}\partial_{\mu}\left(\sqrt{g}B^{\mu}\right)\right)\\\
=&\partial_{\mu}\left[\delta_{\xi}\left(\sqrt{g}B^{\mu}\right)\right]=\sqrt{g}\nabla_{\mu}\left[\nabla_{\alpha}\left(B^{\mu}\xi^{\alpha}\right)-B^{\alpha}\nabla_{\alpha}\xi^{\mu}\right].\end{split}$
(19)

Equating the results of (18) and (19) gives:

$\displaystyle\begin{split}\sqrt{g}\nabla_{\mu}\left(\mathcal{L}\xi^{\mu}\right)=&\sqrt{g}\nabla_{\mu}\left[\nabla_{\alpha}\left(B^{\mu}\xi^{\alpha}\right)-B^{\alpha}\nabla_{\alpha}\xi^{\mu}\right]\\\
\Rightarrow
0=&\nabla_{\mu}\left[\left(\mathcal{L}\xi^{\mu}\right)-\nabla_{\alpha}\left(B^{\mu}\xi^{\alpha}\right)+B^{\alpha}\nabla_{\alpha}\xi^{\mu}\right]=\nabla_{\mu}J^{\mu}\end{split}$
(20)

and thus, the Nöther current is given by:

$\displaystyle
J^{\mu}\left[\xi\right]=\left(\mathcal{L}\xi^{\mu}\right)-\nabla_{\alpha}\left(B^{\mu}\xi^{\alpha}\right)+B^{\alpha}\nabla_{\alpha}\xi^{\mu}.$
(21)

Referring back to (17) we see that the Nöther current can also be written as
the total divergence of a second rank tensor given by:

$\displaystyle\begin{split}J^{\mu}\left[\xi\right]=\nabla_{\alpha}\left(B^{\alpha}\xi^{\mu}\right)-\nabla_{\alpha}\left(B^{\mu}\xi^{\alpha}\right)=\nabla_{\alpha}J^{\mu\alpha}\\\
\Rightarrow J^{\mu\alpha}=B^{\alpha}\xi^{\mu}-B^{\mu}\xi^{\alpha}\end{split}$
(22)

From the above results we define the boundary Nöther charge in terms of (21):

$\displaystyle\mathcal{Q}[\xi]=\int_{\Sigma}d\Sigma_{\mu}J^{\mu},$ (23)

where $\Sigma$ is the timelike hyper surface with unit normal $m^{\alpha}$.
Using Stoke’s Theorem [54] we can rewrite $\mathcal{Q}[\xi]$ in terms of
$J^{\alpha\beta}$ as:

$\displaystyle\mathcal{Q}[\xi]=\frac{1}{2}\int_{\partial\Sigma}\sqrt{h}d\Sigma_{\alpha\beta}J^{\alpha\beta},$
(24)

where
$\sqrt{h}d\Sigma_{\alpha\beta}=-\sqrt{h}d^{2}x\left(n_{\alpha}m_{\beta}-n_{\beta}m_{\alpha}\right)$
is the area element of $\partial\Sigma$ and $n^{\alpha}$ is its spacelike unit
vector.

At this point we can demonstrate how the above formalism thus far relates to
black hole entropy by applying it to a previously unexplored (within the
respective formalism) example spacetime of general relativity. The specific
spacetime we are interested in is the near horizon near extremal Kerr metric
[55, 56] in the finite mass/temperature gauge [27, 57] given by the line
element:

$\displaystyle\begin{split}ds^{2}=\frac{1+\cos^{2}\theta}{2}\left(-\frac{r^{2}-2GMr-a^{2}}{\ell^{2}}dt^{2}+\frac{\ell^{2}}{r^{2}-2GMr-a^{2}}dr^{2}+\ell^{2}d\theta^{2}\right)+&\\\
\frac{2\ell^{2}\sin^{2}\theta}{1+\cos^{2}\theta}\left(d\phi+\frac{r-2GM}{\ell^{2}}dt\right)^{2},&\end{split}$
(25)

where $a$ is the angular momentum per unit mass parameter and
$\ell^{2}=r_{+}^{2}+a^{2}$. This particular gauge is relevant for our analysis
since it has non-zero Hawking temperature proportional in its usual way to its
surface gravity $\kappa=f^{\prime}\left(r_{+}\right)/2$, where
$f(r)=\frac{r^{2}-2GMr-a^{2}}{\ell^{2}}$. Additionally, since $\kappa$ is non-
zero, this particular gauge provides us with a non-zero finite time regulator
for $\mathcal{Q}$ and specifically, the respective Hawking temperature
$T_{H}=\kappa/2\pi$ coincides exactly with the non-extremal general Kerr black
hole temperature in standard Boyer-Lindquist coordinates [27, 57]:

For Einstein’s general relativity we can write the general surface term as
[58, 50, 53]

$\displaystyle
B^{\alpha}=\frac{2Q\indices{{}^{\alpha}_{\lambda}{}^{\sigma}_{\gamma}}g^{\beta\gamma}\Gamma\indices{{}^{\lambda}_{\beta}{}_{\sigma}}}{16\pi
G},~{}\text{where}~{}Q\indices{{}^{\alpha}_{\lambda}{}^{\sigma}_{\gamma}}=\frac{1}{2}\left(\delta\indices{{}^{\alpha}_{\lambda}}\delta\indices{{}^{\sigma}_{\gamma}}-\delta\indices{{}^{\alpha}_{\gamma}}\delta\indices{{}^{\sigma}_{\lambda}}$

and $\Gamma\indices{{}^{\lambda}_{\beta}{}_{\sigma}}$ is the usual metric
compatible Levi-Civita connection. Next, the spacetime in (25) has the
standard temporal Killing isometry $\xi_{(t)}=\partial_{t}$. For this
particular isometry choice the charge $\mathcal{Q}[\xi_{(t)}]$ (called the
Nöther energy) has the specific physical interpretation as the product of the
entropy and temperature of $\partial\Sigma$ [53, 59, 60]. For the metric in
(25), $\partial\Sigma$ corresponds to the black hole horizon and thus
$\left.\mathcal{Q}[\xi_{(t)}]\right|_{r_{+}}$ should be equal to the product
of $T_{H}$ and $S_{BH}$ of (2). Using (25), (2) and (24) to compute the
charge, we have:

$\displaystyle\begin{split}B^{r}=&-\frac{2f^{\prime}\left(r_{+}\right)}{16\pi
G\left(1+\cos^{2}\theta\right)},~{}B^{\theta}=-\frac{8\cos^{2}\theta\cot\theta}{16\pi
G\ell^{2}\left(1+\cos^{2}\theta\right)^{2}},~{}\text{and}~{}d\Sigma_{tr}=-\frac{1+\cos^{2}\theta}{2}\\\
&\Rightarrow\left.\mathcal{Q}[\xi]\right|_{r_{+}}=\frac{1}{2}\int_{\partial\Sigma}\sqrt{h}d\Sigma_{\alpha\beta}J^{\alpha\beta}=\frac{1}{32\pi
G}\int\sqrt{h}d^{2}x4\kappa=\frac{A\kappa}{8\pi G},\end{split}$ (26)

where $A$ is the black hole horizon area, as aforementioned. Multiplying the
above by the finite time regulator $\beta=1/T_{H}=2\pi/\kappa$ gives:

$\displaystyle\begin{split}\mathcal{Q}[\xi]\beta=\frac{A\kappa}{8\pi
G}\frac{2\pi}{\kappa}=\frac{A}{4G}\end{split}$ (27)

and thus, reproducing the Bekenstein-Hawking entropy formula for the near
horizon near extremal Kerr metric from the boundary Nöther current, within
Einstein gravity.

The above thermodynamic analysis infers that the black hole entropy is encoded
within the boundary term of the total action. We will now explore
fundamentally how this inference can be established from a statistical
counting of canonical quantum microstates via the general definition of the
generator algebra:

$\displaystyle\left[\mathcal{Q}_{1},\mathcal{Q}_{2}\right]=\frac{1}{2}\left[\delta_{\xi_{1}}\mathcal{Q}\left[\xi_{2}\right]-\delta_{\xi_{2}}\mathcal{Q}\left[\xi_{1}\right]\right].$
(28)

Next starting with (23), let us look at
$\delta_{\xi_{1}}\mathcal{Q}\left[\xi_{2}\right]$:

$\displaystyle\begin{split}\delta_{\xi_{1}}\mathcal{Q}\left[\xi_{2}\right]=&\int
d\Sigma_{\mu}\delta_{\xi_{1}}\left(\sqrt{g}J^{\mu}\left[\xi_{2}\right]\right)\\\
=&\int\sqrt{g}d\Sigma_{\mu}\left\\{\nabla_{\alpha}\left(\xi_{1}^{\alpha}J^{\mu}\left[\xi_{2}\right]-\xi_{1}^{\mu}J^{\alpha}\left[\xi_{2}\right]\right)\right\\}\\\
=&\int_{\partial\Sigma}\sqrt{h}d\Sigma_{\mu\nu}\xi_{1}^{\nu}J^{\mu}\left[\xi_{2}\right],\end{split}$
(29)

which implies;

$\displaystyle\begin{split}\left[\mathcal{Q}_{1},\mathcal{Q}_{2}\right]=&\frac{1}{2}\left(\int_{\partial\Sigma}\sqrt{h}d\Sigma_{\mu\nu}\xi_{1}^{\nu}J^{\mu}\left[\xi_{2}\right]-\int_{\partial\Sigma}\sqrt{h}d\Sigma_{\mu\nu}\xi_{2}^{\nu}J^{\mu}\left[\xi_{1}\right]\right)\\\
=&\frac{1}{2}\int_{\partial\Sigma}\sqrt{h}d\Sigma_{\mu\nu}\left(\xi_{2}^{\mu}J^{\nu}\left[\xi_{1}\right]-\xi_{1}^{\mu}J^{\nu}\left[\xi_{2}\right]\right).\end{split}$
(30)

##  3 Horizon Microstates and CWG: Vacuum

In the next two sections we will compute the generator algebra of specific
black hole solutions of CWG. A natural starting point is the near horizon near
extremal Kerr metric in (25), since it is a vacuum solution of general
relativity and thus, also a vacuum solution of CWG. To begin, we rewrite (25)
in terms of the usual general functions:

$\displaystyle\begin{split}ds^{2}=\frac{1+\cos^{2}\theta}{2}\left(-f\left(r_{+}+\rho\right)dt^{2}+\frac{1}{f\left(r_{+}+\rho\right)}d\rho^{2}+\ell^{2}d\theta^{2}\right)+&\\\
\frac{2\ell^{2}\sin^{2}\theta}{1+\cos^{2}\theta}\left(d\phi-A\left(r_{+}+\rho\right)dt\right)^{2},&\end{split}$
(31)

where we have employed a Bondi-like radial shift $r\to r_{+}+\rho$ thus
placing the horizon limit at $\rho=0$. Next, we imposed the following near
horizon metric fall off conditions [14]:

$\displaystyle\delta
g_{\mu\nu}=\left(\begin{array}[]{cccc}\mathcal{O}\left(\rho\right)&\mathcal{O}\left(1\right)&\mathcal{O}\left(1\right)&\mathcal{O}\left(1\right)\\\
&\mathcal{O}\left(1\right)&\mathcal{O}\left(1\right)&\mathcal{O}\left(\frac{1}{\rho}\right)\\\
&&\mathcal{O}\left(1\right)&\mathcal{O}\left(1\right)\\\
&&&\mathcal{O}\left(1\right)\end{array}\right),$ (36)

where we have indexed $\mu$ and $\nu$ over $\\{t,\rho,\theta,\phi\\}$. The
above boundary conditions are preserved by a copy of the conformal group
generated by the diffeomorphism:

$\displaystyle\xi_{\epsilon}=\left(\epsilon(t)-\frac{\rho}{f\left(r_{+}+\rho\right)}\epsilon^{\prime}(t)\right)\partial_{t}-\rho\epsilon^{\prime}(t)\partial_{\rho},$
(37)

which is evident when choosing $\epsilon(t)$ to be normalized circle
diffeomorphisms; $\frac{e^{in\kappa t}}{\kappa}$ and computing the classical
Lie bracket in the horizon limit $\rho=0$, which yields the centrally extended
Virasoro algebra:

$\displaystyle
i\left\\{\xi_{m},\xi_{n}\right\\}=(m-n)\xi_{m+n}-\frac{m^{3}}{2\kappa}\delta_{m+n,0}\partial_{t}.$
(38)

The boundary action for CWG is given by [38, 39, 43]:

$\displaystyle S_{B}=\frac{\alpha_{c}}{4\pi}\int
d^{3}x\sqrt{\sigma}W^{\mu\nu\alpha\beta}n_{\mu}n_{\nu}\nabla_{\alpha}n_{\beta},$
(39)

where $n_{\mu}=\left\\{0,\sqrt{\frac{1+\cos^{2}\theta}{2f(r)}},0,0\right\\}$
is the previously defined spacelike unit normal. From the action (39) we have
the boundary Lagrangian density:

$\displaystyle\mathcal{L}=\frac{\alpha}{4\pi}W^{\mu\nu\alpha\beta}n_{\mu}n_{\nu}\nabla_{\alpha}n_{\beta},$
(40)

from which we obtain $B^{\alpha}=n^{\alpha}\mathcal{L}$. Now substituting
these results and (37) into (22) through (24) and (30), where
$m_{\mu}=\left\\{\sqrt{\frac{\left(1+\cos^{2}\theta\right)f(r)}{2}},0,0\right\\}$,
we obtain in the horizon limit:

$\displaystyle\begin{split}\mathcal{Q}=\frac{\alpha_{c}}{3\pi\ell^{2}}\int_{\partial\Sigma}\sqrt{h}d^{2}x&\left[\left(1+\cos^{2}\theta\right)\left(-32\ell^{4}\sin^{2}\theta
A^{\prime}(r_{+})^{2}+\ell^{2}(\cos(2\theta)+3)^{2}f^{\prime\prime}(r_{+})\right.\right.\\\
&\left.\left.+44\cos(2\theta)-\cos(4\theta)+21\right)\right]/\left[(\cos(2\theta)+3)^{4}\right]\\\
&\left(\kappa\epsilon(t)-\frac{\epsilon^{\prime}(t)}{2}\right)\end{split}$
(41)

and

$\displaystyle\begin{split}\left[\mathcal{Q}_{1},\mathcal{Q}_{2}\right]=\frac{\alpha_{c}}{3\pi\ell^{2}}\int_{\partial\Sigma}\sqrt{h}d^{2}x&\left[\left(1+\cos^{2}\theta\right)\left(-32\ell^{4}\sin^{2}\theta
A^{\prime}(r_{+})^{2}+\ell^{2}(\cos(2\theta)+3)^{2}f^{\prime\prime}(r_{+})\right.\right.\\\
&\left.\left.+44\cos(2\theta)-\cos(4\theta)+21\right)\right]/\left[(\cos(2\theta)+3)^{4}\right]\\\
&\left[\kappa\left(\epsilon_{1}\epsilon^{\prime}_{2}-\epsilon_{2}\epsilon^{\prime}_{1}\right)-\frac{\epsilon_{1}\epsilon^{\prime\prime}_{2}-\epsilon_{2}\epsilon^{\prime\prime}_{1}}{2}+\frac{\epsilon^{\prime}_{1}\epsilon^{\prime\prime}_{2}-\epsilon^{\prime}_{2}\epsilon^{\prime\prime}_{1}}{4\kappa}\right].\end{split}$
(42)

Next, upon substituting $\epsilon(t)=\frac{e^{in\kappa t}}{\kappa}$ and
performing the integration over $\sqrt{h}d^{2}x$ we have:

$\displaystyle\begin{split}\mathcal{Q}_{m}=\frac{\alpha_{c}}{4\pi}\left[A^{\prime}\left(r_{+}\right)+f^{\prime\prime}\left(r_{+}\right)\right]A\delta_{m,0}\end{split}$
(43)

and

$\displaystyle\begin{split}\left[\mathcal{Q}_{m},\mathcal{Q}_{n}\right]=\frac{\alpha_{c}}{4\pi}\left[A^{\prime}\left(r_{+}\right)+f^{\prime\prime}\left(r_{+}\right)\right]\left[iA(m-n)\delta_{m+n,0}-im^{3}\frac{A}{2}\delta_{m+n,0}\right],\end{split}$
(44)

which we recognize as the quantum Virasoro algebra of the boundary degrees of
freedom. Furthermore we can read off the central charge and lowest Virasoro
mode:

$\displaystyle\begin{split}\mathcal{Q}_{0}=&\frac{\alpha_{c}}{4\pi}\left[A^{\prime}\left(r_{+}\right)+f^{\prime\prime}\left(r_{+}\right)\right]A\\\
\frac{c}{12}=&\frac{\alpha_{c}}{4\pi}\left[A^{\prime}\left(r_{+}\right)+f^{\prime\prime}\left(r_{+}\right)\right]\frac{A}{2},\end{split}$
(45)

which after employing Cardy’s formula yields the black hole entropy of the
near horizon near extremal Kerr black hole within CWG:

$\displaystyle
S_{BH}=2\pi\sqrt{\frac{\mathcal{Q}_{0}c}{6}}=\frac{\alpha_{c}}{2}\left[A^{\prime}\left(r_{+}\right)+f^{\prime\prime}\left(r_{+}\right)\right]A.$
(46)

Comparing the above to the Wald entropy for CWG (6) of the same spacetime (31)
we have:

$\displaystyle S=-\frac{\alpha_{c}}{8}\int
W_{\mu\nu\alpha\beta}\epsilon^{\mu\nu}\epsilon^{\alpha\beta}d\Sigma=\frac{\alpha_{c}}{2}\left[A^{\prime}\left(r_{+}\right)+f^{\prime\prime}\left(r_{+}\right)\right]A,$
(47)

which precisely agrees with the previous result above for the near horizon
near extremal Kerr black hole from counting the canonical quantum microstates
via the boundary Nöther current method.

##  4 Horizon Microstates and CWG: non-Vacuum

Now we turn our attention to non-vacuum solutions of CWG, which provides us
with an additional venue to explore more unique spacetimes. Specifically, we
focus on the near horizon near extremal CWG-Maxwell theory. The CWG-Maxwell
system has been well explored for the Coulomb potential in [13] with solution
given by the line element (15), where:

$\displaystyle f(r)=w+\frac{u}{r}+\gamma
r-kr^{2},~{}w=1-3\beta\gamma,~{}u=-\beta(2-3\beta\gamma)-\frac{Q^{2}}{8\alpha\gamma},$
(48)

and $Q$ is the black hole charge parameter for the $U(1)$ vector potential
$A_{\mu}=\left\\{Q/r,0,0,0\right\\}$. For simplicity and without loss of
generality, we can explore the extremal limit of (48) in the mathematically
simplified case where $\beta=\frac{3k+\gamma^{2}}{9k\gamma}$ and with extremal
limit given by $Q=\sqrt{-\frac{8\alpha}{3}}$. In this scenario the metric (48)
takes the form:

$\displaystyle
ds^{2}=\frac{\left(3rk-\gamma\right)^{3}}{27rk^{2}}dt^{2}-\frac{27rk^{2}}{\left(3rk-\gamma\right)^{3}}dr^{2}+r^{2}d\Omega^{2}$
(49)

and looking at the near horizon, of the above in the standard way, where $r\to
r\lambda+\frac{\gamma}{3k}$, $t\sim t/\lambda$ and further in the limit as
$\lambda\to 0$, we get $g_{tt}\to\infty$. This makes it seem naively unclear
how to investigate the near horizon near extremal limit of the CWG-Maxwell
theory, without killing off degrees of freedom or running into divergences in
the metric.

However, it is well known [44, 45, 46, 47] that in the near horizon near
extremal limit of charged black holes the $U(1)$ gauge potential becomes
linear:

$\displaystyle A_{\mu}=\\{Qr,0,0,0\\}.$ (50)

Additionally, the CWG-Maxwell system is stationary for the Euler-Lagrange
equation:

$\displaystyle-2\alpha_{c}W_{\mu\nu}+\frac{1}{2}T_{\mu\nu}=0,$ (51)

where
$T_{\mu\nu}=F_{\mu\alpha}F\indices{{}_{\nu}^{\alpha}}-\frac{1}{4}g_{\mu\nu}F^{2}$
is the Maxwell stress tensor and
$W_{\mu\nu}=2\nabla_{\alpha}\nabla_{\beta}W\indices{{}_{\mu}^{\alpha}{}^{\beta}_{\nu}}+R_{\alpha_{\beta}}W\indices{{}_{\mu}^{\alpha}{}^{\beta}_{\nu}}$
is the Bach tensor, both of which stem from the inverse metric variation of
the CWG-Maxwell theory. Consequently and employing the ansatz (11), we are
free to forge for near horizon near extremal CWG-Maxwell solutions directly
from the above field equation with linear $U(1)$ potential, to aid in our
analysis. Our current investigation includes the two following general
solutions:

$\displaystyle\begin{split}ds^{2}=&-B(r)dt^{2}+\frac{1}{B(r)}dr^{2}+\ell^{2}d\Omega^{2}\\\
=&-\left(\frac{r^{3}\left(2\alpha\left(c_{1}^{2}\ell^{4}-4\right)-3Q^{2}\ell^{4}\right)}{24\alpha
c_{2}\ell^{4}}+\frac{r^{2}c_{1}}{2}+rc_{2}\right)dt^{2}\\\
&+\left(\frac{r^{3}\left(2\alpha\left(c_{1}^{2}\ell^{4}-4\right)-3Q^{2}\ell^{4}\right)}{24\alpha
c_{2}\ell^{4}}+\frac{r^{2}c_{1}}{2}+rc_{2}\right)^{-1}dr^{2}+\ell^{2}d\Omega^{2}\end{split}$
(52)

and

$\displaystyle\begin{split}ds_{c_{3}}^{2}=&\sin\theta\left(-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+\ell^{2}d\theta^{2}\right)+\ell^{2}\sin\theta
d\phi^{2}\\\
=&\sin\theta\left(-\frac{r^{2}+c_{3}r}{\ell^{2}}dt^{2}+\frac{\ell^{2}}{r^{2}+c_{3}r}dr^{2}+\ell^{2}d\theta^{2}\right)+\ell^{2}\sin\theta
d\phi^{2},\end{split}$ (53)

where the $c_{i}$’s are constants. The above solutions classify (general
relativisticly) globally both as $Petrov~{}type~{}D$ and additionally as
$Segre~{}type~{}[(11)(1,1)]~{}(IID-M2(11))$ and
$[(11),Z\bar{Z}]~{}(II-M3(111))$ respectively. Without loss of generality and
to ensure local $AdS_{2}\times S^{2}$ topology, we can impose that
$c_{1}=\sqrt{\frac{3Q^{2}\ell^{4}+8\alpha_{c}}{2\ell^{4}\alpha_{c}}}$, which
implies for (52) that:

$\displaystyle\begin{split}ds_{c_{2}}^{2}=&-\left(r^{2}\sqrt{\frac{3Q^{2}\ell^{4}+8\alpha_{c}}{8\ell^{4}\alpha_{c}}}+rc_{2}\right)dt^{2}+\left(r^{2}\sqrt{\frac{3Q^{2}\ell^{4}+8\alpha_{c}}{8\ell^{4}\alpha_{c}}}+rc_{2}\right)^{-1}dr^{2}+\ell^{2}d\Omega^{2}.\end{split}$
(54)

We have included the subscripts $c_{2}$ and $c_{3}$ on the above line elements
in (53) and (54) in order to distinguish them in subsequent
discussion/analysis. Next, we will compute the black hole entropy of the above
two solutions via the previously outlined Nöther current method and again
compare our results to the Wald entropy formula.

It turns out that the above solutions exhibit the same (or even more relaxed)
near horizon fall off conditions (36) and thus near horizon preserving
isometry given by (37). This can be more readily demonstrated, since the above
solutions are spherically symmetric, via the Bondi-like transformation:

$\displaystyle dt=du+\frac{1}{f\left(r_{+}+\rho\right)}d\rho$ (55)

and solving the Killing equation $\mathcal{L}_{\xi}g_{\mu\nu}=0$, which yields
for both $ds^{2}_{c_{2}}$ and $ds^{2}_{c_{3}}$ cases:

$\displaystyle\xi^{u}=$ $\displaystyle F(u,\theta,\phi)~{}\text{and}$ (56)
$\displaystyle\xi^{\rho}=$ $\displaystyle-\rho\partial_{u}F(u,\theta,\phi).$
(57)

Transforming back to the temporal coordinate $t$, we have:

$\displaystyle\xi^{t}=$
$\displaystyle\epsilon(t,\theta,\phi)-\frac{\rho}{f\left(r_{+}+\rho\right)}\partial_{t}\epsilon(t,\theta,\phi)~{}\text{and}$
(58) $\displaystyle\xi^{\rho}=$
$\displaystyle-\rho\partial_{t}\epsilon(t,\theta,\phi),$ (59)

which matches (37) for an $\epsilon$ that exhibits angular dependence. Now,
for both $ds^{2}_{c_{2}}$ and $ds^{2}_{c_{3}}$ cases we may chose
$n_{\mu}=\left\\{0,\frac{1}{\sqrt{f(r)}},0,0\right\\}$,
$m_{\mu}=\left\\{-\sqrt{f(r)},0,0,0\right\\}$,
$\epsilon(t,\theta,\phi)=\epsilon(t)$ and following the the same procedures
from Sec. 3, we find:

$\displaystyle\mathcal{Q}=\begin{cases}\frac{\alpha_{c}}{4\pi}\frac{2-\ell^{2}f^{\prime\prime}\left(r_{+}\right)}{6\ell^{2}}\int_{\partial\Sigma}\sqrt{h}d^{2}x\left(\kappa\epsilon(t)-\frac{\epsilon^{\prime}(t)}{2}\right)&\text{case}~{}c_{2}\\\
\frac{\alpha_{c}}{4\pi}\frac{-f^{\prime\prime}\left(r_{+}\right)}{6}\int_{\partial\Sigma}\sqrt{h}d^{2}x\left(\kappa\epsilon(t)-\frac{\epsilon^{\prime}(t)}{2}\right)&\text{case}~{}c_{3}\end{cases}$
(60)

and

$\displaystyle\left[\mathcal{Q}_{1},\mathcal{Q}_{2}\right]=\begin{cases}\frac{\alpha_{c}}{4\pi}\frac{2-\ell^{2}f^{\prime\prime}\left(r_{+}\right)}{6\ell^{2}}\int_{\partial\Sigma}\sqrt{h}d^{2}x\left[\kappa\left(\epsilon_{1}\epsilon^{\prime}_{2}-\epsilon_{2}\epsilon^{\prime}_{1}\right)-\frac{\epsilon_{1}\epsilon^{\prime\prime}_{2}-\epsilon_{2}\epsilon^{\prime\prime}_{1}}{2}+\frac{\epsilon^{\prime}_{1}\epsilon^{\prime\prime}_{2}-\epsilon^{\prime}_{2}\epsilon^{\prime\prime}_{1}}{4\kappa}\right]&\text{case}~{}c_{2}\\\
\frac{\alpha_{c}}{4\pi}\frac{-f^{\prime\prime}\left(r_{+}\right)}{6}\int_{\partial\Sigma}\sqrt{h}d^{2}x\left[\kappa\left(\epsilon_{1}\epsilon^{\prime}_{2}-\epsilon_{2}\epsilon^{\prime}_{1}\right)-\frac{\epsilon_{1}\epsilon^{\prime\prime}_{2}-\epsilon_{2}\epsilon^{\prime\prime}_{1}}{2}+\frac{\epsilon^{\prime}_{1}\epsilon^{\prime\prime}_{2}-\epsilon^{\prime}_{2}\epsilon^{\prime\prime}_{1}}{4\kappa}\right]&\text{case}~{}c_{3}\end{cases}.$
(61)

Now, substituting $\epsilon(t)=\frac{e^{in\kappa t}}{\kappa}$ and performing
the integration over $\sqrt{h}d^{2}x$ we obtain:

$\displaystyle\mathcal{Q}_{m}=\begin{cases}\frac{\alpha_{c}}{4\pi}\frac{2-\ell^{2}f^{\prime\prime}\left(r_{+}\right)}{6\ell^{2}}A\delta_{m,0}&\text{case}~{}c_{2}\\\
\frac{\alpha_{c}}{4\pi}\frac{-f^{\prime\prime}\left(r_{+}\right)}{6}A\delta_{m,0}&\text{case}~{}c_{3}\end{cases}$
(62)

and

$\displaystyle\left[\mathcal{Q}_{m},\mathcal{Q}_{n}\right]=\begin{cases}\frac{\alpha_{c}}{4\pi}\frac{2-\ell^{2}f^{\prime\prime}\left(r_{+}\right)}{6\ell^{2}}\left[iA(m-n)\delta_{m+n,0}-im^{3}\frac{A}{2}\delta_{m+n,0}\right]&\text{case}~{}c_{2}\\\
\frac{\alpha_{c}}{4\pi}\frac{-f^{\prime\prime}\left(r_{+}\right)}{6}\left[iA(m-n)\delta_{m+n,0}-im^{3}\frac{A}{2}\delta_{m+n,0}\right]&\text{case}~{}c_{3}\end{cases},$
(63)

which we again recognize as the quantum Virasoro algebra of the boundary
degrees of freedom. Again, reading off the central charge and lowest Virasoro
mode, we have:

$\displaystyle\begin{split}\mathcal{Q}_{0}=&\begin{cases}\frac{\alpha_{c}}{4\pi}\frac{2-\ell^{2}f^{\prime\prime}\left(r_{+}\right)}{6\ell^{2}}A&\text{case}~{}c_{2}\\\
\frac{\alpha_{c}}{4\pi}\frac{-f^{\prime\prime}\left(r_{+}\right)}{6}A&\text{case}~{}c_{3}\end{cases}\\\
\frac{c}{12}=&\begin{cases}\frac{\alpha_{c}}{4\pi}\frac{2-\ell^{2}f^{\prime\prime}\left(r_{+}\right)}{6\ell^{2}}\frac{A}{2}&\text{case}~{}c_{2}\\\
\frac{\alpha_{c}}{4\pi}\frac{-f^{\prime\prime}\left(r_{+}\right)}{6}\frac{A}{2}&\text{case}~{}c_{3}\end{cases}\end{split}$
(64)

and after employing Cardy’s formula, we obtain the black hole entropy for
$ds^{2}_{c_{2}}$ and $ds^{2}_{c_{3}}$ within CWG:

$\displaystyle
S_{BH}=2\pi\sqrt{\frac{\mathcal{Q}_{0}c}{6}}=\begin{cases}\frac{\alpha_{c}\left(2-\ell^{2}f^{\prime\prime}\left(r_{+}\right)\right)}{12\ell^{2}}A&\text{case}~{}c_{2}\\\
\frac{\alpha_{c}f^{\prime\prime}\left(r_{+}\right)}{12}A&\text{case}~{}c_{3}\end{cases}.$
(65)

Finally, comparing the above results to the CWG Wald entropy formula (6) we
have:

$\displaystyle S=-\frac{\alpha_{c}}{8}\int
W_{\mu\nu\alpha\beta}\epsilon^{\mu\nu}\epsilon^{\alpha\beta}d\Sigma=\begin{cases}\frac{\alpha_{c}\left(2-\ell^{2}f^{\prime\prime}\left(r_{+}\right)\right)}{12\ell^{2}}A&\text{case}~{}c_{2}\\\
-\frac{\alpha_{c}f^{\prime\prime}\left(r_{+}\right)}{12}A&\text{case}~{}c_{3}\end{cases},$
(66)

which again precisely agrees with the results for $ds^{2}_{c_{2}}$ and
$ds^{2}_{c_{3}}$ from counting the canonical quantum microstates via the
boundary Nöther current method, modulo the minus sign in the $ds^{2}_{c_{3}}$
case. We should note that, in the $ds^{2}_{c_{3}}$ case, we could have chosen
the minus sign after taking the square root in the above Cardy formula.

##  5 Quantum Fields and $AdS_{2}/CFT_{1}$ in The Background of CWG

In this section we will draw upon semiclassical methods in order to explore
the construction of a near horizon $CFT$ action which holographically encodes
the black hole thermodynamics of the respective black hole choice. We will be
focusing our efforts on the $ds^{2}_{c_{2}}$ case and following methods
outlined in [26, 27, 61, 62]. In order to extract the two dimensional field
content relevant to our construction we will probe the spacetime with a
minimally coupled scalar, then taking the near horizon limit and integrating
out angular degrees of freedom via a Robinson-Wilczek two dimensional
decomposition [63]. The resulting two dimensional theory will exhibit a known
quantum effective action akin to the Polyakov action:

$\displaystyle S_{eff}\sim\bar{\beta}^{\psi}\int
d^{2}x\sqrt{-g^{(2)}}R^{(2)}\frac{1}{\square_{g^{(2)}}}R^{(2)}+\cdots,$ (67)

where $\bar{\beta}^{\psi}$ is proportional to the Weyl anomaly coefficient
[64, 65]. Our next task will be to construct the respective effective theory
and value of $\bar{\beta}^{\psi}$ to $s$-wave approximation within the CWG
paradigm. We will also need to use our results from Sec. 4 in order to tune
$\bar{\beta}^{\psi}$ to reproduce the respective black hole entropy of the
$ds^{2}_{c_{2}}$ spacetime. The $s$-wave approximation is appropriate if we
infer that our scalar probe has gravitational origins, in other words
$ds^{2}_{c_{2}}$ exhibits a Kaluza-Klein decomposition:

$\displaystyle\begin{split}ds^{2}=~{}&-f(r)dt^{2}+f(r)^{-1}dr^{2}+\ell^{2}e^{-2\psi(r)}\left[d\theta^{2}+\sin^{2}\theta\left(d\phi-
Adt\right)^{2}\right]\\\
=~{}&ds^{2}_{2D}+\ell^{2}e^{-2\psi(r)}\left[d\theta^{2}+\sin^{2}\theta\left(d\phi-
A_{\mu}dx^{\mu}\right)^{2}\right],\end{split}$ (68)

where we have introduced a two-dimensional black hole coupled to a two-
dimensional real scalar and two-dimensional $U(1)$ gauge field. In the case of
$ds^{2}_{c_{2}}$, we see that the only allowable gauge field couplings are
linear phase shifts given by $\phi\to\phi-At$, which are trivial. However, for
spacetimes not exhibiting global spherical symmetry, the $s$-wave is still an
appropriate approximation, since the important gravitational dynamics seem to
be contained in this regime [66].

We will begin our construction by extracting the contributing two-dimensional
near-horizon field content by considering the four dimensional free scalar
field in the background of (68):

$\displaystyle S^{(4)}[\varphi,g]=$ $\displaystyle-\frac{1}{2}\int
d^{4}x\sqrt{-g}\nabla_{\mu}\varphi\nabla^{\mu}\varphi.$ (69)

Next, we expand $\varphi$ in terms of spherical harmonics and examine the
regime where $r\sim r_{+}$ in tortoise coordinates. Then, integrating out
angular degrees of freedom, we are left with:

$\displaystyle S^{(4)}[\varphi,g]\overset{r\sim
r_{+}}{\longrightarrow}S^{(2)}[\varphi_{lm},g^{(2)}]=\frac{\ell^{2}}{2}\int
dtdr\varphi^{*}_{lm}\left[\frac{1}{f(r)}\left(\partial
t-iA_{t}\right)^{2}-\partial_{r}f(r)\partial_{r}\right]\varphi_{lm}.$ (70)

This simplified action is the result of to the fact that in the examined
regime, interacting/mixing and potential terms $(\sim l(l+1)\ldots)$ are
weighted by the near horizion exponentially decaying term
$f\left(r\left(r^{*}\right)\right)\sim e^{2\kappa r^{*}}$. Thus, in the
respective regime, (69) simplifies to an action of infinite massless complex
scalar fields given by:

$\displaystyle S=-\frac{\ell^{2}}{2}\int
d^{2}x\varphi^{*}_{lm}D_{\mu}\left[\sqrt{-g^{(2)}}g^{\mu\nu}_{(2)}D_{\nu}\right]\varphi_{lm},$
(71)

where $D_{\mu}=\partial_{\mu}-imA_{\mu}$ is the gauge covariant derivative. We
have now arrived at the Robinson and Wilczek two-dimensional analog (RW2DA)
fields for the (68) spacetime represented by:

$\displaystyle g\indices{{}^{(2)}_{\mu}{}_{\nu}}=~{}$
$\displaystyle\left(\begin{array}[]{cc}-f(r)&0\\\
0&\frac{1}{f(r)}\end{array}\right)$ $\displaystyle
f(r)=r^{2}\sqrt{\frac{3Q^{2}\ell^{4}+8\alpha_{c}}{8\ell^{4}\alpha_{c}}}+rc_{2}$
(74)

and trivial $U(1)$ gauge field given by:

$\displaystyle A=A_{t}dt=constant=0.$ (75)

Now, taking the $s$-wave approximation and implementing the CWG inspired
redefinition $\varphi_{00}=\sqrt{\gamma}\psi$ (where $\psi$ is unitless and
$\gamma$ will be proportional to $\alpha_{c}$), the effective action for the
above two dimensional theory (71) will be given by the two parts [67, 68]:

$\displaystyle\Gamma=$ $\displaystyle\Gamma_{grav}+\Gamma_{U(1)},$ (76)

where

$\displaystyle\begin{split}\Gamma_{grav}=&\frac{\gamma\ell^{2}}{96\pi}\int
d^{2}x\sqrt{-g^{(2)}}R^{(2)}\frac{1}{\square_{g^{(2)}}}R^{(2)}~{}\text{and}\\\
\Gamma_{U(1)}=&\frac{\gamma e^{2}\ell^{2}}{2\pi}\int
F\frac{1}{\square_{g^{(2)}}}F.\end{split}$ (77)

Comparing the above to (67), we see that in our CWG inspired case
$\bar{\beta}^{\psi}=\frac{\gamma\ell^{2}}{96\pi}$. We should note that
$\Gamma_{U(1)}=0$ in (76) above for the $ds^{2}_{c_{2}}$ case, however we will
keep track of it until now for completeness.

The next step will be to restore locality in the quantum effective action
(76). This may be done via the introduction of auxiliary scalar fields $\Phi$
and $B$, where:

$\displaystyle\square_{g^{(2)}}\Phi=R^{(2)}~{}\mbox{and}~{}\square_{g^{(2)}}B=\epsilon^{\mu\nu}\partial_{\mu}A_{\nu}.$
(78)

For general $f(r)$ and $A_{t}(r)$ the above reduces to the following
differential equations:

$\displaystyle\begin{split}-\frac{1}{f(r)}\partial_{t}^{2}\Phi+\partial_{r}f(r)\partial_{r}\Phi=&R^{(2)}\\\
-\frac{1}{f(r)}\partial_{t}^{2}B+\partial_{r}f(r)\partial_{r}B=&-\partial_{r}A_{t},\end{split}$
(79)

with solutions given by:

$\displaystyle\begin{split}\Phi(t,r)=&\alpha_{1}t+\int
dr\frac{\alpha_{2}-f^{\prime}(r)}{f(r)}\\\ B(t,r)=&\beta_{1}t+\int
dr\frac{\beta_{2}-A_{t}(r)}{f(r)},\end{split}$ (80)

where $\alpha_{i}$ and $\beta_{i}$ are integration constants. Now,
substituting (78) into (77) yields our final near horizon $CFT$ in Liouville
form given by:

$\displaystyle\begin{split}S_{NHCFT}=&\frac{\gamma\ell^{2}}{96\pi}\int
d^{2}x\sqrt{-g^{(2)}}\left\\{-\Phi\square_{g^{(2)}}\Phi+2\Phi
R^{(2)}\right\\}\\\ &+\frac{\gamma e^{2}\ell^{2}}{2\pi}\int
d^{2}x\sqrt{-g^{(2)}}\left\\{-B\square_{g^{(2)}}B+2B\left(\frac{\epsilon^{\mu\nu}}{\sqrt{-g^{(2)}}}\right)\partial_{\mu}A_{\nu}\right\\}\end{split}$
(81)

For the next part, we investigate and compute the non trivial asymptotic
symmetries for the two dimensional gravitational part of (74) for large $r$
behavior given by:

$\displaystyle g\indices{{}^{(0)}_{\mu}{}_{\nu}}=$
$\displaystyle\left(\begin{array}[]{cc}-\frac{\sqrt{\frac{3Q^{2}\ell^{4}+8\alpha_{c}}{\ell^{4}\alpha_{c}}}r^{2}}{2\sqrt{2}}-c_{2}r+\mathcal{O}\left(\left(\frac{1}{r}\right)^{3}\right)&0\\\
0&\frac{2\sqrt{2}}{\sqrt{\frac{3Q^{2}\ell^{4}+8\alpha_{c}}{\ell^{4}\alpha_{c}}}r^{2}}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{3}\right)\end{array}\right),$
(84)

which is asymptotically $AdS_{2}$ with Ricci Scalar,
$R=-\frac{2}{l^{2}}+O\left(\left(\frac{1}{r}\right)^{1}\right)$, where
$l^{2}=\frac{2\sqrt{2}}{\sqrt{\frac{3Q^{2}\ell^{4}+8\alpha_{c}}{\ell^{4}\alpha_{c}}}}$.
Coupling the above with the following metric fall-off conditions:

$\displaystyle\delta
g_{\mu\nu}=\left(\begin{array}[]{cc}\mathcal{O}\left(r\right)&\mathcal{O}\left(\left(\frac{1}{r}\right)^{0}\right)\\\
\mathcal{O}\left(\left(\frac{1}{r}\right)^{0}\right)&\mathcal{O}\left(\left(\frac{1}{r}\right)^{3}\right)\end{array}\right),$
(87)

we obtain the following set of asymptotic symmetries:

$\displaystyle\chi=-C_{1}\frac{r\sqrt{\frac{16}{\ell^{4}}+\frac{6Q^{2}}{\alpha_{c}}}\xi(t)}{4c_{2}+r\sqrt{\frac{16}{\ell^{4}}+\frac{6Q^{2}}{\alpha_{c}}}}\partial_{t}+C_{2}r\xi^{\prime}(t)\partial_{r},$
(88)

for arbitrary function $\xi(t)$ and appropriate normalization constants
$C_{i}$. Next, transforming to conformal light cone coordinates, which are
given by:

$\displaystyle x^{\pm}=t\pm r^{*},$ (89)

where $r^{*}$ is the radial tortoise coordinate; we obtain:

$\displaystyle\chi^{\pm}=\frac{\left(-C_{1}r\left(r^{*}\right)\sqrt{\frac{16}{\ell^{4}}+\frac{6Q^{2}}{\alpha_{c}}}\xi(x^{+},x^{-})\pm
4C_{2}\xi^{\prime}(x^{+},x^{-})\right)}{4c_{2}+r\left(r^{*}\right)\sqrt{\frac{16}{\ell^{4}}+\frac{6Q^{2}}{\alpha_{c}}}}.$
(90)

The above diffeomorphisms are smooth on the respective radial boundary and the
arbitrary constants can be normalized in order for $\chi^{\pm}$ to satisfy a
Witt or $Diff$$\left(S^{1}\right)$ subalgebra
$i\left\\{\chi^{\pm}_{m},\chi^{\pm}_{n}\right\\}=(m-n)\chi^{\pm}_{(m+n)}$ when
choosing a circle diffeomorphism representation for $\xi(x^{+},x^{-})$.

The energy-momentum tensor for (81) is given by the standard definition:

$\displaystyle\begin{split}\left\langle
T_{\mu\nu}\right\rangle=&\frac{2}{\sqrt{-g^{(2)}}}\frac{\delta
S_{NHCFT}}{\delta g\indices{{}^{(2)}{}^{\mu}{}^{\nu}}}\\\
=&\frac{\gamma\ell^{2}}{48\pi
G}\left\\{\partial_{\mu}\Phi\partial_{\nu}\Phi-2\nabla_{\mu}\partial_{\nu}\Phi+g\indices{{}^{(2)}_{\mu}{}_{\nu}}\left[2R^{(2)}-\frac{1}{2}\nabla_{\alpha}\Phi\nabla^{\alpha}\Phi\right]\right\\}.\end{split}$
(91)

At this point it will be advantageous to pick a convenient choice for the
$c_{2}$ constant in (74), such that:

$\displaystyle\begin{split}f(r)=&\frac{r^{2}-\ell r}{l^{2}},~{}\text{where}\\\
l^{2}=&\sqrt{\frac{8\ell^{4}\alpha_{c}}{3Q^{2}\ell^{4}+8\alpha_{c}}},~{}\text{as
previously defined and}\\\ c_{2}=&-\frac{\ell}{l^{2}}.\end{split}$ (92)

Now, using the general solution (80) and substituting into (91), and
additionally implementing modified Unruh Vacuum boundary conditions (MUBC)
[69]:

$\displaystyle\begin{cases}\left\langle
T_{++}\right\rangle=0&r\rightarrow\infty,~{}l\rightarrow\infty\\\ \left\langle
T_{--}\right\rangle=0&r\rightarrow r_{+}\end{cases}$ (93)

we are able to determine the integration constants $\alpha_{i}$:

$\displaystyle\begin{split}\alpha_{1}=&-\alpha_{2}=\frac{1}{2}f^{\prime}(r_{+})\end{split}$
(94)

and thus, the EMT. Given the two-dimensional reduction, our above EMT exhibits
a Weyl (trace) anomaly, which is given by:

$\displaystyle\left\langle
T\indices{{}_{\mu}^{\mu}}\right\rangle=-\bar{\beta}^{\psi}{4}R^{(2)},$ (95)

which uniquely determines the value of central charge via [35]:

$\displaystyle\frac{c}{24\pi}=4\bar{\beta}^{\psi}\Rightarrow
c=\ell^{2}\gamma=\frac{\gamma}{4\pi}A.$ (96)

In addition and due to our use of the previously defined MUBC, the specific
asymptotic boundary to $\mathcal{O}(\frac{1}{\ell})^{2}$ (which we denote by
the single limit $x^{+}\to\infty$), implies that the EMT is dominated by one
holomorphic component $\left\langle T_{--}\right\rangle$. When expanding this
respective component in terms of boundary fields (84) and computing the
response to the respective asymptotic symmetries, we obtain:

$\displaystyle\delta_{\chi^{-}}\left\langle
T_{--}\right\rangle=\chi^{-}\left\langle
T_{--}\right\rangle^{\prime}+2\left\langle
T_{--}\right\rangle\left(\chi^{-}\right)^{\prime}+\frac{c}{24\pi}\left(\chi^{-}\right)^{\prime\prime\prime}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{3}\right),$
(97)

where the primes indicate time derivatives. Thus, the above result tells us
that $\left\langle T_{--}\right\rangle$ transforms asymptotically as the EMT
of a $CFT$ of one dimension, with center given by (96).

Given the above, we are now able to compute the generator charge alegra of the
full asymptotic symmetry group, by compactifying (which regulates the
asymptotic charges) the $x^{-}$ coordinate to a circle parametrized from $0\to
2\pi/\kappa$, where we define the respective asymptotic conserved charge as:

$\displaystyle\mathcal{Q}_{n}=\lim_{x^{+}\to\infty}\int dx^{\mu}\left\langle
T_{\mu\nu}\right\rangle\chi^{\nu}_{n}.$ (98)

Next, we replace $\xi(x^{+},x^{-})$ with circle diffieomorphisms
$\frac{e^{-in\kappa x^{\pm}}}{\kappa}$ in (90) and calculate the canonical
response of $\mathcal{Q}_{n}$ with respect to the asymptotic symmetry,
yielding:

$\displaystyle\delta_{\chi^{-}_{m}}\mathcal{Q}_{n}=\left[\mathcal{Q}_{m},\mathcal{Q}_{n}\right]=(m-n)\mathcal{Q}_{n}+\frac{c}{12}m\left(m^{2}-1\right)\delta_{m+n,0}.$
(99)

In other words, the asymptotic quantum generator algebra is given by a
centrally extended Virasoro algebra, with center given by (96) and with lowest
eigen-mode given by:

$\displaystyle\mathcal{Q}_{0}=\frac{\gamma}{96\pi}A.$ (100)

What we have shown thus far, is that the RW2DA fields of $ds^{2}_{c_{2}}$
given in (74) are essentially descried by a one dimensional string theory
given by (81) and whose horizon microstates are holographically dual to a
$CFT_{1}$ with center given by:

$\displaystyle c$ $\displaystyle=\frac{\gamma}{4\pi}A$ (101)

and lowest Virasoro eigen-mode:

$\displaystyle\mathcal{Q}_{0}$ $\displaystyle=\frac{\gamma}{96\pi}A.$ (102)

Using these results in the Cardy Formula (4) we obtain:

$\displaystyle S=2\pi\sqrt{\frac{c\mathcal{Q}_{0}}{6}}=\frac{\gamma A}{24}.$
(103)

Comparing the above to (65) allows us to fix the $\gamma$ constant for the
$ds^{2}_{c_{2}}$ case, which yields:

$\displaystyle\gamma=\frac{8\pi\alpha_{c}\left(2-\ell^{2}f^{\prime\prime}\left(r_{+}\right)\right)}{\ell^{2}}.$
(104)

Next, and now that we have fixed $\gamma$ and have computed the center (96) we
can show how our specific $AdS/CFT$ construction also encodes black hole
temperature by focusing on the EMT, which on the horizon is determined by one
holomorphic component:

$\displaystyle\left\langle
T_{++}\right\rangle=-\frac{c}{48\pi}\left(\frac{f\left(r_{+}\right)}{2}\right)^{2}.$
(105)

The above is precisely the Hawking flux $(HF)$ modulo the central charge:

$\displaystyle\left\langle
T_{++}\right\rangle=cHF=-\frac{c\pi}{12}\left(T_{H}\right)^{2},$ (106)

where $T_{H}=\frac{f\left(r_{+}\right)}{4\pi}$. These are interesting results,
in that it demonstrates that our constructed $AdS_{2}/CFT_{1}$ correspondence
contains both black hole entropy and black hole temperature.

##  6 Discussion and Concluding Remarks

In this paper we explored thermodynamic properties of black hole solutions
within the CWG paradigm. We examined both vacuum and non-vacuum solutions and
computed the respective black hole entropy of several different solutions
within the Nöther current method. We further corroborated our results from the
Nöther current method by comparing them to the Wald entropy formula for CWG.
Despite our results agreeing with Wald’s entropy, we have further shed light
on the non-universality of black hole entropy within CWG, depending on the
specific black hole examined and their symmetries. This is obviously different
compared to Einstein-Hilbert gravity, where the black hole entropy seems to be
given by the standard Bekenstein-Hawking formula irrespective of spacetime.

Our Nöther current analysis implied that the respective quantum black hole
horizon degrees of freedom are dual to a conformal field theory of lesser
dimension. Thus, in our last task, we investigated this duality in more detail
by constructing a two dimensional near horizon quantum effective action via
semiclassical methods. We then showed how the resulting EMT, at the
appropriate boundary, forms one copy of a centrally extended Virasoro algebra
whose center and lowest eigen-mode encode calc hole entropy via Cardy’s
formula. We also demonstrate how our $CFT$ construction also contains black
hole temperature via the quantum holomorphic flux of the $CFT$ on the horizon.

One interesting caveat in our $AdS/CFT$ construction is the presence of a
parameter $\gamma$, which needed to be fixed by comparing the resulting
entropy to that obtained from the Nöther current method. It would be
interesting for future work to perform a renormalization group flow analysis
on this parameter, since it appears as part of the two dimensional $CFT$’s
coupling. This may provide new insight on an analog $c$-theorem [70] for CWG.

We thank Vincent Rodgers and Philip Mannheim for enlightening discussions.

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