††institutetext: Indian Institute of Science Education and Research Bhopal,
Bhopal Bypass, Bhauri, Bhopal 462066, India

# Logarithmic corrections for near-extremal black holes

Nabamita Banerjee  Muktajyoti Saha  and Suthanth Srinivasan
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present the computation of logarithmic corrections to near-extremal black
hole entropy from one-loop Euclidean gravity path integral around the near-
horizon geometry. We extract these corrections employing a suitably modified
heat kernel method, where the near-extremal near-horizon geometry is treated
as a perturbation around the extremal near-horizon geometry. Using this method
we compute the logarithmic corrections to non-rotating solutions in four
dimensional Einstein-Maxwell and $\mathcal{N}=2,4,8$ supergravity theories. We
also discuss the limit that suitably recovers the extremal black hole results.

##  1 Introduction

The universal structure of black hole entropy is a powerful property of
quantum gravitational theories. In the semiclassical regime, the entropy has a
universal form, proportional to the area of the horizon Bekenstein:1973ur ;
Hawking:1976de . The leading-order quantum correction to this area law depends
on the logarithm of horizon size Solodukhin:1994yz ; Mann:1997hm ;
Medved:2004eh ; Cai:2009ua ; Aros:2010jb . These logarithmic corrections are
semi-universal in nature, in the sense that they depend only on the infrared
data of the theory. Euclidean gravity formalism Gibbons:1976ue ; York:1986it
has been proven successful in the computation of black hole entropy. The area
law can be reproduced from a saddle point approximation of the Euclidean path
integral whereas the one-loop contributions to the path integral capture the
logarithmic corrections. A complete microscopic description of black hole
entropy is however not understood yet, which requires the UV completion of
gravity theories. Nevertheless, any sensible UV-complete theory of gravity
should correctly reproduce the area and logarithmic terms in black hole
entropy.

Generic black hole solutions can be categorized as extremal and non-extremal,
depending on whether their temperature is zero or non-zero respectively. In
the near-horizon region of an extremal black hole, an infinitely long AdS2
factor emerges. This results in an enhancement of symmetries which in turn
govern the dynamics of such black holes. This feature does not hold for non-
extremal black holes, where the full geometry is required to understand their
dynamics. In generic theories, the Bekenstein-Hawking entropy of extremal
black holes can be easily computed using Sen’s entropy function formalism
Sen:2005wa ; Sen:2008yk ; Sen:2007qy . Beyond the semiclassical regime, the
idea has also been generalized to a quantum entropy function which can be used
to compute the logarithmic corrections for extremal black holes Sen:2008vm ;
Banerjee:2009af ; Banerjee:2010qc . These formulations depend on the emergent
near-horizon AdS2 factor and its symmetries. Sen and collaborators also
developed a technique to extract the logarithmic contributions to the entropy
of generic (non)-extremal black holes using the heat kernel of the one-loop
action of a gravity theory. For extremal black holes, again the near-horizon
geometry was used Banerjee:2011jp ; Sen:2012cj ; Sen:2012dw ;
Bhattacharyya:2012wz . Whereas for the non-extremal black holes, the results
depend on the full geometry Sen:2012kpz 111The technique was applied to
various theories Karan:2020njm ; Karan:2021teq ; Banerjee:2021pdy ;
Karan:2022dfy .. String theory has provided a microscopic realization of
entropy for a class of supersymmetric extremal black holes Strominger:1996sh ;
Breckenridge:1996is ; David:2006ru ; David:2006ud ; Sen:2008yk ; Gupta:2008ki
; Banerjee:2007ub ; Banerjee:2008ky ; Banerjee:2009uk ; Banerjee:2009af . For
extremal black holes appearing in various string theories Banerjee:2010qc ;
Banerjee:2011jp ; Sen:2012cj , the matching of logarithmic corrections from
gravitational and microscopic perspectives has been achieved. In the recent
works H:2023qko ; Anupam:2023yns , the extremal black hole results are also
correctly reproduced from the limit of a finite temperature geometry
computation.

Black holes with very small temperatures are called near-extremal. Such black
holes with very small temperatures still exhibit some interesting features
that are characteristic of extremal black holes. Thus they are good candidates
to generalize the progress made for extremal black holes in the presence of
small but non-zero temperatures. Recent studies have rendered near-extremal
black holes to have quite distinct properties by their own virtue. Notably, it
has been shown in Nayak:2018qej ; Moitra:2019bub ; Iliesiu:2020qvm  that the
dynamics of such black holes can be obtained from an effective 1D Schwarzian
description (See Heydeman:2020hhw ; Kolekar:2018sba ; Banerjee:2021vjy  for
generalizations to various systems). Iliesiu:2020qvm  shows that the low
temperature dynamics is governed by one-loop corrections proportional to the
logarithm of temperature, which are different than the usual logarithmic
corrections. Recent works Iliesiu:2022onk ; Banerjee:2023quv  trace back the
origin of these terms from a 4D Euclidean gravity computation, in a language
similar to that of the usual logarithmic contributions. It was shown that
these terms appear from the one-loop quantization on the near-horizon near-
extremal background, which is a small deviation of the extremal near-horizon
geometry. The quantization procedure depends on the computation of the
eigenvalues of the kinetic operator of small fluctuations around the classical
background. The idea of Banerjee:2023quv  is to use first-order perturbation
theory to compute these eigenvalues and extract the logarithm of temperature
terms.

The goal of this work would be to analyze the logarithmic corrections for
near-extremal black holes, carefully taking care of the differences with
(non)-extremal solutions. Our approach would be to employ first-order
perturbation theory Banerjee:2023quv  to correctly modify the usual heat-
kernel method for this purpose. The usual approach for (non)-extremal black
holes depends on a scaling property of parameters, where all the length scales
are of the same order and thus scale uniformly. However, for near-extremal
black holes, the issue is subtle as the inverse temperature and charges bring
in large independent length scales. We first separate the logarithmic
contributions coming from these two large parameters. We then use this method
to compute the logarithmic corrections to near-extremal entropy in
$\mathcal{N}=2,4,8$ supergravity theories. We also recover the extremal
logarithmic corrections for both supersymmetric and non-supersymmetric
theories by appropriately taking the extremal limit of our results. Our
modified heat-kernel method correctly reproduces the results for Einstein-
Maxwell theory as earlier found in Iliesiu:2020qvm ; Iliesiu:2022onk ;
Banerjee:2023quv . We also find agreement with the existing results of
Iliesiu:2022onk  for $\mathcal{N}=2,4,8$ supergravity theories for a
particular near-extremal solution. However, to regulate a certain kind of
divergence at a zero temperature limit of the near-extremal result, a sum over
different saddle points was considered in Iliesiu:2022onk . In our computation
of the logarithmic correction for a non-rotating near-extremal black hole,
such a regularization is not required. We elaborate on this issue further in
the discussion section.

The paper is organized as follows: In section 2, we discuss the structure of
the near-horizon geometry of a near-extremal black hole. In section 3, we
first review the heat kernel method to compute logarithmic corrections for
extremal black holes and then discuss how to modify the method for
computations for near-extremal black holes. This prescription is one of the
main results of this paper. We also discuss how to take an appropriate
extremal limit of the near-extremal computation. In section 4, we compute the
logarithmic corrections to near-extremal black hole entropy appearing in
$\mathcal{N}=2$ supergravity theory. Section 5 contains the results for the
same for $\mathcal{N}=4,8$ supergravity theories. Finally, we summarize and
discuss our results in section 6.

##  2 Near-extremal background

In this section, we discuss the properties of a near-extremal black hole
solution near the horizon. We will begin by reviewing the classical geometry
Iliesiu:2020qvm ; Banerjee:2023quv  in Einstein-Maxwell theory and then extend
the idea to generic theories. Let us consider the 4D Einstein-Maxwell theory
with the Euclidean action,

$\displaystyle\mathcal{S}=-\int d^{4}x\sqrt{g}(R-F^{2}).$ (1)

We have set the Newton constant to $1/16\pi$. Therefore, the action has the
dimensions of length-squared. A spherically symmetric black hole solution in
this theory is described by the Reissner-Nördstrom geometry, which is
parametrized by mass and charge,

$\displaystyle
ds^{2}=g_{AB}dx^{A}dx^{B}=f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},\quad
f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}},$ (2) $\displaystyle
A_{t}=iQ\left(\frac{1}{r_{+}}-\frac{1}{r}\right),\quad
F_{rt}=\frac{iQ}{r^{2}}.$ (3)

For $M>Q$, we have a non-extremal black hole that has a finite temperature. At
extremality i.e. for $M=Q$, the solution has a horizon radius given by $Q$ and
it has zero temperature. We are interested in a near-extremal black hole,
which has the same charge but mass slightly greater than the extremal mass. We
parametrize the solution with charge $Q$ and temperature $T$. Since it is a
near-extremal black hole, we work in the regime $QT\ll 1$, which signifies
that we are very close to extremality. The reason for fixing the charge is
that finally, we want to compute the entropy in a microcanonical ensemble. The
quantum entropy function formalism Sen:2005wa ; Sen:2008yk ; Sen:2007qy  for
extremal black holes already produces the microcanonical entropy from the
gravitational side. Therefore, keeping the charges fixed while introducing a
temperature preserves this feature for the charges. In the final expression,
an inverse Laplace transform of variables from inverse temperature to energy
would directly give us the microcanonical entropy of the near-extremal black
hole.

As we move close to the horizon of an extremal black hole, an AdS2 factor
emerges and this results in a symmetry enhancement. The near-horizon throat
being very large, these symmetries govern the dynamics of (near)-extremal
black holes. The near-horizon geometry Iliesiu:2022onk ; Banerjee:2023quv  is
given by ${g_{AB}=g^{0}_{AB}+Tg^{(c)}_{AB},A_{B}={A}^{0}_{B}+TA^{(c)}_{B}}$,
such that $\\{g^{0},A^{0}\\}$ denotes the extremal AdS${}_{2}\times$S2
geometry:

$\displaystyle{g}^{0}_{AB}dx^{A}dx^{B}=Q^{2}(d\eta^{2}+\sinh^{2}{\eta}d\theta^{2})+Q^{2}(d\psi^{2}+\sin^{2}{\psi}d\varphi^{2}),$
$\displaystyle{F}^{0}_{\mu\nu}=\frac{i}{Q}\varepsilon_{\mu\nu},\quad{A}^{0}_{\theta}=iQ(\cosh{\eta}-1).$
(4)

The small temperature causes deviations from this geometry, given by

$\displaystyle g^{(c)}_{AB}dx^{A}dx^{B}=4\pi
Q^{3}(2+\cosh{\eta})\tanh^{2}\left(\frac{\eta}{2}\right)(d\eta^{2}-\sinh^{2}{\eta}d\theta^{2})+4\pi
Q^{3}\cosh{\eta}d\Omega^{2},$ $\displaystyle F^{(c)}_{\mu\nu}=-4\pi
i\cosh{\eta}\varepsilon_{\mu\nu},\quad A^{(c)}_{\theta}=-2\pi
iQ^{2}\sinh^{2}{\eta}.$ (5)

The near-horizon coordinates range from $0<\eta<\eta_{0}$ and $0<\theta<2\pi$
such that the horizon is located at $\eta=0$. We will denote the coordinates
on AdS2 by $x^{\mu}$ and the coordinates on S2 by $x^{i}$.
$\varepsilon_{\mu\nu}$ is the Levi-Civita tensor on AdS2, with the non-zero
component being ${\varepsilon_{\eta\theta}=Q^{2}\sinh{\eta}}$. The near-
horizon geometry is glued to the asymptotic geometry at the boundary located
at $\eta=\eta_{0}$, which is very far away from the horizon. Due to a small
temperature, the throat is large enough to capture the properties of the black
hole. The quantum corrections are large in the near-horizon region but very
much suppressed in the asymptotic region. It essentially motivates us to
quantize the system on the near-horizon background. This is extensively
studied in Iliesiu:2020qvm ; Iliesiu:2022onk ; Banerjee:2023quv .

In generic theories, the extremal black hole solution can be parametrized by
the charges corresponding to various gauge fields. Similar to the Reissner-
Nördstrom scenario, we will keep those charges fixed and introduce a
temperature parameter to describe a near-extremal solution. This will again
cause first-order temperature deviation from the extremal near-horizon
geometry in a way that the flux corresponding to different gauge field
strengths remains the same as the extremal black hole. Hence, schematically we
write the near-horizon geometry as,

$\displaystyle\bar{\Psi}(T,Q_{i})=\bar{\Psi}^{0}(Q_{i})+T\bar{\Psi}^{(c)}(Q_{i}).$
(6)

Here, $\Psi$ denotes all the fields in the theory and $Q_{i}$ denotes the
charges that parametrize the extremal solution. The bar is to signify that we
are considering a classical background. $\bar{\Psi}^{0}$ denotes the field
content of the extremal near-horizon geometry, whereas $\bar{\Psi}^{(c)}$
denotes the corrections in presence of small temperature. The charge
dependence in $\bar{\Psi}^{(c)}$ can be derived from that of $\bar{\Psi}^{0}$.
The charges are of the same order i.e. $Q_{i}\sim a$, where $a$ characterizes
the horizon size of the extremal black hole. The working regime is set by
$aT\ll 1$.

##  3 Heat kernel prescription for logarithmic corrections

Let us consider a theory described by Euclidean action $\mathcal{S}$. We would
like to consider the corresponding Euclidean path integral to one-loop order.
For this, we turn on small fluctuations ($\Psi$) around a classical background
($\bar{\Psi}$) and expand the action to quadratic order in fluctuations. The
zeroth-order term gives the saddle point contribution and the first-order term
vanishes since the background satisfies the equations of motion. Due to the
quadratic term, the one-loop path integral can be expressed as a Gaussian
integral of bosonic and fermionic fluctuations,

$\displaystyle Z=\int\mathcal{D}\Psi\ \text{e}^{-\int
d^{4}x\sqrt{\bar{g}}\Psi^{i}\bar{\Delta}_{ij}\Psi^{j}}.$ (7)

Here, the operator $\bar{\Delta}$ depends on the theory in consideration and
can be obtained on any arbitrary background. The $i,j$ indices capture any
sort of spacetime or internal indices for the fields. For bosonic variables,
the operator is a two-derivative operator whereas for fermionic variables, it
is linear in derivative. The one-loop path integral can in principle be
obtained from the determinant of this operator $\bar{\Delta}$ on a particular
classical background. The determinant may be ill-defined due to the presence
of zero modes of the kinetic operator, but these are dealt with separately.
The non-zero mode contributions can be expressed as,

$\displaystyle\log{Z}_{\text{nz}}=-\frac{1}{2}\sideset{}{{}^{\prime}}{\sum}_{n}\log{\kappa}_{n}+\frac{1}{4}\sideset{}{{}^{\prime}}{\sum}_{n}\log{\kappa}^{f}_{n}.$
(8)

The prime indicates that we are summing over non-zero modes only. We have
written the bosonic and fermionic (denoted by a superscript $f$) contributions
separately since the bosonic Gaussian integral gives
$(\det\bar{\Delta})^{-1/2}$ whereas the fermionic integral222We are
considering Majorana fermions, which have half the degrees of freedom as
compared to Dirac fermions due to reality constraint. Hence the Gaussian
integral gives the square root of determinant. gives
$(\det\bar{\Delta})^{1/2}$. Also, $\\{\kappa_{n}^{f}\\}$ denote the
eigenvalues of the squared fermionic operator so that the eigenvalues
$\kappa_{n}$ and $\kappa_{n}^{f}$ have the same dimensions. This is reflected
by the additional factor of $1/2$ in front of the fermionic contribution.

Without directly computing the individual eigenvalues, the heat kernel
corresponding to the kinetic operator can be used to evaluate (8). We work
with the bosonic and fermionic parts separately. First, we will discuss the
bosonic contributions and later draw an analogy for the fermionic case. The
heat kernel for the kinetic operator of bosonic fluctuations is defined as,

$\displaystyle
K^{ij}(x,x^{\prime};s)=\sum_{n}\text{e}^{-\kappa_{n}s}f^{i}_{n}(x)f^{j}_{n}(x^{\prime}).$
(9)

Here, $\kappa_{n}$ denotes the eigenvalues of the kinetic operator
$\bar{\Delta}$, corresponding to the eigenfunction $f_{n}^{i}(x)$, which we
are taking to be real. For complex eigenfunctions, we should take the complex
conjugate for one of the eigenfunctions in the sum. The eigenfunctions are
orthonormal with respect to the inner product,

$\displaystyle\int
d^{4}x\sqrt{\bar{g}}G_{ij}f^{i}_{n}(x)f^{j}_{m}(x)=\delta_{nm}.$ (10)

Here, $G_{ij}$ is the metric that is induced on the field space. The heat
kernel satisfies a ‘heat equation’ of the form
$(\partial_{s}+\bar{\Delta})K(x,x^{\prime};s)=0$, where we have suppressed the
internal indices. For fermionic fluctuations, we will consider the heat kernel
corresponding to the squared kinetic operator and further define the heat
kernel with an additional factor of $-1/2$ such that333An important point to
note is that we should only consider the eigenfunctions of the linear
derivative kinetic operator and take the corresponding eigenvalues of the
squared kinetic operators.,

$\displaystyle
K^{ij}_{f}(x,x^{\prime};s)=-\frac{1}{2}\sum_{n}\text{e}^{-\kappa_{n}^{f}s}f^{i}_{n}(x)f^{j}_{n}(x^{\prime}).$
(11)

To evaluate the logarithm of partition function (8) in terms of the heat
kernel, the following identity is used,

$\displaystyle\log{A}=-\int_{\epsilon}^{\infty}\frac{ds}{s}\text{e}^{-As}.$
(12)

$\epsilon\to 0$ is a small cutoff introduced to regulate the integral, which
in our case will be provided by the UV cutoff of the theory444In this
integral, $A$ is considered to be much above the cutoff scale $\epsilon$. We
will get back to this point in a later discussion.. Using the above identity,
the bosonic non-zero modes sector of (8) can be expressed in terms of the
trace of the heat kernel as,

$\displaystyle\log{Z}_{\text{nz}}=\frac{1}{2}\sideset{}{{}^{\prime}}{\sum}_{n}\int_{\epsilon}^{\infty}\frac{ds}{s}\text{e}^{-\kappa_{n}s}=\frac{1}{2}\int_{\epsilon}^{\infty}\frac{ds}{s}\int
d^{4}x\sqrt{g}K^{\prime}(x,x;s).$ (13)

Here, we are using the notation $K(x,x^{\prime};s)\equiv
G_{ij}K^{ij}(x,x^{\prime};s)$. Again, the prime denotes the non-zero
contribution only. Since for fermionic fluctuations, the additional factor of
$-1/2$ is absorbed into the definition of the heat kernel, the final form in
terms of the trace remains the same. In particular, the fermionic non-zero
mode contribution is given by,

$\displaystyle\log{Z}_{\text{nz}}=-\frac{1}{4}\sideset{}{{}^{\prime}}{\sum}_{n}\int_{0}^{\infty}\frac{ds}{s}\text{e}^{-\kappa^{f}_{n}s}=\frac{1}{2}\int_{\epsilon}^{\infty}\frac{ds}{s}\int
d^{4}x\sqrt{g}K^{\prime}_{f}(x,x;s).$ (14)

In general, it is very difficult to compute the full one-loop contribution in
an arbitrary theory. In Banerjee:2010qc ; Banerjee:2011jp ; Sen:2012cj ;
Bhattacharyya:2012wz ; Sen:2012kpz ; Sen:2012dw , it was shown how to extract
the logarithmic corrections to (non)-extremal black hole entropy from the heat
kernel prescription. It was argued that the logarithmic corrections appear
from the small $s$-region of the integration. Although the techniques for
extremal and non-extremal black holes are essentially the same in spirit,
there are some important differences between these two cases. One of them is
that for the extremal black hole, the computation is done on the near-horizon
background whereas for the non-extremal black hole, the full geometry must be
used. The analysis for a near-extremal black hole will be a close cousin to
the analysis for an extremal black hole, due to the presence of a large near-
horizon throat. Also, the temperature and charge parameters introduce
different scales in the solution in contrast with a generic non-extremal black
hole where all the parameters have uniform scaling. Hence, we now briefly
review the method for extremal black holes following the works of
Banerjee:2011jp ; Sen:2012kpz .

We first consider the case of $\mathcal{N}=2$ supergravity theory, which
contains the Einstein-Maxwell theory as the bosonic sector. Here the horizon
size is equal to the charge and the extremal black hole can be parametrized by
charge only. The analysis of the subsequent part of this section can be easily
generalized to arbitrary theories with various charges. Since the charges
scale uniformly with the extremal horizon size, we can just work with one of
the charges or the horizon size.

###  3.1 Brief review of the approach for extremal black holes

In this section we briefly review the prescription for extremal black holes.
Let us consider a spherically symmetric extremal black hole solution
parametrized by its electric charge $Q$ only. Its near-horizon geometry555We
will denote the extremal geometry with a superscript 0. is AdS${}_{2}\times$S2
with radii $Q$. The extremal solution has the following scaling dependence,

$\displaystyle g_{\mu\nu}^{0}\sim Q^{2},\quad A_{\mu}^{0}\sim Q.$ (15)

As mentioned earlier, the action must have the dimensions of length squared
such that the quadratic fluctuated action should also satisfy this scaling
property,

$\displaystyle\mathcal{S}^{(1)}\equiv\int
d^{4}x\sqrt{g^{0}}\Psi^{i}\Delta^{0}_{ij}\Psi^{j}\sim Q^{2}.$ (16)

Due to the form of the extremal metric, in 4D we have $\sqrt{g^{0}}\sim
Q^{4}$. Thus we find that the two-derivative kinetic operator must have the
dependence $\sim Q^{-2}$ since it is constructed out of the extremal
background. First, we consider the non-zero mode contribution to the partition
function. From the scaling properties discussed above, we can find that the
non-zero eigenvalues of the kinetic operator must be of the form,

$\displaystyle\kappa^{0}_{n}\sim\frac{1}{Q^{2}}.$ (17)

The dimensions of the eigenfunctions for different fields can be read off from
the orthonormality condition (10). The Schwinger parameter $s$ has dimensions
of length squared. Using these dimensional dependencies, we perform a change
of variable $s\to\hat{s}=s/Q^{2}$ such that (13) becomes,

$\displaystyle\log
Z_{\text{nz}}=\frac{1}{2}\int_{\epsilon/Q^{2}}^{\infty}\frac{d\hat{s}}{\hat{s}}\int
d^{4}x\sqrt{g^{0}}K^{\prime}(0;\hat{s}).$ (18)

The hat on various quantities denote their dimensionless variants, after the
$Q$-dependencies are stripped off. Due to the homogeneity of the extremal
geometry, the coordinate dependence goes away from the trace of the heat
kernel. As argued in Bhattacharyya:2012wz , the logarithmic corrections appear
from the small $s$-regime, where the heat kernel trace over a background can
be expanded in terms of Seeley-de Witt coefficients,

$\displaystyle
K(x,x;s)=\sum_{m=0}^{\infty}K_{-\frac{D}{2}+m}(x)\hat{s}^{-\frac{D}{2}+m}.$
(19)

$K_{-\frac{D}{2}+m}(x)$ is a $2m$-derivative local diffeomorphism and gauge
invariant term constructed out of the background fields. For extremal
background, these coefficients are just constant terms Sen:2012kpz . The
$s$-independent part of the trace, in particular, the integration of the
coefficient $K_{0}$ determines the coefficient of the $\log{Q}$ term in
$\log{Z_{\text{nz}}}$, given by,

$\displaystyle\log{Z_{\text{nz}}}=\frac{1}{2}\int
d^{4}x\sqrt{g^{0}}\left(K_{0}-K(0)\right)\log
Q^{2}=8\pi^{2}Q^{2}(\cosh\eta_{0}-1)\left(K_{0}-K(0)\right)\log Q.$ (20)

Here, $K(0)$ determines the number of zero modes, given by,

$\displaystyle N_{\text{zm}}=\int
d^{4}x\sqrt{g^{0}}K(0)=8\pi^{2}Q^{2}(\cosh\eta_{0}-1)K(0);\quad
K(0)\equiv\sum_{\text{zm}}G_{ij}f^{i}_{n}f^{j}_{n}.$ (21)

Now we consider the contributions coming from the zero modes of the extremal
black hole that arise in the near-horizon geometry. Zero modes are associated
with the symmetries that are spontaneously broken by the black hole solution.
Due to the presence of the AdS2 boundary, the global symmetries of the
solution get enhanced to the infinite-dimensional algebra of asymptotic
symmetries. We have an infinite number of zero modes associated with the
spontaneous breaking of these asymptotic symmetries. These fluctuations are
generated by large gauge transformations and diffeomorphisms, which are non-
normalizable. There is no Gaussian suppression for these modes in the path
integral, hence the integral is typically divergent. However, there can be
non-trivial logarithmic corrections coming from the path integral measure of
these modes and these contributions are fixed by the normalization conditions
imposed on these modes.

To understand the contributions, we make a change of variables from the field
variables to the large gauge transformation parameters that generate the
corresponding fluctuations. These large gauge parameters do not depend on the
charge, hence their integration does not give rise to any charge dependence.
The non-triviality appears when the change of variables from the fluctuations
to the gauge parameters is performed. The corresponding Jacobian due to the
change of variable results in a particular dependence in charge. A factor of
$Q^{\beta_{r}}$ is introduced in the measure from the Jacobian for each zero
mode. The number $\beta_{r}$ depends on the particular type of mode in
consideration. It can be fixed by analyzing the field fluctuations and the
corresponding gauge parameters. We discuss this feature in the later sections
when we consider extremal black hole solutions in particular theories. Hence,
the zero mode contribution to the logarithm of partition function is given as,

$\displaystyle\log
Z_{\text{zm}}=\sum_{r}\beta_{r}N^{r}_{\text{zm}}\log{Q}=8\pi^{2}Q^{2}(\cosh\eta_{0}-1)\sum_{r}\beta_{r}K^{r}(0)\log{Q}.$
(22)

Here, we have included a factor of $Q^{\beta_{r}}$ for each zero mode, labeled
by $r$, and the sum characterizes a sum over different types of zero modes.
$N^{r}_{\text{zm}}$ denotes the number of such zero modes, given by (21) where
the sum should involve the zero mode eigenfunctions of that particular
category. $\eta=\eta_{0}$ denotes the boundary of the near-horizon AdS2
throat, which is very large. The boundary cutoff-dependent pieces in the
expression of $\log{Z}$ can be interpreted as an infinite shift in the ground
state energy. The cutoff-independent part will give the actual contribution to
entropy.

#### Fermionic sector

A similar analysis can be performed for the fermionic fluctuations also. Since
we are considering the eigenvalues of the squared kinetic operator, the
dimensional dependencies remain the same as that of the bosonic operator as
both of these are two-derivative terms. The form of the non-zero mode
contribution remains exactly the same i.e.

$\displaystyle\log{Z_{\text{nz}}}=\frac{1}{2}\int
d^{4}x\sqrt{g^{0}}\left(K^{f}_{0}-K^{f}(0)\right)\log Q^{2}.$ (23)

Now we consider the zero modes contribution. The number of zero modes in terms
of the trace of the heat kernel is given as,

$\displaystyle N^{f}_{\text{zm}}=-2\int
d^{4}x\sqrt{g^{0}}K^{f}(0)=-16\pi^{2}Q^{2}(\cosh\eta_{0}-1)K^{f}(0);\quad
K^{f}(0)\equiv-\frac{1}{2}\sum_{\text{zm}}G_{ij}f^{i}_{n}f^{j}_{n}.$ (24)

The fermionic zero modes are spin-$3/2$ fluctuations, associated with the
breaking of the supergroup of asymptotic symmetries and these are generated by
non-normalizable spin-$1/2$ parameters. Since there is no Gaussian
suppression, these integrals typically go to zero. To extract the charge
dependence, we again perform a change of variable from the fields to the gauge
parameters, which introduces a factor of $Q^{-\beta_{f}/2}$ per Majorana zero
mode in the measure. Hence, the fermionic zero modes contribution to the
logarithm of partition function is given as,

$\displaystyle\log
Z_{\text{zm}}=-\frac{1}{2}\sum_{f}\beta_{f}N^{f}_{\text{zm}}\log{Q}=8\pi^{2}Q^{2}(\cosh\eta_{0}-1)\sum_{r}\beta_{f}K^{f}(0)\log{Q}.$
(25)

The total logarithm of charge contribution to the extremal black hole
partition function is given as,

$\displaystyle\log Z\sim(c_{0}+c_{0}^{f})\log Q,$ (26) $\displaystyle
c_{0}=-8\pi^{2}Q^{2}[K_{0}+\sum_{r\in\text{zm}}(\beta_{r}-1)K^{r}(0)],$ (27)
$\displaystyle
c^{f}_{0}=-8\pi^{2}Q^{2}[K^{f}_{0}+\sum_{f\in\text{zm}}(\beta_{f}-1)K^{f}(0)].$
(28)

We will compare this coefficient with the corresponding coefficient for the
near-extremal black hole.

###  3.2 Approach for near-extremal black holes

In this section, we will consider the one-loop path integral around a
spherically symmetric near-extremal black hole solution parametrized by
temperature $T$ and charge $Q$ such that $QT\ll 1$. As discussed earlier, this
solution also has a finite but very large near-horizon throat having a
geometry that can be described as a linear-order temperature correction to the
extremal AdS${}_{2}\times$S2 geometry. We will quantize the system on the
near-horizon background following the approach of first-order perturbation
theory Banerjee:2023quv . However, we will resort to a heat kernel method that
does not depend on explicit eigenvalue computations, in contrast with
Banerjee:2023quv .

The dogma of the first-order perturbation theory technique is that it recasts
the quantization procedure around the near-extremal background into finding
the spectrum (or in this case, the heat kernel) of a modified operator on the
AdS${}_{2}\times$S2 geometry itself. Since the near-extremal background can be
described as a linear order temperature deviation from the extremal one, the
quadratic action can be rewritten as,

$\displaystyle S^{(2)}=-\int d^{4}x\sqrt{g^{0}}\
\Psi^{i}(\Delta^{0}_{ij}+T\Delta^{(c)}_{ij})\Psi^{j}.$ (29)

We have clubbed all the temperature-dependent corrections into the operator
$\Delta^{(c)}$ and our interest lies in finding the heat kernel for the
modified operator $\Delta\equiv\Delta^{0}+T\Delta^{(c)}$. In particular, the
eigenvalue equation of this operator is given by,

$\displaystyle\Delta_{ij}f^{j}_{n}=\kappa_{n}f_{n,i},\quad\kappa_{n}=\kappa_{n}^{0}+T\kappa_{n}^{c},\quad
f_{n}=f_{n}^{0}+Tf_{n}^{c}.$ (30)

Here $\Delta^{0}_{ij}f^{0j}_{n}=\kappa^{0}_{n}f^{0}_{n,i}$. Using first-order
perturbation theory, we have,

$\displaystyle\kappa^{c}_{n}=\int d^{4}x\sqrt{g^{0}}\ f^{0i}_{n}(x)\
\Delta^{(c)}_{ij}\ f^{0j}_{n}(x)$ (31) $\displaystyle
f_{n}^{ci}(x)=\sum_{m\neq n}\frac{1}{\kappa^{0}_{n}-\kappa^{0}_{m}}\left(\int
d^{4}x^{\prime}\sqrt{g^{0}}\ f^{0j}_{m}(x^{\prime})\ \Delta^{(c)}_{jk}\
f^{0k}_{n}(x^{\prime})\right)\ f_{m}^{0i}(x)$ (32)

The heat kernel is again defined as (9) for bosonic fluctuations and as (11)
for fermionic fluctuations using these eigenvalues and
eigenfunctions666Although we will never explicitly require the eigenfunction
corrections..

Our goal would be to extract the logarithmic corrections in $Q$ and $T$ since
they bring different scales to the solution. For this, we again consider the
non-zero and zero modes contributions separately. Further, we have two kinds
of non-zero modes, namely those that have have non-zero and slightly non-zero
eigenvalues. The proper non-zero modes are the ones that are also non-zero on
the extremal background. The proper zero modes are the ones that remain zero
modes on the near-extremal background. The contributions are schematically
divided into three parts,

$\displaystyle\log Z=\log Z_{\text{nz}}+\log Z_{\text{snz}}+\log
Z_{\text{zm}}.$ (33)

The treatment of the proper non-zero modes will be done in the same way as the
non-zero modes of the extremal black hole, since these eigenvalues still scale
like $1/Q^{2}$. As discussed in Banerjee:2023quv , these modes give rise to
$\log{Q}$ terms only. The coefficient will also remain the same as the non-
zero part of the extremal computation. The same is true for the proper zero
modes, which give rise to only $\log Q$ corrections through the path integral
measure. Hence, the coefficient of $\log Q$ terms coming from the proper non-
zero and proper zero modes will remain equal to their counterparts in the
extremal case.

#### Bosonic slightly non-zero sector

We will consider the contribution of the slightly non-zero modes separately
since these modes will give rise to both $\log Q$ and $\log T$ contributions.
Although the proper non-zero and proper zero modes computations are quite
similar for bosonic and fermionic fluctuations, the fermionic slightly non-
zero modes have important differences from the bosonic ones. Hence, first we
will consider the bosonic sector and deal with the fermionic sector later. The
slightly non-zero modes were zero modes on the extremal background but got
lifted in presence of temperature. Evidently, the eigenvalues of these modes
are $\mathcal{O}(T)$ i.e. there is no $\mathcal{O}(1)$ piece. The relevant
contribution in the partition function is given as,

$\displaystyle\log
Z_{\text{snz}}=\frac{1}{2}\int_{\epsilon}^{\infty}\frac{ds}{s}\left(\overline{\sum}_{n}e^{-T\kappa_{n}^{c}s}\right).$
(34)

We introduce the notation $\overline{\sum}_{n}$ to indicate that we are
summing over the zero modes of the extremal solution, which get promoted to
slightly non-zero modes. From dimensional analysis, we find that,

$\kappa_{n}^{c}=\frac{\hat{\kappa}_{n}^{c}}{Q},$ (35)

where hat denotes a dimensionless object. Similar to the earlier case, we make
the transformation $s\to\hat{s}=\frac{Ts}{Q}$ such that,

$\displaystyle\log
Z_{\text{snz}}=\frac{1}{2}\int_{T\epsilon/Q}^{\infty}\frac{d\hat{s}}{\hat{s}}\left(\overline{\sum}_{n}e^{-\hat{\kappa}_{n}^{c}\hat{s}}\right).$
(36)

We can recognize that again the logarithmic corrections arise from the
small-$\hat{s}$ integration of the $\hat{s}$-independent part of the term in
parentheses. Thus we get,

$\log
Z_{\text{snz}}=-\frac{1}{2}N_{\text{snz}}\log\left(\frac{T\epsilon}{Q}\right),\quad
N_{\text{snz}}\equiv\overline{\sum}_{n}1.$ (37)

We conclude that the coefficient of this correction is given by the number of
zero modes777We have used the fact that the corrected eigenfunctions are
orthonormal at order $T$ and the orthonormality condition is exactly equal to
that of the extremal eigenfunctions. of the extremal solution, which get
promoted to non-zero modes when we switch on a small temperature. In terms of
the trace of the heat kernel, we find that,

$\displaystyle N_{\text{snz}}=8\pi^{2}Q^{4}(\cosh{\eta_{0}}-1)\bar{K}(0).$
(38)

The bar again denotes a sum over the slightly non-zero modes. Since we are
considering the quantization problem as finding the spectrum of a modified
operator on the extremal background, the boundary cutoff-dependent piece can
again be dropped off. Hence, the logarithmic corrections from the cutoff-
independent part are given as,

$\displaystyle\log Z_{\text{snz}}\sim
4\pi^{2}Q^{4}\bar{K}(0)\log\left(\frac{T\epsilon}{Q}\right).$ (39)

#### Fermionic slightly non-zero sector

Now we consider the fermionic slightly non-zero modes. As mentioned earlier,
the treatment of proper non-zero and proper zero modes will still remain the
same. Although these modes have non-zero eigenvalues, we will show that we
cannot treat them in a way similar to the bosonic slightly non-zero modes by
considering the squared kinetic operator. Instead, we will need to work with
the linear derivative kinetic operator itself. To understand the issue better,
let us consider the generic form of the eigenvalues of the fermionic kinetic
operator, given as888For slightly non-zero modes, we are denoting the
eigenvalues of the kinetic operator with $\kappa$. It is not to be confused
with the eigenvalues of the squared operator. $\kappa_{n}^{f}=A+BT$. For
slightly non-zero modes, the $\mathcal{O}(1)$ piece is zero i.e. $A=0$. Thus
the eigenvalues of the squared operator have no $\mathcal{O}(T)$ piece since
that is proportional to the $\mathcal{O}(1)$ piece itself. Hence the squared
kinetic operator has eigenvalues of $\mathcal{O}(T^{2})$. Although we could
work with these squared eigenvalues and later take their square root, it would
be convenient to work with the eigenvalues of the fermionic kinetic operator
and not its square while dealing with these slightly non-zero modes. The
contribution of these modes to the partition function is given by,

$\displaystyle\log
Z^{f}_{\text{snz}}=\frac{1}{2}\overline{\sum}_{\text{snz}}\log{\kappa}^{f}_{n}.$
(40)

We again bring in an auxiliary variable to express this like (12). However,
this variable has different dimensions than that of the variable used for the
bosonic case. Thus we have,

$\log
Z^{f}_{\text{snz}}=-\frac{1}{2}\int_{\epsilon^{\prime}}^{\infty}\frac{ds}{s}\left(\overline{\sum}_{n}e^{-T\kappa_{n}^{c}s}\right),$
(41)

where $\epsilon^{\prime}$ is again a small cutoff which is related to the UV
cutoff of the theory. $\kappa_{n}^{c}$ now denotes the shifts in the
eigenvalues of the fermionic kinetic operator (and not its square, to be
emphatic) corresponding to the slightly non-zero modes. These corrections are
dimensionless since the eigenvalues of the fermionic kinetic operator scale as

$\kappa_{n}\sim\frac{1}{Q},$ (42)

by virtue of it being a one-derivative operator. To extract the logarithmic
contributions, we perform the rescaling $s\to\hat{s}=Ts$ so that,

$\log
Z^{f}_{\text{snz}}=-\frac{1}{2}\int_{T\epsilon^{\prime}}^{\infty}\frac{d\hat{s}}{\hat{s}}\left(\overline{\sum}_{n}e^{-\kappa_{n}^{c}\hat{s}}\right)=\frac{1}{2}N^{f}_{\text{snz}}\log\left(T\epsilon^{\prime}\right).$
(43)

Here, $N^{f}_{\text{snz}}$ denotes the number of fermionic slightly non-zero
modes, defined through (24) and summing over the appropriate modes. It can be
easily seen that if we used the eigenvalues of the squared operator, the lower
limit of the $s$-integral would be given by $T^{2}\epsilon$, where $\epsilon$
is the UV cutoff. This is because the eigenvalues of the squared
operator999This should be thought of as a formal square and not a second-order
perturbation theory computation. scale like $1/Q^{2}$. From this comparison,
we find that $\epsilon^{\prime}\sim\sqrt{\epsilon}$. Thus we do not get any
$\log Q$ dependence from this sector.

Now, we will write down the complete logarithmic contributions in charge and
temperature. The only distinction between extremal and near-extremal results
appears from the slightly non-zero modes sector. On the extremal background,
these modes would have contributed like ordinary zero modes where the
logarithmic contribution comes from the path integral measure. To make a clean
comparison, we add and subtract these contributions of the strictly extremal
background. Then the logarithmic contribution in the logarithm of partition
function can be written as,

$\displaystyle\log Z\sim$
$\displaystyle\left[c_{0}+8\pi^{2}Q^{4}\sum_{r\in\text{snz}}\left(\beta_{r}-\frac{1}{2}\right)K^{r}(0)\right]\log
Q+8\pi^{2}Q^{4}\sum_{r\in\text{snz}}\frac{1}{2}K^{r}(0)\log T$
$\displaystyle+$
$\displaystyle\left[c^{f}_{0}+8\pi^{2}Q^{4}\sum_{f\in\text{snz}}\beta_{f}\
K^{f}(0)\right]\log Q+8\pi^{2}Q^{4}\sum_{f\in\text{snz}}K^{f}(0)\log T.$ (44)

We have explicitly brought out the difference in the coefficients of $\log Q$
terms for the extremal and the near-extremal black holes to draw a proper
analogy. Here $r,f$ label different types of bosonic and fermionic slightly
non-zero modes respectively. As discussed earlier, $\beta_{r}$ and $\beta_{f}$
are the numbers determining the logarithmic contribution for these modes for
zero temperature i.e. when these are zero modes. This equation (44) is one of
the main results of this paper and can be applied to compute the near-extremal
logarithmic corrections in arbitrary theories of gravity, an analysis which
was so far missing in the literature.

###  3.3 Regime of validity

We analyze the temperature regime where the formula (44) holds. Since we are
considering a near-extremal black hole, we already imposed an upper bound on
the temperature as $QT\ll 1$. In this section, we show that there is also a
lower bound in temperature till which the final near-extremal partition
function is valid.

As discussed earlier, the non-zero and zero mode contributions for a near-
extremal black hole can be evaluated similarly to those of an extremal black
hole. The difference appears due to the presence of slightly non-zero modes.
We want to understand the lowest temperature, till which we can consider these
modes to be ‘slightly’ non-zero and below which the currently available
methods start treating them as zero modes. Intuitively, this lowest
temperature will signify a point where the Gaussian integral flattens enough
that it can no longer be treated as a suppressed integral. For a simple
Gaussian integral, this regime can be thought of as a limit where the width of
the Gaussian becomes much larger than the domain of the integration. To
understand this in the heat kernel formalism, let us note that in the identity
(12), the parameter $A$ should be much larger than the cutoff $\epsilon$. If
$A$ becomes comparable with $\epsilon$, we could expand the integrand in a
series in $\epsilon$. Under an $\epsilon\to 0$ limit, the integral would be
typically divergent depending on the cutoff only, and the logarithmic
dependence on the parameter $A$ will be lost. This is equivalent to the
flattening of the Gaussian. Similar arguments hold for the partition function
computation, where $A$ is replaced by appropriate black hole parameters. This
behavior sets the lower bound on temperature through (36) such that it does
not become comparable with the UV cutoff of the theory. Hence, our
result101010In such a small scale of temperature, the nonperturbative
corrections to the path integral might become important as suggested in
Iliesiu:2022onk . These effects are not completely understood yet and out of
the scope of our current perturbative computations. (44) is valid in the
following temperature range,

$\displaystyle y\left(\frac{\epsilon}{Q^{2}}\right)\ll QT\ll 1.$ (45)

Here, $y$ can be a certain power of the dimensionless combination
$\frac{\epsilon}{Q^{2}}$. For different kinds of field fluctuations, the
explicit power law dependence might change. Thus the lower limit should be the
largest of all these bounds. However, the temperature must be much above this
bound. Below this temperature bound, we should start treating these modes as
zero modes again in the heat kernel prescription. This indicates that the
current machinery is not enough fine-tuned to identify a slightly non-zero
mode in this regime. Thus in such a low temperature range, the extremal black
hole result (28) tends to hold.

###  3.4 Extremal limit

Let us consider a near-extremal black hole in the temperature regime where we
can distinguish between zero and slightly non-zero modes. We have shown that a
near-extremal and an extremal black hole having the same charges may have
different $\log Q$ coefficients, which is a physically consistent observation.
Now we will show that a systematic extremal limit of the near-extremal
computations is possible and under this limit, we can recover the $\log Q$
coefficient as predicted from the computations on the extremal geometry.

Let us consider the example of a simple Gaussian integral to illustrate the
limit,

$\displaystyle I_{\alpha}=\int_{-\infty}^{\infty}\alpha dx\
\text{e}^{-b\alpha^{2}x^{2}}=\frac{\sqrt{\pi}}{\sqrt{b}}.$ (46)

We want to match the values of $I_{\alpha}$ under independent $b\to 0$ limits
on both sides of the equation. In particular, we would like to compare the
$\alpha$-dependence of the integral111111We have introduced the explicit
scaling factors of $\alpha$ to point out that the result is independent of any
rescaling of the variable $x$. This feature will be important for slightly
non-zero modes.. Naively it seems that the right hand side is (i.e. after
performing the Gaussian integral) independent of $\alpha$. However if we take
$b=0$ on left hand side, the integral seems to pick up a factor of $\alpha$.
Thus, there seems to be an apparent discrepancy in the result. Here we show
that it is possible to extract this factor of $\alpha$ even after performing
the integral. For this, we need to regulate the integral by evaluating it on
an interval instead and the divergences will be coming from the length of this
interval. We consider the Gaussian error function, defined as,

$\displaystyle\text{erf}(y)=\frac{2}{\sqrt{\pi}}\int_{0}^{y}\text{e}^{-x^{2}}dx.$
(47)

Evaluating $I_{\alpha}$ on an interval gives,

$\displaystyle I_{\alpha}=\int_{-L}^{L}\alpha dx\
\text{e}^{-b\alpha^{2}x^{2}}=\frac{\sqrt{\pi}}{\sqrt{b}}\
\text{erf}(\sqrt{b}\alpha L).$ (48)

If we take a $b\rightarrow 0$ limit on the left side (i.e. in the integrand),
we get,

$\displaystyle I_{\alpha}=\int\alpha dx=\alpha\cdot 2L.$ (49)

The $b\to 0$ limit on the right side (i.e. after performing the integral)
amounts to studying the behavior of the error function under $b\to 0$ and
$L\to\infty$ limits, which gives

$\displaystyle I_{\alpha}=\alpha\cdot
2L-\frac{2}{3}L^{3}\alpha^{3}b+\mathcal{O}(b^{2}).$ (50)

Hence, in a strict $b\to 0$ limit, both sides agree on the dependence of
$\alpha$. A similar analysis of such a limit can be performed for fermionic
Gaussian integrals, where the results are not divergent. Instead, the result
goes to zero when the integrand is 1, as is typical for Grassmann variables.
We will discuss the scaling dependence in the next few lines, skipping an
explicit derivation. In presence of a Gaussian suppression, any arbitrary
scaling of the integration variable keeps the result unchanged. But if we take
the Gaussian exponent to zero, the scaling factor in the measure can no longer
be canceled from the Gaussian.

These features of Gaussian integrals are crucial in understanding a zero
temperature limit of the integral over the slightly non-zero modes. As
discussed earlier, the zero modes of an extremal black hole are generated by
large gauge transformation parameters that are independent of the black hole
scale. To understand the $\log{Q}$ contribution coming from these zero modes,
a change of variable is performed from the fields to the gauge parameters
Sen:2012kpz . The gauge parameters being $Q$-independent, the measure acquires
a factor of $Q^{\beta_{r}}$, coming from the Jacobian of change of variable.
The number $\beta_{r}$ depends on the type of zero mode in consideration. This
factor nontrivially contributes since there is no Gaussian suppression for
zero modes on the extremal background. However, as discussed above, if there
is a Gaussian suppression, the result of the integral is independent of the
scale factor. To understand this, we draw an analogy with the integral
$I_{\alpha}$, where $Q^{\beta_{r}}$ is playing the role of $\alpha$ in our
case. Hence, for slightly non-zero modes integral, this normalization does not
affect the Gaussian integral result. Since we want to consider an extremal
limit, we would first like to make the change of variable to the gauge
parameters so that integration limits become $Q$-independent. Then for each
slightly non-zero mode integral, we will get a factor of $Q^{\beta_{r}}$ in
the measure and a factor of $Q^{2\beta_{r}}$ in the Gaussian exponent. This
does not change the result of the Gaussian integral when $T\neq 0$. But when
we take the $T=0$ limit, our analysis shows that we get a factor of
$Q^{\beta_{r}}$ coming from each mode, resulting in a coefficient:

$\displaystyle Q^{\beta_{r}(\text{Number of modes})}.$ (51)

This precisely agrees with the computations on an extremal black hole, where
these modes were treated as zero modes. The argument for fermionic integrals
also goes through similarly.

This simple analysis indicates that if we consider the result of Gaussian
integral of slightly non-zero modes in the presence of temperature and take a
zero temperature limit of the result, it should boil down to the extremal
result. The caveat here is that we can only take such a limit insofar as we
have not made use of any techniques that necessarily require the temperature
of our system to be non-zero. For instance, even when the temperature is non-
zero, the contributions coming from an infinite number of slightly non-zero
modes typically give divergent factors. To regulate this divergence, a Zeta
function regularization was performed in Banerjee:2023quv , which finally gave
a non-zero $\log T$ contribution. Similar regularization is considered even in
the one-loop partition function of the Schwarzian theory Saad:2019lba , which
in turn is used to draw the conclusions in Iliesiu:2020qvm ; Iliesiu:2022onk .
This regularization explicitly uses that $T\neq 0$, however small it is.
Hence, it is not correct to take an extremal limit beyond this regularization.
In the heat kernel approach, we cannot take a zero temperature limit after we
have extracted the scale dependence of the slightly non-zero eigenvalues in
(36), which in turn changes the lower limit of the $s$-integral. After this
rescaling, it is not possible to take a strict $T=0$ limit. The limits of
applicability of our analysis was discussed in the earlier section.

The main takeaway point of this analysis is that it is possible to recover the
extremal $\log Q$ correction from the near-extremal result in any arbitrary
theory as long as we do not explicitly assume $T\neq 0$, in performing some
kind of regularization. Hence, the near-extremal results are perfectly
consistent with the extremal black hole result, which are indeed the complete
logarithmic corrections at extremality. This agreement is based on an
appropriate extremal limit and after the regularization procedures, it is not
possible to smoothly go to extremality. Our work provides a systematic
computation of logarithmic corrections to near-extremal entropy, which may be
different than the corresponding extremal results.

##  4 Pure $\mathcal{N}=2$ supergravity

In this section, we will compute the logarithmic corrections for a near-
extremal black hole in pure $\mathcal{N}=2$ supergravity theory in four
dimensions using the formula (44). Our task will be to identify the slightly
non-zero modes by analyzing the structure of the near-extremal background,
without ever explicitly computing the eigenvalues. The bosonic sector of this
theory is the Einstein-Maxwell theory. Hence, the non-rotating black hole
solution is described by Reissner-Nördstrom geometry. We will first understand
the bosonic sector contributions.

###  4.1 Bosonic sector: Einstein-Maxwell theory

We need to understand which of the bosonic zero modes of the extremal black
hole will get lifted in the presence of temperature. As mentioned earlier,
these zero modes are associated with the spontaneous breaking of certain
asymptotic symmetries near the AdS2 boundary. To be precise, the $SL(2,R)$
global isometry enhances to the asymptotic symmetries of AdS2, which are large
diffeomorphisms forming the Virasoro symmetry algebra. The $U(1)$ and $SO(3)$
symmetries enhance to large gauge transformations near the boundary. These
asymptotic symmetries are spontaneously broken to the global part, which is
the isometry group of the extremal solution. Hence, we have infinite number of
zero modes associated with this breaking. These modes are the tensor modes,
$l=0$ vector modes and $l=1$ vector modes Banerjee:2010qc ; Banerjee:2011jp ;
Sen:2012kpz . Now we will discuss their fate in presence of small temperature
correction to the background.

#### Tensor modes on near-extremal background

As discussed above, the tensor modes are associated with large diffeomorphisms
on AdS2. These fluctuations are asymptotically AdS2, following a particular
falloff behavior near the boundary located at large radial distance
$\eta=\eta_{0}$. As a result, the corresponding Ricci scalars are well
approximated to the negative constant value of the extremal AdS2 geometry. In
fact, deviations from this value are much suppressed in the radial coordiante
$\text{e}^{-\eta}$. The large diffeomorphisms preserve these boundary
behaviors, so that they are asymptotic symmetries.

The near-extremal background is not asymptotically AdS2, which can be realized
from the expression of Ricci scalar,

$\displaystyle R=-\frac{2}{Q^{2}}+\frac{24\pi
T}{Q}\cosh{\eta}+\mathcal{O}(T)^{2}.$ (52)

In fact, near the boundary the linear-order temperature correction grows.
Hence, the asymptotic symmetries are lost in presence of temperature. As a
result, there will be no zero modes in this sector and we conclude that the
tensor modes will become slightly non-zero modes on the near-extremal
background. This can be thought of as a consequence of the way the near-
extremal solution was constructed by introducing additional mass above the
extremal mass.

####  $U(1)$ vector modes on near-extremal background

These fluctuations are generated by large gauge transformations, which are
symmetries on the extremal background. Since these modes are pure gauge, their
corresponding field strengths are zero. Therefore, the flux in presence of
these fluctuations is equal to the flux of the extremal background i.e.
proportional to the electric charge of the extremal black hole. While
constructing the near-extremal solution, we kept the electric charge fixed.
From the near-horizon perspective, we find that the flux of the near-extremal
solution is still equal to the flux of the extremal solution,

$\displaystyle F_{U(1)}=\int d\psi d\varphi\sqrt{g}F^{\eta\theta}\propto
Q+\mathcal{O}(T^{2})$ (53)

Note that, if we had considered the near-horizon geometry till deviations of
order $T^{2}$, the flux should be corrected at $\mathcal{O}(T^{3})$ since the
charge was kept fixed in the full solution.

Therefore we find that there is an infinite number of field configurations
having the same charge as the extremal and near-extremal background. This
implies that the large $U(1)$ gauge transformations are still symmetries of
the background. Hence, the associated modes are still zero modes.

####  $SO(3)$ vector modes on near-extremal background

To understand the fate of the cross-component fluctuations $h_{\mu i}$, we
will first put them into the canonical form of Kaluza-Klein ansatz described
in Iliesiu:2020qvm . Here $\mu$ refers to the indices of AdS2 and $i$ labels
the $S^{2}$ indices. The form of these fluctuations is given by,

$\displaystyle h_{\mu i}\equiv v_{\mu}^{m}\xi^{m}_{i}=\partial_{\mu}\Phi\
\xi^{m}_{i}.$ (54)

Here $\xi^{m}_{i}$ are the vectors on S2 that generate $SO(3)$ algebra for
$m=0,\pm 1$. $\Phi$ is the large gauge transformation parameter having a
discrete label, that we have suppressed here. On a spherically symmetric
background, these fluctuations can be expressed as,

$\displaystyle ds^{2}$
$\displaystyle=g_{\mu\nu}dx^{\mu}dx^{\nu}+g_{ij}dx^{i}dx^{j}+2h_{\mu
i}dx^{\mu}dx^{j}$ (55)
$\displaystyle=(g_{\mu\nu}-X(\eta)\partial_{\mu}\Phi\partial_{\nu}\Phi)+g_{ij}(dx^{i}+V^{m}\xi_{m}^{i})(dx^{j}+V^{m}\xi_{m}^{j})$
(56)

$X(\eta)=\sum_{m,p}g_{ij}\xi^{i}_{m}\xi^{j}_{m}$ can be computed from the
knowledge of the background. The reason behind considering this ansatz is that
now the fields $V=V^{m}\xi_{m}\equiv
v^{m}_{\mu}dx^{\mu}\xi^{i}_{m}\partial_{i}$ can be realized as $SO(3)$-valued
gauge fields living on a 2D geometry specified by metric
$\tilde{g}_{\mu\nu}=g_{\mu\nu}-X(\eta)\partial_{\mu}\Phi\partial_{\nu}\Phi$.

The field strength $H=dV-V\wedge V$ corresponding to the $SO(3)$ gauge field
is again zero. Hence in presence of these fluctuations, the value of flux
should not change from the value of flux of the spherically symmetric
background. The question now boils down to whether the modified metric
$\tilde{g}$ can be well approximated to the actual background $g$ from an
asymptotic point of view. Analyzing the Ricci scalar, it can be shown that
when the background $g$ is extremal, the modified 2D metric is still
asymptotically AdS2. Hence in this way, it can be argued that these
fluctuations correspond to certain asymptotic symmetries. But when the
background is near-extremal, the asymptotic AdS2 structure is lost, as shown
earlier. Therefore, this analysis shows that the fluctuations generated by
large $SO(3)$ transformations are no longer symmetries in presence of
temperature and there are no associated zero modes. The $l=1$ vector modes are
then expected to be slightly non-zero modes on the near-extremal background.

Therefore, we have shown that out of the extremal bosonic zero modes, only the
tensor and $l=1$ vector modes i.e. all the metric zero modes get promoted to
slightly non-zero modes on the near-extremal background. The $U(1)$ vector
modes are still zero modes of the black hole. This is in agreement with the
direct eigenvalue computations of Banerjee:2023quv . This result can be traced
back to the construction of the near-extremal solution by adding a small mass
above extremality while the electric charge was held fixed. Therefore,
whenever we keep the charges corresponding to the gauge fields fixed, the
corresponding zero modes do not get lifted even in the presence of
temperature.

#### Coefficient of logarithmic terms in Einstein-Maxwell sector

Since the metric zero modes are lifted, we would like to compute the
corresponding trace of the heat kernel which in turn gives the number of such
modes. As shown in Sen:2012kpz , for these modes $\beta_{m}=2$, where we are
denoting the zero mode label $r$ by $m$ to indicate metric zero modes. The
trace formula gives

$\displaystyle K^{m}(0)=\frac{3}{4\pi^{2}Q^{4}}.$ (57)

Using (44), we find the logarithmic corrections coming from the bosonic sector
to be,

$\displaystyle\log Z\sim(c_{0}+9)\log Q+3\log T.$ (58)

This dependence exactly matches with the explicit eigenvalue computations of
Banerjee:2023quv .

###  4.2 Full logarithmic corrections

The extremal black hole solution is half-BPS i.e. it preserves half of the
supersymmetry. The preserved supersymmetry gets an infinite extension near the
boundary, which in turn results in an infinite number of Goldstone modes.
Since the near-extremal black hole has mass above the BPS limit, it does not
preserve the supersymmetry that was earlier present in the extremal
background. This uplifts the degeneracy of the zero modes. Hence we consider
their contribution in the slightly non-zero part. For these modes,
$\beta_{f}=3$ and the trace of the heat kernel is given as,

$\displaystyle K^{f}(0)=-\frac{1}{2\pi^{2}Q^{2}}.$ (59)

The fermionic contribution to the logarithmic corrections is given from (44),

$\displaystyle\log Z\sim(c^{f}_{0}-12)\log Q-4\log T.$ (60)

Combining the contributions of (58) and (60), we get the full logarithmic
corrections for a near-extremal black hole in pure $\mathcal{N}=2$
supergravity theory as follows,

$\displaystyle\log Z\sim(c_{\text{ext}}-3)\log Q-\log T.$ (61)

Here $c_{\text{ext}}=c_{0}+c^{f}_{0}$ is the coefficient of the logarithmic
term for the extremal black hole.

Within the regime of validity of the near-extremal partition function, the
density of states can be computed from the inverse Laplace transform of the
partition function,

$\displaystyle\rho(E)=\text{e}^{S_{0}}\int d\beta\text{e}^{\beta
E+\frac{S_{1}}{\beta}}Z(\beta).$ (62)

Here, we have included the saddle point contribution to the partition
function, where $S_{0}=16\pi^{2}Q^{2}$ denotes the extremal part and
$S_{1}=32\pi^{3}Q^{3}$ signifies the near-extremal correction. The saddle
point contribution can be computed using the standard Gibbons-Hawking-York
prescription as described in Banerjee:2023quv . $E$ denotes the energy above
extremality. $Z(\beta)$ denotes the contribution to the one-loop partition
function coming from the logarithmic corrections. Substituting
$Z(\beta)=Q^{c_{\text{ext}}-3}\beta$, we get the expression for the density of
states as,

$\displaystyle\rho(E)\sim
Q^{c_{\text{ext}}+3}\leftindex_{0}F_{1}(3;32\pi^{3}Q^{3}).$ (63)

The small $E$ expansion of the same goes as,

$\rho(E)\sim\text{e}^{S_{0}}Q^{c_{\text{ext}}+3}\left[1+\frac{32}{3}(\pi
Q)^{3}E+\mathcal{O}(E^{2})\right].$ (64)

Hence, the logarithmic correction to entropy will be given by the leading
term, whereas the subleading terms will give a polynomial contribution. For
the $\mathcal{N}=2$ theory, the half-BPS extremal solution has a coefficient
$c_{\text{ext}}=\frac{23}{12}$ for the logarithmic contribution Sen:2012kpz .
Thus for the near-extremal solution that we are considering, we have
$Z(\beta)\sim Q^{-\frac{13}{12}}\beta$. Corresponding to this, we obtain the
density of states as follows.

$\displaystyle\rho(E)\sim\text{e}^{S_{0}}Q^{\frac{59}{12}}\left[1+\frac{32}{3}(\pi
Q)^{3}E+\mathcal{O}(E^{2})\right].$ (65)

This result is valid in the following regime of energy above extremality owing
to (45),

$\displaystyle y\left(\frac{\epsilon}{Q^{2}}\right)\ll QE\ll 1.$ (66)

Thus the $\log Q$ correction to the entropy of a near-extremal, non-rotating
black hole in $\mathcal{N}=2$ supergravity is given by,

$\Delta S_{\log}=\frac{59}{12}\log Q.$ (67)

Our result (61) agrees with the findings of Iliesiu:2022onk  for a particular
saddle point.

##  5 $\mathcal{N}=4,8$ Supergravity

In this section, we will discuss the logarithmic corrections to the entropy of
non-rotating, near-extremal black holes in $\mathcal{N}=4,8$ supergravity
theories. We will consider the near-extremal solutions to be small temperature
deviations of the $\frac{1}{4}$-BPS and $\frac{1}{8}$-BPS solutions. As
discussed earlier, the extremal solutions are parametrized by multiple
charges, unlike the $\mathcal{N}=2$ case. All these charges scale uniformly
with the horizon size $a$. Thus we will consider a near-extremal solution
having temperature such that $aT\ll 1$. In this temperature regime, the near-
horizon geometry can be expressed as in (6).

Hence, the analysis of section 3 can be carried out similarly by replacing the
charge parameter $Q$ by the extremal horizon size $a$. As discussed, the
coefficient of $\log a$ contribution from fully non-zero and fully zero modes
should remain the same as the corresponding extremal results presented in
Banerjee:2010qc ; Banerjee:2011jp . In order to find the complete logarithmic
contribution, we focus on the slightly non-zero mode sector.

We begin with the zero modes of the extremal solutions in these theories. The
$\mathcal{N}=4$ theory consists of gravity and matter multiplets. The matter
multiplet is comprised of $U(1)$ gauge fields, scalars and spin-$1/2$ fields.
Thus zero modes only appear from the gauge fields Banerjee:2010qc . The
gravity multiplet has metric and Rarita-Schwinger fields which coincide with
the $\mathcal{N}=2$ theory field content such that the zero mode structure of
this sector is also identical. In addition, there are some $U(1)$ gauge fields
that give rise to zero modes Banerjee:2011jp . The $\mathcal{N}=8$ theory
contains additional gauge fields and scalars compared to the $\mathcal{N}=4$
field content Banerjee:2011jp .

Therefore, the considerations of Banerjee:2010qc ; Banerjee:2011jp  show that
the metric and fermionic zero mode spectra of $\mathcal{N}=4,8$ theories are
identical to that of the $\mathcal{N}=2$ theory Sen:2012kpz . The only
difference appears due to additional gauge field zero modes. Since we are
keeping the charges corresponding to the gauge fields to be fixed to their
extremal values, these gauge field zero modes should remain degenerate even in
the presence of a non-zero temperature. Thus although the extremal logarithmic
corrections are different in these theories, the slightly non-zero mode
contribution remains the same. The density of states, and hence the entropy
correction again has the form of (64), with a coefficient $c_{\text{ext}}+3$
for the logarithmic correction. Here, $c_{\text{ext}}$ is to be substituted by
the appropriate value of logarithmic correction coefficient for the
corresponding extremal solutions in $\mathcal{N}=4,8$ theories Banerjee:2010qc
; Banerjee:2011jp .

##  6 Discussions

In this paper, we have constructed a methodology to suitably modify the heat
kernel prescription to compute the logarithmic corrections for near-extremal
black hole entropy. Since the usual methods for (non)-extremal black holes
cannot be directly applied in the near-extreme regime, our work provides an
important framework to study the logarithmic corrections for these black
holes. Unlike (non)-extremal black holes, the parameters of a near-extremal
black hole do not scale uniformly. Using our method, we are able to suitably
separate the contributions coming from inverse temperature and the charges. We
show that such a separation depends on appropriately extracting the scale
dependence of the contribution to the heat kernel coming from slightly non-
zero modes. These modes are zero modes of the extremal black hole that become
slightly non-degenerate on the near-extremal background. We show that our
computations are valid in a regime where the temperature parameter not only
has an upper cutoff in terms of the charges, but also a lower limit set by the
UV cutoff of the theory. The lower cutoff appears to be a limitation of the
current techniques. The method can be applied to compute the logarithmic
correction to the entropy of a near-extremal solution in any arbitrary theory
of gravity.

We also show that it is possible to correctly obtain the extremal black hole
result as a limit of the near-extremal black hole computation. The limit
should be taken systematically at the level of the Gaussian integrals of
slightly non-zero fluctuations as long as the near-extremal results are not
regularized. Such non-zero temperature regularization is typically performed
in (super)-Schwarzian theories. In the heat kernel formalism, the lower limit
of the Schwinger parameter integration is rescaled by a non-zero temperature
and this limit cannot be set strictly to zero. Such regularizations or
rescalings explicitly impose a non-zero temperature condition. Thus it is not
appropriate to take a zero temperature limit of the final result. This proves
that the logarithmic corrections computed from extremal near-horizon geometry
are robust results in any arbitrary theory. Thus there is no apparent
contradiction between the extremal and near-extremal logarithmic corrections
as long as an ill-defined zero temperature limit of the near-extremal result
is not taken.

Our current analysis can be naturally extended to higher dimensional
spherically symmetric near-extremal black holes as well. The scaling property
of the eigenvalues of the kinetic operator remains the same. Thus the generic
expression (44) for the partition function should remain to hold. Depending on
the theories in consideration, the coefficient of the logarithmic correction
will change due to the count of slightly non-zero modes. Since the extremal
black hole solution always has an AdS2 factor, the large diffeomorphism
symmetries will always get lifted in the presence of temperature. The AdS2
factor is present even for rotating black holes such that this argument again
goes through. These black holes do not have rotational symmetries at
extremality, meaning the only slightly non-zero modes coming from the metric
will be the large diffeomorphisms of AdS2. Implementing the logic of section
4.1, we can conclude that the logarithm of temperature correction in the
logarithm of partition function for 4D Kerr and Kerr-Newman black holes will
have a coefficient $\frac{3}{2}$. This is in agreement with the results found
in the recent works Kapec:2023ruw ; Rakic:2023vhv .

Using the modified heat-kernel method we compute the logarithmic corrections
to the entropy for near-extremal, non-rotating black hole solutions in 4D
Einstein-Maxwell theory and in $\mathcal{N}=2,4,8$ supergravity theories. We
find a perfect agreement of our result with the existing one in the literature
for the Einstein-Maxwell theory. For the supergravity theories, the
logarithmic contributions agree with that of Iliesiu:2020qvm ; Iliesiu:2022onk
when they consider only a particular black hole saddle. However as per the
results of Iliesiu:2020qvm ; Iliesiu:2022onk , which is derived from an
effective theory prescription of these black holes, the partition function of
each saddle diverges at $T=0$ and they propose to regulate the same by taking
contributions from an infinite number of saddle points, that modifies the net
logarithmic contributions to the entropy. We essentially differ from them in
this interpretation. As elaborated in section 3.4, a zero temperature limit
cannot be taken from the regularized near-extremal result. Thus there is no
apparent divergence in the result and a regularization is not required.
However, it would be interesting to compare the computations of
Iliesiu:2020qvm ; Iliesiu:2022onk  with a direct 4D Euclidean action
computations vide Banerjee:2023quv  for the same systems. One may hope to try
to understand the implications, if any, of the sum over saddle points of the
effective theory in the 4D picture in the context of the black hole solutions
under consideration. This could help us resolve the apparent disagreement over
interpretation that currently seems to be there. We leave this for our future
study.

###### Acknowledgements.

We are extremely grateful to Ashoke Sen for discussions and his comments on
the work. MS would like to thank ICTS, Bengaluru for their hospitality during
an important stage of the work. MS would also like to thank Arindam
Bhattacharjee and Rajesh Kumar Gupta for useful discussions. The work of NB is
partially supported by SERB POWER grant. Finally, we are thankful to the
people of India for their generous support towards fundamental research.

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