††institutetext: Department of Physics,
Indian Institute of Technology (Indian School of Mines),
Dhanbad, Jharkhand-826004, India

# Logarithmic corrections to black hole entropy in matter coupled
$\mathcal{N}\geq 1$ Einstein-Maxwell supergravity

Sudip Karan  and Binata Panda<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We calculate the first three Seeley-DeWitt coefficients for fluctuation of the
massless fields of a $\mathcal{N}=2$ Einstein-Maxwell supergravity theory
(EMSGT) distributed into different multiplets in $d=4$ space-time dimensions.
By utilizing the Seeley-DeWitt data in the quantum entropy function formalism,
we then obtain the logarithmic correction contribution of individual
multiplets to the entropy of extremal Kerr-Newman family of black holes. Our
results allow us to find the logarithmic entropy corrections for the extremal
black holes in a fully matter coupled $\mathcal{N}=2,d=4$ EMSGT, in a
particular class of $\mathcal{N}=1,d=4$ EMSGT as consistent decomposition of
$\mathcal{N}=2$ multiplets ($\mathcal{N}=2\to\mathcal{N}=1$) and in
$\mathcal{N}\geq 3,d=4$ EMSGTs by decomposing them into $\mathcal{N}=2$
multiplets ($\mathcal{N}\geq 3\to\mathcal{N}=2$). For completeness, we also
obtain logarithmic entropy correction results for the non-extremal Kerr-Newman
black holes in the matter coupled $\mathcal{N}\geq 1,d=4$ EMSGTs by employing
the same Seeley-DeWitt data into a different Euclidean gravity approach
developed in Sen:2013ns .

##  1 Introduction

In gravity theories, black holes are awarded an intrinsic entropy given by the
eminent semi-classical Bekenstein-Hawking entropy formula Bekenstein:1973jb ;
Hawking:1975sh . To test the visibility of any putative quantum theory of
gravity, such as string theory, we need to investigate black holes, especially
their entropies from both macroscopic and microscopic sides. On the
macroscopic side (or gravity side), black holes are described as classical
solutions to some low-energy effective field theory of gravity where the
Bekenstein-Hawking entropy formula receives classical higher derivative
corrections ($\alpha^{\prime}$) and quantum or string loop corrections
($g_{s}$). The higher derivative corrections are encoded by the familiar Wald
formula Wald:1993rw . The most general quantum corrected form of the
Bekenstein-Hawking entropy can be obtained as a series expansion in the
inverse power of a large horizon area $\mathcal{A}_{H}$ (the large-charge
limit),111The fundamental constants like the Boltzmann constant ($k_{B}$),
reduced Planck’s constant ($\hbar$) and speed of light ($c$) are considered as
a unity.

$S_{\text{BH}}(\mathcal{A}_{H})=\frac{\mathcal{A}_{H}}{4G_{N}}+\mathcal{C}\thinspace\text{ln}\thinspace\left(\frac{\mathcal{A}_{H}}{G_{N}}\right)+\sum_{n=1}^{\infty}C_{n}{\mathcal{A}_{H}}^{-n+1},$
(1)

where $G_{N}$ is the gravitational constant and $\mathcal{C},C_{n}$ are
numerical constants that generally depend on different black hole parameters.
In the expansion (1), the leading part is the Bekenstein-Hawking entropy
$\frac{\mathcal{A}_{H}}{4G_{N}}$, while the subleading parts are quantum
corrections to it. The second term with the universal form proportional to
$\text{ln}\thinspace\frac{\mathcal{A}_{H}}{G_{N}}$ is the principal quantum
correction, called the logarithmic correction. This particular one-loop
correction222The n-loop quantum corrections to black hole entropy are of the
order $g_{s}\sim\mathcal{A}_{H}^{-n+1}$, where the one-loop correction is
proportional to $\mathcal{A}_{H}^{0},\text{ln}\thinspace\mathcal{A}_{H}$, etc.
can be computed using just the massless fluctuations (IR data) running in the
one-loop Banerjee:2011oo ; Banerjee:2011pp ; Sen:2012qq ; Sen:2012rr ;
Bhattacharyya:2012ss ; Chowdhury:2014np ; Gupta:2014ns ; Jeon:2017ij ;
Karan:2019sk ; Sen:2013ns ; Keeler:2014nn ; Charles:2015nn ; Larsen:2015nx ;
Castro:2018tg ; Ferrara:2012bp ; Banerjee:2020wbr and is independent of the UV
completion of the theory. Naturally, the logarithmic correction can serve as a
laboratory for testing any proposed quantum completion of the classical
gravity theory.

In the extremal (zero-temperature) limit, black holes cease to evaporate and
become a stable platform for investigating the logarithmic entropy correction
program. Often but not always, extremal black holes resemble BPS states in
supergravity (e.g., extremal Reissner-Nordström black holes in four-
dimensional supergravities Mohaupt:2000mj ) and add more stability for being
the lightest charged black holes. Remarkably, the full quantum entropy of an
extremal black hole can be obtained from just the near-horizon analysis
without knowing details of the entire black hole geometry. The near-horizon
geometry of every extremal black hole accommodates an $AdS_{2}$ part. Hence,
the rules of $AdS_{2}/CFT_{1}$ conjecture prescribe that the entropy of
extremal black holes is equal to the entropy calculated from the full string
partition function characterizing the near-horizon geometry. This prescription
is recognized as the famous quantum entropy function formalism Sen:2008wa ;
Sen:2009wb ; Sen:2009wc , which turns out to be a smart trick to compute
quantum corrections to the extremal black hole entropy. So far, quantum
entropy function formalism gained immense success by providing logarithmic
entropy corrections for many supersymmetric and non-supersymmetric extremal
black holes Banerjee:2011oo ; Banerjee:2011pp ; Sen:2012rr ; Sen:2012qq ;
Chowdhury:2014np ; Gupta:2014ns ; Bhattacharyya:2012ss ; Karan:2019sk ;
Jeon:2017ij ; Keeler:2014nn ; Ferrara:2012bp ; Banerjee:2020wbr ;
Larsen:2015nx . Readers are also referred to Solodukhin:1995na ;
Solodukhin:1995nb ; Fursaev:1995df ; Mavromatos:1996kc ; Mann:1996bi ;
Mann:1998hm ; Kaul:2000rk ; Carlip:2000nv ; Govindarajan:2001ee ; Gupta:2002bg
; Medved:2004eh ; Page:2005xp ; Banerjee:2008cf ; Banerjee:2009fz ;
Majhi:2009gi ; Cai:2010ua ; Aros:2010jb ; Solodukhin:2010pk for some earlier
techniques to obtain the logarithmic entropy corrections.

At present, it is impossible to compute the full quantum partition function of
any gravity theory. One can try to calculate it using supersymmetric
localization Dabholkar:2011ad ; Dabholkar:2013ad ; Gupta:2013rkg ;
Dabholkar:2015ad ; Murthy:2015sm ; Gupta:2015rkg ; Murthy:2016sm as well as
perturbatively using saddle point or loop expansion Banerjee:2011oo ;
Banerjee:2011pp ; Sen:2012rr ; Sen:2012qq ; Chowdhury:2014np ; Gupta:2014ns ;
Bhattacharyya:2012ss ; Karan:2019sk ; Jeon:2017ij ; Banerjee:2020wbr . In this
paper, we opt for the latter approach where the leading saddle point is
nothing but the classical black hole solution and the subleading loop-terms
around the saddle point serve as quantum corrections. Entropy computed on the
saddle point is the Bekenstein-Hawking entropy or Wald’s entropy if the higher
derivative corrections are included. At one-loop (next to saddle point),
filtering out only massless modes leads to the desired logarithmic entropy
correction to the classical black hole entropy. Necessarily, evaluation of the
logarithmic entropy corrections requires the analysis of one-loop quantum
effective action characterizing only massless fluctuations. The heat kernel
method can guide us from here by describing the one-loop effective action in
terms of heat kernel of the kinetic operator characterizing the fluctuations.
The heat kernel can be further expanded perturbatively in Seeley-DeWitt
coefficients Seeley:1966tt ; Seeley:1969uu ; DeWitt:1965ff ; DeWitt:1967gg ;
DeWitt:1967hh ; DeWitt:1967ii , which we can compute in terms of background
field and geometric invariants following a standard approach
Vassilevich:2003ll . Most of the other recognized methods333The current paper
only discusses the heat kernel methods that are more relevant and successful
in the logarithmic entropy correction calculations. However, in general, the
heat kernel can be evaluated by various other available perturbative expansion
methods, quasinormal-mode approach Denef:2010nc , group-theoretic method
David:2010xn ; Gopakumar:2011xi ; Lovrekovic:2016ni , etc. that compute the
heat kernel for black hole entropy corrections are either restricted to fixed
backgrounds (constant fields, presence of rotational symmetry, homogeneous
space-time, etc.) or involve some complex computations. The eigenfunction
expansion method employed in Banerjee:2011oo ; Banerjee:2011pp ; Sen:2012qq ;
Gupta:2014ns earns excellent success in this direction. However, that is
constrained to only the Bertotti-Robinson extremal near-horizon geometry
$AdS_{2}\times S^{2}$. This eigenfunction expansion method is further
customized in from of the on-shell Keeler:2014nn and off-shell Larsen:2015nx
methods. The hybrid method advocated in Charles:2015nn ; Castro:2018tg is
analogous to the standard Seeley-DeWitt computation method Vassilevich:2003ll
but modified with field redefinition. One needs specialized skills to redefine
fields in different gravity theories where it is also impossible to keep track
of individual field contributions. This, in turn, destroys the generality
within the hybrid method Charles:2015nn ; Castro:2018tg .

Supergravity theories are particular class of vacuums that can be realized as
the low-energy truncation of string theories compactified down to four space-
time dimensions. Logarithmic entropy corrections have been extensively
investigated in supergravity theories, which already have a well-established
microscopic counterpart within string theory Mandal:2010im ; Sen:2014nvt .
But, most of the popularly known examples Banerjee:2011oo ; Banerjee:2011pp ;
Sen:2012qq ; Gupta:2014ns ; Larsen:2015nx ; Keeler:2014nn ; Ferrara:2012bp of
logarithmic entropy corrections in supergravities are for extremal black holes
with non-rotating geometry. So far, not many macroscopic results are available
for the extremal rotating black holes in $\mathcal{N}\geq 1$ supergravities.
This is because the technical approaches used to analyze the one-loop
contribution to partition function are mostly best suited for extremal non-
rotating backgrounds with the near-horizon geometry $AdS_{2}\times S^{2}$.
However, there are some reports Sen:2012rr ; Bhattacharyya:2012ss of
logarithmic corrections for extremal rotating black holes in four-dimensional
non-supersymmetric Einstein-Maxwell theories. Especially in
Bhattacharyya:2012ss , the authors have utilized the standard Seeley-DeWitt
computation technique Vassilevich:2003ll into the framework of quantum entropy
function formalism that surprisingly provides a global platform to manage the
logarithmic correction results for extremal Kerr-Newman, Kerr and Reissner-
Nordström black holes. Motivated by this success, the current paper aims to
extend the logarithmic correction program for all types of extremal black
holes (rotating or non-rotating, charged or uncharged) in fully matter-coupled
$\mathcal{N}\geq 1$ supergravity theories. We also receive a strong hint from
the fact that both the quantum entropy function formalism Sen:2008wa ;
Sen:2009wb ; Sen:2009wc and the calculation technique Vassilevich:2003ll do
not at all rely on supersymmetry. As a way forward, we have already gained
some initial success by testing the global framework for extremal black holes
in ‘minimal’ (or pure) $\mathcal{N}=2$ Karan:2019sk and ‘non-minimal’
$\mathcal{N}=1$ Banerjee:2020wbr supergravity theories. But, both of them are
not fully matter-coupled, rather special cases.444The ‘minimal’
$\mathcal{N}=2$ contains only supergravity multiplet, while in the ‘non-
minimal’ $\mathcal{N}=1$ a vector multiplet is non-minimally coupled to the
supergravity multiplet. Note that similar logarithmic entropy corrections for
non-extremal black holes are also reported in Charles:2015nn ; Castro:2018tg
via the conventional Euclidean gravity approach Sen:2013ns .

The $\mathcal{N}=2,d=4$ Einstein-Maxwell supergravity theory (EMSGT) is
significant and arises as an IR limit of type-II string theory compactified on
a Calabi–Yau three-fold (e.g., see Grana:2006mg ; Freedman:2012xp ). In such a
theory, the backgrounds of a four-dimensional Einstein-Maxwell theory are
embedded into a $\mathcal{N}=2$ supergravity so that one can interpret the
Kerr-Newman family of black holes555The Kerr-Newman black holes are general
solutions to the Einstein-Maxwell theory of gravity Adamo:2014lk . as
solutions of the resultant $\mathcal{N}=2,d=4$ EMSGT Charles:2015nn ;
Karan:2019sk ; Sen:2012qq . A general $\mathcal{N}=2,d=4$ EMSGT contains four
massless multiplets (supergravity, gravitino, vector and hyper), and the
logarithmic correction contributions of these multiplets to the entropy of
extremal black holes can be extracted by analyzing quadratic field
fluctuations. As discussed, we particularly interested to do this by
calculating the Seeley-DeWitt coefficients. The corresponding calculations are
unwieldy but manageable if one proceeds systematically. Any motive to extend
the logarithmic correction program for the higher-extended EMSGTs will
gradually increase the technical complexities, requiring tremendous skills in
managing the Seeley-DeWitt trace calculations. But, the contents of
supergravities are such that they are always kinematically reducible to lower
supergravities. Based on this feature, the $\mathcal{N}=2,d=4$ EMSGT results
are found to be crucial for carrying out the logarithmic correction program of
the extremal Kerr-Newman family of black holes in all the matter coupled
$\mathcal{N}\geq 1,d=4$ EMSGTs. This is the central objective of this paper.

The prime technical results of this paper are recorded in two parts. In the
first part (see section 3), we calculate the Seeley-DeWitt coefficients of the
supergravity, gravitino, vector, and hyper multiplet of the
$\mathcal{N}=2,d=4$ EMSGT by analyzing quadratic fluctuations of their
content. The supergravity multiplet results are borrowed from our previous
work Karan:2019sk , while other multiplet results are freshly calculated by
employing the standard manual Vassilevich:2003ll . The calculated Seeley-
DeWitt coefficients agree with the results of both the eigenfunction expansion
method Sen:2012qq ; Keeler:2014nn and the hybrid field-redefinition approach
Charles:2015nn . We then make use of the Seeley-DeWitt data in the quantum
entropy function framework and derive the logarithmic correction contributions
of individual multiplets to the entropy of extremal Kerr-Newman, Kerr and
Reissner-Nordström black holes (see eqs. 50, 68, 81 and 92). In the second
part (refer section 4), we utilize the $\mathcal{N}=2$ multiplet data to
obtain logarithmic correction results for the extremal Kerr-Newman family of
black holes in three different cases –  (i) a matter coupled
$\mathcal{N}=2,d=4$ EMSGT by coupling the supergravity multiplet with any
arbitrary number of vector and hyper multiplets,  (ii) a particular class of
matter coupled $\mathcal{N}=1,d=4$ EMSGT obtained as consistent decomposition
of $\mathcal{N}=2$ multiplets, and  (iii) the matter coupled $\mathcal{N}\geq
3,d=4$ EMSGTs by decomposing them into $\mathcal{N}=2$ multiplets.  The
results for all three cases are recorded in eqs. 94, 96 and 100, respectively.
The entropy corrections for Reissner-Nordström black holes match with the
results available in Sen:2012qq ; Banerjee:2011pp ; Keeler:2014nn ;
Ferrara:2012bp , while the Kerr-Newman and Kerr corrections are new reports to
the literature. We derive a combined and compact logarithmic correction
formula (101) for the extremal Kerr-Newman black holes in the matter coupled
$\mathcal{N}\geq 2,d=4$ EMSGTs. In comparison, we also extract the logarithmic
entropy corrections for non-extremal Kerr-Newman family of black holes in the
matter coupled $\mathcal{N}\geq 1,d=4$ EMSGTs using the same $\mathcal{N}=2$
Seeley-DeWitt data in the Euclidean gravity approach Sen:2013ns . The non-
extremal results are recorded in eqs. 112, 113 and 114, where the
$\mathcal{N}\geq 2$ part is a reproduction of Charles:2015nn and the
$\mathcal{N}=1$ part is new results.

The rest of this paper is planned in the following way. Section 2 reviews a
general framework of evaluating logarithmic correction to the entropy of
extremal Kerr-Newman family of black holes using Seeley-DeWitt coefficients in
the quantum entropy function formalism. In section 3, we analyze quadratic
fluctuations in the multiplets of $\mathcal{N}=2,d=4$ EMSGT and find their
logarithmic correction contributions to the entropy of extremal black holes by
calculating corresponding Seeley-DeWitt coefficients. In section 4, we obtain
the logarithmic entropy correction results for the extremal Kerr-Newman family
of black holes in the matter coupled $\mathcal{N}\geq 1,d=4$ EMSGTs. We also
revisit the same for the non-extremal Kerr-Newman family of black holes via
the Euclidean gravity approach Sen:2013ns . We end up discussing some
highlights and concluding remarks about the work presented in this paper.
Appendix A records the equations of motion and identities used in this paper.
In appendix B, we present a brief flow chart of comparatively lengthy trace
calculations in the non-minimally coupled sectors of $\mathcal{N}=2$
gravitino, vector and hyper multiplets.

##  2 The general framework

In this section, we start by briefly reviewing the basic algorithm for
computing logarithmic correction to the entropy of extremal black holes
following the quantum entropy function formalism Sen:2008wa ; Sen:2009wb ;
Sen:2009wc . We then analyze the relation between logarithmic entropy
correction and the one-loop effective action for massless fields propagating
in the near-horizon geometry of extremal black holes. We also shed light on
the evaluation of the one-loop effective action from the knowledge of Seeley-
DeWitt coefficients for these massless fluctuations. We outline a standard and
efficient approach for calculating the Seeley-DeWitt coefficients following
the manual Vassilevich:2003ll , where the coefficients are demonstrated in
terms of different background fields and geometric invariants. Finally, we set
up for the near-horizon extremal limits of the Kerr-Newman, Kerr and Reissner-
Nordström black holes after clearly highlighting the key working formula of
the logarithmic entropy corrections.

###  2.1 Heat kernel treatment of one-loop effective action and Seeley-DeWitt
coefficients

We consider a 4D theory of gravity characterized by the Euclideanized action
$\mathcal{S}$ (with Lagrangian density $\mathscr{L}$),

$\mathcal{S}[g,\xi]=\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace
g}\thinspace\mathscr{L},$ (2)

where $\\{\xi\\}$ is the matter field configuration propagating through a
geometry represented by the metric $g$. One can choose ($\bar{g},\bar{\xi}$)
as an arbitrary solution to the classical equations of motion of the theory
and fluctuate the metric $g$ and fields $\\{\xi\\}$ around the classical
background ($\bar{g},\bar{\xi}$) for small quantum fluctuations
$\tilde{\xi}_{m}=\\{\tilde{{g}},\tilde{\xi}\\}$,

${g}=\bar{g}+\tilde{{g}},\enspace\xi=\bar{\xi}+\tilde{\xi}.$ (3)

If $\Lambda$ is the differential operator that governs the quadratic
fluctuation of the action (2), then we can express the quadratic fluctuated
action form $\delta^{2}\mathcal{S}$ as

$\delta^{2}\mathcal{S}[\tilde{{g}},\tilde{\xi}]=\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\tilde{\xi}_{m}\Lambda\tilde{\xi}_{n},$
(4)

which necessarily evaluates the quantum corrected Euclidean one-loop effective
action $\mathcal{W}$ as Gibbons:1977ta ; Hawking:1978td ; Hawking:1977te ;
Avramidi:1994th ; Denardo:1982tb ; Peixoto:2001wx

$e^{-\mathcal{W}}=\int\mathcal{D}[\tilde{g},\tilde{\xi}]\exp(-\delta^{2}\mathcal{S}[\tilde{{g}},\tilde{\xi}])=(\text{det}\thinspace\Lambda)^{-\chi/2},$
(5)

where $\chi=+1$ for boson fluctuations and $\chi=-1$ for fermion fluctuations.
$\mathcal{D}[\tilde{g},\tilde{\xi}]$ suggests that the integration (5) is
acting over all possible fluctuations of the metric and matter fields. Let us
now assume that $\\{f_{i}\\}$ and $\\{\lambda_{i}\\}$ are respectively the
complete basis of eigenfunctions and eigenvalues of the operator $\Lambda$
satisfying the eigenvalue equation $\Lambda f_{i}(x)=\lambda_{i}f_{i}(x)$ with
the normalization conditions,

$\begin{gathered}\sum_{i}f_{i}(x)f_{i}(y)=\delta^{4}(x,y),\\\
\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace
f_{i}(x)f_{j}(x)=\delta_{ij},\end{gathered}$ (6)

where $\delta^{4}(x,y)$ and $\delta_{ij}$ are appropriate delta functions. The
heat kernel $K(x,y;s)$ of the operator $\Lambda$ and its trace $D(s)$ (dubbed
as the heat trace) are respectively defined as Vassilevich:2003ll

$\begin{gathered}K(x,y;s)=\sum_{i}e^{-\lambda_{i}s}f_{i}(x)f_{i}(y),\\\
D(s)=\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace
K(x,x;s)=\sum_{i}e^{-\lambda_{i}s},\end{gathered}$ (7)

where $s$ is an auxiliary parameter (dimension of length square), called the
heat kernel parameter or proper time. At this point, the Schwinger-DeWitt
proper time representation Schwinger:1951sp ; DeWitt:1975ps allows us to
express the one-loop effective action $\mathcal{W}$ and its Lagrangian density
$\Delta\mathscr{L}_{\text{eff}}$ in terms of the heat trace and heat kernel
respectively as

$\begin{gathered}\mathcal{W}=-\frac{1}{2}\int_{\epsilon}^{\infty}\frac{\mathrm{d}s}{s}\chi
D(s),\\\
\Delta\mathscr{L}_{\text{eff}}=-\frac{1}{2}\int_{\epsilon}^{\infty}\frac{\mathrm{d}s}{s}\chi
K(x,x;s).\end{gathered}$ (8)

where $\epsilon$ is a cut-off ($\epsilon\sim{l_{p}}^{2}\sim G_{N}$) introduced
to regulate the UV divergence of $\mathcal{W}$. This is the central idea about
the heat kernel method, where one can evaluate the one-loop effective action
via heat kernel (precisely the trace of heat kernel) of the kinetic operator
$\Lambda$ without worrying about its eigenfunctions. Furthermore, the heat
kernel $K(x,y;s)$ is associated with a power-law asymptotic expansion for
small $s$, called the Seeley-DeWitt expansion,

$D(s)\cong\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\sum_{n=0}^{\infty}s^{n-2}a_{2n}(x),$
(9)

where $a_{2n}(x)$ are known as Seeley-DeWitt coefficientsSeeley:1966tt ;
Seeley:1969uu ; DeWitt:1965ff ; DeWitt:1967gg ; DeWitt:1967hh ; DeWitt:1967ii
. A standard computation approach of these coefficients is outlined in section
2.2.

It is also essential to define the zero modes of the operator $\Lambda$ in the
present heat kernel treatment. Among the eigenfunctions $f_{i}(x)$ of
$\Lambda$, there may be a particular group of modes for which the eigenvalues
$\lambda_{i}$ vanish, i.e., $\lambda_{i}=0$. These modes are recognized as
zero modes of the operator $\Lambda$ and denoted by the special notation
$f^{0}_{i}(x)$ with the eigenvalue equation,

$\Lambda f^{0}_{i}(x)=0.$ (10)

For all possible fluctuations of the operator $\Lambda$, the total number of
zero modes $n_{\text{zm}}$ can be defined as

$\begin{gathered}D(s)=\sum_{i}e^{-s\lambda_{i}}=\sideset{}{{}^{\prime}}{\sum}_{i\atop(\lambda_{i}\neq
0)}e^{-s\lambda_{i}}+n_{\text{zm}},\\\
n_{\text{zm}}=\sum_{i}\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace
f^{0}_{i}(x)f^{0}_{i}(x),\end{gathered}$ (11)

where $\sideset{}{{}^{\prime}}{\sum}$ represents summation over the non-zero
modes only.

###  2.2 A standard computation strategy for the Seeley-DeWitt coefficients

In this paper, our motive is to express the Seeley-DeWitt coefficients in
terms of different invariants induced from the background metric, fields and
their covariant derivatives present in a particular 4D gravity theory. To
fulfill this demand, we cast a standard approach following the manual
Vassilevich:2003ll .666For a detailed review, please refer to Karan:2019sk .
The present approach assumes that the quadratic order fluctuated action (4)
must be customized up to a total derivative for providing the kinetic
differential operator $\Lambda$ a minimal and Laplace-type form,

$\displaystyle\tilde{\xi}_{m}\Lambda^{mn}\tilde{\xi}_{n}=\pm\tilde{\xi}_{m}\big{(}D^{\rho}D_{\rho}I^{mn}+(N^{\rho}D_{\rho})^{mn}+P^{mn}\big{)}\tilde{\xi}_{n},$
(12)

where $m$ is a mixed index that includes both the tensor indices and the
labeling for any arbitrary fluctuation $\xi_{m}$; $D_{\rho}$ is the ordinary
covariant derivative with Christoffel and spin connections; $I$ is the
identity operator in fields space; $N^{\rho},P$ are arbitrary matrices
constructed from background fields. In any generic theory of gravity, the
fields are interacting and hence we need to redefine the covariant derivative
$D_{\rho}$ by embedding a gauge connection $\omega_{\rho}$,

$\mathcal{D}_{\rho}\tilde{\xi}_{m}=D_{\rho}\tilde{\xi}_{m}+{(\omega_{\rho})_{m}}^{n}\tilde{\xi}_{n}\quad\forall
m\neq n.$ (13)

Naturally, the connection $\omega_{\rho}$ vanishes for free or minimally-
coupled777A free or minimal-coupled fluctuation is only coupled to background
gravity via the $\sqrt{\text{det}\thinspace\bar{g}}$ term in the fluctuated
action form (4). fluctuations. The curvature corresponding to the new
covariant derivative $\mathcal{D}_{\rho}$ is

$(\Omega_{\rho\sigma})^{mn}\equiv[\mathcal{D}_{\rho},\mathcal{D}_{\sigma}]^{mn}=[D_{\rho},D_{\sigma}]^{mn}+{D_{[\rho}\omega_{\sigma]}}^{mn}+[\omega_{\rho},\omega_{\sigma}]^{mn}.$
(14)

The schematic (12) therefore reduces to a more generalized and standard form,

$\tilde{\xi}_{m}\Lambda^{mn}\tilde{\xi}_{n}=\pm\tilde{\xi}_{m}\big{(}\mathcal{D}^{\rho}\mathcal{D}_{\rho}I^{mn}+E^{mn}\big{)}\tilde{\xi}_{n},$
(15)

where

$(\omega_{\rho})^{mn}=\frac{1}{2}(N_{\rho})^{mn},\thinspace
E^{mn}=P^{mn}-(D^{\rho}\omega_{\rho})^{mn}-(\omega^{\rho})^{mp}{(\omega_{\rho})_{p}}^{n}.$
(16)

With this structure, the first three Seeley-DeWitt coefficients for the
quadratic fluctuations of any arbitrary 4D theory are obtained by computing
$\text{tr}(I)$, $\text{tr}(E)$, $\text{tr}(E^{2})$ and
$\text{tr}(\Omega_{\rho\sigma}\Omega^{\rho\sigma})$ in the following formulae
Vassilevich:2003ll

$\begin{split}(4\pi)^{2}a_{0}(x)&=\chi\,\mathrm{tr}(I),\\\
(4\pi)^{2}a_{2}(x)&=\frac{\chi}{6}\,\mathrm{tr}(RI+6E),\\\
(4\pi)^{2}a_{4}(x)&=\frac{\chi}{360}\mathrm{tr}\Big{(}60RE+180E^{2}+30\Omega_{\rho\sigma}\Omega^{\rho\sigma}\\\
&\quad+(5R^{2}+2R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-2R_{\mu\nu}R^{\mu\nu})I\Big{)}.\end{split}$
(17)

where $R$, $R_{\mu\nu}$ and $R_{\mu\nu\rho\sigma}$ are respectively the Ricci
scalar, Ricci tensor and Riemann tensor corresponding to the background metric
$\bar{g}_{\mu\nu}$. It is essential to mention that the quadratic fluctuated
actions $\eqref{hk3}$ of fermionic fluctuations are generally controlled by
first-order operators (e.g., see Karan:2018ac ). But the special technique of
Sen:2012qq ; Peixoto:2001wx allows us to revise the one-loop determinant of
the first-order fermionic operators, denoted by $\not{D}$, via the following
way

$\text{ln det}\thinspace\not{D}=\text{ln
det}\thinspace\not{D}^{\dagger}=\frac{1}{2}\thinspace\text{ln
det}\thinspace\not{D}^{\dagger}\not{D},$ (18)

where $\not{D}^{\dagger}\not{D}$ can be structured further to achieve the
typical Laplace-type form (12). The treatment (18) will double up the
integration variables for a complex Dirac spinor operator, which in return
compensates the additional 1/2 factor. Consequently, we can state that the
heat kernel treatment primarily considers any fermion as Dirac. But for a real
Majorana spinor operator, one needs to include the extra 1/2 factor in the
Seeley-DeWitt formulae (17). Weyl spinors can not fit into this prescription
due to the presence of both left and right chiral states, providing
$\text{det}(\not{D}^{\dagger}\not{D})\neq(\text{det}\thinspace\not{D})^{2}\thinspace\text{or}\thinspace(\text{det}\thinspace\not{D}^{\dagger})^{2}$.
Therefore, the exclusive values of $\chi$ in the formulae (17) for different
fields are

$\chi=\begin{cases}-1&\text{for Dirac spinors},\\\ -\frac{1}{2}&\text{for
Majorana spinors},\\\ +1&\text{for bosons and scalars}.\end{cases}$ (19)

Again, the ghost fields that appear in gauge-fixing of the theory follow the
reverse of above.

In this paper, we will mainly deal with Einstein-Maxwell supergravity theories
that characterize a $R=0$ manifold (see eq. 47). For scalar ($\phi$), vector
($a_{\mu}$), spin-1/2 ($\lambda$) and spin-3/2 ($\psi_{\mu}$) field
fluctuations, the covariant derivative commutation relations have the
following standard forms

$\displaystyle[D_{\rho},D_{\sigma}]\phi$ $\displaystyle=0,$ (20a)
$\displaystyle[D_{\rho},D_{\sigma}]a_{\mu}$
$\displaystyle=-{R^{\nu}}_{\mu\rho\sigma}a_{\nu},$ (20b)
$\displaystyle[D_{\rho},D_{\sigma}]\lambda$
$\displaystyle=\frac{1}{8}[\gamma^{\alpha},\gamma^{\beta}]R_{\rho\sigma\alpha\beta}\lambda,$
(20c) $\displaystyle[D_{\rho},D_{\sigma}]\psi_{\mu}$
$\displaystyle=-{R^{\nu}}_{\mu\rho\sigma}\psi_{\nu}+\frac{1}{8}[\gamma^{\alpha},\gamma^{\beta}]R_{\rho\sigma\alpha\beta}\psi_{\mu}.$
(20d)

With all these data and following the standard manual provided in this
section, the Seeley-DeWitt contributions for the free or minimally-coupled
field fluctuations are relatively less complicated and straightforward
compute.888Readers are referred to Karan:2018ac for revising the free spin-1/2
and spin-3/2 calculations. The results for the first three Seeley-DeWitt
coefficients can be gathered and combined into a single frame as

$\begin{split}(4\pi)^{2}{a_{0}}^{\text{min-
coupled}}(x)&=(n_{0}+2n_{1}-2n_{1/2}-2n_{3/2}),\\\
(4\pi)^{2}{a_{2}}^{\text{min-coupled}}(x)&=0,\\\ (4\pi)^{2}{a_{4}}^{\text{min-
coupled}}(x)&=\frac{1}{360}\Big{\\{}\big{(}2n_{0}-26n_{1}+\frac{7}{2}n_{1/2}-\frac{233}{2}n_{3/2}\big{)}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\\\
&\qquad+\big{(}-2n_{0}+176n_{1}+4n_{1/2}+4n_{3/2}\big{)}R_{\mu\nu}R^{\mu\nu}\Big{\\}},\end{split}$
(21)

where $n_{0}$, $n_{1}$, $n_{1/2}$ and $n_{3/2}$ are respectively the
multiplicities of the real massless free scalar, vector, spin-1/2 Majorana and
spin-3/2 Majorana field fluctuations. The results (21) play crucial side roles
in the calculation of section 3.

###  2.3 Seeley-DeWitt coefficient “recipe” for logarithmic correction to the
entropy of extremal Kerr-Newman black holes

For extremal black holes, the prescription of quantum entropy function
formalism Sen:2008wa ; Sen:2009wb ; Sen:2009wc is a comfortable and efficient
method to determine the quantum corrections to the black hole entropy. With
the help of this formalism, our aim is to provide a clear structure for
computing logarithmic correction to the entropy of extremal Kerr-Newman family
of black holes. The Seeley-DeWitt coefficients play a vital role in these
logarithmic correction computations.

####  2.3.1 General set-up via the quantum entropy function formalism

The basic algorithm of calculating logarithmic entropy corrections for
extremal black holes via the quantum entropy function formalism is presented
as follows –

  1. 1.

The entropy of extremal black holes can be computed from their near-horizon
analysis and we do not need to know the details of full space-time. The near-
horizon geometry of every extremal black hole can be reshaped into the form
$AdS_{2}\times\mathcal{K}$,

${\mathrm{d}s^{2}}_{\text{horizon}}=\bar{g}_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=\mathrm{d}s^{2}_{AdS_{2}}+\mathrm{d}s^{2}_{\mathcal{K}},$
(22)

where $\mathcal{K}$ is a space described by the angular coordinates or compact
coordinates and $\bar{g}_{\mu\nu}$ is the metric describing the extremal near-
horizon geometry with coordinates $x^{\mu}$.

  2. 2.

The Euclidean string partition function
$\mathcal{Z}_{AdS_{2}\times\mathcal{K}}$ describing the extremal near-horizon
geometry is defined as

$\mathcal{Z}_{AdS_{2}\times\mathcal{K}}=\int\mathcal{D}[\bar{g},\xi]\exp\left(-\mathcal{S}[\bar{g},\xi]\right),$
(23)

where $\xi$ collectively denotes all the string fields and $\mathcal{S}$ is
the action with Euclideanized time ($t\to-i\theta$). The string fields $\xi$
asymptotically approach to the classical near-horizon geometry (22), which
serves as a boundary condition for this formalism. The boundary condition
modifies the partition function (23) into a standard quantum entropy function
form by accommodating a Gibbons-Hawking type boundary term
$\text{exp}\left[-iq_{k}\oint d\theta\mathcal{A}_{\theta}^{(k)}\right]$, where
$\mathcal{A}_{\theta}^{(k)}$ are gauge fields on $AdS_{2}$ and $q_{k}$ are the
corresponding charges Sen:2008wa ; Sen:2009wb ; Sen:2009wc ; Sen:2012qq ;
Sen:2012rr .

  3. 3.

$AdS_{2}/CFT_{1}$ conjecture suggests that the string partition function in
$AdS_{2}$ is equal to the partition function of the boundary $CFT_{1}$, i.e.,
$\mathcal{Z}_{AdS_{2}\times\mathcal{K}}=\mathcal{Z}_{CFT_{1}}$. The dual
$CFT_{1}$ lives at the one-dimensional boundary of $AdS_{2}$ and can be
treated as the IR limit of the quantum mechanics associated with the black
hole microstates. Therefore, the entropy of the extremal black hole is equal
to the entropy calculated from the full near-horizon partition function
$\mathcal{Z}_{AdS_{2}\times\mathcal{K}}$.

  4. 4.

$\mathcal{Z}_{AdS_{2}\times\mathcal{K}}$ is divergent due to the infinite
volume of $AdS_{2}$. But this divergence can be regulated by inserting an IR
cut-off, which limits the boundary of $AdS_{2}$ by a finite length $\ell$. On
the $CFT_{1}$ side (i.e., microscopic side) if we take $\ell\to\infty$ limit,
only the ground state will contribute:

$\lim_{\ell\to\infty}\mathcal{Z}_{CFT_{1}}=d_{\text{hor}}e^{-\ell E_{0}},$
(24)

where $d_{\text{hor}}$ and $E_{0}$ are respectively the degeneracy and energy
of the states of $CFT_{1}$. On the $AdS_{2}$ side (i.e., macroscopic or
gravity side), the trick is to choose appropriate coordinates so that the
partition function $\mathcal{Z}_{AdS_{2}\times\mathcal{K}}$ can be structured
in the form of $\mathcal{Z}_{CFT_{1}}$:

$\lim_{\ell\to\infty}\mathcal{Z}_{AdS_{2}\times\mathcal{K}}=\mathcal{Z}_{AdS_{2}\times\mathcal{K}}^{\text{finite}}\cdot
e^{-{C}\ell+\mathcal{O}(\ell^{-1})},$ (25)

where ${C}$ is some arbitrary constant and
$\mathcal{Z}_{AdS_{2}\times\mathcal{K}}^{\text{finite}}$ is the cutoff-
insensitive (i.e., $\ell$ independent) finite part. Then using
$AdS_{2}/CFT_{1}$ conjecture one identifies
$\mathcal{Z}_{AdS_{2}\times\mathcal{K}}^{\text{finite}}$ with
$d_{\text{hor}}$, i.e.,
$\mathcal{Z}_{AdS_{2}\times\mathcal{K}}^{\text{finite}}$ can be recognized as
the macroscopic horizon degeneracy of the extremal black hole. This kind of
prescription of horizon degeneracy is termed as the quantum entropy function
Sen:2008wa ; Sen:2009wb ; Sen:2009wc .
$\mathcal{Z}_{AdS_{2}\times\mathcal{K}}^{\text{finite}}$ reduces to the Wald
entropy equality in the classical limit, which checks the fact that the
quantum corrected entropy ($S_{\text{BH}}$) of the extremal black hole can be
identified as

$S_{\text{BH}}=\ln\mathcal{Z}_{AdS_{2}\times\mathcal{K}}^{\text{finite}}.$
(26)

  5. 5.

Computation of the full path integral of any gravity theory is almost an
impossible task. But one can carry out saddle point expansion Banerjee:2011oo
; Banerjee:2011pp ; Sen:2012rr ; Sen:2012qq ; Chowdhury:2014np ; Gupta:2014ns
; Bhattacharyya:2012ss ; Karan:2019sk ; Jeon:2017ij ; Keeler:2014nn ;
Larsen:2015nx ; Banerjee:2020wbr of the string partition function (23) for
small fluctuations around the classical near-horizon background (22). At one
loop, the correction to $\mathcal{Z}_{AdS_{2}\times\mathcal{K}}$ can be
obtained as

$\mathcal{Z}_{AdS_{2}\times\mathcal{K}}^{\text{1-loop}}=e^{-\mathcal{W}},$
(27)

where the one-loop effective action $\mathcal{W}$ is expressed via the kinetic
operator for quadratic fluctuations of string fields around the classical
near-horizon background.

  6. 6.

One-loop corrections generally have contributions from all the massive stringy
states running into the loop, but the logarithmic corrections are typical one-
loop quantum corrections that only depend on the massless states. Hence, the
logarithmic corrections to the extremal black holes, denoted as $\Delta
S_{\text{BH}}$, can be computed by using the massless fluctuations in the
finite part of the one-loop effective action $\mathcal{W}$ Banerjee:2011oo ;
Banerjee:2011pp ; Sen:2012rr ; Sen:2012qq ; Chowdhury:2014np ; Gupta:2014ns ;
Bhattacharyya:2012ss ; Karan:2019sk ; Jeon:2017ij ; Keeler:2014nn ;
Larsen:2015nx ; Banerjee:2020wbr .

The heat kernel treatment as discussed in section 2.1 is an effective path to
evaluate the one-loop effective action $\mathcal{W}$. However, the zero mode
contribution to $\mathcal{W}$ can not be assessed correctly by the heat kernel
treatment because $\mathcal{W}$ cannot preserve its Gaussian structure (5) for
zero modes. As a remedy, one can remove zero modes from the heat kernel and
substitute them back via a volume factor of different asymptotic symmetry
groups responsible for these zero modes Banerjee:2011oo ; Banerjee:2011pp ;
Sen:2012rr ; Sen:2012qq ; Gupta:2014ns ; Bhattacharyya:2012ss ;
Chowdhury:2014np ; Jeon:2017ij ; Sen:2013ns . This process splits the
Euclidean one-loop partition function (5) into two parts,

$e^{-\mathcal{W}}=(\text{det}^{\prime}\thinspace\Lambda)^{-\chi/2}\cdot\mathcal{Z}_{\mathrm{zero}}(L),$
(28)

where the primed part describes determinant over non-zero modes of the
operator $\Lambda$ and $\mathcal{Z}_{\mathrm{zero}}(L)$ is the zero-mode
integral with an overall length scale $L$. In this setup, $L$ is defined in a
way that the non-zero eigenvalues of the Laplace-type operator $\Lambda$ scale
as $1/L^{2}$ and consequently, the heat kernel parameter is reset as
$\bar{s}=s/L^{2}$. The non-zero mode part of the one-loop effective action is
then extracted as

$\begin{split}\mathcal{W}^{\prime}=\frac{\chi}{2}\thinspace\text{ln}\thinspace\text{det}^{\prime}\Lambda&=-\frac{\chi}{2}\int_{\epsilon/L^{2}}^{\infty}\frac{\mathrm{d}\bar{s}}{\bar{s}}\Big{(}D(\bar{s})-n_{\text{zm}}\Big{)}\\\
&=-\frac{1}{2}\bigg{(}\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace
a_{4}(x)-\chi
n_{\text{zm}}\bigg{)}\ln\left(\frac{L^{2}}{G_{N}}\right)+\cdots,\end{split}$
(29)

where the logarithmic term obtained after integrating out only $\bar{s}$
independent part in the perturbative expansion (9) of $D(\bar{s})$ and
$\cdots$ indicates the non-logarithmic terms. On the other hand, the zero-mode
part $\mathcal{Z}_{\mathrm{zero}}(L)$ can be analyzed by evaluating the
Jacobian for the change of integration variables from fields to the parameters
labeling zero-mode asymptotic symmetries Banerjee:2011oo ; Banerjee:2011pp ;
Sen:2012rr ; Sen:2012qq ; Gupta:2014ns ; Bhattacharyya:2012ss ;
Chowdhury:2014np ; Jeon:2017ij ; Sen:2013ns . This analysis assigns a factor
of $L^{\beta_{\tilde{\xi}}}$ corresponding to each zero mode, providing

$\mathcal{Z}_{\mathrm{zero}}(L)=L^{\sum_{\tilde{\xi}}\chi\beta_{\tilde{\xi}}n^{0}_{\tilde{\xi}}}\cdot\mathcal{Z}_{0},$
(30)

where $\beta_{\tilde{\xi}}$ are constant numbers computed individually for
different fluctuations; $n^{0}_{\tilde{\xi}}$ are the number of zero modes of
any arbitrary fluctuation $\tilde{\xi}$, giving the total zero-mode
$n_{\text{zm}}=\sum_{\tilde{\xi}}n^{0}_{\tilde{\xi}}$; $\mathcal{Z}_{0}$ is
independent of $L$. By putting the contributions (29) and (30) together, we
present the revised form of one-loop effective action as

$\mathcal{W}=-\frac{1}{2}\bigg{(}\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace
a_{4}(x)+\sum_{\tilde{\xi}}\chi(\beta_{\tilde{\xi}}-1)n^{0}_{\tilde{\xi}}\bigg{)}\ln\left(\frac{L^{2}}{G_{N}}\right)+\cdots,$
(31)

which yields the following central formula for finding the logarithmic
correction to the entropy of large999Logarithmic corrections are quantum
corrections in the large-charge limit. In this limit, the horizon area of the
black holes are scaled as $\mathcal{A}_{H}\sim L^{2}$, where $L$ is large
Bhattacharyya:2012ss . extremal black holes

$\begin{split}\Delta
S_{\text{BH}}&=\frac{1}{2}(\mathcal{C}_{\text{local}}+\mathcal{C}_{\text{zm}})\ln\left(\frac{\mathcal{A}_{H}}{G_{N}}\right),\\\
\mathcal{C}_{\text{local}}&=\int_{\text{horizon}}\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace
a_{4}(x),\\\
\mathcal{C}_{\text{zm}}&=\sum_{\tilde{\xi}}\chi(\beta_{\tilde{\xi}}-1)n^{0}_{\tilde{\xi}}.\end{split}$
(32)

Therefore, the local contribution in logarithmic correction formula, denoted
as $\mathcal{C}_{\text{local}}$, is obtained by calculating the $a_{4}(x)$
coefficient for the massless fluctuations and then integrating it over the
near-horizon geometry of extremal black holes with the aid of appropriate
limits (e.g., see eqs. 43, 44 and 45 for the extremal Kerr-Newman family of
black holes). On the other hand, the correction due to zero-modes101010One
should remember that only the $\beta_{\tilde{\xi}}$ terms are related to zero
modes in the structure of $\mathcal{C}_{\text{zm}}$, while the (-1) terms
arise for subtracting zero modes from the heat kernel.
$\mathcal{C}_{\text{zm}}$ has been enumerated in a variety of works
Banerjee:2011pp ; Sen:2012rr ; Sen:2012qq ; Sen:2013ns that allow us to use
the following compact formula Charles:2015nn

$\mathcal{C}_{\text{zm}}=-(3+\mathbb{K})+2N_{\text{SUSY}}+3\delta,$ (33)

where $\mathbb{K}$ is 1 for rotating black holes (e.g., Kerr-Newman, Kerr), 3
otherwise (e.g., Reissner-Nordström, Schwarzschild); $N_{\text{SUSY}}$ is 4
for BPS black holes, 0 otherwise; $\delta$ is 1 for non-extremal black holes,
0 otherwise.

####  2.3.2 Extremal near-horizon structures of Kerr-Newman family of black
holes

Our ultimate motive is to investigate the extremal near-horizon structures of
Kerr-Newman family (Kerr-Newman, Kerr and Reissner-Nordström) of black holes
and extract the integration limits required for their logarithmic entropy
corrections. Therefore, we start with the metric describing a general Kerr-
Newman solution Adamo:2014lk to the 4D Einstein-Maxwell theories,

$\begin{split}\mathrm{d}s^{2}&=\bar{g}_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}\\\
&=-\frac{r^{2}+b^{2}\thinspace\text{cos}^{2}\thinspace\psi-2Mr+Q^{2}}{r^{2}+b^{2}\thinspace\text{cos}^{2}\thinspace\psi}\,\mathrm{d}t^{2}+\frac{r^{2}+b^{2}\thinspace\text{cos}^{2}\thinspace\psi}{r^{2}+b^{2}-2Mr+Q^{2}}\,\mathrm{d}r^{2}\\\
&\quad+\frac{(r^{2}+b^{2}\thinspace\text{cos}^{2}\thinspace\psi)(r^{2}+b^{2})+(2Mr-Q^{2})b^{2}\thinspace\text{sin}^{2}\thinspace\psi}{r^{2}+b^{2}\thinspace\text{cos}^{2}\thinspace\psi}\thinspace\text{sin}^{2}\thinspace\psi\,\mathrm{d}\phi^{2}\\\
&\quad+(r^{2}+b^{2}\thinspace\text{cos}^{2}\thinspace\psi)\,\mathrm{d}\psi^{2}+\frac{2(Q^{2}-2Mr)b}{r^{2}+b^{2}\thinspace\text{cos}^{2}\thinspace\psi}\thinspace\text{sin}^{2}\thinspace\psi\,\mathrm{d}t\,\mathrm{d}\phi,\end{split}$
(34)

where $Q$, $M$, $J$ and $b=J/M$ are respectively the charge, mass, angular
momentum and Kerr parameter. The above Kerr-Newman metric $\bar{g}_{\mu\nu}$
is described in terms of standard spherical coordinates ($t,r,\psi,\phi$) and
serves the following Riemannian and Ricci invariant forms Henry:2000wd ;
Cherubini:2002we

$\displaystyle\begin{split}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}&=\frac{8}{(r^{2}+b^{2}\thinspace\text{cos}^{2}\thinspace\psi)^{6}}\Big{(}6M^{2}(r^{6}-15b^{2}r^{4}\thinspace\text{cos}^{2}\thinspace\psi+15b^{4}r^{2}\thinspace\text{cos}^{4}\thinspace\psi\\\
&\quad-b^{6}\thinspace\text{cos}^{6}\thinspace\psi)-12MQ^{2}r(r^{4}-10r^{2}b^{2}\thinspace\text{cos}^{2}\thinspace\psi+5b^{4}\thinspace\text{cos}^{4}\thinspace\psi)\\\
&\quad+Q^{4}(7r^{4}-34r^{2}b^{2}\thinspace\text{cos}^{2}\thinspace\psi+7b^{4}\thinspace\text{cos}^{4}\thinspace\psi)\Big{)},\\\
R_{\mu\nu}R^{\mu\nu}&=\frac{4Q^{4}}{(r^{2}+b^{2}\thinspace\text{cos}^{2}\thinspace\psi)^{4}}.\end{split}$
(35)

The apparent singularities for the metric (34) are described as

$r^{2}+b^{2}-2Mr+Q^{2}=0,$ (36)

which provides the locations of event horizons (outer and inner, respectively)
at

$r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}-b^{2}}.$ (37)

With our choice,111111$\hbar=c=k_{B}=1,G_{N}=1/16\pi$. Also, it is impossible
to access the interior of a black hole and we consider only the outer horizon
$r_{+}$ of area $\mathcal{A}_{H}=4\pi({r_{+}}^{2}+b^{2})$ for the Kerr-Newman
metric Bekenstein:2008bk . the Bekenstein-Hawking entropy ($S_{\text{BH}}$)
and Hawking temperature ($T_{\text{BH}}$) for the Kerr-Newman black hole take
the forms,

$\begin{split}&S_{\text{BH}}=\frac{\mathcal{A}_{H}}{4G_{N}}=16\pi^{2}\left(2M^{2}-Q^{2}+2M\sqrt{M^{2}-Q^{2}-b^{2}}\right),\\\
&T_{\text{BH}}=\frac{1}{\frac{\partial S_{\text{BH}}}{\partial
M}}=\frac{1}{32\pi^{2}}\frac{\sqrt{M^{2}-Q^{2}-b^{2}}}{\left(2M^{2}-Q^{2}+2M\sqrt{M^{2}-Q^{2}-b^{2}}\right)}.\end{split}$
(38)

In the extremal limit, a black hole ceases to evaporate and the Hawking
temperature vanishes ($T_{\text{BH}}=0$). This leads us to the extremality
condition for the Kerr-Newman black hole (34) as

$M^{2}=Q^{2}+b^{2},$ (39)

which yields $r_{+}=r_{-}=M$, i.e., the two horizons merge into a single
horizon of radius $r_{H}=M$. The extremality condition (39) also explains the
fact that for a vanishing temperature ($T_{\text{BH}}=0$), the black hole
still possesses a finite entropy which can be expressed in terms of its
parameters as

$S_{\text{BH}}=16\pi^{2}\left(Q^{2}+2b^{2}\right).$ (40)

To structure the extremal near-horizon geometry of the Kerr-Newman black hole,
we now review the standard trick used in Bhattacharyya:2012ss . We consider a
new set coordinates ($\tau,\eta,\psi,\zeta$) and set up the following
coordinate transformations in terms of a new parameter $\sigma$

$\begin{gathered}M^{2}=Q^{2}+b^{2}+\sigma^{2},\thinspace
r=M+\sigma\thinspace\text{cosh}\thinspace\eta,\thinspace
t=\frac{(2b^{2}+Q^{2})\tau}{\sigma},\\\
\phi=\zeta+\frac{b}{Q^{2}+2b^{2}}\left(1-\frac{2M\sigma}{Q^{2}+2b^{2}}\right)t.\end{gathered}$
(41)

The parameter $\sigma$ serves the desired extremal near-horizon condition as
$\sigma\to 0$ (for fixed $\eta$, $\tau$) and therefore reduces the metric (34)
to the Euclidean near-horizon geometry of the extremal Kerr-Newman black hole,

$\begin{split}{\mathrm{d}s^{2}}_{\text{horizon}}&=\bar{g}_{\mu\nu}\,\mathrm{d}x^{\mu}\,\mathrm{d}x^{\nu}\\\
&=\Big{(}Q^{2}+b^{2}(1+\text{cos}^{2}\thinspace\psi)\Big{)}(\mathrm{d}\eta^{2}+\text{sinh}^{2}\thinspace\eta\,\mathrm{d}\theta^{2}+\mathrm{d}\psi^{2})\\\
&\quad+\frac{(Q^{2}+2b^{2})^{2}}{\Big{(}Q^{2}+b^{2}(1+\text{cos}^{2}\thinspace\psi)\Big{)}}\left(\mathrm{d}\zeta-i\frac{2Mb}{(Q^{2}+2b^{2})}(\text{cosh}\thinspace\eta-1)\mathrm{d}\theta\right)^{2}\text{sin}^{2}\thinspace\psi,\end{split}$
(42)

where $\tau=-i\theta$ is set for Euclidean continuation with $\theta$ as the
Euclideanized time of period $2\pi$. As expected, the above extremal near-
horizon geometry is now structured in the form $AdS_{2}\times\mathcal{K}$,
where the $AdS_{2}$ part is parametrized by ($\eta,\theta$) for
$0\leq\eta<\infty;\theta\equiv\theta+2\pi$. For the extremal near-horizon
Kerr-Newman metric (42), one obtains the following finite121212The finite word
depicts the fact that the results (43) dropped all the terms proportional to
the IR regulated $\eta$ coordinate as per the requirement in the quantum
entropy function formalism Bhattacharyya:2012ss ; Banerjee:2020wbr . limits
Bhattacharyya:2012ss

$\begin{split}\int_{\text{horizon}}\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}&=-16\pi^{2}\Big{(}3\mathcal{B}-(8{b^{\prime}}^{6}+20{b^{\prime}}^{4}+8{b^{\prime}}^{2}-1)\mathcal{B}^{\prime}\Big{)},\\\
\int_{\text{horizon}}\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}R_{\mu\nu}R^{\mu\nu}&=-4\pi^{2}\Big{(}3\mathcal{B}+(8{b^{\prime}}^{2}+5)\mathcal{B}^{\prime}\Big{)},\end{split}$
(43)

where

${b^{\prime}}=J/MQ,\mathcal{B}=\frac{2{b^{\prime}}^{2}+1}{{b^{\prime}}({b^{\prime}}^{2}+1)^{5/2}}\text{tan}^{-1}\left(\frac{{b^{\prime}}}{\sqrt{{b^{\prime}}^{2}+1}}\right),\thinspace\text{and}\thinspace\mathcal{B}^{\prime}=\frac{1}{({b^{\prime}}^{2}+1)^{2}(2{b^{\prime}}^{2}+1)}.$

We can also achieve similar limits for the extremal near-horizon Kerr and
Reissner-Nordström black hole backgrounds as special cases of the Kerr-Newman
geometry (42):

  * $\blacksquare$

For the extremal Kerr black holes, we need to set $J\neq 0,Q=0$ (i.e.,
${b^{\prime}}\to\infty$) in eq. 43, which provides

$\begin{split}\int_{\text{horizon}}\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}&=64\pi^{2},\\\
\int_{\text{horizon}}\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}R_{\mu\nu}R^{\mu\nu}&=0.\end{split}$
(44)

  * $\blacksquare$

For the extremal Reissner-Nordström black holes, we need to set $J=0,Q\neq 0$
(i.e., ${b^{\prime}}\to 0$) in eq. 43, which gives

$\begin{split}\int_{\text{horizon}}\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}&=-64\pi^{2},\\\
\int_{\text{horizon}}\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}R_{\mu\nu}R^{\mu\nu}&=-32\pi^{2}.\end{split}$
(45)

The above near-horizon extremal limits (eqs. 43, 44 and 45) of the Riemannian
and Ricci invariants are crucial ingredients for computing logarithmic
correction (especially the $\mathcal{C}_{\text{local}}$ contribution (32)) to
the entropy of Kerr-Newman family of black holes in different 4D Einstein-
Maxwell theories.

##  3 Seeley-DeWitt coefficients and logarithmic correction contributions of
the multiplets of $\mathcal{N}=2,d=4$ EMSGT

Multiplets |  $s_{\text{max}}$ |  $\bm{s=2}$ |  $\bm{s=\frac{3}{2}}$ |  $\bm{s=1}$ |  $\bm{s=\frac{1}{2}}$ |  $\bm{s=0}$
---|---|---|---|---|---|---
supergravity ($\mathcal{G}_{2}$) |  2 |  1 |  2 |  1 |  |
gravitino ($\mathcal{G}_{3/2}$) |  $\frac{3}{2}$ |  |  1 |  2 |  1 |
vector ($\mathcal{G}_{1}$) |  1 |  |  |  1 |  2 |  1+1
hyper ($\mathcal{G}_{1/2}$) |  $\frac{1}{2}$ |  |  |  |  2 |  2+2
Table 1: Different multiplets and spin content of a $\mathcal{N}=2,d=4$ EMSGT.

Supergravity theories are field theories that emerge as the IR (low energy)
limit of superstring theories where supersymmetric matters are coupled to
gravity. The field content of these theories is determined by the space-time
dimension ($d$) and the amount of supersymmetry ($\mathcal{N}$) they embed.
$\mathcal{N}=2,d=4$ EMSGT is the most simple and significant extended
supergravity embedded Einstein-Maxwell theory, which arises as an IR limit of
type-II string theory compactified on a Calabi–Yau three-fold Grana:2006mg ;
Freedman:2012xp . One can structure such theory by embedding the general
background solutions (Kerr-Newman family of black holes Adamo:2014lk ) of a 4D
Einstein-Maxwell theory into a $\mathcal{N}=2$ supergravity with the condition
that the supergravity fields fluctuate around the background of the Einstein-
Maxwell theory. This ensures that both the Einstein-Maxwell theory and the
$\mathcal{N}=2$ supergravity share common equations of motion and solutions,
which in turn justifies the embedding Charles:2015nn . Such embedding induces
both minimal and non-minimal couplings between the background and matter
fluctuations distributed in different multiplets of the resultant theory.

For a general $\mathcal{N}=2,d=4$ EMSGT, the massless fields are distributed
in four kinds of multiplets – supergravity multiplet ($\mathcal{G}_{2}$),
gravitino multiplet ($\mathcal{G}_{3/2}$), vector multiplet
($\mathcal{G}_{1}$) and hyper multiplet ($\mathcal{G}_{1/2}$). The spin
contents of these multiplets are summarized in table 1, where each multiplet
has equal bosonic and fermionic on-shell degrees of freedom. Interestingly,
the quadratic field fluctuations of any particular multiplet do not mix with
fluctuations of other multiplets. This permits us to study the quadratic
fluctuations of each $\mathcal{N}=2$ multiplet (divided into bosonic and
fermionic sectors) separately and calculate the necessary Seeley-DeWitt
coefficients. We then employ the Seeley-DeWitt data to obtain the logarithmic
correction contributions of individual multiplets to the entropy of extremal
Kerr-Newman, Kerr and Reissner-Nordström black holes. Note that the
supergravity multiplet results are collected from Karan:2019sk , while
calculations for other multiplets are thoroughly executed by pursuing the
standard procedure mentioned in section 2.

###  3.1 $\mathcal{N}=2,d=4$ EMSGT: supergravity multiplet

The field contents of the supergravity multiplet ($\mathcal{G}_{2}$) of a
$\mathcal{N}=2,d=4$ EMSGT can be divided into two sectors –

  * Bosonic sector ($\mathcal{G}^{b}_{2}$): one vector field $A_{\mu}$, dubbed as graviphoton, is minimally coupled to one graviton $g_{\mu\nu}$. This part is identical to the pure 4D Einstein-Maxwell theory;

  * Fermionic sector ($\mathcal{G}^{f}_{2}$): two massless Majorana gravitini $\\{\psi_{\mu}$, $\phi_{\mu}\\}$ are coupled non-minimally with their Hermitian conjugates $\\{\bar{\psi}_{\mu}$, $\bar{\phi}_{\mu}\\}$ through the background graviphoton field strength $\bar{F}_{\mu\nu}$ and its dual $\bar{H}_{\mu\nu}=-\varepsilon_{\mu\nu\rho\sigma}\bar{F}^{\rho\sigma}$.

The action of a pure 4D Einstein-Maxwell theory describing the bosonic-sector
of the supergravity multiplet is

$\mathcal{S}_{\text{EM}}=\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace{g}}\thinspace\left(\mathcal{R}-F_{\mu\nu}F^{\mu\nu}\right),$
(46)

where ${F}_{\mu\nu}\equiv\partial_{\mu}{A}_{\nu}-\partial_{\nu}{A}_{\mu}$ is
the graviphoton field strength tensor and $\mathcal{R}$ is Ricci scalar
corresponding to the metric $g_{\mu\nu}$. Any arbitrary classical solution
$(\bar{g}_{\mu\nu},\bar{A}_{\mu})$ to the theory (46) satisfies the following
equations of motion

$R_{\mu\nu}=2\bar{F}_{\mu\rho}{\bar{F_{\nu}}}^{\rho}-\frac{1}{2}\bar{g}_{\mu\nu}\bar{F}_{\rho\sigma}\bar{F}^{\rho\sigma},R=0,$
(47)

where $\bar{F}_{\mu\nu}$ is the strength of the background graviphoton field
$\bar{A}_{\mu}$; $R_{\mu\nu}$ and $R$ are respectively Ricci tensor and Ricci
scalar corresponding to the background metric $\bar{g}_{\mu\nu}$. All the
equations of motion, Maxwell equations, Bianchi identities satisfied by
$\bar{F}_{\mu\nu}$ and it’s $\bar{H}_{\mu\nu}$ are recorded in appendix A. As
mentioned, the $\mathcal{N}=2,d=4$ EMSGT is structured in such a manner that
all the matter fields should fluctuate around the background
$(\bar{g}_{\mu\nu},\bar{A}_{\mu})$. For that, the $\mathcal{N}=2,d=4$ EMSGT
shares the same equations of motion, solutions, identities as the 4D Einstein-
Maxwell theory (or the bosonic-sector ($\mathcal{G}^{b}_{2}$)) (46). Also,
there will be various non-minimal couplings in the fluctuation of
$\mathcal{N}=2$ matter fields via $\bar{F}_{\mu\nu}$ and $\bar{H}_{\mu\nu}$.
For the fluctuations,

$g_{\mu\nu}=\bar{g}_{\mu\nu}+\sqrt{2}h_{\mu\nu},\thinspace
A_{\mu}=\bar{A}_{\mu}+\frac{1}{2}a_{\mu},$ (48)

the Seeley-DeWitt coefficients and the logarithmic correction results of the
quadratic fluctuated $\mathcal{N}=2$ supergravity multiplet131313For the
quadratic fluctuated actions of the bosonic ($\mathcal{G}^{b}_{2}$) and
fermionic ($\mathcal{G}^{f}_{2}$) sectors of the $\mathcal{N}=2$ supergravity
multiplet, readers are referred to eqs. (5.4) and (5.21) of the work
Karan:2019sk . ($\mathcal{G}_{2}$) have already been achieved in our earlier
work Karan:2019sk . The first three Seeley-DeWitt coefficients are
Karan:2019sk ,

$\displaystyle\begin{split}&(4\pi)^{2}a^{\mathcal{G}_{2}}_{0}(x)=0,\\\
&(4\pi)^{2}a^{\mathcal{G}_{2}}_{2}(x)=14\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&(4\pi)^{2}a^{\mathcal{G}_{2}}_{4}(x)=\frac{11}{24}\left(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}\right),\end{split}$
(49)

and using them in the formulae (32), the logarithmic correction contributions
of the supergravity multiplet ($\mathcal{G}_{2}$) to the entropy of extremal
Kerr-Newman family of black holes are obtained as

$\Delta
S^{\mathcal{G}_{2}}_{\text{BH}}=\begin{cases}\left(\frac{11}{12}\frac{(2{b^{\prime}}^{6}+5{b^{\prime}}^{4}+4{b^{\prime}}^{2}+1)}{({b^{\prime}}^{2}+1)^{2}(2{b^{\prime}}^{2}+1)}-2\right)\text{ln}\thinspace\mathcal{A}_{H}&\text{for
Kerr-Newman},\\\
-\frac{13}{12}\thinspace\text{ln}\thinspace\mathcal{A}_{H}&\text{for Kerr},\\\
\frac{23}{12}\thinspace\text{ln}\thinspace\mathcal{A}_{H}&\text{for Reissner-
Nordstr\"{o}m}.\end{cases}$ (50)

The above results contain $\mathcal{C}_{\text{zm}}=-4,-4$ and 2 respectively
for the extremal Kerr-Newman, Kerr and Reissner-Nordström black holes,
according to the formula (33). Here extremal Reissner-Nordström black holes
are the only example of a BPS solution in the $\mathcal{N}=2,d=4$ EMSGT.

###  3.2 $\mathcal{N}=2,d=4$ EMSGT: gravitino multiplet

The field contents of the gravitino multiplet ($\mathcal{G}_{3/2}$) of a
$\mathcal{N}=2,d=4$ EMSGT can be split into the following sectors –

  * Bosonic sector ($\mathcal{G}^{b}_{3/2}$): two vector fields are minimally coupled to the background gravity,

  * Fermionic sector ($\mathcal{G}^{f}_{3/2}$): one Majorana gravitino $\psi_{\mu}$, one Majorana gaugino $\lambda$ and their Hermitian conjugates ($\bar{\psi}_{\mu}$, $\bar{\lambda}$) are non-minimally coupled via the background graviphoton field strength $\bar{F}_{\mu\nu}$.

The quadratic order fluctuated action describing the fermionic-sector
($\mathcal{G}^{f}_{3/2}$) is

$\begin{split}\delta^{2}\mathcal{S}_{\mathcal{G}^{f}_{3/2}}&=\frac{2}{\kappa^{2}}\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\mathscr{L}_{\mathcal{G}^{f}_{3/2}},\\\
\mathscr{L}_{\mathcal{G}^{f}_{3/2}}&=-\bar{\psi}_{\mu}\gamma^{\mu\rho\nu}D_{\rho}\psi_{\nu}-\bar{\lambda}\gamma^{\rho}D_{\rho}\lambda-\frac{1}{2\sqrt{2}}\bar{\psi}_{\mu}\hat{\mathcal{F}}\gamma^{\mu}\lambda-\frac{1}{2\sqrt{2}}\bar{\lambda}\gamma^{\nu}\hat{\mathcal{F}}\psi_{\nu},\end{split}$
(51)

where $\kappa=1/\sqrt{2}$ and
$\hat{\mathcal{F}}=\bar{F}_{\alpha\beta}\gamma^{\alpha}\gamma^{\beta}$.141414The
useful definitions and identities related to the 4D gamma matrices
$\gamma^{\mu}$ are listed in appendix A. We adopted the action from151515See
eq. (2.41) of Charles:2015nn . Charles:2015nn and rescaled the gravitino field
as $\psi_{\mu}\to\sqrt{2}\psi_{\mu}$ so that both the gravitino and gaugino
fields have the same normalization state in their kinetic parts of the
operator form (3.2). We then cast the harmonic gauge
$\gamma^{\mu}\psi_{\mu}=0$ and gauge fix the action (51) by inserting the
gauge-fixing term,

$\displaystyle\frac{1}{\kappa^{2}}\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace(\bar{\psi}_{\mu}\gamma^{\mu})\gamma^{\rho}D_{\rho}(\gamma^{\nu}\psi_{\nu}),$
(52)

followed by the ghost action Banerjee:2011oo ,

$\displaystyle\mathcal{S}_{\mathcal{G}^{f}_{3/2},\text{ghost}}=\frac{1}{\kappa^{2}}\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace(\bar{\tilde{b}}\gamma^{\mu}D_{\mu}\tilde{c}+\bar{\tilde{e}}\gamma^{\mu}D_{\mu}\tilde{e}),$
(53)

where $\tilde{b}$, $\tilde{c}$, and $\tilde{e}$ are three bosonic ghosts
introduced to the system due to the choice of the particular gauge fixing
(52). These ghost fields have no interaction with any other fields of the
fermionic-sector ($\mathcal{G}^{f}_{3/2}$), and hence we will evaluate their
contributions separately. The standard prescription of section 2.2 assists us
from here. Essentially the gauge fixed action (without accounting for the
ghost action (53)) provides a first-order Dirac-type operator
$\not{D}$,161616In eq. 56, the fermionic operator is set as Dirac to proceed
with the standard set-up described in section 2.2. But in the end, we extract
back the Seeley-DeWitt coefficients (see the results (64)) for the actual
Majorana spinors described in the action (51) by setting $\chi=-1/2$ according
to the relation (19). The same treatment is followed throughout this paper for
other Majorana fermions present in the $\mathcal{N}=2,d=4$ EMSGT.

$\displaystyle\begin{gathered}\delta^{2}\mathcal{S}_{\mathcal{G}^{f}_{3/2}}=\frac{2}{\kappa^{2}}\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\tilde{\xi}_{m}\not{D}^{mn}\tilde{\xi}_{n},\\\
\tilde{\xi}_{m}\not{D}^{mn}\tilde{\xi}_{n}=\frac{i}{2}\bar{\psi}_{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\mu}D_{\rho}\psi_{\nu}-i\bar{\lambda}\gamma^{\rho}D_{\rho}\lambda-\frac{i}{2\sqrt{2}}\bar{\psi}_{\mu}\hat{\mathcal{F}}\gamma^{\mu}\lambda-\frac{i}{2\sqrt{2}}\bar{\lambda}\gamma^{\nu}\hat{\mathcal{F}}\psi_{\nu},\end{gathered}$
(56)

from which one needs to structure the necessary operator $\Lambda$ in the
following manner:

$\displaystyle\tilde{\xi}_{m}\Lambda^{mn}\tilde{\xi}_{n}$
$\displaystyle=\tilde{\xi}_{m}(\not{D}^{mp})^{\dagger}{\not{D}_{p}}^{n}\tilde{\xi}_{n}$
$\displaystyle=-\bar{\psi}_{\mu}\bigg{\\{}\frac{1}{4}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\gamma^{\nu}\gamma^{\sigma}\gamma_{\alpha}D_{\rho}D_{\sigma}-\frac{1}{8}\hat{\mathcal{F}}\gamma^{\mu}\gamma^{\nu}\hat{\mathcal{F}}\bigg{\\}}\psi_{\nu}$
$\displaystyle\qquad-\bar{\lambda}\bigg{\\{}\gamma^{\rho}\gamma^{\sigma}D_{\rho}D_{\sigma}-\frac{1}{8}\gamma^{\alpha}\hat{\mathcal{F}}\hat{\mathcal{F}}\gamma_{\alpha}\bigg{\\}}\lambda$
$\displaystyle\qquad+\bar{\psi}_{\mu}\bigg{\\{}\frac{1}{4\sqrt{2}}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}(D_{\rho}\hat{\mathcal{F}}+\hat{\mathcal{F}}D_{\rho})\gamma_{\alpha}+\frac{1}{2\sqrt{2}}\hat{\mathcal{F}}\gamma^{\mu}\gamma^{\rho}D_{\rho}\bigg{\\}}\lambda$
$\displaystyle\qquad-\bar{\lambda}\bigg{\\{}\frac{1}{4\sqrt{2}}\gamma^{\alpha}\hat{\mathcal{F}}\gamma^{\nu}\gamma^{\rho}\gamma_{\alpha}D_{\rho}+\frac{1}{2\sqrt{2}}\gamma^{\rho}\gamma^{\nu}(D_{\rho}\hat{\mathcal{F}}+\hat{\mathcal{F}}D_{\rho})\bigg{\\}}\psi_{\nu}.$
(57)

Make sure to consider the fact that in our choice of 4D Euclidean space-time,
$\gamma^{\mu}$ matrices are Hermitian,
$(\gamma^{\rho}D_{\rho})^{\dagger}=-\gamma^{\rho}D_{\rho}$,
$(\gamma^{\nu}\gamma^{\rho}\gamma^{\mu}D_{\rho})^{\dagger}=-\gamma^{\nu}\gamma^{\rho}\gamma^{\mu}D_{\rho}$,
$({\hat{\mathcal{F}}})^{\dagger}=\hat{\mathcal{F}}$. The operator form (3.2)
is not of Laplace-type as prescribed in the schematic (12) because the
principal kinetic terms are not appropriately contracted to provide the
Laplacian $D^{\rho}D_{\rho}$. To rectify this issue, we set up the following
relations for the gravitino $\psi_{\mu}$ and gaugino $\lambda$ fluctuations

$\displaystyle\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\gamma^{\nu}\gamma^{\sigma}\gamma_{\alpha}D_{\rho}D_{\sigma}\psi_{\mu}$
$\displaystyle=(4\mathbb{I}_{4}\bar{g}^{\mu\nu}D^{\rho}D_{\rho}+2\gamma^{\alpha}\gamma^{\beta}{R^{\mu\nu}}_{\alpha\beta}$
$\displaystyle\qquad-2\gamma^{\mu}\gamma^{\alpha}{R^{\nu}}_{\alpha}+2\gamma^{\nu}\gamma^{\alpha}{R^{\mu}}_{\alpha}-4\mathbb{I}_{4}R^{\mu\nu})\psi_{\mu},$
(58a) $\displaystyle\gamma^{\rho}\gamma^{\sigma}D_{\rho}D_{\sigma}\lambda$
$\displaystyle=\mathbb{I}_{4}D^{\rho}D_{\rho}\lambda,$ (58b)

and restructure $\Lambda$ into the required Laplace-type form,

$\displaystyle\tilde{\xi}_{m}\Lambda^{mn}\tilde{\xi}_{n}$
$\displaystyle=-\mathbb{I}_{4}\bar{g}^{\mu\nu}\bar{\psi}_{\mu}D^{\rho}D_{\rho}\psi_{\nu}-\mathbb{I}_{4}\bar{\lambda}D^{\rho}D_{\rho}\lambda$
$\displaystyle\quad-\bar{\psi}_{\mu}\bigg{\\{}\frac{1}{2}\gamma^{\alpha}\gamma^{\beta}{R^{\mu\nu}}_{\alpha\beta}-\frac{1}{2}\gamma^{\mu}\gamma^{\alpha}{R^{\nu}}_{\alpha}+\frac{1}{2}\gamma^{\nu}\gamma^{\alpha}{R^{\mu}}_{\alpha}-\mathbb{I}_{4}R^{\mu\nu}$
$\displaystyle\quad-\frac{1}{8}\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma^{\nu}\gamma^{\theta}\gamma^{\phi}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}\bigg{\\}}\psi_{\nu}+\frac{1}{8}\bar{\lambda}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{\theta}\gamma^{\phi}\gamma_{\sigma}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}\lambda$
$\displaystyle\quad+\frac{1}{4\sqrt{2}}\bar{\psi}_{\mu}\gamma^{\sigma}\gamma^{\rho}\gamma^{\mu}\gamma^{\alpha}\gamma^{\beta}\gamma_{\sigma}(D_{\rho}\bar{F}_{\alpha\beta})\lambda-\frac{1}{2\sqrt{2}}\bar{\lambda}\gamma^{\rho}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta}(D_{\rho}\bar{F}_{\alpha\beta})\psi_{\nu}$
$\displaystyle\quad+\bar{\psi}_{\mu}\bigg{\\{}\frac{1}{4\sqrt{2}}\gamma^{\sigma}\gamma^{\rho}\gamma^{\mu}\gamma^{\alpha}\gamma^{\beta}\gamma_{\sigma}\bar{F}_{\alpha\beta}+\frac{1}{2\sqrt{2}}\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma^{\rho}\bar{F}_{\alpha\beta}\bigg{\\}}(D_{\rho}\lambda)$
$\displaystyle\quad-\bar{\lambda}\bigg{\\{}\frac{1}{4\sqrt{2}}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{\nu}\gamma^{\rho}\gamma_{\sigma}\bar{F}_{\alpha\beta}+\frac{1}{2\sqrt{2}}\gamma^{\rho}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta}\bar{F}_{\alpha\beta}\bigg{\\}}(D_{\rho}\psi_{\nu}).$
(59)

The derivations (58a) and (58b) are executed by employing the commutation
relations (20d) and (20c) and also considering up to total derivative terms.
Since the operator form (3.2) is now comparable with the Laplace-type
schematic (12), we extract out the matrices $I$, $P$ and
$\omega^{\rho}(=\frac{1}{2}N^{\rho})$ as

$\begin{split}\tilde{\xi}_{m}I^{mn}\tilde{\xi}_{n}&=\bar{\psi}_{\mu}\mathbb{I}_{4}\bar{g}^{\mu\nu}\psi_{\nu}+\bar{\lambda}\mathbb{I}_{4}\lambda,\\\
\tilde{\xi}_{m}P^{mn}\tilde{\xi}_{n}&=\bar{\psi}_{\mu}\bigg{\\{}\frac{1}{2}\gamma^{\alpha}\gamma^{\beta}{R^{\mu\nu}}_{\alpha\beta}-\frac{1}{2}\gamma^{\mu}\gamma^{\alpha}{R^{\nu}}_{\alpha}+\frac{1}{2}\gamma^{\nu}\gamma^{\alpha}{R^{\mu}}_{\alpha}-\mathbb{I}_{4}R^{\mu\nu}\\\
&\quad-\frac{1}{8}\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma^{\nu}\gamma^{\theta}\gamma^{\phi}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}\bigg{\\}}\psi_{\nu}-\frac{1}{8}\bar{\lambda}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{\theta}\gamma^{\phi}\gamma_{\sigma}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}\lambda\\\
&\quad-\frac{1}{4\sqrt{2}}\bar{\psi}_{\mu}\gamma^{\sigma}\gamma^{\rho}\gamma^{\mu}\gamma^{\alpha}\gamma^{\beta}\gamma_{\sigma}(D_{\rho}\bar{F}_{\alpha\beta})\lambda+\frac{1}{2\sqrt{2}}\bar{\lambda}\gamma^{\rho}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta}(D_{\rho}\bar{F}_{\alpha\beta})\psi_{\nu},\\\
\tilde{\xi}_{m}{(\omega^{\rho})}^{mn}\tilde{\xi}_{n}&=-\bar{\psi}_{\mu}\bigg{\\{}\frac{1}{4\sqrt{2}}\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma^{\rho}\bar{F}_{\alpha\beta}+\frac{1}{8\sqrt{2}}\gamma^{\sigma}\gamma^{\rho}\gamma^{\mu}\gamma^{\alpha}\gamma^{\beta}\gamma_{\sigma}\bar{F}_{\alpha\beta}\bigg{\\}}\lambda\\\
&\quad+\bar{\lambda}\bigg{\\{}\frac{1}{4\sqrt{2}}\gamma^{\rho}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta}\bar{F}_{\alpha\beta}+\frac{1}{8\sqrt{2}}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{\nu}\gamma^{\rho}\gamma_{\sigma}\bar{F}_{\alpha\beta}\bigg{\\}}\psi_{\nu}.\end{split}$
(60)

All the above data are sufficient to express $\Lambda$ in the standard form
(15), where the matrices $E$ and $\Omega_{\rho\sigma}$ hold the following
expressions (according to the formulae (16) and (14))

$\displaystyle\tilde{\xi}_{m}E^{mn}\tilde{\xi}_{n}$
$\displaystyle=\bar{\psi}_{\mu}\Bigg{\\{}\frac{1}{2}\gamma^{\alpha}\gamma^{\beta}{R^{\mu\nu}}_{\alpha\beta}-\frac{1}{2}\gamma^{\mu}\gamma^{\alpha}{R^{\nu}}_{\alpha}+\frac{1}{2}\gamma^{\nu}\gamma^{\alpha}{R^{\mu}}_{\alpha}-\mathbb{I}_{4}R^{\mu\nu}$
$\displaystyle\qquad-\frac{1}{8}\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma^{\nu}\gamma^{\theta}\gamma^{\phi}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}+\frac{1}{32}\Big{(}\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma^{\rho}+\frac{1}{2}\gamma^{\sigma}\gamma^{\rho}\gamma^{\mu}\gamma^{\alpha}\gamma^{\beta}\gamma_{\sigma}\Big{)}$
$\displaystyle\qquad\quad\times\Big{(}\gamma_{\rho}\gamma^{\nu}\gamma^{\theta}\gamma^{\phi}+\frac{1}{2}\gamma^{\tau}\gamma^{\theta}\gamma^{\phi}\gamma^{\nu}\gamma_{\rho}\gamma_{\tau}\Big{)}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}\Bigg{\\}}\psi_{\nu}$
$\displaystyle\quad+\bar{\lambda}\Bigg{\\{}-\frac{1}{8}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{\theta}\gamma^{\phi}\gamma_{\sigma}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}+\frac{1}{32}\Big{(}\gamma^{\rho}\gamma^{\mu}\gamma^{\alpha}\gamma^{\beta}+\frac{1}{2}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma^{\rho}\gamma_{\sigma}\Big{)}$
$\displaystyle\qquad\quad\times\Big{(}\gamma^{\theta}\gamma^{\phi}\gamma_{\mu}\gamma_{\rho}+\frac{1}{2}\gamma^{\tau}\gamma_{\rho}\gamma_{\mu}\gamma^{\theta}\gamma^{\phi}\gamma_{\tau}\Big{)}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}\Bigg{\\}}\lambda$
$\displaystyle\quad+\bar{\psi}_{\mu}\Bigg{\\{}\frac{1}{4\sqrt{2}}\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma^{\rho}(D_{\rho}\bar{F}_{\alpha\beta})-\frac{1}{8\sqrt{2}}\gamma^{\sigma}\gamma^{\rho}\gamma^{\mu}\gamma^{\alpha}\gamma^{\beta}\gamma_{\sigma}(D_{\rho}\bar{F}_{\alpha\beta})\Bigg{\\}}\lambda$
$\displaystyle\quad+\bar{\lambda}\Bigg{\\{}\frac{1}{4\sqrt{2}}\gamma^{\rho}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta}(D_{\rho}\bar{F}_{\alpha\beta})-\frac{1}{8\sqrt{2}}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{\nu}\gamma^{\rho}\gamma_{\sigma}(D_{\rho}\bar{F}_{\alpha\beta})\Bigg{\\}}\psi_{\nu},$
(61) $\displaystyle\tilde{\xi}_{m}(\Omega_{\rho\sigma})^{mn}\tilde{\xi}_{n}$
$\displaystyle=\bar{\psi}_{\mu}\Bigg{\\{}\mathbb{I}_{4}{R^{\mu\nu}}_{\rho\sigma}+\frac{1}{4}\bar{g}^{\mu\nu}\gamma^{\alpha}\gamma^{\beta}R_{\rho\sigma\alpha\beta}-\frac{1}{32}\left(\Big{(}\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma_{\rho}+\frac{1}{2}\gamma^{\tau}\gamma_{\rho}\gamma^{\mu}\gamma^{\alpha}\gamma^{\beta}\gamma_{\tau}\Big{)}\right.$
$\displaystyle\qquad\quad\left.\times\Big{(}\gamma_{\sigma}\gamma^{\nu}\gamma^{\theta}\gamma^{\phi}+\frac{1}{2}\gamma^{\lambda}\gamma^{\theta}\gamma^{\phi}\gamma^{\nu}\gamma_{\sigma}\gamma_{\lambda}\Big{)}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}-(\rho\leftrightarrow\sigma)\right)\Bigg{\\}}\psi_{\nu}$
$\displaystyle\quad+\bar{\lambda}\Bigg{\\{}\frac{1}{4}\gamma^{\alpha}\gamma^{\beta}R_{\rho\sigma\alpha\beta}-\frac{1}{32}\left(\Big{(}\gamma_{\rho}\gamma^{\mu}\gamma^{\alpha}\gamma^{\beta}+\frac{1}{2}\gamma^{\tau}\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma_{\rho}\gamma_{\tau}\Big{)}\right.$
$\displaystyle\qquad\quad\left.\times\Big{(}\gamma^{\theta}\gamma^{\phi}\gamma_{\mu}\gamma_{\sigma}+\frac{1}{2}\gamma^{\lambda}\gamma_{\sigma}\gamma_{\mu}\gamma^{\theta}\gamma^{\phi}\gamma_{\lambda}\Big{)}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}-(\rho\leftrightarrow\sigma)\right)\Bigg{\\}}\lambda$
$\displaystyle\quad-\bar{\psi}_{\mu}\Bigg{\\{}\frac{1}{4\sqrt{2}}\Big{(}\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma_{\sigma}+\frac{1}{2}\gamma^{\tau}\gamma_{\sigma}\gamma^{\mu}\gamma^{\alpha}\gamma^{\beta}\gamma_{\tau}\Big{)}(D_{\rho}\bar{F}_{\alpha\beta})-(\rho\leftrightarrow\sigma)\Bigg{\\}}\lambda$
$\displaystyle\quad+\bar{\lambda}\Bigg{\\{}\frac{1}{4\sqrt{2}}\Big{(}\gamma_{\sigma}\gamma^{\nu}\gamma^{\alpha}\gamma^{\beta}+\frac{1}{2}\gamma^{\tau}\gamma^{\alpha}\gamma^{\beta}\gamma^{\nu}\gamma_{\sigma}\gamma_{\tau}\Big{)}(D_{\rho}\bar{F}_{\alpha\beta})-(\rho\leftrightarrow\sigma)\Bigg{\\}}\psi_{\nu}.$
(62)

At this point, our major task is to calculate $\text{tr}(E)$, $\text{tr}(E)$,
$\text{tr}(E^{2})$ and $\text{tr}(\Omega_{\rho\sigma}\Omega^{\rho\sigma})$.
These trace calculations are quite lengthy but straightforward; hence one
needs to proceed systematically. The required trace results are (see appendix
B for calculation details),

$\begin{split}\text{tr}(I)&=4\times 4+4=20,\\\
\text{tr}(E)&=-8\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
\text{tr}(E^{2})&=2R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-5R_{\mu\nu}R^{\mu\nu}-2R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}+10(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2},\\\
\text{tr}(\Omega_{\rho\sigma}\Omega^{\rho\sigma})&=-\frac{13}{2}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+10R_{\mu\nu}R^{\mu\nu}+12R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-60(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}.\end{split}$
(63)

Putting all these trace data in the formulae (17) and also setting $\chi=-1/2$
to account for the Majorana gravitino and gaugino, we obtain the first three
Seeley-DeWitt coefficients for the fermionic-sector (without the ghost
contribution),

$\begin{split}(4\pi)^{2}{a_{0}}^{\mathcal{G}^{f}_{3/2},\text{no-
ghost}}(x)&=-10,\\\ (4\pi)^{2}{a_{2}}^{\mathcal{G}^{f}_{3/2},\text{no-
ghost}}(x)&=4\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
(4\pi)^{2}{a_{4}}^{\mathcal{G}^{f}_{3/2},\text{no-
ghost}}(x)&=-\frac{1}{144}\left(41R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-128R_{\mu\nu}R^{\mu\nu}\right).\end{split}$
(64)

Additionally, one needs to include the contribution of the three minimally-
coupled spin-1/2 Majorana ghosts in the action (53). The combined Seeley-
DeWitt coefficients for the ghosts fields are
$a_{2n}^{\mathcal{G}^{f}_{3/2},\text{ghost}}=-3a_{2n}^{1/2}$, where the single
spinor contribution $a_{2n}^{1/2}$ can be accessed from the formulae (21) (set
$n_{1/2}=1,n_{0}=n_{1}=n_{3/2}=0$). Here the minus sign reflects that the
ghost fields counter the conventional spin-statistics of real fields. Thus,
the net contribution of the fermionic-sector ($\mathcal{G}^{f}_{3/2}$) is

$a_{2n}^{\mathcal{G}^{f}_{3/2}}=a_{2n}^{\mathcal{G}^{f}_{3/2},\text{no-
ghost}}+a_{2n}^{\mathcal{G}^{f}_{3/2},\text{ghost}}.$ (65)

On the other hand, the Seeley-DeWitt coefficients of the bosonic-sector
($\mathcal{G}^{b}_{3/2}$) with two minimally-coupled vector fields are
obtained as

$\begin{split}&(4\pi)^{2}{a_{0}}^{\mathcal{G}^{b}_{3/2}}(x)=4,\\\
&(4\pi)^{2}{a_{2}}^{\mathcal{G}^{b}_{3/2}}(x)=0,\\\
&(4\pi)^{2}{a_{4}}^{\mathcal{G}^{b}_{3/2}}(x)=-\frac{1}{90}\left(13R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-88R_{\mu\nu}R^{\mu\nu}\right).\end{split}$
(66)

Adding up the contributions $a_{2n}^{\mathcal{G}^{f}_{3/2}}$ and
$a_{2n}^{\mathcal{G}^{b}_{3/2}}$ necessarily lead to the first three Seeley-
DeWitt coefficients of the whole gravitino multiplet ($\mathcal{G}_{3/2}$),

$\begin{split}&(4\pi)^{2}a^{\mathcal{G}_{3/2}}_{0}(x)=0,\\\
&(4\pi)^{2}a^{\mathcal{G}_{3/2}}_{2}(x)=4\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&(4\pi)^{2}a^{\mathcal{G}_{3/2}}_{4}(x)=-\frac{11}{24}\left(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}\right).\end{split}$
(67)

Finally, we employ the $a^{\mathcal{G}_{3/2}}_{4}(x)$ data into the formula
(32) and find the logarithmic correction contribution of a single
$\mathcal{N}=2$ gravitino multiplet to the entropy of extremal Kerr-Newman,
Kerr and Reissner-Nordström black holes (for the limits (43), (44) and (45),
respectively). The results are

$\Delta
S^{\mathcal{G}_{3/2}}_{\text{BH}}=\begin{cases}-\frac{11}{12}\frac{(2{b^{\prime}}^{6}+5{b^{\prime}}^{4}+4{b^{\prime}}^{2}+1)}{({b^{\prime}}^{2}+1)^{2}(2{b^{\prime}}^{2}+1)}\thinspace\text{ln}\thinspace\mathcal{A}_{H}&\text{for
Kerr-Newman},\\\
-\frac{11}{12}\thinspace\text{ln}\thinspace\mathcal{A}_{H}&\text{for Kerr},\\\
-\frac{11}{12}\thinspace\text{ln}\thinspace\mathcal{A}_{H}&\text{for Reissner-
Nordstr\"{o}m}.\end{cases}$ (68)

###  3.3 $\mathcal{N}=2,d=4$ EMSGT: vector multiplet

The field contents of the vector multiplet ($\mathcal{G}_{1}$) of a
$\mathcal{N}=2,d=4$ EMSGT can be distributed into two sectors –

  * Bosonic sector ($\mathcal{G}^{b}_{1}$): one complex scalar field $\phi_{c}$ is non-minimally coupled to a vector field $a_{\mu}$ through the background graviphoton field strength $\bar{F}^{\mu\nu}$,

  * Fermionic sector ($\mathcal{G}^{f}_{1}$): two minimally-coupled Majorana gauginos.

To begin with the bosonic-sector, our immediate task is to open up the complex
scalar field $\phi_{c}$ into its real and imaginary parts ($\phi_{R}$ and
$\phi_{I}$, respectively) Sen:2012qq ,

$\phi_{c}=\frac{1}{2}(\phi_{R}-i\phi_{I}).$ (69)

The quadratic order fluctuated bosonic-sector action for the fluctuations
$\tilde{\xi}_{m}=\\{a_{\mu},\phi_{R},\phi_{I}\\}$ is expressed as

$\begin{split}\delta^{2}\mathcal{S}_{\mathcal{G}^{b}_{1}}&=\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\mathscr{L}_{\mathcal{G}^{b}_{1}},\\\
\mathscr{L}_{\mathcal{G}^{b}_{1}}&=-\frac{1}{32}f_{\mu\nu}f^{\mu\nu}-\frac{1}{16}D_{\rho}\phi_{R}D^{\rho}\phi_{R}-\frac{1}{16}D_{\rho}\phi_{I}D^{\rho}\phi_{I}+\frac{1}{64}{\phi_{R}}^{2}\bar{H}_{\mu\nu}\bar{H}^{\mu\nu}\\\
&\quad-\frac{1}{16}{\phi_{I}}^{2}\bar{F}_{\mu\nu}\bar{F}^{\mu\nu}-\frac{i}{16}\phi_{R}\phi_{I}\bar{F}_{\mu\nu}\bar{H}^{\mu\nu}+\frac{i}{16}\phi_{R}f_{\mu\nu}\bar{H}^{\mu\nu}+\frac{1}{8}\phi_{I}f_{\mu\nu}\bar{F}^{\mu\nu},\end{split}$
(70)

where $f_{\mu\nu}=\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}$ is the strength
tensor of the vector fluctuation $a_{\mu}$. To proceed with the strategy of
section 2.2, we customize the quadratic action (70) further up to total
derivatives. This involves the use of commutation relations (20a) and (20b) as
well as the inclusion of a gauge-fixing term to gauge fix the vector field
fluctuated part $f_{\mu\nu}f^{\mu\nu}$,

$\displaystyle-\frac{1}{16}\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace(D_{\mu}a^{\mu})(D_{\nu}a^{\nu}).$
(71)

The relevant Laplace-type kinetic operator $\Lambda$ holds the form,

$\begin{split}\delta^{2}\mathcal{S}_{\mathcal{G}^{b}_{1}}&=\frac{1}{16}\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\tilde{\xi}_{m}\Lambda^{mn}\tilde{\xi}_{n},\\\
\tilde{\xi}_{m}\Lambda^{mn}\tilde{\xi}_{n}&=\bar{g}^{\mu\nu}a_{\mu}D^{\rho}D_{\rho}a_{\nu}+\phi_{R}D^{\rho}D_{\rho}\phi_{R}+\phi_{I}D^{\rho}D_{\rho}\phi_{I}-a_{\mu}R^{\mu\nu}a_{\nu}\\\
&\quad+\frac{1}{4}\phi_{R}\bar{H}_{\mu\nu}\bar{H}^{\mu\nu}\phi_{R}-\phi_{I}\bar{F}_{\mu\nu}\bar{F}^{\mu\nu}\phi_{I}-\frac{i}{2}\phi_{R}\bar{F}_{\mu\nu}\bar{H}^{\mu\nu}\phi_{I}-\frac{i}{2}\phi_{I}\bar{F}_{\mu\nu}\bar{H}^{\mu\nu}\phi_{R}\\\
&\quad+ia_{\mu}\bar{H}^{\mu\rho}(D_{\rho}\phi_{R})+i\phi_{R}\bar{H}^{\rho\nu}(D_{\rho}a_{\nu})+2a_{\mu}\bar{F}^{\mu\rho}(D_{\rho}\phi_{I})+2\phi_{I}\bar{F}^{\rho\nu}(D_{\rho}a_{\nu}),\end{split}$
(72)

and provides the results for the matrices $I$, $P$ and
$\omega^{\rho}\thinspace(=\frac{1}{2}N^{\rho})$,

$\begin{split}\tilde{\xi}_{m}I^{mn}\tilde{\xi}_{n}&=a_{\mu}\bar{g}^{\mu\nu}a_{\nu}+\phi_{R}\phi_{R}+\phi_{I}\phi_{I},\\\
\tilde{\xi}_{m}P^{mn}\tilde{\xi}_{n}&=-a_{\mu}R^{\mu\nu}a_{\nu}+\frac{1}{4}\phi_{R}\bar{H}_{\mu\nu}\bar{H}^{\mu\nu}\phi_{R}-\phi_{I}\bar{F}_{\mu\nu}\bar{F}^{\mu\nu}\phi_{I}\\\
&\qquad-\frac{i}{2}\phi_{R}\bar{F}_{\mu\nu}\bar{H}^{\mu\nu}\phi_{I}-\frac{i}{2}\phi_{I}\bar{F}_{\mu\nu}\bar{H}^{\mu\nu}\phi_{R},\\\
\tilde{\xi}_{m}{(\omega^{\rho})}^{mn}\tilde{\xi}_{n}&=\frac{i}{2}a_{\mu}\bar{H}^{\mu\rho}\phi_{R}+\frac{i}{2}\phi_{R}\bar{H}^{\rho\nu}a_{\nu}+a_{\mu}\bar{F}^{\mu\rho}\phi_{I}+\phi_{I}\bar{F}^{\rho\nu}a_{\nu}.\end{split}$
(73)

Consequently, we can now utilize the above data into the formulae (16) and
(14) for deriving the expressions of $E$ and $\Omega_{\rho\sigma}$. The
results are

$\begin{split}&\tilde{\xi}_{m}E^{mn}\tilde{\xi}_{n}=a_{\mu}\Big{\\{}-R^{\mu\nu}+\bar{F}^{\mu\rho}\bar{F^{\nu}}_{\rho}-\frac{1}{4}\bar{H}^{\mu\rho}\bar{H^{\nu}}_{\rho}\Big{\\}}a_{\nu},\\\
&\tilde{\xi}_{m}(\Omega_{\rho\sigma})^{mn}\tilde{\xi}_{n}=a_{\mu}\Big{\\{}{R^{\mu\nu}}_{\rho\sigma}+\left({\bar{F^{\mu}}}_{\rho}{\bar{F_{\sigma}}}^{\nu}-{\bar{F^{\mu}}}_{\sigma}{\bar{F_{\rho}}}^{\nu}\right)-\frac{1}{4}\left({\bar{H^{\mu}}}_{\rho}{\bar{H_{\sigma}}}^{\nu}-{\bar{H^{\mu}}}_{\sigma}{\bar{H_{\rho}}}^{\nu}\right)\Big{\\}}a_{\nu}\\\
&\qquad\qquad\enspace+\frac{i}{2}a_{\mu}(D^{\mu}\bar{H}_{\rho\sigma})\phi_{R}-\frac{i}{2}\phi_{R}(D^{\nu}\bar{H}_{\rho\sigma})a_{\nu}+a_{\mu}(D^{\mu}\bar{F}_{\rho\sigma})\phi_{I}-\phi_{I}(D^{\nu}\bar{F}_{\rho\sigma})a_{\nu},\end{split}$
(74)

where simplifications were done with the help of Maxwell-Bianchi
equations.171717See appendix A. We, therefore, compute the following needful
trace values (see appendix B for details)

$\begin{split}&\text{tr}(I)=4+1+1=6,\\\ &\text{tr}(E)=0,\\\
&\text{tr}(E^{2})=0,\\\
&\text{tr}(\Omega_{\rho\sigma}\Omega^{\rho\sigma})=-R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+2R_{\mu\nu}R^{\mu\nu},\end{split}$
(75)

which allow us to find the first three Seeley-DeWitt coefficients for the
bosonic-sector (without the ghost contribution) with the help of the formulae
(17),

$\begin{split}(4\pi)^{2}{a_{0}}^{\mathcal{G}^{b}_{1},\text{no-
ghost}}(x)&=6,\\\ (4\pi)^{2}{a_{2}}^{\mathcal{G}^{b}_{1},\text{no-
ghost}}(x)&=0,\\\ (4\pi)^{2}{a_{4}}^{\mathcal{G}^{b}_{1},\text{no-
ghost}}(x)&=-\frac{1}{60}\left(3R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-8R_{\mu\nu}R^{\mu\nu}\right).\end{split}$
(76)

Here $\chi=1$ is chosen to account for the vector and scalar fluctuations.
Also, the gauge-fixing term (71) produces two scalar ghost fields $b$ and $c$
via the Lagrangian Banerjee:2011oo ,

$\displaystyle\mathscr{L}_{\mathcal{G}^{b}_{1},\thinspace\text{ghost}}=bD^{\rho}D_{\rho}c,$
(77)

The above scalar ghosts are minimally-coupled and their Seeley-DeWitt
contribution is thus $a_{2n}^{\mathcal{G}^{b}_{1},\text{ghost}}=-2a_{2n}^{s}$,
where the single scalar field results $a_{2n}^{s}$ can be accessed from the
formulae (21) (set $n_{0}=1,n_{1}=n_{1/2}=n_{3/2}=0$). Hence, the net Seeley-
DeWitt contribution of the bosonic-sector ($\mathcal{G}^{b}_{1}$) becomes

$a_{2n}^{\mathcal{G}^{b}_{1}}=a_{2n}^{\mathcal{G}^{b}_{1},\text{no-
ghost}}+a_{2n}^{\mathcal{G}^{b}_{1},\text{ghost}}.$ (78)

On the other hand, the fermionic-sector ($\mathcal{G}^{f}_{1}$) casts two
minimally-coupled spin-1/2 gaugino fields and provides,

$\begin{split}(4\pi)^{2}{a_{0}}^{\mathcal{G}^{f}_{1}}(x)&=-4,\\\
(4\pi)^{2}{a_{2}}^{\mathcal{G}^{f}_{1}}(x)&=0,\\\
(4\pi)^{2}{a_{4}}^{\mathcal{G}^{f}_{1}}(x)&=\frac{1}{360}\left(7R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+8R_{\mu\nu}R^{\mu\nu}\right).\end{split}$
(79)

By combining the contributions of bosonic and fermionic sectors, the Seeley-
DeWitt coefficients of the full vector multiplet ($\mathcal{G}_{1}$) are
obtained as

$\begin{split}&(4\pi)^{2}a^{\mathcal{G}_{1}}_{0}(x)=0,\\\
&(4\pi)^{2}a^{\mathcal{G}_{1}}_{2}(x)=0,\\\
&(4\pi)^{2}a^{\mathcal{G}_{1}}_{4}(x)=-\frac{1}{24}\left(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}\right).\end{split}$
(80)

Finally, the $a^{\mathcal{G}_{1}}_{4}(x)$ result needs to be utilized in the
set-up of section 2.3 for calculating the logarithmic entropy correction
contributions of a single $\mathcal{N}=2$ vector multiplet. The results are

$\Delta
S^{\mathcal{G}_{1}}_{\text{BH}}=\begin{cases}-\frac{1}{12}\frac{(2{b^{\prime}}^{6}+5{b^{\prime}}^{4}+4{b^{\prime}}^{2}+1)}{({b^{\prime}}^{2}+1)^{2}(2{b^{\prime}}^{2}+1)}\thinspace\text{ln}\thinspace\mathcal{A}_{H}&\text{for
Kerr-Newman},\\\
-\frac{1}{12}\thinspace\text{ln}\thinspace\mathcal{A}_{H}&\text{for Kerr},\\\
-\frac{1}{12}\thinspace\text{ln}\thinspace\mathcal{A}_{H}&\text{for Reissner-
Nordstr\"{o}m}.\end{cases}$ (81)

###  3.4 $\mathcal{N}=2,d=4$ EMSGT: hyper multiplet

The field contents of the hyper multiplet ($\mathcal{G}_{1/2}$) of a
$\mathcal{N}=2,d=4$ EMSGT is also distributed into bosonic and fermionic
sectors –

  * Bosonic sector ($\mathcal{G}^{b}_{1/2}$): four minimally-coupled real scalars,

  * Fermionic sector ($\mathcal{G}^{f}_{1/2}$): two spin-1/2 Weyl spinors $\\{\zeta_{1},\zeta_{2}\\}$ are coupled non-minimally to their opposite chiralities $\\{\zeta^{1},\zeta^{2}\\}$ via the background graviphoton strength $\bar{F}^{\mu\nu}$.

We begin with the fermionic-sector and define a Majorana spinor corresponding
to each Weyl fermions by combining with their chiral partner Sen:2012qq ,

$\lambda_{1}=\zeta_{1}+\zeta^{1},\thinspace\lambda_{2}=\zeta_{2}+\zeta^{2},$
(82)

where the newly formed Majorana spinors $\lambda_{1}$ and $\lambda_{2}$ are
coupled through the following action (up to quadratic order)

$\begin{gathered}\delta^{2}\mathcal{S}_{\mathcal{G}^{f}_{1/2}}=-\frac{1}{2}\int\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\mathscr{L}_{\mathcal{G}^{f}_{1/2}},\\\
\mathscr{L}_{\mathcal{G}^{f}_{1/2}}=\bar{\lambda}_{1}\gamma^{\rho}D_{\rho}\lambda_{1}+\bar{\lambda}_{2}\gamma^{\rho}D_{\rho}\lambda_{2}-\frac{1}{4}\bar{\lambda}_{1}\bar{F}_{\alpha\beta}\gamma^{\alpha}\gamma^{\beta}\lambda_{2}+\frac{1}{4}\bar{\lambda}_{2}\gamma^{\alpha}\gamma^{\beta}\bar{F}_{\alpha\beta}\lambda_{1}.\end{gathered}$
(83)

By expressing
$\mathscr{L}_{\mathcal{G}^{f}_{1/2}}=\tilde{\xi}_{m}\not{D}^{mn}\tilde{\xi}_{n}$,
one finds the form of relevant first-order Dirac-type operator $\not{D}$ as

$\tilde{\xi}_{m}\not{D}^{mn}\tilde{\xi}_{n}=i\bar{\lambda}_{1}\gamma^{\rho}D_{\rho}\lambda_{1}+i\bar{\lambda}_{2}\gamma^{\rho}D_{\rho}\lambda_{2}-\frac{i}{4}\bar{\lambda}_{1}\hat{\mathcal{F}}\lambda_{2}+\frac{i}{4}\bar{\lambda}_{2}\hat{\mathcal{F}}\lambda_{1}.$
(84)

Note that the operator $\not{D}$ is Hermitian since $\gamma^{\mu}$,
$\hat{\mathcal{F}}=\bar{F}_{\alpha\beta}\gamma^{\alpha}\gamma^{\beta}$ are
Hermitian and $(\gamma^{\rho}D_{\rho})^{\dagger}=-\gamma^{\rho}D_{\rho}$. From
here, one needs to follow up the typical fermionic strategy of structuring a
second-order operator $\Lambda$ by squaring the Dirac-type operator form (84):

$\displaystyle\tilde{\xi}_{m}\Lambda^{mn}\tilde{\xi}_{n}$
$\displaystyle=\tilde{\xi}_{m}(\not{D}^{mp})^{\dagger}{\not{D}_{p}}^{n}\tilde{\xi}_{n}$
$\displaystyle=-\bar{\lambda}_{1}\bigg{\\{}\gamma^{\rho}\gamma^{\sigma}D_{\rho}D_{\sigma}-\frac{1}{4}\hat{\mathcal{F}}\hat{\mathcal{F}}\bigg{\\}}\lambda_{1}-\bar{\lambda}_{2}\bigg{\\{}\gamma^{\rho}\gamma^{\sigma}D_{\rho}D_{\sigma}-\frac{1}{4}\hat{\mathcal{F}}\hat{\mathcal{F}}\bigg{\\}}\lambda_{2}$
$\displaystyle\qquad+\bar{\lambda}_{1}\bigg{\\{}\frac{1}{2}\gamma^{\rho}(D_{\rho}\hat{\mathcal{F}}+\hat{\mathcal{F}}D_{\rho})+\frac{1}{2}\hat{\mathcal{F}}\gamma^{\rho}D_{\rho}\bigg{\\}}\lambda_{2}$
$\displaystyle\qquad-\bar{\lambda}_{2}\bigg{\\{}\frac{1}{2}\hat{\mathcal{F}}\gamma^{\rho}D_{\rho}+\frac{1}{2}\gamma^{\rho}(D_{\rho}\hat{\mathcal{F}}+\hat{\mathcal{F}}D_{\rho})\bigg{\\}}\lambda_{1}.$
(85)

The above operator form is not of Laplace-type as depicted in the schematic
(12). But we can repair it using the operator identity (58b) and achieve the
desired Laplace-type form of $\Lambda$ as

$\displaystyle\tilde{\xi}_{m}\Lambda^{mn}\tilde{\xi}_{n}$
$\displaystyle=-\mathbb{I}_{4}\bar{\lambda}_{1}D^{\rho}D_{\rho}\lambda_{1}-\mathbb{I}_{4}\bar{\lambda}_{2}D^{\rho}D_{\rho}\lambda_{2}+\frac{1}{4}\bar{\lambda}_{1}\hat{\mathcal{F}}\hat{\mathcal{F}}\lambda_{1}$
$\displaystyle\quad+\frac{1}{4}\bar{\lambda}_{2}\hat{\mathcal{F}}\hat{\mathcal{F}}\lambda_{2}+\frac{1}{2}\bar{\lambda}_{1}\gamma^{\rho}(D_{\rho}\hat{\mathcal{F}})\lambda_{2}-\frac{1}{2}\bar{\lambda}_{2}\gamma^{\rho}(D_{\rho}\hat{\mathcal{F}})\lambda_{1}$
$\displaystyle\quad+\frac{1}{2}\bar{\lambda}_{1}(\gamma^{\rho}\hat{\mathcal{F}}+\hat{\mathcal{F}}\gamma^{\rho})D_{\rho}\lambda_{2}-\frac{1}{2}\bar{\lambda}_{2}(\gamma^{\rho}\hat{\mathcal{F}}+\hat{\mathcal{F}}\gamma^{\rho})D_{\rho}\lambda_{1}.$
(86)

It is now quite easy to extract out $I$, $P$ and $\omega^{\rho}$
($=\frac{1}{2}N^{\rho}$),

$\displaystyle\tilde{\xi}_{m}I^{mn}\tilde{\xi}_{n}$
$\displaystyle=\bar{\lambda}_{1}\mathbb{I}_{4}\lambda_{1}+\bar{\lambda}_{2}\mathbb{I}_{4}\lambda_{2},$
$\displaystyle\tilde{\xi}_{m}P^{mn}\tilde{\xi}_{n}$
$\displaystyle=-\frac{1}{4}\bar{\lambda}_{1}\gamma^{\alpha}\gamma^{\beta}\gamma^{\theta}\gamma^{\phi}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}\lambda_{1}-\frac{1}{4}\bar{\lambda}_{2}\gamma^{\alpha}\gamma^{\beta}\gamma^{\theta}\gamma^{\phi}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}\lambda_{2}$
$\displaystyle\quad-\frac{1}{2}\bar{\lambda}_{1}\gamma^{\rho}\gamma^{\alpha}\gamma^{\beta}(D_{\rho}\bar{F}_{\alpha\beta})\lambda_{2}+\frac{1}{2}\bar{\lambda}_{2}\gamma^{\rho}\gamma^{\alpha}\gamma^{\beta}(D_{\rho}\bar{F}_{\alpha\beta})\lambda_{1},$
(87) $\displaystyle\tilde{\xi}_{m}{(\omega^{\rho})}^{mn}\tilde{\xi}_{n}$
$\displaystyle=-\bar{\lambda}_{1}\bigg{\\{}\frac{1}{4}\gamma^{\rho}\gamma^{\alpha}\gamma^{\beta}\bar{F}_{\alpha\beta}+\frac{1}{4}\gamma^{\alpha}\gamma^{\beta}\gamma^{\rho}\bar{F}_{\alpha\beta}\bigg{\\}}\lambda_{2}$
$\displaystyle\quad+\bar{\lambda}_{2}\bigg{\\{}\frac{1}{4}\gamma^{\rho}\gamma^{\alpha}\gamma^{\beta}\bar{F}_{\alpha\beta}+\frac{1}{4}\gamma^{\alpha}\gamma^{\beta}\gamma^{\rho}\bar{F}_{\alpha\beta}\bigg{\\}}\lambda_{1}.$

and then jot down the expressions for the matrices $\Omega_{\rho\sigma}$ and
$E$ (using eqs. 14 and 16). The simplified results are

$\displaystyle\tilde{\xi}_{m}E^{mn}\tilde{\xi}_{n}$
$\displaystyle=\frac{1}{16}\bar{\lambda}_{1}\gamma^{\rho}\gamma^{\alpha}\gamma^{\beta}\gamma^{\theta}\gamma^{\phi}\gamma_{\rho}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}\lambda_{1}+\frac{1}{16}\bar{\lambda}_{2}\gamma^{\rho}\gamma^{\alpha}\gamma^{\beta}\gamma^{\theta}\gamma^{\phi}\gamma_{\rho}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}\lambda_{2},$
$\displaystyle\tilde{\xi}_{m}{(\Omega_{\rho\sigma})}^{mn}\tilde{\xi}_{n}$
$\displaystyle=\bar{\lambda}_{1}\bigg{\\{}\frac{1}{4}\gamma^{\alpha}\gamma^{\beta}R_{\rho\sigma\alpha\beta}-\frac{1}{16}\Big{(}(\gamma_{\rho}\gamma^{\alpha}\gamma^{\beta}+\gamma^{\alpha}\gamma^{\beta}\gamma_{\rho})$
$\displaystyle\qquad\quad\times(\gamma_{\sigma}\gamma^{\theta}\gamma^{\phi}+\gamma^{\theta}\gamma^{\phi}\gamma_{\sigma})\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}-(\rho\leftrightarrow\sigma)\Big{)}\bigg{\\}}\lambda_{1}$
$\displaystyle\quad+\bar{\lambda}_{2}\bigg{\\{}\frac{1}{4}\gamma^{\alpha}\gamma^{\beta}R_{\rho\sigma\alpha\beta}-\frac{1}{16}\Big{(}(\gamma_{\rho}\gamma^{\alpha}\gamma^{\beta}+\gamma^{\alpha}\gamma^{\beta}\gamma_{\rho})$
(88)
$\displaystyle\qquad\quad\times(\gamma_{\sigma}\gamma^{\theta}\gamma^{\phi}+\gamma^{\theta}\gamma^{\phi}\gamma_{\sigma})\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}-(\rho\leftrightarrow\sigma)\Big{)}\bigg{\\}}\lambda_{2}$
$\displaystyle\quad-\bar{\lambda}_{1}\bigg{\\{}\frac{1}{4}(\gamma_{\sigma}\gamma^{\alpha}\gamma^{\beta}+\gamma^{\alpha}\gamma^{\beta}\gamma_{\sigma})(D_{\rho}\bar{F}_{\alpha\beta})-(\rho\leftrightarrow\sigma)\bigg{\\}}\lambda_{2}$
$\displaystyle\quad+\bar{\lambda}_{2}\bigg{\\{}\frac{1}{4}(\gamma_{\sigma}\gamma^{\alpha}\gamma^{\beta}+\gamma^{\alpha}\gamma^{\beta}\gamma_{\sigma})(D_{\rho}\bar{F}_{\alpha\beta})-(\rho\leftrightarrow\sigma)\bigg{\\}}\lambda_{1}.$

The absence of any cross-terms involving $\lambda_{1}$ and $\lambda_{2}$ in
the expression of $E$ is not a direct consequence – the type of coupling
between $\lambda_{1}$ and $\lambda_{2}$ makes the cross-terms to vanish while
simplifying $E$ with the help of the relations (3.4). As usual, our ultimate
task is to calculate the trace values (refer to appendix B),

$\begin{split}\text{tr}(I)&=4+4=8,\\\
\text{tr}(E)&=-4\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
\text{tr}(E^{2})&=4(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}-2R_{\mu\nu}R^{\mu\nu},\\\
\text{tr}(\Omega_{\rho\sigma}\Omega^{\rho\sigma})&=-R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+16R_{\mu\nu}R^{\mu\nu}-24(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2},\end{split}$
(89)

and insert them into the formulae (17) for finding the Seeley-DeWitt
coefficients of the fermionic-sector ($\mathcal{G}^{f}_{1/2}$):

$\begin{split}(4\pi)^{2}a^{\mathcal{G}^{f}_{1/2}}_{0}(x)&=-4,\\\
(4\pi)^{2}a^{\mathcal{G}^{f}_{1/2}}_{2}(x)&=2\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
(4\pi)^{2}a^{\mathcal{G}^{f}_{1/2}}_{4}(x)&=\frac{1}{360}\left(7R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-52R_{\mu\nu}R^{\mu\nu}\right).\end{split}$
(90)

The above result considered $\chi=-1/2$ to account for the Majorana fermions
$\lambda_{1}$ and $\lambda_{2}$. On the other hand, the bosonic-sector
($\mathcal{G}^{b}_{1/2}$) has four minimally-coupled scalars and the
corresponding Seeley-DeWitt contributions $a_{2n}^{\mathcal{G}^{b}_{1/2}}$ can
be accessed by setting $n_{0}=4,n_{1}=n_{1/2}=n_{3/2}=0$ in the formulae (21).
Adding up the $\mathcal{G}^{f}_{1/2}$ and $\mathcal{G}^{b}_{1/2}$
contributions, the first three Seeley-DeWitt coefficients of the hyper
multiplet ($\mathcal{G}_{1/2}$) are presented as

$\begin{split}&(4\pi)^{2}a^{\mathcal{G}_{1/2}}_{0}(x)=0,\\\
&(4\pi)^{2}a^{\mathcal{G}_{1/2}}_{2}(x)=2\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&(4\pi)^{2}a^{\mathcal{G}_{1/2}}_{4}(x)=\frac{1}{24}\left(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}\right).\end{split}$
(91)

The above Seeley-DeWitt data, especially the $a^{\mathcal{G}_{1/2}}_{4}(x)$,
are all we need (in the structure of section 2.3) for calculating the
contribution of a single $\mathcal{N}=2$ hyper multiplet in logarithmic
corrections to the entropy of extremal Kerr-Newman, Kerr and Reissner-
Nordström black holes. The results are

$\Delta
S^{\mathcal{G}_{1/2}}_{\text{BH}}=\begin{cases}\frac{1}{12}\frac{(2{b^{\prime}}^{6}+5{b^{\prime}}^{4}+4{b^{\prime}}^{2}+1)}{({b^{\prime}}^{2}+1)^{2}(2{b^{\prime}}^{2}+1)}\thinspace\text{ln}\thinspace\mathcal{A}_{H}&\text{for
Kerr-Newman},\\\
\frac{1}{12}\thinspace\text{ln}\thinspace\mathcal{A}_{H}&\text{for Kerr},\\\
\frac{1}{12}\thinspace\text{ln}\thinspace\mathcal{A}_{H}&\text{for Reissner-
Nordstr\"{o}m}.\end{cases}$ (92)

All the Seeley-DeWitt results recorded in eqs. 49, 67, 80 and 91 are
consistent with the eigenfunction expansion technique results Sen:2012qq ;
Keeler:2014nn for $AdS_{2}\times S^{2}$ background and also with the results
of hybrid field redefinition approach Charles:2015nn . Interestingly, the
$a_{4}(x)$ results are noticed to be fully electro-magnetic duality invariant,
i.e., they do not include the terms proportional to
$R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}$ and
$(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}$. It is also essential to mention that
the logarithmic corrections (68), (81) and (92) are obtained by incorporating
only the $\mathcal{C}_{\text{local}}$ contribution for the corresponding
multiplets. This is because the $\mathcal{C}_{\text{zm}}$ contribution (33) is
fixed for a particular $\mathcal{N}=2,d=4$ EMSGT and already added in the
supergravity multiplet results (50), while the $\mathcal{C}_{\text{local}}$
contribution is different for individual multiplets. The extremal Reissner-
Nordström logarithmic correction results are in accordance with Sen:2012qq ;
Keeler:2014nn .

##  4 Discussions

In this section, we aim to provide some crucial applications of the results
obtained in section 3 for different $\mathcal{N}=2$ multiplets. This includes
finding logarithmic corrections to the entropy of extremal Kerr-Newman, Kerr
and Reissner-Nordström black holes in the matter coupled $\mathcal{N}=2$
EMSGT, $\mathcal{N}=1$ EMSGT as truncation of $\mathcal{N}=2$ multiplets and
$\mathcal{N}\geq 3$ EMSGTs by decomposing into $\mathcal{N}=2$ multiplets. In
this way, we achieve our central goal of carrying out the logarithmic entropy
correction program for extremal black holes in all $\mathcal{N}\geq 1$ matter
coupled EMSGTs. Additionally, we explore the same results for the non-extremal
Kerr-Newman family of black holes and compare them with the extremal results.
To achieve this goal, we follow the Euclidean gravity approach Sen:2013ns
where one needs to analyze the coefficients of trace anomalies from the
$a_{4}(x)$ results. Note that the extremal Reissner-Nordström black holes are
$\frac{1}{\mathcal{N}}$-BPS solutions in the $\mathcal{N}\geq 2,d=4$ EMSGTs,
while all other black holes are non-BPS in the $\mathcal{N}\geq 1,d=4$ EMSGTs.
We end up discussing some concluding remarks about our results.

###  4.1 Logarithmic correction to the entropy of extremal black holes in the
matter coupled $\mathcal{N}=2,d=4$ EMSGT

A pure or ‘minimal’ $\mathcal{N}=2,d=4$ EMSGT includes only a supergravity
multiplet Karan:2019sk . At any point, one can generalize it further by
coupling additional matter multiplets such as $n_{V}$ number of vector
multiplets and $n_{H}$ number of hyper multiplets to the supergravity
multiplet. This generalized theory is called the matter coupled
$\mathcal{N}=2,d=4$ EMSGT in which several non-minimal couplings are
distributed so that each multiplet behaves as a minimal one. The Seeley-DeWitt
coefficients and logarithmic correction contributions of the $\mathcal{N}=2$
supergravity, vector and hyper multiplets have been evaluated and recorded in
sections 3.1, 3.3 and 3.4, respectively. Putting these results together, we
present the form of $a_{4}(x)$ coefficient in the matter coupled
$\mathcal{N}=2,d=4$ EMSGT as

$(4\pi)^{2}a^{\mathcal{N}=2}_{4}(x)=\frac{1}{24}\big{(}11-n_{V}+n_{H}\big{)}\big{(}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}\big{)},$
(93)

followed by the logarithmic correction to the entropy of extremal Kerr-Newman,
Kerr and Reissner-Nordström black holes,

$\Delta
S^{\mathcal{N}=2}_{\text{BH}}=\begin{cases}\left(\frac{1}{12}\big{(}11-n_{V}+n_{H}\big{)}\frac{(2{b^{\prime}}^{6}+5{b^{\prime}}^{4}+4{b^{\prime}}^{2}+1)}{({b^{\prime}}^{2}+1)^{2}(2{b^{\prime}}^{2}+1)}-2\right)\text{ln}\thinspace\mathcal{A}_{H}&\text{for
Kerr-Newman},\\\
\frac{1}{12}\big{(}-13-n_{V}+n_{H}\big{)}\thinspace\text{ln}\thinspace\mathcal{A}_{H}&\text{for
Kerr},\\\
\frac{1}{12}\big{(}23-n_{V}+n_{H}\big{)}\text{ln}\thinspace\mathcal{A}_{H}&\text{for
Reissner-Nordstr\"{o}m}.\end{cases}$ (94)

The extremal Reissner-Nordström black hole result agrees with Sen:2012qq ,
where calculations were done using the eigenfunction expansion method for
$AdS_{2}\times S^{2}$ background. The formulae (94) are actually the
generalization of the ‘minimal’ $\mathcal{N}=2,d=4$ EMSGT results provided in
Karan:2019sk .

###  4.2 Logarithmic correction to the entropy of extremal black holes in the
matter coupled $\mathcal{N}=1,d=4$ EMSGT

Any general $\mathcal{N}=1$ EMSGT consists of four massless multiplets – the
graviton (${g}_{2}$), gravitino (${g}_{3/2}$), vector (${g}_{1}$) and chiral
(${g}_{1/2}$) multiplets. The spin contents of these multiplets in $d=4$
space-time dimensions are recorded in table 2. Analyzing the spin content of
both table 1 and table 2, it is possible to disintegrate the
$\mathcal{N}=2,d=4$ EMSGT multiplets into the multiplets of
$\mathcal{N}=1,d=4$ EMSGT as following

$\displaystyle\begin{split}\mathcal{G}_{2}&=g_{2}+g_{3/2},\\\
\mathcal{G}_{3/2}&=g_{3/2}+g_{1},\\\ \mathcal{G}_{1}&=g_{1}+g_{1/2},\\\
\mathcal{G}_{1/2}&=2g_{1/2}.\end{split}$ (95)

This particular class of $\mathcal{N}=1$ EMSGT is purely obtained as
consistent kinematic decomposition of $\mathcal{N}=2$ multiplets, where the
interactions in different multiplets are constrained by some assumptions.
However, there is no restriction on the multiplicity of any multiplet content.
There must be at least one vector ($g_{1}$) and chiral ($g_{1/2}$) multiplets.
Readers are referred to Ferrara:2012bp ; Andrianopoli:2002gm and references
therein for all technical details and related assumptions for the reduction
$\mathcal{N}=2\to\mathcal{N}=1$. The $\mathcal{N}=2$ data from eqs. 50, 68, 81
and 92 can help us solve the equalities (95) and lead to logarithmic entropy
correction contributions of the $\mathcal{N}=1,d=4$ multiplets as recorded in
table 3. In a matter coupled $\mathcal{N}=1$ EMSGT, a single supergravity
multiplet is minimally coupled to $n_{v}$ number of vector multiplets and
$n_{c}$ number of chiral multiplets. Therefore, the logarithmic corrections to
the entropy of Kerr-Newman, Kerr and Reissner-Nordström black holes in the
matter coupled $\mathcal{N}=1,d=4$ EMSGT (accordant with the truncation (95))
are obtained as

$\Delta
S^{\mathcal{N}=1}_{\text{BH}}=\begin{cases}\left(\frac{1}{24}\big{(}41-3n_{v}+n_{c}\big{)}\frac{(2{b^{\prime}}^{6}+5{b^{\prime}}^{4}+4{b^{\prime}}^{2}+1)}{({b^{\prime}}^{2}+1)^{2}(2{b^{\prime}}^{2}+1)}-2\right)\text{ln}\thinspace\mathcal{A}_{H}&\text{for
Kerr-Newman},\\\
\frac{1}{24}\big{(}-7-3n_{v}+n_{c}\big{)}\thinspace\text{ln}\thinspace\mathcal{A}_{H}&\text{for
Kerr},\\\
\frac{1}{24}\big{(}65-3n_{v}+n_{c}\big{)}\thinspace\text{ln}\thinspace\mathcal{A}_{H}&\text{for
Reissner-Nordstr\"{o}m}.\end{cases}$ (96)

The above extremal Reissner-Nordström result is consistent with Ferrara:2012bp
.

Multiplets |  $s_{\text{max}}$ |  $\bm{s=2}$ |  $\bm{s=\frac{3}{2}}$ |  $\bm{s=1}$ |  $\bm{s=\frac{1}{2}}$ |  $\bm{s=0}$
---|---|---|---|---|---|---
supergravity (${g}_{2}$) |  2 |  1 |  1 |  |  |
gravitino (${g}_{3/2}$) |  $\frac{3}{2}$ |  |  1 |  1 |  |
vector (${g}_{1}$) |  1 |  |  |  1 |  1 |
chiral (${g}_{1/2}$) |  $\frac{1}{2}$ |  |  |  |  1 |  1+1
Table 2: Different multiplets and spin content of a $\mathcal{N}=1,d=4$ EMSGT. $\mathcal{N}=1$ multiplet contributions |  Kerr-Newman |  Kerr |  Reissner-Nordström
---|---|---|---
$\Delta S^{g_{2}}_{\text{BH}}/\ln\mathcal{A}_{H}$ |  $\frac{41}{24}\frac{(2{b^{\prime}}^{6}+5{b^{\prime}}^{4}+4{b^{\prime}}^{2}+1)}{({b^{\prime}}^{2}+1)^{2}(2{b^{\prime}}^{2}+1)}-2$ |  $-\frac{7}{24}$ |  $\frac{65}{24}$
$\Delta S^{g_{3/2}}_{\text{BH}}/\ln\mathcal{A}_{H}$ |  $-\frac{19}{24}\frac{(2{b^{\prime}}^{6}+5{b^{\prime}}^{4}+4{b^{\prime}}^{2}+1)}{({b^{\prime}}^{2}+1)^{2}(2{b^{\prime}}^{2}+1)}$ |  $-\frac{19}{24}$ |  $-\frac{19}{24}$
$\Delta S^{g_{1}}_{\text{BH}}/\ln\mathcal{A}_{H}$ |  $-\frac{1}{8}\frac{(2{b^{\prime}}^{6}+5{b^{\prime}}^{4}+4{b^{\prime}}^{2}+1)}{({b^{\prime}}^{2}+1)^{2}(2{b^{\prime}}^{2}+1)}$ |  $-\frac{1}{8}$ |  $-\frac{1}{8}$
$\Delta S^{g_{1/2}}_{\text{BH}}/\ln\mathcal{A}_{H}$ |  $\frac{1}{24}\frac{(2{b^{\prime}}^{6}+5{b^{\prime}}^{4}+4{b^{\prime}}^{2}+1)}{({b^{\prime}}^{2}+1)^{2}(2{b^{\prime}}^{2}+1)}$ |  $\frac{1}{24}$ |  $\frac{1}{24}$
Table 3: Logarithmic correction contributions of the multiplets of the
decomposed $\mathcal{N}=1,d=4$ EMSGT.

###  4.3 Logarithmic correction to the entropy of extremal black holes in the
matter coupled $\mathcal{N}\geq 3,d=4$ EMSGTs

For any general $\mathcal{N}\geq 3,d=4$ EMSGT, the massless content of
supergravity multiplet (${\mathcal{G}_{2}}^{\mathcal{N}\geq 3}$) includes one
graviton, $\mathcal{N}$ number of gravitini and
$\frac{1}{2}\mathcal{N}(\mathcal{N}-1)$ number of vector fields
Freedman:2012xp . At any point, it is possible to decompose them by
distributing the graviton, two gravitini and one vector field into a
$\mathcal{N}=2$ supergravity multiplet ($\mathcal{G}_{2}$). Then the remaining
content of the $\mathcal{N}\geq 3$ supergravity multiplet
(${\mathcal{G}_{2}}^{\mathcal{N}\geq 3}$), vector multiplet
(${\mathcal{G}_{1}}^{\mathcal{N}\geq 3}$) and hyper multiplet
(${\mathcal{G}_{1/2}}^{\mathcal{N}\geq 3}$) can be decomposed into
$(\mathcal{N}-2)$ number of $\mathcal{N}=2$ gravitino multiplets
($\mathcal{G}_{3/2}$) and a particular combination of $\mathcal{N}=2$ vector
($\mathcal{G}_{1}$) and $\mathcal{N}=2$ hyper ($\mathcal{G}_{1/2}$)
multiplets. Collectively, one can report the following multiplet
decompositions for the matter coupled $\mathcal{N}\geq 3,d=4$
EMSGTs181818Since $\mathcal{N}\geq 9,d=4$ EMSGTs are deserts of more than one
graviton and higher spin ($s\geq$5/2) interactions, the decomposition (98) is
valid for the $\mathcal{N}<9,d=4$ EMSGTs only.

$\displaystyle\begin{gathered}{\mathcal{G}_{2}}^{\mathcal{N}\geq
3}+{\mathcal{G}_{1}}^{\mathcal{N}\geq 3}+{\mathcal{G}_{1/2}}^{\mathcal{N}\geq
3}=\mathcal{G}_{2}+(\mathcal{N}-2)\mathcal{G}_{3/2}+n_{V}\mathcal{G}_{1}+n_{H}\mathcal{G}_{1/2},\end{gathered}$
(98)

where $n_{V}$ and $n_{H}$ are respectively multiplicities of the
$\mathcal{N}=2$ vector ($\mathcal{G}_{1}$) and $\mathcal{N}=2$ hyper
($\mathcal{G}_{1/2}$) multiplets for any particular $\mathcal{N}\geq
3\to\mathcal{N}=2$ decomposition. For example Keeler:2014nn ; Charles:2015nn ;
Ferrara:2012bp , $\mathcal{N}=3\to\mathcal{N}=2$ has $n_{V}=n_{H}$,
$\mathcal{N}=4\to\mathcal{N}=2$ has $n_{V}=n_{H}+1$,
$\mathcal{N}=5\to\mathcal{N}=2$ has $n_{V}=3,n_{H}=1$,
$\mathcal{N}=6\to\mathcal{N}=2$ has $n_{V}=7,n_{H}=4$,
$\mathcal{N}=8\to\mathcal{N}=2$ has $n_{V}=15,n_{H}=10$, etc. However, we
speculate that the numbers $n_{V}$ and $n_{H}$ for the decomposed matter
coupled $\mathcal{N}\geq 3,d=4$ EMSGTs are related by a general formula,

$n_{H}-n_{V}=3-\mathcal{N}.$ (99)

The $\mathcal{N}=2$ logarithmic correction data of eqs. 50, 81 and 92, along
with the relation (99), help us to solve the decomposition (98). This
essentially provides the logarithmic correction to the entropy of extremal
Kerr-Newman, Kerr and Reissner-Nordström black holes in the matter coupled
$\mathcal{N}\geq 3,d=4$ EMSGTs,

$\Delta S^{\mathcal{N}\geq
3}_{\text{BH}}=\begin{cases}\left((3-\mathcal{N})\frac{(2{b^{\prime}}^{6}+5{b^{\prime}}^{4}+4{b^{\prime}}^{2}+1)}{({b^{\prime}}^{2}+1)^{2}(2{b^{\prime}}^{2}+1)}-2\right)\text{ln}\thinspace\mathcal{A}_{H}&\text{for
Kerr-Newman},\\\
(1-\mathcal{N})\thinspace\text{ln}\thinspace\mathcal{A}_{H}&\text{for
Kerr},\\\
(4-\mathcal{N})\thinspace\text{ln}\thinspace\mathcal{A}_{H}&\text{for
Reissner-Nordstr\"{o}m}.\end{cases}$ (100)

The extremal Reissner-Nordström corrections are in alliance with
Banerjee:2011pp where calculations were encountered by the eigenfunction
expansion method for $\mathcal{N}=4,8$ EMSGTs. Also, the Reissner-Nordström
formula for all $\mathcal{N}\geq 3$ EMSGTs is the same as estimated in
Keeler:2014nn ; Ferrara:2012bp .

It is also possible to fit both the matter coupled $\mathcal{N}=2,d=4$ EMSGT
and $\mathcal{N}\geq 3,d=4$ EMSGTs into a single frame. By merging the results
(94) and (100), the combined logarithmic correction formula for extremal Kerr-
Newman family of black holes in the matter coupled $\mathcal{N}\geq 2,d=4$
EMSGTs is presented as

$\Delta S^{\mathcal{N}\geq
2}_{\text{BH}}=\left(\frac{1}{12}\big{(}11-11(\mathcal{N}-2)-n_{V}+n_{H}\big{)}\frac{(2{b^{\prime}}^{6}+5{b^{\prime}}^{4}+4{b^{\prime}}^{2}+1)}{({b^{\prime}}^{2}+1)^{2}(2{b^{\prime}}^{2}+1)}+\frac{\mathcal{C}_{\text{zm}}}{2}\right)\text{ln}\thinspace\mathcal{A}_{H}.$
(101)

The above formula is general, compact, and one of the central interest of this
paper. It provides the logarithmic correction results of extremal Kerr-Newman
(${b^{\prime}}\to$ finite and non-zero), Kerr (${b^{\prime}}\to\infty$) and
Reissner-Nordström (${b^{\prime}}\to 0$) black holes in the matter coupled
$\mathcal{N}\geq 2,d=4$ EMSGTs. Here the $\mathcal{C}_{\text{zm}}$
contribution for a particular choice of extremal black hole is obtained by
setting $\delta=0$ in the formula (33).

###  4.4 Logarithmic correction to the entropy of non-extremal black holes in
the matter coupled $\mathcal{N}\geq 1,d=4$ EMSGTs

The Euclidean gravity approach Sen:2013ns (also reviewed in Charles:2015nn )
aids in extending the logarithmic entropy correction program for non-extremal
black holes via the following formula

$\Delta S_{\text{BH}}^{\text{non-
ext}}=\frac{1}{2}(\mathcal{C}_{\text{local}}+\mathcal{C}_{\text{zm}})\,\text{ln}\thinspace\left(\frac{\mathcal{A}_{H}}{G_{N}}\right),$
(102)

where the local contribution can be described as a density of the Seeley-
DeWitt coefficient $a_{4}(x)$ over the whole black hole geometry including the
near-horizon,

$\mathcal{C}_{\text{local}}=\int_{\text{full
geometry}}\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace
a_{4}(x).$ (103)

It is also well understood that the logarithmic correction formula (102)
stores all trace anomaly data of the stress-energy tensor into the Seeley-
DeWitt coefficient $a_{4}(x)$ (see Keeler:2014nn , for instance). And one can
express Charles:2015nn ,

$\displaystyle(4\pi)^{2}a_{4}(x)=cW_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}-aE_{4},$
(104)

where the Weyl tensor square $W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}$ and
the 4D Euler density $E_{4}$ are particular trace anomalies with coefficients
$c$ and $a$, respectively. For any background geometry,
$W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}$ and $E_{4}$ have the typical forms,

$\displaystyle\begin{gathered}W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-2R_{\mu\nu}R^{\mu\nu}+\frac{1}{3}R^{2},\\\
E_{4}=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}.\end{gathered}$
(107)

The coefficients of the trace anomalies ($c,a$) are numerical constants that
depend on the type of field content and their coupling to the background. With
the help of the data available in Sen:2013ns , we present the following
necessary limits of the trace anomalies over the whole non-extremal Kerr-
Newman geometry (34) (in the same Euclidean continuation $t\to-i\theta$)

$\begin{split}\int_{\text{full
geometry}}\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace
E_{4}&=64\pi^{2},\\\ \int_{\text{full
geometry}}\mathrm{d}^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace
W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}&=64\pi^{2}+\frac{\pi\beta
Q^{4}}{b^{5}r_{H}^{4}(b^{2}+r_{H}^{2})}\bigg{(}3b^{5}r_{H}+2b^{3}r_{H}^{3}+3br_{H}^{5}\\\
&\qquad+3(b^{2}-r_{H}^{2})(b^{2}+r_{H}^{2})^{2}\thinspace\text{tan}^{-1}\Big{(}\frac{b}{r_{H}}\Big{)}\bigg{)},\end{split}$
(108)

where $b=J/M$. The period $\beta$ of the periodic coordinate $\theta$ and the
outer horizon radius $r_{H}$ hold the forms,

$\begin{split}r_{H}&=r_{+}=M+\sqrt{M^{2}-Q^{2}-b^{2}},\\\
\beta&=\frac{1}{T_{BH}}=\frac{32\pi^{2}}{\sqrt{M^{2}-Q^{2}-b^{2}}}\left(2M^{2}-Q^{2}+2M\sqrt{M^{2}-Q^{2}-b^{2}}\right).\end{split}$
(109)

To accomplish the logarithmic correction program for the non-extremal Kerr-
Newman family of black holes, one needs to extract the trace anomaly data
$(c,a)$ from the particular $a_{4}(x)$ form (104) and then insert them in the
$\mathcal{C}_{\text{local}}$ formula (103) along with the non-extremal limits
(108). For the $\mathcal{C}_{\text{zm}}$ contributions, we use the same
general formula (33).

EMSGTs |  $\bm{c}$ |  $\bm{a}$
---|---|---
$\mathcal{N}=1,d=4$ EMSGT |  0 |  $-\frac{1}{48}\big{(}41-3n_{v}+n_{c}\big{)}$
$\mathcal{N}\geq 2,d=4$ EMSGTs |  0 |  $-\frac{1}{24}\big{(}11-11(\mathcal{N}-2)-n_{V}+n_{H}\big{)}$
Table 4: Coefficients of trace anomalies $(c,a)$ data for the matter coupled
$\mathcal{N}\geq 1,d=4$ EMSGTs.

The $a_{4}(x)$ coefficients for the matter coupled $\mathcal{N}=1$ EMSGT
(section 4.2) and $\mathcal{N}\geq 2$ EMSGTs (section 4.3) are extracted as

$\displaystyle(4\pi)^{2}a^{\mathcal{N}=1}_{4}(x)$
$\displaystyle=\frac{1}{48}\big{(}41-3n_{v}+n_{c}\big{)}\big{(}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}\big{)},$
(110) $\displaystyle(4\pi)^{2}a^{\mathcal{N}\geq 2}_{4}(x)$
$\displaystyle=\frac{1}{24}\big{(}11-11(\mathcal{N}-2)-n_{V}+n_{H}\big{)}\big{(}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}\big{)},$
(111)

which yields the $c$ and $a$ values as recorded in table 4. Therefore, the
logarithmic entropy corrections for the non-extremal Kerr-Newman family of
black holes in the matter coupled $\mathcal{N}\geq 1,d=4$ EMSGTs are obtained
as

$\displaystyle\Delta S^{\mathcal{N}=1\thinspace\text{non-ext}}_{\text{BH}}$
$\displaystyle=\begin{cases}\frac{1}{24}(29-3n_{v}+n_{c})\ln\mathcal{A}_{H}&\text{for
rotating BHs},\\\ \frac{1}{24}(5-3n_{v}+n_{c})\ln\mathcal{A}_{H}&\text{for
non-rotating BHs}.\end{cases}$ (112) $\displaystyle\Delta
S^{\mathcal{N}=2\thinspace\text{non-ext}}_{\text{BH}}$
$\displaystyle=\begin{cases}\frac{1}{12}\big{(}5-n_{V}+n_{H}\big{)}\ln\mathcal{A}_{H}&\text{for
rotating BHs},\\\
\frac{1}{12}\big{(}-7-n_{V}+n_{H}\big{)}\ln\mathcal{A}_{H}&\text{for non-
rotating BHs},\end{cases}$ (113) $\displaystyle\Delta S^{\mathcal{N}\geq
3\thinspace\text{non-ext}}_{\text{BH}}$
$\displaystyle=\begin{cases}\left(\frac{5}{2}-\mathcal{N}\right)\ln\mathcal{A}_{H}&\text{for
rotating BHs},\\\
\left(\frac{3}{2}-\mathcal{N}\right)\ln\mathcal{A}_{H}&\text{for non-rotating
BHs}.\end{cases}$ (114)

Here $\mathcal{C}_{\text{zm}}$ contributions for the particular choice of non-
extremal rotating (Kerr-Newman and Kerr) and non-rotating (Reissner-Nordström
and Schwarzschild) black holes are considered by setting $N_{\text{SUSY}}=0$
and $\delta=1$ in the formula (33). The $\mathcal{N}\geq 2$ part (eqs. 113 and
114) accurately reproduces the results of Charles:2015nn .

###  4.5 Conclusions and remarks

The particular technical set-up employed in the current paper successfully
provided logarithmic entropy corrections for all types (rotating, non-
rotating, charged and uncharged) of extremal black holes in the matter coupled
$\mathcal{N}\geq 1,d=4$ EMSGTs. For this achievement, the key role is played
by the third Seeley-DeWitt coefficient $a_{4}(x)$. The $a_{4}(x)$ results of
different multiplets of the $\mathcal{N}=2,d=4$ EMSGT, calculated via the
standard computation approach of section 2.2, are found to be manageable only
in terms of background Riemannian and Ricci invariants. Therefore, the results
are applicable for all back hole backgrounds of the theory in proper limits of
black hole parameters. Utilization of the background-geometric-invariant form
of $a_{4}(x)$ data into the quantum entropy function formalism (described in
section 2.3) serves as a global platform to understand the logarithmic entropy
corrections for all extremal Kerr-Newman family (Kerr-Newman, Kerr and
Reissner-Nordström) of black holes. Also, the $\mathcal{N}\geq
3\to\mathcal{N}=2$ decompositions are executed and managed so that we ended up
with a generalized and compact logarithmic correction formula (only in terms
of $\mathcal{N}$) for each black hole in all matter coupled $\mathcal{N}\geq
3,d=4$ EMSGTs. Quantum corrections to the entropy of extremal rotating black
holes in matter-coupled supergravities were previously unknown but
successfully investigated in this paper. The results of extremal Kerr-Newman
and Kerr black holes are new reports to the literature. All these depict the
main highlights and novelty of the work presented in this paper.

In a very similar fashion, we also extracted logarithmic entropy corrections
for the full non-extremal Kerr-Newman family (Kerr-Newman, Kerr, Reissner-
Nordström and Schwarzschild) of black holes in the matter coupled
$\mathcal{N}\geq 1,d=4$ EMSGTs. The used $a_{4}(x)$ data are common, but the
Euclidean gravity approach is different (see section 4.4). Most of these non-
extremal results are previously known; however, the corrections in the
$\mathcal{N}=1,d=4$ EMSGT are novel. Comparison between the extremal (eqs. 94,
96 and 100) and non-extremal (eqs. (112), (113) and (114)) results leads us to
an interesting remark. When the non-extremal Kerr-Newman family of black holes
are interpreted as solutions of matter coupled $\mathcal{N}\geq 1,d=4$ EMSGTs,
the logarithmic corrections to their entropy become fully universal. But in
the extremal limit, only the Kerr and Reissner-Nordström results are
universal, while entropy corrections for the Kerr-Newman black holes become
dependent on black hole parameters (via $b^{\prime}=\frac{J}{MQ}$). This seems
quite surprising and may prove to be vital information for understanding the
quantum entropy of black holes inside microscopic theories in the future. For
any quantum theory of gravity, it is always challenging to serve microscopic
answers to the quantum corrected entropy of black holes. Moreover, the complex
nature of the calculated logarithmic correction results will enhance the
technical difficulties a lot more in their microscopic reproduction. The
calculated macroscopic logarithmic correction to the Bekenstein-Hawking
entropy of extremal Reissner-Nordström black holes in $\mathcal{N}=4$ and
$\mathcal{N}=8$ EMSGTs (originally reported in Banerjee:2011pp ) already have
an exact microscopic agreement in string theory (see Mandal:2010im ;
Sen:2014nvt and references therein). No such concrete microscopic answers
(particularly in string theory) for any other calculated non-extremal and
remaining extremal corrections are reported to date. However, it will always
be interesting to overcome the technical barriers and complexities and explore
the microscopic results as future progress.

###### Acknowledgements.

The authors are grateful to Rajesh Kumar Gupta, Ashoke Sen, Anthony Charles
and Finn Larsen for sharing their expertise during the course of this
research. We also want to thank Gourav Banerjee for his fruitful comments on
this manuscript.

##  Appendix A Gamma matrices, EOMs and identities

In this paper, all the gamma matrices $\gamma^{\mu}$ are Euclidean, Hermitian,
and satisfy the following relations

$\begin{split}\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}&=2\bar{g}^{\mu\nu}\mathbb{I}_{4},\thinspace\left\\{\gamma^{\mu},\gamma^{5}\right\\}=0,\\\
\gamma^{\mu\nu\rho\sigma}&=-\gamma^{5}\varepsilon^{\mu\nu\rho\sigma},\thinspace\gamma^{5}=-\frac{1}{4!}\varepsilon^{\mu\nu\rho\sigma}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma},\\\
\gamma^{\mu\nu}&=\frac{1}{2}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu}),\thinspace\gamma^{\mu\rho\nu}=\frac{1}{2}(\gamma^{\mu}\gamma^{\rho}\gamma^{\nu}-\gamma^{\nu}\gamma^{\rho}\gamma^{\mu}).\end{split}$
(115)

$\mathbb{I}_{4}$ is the identity matrix in the Clifford algebra satisfied by
the 4D gamma matrices. The equations of motion and Bianchi identities,
satisfied by the background graviphoton field strength tensor
$\bar{F}_{\mu\nu}$ and its dual
$\bar{H}_{\mu\nu}=-\varepsilon_{\mu\nu\rho\sigma}\bar{F}^{\rho\sigma}$, are
expressed as

$\begin{split}\bar{F}_{\mu\rho}{\bar{F_{\nu}}}^{\rho}&=\frac{1}{2}R_{\mu\nu}+\frac{1}{4}\bar{g}_{\mu\nu}\bar{F}_{\rho\sigma}\bar{F}^{\rho\sigma},\thinspace
R=0,\\\
\bar{H}_{\mu\rho}{\bar{H_{\nu}}}^{\rho}&=-2R_{\mu\nu}+\bar{g}_{\mu\nu}\bar{F}_{\rho\sigma}\bar{F}^{\rho\sigma},\\\
\bar{H}_{\mu\nu}\bar{H}^{\mu\nu}&=4\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
D_{\mu}\bar{F}^{\mu\nu}&=0,\thinspace D_{[\mu}\bar{F}_{\nu\rho]}=0,\\\
D_{\mu}\bar{H}^{\mu\nu}&=0,\thinspace
D_{[\mu}\bar{H}_{\nu\rho]}=0.\end{split}$ (116)

Starting with (116) and using the gravitational Bianchi
$R_{\mu[\nu\rho\sigma]}=0$, we formulate a list of identities,

$\begin{split}\bar{F}^{\mu\rho}\bar{F^{\nu}}_{\rho}\bar{H}_{\mu\sigma}{\bar{H_{\nu}}}^{\sigma}&=\bar{F}^{\mu\rho}{\bar{H^{\nu}}}_{\rho}\bar{H}_{\mu\sigma}\bar{F_{\nu}}^{\sigma}=(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}-R_{\mu\nu}R^{\mu\nu},\\\
\bar{F}_{\mu\nu}\bar{H}^{\mu\nu}\bar{F}_{\rho\sigma}\bar{H}^{\rho\sigma}&=4(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}-4R_{\mu\nu}R^{\mu\nu},\\\
R_{\mu\rho\nu\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}&=\frac{1}{2}R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma},\\\
R_{\mu\rho\nu\sigma}\bar{H}^{\mu\nu}\bar{H}^{\rho\sigma}&=\frac{1}{2}R_{\mu\nu\rho\sigma}\bar{H}^{\mu\nu}\bar{H}^{\rho\sigma}=2R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-4R_{\mu\nu}R^{\mu\nu},\end{split}$
(117)

followed by the following derivative relations (up to a total derivative)

$\begin{split}(D_{\rho}\bar{F}_{\mu\nu})(D^{\rho}\bar{F}^{\mu\nu})=2(D_{\mu}\bar{{F}_{\rho}}^{\nu})(D_{\nu}\bar{F}^{\rho\mu})&=R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-R_{\mu\nu}R^{\mu\nu},\\\
(D_{\rho}\bar{H}_{\mu\nu})(D^{\rho}\bar{H}^{\mu\nu})=2(D_{\mu}{\bar{H_{\rho}}}^{\nu})(D_{\nu}\bar{H}^{\rho\mu})&=4R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}.\end{split}$
(118)

##  Appendix B Seeley-DeWitt trace calculations for multiplets of
$\mathcal{N}=2,d=4$ EMSGT

In this section, we will present a brief list of comparatively lengthy trace
calculations of the non-minimally coupled cases191919For the minimally-coupled
cases, we have already provided a combined Seeley-DeWitt formula (21). of the
$\mathcal{N}=2,d=4$ EMSGT. This includes the fermionic-sector
($\mathcal{G}^{f}_{3/2}$) of gravitino multiplet, the bosonic-sector
($\mathcal{G}^{b}_{1}$) of vector multiplet and the fermionic-sector
($\mathcal{G}^{f}_{1/2}$) of hyper multiplet. These trace calculations are
undoubtedly lengthy and tedious but not much difficult to manage. One needs to
proceed systematically by executing every term step-by-step with the help of
gamma identities, Maxwell-Bianchi identities, background equations of motion
mentioned in appendix A. The trace calculations also need a distinctive trick
to follow: if $I^{mn}$ is the effective metric corresponding to any two
arbitrary matrices $A^{mn}$ and $B^{mn}$, then ${A^{m}}_{n}=A^{mp}I_{pn}$,
$\text{tr}(A)={A^{m}}_{m}$, $\text{tr}(AB)={A^{m}}_{n}{B^{n}}_{m}$ and
$\text{tr}(AB)=\text{tr}(BA)$. As mentioned, our ultimate goal is to compute
and express the traces in terms of the background invariants like
$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$, $R_{\mu\nu}R^{\mu\nu}$,
$R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}$,
$\bar{F}_{\mu\nu}\bar{F}^{\mu\nu}$, $(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}$,
etc.

###  $\mathcal{N}=2,d=4$ EMSGT: gravitino multiplet

For the fermionic-sector fluctuations
$\tilde{\xi}_{m}=\\{\psi_{\mu},\lambda\\}$ of the $\mathcal{N}=2$ gravitino
multiplet (section 3.2), we define the following traces

$\displaystyle\text{tr}(E)=\text{tr}\Big{(}{E^{\psi_{\mu}}}_{\psi_{\nu}}+{E^{\lambda}}_{\lambda}+{E^{\psi_{\mu}}}_{\lambda}+{E^{\lambda}}_{\psi_{\nu}}\Big{)},$
(119a)
$\displaystyle\text{tr}(E^{2})=\text{tr}\Big{(}{E^{\psi_{\mu}}}_{\psi_{\nu}}{E^{\psi_{\nu}}}_{\psi_{\mu^{\prime}}}+{E^{\lambda}}_{\lambda}{E^{\lambda}}_{\lambda}+{E^{\psi_{\mu}}}_{\lambda}{E^{\lambda}}_{\psi_{\nu}}+{E^{\lambda}}_{\psi_{\nu}}{E^{\psi_{\nu}}}_{\lambda}\Big{)},$
(119b)
$\displaystyle\text{tr}(\Omega_{\rho\sigma}\Omega^{\rho\sigma})=\text{tr}\Big{(}{(\Omega_{\rho\sigma})^{\psi_{\mu}}}_{\psi_{\nu}}{(\Omega^{\rho\sigma})^{\psi_{\nu}}}_{\psi_{\mu^{\prime}}}+{(\Omega_{\rho\sigma})^{\lambda}}_{\lambda}{(\Omega^{\rho\sigma})^{\lambda}}_{\lambda}$
$\displaystyle\qquad\qquad\qquad\quad+{(\Omega_{\rho\sigma})^{\psi_{\mu}}}_{\lambda}{(\Omega^{\rho\sigma})^{\lambda}}_{\psi_{\nu}}+{(\Omega_{\rho\sigma})^{\lambda}}_{\psi_{\nu}}{(\Omega^{\rho\sigma})^{\psi_{\nu}}}_{\lambda}\Big{)}.$
(119c)

Simplifying section 3.2 further, all the components of $E$ are expressed as

$\displaystyle{E^{\psi_{\mu}}}_{\psi_{\nu}}$
$\displaystyle=\underbrace{\frac{1}{8}\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma_{\nu}\gamma^{\theta}\gamma^{\phi}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}}_{X_{1}}\underbrace{+\frac{1}{2}\gamma^{\theta}\gamma^{\phi}\gamma^{\alpha}\gamma^{\mu}\bar{F}_{\theta\phi}\bar{F}_{\nu\alpha}}_{X_{2}}$
$\displaystyle\quad\underbrace{-\frac{1}{8}\bar{g}^{\mu}_{\nu}\gamma^{\alpha}\gamma^{\beta}\gamma^{\theta}\gamma^{\phi}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}}_{X_{3}}\underbrace{+\gamma^{\theta}\gamma^{\phi}\bar{F^{\mu}}_{\nu}\bar{F}_{\theta\phi}+\frac{1}{2}\gamma^{\alpha}\gamma^{\beta}{R^{\mu}}_{\nu\alpha\beta}}_{X_{4}}$
$\displaystyle\quad\underbrace{-\frac{1}{4}\gamma^{\mu}\gamma^{\alpha}R_{\nu\alpha}+\frac{3}{4}\gamma_{\nu}\gamma^{\alpha}{R^{\mu}}_{\alpha}}_{X_{5}}\underbrace{-\frac{3}{2}\mathbb{I}_{4}{R^{\mu}}_{\nu}-\frac{1}{2}\mathbb{I}_{4}\bar{g^{\mu}}_{\nu}\bar{F}_{\alpha\beta}\bar{F}^{\alpha\beta}}_{X_{6}},$
(120a) $\displaystyle{E^{\lambda}}_{\lambda}$
$\displaystyle=\underbrace{\frac{1}{2}\gamma^{\alpha}\gamma^{\beta}\gamma^{\theta}\gamma^{\phi}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}}_{X_{7}},$
(120b) $\displaystyle{E^{\psi_{\mu}}}_{\lambda}$
$\displaystyle=\underbrace{\frac{1}{4\sqrt{2}}\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma^{\rho}D_{\rho}\bar{F}_{\alpha\beta}}_{X_{8}}\underbrace{-\frac{1}{8\sqrt{2}}\gamma^{\sigma}\gamma^{\rho}\gamma^{\mu}\gamma^{\alpha}\gamma^{\beta}\gamma_{\sigma}D_{\rho}\bar{F}_{\alpha\beta}}_{X_{9}},$
(120c) $\displaystyle{E^{\lambda}}_{\psi_{\nu}}$
$\displaystyle=\underbrace{\frac{1}{4\sqrt{2}}\gamma^{\rho}\gamma_{\nu}\gamma^{\alpha}\gamma^{\beta}D_{\rho}\bar{F}_{\alpha\beta}}_{X_{10}}\underbrace{-\frac{1}{8\sqrt{2}}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma_{\nu}\gamma^{\rho}\gamma_{\sigma}D_{\rho}\bar{F}_{\alpha\beta}}_{X_{11}}$
(120d)

Then,

$\begin{split}&\text{tr}(X_{1})=-4\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&\text{tr}(X_{2})=4\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&\text{tr}(X_{3})=4\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&\text{tr}(X_{4})=0,\end{split}\hskip
36.135pt\begin{split}&\text{tr}(X_{5})=0,\\\
&\text{tr}(X_{6})=-8\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&\text{tr}(X_{7})=-4\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&\text{tr}(X_{8})=0,\end{split}\hskip
36.135pt\begin{split}&\text{tr}(X_{9})=0,\\\ &\text{tr}(X_{10})=0,\\\
&\text{tr}(X_{11})=0.\end{split}$ (121)

Therefore,

$\begin{split}&\text{tr}({E^{\psi_{\mu}}}_{\psi_{\nu}})=\sum_{i=1}^{6}\text{tr}(X_{i})=-4\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&\text{tr}({E^{\lambda}}_{\lambda})=\text{tr}(X_{7})=-4\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\end{split}\hskip
36.135pt\begin{split}&\text{tr}({E^{\psi_{\mu}}}_{\lambda})=\sum_{i=8}^{9}\text{tr}(X_{i})=0,\\\
&\text{tr}({E^{\lambda}}_{\psi_{\nu}})=\sum_{i=10}^{11}\text{tr}(X_{i})=0.\end{split}$
(122)

Also,

$\begin{split}\text{tr}(X_{1}X_{1})&=8(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}-4R_{\mu\nu}R^{\mu\nu},\\\
\text{tr}(X_{2}X_{2})&=8(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}-4R_{\mu\nu}R^{\mu\nu},\\\
\text{tr}(X_{3}X_{3})&=2(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}-R_{\mu\nu}R^{\mu\nu},\\\
\text{tr}(X_{4}X_{4})&=2R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+8(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}\\\
&\qquad+8R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma},\\\
\text{tr}(X_{5}X_{5})&=3R_{\mu\nu}R^{\mu\nu},\\\
\text{tr}(X_{6}X_{6})&=4(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}+9R_{\mu\nu}R^{\mu\nu},\\\
\text{tr}(X_{7}X_{7})&=8(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}-4R_{\mu\nu}R^{\mu\nu},\\\
\text{tr}(X_{1}X_{2})&=2R_{\mu\nu}R^{\mu\nu}-4(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2},\\\
\text{tr}(X_{1}X_{3})&=R_{\mu\nu}R^{\mu\nu}-2(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2},\\\
\text{tr}(X_{1}X_{4})&=6R_{\mu\nu}R^{\mu\nu}-4(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}\\\
&\qquad-4R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma},\\\
\text{tr}(X_{1}X_{5})&=0,\\\
\text{tr}(X_{1}X_{6})&=2(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2},\end{split}\hskip
10.84006pt\begin{split}\text{tr}(X_{2}X_{3})&=2(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}-R_{\mu\nu}R^{\mu\nu},\\\
\text{tr}(X_{2}X_{4})&=-2R_{\mu\nu}R^{\mu\nu}-4(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2},\\\
\text{tr}(X_{2}X_{5})&=5R_{\mu\nu}R^{\mu\nu},\\\
\text{tr}(X_{2}X_{6})&=-3R_{\mu\nu}R^{\mu\nu}-2(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2},\\\
\text{tr}(X_{3}X_{4})&=0,\\\ \text{tr}(X_{3}X_{5})&=0,\\\
\text{tr}(X_{3}X_{6})&=-2(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2},\\\
\text{tr}(X_{4}X_{5})&=-8R_{\mu\nu}R^{\mu\nu},\\\ \text{tr}(X_{4}X_{6})&=0,\\\
\text{tr}(X_{5}X_{6})&=-3R_{\mu\nu}R^{\mu\nu},\\\
\text{tr}(X_{8}X_{10})&=R_{\mu\nu}R^{\mu\nu}-R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma},\\\
\text{tr}(X_{8}X_{11})&=0,\\\ \text{tr}(X_{9}X_{10})&=0,\\\
\text{tr}(X_{9}X_{11})&=0.\end{split}$ (123)

Hence,

$\begin{split}&\text{tr}({E^{\psi_{\mu}}}_{\psi_{\nu}}{E^{\psi_{\nu}}}_{\psi_{\mu^{\prime}}})=\sum_{i,j=1}^{6}\text{tr}(X_{i}X_{j})=2R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-3R_{\mu\nu}R^{\mu\nu}+2(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2},\\\
&\text{tr}({E^{\lambda}}_{\lambda}{E^{\lambda}}_{\lambda})=\text{tr}(X_{7}X_{7})=8(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}-4R_{\mu\nu}R^{\mu\nu},\\\
&\text{tr}({E^{\psi_{\mu}}}_{\lambda}{E^{\lambda}}_{\psi_{\nu}})=\text{tr}({E^{\lambda}}_{\psi_{\nu}}{E^{\psi_{\nu}}}_{\lambda})=\sum_{i=8}^{9}\sum_{j=10}^{11}\text{tr}(X_{i}X_{j})=R_{\mu\nu}R^{\mu\nu}-R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}.\end{split}$
(124)

Similarly, section 3.2 provides all the components of $\Omega_{\rho\sigma}$ as

$\displaystyle{(\Omega_{\rho\sigma})^{\psi_{\mu}}}_{\psi_{\nu}}$
$\displaystyle=\underbrace{\mathbb{I}_{4}{R^{\mu}}_{\nu\rho\sigma}}_{Y_{1}}+\underbrace{\frac{1}{4}\bar{g^{\mu}}_{\nu}\gamma^{\alpha}\gamma^{\beta}R_{\rho\sigma\alpha\beta}}_{Y_{2}}$
$\displaystyle\quad\underbrace{-\frac{1}{32}\Big{(}\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma_{\rho}+\frac{1}{2}\gamma^{\tau}\gamma_{\rho}\gamma^{\mu}\gamma^{\alpha}\gamma^{\beta}\gamma_{\tau}\Big{)}\Big{(}\gamma_{\sigma}\gamma_{\nu}\gamma^{\theta}\gamma^{\phi}+\frac{1}{2}\gamma^{\lambda}\gamma^{\theta}\gamma^{\phi}\gamma_{\nu}\gamma_{\sigma}\gamma_{\lambda}\Big{)}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}}_{Y_{3}}$
$\displaystyle\quad\underbrace{+\frac{1}{32}\Big{(}\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma_{\sigma}+\frac{1}{2}\gamma^{\tau}\gamma_{\sigma}\gamma^{\mu}\gamma^{\alpha}\gamma^{\beta}\gamma_{\tau}\Big{)}\Big{(}\gamma_{\rho}\gamma_{\nu}\gamma^{\theta}\gamma^{\phi}+\frac{1}{2}\gamma^{\lambda}\gamma^{\theta}\gamma^{\phi}\gamma_{\nu}\gamma_{\rho}\gamma_{\lambda}\Big{)}\bar{F}_{\alpha\beta}\bar{F}_{\theta\phi}}_{Y_{4}},$
(125a) $\displaystyle{(\Omega_{\rho\sigma})^{\lambda}}_{\lambda}$
$\displaystyle=\underbrace{\frac{1}{4}\gamma^{\alpha}\gamma^{\beta}R_{\rho\sigma\alpha\beta}}_{Y_{5}}$