Seeley-DeWitt coefficients in $\mathcal{N}=2$ Einstein-Maxwell supergravity
theory and logarithmic corrections to $\mathcal{N}=2$ extremal black hole
entropy

Sudip Karan1, Gourav Banerjee2 and Binata Panda3
1,2,3Department of Physics,
Indian Institute of Technology (Indian School of Mines),
Dhanbad, Jharkhand-826004, INDIA
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
We investigate the heat kernel method for one-loop effective action following
the Seeley-DeWitt expansion technique of heat kernel with Seeley-DeWitt
coefficients. We also review a general approach of computing the Seeley-DeWitt
coefficients in terms of background or geometric invariants. We, then consider
the Einstein-Maxwell theory embedded in minimal $\mathcal{N}=2$ supergravity
in four dimensions and compute the first three Seeley-DeWitt coefficients of
the kinetic operator of the bosonic and the fermionic fields in an arbitrary
background field configuration. We find the applications of these results in
the computation of logarithmic corrections to Bekenstein-Hawking entropy of
the extremal Kerr-Newman, Kerr and Reissner-Nordström black holes in minimal
$\mathcal{N}=2$ Einstein-Maxwell supergravity theory following the quantum
entropy function formalism.

###### Contents

  1. 1 Introduction
  2. 2 Heat kernel in 4D Euclidean gravity
    1. 2.1 Euclidean path integrals: the set-up
    2. 2.2 The one-loop effective action
    3. 2.3 The heat kernel method
    4. 2.4 Heat kernel expansion and Seeley-DeWitt coefficients
  3. 3 General approach for computing Seeley-DeWitt coefficients
  4. 4 Minimal $\mathcal{N}=2,d=4$ Einstein-Maxwell supergravity theory
    1. 4.1 Equations of motion and identities
  5. 5 Seeley-DeWitt coefficients in minimal $\mathcal{N}=2,d=4$ Einstein-Maxwell supergravity theory
    1. 5.1 $\mathcal{N}=2$ SUGRA: Bosons
    2. 5.2 $\mathcal{N}=2$ SUGRA: Fermions
    3. 5.3 Combined Seeley-DeWitt coefficients in minimal $\mathcal{N}=2$ EMSGT
  6. 6 Application of Seeley-DeWitt coefficients: logarithmic corrections to the black hole entropy
    1. 6.1 Logarithmic corrections to $\mathcal{N}=2$ extremal black hole entropy
  7. A Trace calculations

##  1 Introduction

Bekenstein-Hawking entropy formula is an approximated formula in two-
derivative classical general relativity where one ignores the higher
derivative terms in the action. Remarkably, this formula is very simple and
universal in the sense that it is completely governed by horizon area, does
not bother about the past history or detailed matter content of the particular
black hole. In string theory, the Bekenstein-Hawking formula is subjected to
two types of corrections — higher curvature corrections and quantum
corrections [1]. The higher curvature corrections arise due to considering
higher derivative terms in the classical action, are captured by the Wald’s
generalization of Bekenstein-Hawking entropy. On the other hand, the quantum
corrections are important for black holes of a size comparable to Planck size.
The Bekenstein-Hawking formula works perfectly for macroscopically large black
holes ($A_{H}\gg 1$)111Actually, large black holes have horizon area $A_{H}\gg
l_{P}^{2}$ ($l_{P}$ is the Planck length). But in this paper we are
considering the units $\hbar=c=k_{B}=1$. and is expected to receive quantum
corrections as one lowers the size of the black hole. In this work, we are
interested in a certain class of quantum corrections as discussed below.

Recent years there have been remarkable research activities222See [2] for a
list of references. in order to find quantum corrections to black hole entropy
formula and the quantum-corrected entropy received the following form,

$S_{\text{BH}}=\frac{A_{H}}{4G_{N}}+c\thinspace\text{ln}\thinspace\frac{A_{H}}{G_{N}}+\text{constant}+\mathcal{O}(A_{H}^{-1}).$
(1.1)

Here $c\thinspace\text{ln}\thinspace\frac{A_{H}}{G_{N}}$ is the logarithmic
correction, proportional to the logarithmic of the size of the black hole,
with a proportionality constant $c$ and $\mathcal{O}(A^{-1})$ are power-law
corrections to the leading order term in the black hole entropy formula.
Although the total quantum corrections will depend on the details of the UV
completion of the low energy theory, the logarithmic corrections can be
determined from the knowledge of only low energy modes. These corrections can
be computed from one-loop effective action of the massless fields in the
particular theory. These logarithmic corrections are universal in this sense
that they appear in any generic theory of gravity. But $c$ does not have the
same sort of universality: it can be determined only from low energy modes,
i.e., the spectrum of massless fields present in the theory, and are
unconcerned about UV completion of the theory [3, 4, 5, 6, 9, 7, 8, 10, 11,
12, 13]. These correction terms are important as they provide us with the non-
trivial information about the microstates of the black holes and hence
providing a valuable testing ground for any candidate theory of quantum
gravity. We still consider macroscopically large black holes to make the
logarithmic terms as leading order corrections [3, 4].

Quantum entropy function formalism [14, 15, 16] is used in [3, 4, 6, 5, 7, 8]
to compute the logarithmic corrections to the entropy of supersymmetric
extremal black holes. This procedure uses the presence of $AdS_{2}$ factors in
the near horizon geometry of extremal black holes and rules of
$AdS_{2}/CFT_{1}$ correspondence. However, it does not depend on supersymmetry
and can be applied to any extremal black holes [9]. Computation of the
logarithmic corrections involves finding the one-loop effective action of the
massless fields in the near horizon geometry of black hole. Evaluation of
effective action was investigated by many different methods, but we consider
the most potent and traditional heat kernel method (discussed in § 2.3). The
main advantage of this method is that it is explicitly covariant, as well as
the most suitable tool to regularize the UV divergence of the one-loop
effective action. Further, in order to compute the heat kernel for one-loop
effective action we pursue the Seeley-DeWitt expansion technique [21, 22, 17,
18, 19, 20, 23, 24, 25, 26, 27, 28, 29], followed by a general approach to
compute the Seeley-DeWitt coefficients for fluctuations of gauge, graviton and
matter fields around an arbitrary background field configuration demonstrated
in [30]. In this approach, the Seeley-DeWitt coefficients are computed for an
arbitrary field configuration without imposing any limitations on the
background geometry. Moreover, they are expressed in terms of local invariants
calculated from background fields and their covariant derivatives. So one can
employ these results for any arbitrary background field configuration.

In comparison, an incomplete list of many other approaches to compute heat
kernel includes some perturbative expansion methods,333 E.g., see section 8 of
[30] eigenfunction expansion method [3, 4, 5], a group theoretic approach [31,
32, 33], quasinormal mode approach [34], etc. But most of these approaches are
either background geometry dependent or model dependent or associated with
some complex computations. Of these the methods followed in [3, 4, 5] and [11,
12, 13] are closest to the one advocated in this work; so below we have given
a detailed comparison between the methods. The advantages of the current
approach are also outlined below eq. 3.8. The eigenfunction method to
calculate the Seeley-DeWitt coefficients followed in [3, 4, 5] involves
explicit knowledge of all the eigenfunctions and eigenvalues of the kinetic
operator operating on the bosonic and the fermionic fields of the
corresponding matter multiplet in the $AdS_{2}\times S^{2}$ background
geometry. This is quite complicated and restricted to this type of geometry
only. In the absence of rotational symmetry of the background geometry, the
computation of eigenfunctions of the kinetic operator is indeed a difficult
task. Whereas the Seeley-DeWitt expansion approach involves expansion of the
heat kernel in powers of small proper time and is applicable for any arbitrary
background geometry beyond $AdS_{2}\times S^{2}$. Here the Seeley-DeWitt
coefficients are expressed in terms of geometric invariants and can be applied
for any arbitrary background. Again, the technique followed by Larsen et al.
in [11, 12, 13] is kind of hybrid — it is an extension of the general Seeley-
DeWitt approach, where computation has been done by redefining fields present
in the theory which induces loss of generality within the approach.

The technical aim of this paper is twofold — first, following the general
Seeley-DeWitt approach [30] (reviewed in § 3) we compute necessary Seeley-
DeWitt coefficients of the quadratic order field fluctuations in the minimal
$\mathcal{N}=2$ Einstein-Maxwell supergravity theory (EMSGT) and then find
their useful applications in the logarithmic correction to the Bekenstein-
Hawking entropy formula for extremal black holes. The results for the
fermionic sector of the minimal $\mathcal{N}=2$ EMSGT are newly computed for
an arbitrary background, while the bosonic sector results are reviewed and
recalculated from [9]. These Seeley-DeWitt coefficient results for the minimal
$\mathcal{N}=2$ EMSGT in four dimensions agree with eigenfunction approach
results of [5] for $AdS_{2}\times S^{2}$ background, as well as with the
hybrid approach results of [11]. Then the logarithmic corrections to the
entropy of extremal black holes in minimal $\mathcal{N}=2$ EMSGT are computed
using the quantum entropy function formalism. The result is stated in eq.
6.15. The result is new in the literature for extremal Kerr- Newman black
holes. Any microscopic theory explaining the entropy of these black holes
should reproduce the result specified in eq. 6.15. With appropriate limits eq.
6.15 produces the Kerr and Reissner-Nordström results which are consistent
with [6] and [5] respectively.

The rest of this paper is structured as follows. In § 2, we will discuss the
heat kernel method in four-dimensional Euclidean gravity. In this context, we
will express the one-loop effective action of our theory in terms of heat
kernels of the corresponding kinetic operator, and also introduce Seeley-
DeWitt coefficients in the short-time expansion of these heat kernels. In § 3,
we will review the general approach, necessary formulae and steps for
calculating the Seeley-DeWitt coefficients. In § 4, we consider Einstein-
Maxwell theory embedded in minimally coupled $\mathcal{N}=2$ supergravity in
4D and briefly explore the basic structure, equations of motion and important
identities of the resulting minimally coupled $\mathcal{N}=2,d=4$ EMSGT. In §
5, we divide the total field contents of the minimally coupled
$\mathcal{N}=2,d=4$ EMSGT into bosonic and fermionic sectors, then employ the
general approach to calculate the first three Seeley-DeWitt coefficients for
the two sectors separately.444We actually compute results for the fermionic
sector, while we review [9] for the bosonic sector. In § 6, we will use the
results for Seeley-DeWitt coefficients to compute logarithmic corrections to
the entropy of extremal Kerr-Newman, Reissner-Nordström and Kerr black holes
in minimally coupled $\mathcal{N}=2$ EMSGT following quantum entropy function
formalism, and comment on the consistency of the results.

##  2 Heat kernel in 4D Euclidean gravity

In this section, we briefly summarize the heat kernel approach in the context
of 4D Euclidean gravity theory. Effective action plays a crucial theoretical
role in quantum field theory and quantum gravity since it encodes almost all
information about a quantum system. Unfortunately, it is associated with some
mathematical pathologies and hence very difficult to compute. Heat kernel
method is found to be a useful tool in this purpose.

###  2.1 Euclidean path integrals: the set-up

We start with the partition function $Z$ in Euclidean path integral
representation [35],

$\displaystyle Z=\int D[g,\xi]\text{exp}\left(-\mathcal{S}[g,\xi]\right),$
(2.1)
$\displaystyle\mathcal{S}[g,\xi]=\int_{\mathcal{M}}d^{4}x\sqrt{\text{det}\thinspace
g}\thinspace\mathcal{L}[g,\xi].$ (2.2)

$\mathcal{S}[{g},\xi]$ is the Euclidean action of the matter fields $\xi$
propagating through a geometry described by the metric ${g}$ over an arbitrary
manifold $\mathcal{M}$; $D[{g},\xi]$ suggests that we have to carry out the
functional integration (2.1) over all possible matter fields $\xi$ and metric
$g$.

The leading contribution to the path integral (2.1) is expected to arise from
the metric and the matter fields that are near the background fields $\bar{g}$
and $\bar{\xi}$. Here $(\bar{g},\bar{\xi})$ extremize our action (2.2), i.e.
they are the solution of the classical field equations with proper boundary
conditions and periodicity. At this stage, one can express ${g}$ and $\xi$ in
terms of their fluctuations around the background as

${g}=\bar{g}+\tilde{{g}},\enspace\xi=\bar{\xi}+\tilde{\xi},$ (2.3)

and expand the action in Taylor series around the classical solution
$(\bar{g},\bar{\xi})$ ,

$\mathcal{S}[{g},\xi]=\mathcal{S}[\bar{g},\bar{\xi}]+\mathcal{S}_{2}[\tilde{{g}},\tilde{\xi}]+\text{higher
order terms},$ (2.4)

where $\mathcal{S}[\bar{g},\bar{\xi}]$ is the action of the classical
background fields $\bar{g}$ and $\bar{\xi}$;
$\mathcal{S}_{2}[\tilde{{g}},\tilde{\xi}]$ is the quadratic order fluctuated
action in the fluctuations $\tilde{{g}}$ and $\tilde{\xi}$. Note that there is
no linear term in eq. 2.4 as we consider the action is not coupled to any
external sources of $\xi$. Replacing eq. 2.4 into eq. 2.1 and ignoring the
higher order terms, we have

$\text{ln}\thinspace Z\approx-\mathcal{S}[\bar{g},\bar{\xi}]+\text{ln}\int
D[\tilde{g},\tilde{\xi}]\text{exp}(-\mathcal{S}_{2}[\tilde{{g}},\tilde{\xi}])$
(2.5)

$\mathcal{S}[\bar{g},\bar{\xi}]$ is the background field contribution to the
partition function while the second term of eq. 2.5 is regarded as the
contribution of the matter quanta and thermal gravitons (fluctuation of
metric) on the background geometry [36, 37].

###  2.2 The one-loop effective action

The quantum corrected one-loop effective action $W$ [39, 38] is expressed as

$W=-\text{ln}\thinspace\int
D[\tilde{g},\tilde{\xi}]\text{exp}(-\mathcal{S}_{2}[\tilde{{g}},\tilde{\xi}])$
(2.6)

The quadratic term of action $\mathcal{S}_{2}$ in eq. 2.4 can be expressed as

$\mathcal{S}_{2}[\tilde{{g}},\tilde{\xi}]=\int_{\mathcal{M}}d^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\tilde{\xi}\Lambda\tilde{\xi},$
(2.7)

where $\Lambda$ is a differential operator which controls the quadratic order
quantum fluctuations around the classical background. For bosonic fields,
$\Lambda$ is a second-order differential operator. Again, it is real, elliptic
and self-adjoint since our background metric $\bar{g}$ has Euclidean signature
(i.e. positive and real). But for fermionic fields, $\Lambda$ is of first-
order. We will derive all the important results for bosonic fields first and
then extend them for fermionic fields. If $\tilde{\xi}$ is a graviton or gauge
field then $\Lambda$ will receive a complicated tensor structure, which has
been suppressed here for simplicity.

Considering $\Lambda$ has a discrete spectrum of eigenvalues $\lambda_{i}$,
the one-loop effective action given eq. 2.6 will take the following form [40]

$W=\frac{\chi}{2}\thinspace\text{ln}\thinspace\text{det}\thinspace\Lambda=\frac{\chi}{2}\sum_{i}\text{ln}\thinspace\lambda_{i},$
(2.8)

where $\chi=\pm 1$ for bosons and fermions respectively. Note that the
spectrum of $\Lambda$ may in practice also be continuous, and our basic
structure of the heat kernel method and the general Seeley-DeWitt computation
approach (§§ 2.3 and 3) are valid for both the discrete and continuous
spectrums.

###  2.3 The heat kernel method

Practically, one finds it difficult to compute $W$ directly, especially when
fluctuations are gauge fields and gravitons. The most straightforward method
is to compute the summation in eq. 2.8 provided we have the knowledge about
the complete basis of normalized eigenfunctions $\\{f_{i}\\}$ and eigenvalues
$\\{\lambda_{i}\\}$ obeying $\Lambda f_{i}=\lambda_{i}f_{i}$. But this is a
challenging task. Also, from the series in eq. 2.8 it is evident that the one
loop determinant det$\Lambda$ diverges (UV-divergence in quantum field theory)
and traditional way to regulate this is the heat kernel method: expressing $W$
in terms of diagonal matrix element of heat kernel via a special integral
representation, called proper time representation (2.15).

In terms of eigenfunctions $\\{f_{i}^{m}\\}$ and eigenvalues
$\\{\lambda_{i}\\}$ of the kinetic operator $\Lambda$, heat kernel (of the
operator $e^{-s\Lambda}$) can be defined as [6, 5]

$K^{mn}(x,y;s)=\sum_{i}e^{-\lambda_{i}s}f_{i}^{m}(x)f_{i}^{n}(y),$ (2.9)

where $m$, $n$ are set of indices of the eigenfunctions at two points $x$, $y$
respectively on $\mathcal{M}$; $s$ is an auxiliary time variable, called
proper time. The eigenfunctions $\\{f_{i}^{m}\\}$ have been normalized so that

$\sum_{i}G_{mn}f_{i}^{m}(x)f_{i}^{n}(y)=\delta^{4}(x,y),\enspace\int_{\mathcal{M}}d^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace
G_{mn}f_{i}^{m}(x)f_{j}^{n}(x)=\delta_{ij},$ (2.10)

where $G_{mn}$ is a metric in the space of eigenfunctions (fields) induced by
$\bar{g}$. Contracting over indices $m,n$, let us define the heat kernel
$K(x,y;s)$

$K(x,y;s)=G_{mn}K^{mn}(x,y;s),$ (2.11)

which satisfies the diffusion equation

$\left(\frac{\partial}{\partial s}+\Lambda_{x}\right)K(x,y;s)=0,$ (2.12)

with boundary conditions

$K(x,y;0)=\delta(x,y),$ (2.13)

where $\Lambda_{x}$ describes the fact that in eq. 2.12 the operator $\Lambda$
is supposed to act only on the first argument of $K(x,y;s)$. The trace of heat
kernel $D(s)$, called heat trace, can be obtained by setting $x=y$ (for
avoiding the dependence of eigenfunctions) in eq. 2.11 and then integrating
over all the $\mathcal{M}$:

$D(s)=\int_{\mathcal{M}}d^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace
K(x,x;s)=\sum_{i}e^{-\lambda_{i}s}.$ (2.14)

Heat trace $D(s)$ is important because it stores information about the
spectrum of the kinetic operator $\Lambda$. For example, one may evaluate555In
deriving eq. 2.15, we have used the identity $\lim_{\epsilon\to
0}\int_{\epsilon}^{\infty}\frac{ds}{s}(e^{-As}-e^{-Bs})=\text{ln}\thinspace\frac{B}{A}$

$W=-\frac{\chi}{2}\int_{\epsilon}^{\infty}\frac{ds}{s}\sum_{i}e^{-\lambda_{i}s}=-\frac{1}{2}\int_{\epsilon}^{\infty}\frac{ds}{s}\chi
D(s).$ (2.15)

This is called Schwinger-DeWitt proper time representation [41, 42] of the
effective action, where $\epsilon$ is an ultraviolet cutoff, naturally
restricted by the Planck length, i.e. $\epsilon\sim{l_{p}}^{2}\sim G_{N}$.
Also, one may identify the one-loop corrected effective Lagrangian density
$\Delta\mathcal{L}_{\text{eff}}$ from eq. 2.15 as

$\Delta\mathcal{L}_{\text{eff}}=-\frac{1}{2}\int_{\epsilon}^{\infty}\frac{ds}{s}\chi
K(x,x;s).$ (2.16)

The main benefit of this approach lies in the fact that heat kernels are
ultraviolet finite for all positive proper time ($s$).

###  2.4 Heat kernel expansion and Seeley-DeWitt coefficients

The conventional way of computing heat kernel is expanding them in powers of
proper time $s$, called heat kernel expansion. It is noticed that $K(x,x;s)$
is analytic for $s>0$, but as $s\to 0$ there is a power-law asymptotic
expansion, known as Seeley-DeWitt expansion666$\chi$ has been absorbed in the
definition (3.8) of $a_{2n}$ by manually putting proper signature following
the spin-statistics. [21, 22, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29],

$\chi K(x,x;s)\cong\sum_{n=0}^{\infty}s^{n-2}a_{2n}(x),$ (2.17)

giving

$\chi
D(s)=\int_{\mathcal{M}}d^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\left\\{\frac{1}{s^{2}}a_{0}(x)+\frac{1}{s}a_{2}(x)+a_{4}(x)+\cdots\right\\},$
(2.18)

where the expansion coefficients $\\{a_{2n}(x)\\}$ are heat trace asymptotics,
called Seeley-DeWitt coefficients [21, 22, 17, 18, 19, 20] or heat kernel
coefficients. A systematic way of computing these coefficients is reviewed in
the next section, where $\\{a_{2n}(x)\\}$ are expressed in terms of background
fields and their covariant derivatives associated with the kinetic
differential operator $\Lambda$.

##  3 General approach for computing Seeley-DeWitt coefficients

The most common method for calculating Seeley-DeWitt coefficients in physics
is the DeWitt iterative technique (reviewed in sec. 4.3 of [30]). But, we will
follow the method of Gilkey [43] (explained in sec. 4.1 of [30]) which seems
to be more efficient for a curved background with arbitrary spin. In this
section, we will review the main ideas, the basic structure and useful
formulae to compute the Seeley-DeWitt coefficients based on the manual [30].
We now start by considering the quadratic order fluctuated action (2.7)
controlled by the kinetic differential operator $\Lambda$ and rewrite its
schematic form in terms of field indices as

$\mathcal{S}_{2}=\int
d^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\tilde{\xi}_{m}\Lambda^{mn}\tilde{\xi}_{n},$
(3.1)

where {$\tilde{\xi}_{m}$} is the set of all fluctuating fields. In order to
successfully apply the procedure, $\Lambda$ is constrained by the following
conditions:

  * $\blacksquare$

$\Lambda$ must be of a second order differential operator. It is useful to
note that the quadratic order fluctuations of the fermions are governed by
first order operators $\not{D}$, i.e.
$\delta^{2}\mathcal{L}=\tilde{\xi}\not{D}\tilde{\xi}$. There we first
restructure them as Dirac-type and Hermitian one, and define [5]

$\text{det}\thinspace\not{D}=\sqrt{\text{det}\not{D}^{\dagger}\not{D}},\enspace\text{ln
det}\thinspace\not{D}=\frac{1}{2}\text{ln det}\thinspace\Lambda,$ (3.2)

where $\Lambda=\not{D}^{\dagger}\not{D}=\not{D}^{2}$. This will include an
additional $1/2$ factor in the description of heat kernel (or in the formula
of Seeley-DeWitt coefficients), which is compensated by a factor of 2 arises
due to the fact that squaring the complex operator doubles the integration
variables.

  * $\blacksquare$

$\Lambda$ must be made up of pseudo-differential operators which excludes
negative and fractional powers of the derivatives.

  * $\blacksquare$

It should be elliptic and self-adjoint, i.e.
$\langle\tilde{\xi}_{m},\Lambda\tilde{\xi}_{n}\rangle=\langle\Lambda\tilde{\xi}_{m},\tilde{\xi_{n}}\rangle$.

  * $\blacksquare$

$\Lambda$ should be in a minimal form of type $g^{\mu\nu}D_{\mu}D_{\nu}+Q$,
where it has a scalar principal part, i.e. second order derivatives are
contracted with the metric.

In summary, we conclude that the present method demands $\Lambda$ to be a
Laplace-type operator having the following schematic form

$\Lambda^{mn}=\pm\big{\\{}(D^{\rho}D_{\rho})G^{mn}+(N^{\rho}D_{\rho})^{mn}+P^{mn}\big{\\}},$
(3.3)

where $G^{mn}$ is some arbitrary matrix,777Precisely, $G$ is the effective
metric tensor which contracts indices of an operator acting on
$\tilde{\xi}_{m}$ and we can also use $G$ to find trace values of these
operators. For example, if $G^{mn}$ be the metric tensor related to an
arbitrary two rank tensor $B^{mn}$, then ${B^{m}}_{n}=B^{mp}G_{pn}$,
$\text{tr}(B)={B^{m}}_{m}$ and $\text{tr}(B^{2})={B^{m}}_{n}{B^{n}}_{m}$.
$D_{\rho}$ is the ordinary covariant derivative with connections determined by
the background metric, and $N^{\rho}$, $P$ are arbitrary matrices constructed
from the background fluctuations. Let us define a new covariant derivative
$\mathcal{D}_{\rho}$ including the field connection $\omega_{\rho}$,

$\mathcal{D}_{\rho}\tilde{\xi}_{m}=D_{\rho}\tilde{\xi}_{m}+{(\omega_{\rho})_{m}}^{n}\tilde{\xi}_{n}\quad\forall
m\neq n,$ (3.4)

and rewrite eq. 3.3 by replacing $D_{\rho}$ with $\mathcal{D}_{\rho}$ as

$\Lambda^{mn}=\pm\left\\{(\mathcal{D}^{\rho}\mathcal{D}_{\rho})G^{mn}+P^{mn}-(D^{\rho}\omega_{\rho})^{mn}-(\omega^{\rho})^{mp}{(\omega_{\rho})_{p}}^{n}+(N^{\rho}-2\omega^{\rho})^{mn}D_{\rho}\right\\}.$
(3.5)

We aim to express eq. 3.5 in the standard form

$\Lambda=\pm\big{\\{}(\mathcal{D}^{\rho}\mathcal{D}_{\rho})I+E\big{\\}},$
(3.6)

and extract the following necessary matrices,

$\displaystyle\begin{split}&I^{mn}=G^{mn},(\omega_{\rho})^{mn}=\frac{1}{2}(N_{\rho})^{mn},\\\
&(E)^{mn}=P^{mn}-(D^{\rho}\omega_{\rho})^{mn}-(\omega^{\rho})^{mp}{(\omega_{\rho})_{p}}^{n},\\\
&(\Omega_{\rho\sigma})^{mn}=[D_{\rho},D_{\sigma}]^{mn}+{D_{[\rho}\omega_{\sigma]}}^{mn}+[\omega_{\rho},\omega_{\sigma}]^{mn},\end{split}$
(3.7)

where $\Omega_{\rho\sigma}\equiv[\mathcal{D}_{\rho},\mathcal{D}_{\sigma}]$ is
the curvature associated with $\mathcal{D}_{\rho}$. So, one needs to identify
the matrices $G$, $P$ and $N$ from the relation (3.5) and use those results in
the formula (3.7) for obtaining $\omega,E$ and $\Omega$. With these
definitions, the formulae for the first three Seeley-DeWitt coefficients [30,
44] are summarized as following

$\displaystyle\begin{split}(4\pi)^{2}a_{0}(x)&=\text{tr}(I),\\\
(4\pi)^{2}a_{2}(x)&=\frac{1}{6}\text{tr}(6E+RI),\\\
(4\pi)^{2}a_{4}(x)&=\frac{1}{360}\text{tr}\big{(}60RE+180E^{2}+30\Omega_{\rho\sigma}\Omega^{\rho\sigma}+(5R^{2}+2R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-2R_{\mu\nu}R^{\mu\nu})I\big{)}.\end{split}$
(3.8)

The formulae given in eq. 3.8 are applicable only for manifolds having no
boundary.888There are Seeley-DeWitt coefficient formulae for manifold with
boundary (see eqs. (5.29)-(5.33) of [30]) also. But in this paper, our main
motive is to compute logarithmic corrections of black hole entropy which
demands to choose the no boundary formulae (discussed in § 6). Hence, all the
total derivative terms have been ignored as they will appear as boundary terms
in the integral (2.17) and hence will disappear after integration. Again for
manifolds satisfying the condition $R=0$ one can neglect terms proportional to
$R$ in this formulae. We consider free999“Free” means we are considering only
interaction with background geometry, all other interactions are turned off.
quantum fields on a Riemannian manifold having no boundaries so that the
Seeley-DeWitt coefficients can be described by local invariants induced from
the background fields and metric. The main advantage of calculating the
Seeley-DeWitt coefficients following this general Seeley-DeWitt approach is —
(a) after considering quadratic fluctuations for different fields in a given
theory we have simple and straightforward formulae to compute the Seeley-
DeWitt coefficients, (b) the coefficients are expressed in terms of geometric
invariants, hence one can employ these results for any arbitrary background
field configuration without imposing any limitation on the background
geometry. It is important to mention that for the fermionic case, we need to
multiply an additional minus sign (correspond to $\chi=-1$) in the above
formulae of Seeley-DeWitt coefficients. We also note that the present method
assumes the fermionic field fluctuations as Dirac spinors, and hence for
casting Majorana and Weyl (having half the degrees of freedom of Dirac spinor)
fermion contribution, one needs to multiply a factor of $-1/2$ in the formulae
(3.8).

Finally, we summarize the key steps of computing Seeley-DeWitt coefficients as
follows:101010For a better understanding of this method and some elementary
examples, readers may refer [44, 11].

  1. 1.

Consider fluctuations for all possible fields present in the theory around a
classical background and express the action in quadratic order of
fluctuations.

  2. 2.

Gauge fix the theory of choice by adding appropriate gauge fixing term to the
quadratic action and find the quadratic fluctuation controlling operator
$\Lambda$ (defined in the eq. 3.1). Identify the appropriate ghosts and keep
them separate.

  3. 3.

With all the restrictions mentioned, restructure $\Lambda$ so that we can have
the canonical form (3.3). Extract the results for the matrices $G$, $P$ and
$N$.

  4. 4.

Define a new effective covariant derivative $\mathcal{D}_{\rho}$ with field
connection $\omega$ as eq. 3.4 and making use of it, modify $\Lambda$ further
to have the form (3.5).

  5. 5.

Comparing the new form of $\Lambda$ with standard form (3.6) read off the
results for matrices $I,E$ and $\Omega$ using eq. 3.7.

  6. 6.

Compute $\text{tr}\thinspace(I)$, $\text{tr}\thinspace(E)$,
$\text{tr}\thinspace(E^{2})$ and $\text{tr}\thinspace(\Omega^{2})$. Use this
data in the formulae (3.8) to find the result for the first three Seeley-
DeWitt coefficients (without ghosts) $a_{2n}^{\text{no ghost}}$.

  7. 7.

Making use of background equations of motion, gamma and tensor identities,
simplify $a_{2n}^{\text{no ghost}}$.

  8. 8.

Repeat the same steps ($3-7$) starting from ghost action to have Seeley-DeWitt
coefficients for ghosts ($a_{2n}^{\text{ghost}}$).

  9. 9.

Obtain the total Seeley-DeWitt coefficients of the theory by
$a_{2n}^{\text{total}}=a_{2n}^{\text{no ghost}}+a_{2n}^{\text{ghost}}$.

##  4 Minimal $\mathcal{N}=2,d=4$ Einstein-Maxwell supergravity theory

In this section, our motive is to briefly summarize a minimally coupled
$\mathcal{N}=2$ supergravity embedded with the solutions of Einstein-Maxwell
theory in four dimensions. There are a class of $\mathcal{N}=2$ supergravity
solutions to the equations of motion, determined by certain conditions [45].
These conditions reduce the full $\mathcal{N}=2$ supergravity equations of
motion to a much simpler form of equations of motion for Einstein-Maxwell
theory. The resultant minimal $\mathcal{N}=2,d=4$ Einstein-Maxwell
supergravity theory (EMSGT) plays a significant role in string phenomenology,
black holes, and holography. In general, $\mathcal{N}=2$ supergravity theories
are coupled to additional matter fields like $n_{V}$ vector multiplets and
$n_{H}$ hyper multiplets [46, 45]. But in this work, we will focus on the
simpler case of minimally coupled $\mathcal{N}=2$ supergravity where
$n_{V}=n_{H}=0$. However, both the theory have the same type of classical or
black hole solutions. At any point, one can compute the Seeley-DeWitt
coefficients and the logarithmic contributions for $n_{V}$ and $n_{H}$
fluctuations separately, and can extend the minimal supergravity results to
these matter coupled theories. The minimal $\mathcal{N}=2$ EMSGT is associated
with the field content provided in table 1. Both the bosonic and fermionic
sectors of the field content are described by the Lagrangians
$\mathcal{L}_{\text{EM}}$ and $\mathcal{L}_{\text{gravitini}}$ respectively in
the action (4.1). Note that all these fields are free in the sense that they
are only coupled to the metric (or graviton).

Physical Field Content
---
Bosonic sector: one graviton ($g_{\mu\nu}$), one graviphoton ($A_{\mu}$)
Fermionic sector: two gravitino ($\psi_{\mu},\phi_{\mu}$)
($\mu,\nu=1\cdots 4$)
Table 1: Field content in minimally coupled $\mathcal{N}=2$ EMSGT.

We consider standard $d=4$ Einstein-Maxwell theory embedded in minimal
$\mathcal{N}=2$ supergravity described by the following action,

$\mathcal{S}=\int
d^{4}x\sqrt{\text{det}\thinspace{g}}\thinspace(\mathcal{L}_{\text{EM}}+\mathcal{L}_{\text{gravitini}}),$
(4.1)

with

$\displaystyle\begin{split}&\mathcal{L}_{\text{EM}}=\left(R-F_{\mu\nu}F^{\mu\nu}\right),\\\
&\mathcal{L}_{\text{gravitini}}=-\frac{1}{2}\bar{\psi}_{\mu}\gamma^{\mu\rho\nu}D_{\rho}\psi_{\nu}-\frac{1}{2}\bar{\phi}_{\mu}\gamma^{\mu\rho\nu}D_{\rho}\phi_{\nu}\\\
&\qquad\qquad\quad+\frac{1}{2}\bar{\psi}_{\mu}\big{(}{F}^{\mu\nu}+\frac{1}{2}\gamma^{5}{H}^{\mu\nu}\big{)}\phi_{\nu}-\frac{1}{2}\bar{\phi}_{\mu}\big{(}{F}^{\mu\nu}+\frac{1}{2}\gamma^{5}{H}^{\mu\nu}\big{)}\psi_{\nu},\end{split}$
(4.2)

where $R$ is the Ricci scalar induced from metric $g_{\mu\nu}$, $F_{\mu\nu}$
the electromagnetic field strength of gauge field $A_{\mu}$ in Einstein-
Maxwell theory, $\psi_{\mu},\phi_{\mu}$ are two gravitini species and
${H}^{\mu\nu}=-\varepsilon^{\mu\nu\rho\sigma}{F}_{\rho\sigma}$ is the dual
field strength.111111In particular, the identity
$\gamma^{\mu\nu\rho\sigma}=-\gamma^{5}\varepsilon^{\mu\nu\rho\sigma}$ is very
much useful, where
$\gamma^{5}=-\frac{1}{4!}\varepsilon^{\mu\nu\rho\sigma}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}$
in four dimensions. $\psi_{\mu},\phi_{\mu}$ are Majorana spinors in 4D and the
action (4.1) describes black holes in pure $\mathcal{N}=2$ supergravity. It is
useful to note that we have set121212We are following the structure and
conventions of [9]. $G_{N}=1/{16\pi}$.

###  4.1 Equations of motion and identities

Let us denote $(\bar{g}_{\mu\nu},\bar{A}_{\mu})$ as an arbitrary solution to
the classical equations of motion of our theory and define corresponding
background field strength tensor
$\bar{F}_{\mu\nu}\equiv\partial_{\mu}\bar{A}_{\nu}-\partial_{\nu}\bar{A}_{\mu}$.
Then the geometry and matter fields are related by Einstein equation,

$R_{\mu\nu}-\frac{1}{2}\bar{g}_{\mu\nu}R=2\bar{F}_{\mu\rho}{\bar{F_{\nu}}}^{\rho}-\frac{1}{2}\bar{g}_{\mu\nu}\bar{F}_{\rho\sigma}\bar{F}^{\rho\sigma}.$
(4.3)

The background gauge field strength $\bar{F}_{\mu\nu}$ satisfies Maxwell
equations and Bianchi identity,

$D^{\mu}\bar{F}_{\mu\nu}=0,\\\ \\\ \enspace D_{[\mu}\bar{F}_{\nu\rho]}=0.$
(4.4)

Eq.(4.3) and eq.(4.4) represents the basic equations of motion of
$\mathcal{N}=2,d=4$ EMSGT. Again, the trace of Eq.(4.3) yields

$R=0,$ (4.5)

Hence for the remaining parts of this paper, we will use this condition and
ignore the terms proportional to $R$. Again using (4.5), (4.3) reduces to,

$\bar{F}_{\mu\rho}{\bar{F_{\nu}}}^{\rho}=\frac{1}{2}R_{\mu\nu}+\frac{1}{4}\bar{g}_{\mu\nu}\bar{F}_{\rho\sigma}\bar{F}^{\rho\sigma}.$
(4.6)

In addition, the dual gauge field strength $\bar{H}_{\mu\nu}$ satisfies the
Maxwell-Bianchi equations of the form,

$D^{\mu}\bar{H}_{\mu\nu}=0,\enspace D_{[\mu}\bar{H}_{\nu\rho]}=0.$ (4.7)

It is also important to mention the gravitational Bianchi identity

$R_{\mu[\nu\rho\sigma]}=0.$ (4.8)

We have used eqs. 4.4, 4.6, 4.7 and 4.8 to simplify our calculations and
express them in closed forms.

##  5 Seeley-DeWitt coefficients in minimal $\mathcal{N}=2,d=4$ Einstein-
Maxwell supergravity theory

In this section, we will calculate the first three Seeley-DeWitt coefficients
of the quadratic order fluctuations of minimally coupled $\mathcal{N}=2,d=4$
EMSGT fields discussed in § 4. The field content is divided into two sectors —
fermions and bosons, for which the contributions will be calculated
separately. We actually compute results for the fermionic sector, while the
bosonic sector results are reviewed and recalculated from [9]. Finally, we
will sum up these results in order to find the complete Seeley-DeWitt
coefficients of the theory.

###  5.1 $\mathcal{N}=2$ SUGRA: Bosons

Let’s now consider the Einstein-Maxwell theory described by the Lagrangian
$\mathcal{L}_{\text{EM}}$ in (4.1) with the action,

$\mathcal{S}_{\text{EM}}=\int
d^{4}x\sqrt{\text{det}\thinspace{g}}\thinspace(R-F_{\mu\nu}F^{\mu\nu}).$ (5.1)

We aim to calculate the Seeley-DeWitt coefficients for this theory by
considering quadratic order fluctuations around a classical background. This
methodology has been developed in [9], where the authors have computed only
the $a_{4}$ coefficient for calculating the logarithmic correction to extremal
Kerr-Newman black hole entropy. But we find it worthful to briefly review this
work and calculate all the first three Seeley-DeWitt coefficients for the
theory, which may be useful in other applications.

Let $(\bar{g}_{\mu\nu},\bar{A}_{\alpha})$ be any arbitrary classical solution
to the equations of motion of Einstein-Maxwell theory, and around this
classical background we consider following fluctuations of metric and gauge
field131313The graviton and graviphoton have been scaled in such a manner that
the kinetic terms for them have the same normalization in the quadratic order
fluctuated action (5.4).

$\tilde{g}_{\mu\nu}=\sqrt{2}h_{\mu\nu},\enspace\tilde{A}_{\alpha}=\frac{1}{2}a_{\alpha},$
(5.2)

where $h_{\mu\nu}$ and $a_{\alpha}$ are the graviton and graviphoton
respectively. Moreover, the theory is gauge fixed by adding the following
gauge-fixing term,

$\displaystyle\begin{split}&-\int
d^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\Big{\\{}(D^{\mu}h_{\mu\rho}-\frac{1}{2}D_{\rho}h)(D^{\nu}{h_{\nu}}^{\rho}-\frac{1}{2}D^{\rho}h)+\frac{1}{2}(D^{\mu}a_{\mu})(D^{\nu}a_{\nu})\Big{\\}}\end{split}$
(5.3)

and the corresponding ghost action is added to the action (5.1). Up to
quadratic order in the fluctuations the gauge fixed action excluding the ghost
part is given by[9],

$\displaystyle\begin{split}&\mathcal{S}_{2}=\frac{1}{2}\int
d^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\tilde{\xi}_{m}\Lambda^{mn}\tilde{\xi}_{n},\\\
&\tilde{\xi}_{m}\Lambda^{mn}\tilde{\xi}_{n}=h_{\mu\nu}\Box
h^{\mu\nu}-\frac{1}{2}h\Box h+a_{\alpha}\Box
a^{\alpha}-a_{\alpha}R^{\alpha\beta}a_{\beta}\\\
&\qquad\qquad\quad+h_{\mu\nu}\Big{\\{}2R^{\mu\alpha\nu\beta}+\bar{g}^{\mu\alpha}R^{\nu\beta}+\bar{g}^{\alpha\beta}R^{\mu\nu}-\bar{g}^{\mu\nu}R^{\alpha\beta}-3\bar{g}^{\nu\beta}R^{\mu\alpha}\\\
&\qquad\qquad\quad-4\bar{F}^{\mu\alpha}\bar{F}^{\nu\beta}+(\frac{1}{2}\bar{g}^{\mu\nu}\bar{g}^{\alpha\beta}-\bar{g}^{\mu\alpha}\bar{g}^{\nu\beta})(\bar{F}_{\theta\varphi}\bar{F}^{\theta\varphi})\Big{\\}}h_{\alpha\beta}\\\
&\qquad\qquad\quad+h_{\mu\nu}\left\\{4\sqrt{2}\bar{g}^{\alpha\nu}\bar{F}^{\rho\mu}-4\sqrt{2}\bar{g}^{\nu\rho}\bar{F}^{\alpha\mu}-2\sqrt{2}\bar{g}^{\mu\nu}\bar{F}^{\rho\alpha}\right\\}(D_{\rho}a_{\alpha}),\end{split}$
(5.4)

where we denote $\Box\equiv D_{\mu}D^{\mu}$,
$h={h^{\mu}}_{\mu}=\text{tr}(h_{\mu\nu})$,
$\bar{F}_{\mu\nu}\equiv\partial_{\mu}\bar{A}_{\nu}-\partial_{\nu}\bar{A}_{\mu}$
and $\tilde{\xi}_{m}=\\{h_{\mu\nu},a_{\alpha}\\}$. Kindly note that $\Lambda$
is not in Laplace-type form due to the presence of the kinetic term $h\Box h$.
We need to restructure $\Lambda$ and express it in the canonical form. In
order to proceed further, we introduce the operator
$G^{\mu\nu\thinspace\alpha\beta}$141414The general form of $G$ is
$\tilde{\xi}_{m}G^{mn}\tilde{\xi}_{n}=h_{\mu\nu}G^{\mu\nu\thinspace\alpha\beta}h_{\alpha\beta}+a_{\alpha}G^{\alpha\beta}a_{\beta}$,
where the form of $G^{\mu\nu\thinspace\alpha\beta}$ is given in eq. 5.5 and
$G^{\alpha\beta}=\bar{g}^{\alpha\beta}$.

$G^{\mu\nu\thinspace\alpha\beta}=\frac{1}{2}(\bar{g}^{\mu\alpha}\bar{g}^{\nu\beta}+\bar{g}^{\mu\beta}\bar{g}^{\nu\alpha}-\bar{g}^{\mu\nu}\bar{g}^{\alpha\beta}),$
(5.5)

and rewrite the kinetic part as $h_{\mu\nu}\Box h^{\mu\nu}-\frac{1}{2}h\Box
h+a_{\alpha}\Box a^{\alpha}=G^{\mu\nu\thinspace\alpha\beta}h_{\mu\nu}\Box
h_{\alpha\beta}+\bar{g}^{\alpha\beta}a_{\alpha}\Box a_{\beta}$. With necessary
modifications of the total derivative terms to make $\Lambda$ Hermitian, the
resulting form of $\Lambda$ becomes

$\displaystyle\begin{split}\tilde{\xi}_{m}\Lambda^{mn}\tilde{\xi}_{n}&=G^{\mu\nu\thinspace\alpha\beta}h_{\mu\nu}\Box
h_{\alpha\beta}+\bar{g}^{\alpha\beta}a_{\alpha}\Box
a_{\beta}-a_{\alpha}R^{\alpha\beta}a_{\beta}\\\
&\quad+h_{\mu\nu}\Big{\\{}R^{\mu\alpha\nu\beta}+R^{\mu\beta\nu\alpha}-\frac{1}{2}(\bar{g}^{\mu\alpha}R^{\nu\beta}+\bar{g}^{\mu\beta}R^{\nu\alpha}\\\
&\quad+\bar{g}^{\nu\alpha}R^{\mu\beta}+\bar{g}^{\nu\beta}R^{\mu\alpha})-2(\bar{F}^{\mu\alpha}\bar{F}^{\nu\beta}+\bar{F}^{\mu\beta}\bar{F}^{\nu\alpha})\\\
&\quad+\frac{1}{2}(\bar{g}^{\mu\nu}\bar{g}^{\alpha\beta}-\bar{g}^{\mu\alpha}\bar{g}^{\nu\beta}-\bar{g}^{\mu\beta}\bar{g}^{\nu\alpha})(\bar{F}_{\theta\varphi}\bar{F}^{\theta\varphi})\Big{\\}}h_{\alpha\beta}\\\
&\quad-
h_{\mu\nu}\Big{\\{}\frac{1}{4}(D_{\rho}K^{\rho})^{\mu\nu\thinspace\alpha}-\frac{1}{2}(K^{\rho})^{\mu\nu\thinspace\alpha}D_{\rho}\Big{\\}}a_{\alpha}\\\
&\quad-
a_{\alpha}\Big{\\{}\frac{1}{4}(D_{\rho}K^{\rho})^{\mu\nu\thinspace\alpha}+\frac{1}{2}(K^{\rho})^{\mu\nu\thinspace\alpha}D_{\rho}\Big{\\}}h_{\mu\nu},\end{split}$
(5.6)

where

$(K^{\rho})^{\mu\nu\thinspace\alpha}=2\sqrt{2}(\bar{g}^{\alpha\mu}\bar{F}^{\rho\nu}+\bar{g}^{\alpha\nu}\bar{F}^{\rho\mu}-\bar{g}^{\mu\rho}\bar{F}^{\alpha\nu}-\bar{g}^{\nu\rho}\bar{F}^{\alpha\mu}-\bar{g}^{\mu\nu}\bar{F}^{\rho\alpha}).$
(5.7)

We identify the two matrices $P$ and $N$ (defined in (3.3)) as

$\displaystyle\begin{split}&\tilde{\xi}_{m}{(N^{\rho})}^{mn}\tilde{\xi}_{n}=\frac{1}{2}h_{\mu\nu}(K^{\rho})^{\mu\nu\thinspace\alpha}a_{\alpha}-\frac{1}{2}a_{\alpha}(K^{\rho})^{\mu\nu\thinspace\alpha}h_{\mu\nu},\\\
&\tilde{\xi_{m}}P^{mn}\tilde{\xi_{n}}=h_{\mu\nu}\Big{\\{}R^{\mu\alpha\nu\beta}+R^{\mu\beta\nu\alpha}-\frac{1}{2}(\bar{g}^{\mu\alpha}R^{\nu\beta}+\bar{g}^{\mu\beta}R^{\nu\alpha}\\\
&\qquad\qquad\quad+\bar{g}^{\nu\alpha}R^{\mu\beta}+\bar{g}^{\nu\beta}R^{\mu\alpha})-2(\bar{F}^{\mu\alpha}\bar{F}^{\nu\beta}+\bar{F}^{\mu\beta}\bar{F}^{\nu\alpha})\\\
&\qquad\qquad\quad+\frac{1}{2}(\bar{g}^{\mu\nu}g^{\alpha\beta}-\bar{g}^{\mu\alpha}g^{\nu\beta}-\bar{g}^{\mu\beta}\bar{g}^{\nu\alpha})(\bar{F}_{\theta\varphi}\bar{F}^{\theta\varphi})\Big{\\}}h_{\alpha\beta}\\\
&\qquad\qquad\quad-
a_{\alpha}R^{\alpha\beta}a_{\beta}+\frac{\sqrt{2}}{2}h_{\mu\nu}\Big{\\{}D^{\mu}\bar{F}^{\alpha\nu}+D^{\nu}\bar{F}^{\alpha\mu}\Big{\\}}a_{\alpha}\\\
&\qquad\qquad\quad+\frac{\sqrt{2}}{2}a_{\alpha}\Big{\\{}D^{\mu}\bar{F}^{\alpha\nu}+D^{\nu}\bar{F}^{\alpha\mu}\Big{\\}}h_{\mu\nu}.\end{split}$
(5.8)

By comparing $\Lambda$ from eq. 5.6 with the standard form (3.6), one can
extract the useful matrices $I$, $\omega$, $E$ and $\Omega$, and find their
traces in order to compute the Seeley-DeWitt coefficients. It is now
straightforward to express the identity matrix as

$\tilde{\xi_{m}}I^{mn}\tilde{\xi_{n}}=h_{\mu\nu}G^{\mu\nu\thinspace\alpha\beta}h_{\alpha\beta}+a_{\alpha}\bar{g}^{\alpha\beta}a_{\beta}.$
(5.9)

Making use of the results of $N$ and $P$ from eq. 5.8 in the formulae (3.7),
we express

$\displaystyle\begin{split}&\tilde{\xi}_{m}(\omega^{\rho})^{mn}\tilde{\xi}_{n}=\frac{1}{4}h_{\mu\nu}(K^{\rho})^{\mu\nu\thinspace\alpha}a_{\alpha}-\frac{1}{4}a_{\alpha}(K^{\rho})^{\mu\nu\thinspace\alpha}h_{\mu\nu},\\\
&\tilde{\xi}_{m}E^{mn}\tilde{\xi}_{n}=h_{\mu\nu}(R^{\mu\alpha\nu\beta}+R^{\mu\beta\nu\alpha}-\bar{g}^{\mu\nu}R^{\alpha\beta}-\bar{g}^{\alpha\beta}R^{\mu\nu})h_{\alpha\beta}\\\
&\qquad\qquad\quad+\frac{3}{2}a_{\alpha}\bar{g}^{\alpha\beta}\bar{F}_{\mu\nu}\bar{F}^{\mu\nu}a_{\beta}+\frac{\sqrt{2}}{2}h_{\mu\nu}(D^{\mu}\bar{F}^{\alpha\nu}+D^{\nu}\bar{F}^{\alpha\mu})a_{\alpha}\\\
&\qquad\qquad\quad+\frac{\sqrt{2}}{2}a_{\alpha}(D^{\mu}\bar{F}^{\alpha\nu}+D^{\nu}\bar{F}^{\alpha\mu})h_{\mu\nu},\\\
&\tilde{\xi}_{m}(\Omega^{\rho\sigma})^{mn}\tilde{\xi}_{n}=h_{\mu\nu}\Big{\\{}\frac{1}{2}(\bar{g}^{\nu\beta}R^{\mu\alpha\rho\sigma}+\bar{g}^{\nu\alpha}R^{\mu\beta\rho\sigma}+\bar{g}^{\mu\beta}R^{\nu\alpha\rho\sigma}+\bar{g}^{\mu\alpha}R^{\nu\beta\rho\sigma})\\\
&\qquad\qquad\qquad\quad+[\omega^{\rho},\omega^{\sigma}]^{\mu\nu\thinspace\alpha\beta}\Big{\\}}h_{\alpha\beta}+a_{\alpha}\Big{\\{}R^{\alpha\beta\rho\sigma}+[\omega^{\rho},\omega^{\sigma}]^{\alpha\beta}\Big{\\}}a_{\beta}\\\
&\qquad\qquad\qquad\quad+h_{\mu\nu}{D^{[\rho}\omega^{\sigma]}}^{\mu\nu\thinspace\alpha}a_{\alpha}+a_{\alpha}{D^{[\rho}\omega^{\sigma]}}^{\alpha\thinspace\mu\nu}h_{\mu\nu}\end{split}$
(5.10)

Note that in writing the expression for $\Omega$, we have used the following
commutation relation,

$\displaystyle\begin{split}\tilde{\xi}_{m}[D^{\rho},D^{\sigma}]^{mn}\tilde{\xi}_{n}&=h_{\mu\nu}[D^{\rho},D^{\sigma}]h^{\mu\nu}+a_{\alpha}[D^{\rho},D^{\sigma}]a^{\alpha}\\\
&=h_{\mu\nu}\Big{\\{}\bar{g}^{\nu\beta}R^{\mu\alpha\rho\sigma}+\bar{g}^{\mu\beta}R^{\nu\alpha\rho\sigma}\Big{\\}}h_{\alpha\beta}+a_{\alpha}R^{\alpha\beta\rho\sigma}a_{\beta}.\end{split}$
(5.11)

The useful traces for the matrices are finally written down in (5.12). We skip
the details of the intermediate steps as the calculation is straightforward
although tedious.

$\displaystyle\begin{split}&\text{tr}(I)=10+4=14,\\\
&\text{tr}(E)=6\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&\text{tr}(E^{2})=3R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-7R_{\mu\nu}R^{\mu\nu}+3R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}+9(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2},\\\
&\text{tr}(\Omega_{\rho\sigma}\Omega^{\rho\sigma})=-7R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+56R_{\mu\nu}R^{\mu\nu}-18R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}\\\
&\hskip 72.26999pt-54(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}.\end{split}$
(5.12)

Substituting these trace values in the formulae (3.8), we find the Seeley-
DeWitt coefficients of Einstein-Maxwell theory (without ghost). The results
are

$\displaystyle\begin{split}&(4\pi)^{2}a_{0}^{\text{EM}}=14,\\\
&(4\pi)^{2}a_{2}^{\text{EM}}=6\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&(4\pi)^{2}a_{4}^{\text{EM}}=\frac{1}{180}\left(179R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+196R_{\mu\nu}R^{\mu\nu}\right).\end{split}$
(5.13)

The ghost fields are non-interacting and hence can be treated separately. The
corresponding ghost action is given by,

$\mathcal{S}_{\text{ghost},b}=\frac{1}{2}\int
d^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\left\\{2b_{\mu}(\bar{g}^{\mu\nu}\Box+R^{\mu\nu})c_{\nu}+2b\Box
c-4b\bar{F}^{\rho\nu}D_{\rho}c_{\nu}\right\\},$ (5.14)

where $b_{\mu},c_{\mu}$ (vector fields) are diffeomorphism ghosts related to
the graviton; $b,c$ (scalar fields) are the ghosts associated with the
graviphoton. Adjusting this action up to total derivative, we express the
corresponding $\Lambda$ as,

$\displaystyle\begin{split}\tilde{\xi}_{m}\Lambda^{mn}\tilde{\xi}_{n}&=b_{\mu}g^{\mu\nu}\Box
c_{\nu}+c_{\mu}g^{\mu\nu}\Box b_{\nu}+b\Box c+c\Box b\\\
&\quad+b_{\mu}R^{\mu\nu}c_{\nu}+c_{\mu}R^{\mu\nu}b_{\nu}-2b\bar{F}^{\rho\nu}D_{\rho}c_{\nu}+2c_{\mu}\bar{F}^{\rho\mu}D_{\rho}b,\end{split}$
(5.15)

and comparing with (3.3) identify the matrices $N$ and $P$ as

$\displaystyle\begin{split}&\tilde{\xi}_{m}(N^{\rho})^{mn}\tilde{\xi}_{n}=-2b\bar{F}^{\rho\nu}c_{\nu}+2c_{\mu}\bar{F}^{\rho\mu}b,\\\
&\tilde{\xi}_{m}P^{mn}\tilde{\xi}_{n}=b_{\mu}R^{\mu\nu}c_{\nu}+c_{\mu}R^{\mu\nu}b_{\nu}.\end{split}$
(5.16)

We follow the general approach described in § 3 and making use of formulae
(3.7) we extract the following necessary matrices

$\displaystyle\begin{split}&\tilde{\xi}_{m}I^{mn}\tilde{\xi}_{n}=b_{\mu}\bar{g}^{\mu\nu}c_{\nu}+c_{\mu}\bar{g}^{\mu\nu}b_{\nu}+bc+cb,\\\
&\tilde{\xi}_{m}(\omega_{\rho})^{mn}\tilde{\xi}_{n}=-b{\bar{F_{\rho}}}^{\nu}c_{\nu}+c_{\mu}\bar{F_{\rho}}^{\mu}b,\\\
&\tilde{\xi}_{m}E^{mn}\tilde{\xi}_{n}=b_{\mu}R^{\mu\nu}c_{\nu}+c_{\mu}R^{\mu\nu}b_{\nu},\\\
&\tilde{\xi}_{m}(\Omega_{\rho\sigma})^{mn}\tilde{\xi}_{n}=b_{\mu}{R^{\mu\nu}}_{\rho\sigma}c_{\nu}+c_{\mu}{R^{\mu\nu}}_{\rho\sigma}b_{\nu}+b(D^{\nu}\bar{F}_{\rho\sigma})c_{\nu}-c_{\mu}(D^{\mu}\bar{F}_{\rho\sigma})b.\end{split}$
(5.17)

Using (5.17), one can compute the following traces for the ghost fields,

$\displaystyle\begin{split}&\text{tr}(I)=4+4+1+1=10,\\\
&\text{tr}(E)=0,\enspace\text{tr}(E^{2})=2R_{\mu\nu}R^{\mu\nu},\\\
&\text{tr}(\Omega_{\rho\sigma}\Omega^{\rho\sigma})=-2R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma},\end{split}$
(5.18)

and can use them in order to find Seeley-DeWitt coefficients for ghost fields
corresponding to the bosonic action of the theory. This yields

$\displaystyle\begin{split}&(4\pi)^{2}a_{0}^{\text{ghost},b}=-10,\\\
&(4\pi)^{2}a_{2}^{\text{ghost},b}=0,\\\
&(4\pi)^{2}a_{4}^{\text{ghost},b}=\frac{1}{18}\left(2R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-17R_{\mu\nu}R^{\mu\nu}\right).\end{split}$
(5.19)

Note that an overall minus sign is put on each coefficient since ghosts fields
follow the reverse of usual spin-statistics.

###  5.2 $\mathcal{N}=2$ SUGRA: Fermions

The fermionic sector of $\mathcal{N}=2$ includes two Majorana gravitini fields
described by the Lagrangian $\mathcal{L}_{\text{gravitini}}$ expressed in
(4.2). To gauge fix the fermionic action we add the following term to
$\mathcal{L}_{\text{gravitini}}$,

$\mathcal{L}_{\text{gf}}=\frac{1}{4}\bar{\psi}_{\mu}\gamma^{\mu}\gamma^{\rho}D_{\rho}\gamma^{\nu}\psi_{\nu}+\frac{1}{4}\bar{\phi}_{\mu}\gamma^{\mu}\gamma^{\rho}D_{\rho}\gamma^{\nu}\phi_{\nu}$
(5.20)

and we can express the quadratic part of the corresponding gauge fixed action
in Dirac form as

$\begin{gathered}\mathcal{S}_{{f}}=\int
d^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\mathcal{L}_{{f}},\\\
\mathcal{L}_{{f}}=\mathcal{L}_{\text{gravitini}}+\mathcal{L}_{\text{gf}}=\frac{i}{4}\bar{\psi}_{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\mu}D_{\rho}\psi_{\nu}+\frac{i}{4}\bar{\phi}_{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\mu}D_{\rho}\phi_{\nu}\\\
\hskip
130.08621pt+\frac{i}{2}\bar{\psi}_{\mu}\big{(}\bar{F}^{\mu\nu}+\frac{1}{2}\gamma^{5}\bar{H}^{\mu\nu}\big{)}\phi_{\nu}-\frac{i}{2}\bar{\phi}_{\mu}\big{(}\bar{F}^{\mu\nu}+\frac{1}{2}\gamma^{5}\bar{H}^{\mu\nu}\big{)}\psi_{\nu},\end{gathered}$
(5.21)

Now, we can rewrite the action for gravitini fluctuations
$\tilde{\xi}_{m}=\\{\psi_{\mu},\phi_{\nu}\\}$ as

$\begin{gathered}\mathcal{S}_{2}=\frac{1}{2}\int
d^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\tilde{\xi}_{m}\not{D}^{mn}\tilde{\xi}_{n},\\\
\tilde{\xi}_{m}\not{D}^{mn}\tilde{\xi}_{n}=\frac{i}{2}\bar{\psi}_{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\mu}D_{\rho}\psi_{\nu}+\frac{i}{2}\bar{\phi}_{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\mu}D_{\rho}\phi_{\nu}\\\
\hskip
130.08621pt+i\bar{\psi}_{\mu}\big{(}\bar{F}^{\mu\nu}+\frac{1}{2}\gamma^{5}\bar{H}^{\mu\nu}\big{)}\phi_{\nu}-i\bar{\phi}_{\mu}\big{(}\bar{F}^{\mu\nu}+\frac{1}{2}\gamma^{5}\bar{H}^{\mu\nu}\big{)}\psi_{\nu}.\end{gathered}$
(5.22)

The action contains first-order Dirac-type differential operator $\not{D}$,
but for computing the Seeley-DeWitt coefficients following general approach §
3 we need to make it quadratic in order. With our choice of four-dimensional
Euclidean space-time, gamma matrices are Hermitian, the gravitino conjugates
are
$\bar{\psi}_{\mu}=\psi_{\mu}^{\dagger},\bar{\phi}_{\mu}=\phi_{\mu}^{\dagger}$,
and
$\left(\gamma^{\nu}\gamma^{\rho}\gamma^{\mu}D_{\rho}\right)^{\dagger}=-\gamma^{\nu}\gamma^{\rho}\gamma^{\mu}D_{\rho}$.
We will consider $\bar{F}^{\mu\nu}$ to be real. Hence $\not{D}$ turns out to
be Hermitian, i.e. $\not{D}^{\dagger}=\not{D}$. Now one can follow the method
mentioned in § 3 to obtain the necessary Laplace-type operator $\Lambda$ of
the fermionic sector as

$\begin{split}&\tilde{\xi}_{m}\Lambda^{mn}\tilde{\xi}_{n}=\tilde{\xi}_{m}\not{D}^{mp}{\not{D}_{p}}^{n}\tilde{\xi}_{n}\\\
&\quad=-\bar{\psi}_{\mu}\Big{\\{}\frac{1}{4}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\gamma^{\nu}\gamma^{\sigma}\gamma_{\alpha}D_{\rho}D_{\sigma}-\big{(}\bar{F}^{\mu\alpha}+\frac{1}{2}\gamma^{5}\bar{H}^{\mu\alpha}\big{)}\big{(}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}{\bar{H_{\alpha}}}^{\nu}\big{)}\Big{\\}}\psi_{\nu}\\\
&\qquad-\bar{\phi}_{\mu}\Big{\\{}\frac{1}{4}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\gamma^{\nu}\gamma^{\sigma}\gamma_{\alpha}D_{\rho}D_{\sigma}-\big{(}\bar{F}^{\mu\alpha}+\frac{1}{2}\gamma^{5}\bar{H}^{\mu\alpha}\big{)}\big{(}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}{\bar{H_{\alpha}}}^{\nu}\big{)}\Big{\\}}\phi_{\nu}\\\
&\qquad-\frac{1}{2}\bar{\psi}_{\mu}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}D_{\rho}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}D_{\rho}{\bar{H_{\alpha}}}^{\nu}\big{)}\phi_{\nu}+\frac{1}{2}\bar{\phi}_{\mu}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}D_{\rho}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}D_{\rho}{\bar{H_{\alpha}}}^{\nu}\big{)}\psi_{\nu}\\\
&\qquad-\frac{1}{2}\bar{\psi}_{\mu}\Big{\\{}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}{\bar{H_{\alpha}}}^{\nu}\big{)}+\big{(}\bar{F^{\mu}}_{\alpha}+\frac{1}{2}\gamma^{5}{\bar{H^{\mu}}}_{\alpha}\big{)}\gamma^{\nu}\gamma^{\rho}\gamma^{\alpha}\Big{\\}}D_{\rho}\phi_{\nu}\\\
&\qquad+\frac{1}{2}\bar{\phi}_{\mu}\Big{\\{}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}{\bar{H_{\alpha}}}^{\nu}\big{)}+\big{(}\bar{F^{\mu}}_{\alpha}+\frac{1}{2}\gamma^{5}{\bar{H^{\mu}}}_{\alpha}\big{)}\gamma^{\nu}\gamma^{\rho}\gamma^{\alpha}\Big{\\}}D_{\rho}\psi_{\nu}.\end{split}$
(5.23)

The second derivative terms of eq. 5.23 can be simplified to the following
form [44],

$\displaystyle\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\gamma^{\nu}\gamma^{\sigma}\gamma_{\alpha}D_{\rho}D_{\sigma}=4\mathbb{I}_{4}\bar{g}^{\mu\nu}D^{\rho}D_{\rho}-(\gamma^{\alpha}\gamma^{\beta}\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}\gamma^{\beta}\gamma^{\alpha})[D_{\alpha},D_{\beta}],$
$\displaystyle\qquad\qquad=4\mathbb{I}_{4}\bar{g}^{\mu\nu}D^{\rho}D_{\rho}-4\mathbb{I}_{4}R^{\mu\nu}+2\gamma^{\alpha}\gamma^{\beta}{R^{\mu\nu}}_{\alpha\beta}-2\gamma^{\mu}\gamma^{\alpha}{R^{\nu}}_{\alpha}+2\gamma^{\nu}\gamma^{\alpha}{R^{\mu}}_{\alpha}.$
(5.24)

Using (5.2) and re-arranging the terms (5.23) can be expressed as,

$\displaystyle\begin{split}&\tilde{\xi}_{m}\Lambda^{mn}\tilde{\xi}_{n}=-\mathbb{I}_{4}\bar{g}^{\mu\nu}\bar{\psi}_{\mu}D^{\rho}D_{\rho}\psi_{\nu}-\mathbb{I}_{4}\bar{g}^{\mu\nu}\bar{\phi}_{\mu}D^{\rho}D_{\rho}\phi_{\nu}+\bar{\psi}_{\mu}\Big{\\{}\mathbb{I}_{4}R^{\mu\nu}-\frac{1}{2}\gamma^{\alpha}\gamma^{\beta}{R^{\mu\nu}}_{\alpha\beta}\\\
&\qquad\qquad\quad+\frac{1}{2}\gamma^{\mu}\gamma^{\alpha}{R^{\nu}}_{\alpha}-\frac{1}{2}\gamma^{\nu}\gamma^{\alpha}{R^{\mu}}_{\alpha}+\big{(}\bar{F}^{\mu\alpha}+\frac{1}{2}\gamma^{5}\bar{H}^{\mu\alpha}\big{)}\big{(}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}{\bar{H_{\alpha}}}^{\nu}\big{)}\Big{\\}}\psi_{\nu}\\\
&\qquad\qquad\quad+\bar{\phi}_{\mu}\Big{\\{}\mathbb{I}_{4}R^{\mu\nu}-\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}\\\
&\qquad\qquad\quad+\big{(}\bar{F}^{\mu\alpha}+\frac{1}{2}\gamma^{5}\bar{H}^{\mu\alpha}\big{)}\big{(}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}{\bar{H_{\alpha}}}^{\nu}\big{)}\Big{\\}}\phi_{\nu}\\\
&\qquad\qquad\quad-\frac{1}{2}\bar{\psi}_{\mu}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}D_{\rho}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}D_{\rho}{\bar{H_{\alpha}}}^{\nu}\big{)}\phi_{\nu}\\\
&\qquad\qquad\quad+\frac{1}{2}\bar{\phi}_{\mu}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}D_{\rho}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}D_{\rho}{\bar{H_{\alpha}}}^{\nu}\big{)}\psi_{\nu}\\\
&\qquad\qquad\quad-\frac{1}{2}\bar{\psi}_{\mu}\Big{\\{}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}{\bar{H_{\alpha}}}^{\nu}\big{)}+\big{(}\bar{F^{\mu}}_{\alpha}+\frac{1}{2}\gamma^{5}{\bar{H^{\mu}}}_{\alpha}\big{)}\gamma^{\nu}\gamma^{\rho}\gamma^{\alpha}\Big{\\}}D_{\rho}\phi_{\nu}\\\
&\qquad\qquad\quad+\frac{1}{2}\bar{\phi}_{\mu}\Big{\\{}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}{\bar{H_{\alpha}}}^{\nu}\big{)}+\big{(}\bar{F^{\mu}}_{\alpha}+\frac{1}{2}\gamma^{5}{\bar{H^{\mu}}}_{\alpha}\big{)}\gamma^{\nu}\gamma^{\rho}\gamma^{\alpha}\Big{\\}}D_{\rho}\psi_{\nu}.\end{split}$
(5.25)

So we have adjusted $\Lambda$ as the prescribed form (3.5), where the matrices
$N$ and $P$ hold the following form

$\displaystyle\begin{split}&\tilde{\xi}_{m}{(N^{\rho})}^{mn}\tilde{\xi}_{n}=\frac{1}{2}\bar{\psi}_{\mu}\Big{\\{}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}{\bar{H_{\alpha}}}^{\nu}\big{)}+\big{(}\bar{F^{\mu}}_{\alpha}+\frac{1}{2}\gamma^{5}{\bar{H^{\mu}}}_{\alpha}\big{)}\gamma^{\nu}\gamma^{\rho}\gamma^{\alpha}\Big{\\}}\phi_{\nu}\\\
&\qquad\qquad\qquad-\frac{1}{2}\bar{\phi}_{\mu}\Big{\\{}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}{\bar{H_{\alpha}}}^{\nu}\big{)}+\big{(}\bar{F^{\mu}}_{\alpha}+\frac{1}{2}\gamma^{5}{\bar{H^{\mu}}}_{\alpha}\big{)}\gamma^{\nu}\gamma^{\rho}\gamma^{\alpha}\Big{\\}}\psi_{\nu},\\\
&\tilde{\xi}_{m}P^{mn}\tilde{\xi}_{n}=-\bar{\psi}_{\mu}\Big{\\{}\mathbb{I}_{4}R^{\mu\nu}-\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}\\\
&\qquad\qquad\qquad+\big{(}\bar{F}^{\mu\alpha}+\frac{1}{2}\gamma^{5}\bar{H}^{\mu\alpha}\big{)}\big{(}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}{\bar{H_{\alpha}}}^{\nu}\big{)}\Big{\\}}\psi_{\nu}\\\
&\qquad\qquad\qquad-\bar{\phi}_{\mu}\Big{\\{}\mathbb{I}_{4}R^{\mu\nu}-\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}\\\
&\qquad\qquad\qquad+\big{(}\bar{F}^{\mu\alpha}+\frac{1}{2}\gamma^{5}\bar{H}^{\mu\alpha}\big{)}\big{(}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}{\bar{H_{\alpha}}}^{\nu}\big{)}\Big{\\}}\phi_{\nu}\\\
&\qquad\qquad\qquad+\frac{1}{2}\bar{\psi}_{\mu}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}D_{\rho}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}D_{\rho}{\bar{H_{\alpha}}}^{\nu}\big{)}\phi_{\nu}\\\
&\qquad\qquad\qquad-\frac{1}{2}\bar{\phi}_{\mu}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}D_{\rho}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}D_{\rho}{\bar{H_{\alpha}}}^{\nu}\big{)}\psi_{\nu}.\end{split}$
(5.26)

As mentioned in § 3, now our task is to express $\Lambda$ as the standard form
(3.6). Then, using the formulae (3.7) we find

$\displaystyle\begin{split}&\tilde{\xi}_{m}{(\omega^{\rho})}^{mn}\tilde{\xi}_{n}=\frac{1}{4}\bar{\psi}_{\mu}\Big{\\{}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}{\bar{H_{\alpha}}}^{\nu}\big{)}+\big{(}\bar{F^{\mu}}_{\alpha}+\frac{1}{2}\gamma^{5}{\bar{H^{\mu}}}_{\alpha}\big{)}\gamma^{\nu}\gamma^{\rho}\gamma^{\alpha}\Big{\\}}\phi_{\nu}\\\
&\qquad\qquad\qquad-\frac{1}{4}\bar{\phi}_{\mu}\Big{\\{}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}{\bar{H_{\alpha}}}^{\nu}\big{)}+\big{(}\bar{F^{\mu}}_{\alpha}+\frac{1}{2}\gamma^{5}{\bar{H^{\mu}}}_{\alpha}\big{)}\gamma^{\nu}\gamma^{\rho}\gamma^{\alpha}\Big{\\}}\psi_{\nu},\end{split}$
(5.27)

and also identify the corresponding $I$, $E$ and $\Omega$ as the following
compact form

$\displaystyle\begin{split}&\tilde{\xi}_{m}I^{mn}\tilde{\xi}_{n}=\bar{\psi}_{\mu}\mathbb{I}_{4}\bar{g}^{\mu\nu}\psi_{\nu}+\bar{\phi}_{\mu}\mathbb{I}_{4}\bar{g}^{\mu\nu}\phi_{\nu},\\\
&\tilde{\xi}_{m}E^{mn}\tilde{\xi}_{n}=\tilde{\xi}_{m}P^{mn}\tilde{\xi}_{n}-\tilde{\xi}_{m}(D^{\rho}\omega_{\rho})^{mn}\tilde{\xi}_{n}-\tilde{\xi}_{m}(\omega^{\rho})^{mp}{(\omega_{\rho})_{p}}^{n}\tilde{\xi}_{n},\\\
&\tilde{\xi}_{m}(\Omega_{\rho\sigma})^{mn}\tilde{\xi}_{n}=\bar{\psi}_{\mu}\big{(}\mathbb{I}_{4}{R^{\mu\nu}}_{\rho\sigma}+\frac{1}{4}\bar{g}^{\mu\nu}\gamma^{\alpha}\gamma^{\beta}R_{\rho\sigma\alpha\beta}\big{)}\psi_{\nu}\\\
&\qquad\qquad\qquad\quad+\bar{\phi}_{\mu}\big{(}\mathbb{I}_{4}{R^{\mu\nu}}_{\rho\sigma}+\frac{1}{4}\bar{g}^{\mu\nu}\gamma^{\alpha}\gamma^{\beta}R_{\rho\sigma\alpha\beta}\big{)}\phi_{\nu}\\\
&\qquad\qquad\qquad\quad+\tilde{\xi}_{m}{D_{[\rho}\omega_{\sigma]}}^{mn}\tilde{\xi}_{n}+\tilde{\xi}_{m}[\omega_{\rho},\omega_{\sigma}]^{mn}\tilde{\xi}_{n}.\end{split}$
(5.28)

We then computed all the necessary trace values using the information (5.28)
to obtain the Seeley-DeWitt coefficients for the gravitini fields,

$\displaystyle\begin{split}&\text{tr}(I)=16+16=32,\\\
&\text{tr}(E)=-16\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&\text{tr}(E^{2})=4R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+32R_{\mu\nu}R^{\mu\nu}-8R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}+16(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2},\\\
&\text{tr}(\Omega_{\rho\sigma}\Omega^{\rho\sigma})=-12R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-144R_{\mu\nu}R^{\mu\nu}+48R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-96(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}.\end{split}$
(5.29)

Although these computations are extremely lengthy, one can proceed in a
systematic way (refer Appendix A for details of this computation). Making use
of these trace results (5.29), one computes the first three Seeley-DeWitt
coefficients (3.8) for the fermions (two gravitini) in our $\mathcal{N}=2$
SUGRA. The results are

$\displaystyle\begin{split}&(4\pi)^{2}a_{0}^{\text{gravitini}}=-16,\\\
&(4\pi)^{2}a_{2}^{\text{gravitini}}=8\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&(4\pi)^{2}a_{4}^{\text{gravitini}}=-\frac{1}{90}\left(53R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+172R_{\mu\nu}R^{\mu\nu}\right).\end{split}$
(5.30)

Note that we have put an overall $-1/2$ factor by hand on each coefficient to
justify the fermion and Majorana condition, as discussed in § 3.

Next, the choice of gauge fixing in eq. 5.21 introduces six bosonic ghosts in
our system with the action,

$\mathcal{L}_{\text{ghost},f}=\bar{\tilde{b}}_{A}\gamma^{\mu}D_{\mu}\tilde{c}_{A}+\bar{\tilde{e}}_{A}\gamma^{\mu}D_{\mu}\tilde{e}_{A},$
(5.31)

where $\tilde{b}_{A}$, $\tilde{c}_{A}$, and $\tilde{e}_{A}$ are spin half
ghosts with $A=1,2$ for the two gravitini species. These Majorana ghosts are
minimally coupled, hence one can obtain the Seeley-DeWitt coefficients as $6$
times the spin-$\frac{1}{2}$ Seeley-DeWitt coefficients with an opposite sign,
i.e. ${a_{2n}}^{ghost,f}=-6a_{2n}^{1/2}$. The detailed calculations for
$a_{2n}^{1/2}$ have been described in [44]151515See § 2.2.1 of [44] and
finally, use its eq. 2.19 with $R=0$ for $a_{2n}^{1/2}$. for a general
background, and it is very much straightforward. This led us to express the
results for ghost part of fermionic fields as

$\displaystyle\begin{split}&(4\pi)^{2}a_{0}^{\text{ghost},f}=12,\\\
&(4\pi)^{2}a_{2}^{\text{ghost},f}=0,\\\
&(4\pi)^{2}a_{4}^{\text{ghost},f}=-\frac{1}{120}\left(7R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+8R_{\mu\nu}R^{\mu\nu}\right).\end{split}$
(5.32)

###  5.3 Combined Seeley-DeWitt coefficients in minimal $\mathcal{N}=2$ EMSGT

In previous subsections, we have calculated Seeley-DeWitt coefficients for the
field fluctuations of the bosonic and fermionic part of $\mathcal{N}=2$ SUGRA
and also computed the same for corresponding ghost parts separately. These
ghost fields have no interaction with any of graviton, graviphoton and
gravitini fields. Therefore, the contributions from the ghost fields can be
directly added to the results from the gauge fixed part. The total Seeley-
DeWitt coefficients are obtained by adding the bosonic and fermionic
contributions.

Combining the results (5.13) and (5.19) we get the Seeley-DeWitt coefficients
for the bosonic sector of $\mathcal{N}=2$ SUGRA as

$\displaystyle\begin{split}&(4\pi)^{2}a_{0}^{b}=4,\\\
&(4\pi)^{2}a_{2}^{b}=6\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&(4\pi)^{2}a_{4}^{b}=\frac{1}{180}\left(199R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+26R_{\mu\nu}R^{\mu\nu}\right).\end{split}$
(5.33)

This clearly reflects the result derived in [9]. Similarly, for the fermionic
part, we sum up the results (5.30) and (5.32). This yields

$\displaystyle\begin{split}&(4\pi)^{2}a_{0}^{f}=-4,\\\
&(4\pi)^{2}a_{2}^{f}=8\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&(4\pi)^{2}a_{4}^{f}=-\frac{1}{360}\left(233R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+712R_{\mu\nu}R^{\mu\nu}\right).\end{split}$
(5.34)

Our results for the fermionic sector also agrees with eigenfunction approach
results [5] for $AdS_{2}\times S^{2}$ background, as well as with the hybrid
approach results [11].161616Use the identities (4.2) in the results (3.84) of
[11]. We then find the full $\mathcal{N}=2$ SUGRA Seeley-DeWitt coefficients
with the contribution graviton, graviphoton, and gravitini fluctuations as
$a^{\mathcal{N}=2}_{2n}=a_{2n}^{b}+a_{2n}^{f}$. The results are

$\displaystyle\begin{split}&(4\pi)^{2}a^{\mathcal{N}=2}_{0}=0,\\\
&(4\pi)^{2}a^{\mathcal{N}=2}_{2}=14\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&(4\pi)^{2}a^{\mathcal{N}=2}_{4}=\frac{1}{24}\left(11R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-44R_{\mu\nu}R^{\mu\nu}\right).\end{split}$
(5.35)

The following section deals with the use of these coefficients in the
logarithmic correction to the entropy of extremal black holes.

##  6 Application of Seeley-DeWitt coefficients: logarithmic corrections to
the black hole entropy

In this section, we will summarize the general procedure to find the
logarithmic correction to extremal back holes in an arbitrary theory by making
use of the Seeley-DeWitt coefficient $a_{4}(x)$ of the same theory [14, 15,
16, 35]. Then we will use this procedure for extremal black holes in minimal
$\mathcal{N}$=2 Einstein-Maxwell supergravity theory. It has been observed
that Euclidean quantum gravity [35] gives a more detailed picture of the
quantum corrections [3, 4, 6] to the black hole entropy. We will follow the
quantum entropy function formalism [14, 15, 16], in order to calculate the
logarithmic corrections to the entropy of string theoretic black holes. For
extremal black holes, $AdS_{2}/CFT_{1}$ correspondence yields an auxiliary way
of describing the horizon degeneracy: they are equivalently given by finite
(boundary independent) part of the partition function
$Z_{AdS_{2}\times\mathcal{K}}$ of the string theory in the near horizon
geometry structured as $AdS_{2}\times\mathcal{K}$, where $\mathcal{K}$ is
compact space fibered over $AdS_{2}$. The classical limit provides us with the
relation: $Z_{AdS_{2}\times\mathcal{K}}^{\text{finite}}=e^{S_{\text{BH}}}$,
where $S_{\text{BH}}$ is the quantum corrected black hole entropy. The
partition function $Z_{AdS_{2}\times\mathcal{K}}$ computed by carrying out
Euclidean path integral in $AdS_{2}\times\mathcal{K}$ weighted by
$e^{-\mathcal{S}}$ ($\mathcal{S}$ is Euclidean action), over all string fields
that asymptotically approaches to the near horizon geometry of black hole and
for $Z_{AdS_{2}\times\mathcal{K}}^{\text{finite}}$ one needs to consider only
one-loop term which is finite and boundary independent [3, 4, 6, 5].
Logarithmic corrections ($\Delta S_{\text{BH}}$) are special type of one-loop
quantum corrections, requires integrating out non-zero modes of massless
fields171717In this context, we will only turn on massless fields like two-
dimensional metric $g_{\mu\nu}$, U(1) gauge fields $A_{\mu}$, etc. among other
fields present in $AdS_{2}\times\mathcal{K}$. in
$Z_{AdS_{2}\times\mathcal{K}}$, and then find one-loop effective action $W$ or
effective Lagrangian density $\Delta\mathcal{L}_{\text{eff}}$ in quadratic
order fluctuation of $\mathcal{S}$ (see § 2) in the background
$AdS_{2}\times\mathcal{K}$. We find logarithmic corrections as

$\Delta
S_{\text{BH}}=-W=-\int_{AdS_{2}\times\mathcal{K}}d^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\Delta\mathcal{L}_{\text{eff}}^{(\text{nz})}.$
(6.1)

Here $\mathcal{L}_{\text{eff}}^{(\text{nz})}$ is the one-loop corrected
effective Lagrangian density with the contribution of only non-zero modes of
massless fields, i.e. one needs to throw away zero-modes contribution from the
overall result. Then we express [5, 6]

$\displaystyle\Delta\mathcal{L}_{\text{eff}}^{(\text{nz})}$
$\displaystyle=-\frac{\chi}{2}\int_{\epsilon}^{\infty}\frac{ds}{s}\Big{(}K(x,x;s)-\bar{K}(x,x;0)\Big{)}$
(6.2)
$\displaystyle\simeq-\frac{1}{2}\Big{(}a_{4}(x)-\chi\bar{K}(x,x;0)\Big{)}\text{ln}\thinspace
A_{H},$ (6.3)

where $\bar{K}(x,x;0)$ is the zero mode contribution in $K(x,x;s)$.181818We
denote zero mode ($\lambda_{i}=0$) eigenfunctions as $\\{g_{i}^{m}\\}$. Then
following (2.9), we define
$\bar{K}^{mn}(x,x;0)=\sum_{i}g_{i}^{m}(x)g_{i}^{n}(x),\enspace\bar{K}(x,x;0)=G_{mn}\bar{K}^{mn}(x,x;0)$,
with the same orthonormal condition as (2.10). The number of zero modes is
defined as $n_{\text{zm}}=-\int
d^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\chi\bar{K}(x,x;0)$. Note
that logarithmic part arises only from the $s$ independent part of the
expansion (2.17) of $K(x,x;s)$ after integrating over the range $\epsilon\ll
s\ll A_{H}$, where $\epsilon\sim G_{N}=1/{16\pi}$. Finally, we express the key
formula for finding logarithmic correction to extremal black hole entropy as

$\Delta
S_{\text{BH}}=\frac{1}{2}(C_{\text{local}}+C_{\text{zm}})\text{ln}\thinspace
A_{H},$ (6.4)

where the non-zero mode and zero mode contributions are respectively,

$\displaystyle
C_{\text{local}}=\int_{AdS_{2}\times\mathcal{K}}d^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace
a_{4}(x),$ (6.5) $\displaystyle
C_{\text{zm}}=\int_{AdS_{2}\times\mathcal{K}}d^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace\chi(\beta-1)\bar{K}(x,x;0),$
(6.6)

where $\beta$ is the scaling dimension of fields arises for incorporating
corrections due to zero mode contribution in $\Delta S_{\text{BH}}$ [5, 6].
One can evaluate $C_{\text{local}}$ using by computing $a_{4}$ of massless
fields in near horizon geometry of concerned black hole, while $C_{\text{zm}}$
can be obtained from the following special combined formula [11],

$C_{\text{zm}}=-(3+\mathbb{K})+2N_{\text{SUSY}}.$ (6.7)

Here $\mathbb{K}$ (number of rotational isometries) is 3 if for black holes
having $J=0$, otherwise it is 1; $N_{\text{SUSY}}$ (number of preserved
supercharges) is 4 for BPS black holes, 0 otherwise. In order to compute the
logarithmic correction to extremal black hole entropy in any arbitrary theory,
one needs to first compute $a_{4}(x)$ for the quadratic fluctuations of
massless fields in the theory. Then find the results for $C_{\text{local}}$
(using the $a_{4}(x)$ result) and $C_{\text{zm}}$ respectively, and finally
use these results in the formula (6.4). With cognizance of the above
structure, our next aim to compute logarithmic corrections to the entropy of
extremal Kerr-Newman, Reissner-Nordström and Kerr black holes191919Only
extremal Reissner-Nordström black holes are half-BPS in the four-dimensional
$\mathcal{N}=2$ EMSGT. in minimal $\mathcal{N}=2$ EMSGT by making use of
$a_{4}(x)$ result (5.35) of the theory.

###  6.1 Logarithmic corrections to $\mathcal{N}=2$ extremal black hole
entropy

We will start with a Kerr-Newman black hole [9] to calculate associated
logarithmic correction. In appropriate units, the metric of a general Kerr-
Newman black hole characterized by mass $M$, charge $Q$, and angular momentum
$J$ is expressed as

$\begin{split}ds^{2}&=-\frac{r^{2}+a^{2}\text{cos}^{2}\theta-2Mr+Q^{2}}{r^{2}+a^{2}\text{cos}^{2}\theta}dt^{2}+\frac{r^{2}+a^{2}\text{cos}^{2}\theta}{r^{2}+a^{2}-2Mr+Q^{2}}dr^{2}+(r^{2}+a^{2}\text{cos}^{2}\theta)d\theta^{2}\\\
&\enspace+\frac{(r^{2}+a^{2}\text{cos}^{2}\theta)(r^{2}+a^{2})+(2Mr-Q^{2})a^{2}\text{sin}^{2}\theta}{r^{2}+a^{2}\text{cos}^{2}\theta}\text{sin}^{2}\theta
d\varphi^{2}\\\
&\enspace+\frac{2(Q^{2}-2Mr)a}{r^{2}+a^{2}\text{cos}^{2}\theta}\text{sin}^{2}\theta
dtd\varphi,\end{split}$ (6.8)

where $a=J/M$. The horizon is situated at

$r=r_{H}=M+\sqrt{M^{2}-Q^{2}-a^{2}},$ (6.9)

and the corresponding classical Bekenstein-Hawking entropy of black hole
having a horizon area $A_{H}=4\pi({r_{H}}^{2}+a^{2})$ is202020The classical
black hole entropy formula is given as $S_{\text{BH}}=\frac{A_{H}}{4G_{N}}$,
where $G_{N}=1/16\pi$ in our convention.

$S_{\text{BH}}=16\pi^{2}\left(2M^{2}-Q^{2}+2M\sqrt{M^{2}-a^{2}-Q^{2}}\right).$
(6.10)

The extremal limit $M\to\sqrt{a^{2}+Q^{2}}$ yields

$S_{\text{BH}}=16\pi^{2}(2a^{2}+Q^{2}).$ (6.11)

Now for the metric, we have [47, 48]

$\displaystyle\begin{split}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}&=\frac{8}{(r^{2}+a^{2}\text{cos}^{2}\theta)^{6}}\Big{\\{}6M^{2}(r^{6}-15a^{2}r^{4}\text{cos}^{2}\theta+15a^{4}r^{2}\text{cos}^{4}\theta\\\
&\enspace-a^{6}\text{cos}^{6}\theta)-12MQ^{2}r(r^{4}-10r^{2}a^{2}\text{cos}^{2}\theta+5a^{4}\text{cos}^{4}\theta)\\\
&\enspace+Q^{4}(7r^{4}-34r^{2}a^{2}\text{cos}^{2}\theta+7a^{4}\text{cos}^{4}\theta)\Big{\\}},\\\
R_{\mu\nu}R^{\mu\nu}&=\frac{4Q^{4}}{(r^{2}+a^{2}\text{cos}^{2}\theta)^{4}},\\\
\text{det}\thinspace\bar{g}&=(r^{2}+a^{2}\text{cos}^{2}\theta)^{2}\text{sin}^{2}\theta.\end{split}$
(6.12)

In the extremal limit, one can achieve [9]

$\displaystyle\begin{split}\int_{\text{horizon}}d^{4}x\sqrt{\text{det}\thinspace\bar{g}}\thinspace
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}&=-\frac{16\pi^{2}}{b(b^{2}+1)^{5/2}(2b^{2}+1)}\Big{\\{}3(2b^{2}+1)^{2}\text{tan}^{-1}\left(\frac{b}{\sqrt{b^{2}+1}}\right)\\\
&\enspace+b\sqrt{b^{2}+1}(-8b^{6}-20b^{4}-8b^{2}+1)\Big{\\}},\\\
\int_{\text{horizon}}d^{4}x\sqrt{\text{det}\thinspace\bar{g}}R_{\mu\nu}R^{\mu\nu}&=-\frac{4\pi^{2}}{b(b^{2}+1)^{5/2}(2b^{2}+1)}\Big{\\{}3(2b^{2}+1)^{2}\text{tan}^{-1}\left(\frac{b}{\sqrt{b^{2}+1}}\right)\\\
&\enspace+b\sqrt{b^{2}+1}(8b^{2}+5)\Big{\\}},\end{split}$ (6.13)

where $b=a/Q=J/MQ$.

  * $\blacksquare$

Using the $a_{4}(x)$ result (5.35) for minimal $\mathcal{N}=2$ EMSGT and the
limits (6.13) in the relation (6.5), and also making use of the formula (6.7),
for extremal Kerr-Newman black hole ($J\neq 0,Q\neq 0$) we have

$C_{\text{local}}=\frac{11}{6}\frac{(2b^{6}+5b^{4}+4b^{2}+1)}{(b^{2}+1)^{2}(2b^{2}+1)},\enspace
C_{\text{zm}}=-(3+1)+2\times 0=-4.$ (6.14)

Substitution of these in the formula (6.4) yields

$\Delta
S_{\text{BH}}=\frac{11}{12}\frac{(2b^{6}+5b^{4}+4b^{2}+1)}{(b^{2}+1)^{2}(2b^{2}+1)}\thinspace\text{ln}\thinspace
A_{H}-2\thinspace\text{ln}\thinspace A_{H}.$ (6.15)

This is one of our central results. We can get the results for Kerr and
Reissner-Nordström black holes by applying appropriate limits as discussed
below.

  * $\blacksquare$

For an extremal Kerr black hole ($J\neq 0,Q=0$), we have to take $b\to\infty$
in the expression of $C_{\text{local}}$ given in (6.14). This yields

$C_{\text{local}}=\frac{11}{6},\enspace C_{\text{zm}}=-(3+1)+2\times 0=-4,$
(6.16)

hence the logarithmic correction becomes

$\Delta S_{\text{BH}}=-\frac{13}{12}\thinspace\text{ln}\thinspace A_{H}.$
(6.17)

This is in accordance with the result in [6] (set
$n_{V}=1,n_{3/2}=2,n_{S}=n_{F}=0$ in eq.(3.12) and include an additional
$-4\thinspace\text{ln}\thinspace A_{H}$ term for the correction due to zero-
modes of gravitini).

  * $\blacksquare$

For the case of an extremal Reissner-Nordström black hole ($J=0,Q\neq 0$), one
has to put the limit $b\to 0$ in the $C_{\text{local}}$ form given in (6.14).
This gives

$C_{\text{local}}=\frac{11}{6},\enspace C_{\text{zm}}=-(3+3)+2\times 4=2,$
(6.18)

hence the logarithmic correction becomes

$\Delta S_{\text{BH}}=\frac{23}{12}\thinspace\text{ln}\thinspace A_{H}.$
(6.19)

This agrees with the result in [5] (see eq.(5.32)).

To conclude, we have computed the first three Seeley-DeWitt coefficients for
minimal $\mathcal{N}=2$ Einstein-Maxwell supergravity theory following the
general Seeley-DeWitt approach [30] (reviewed in § 3). The fermionic sector
results are newly calculated in this approach, while the bosonic sector
results are reviewed and recalculated from [9]. This computation does not
impose any limitations on the background geometry. The coefficients are
expressed in terms of the local invariants of the theory and can be evaluated
for any arbitrary background. These Seeley-DeWitt coefficient results for the
minimal $\mathcal{N}=2$ Einstein-Maxwell supergravity theory in four
dimensions agree with eigenfunction approach results [5] for $AdS_{2}\times
S^{2}$ background, as well as with the hybrid approach results [11]. As an
application of these results, we have computed the logarithmic corrections to
Bekenstein-Hawking entropy of the extremal Kerr-Newman, Kerr and Reissner-
Nordström black holes in minimal $\mathcal{N}=2$ Einstein-Maxwell supergravity
theory by properly integrating these coefficients in the near horizon geometry
of these black holes. The Kerr and Reissner-Nordström results agree with [6]
and [5] respectively, while the Kerr-Newman result is new to the literature.
Any microscopic theory explaining the entropy of such black holes is expected
to reproduce the subleading logarithmic correction terms specified in eq. 6.15
in addition to the leading Bekenstein-Hawking term.

## Acknowledgment

It is a great pleasure to thank Ashoke Sen for useful discussions at different
stages of this work. The authors would like to thank Finn Larsen for sharing
insight and expertise that greatly improved this manuscript. We would also
like to acknowledge NISER, HRI, BHU and IISER-TVM for their hospitality. BP
acknowledges IIT(ISM), Dhanbad for the Grant (FRS (53)/2013-2014/APH).

##  Appendix A Trace calculations

Using Equations of motion and identities as mentioned in § 4.1, we derive the
following identities212121We have assumed the Euclidean signature for all the
derivations in this paper.

$\displaystyle\begin{split}&\bar{H}_{\mu\nu}\bar{H}^{\mu\nu}=4\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\thinspace\varepsilon^{\mu\nu\rho\sigma}\bar{H}_{\rho\sigma}=-4\bar{F}^{\mu\nu},\\\
&\bar{H}_{\mu\rho}{\bar{H_{\nu}}}^{\rho}=-2R_{\mu\nu}+\bar{g}_{\mu\nu}\bar{F}_{\rho\sigma}\bar{F}^{\rho\sigma},\\\
&\bar{F}^{\mu\rho}\bar{F^{\nu}}_{\rho}\bar{H}_{\mu\sigma}{\bar{H_{\nu}}}^{\sigma}=\bar{F}^{\mu\rho}{\bar{H^{\nu}}}_{\rho}\bar{H}_{\mu\sigma}\bar{F_{\nu}}^{\sigma}=(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}-R_{\mu\nu}R^{\mu\nu},\\\
&\bar{F}_{\mu\nu}\bar{F}_{\rho\sigma}\bar{H}^{\mu\nu}\bar{H}^{\rho\sigma}=4\left((\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}-R_{\mu\nu}R^{\mu\nu}\right),\\\
&R_{\mu\rho\nu\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}=\frac{1}{2}R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma},\\\
&R_{\mu\rho\nu\sigma}\bar{H}^{\mu\nu}\bar{H}^{\rho\sigma}=2\left(R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-2R_{\mu\nu}R^{\mu\nu}\right),\\\
&R_{\mu\nu\rho\sigma}\bar{H}^{\mu\nu}\bar{H}^{\rho\sigma}=4\left(R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-2R_{\mu\nu}R^{\mu\nu}\right),\end{split}$
(A.1)

followed by

$\displaystyle\begin{split}&(D_{\rho}\bar{F}_{\mu\nu})(D^{\rho}\bar{F}^{\mu\nu})=R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-R_{\mu\nu}R^{\mu\nu},\\\
&(D_{\rho}\bar{H}_{\mu\nu})(D^{\rho}\bar{H}^{\mu\nu})=4\left(R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-R_{\mu\nu}R^{\mu\nu}\right),\\\
&(D_{\mu}\bar{{F}_{\rho}}^{\nu})(D_{\nu}\bar{F}^{\rho\mu})=\frac{1}{2}\left(R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-R_{\mu\nu}R^{\mu\nu}\right),\\\
&(D_{\mu}{\bar{H_{\rho}}}^{\nu})(D_{\nu}\bar{H}^{\rho\mu})=2\left(R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-R_{\mu\nu}R^{\mu\nu}\right).\end{split}$
(A.2)

We note that in our analysis we will ignore total derivatives terms as these
will not contribute in heat kernel coefficient integrals. In deriving eq. A.2
we will apply the same, also use Maxwell equation and perform the following
steps

$\begin{split}(D_{\rho}\bar{F}_{\mu\nu})(D^{\rho}\bar{F}^{\mu\nu})&=2\bar{F^{\mu}}_{\nu}D_{\rho}D_{\mu}\bar{F}^{\nu\rho},\\\
&=2\bar{F^{\mu}}_{\nu}[D_{\rho},D_{\mu}]\bar{F}^{\nu\rho}.\end{split}$ (A.3)

We will do the same for dual field strength $\bar{H}_{\mu\nu}$. Finally, the
covariant derivative commutator acting on rank-2 tensor gives the form of the
result (A.2).

Also, using eq. 5.27 for the two gravitini species we write,222222Here only
$\mu,\nu$ are particular tensor indices, labelled by the combination of $\psi$
and $\phi$ fields to denote with which set of field combination is the
particular $\omega$ associated with (or $\mu,\nu$ are contracted with).
Similar notation will be used for $E$ and $\Omega$ also.

$\displaystyle(\omega^{\rho})^{\psi_{\mu}\phi_{\nu}}$
$\displaystyle=\frac{1}{4}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\Big{(}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}{\bar{H_{\alpha}}}^{\nu}\Big{)}+\frac{1}{4}\Big{(}\bar{F^{\mu}}_{\alpha}+\frac{1}{2}\gamma^{5}\bar{H^{\mu}}_{\alpha}\Big{)}\gamma^{\nu}\gamma^{\rho}\gamma^{\alpha},$
(A.4a) $\displaystyle(\omega^{\rho})^{\phi_{\mu}\psi_{\nu}}$
$\displaystyle=-\frac{1}{4}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\Big{(}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}{\bar{H_{\alpha}}}^{\nu}\Big{)}-\frac{1}{4}\Big{(}\bar{F^{\mu}}_{\alpha}+\frac{1}{2}\gamma^{5}\bar{H^{\mu}}_{\alpha}\Big{)}\gamma^{\nu}\gamma^{\rho}\gamma^{\alpha}.$
(A.4b)

####  Calculating $\text{tr}(E)$ and $\text{tr}(E^{2})$

$\displaystyle\text{tr}(E)$
$\displaystyle={E^{\psi_{\mu}}}_{\psi_{\mu}}+{E^{\phi_{\mu}}}_{\phi_{\mu}}+{E^{\psi_{\mu}}}_{\phi_{\mu}}+{E^{\phi_{\mu}}}_{\psi_{\mu}},$
(A.5) $\displaystyle\text{tr}(E^{2})$
$\displaystyle={E^{\psi_{\mu}}}_{\psi_{\nu}}{E^{\psi_{\nu}}}_{\psi_{\mu}}+{E^{\phi_{\mu}}}_{\phi_{\nu}}{E^{\phi_{\nu}}}_{\phi_{\mu}}+{E^{\psi_{\mu}}}_{\phi_{\nu}}{E^{\phi_{\nu}}}_{\psi_{\mu}}+{E^{\phi_{\mu}}}_{\psi_{\nu}}{E^{\psi_{\nu}}}_{\phi_{\mu}}.$
(A.6)

Making use of expressions of $\omega$ and $P$ from eq. 5.27 and eq. 5.26
respectively, one can simplify $E$ to the following form

$\displaystyle\begin{split}&\tilde{\xi}_{m}E^{mn}\tilde{\xi}_{n}=\bar{\psi}_{\mu}{E^{\psi_{\mu}\psi_{\nu}}}\psi_{\nu}+\bar{\phi}_{\mu}{E^{\phi_{\mu}\phi_{\nu}}}\phi_{\nu}+\bar{\psi}_{\mu}{E^{\psi_{\mu}\phi_{\nu}}}\phi_{\nu}+\bar{\phi}_{\mu}{E^{\phi_{\mu}\psi_{\nu}}}\psi_{\nu},\\\
&\qquad\qquad=\bar{\psi}_{\mu}\Big{\\{}-\mathbb{I}_{4}R^{\mu\nu}+\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}\\\
&\quad\qquad\qquad+\frac{1}{2}\gamma^{\alpha}\gamma^{\beta}\big{(}\bar{F}_{\alpha\beta}-\frac{1}{2}\gamma^{5}\bar{H}_{\alpha\beta}\big{)}\big{(}\bar{F}^{\mu\nu}+\frac{1}{2}\gamma^{5}\bar{H}^{\mu\nu}\big{)}\\\
&\quad\qquad\qquad+\frac{1}{4}\bar{g}^{\mu\nu}\gamma^{\alpha}\gamma^{\beta}\big{(}\bar{F_{\alpha}}^{\tau}-\frac{1}{2}\gamma^{5}\bar{H_{\alpha}}^{\tau}\big{)}\big{(}\bar{F}_{\tau\beta}-\frac{1}{2}\gamma^{5}\bar{H}_{\tau\beta}\big{)}\Big{\\}}\psi_{\nu}\\\
&\quad\qquad\qquad+\bar{\phi}_{\mu}\Big{\\{}-\mathbb{I}_{4}R^{\mu\nu}+\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}\\\
&\quad\qquad\qquad+\frac{1}{2}\gamma^{\alpha}\gamma^{\beta}\big{(}\bar{F}_{\alpha\beta}-\frac{1}{2}\gamma^{5}\bar{H}_{\alpha\beta}\big{)}\big{(}\bar{F}^{\mu\nu}+\frac{1}{2}\gamma^{5}\bar{H}^{\mu\nu}\big{)}\\\
&\quad\qquad\qquad+\frac{1}{4}\bar{g}^{\mu\nu}\gamma^{\alpha}\gamma^{\beta}\big{(}\bar{F_{\alpha}}^{\tau}-\frac{1}{2}\gamma^{5}\bar{H_{\alpha}}^{\tau}\big{)}\big{(}\bar{F}_{\tau\beta}-\frac{1}{2}\gamma^{5}\bar{H}_{\tau\beta}\big{)}\Big{\\}}\phi_{\nu}\\\
&\quad\qquad\qquad+\frac{1}{4}\bar{\psi}_{\mu}\Big{\\{}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}D_{\rho}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}D_{\rho}\bar{H_{\alpha}}^{\nu}\big{)}\\\
&\quad\qquad\qquad-\big{(}D_{\rho}\bar{F^{\mu}}_{\alpha}+\frac{1}{2}\gamma^{5}D_{\rho}\bar{H^{\mu}}_{\alpha}\big{)}\gamma^{\nu}\gamma^{\rho}\gamma^{\alpha}\Big{\\}}\phi_{\nu}\\\
&\quad\qquad\qquad-\frac{1}{4}\bar{\phi}_{\mu}\Big{\\{}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}D_{\rho}\bar{F_{\alpha}}^{\nu}+\frac{1}{2}\gamma^{5}D_{\rho}\bar{H_{\alpha}}^{\nu}\big{)}\\\
&\quad\qquad\qquad-\big{(}D_{\rho}\bar{F^{\mu}}_{\alpha}+\frac{1}{2}\gamma^{5}D_{\rho}\bar{H^{\mu}}_{\alpha}\big{)}\gamma^{\nu}\gamma^{\rho}\gamma^{\alpha}\Big{\\}}\psi_{\nu}.\end{split}$
(A.7)

Then we have,

$\displaystyle{E^{\psi_{\mu}}}_{\psi_{\nu}}={E^{\phi_{\mu}}}_{\phi_{\nu}}$
$\displaystyle=\underbrace{-\mathbb{I}_{4}{R^{\mu}}_{\nu}+\frac{1}{2}\gamma^{\alpha}\gamma^{\beta}{R^{\mu}}_{\nu\alpha\beta}-\frac{1}{2}\gamma^{\mu}\gamma^{\alpha}R_{\nu\alpha}+\frac{1}{2}\gamma_{\nu}\gamma^{\alpha}{R^{\mu}}_{\alpha}}_{X_{1}}$
$\displaystyle\quad+\underbrace{\frac{1}{2}\gamma^{\alpha}\gamma^{\beta}\big{(}\bar{F}_{\alpha\beta}-\frac{1}{2}\gamma^{5}\bar{H}_{\alpha\beta}\big{)}\big{(}\bar{F^{\mu}}_{\nu}+\frac{1}{2}\gamma^{5}\bar{H^{\mu}}_{\nu}\big{)}}_{X_{2}}$
$\displaystyle\quad+\underbrace{\frac{1}{4}g^{\mu}_{\nu}\gamma^{\alpha}\gamma^{\beta}\big{(}\bar{F_{\alpha}}^{\tau}-\frac{1}{2}\gamma^{5}\bar{H_{\alpha}}^{\tau}\big{)}\big{(}\bar{F}_{\tau\beta}-\frac{1}{2}\gamma^{5}\bar{H}_{\tau\beta}\big{)}}_{X_{3}},$
(A.8a)
$\displaystyle{E^{\psi_{\mu}}}_{\phi_{\nu}}=-{E^{\phi_{\mu}}}_{\psi_{\nu}}$
$\displaystyle=\underbrace{\frac{1}{4}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}\big{(}D_{\rho}\bar{F}_{\alpha\nu}+\frac{1}{2}\gamma^{5}D_{\rho}\bar{H_{\alpha}}_{\nu}\big{)}}_{X_{4}}$
$\displaystyle\quad\underbrace{-\frac{1}{4}\big{(}D_{\rho}\bar{F^{\mu}}_{\alpha}+\frac{1}{2}\gamma^{5}D_{\rho}\bar{H^{\mu}}_{\alpha}\big{)}\gamma_{\nu}\gamma^{\rho}\gamma^{\alpha}}_{X_{5}},$
(A.8b)
$\displaystyle{E^{\psi_{\nu}}}_{\psi_{\mu}}={E^{\phi_{\nu}}}_{\phi_{\mu}}$
$\displaystyle=\underbrace{-\mathbb{I}_{4}{R^{\nu}}_{\mu}+\frac{1}{2}\gamma^{\theta}\gamma^{\varphi}{R^{\nu}}_{\mu\theta\varphi}-\frac{1}{2}\gamma^{\nu}\gamma^{\theta}R_{\mu\theta}+\frac{1}{2}\gamma_{\mu}\gamma^{\theta}{R^{\nu}}_{\theta}}_{Y_{1}}$
$\displaystyle\quad+\underbrace{\frac{1}{2}\gamma^{\theta}\gamma^{\varphi}\big{(}\bar{F}_{\theta\varphi}-\frac{1}{2}\gamma^{5}\bar{H}_{\theta\varphi}\big{)}\big{(}\bar{F^{\nu}}_{\mu}+\frac{1}{2}\gamma^{5}\bar{H^{\nu}}_{\mu}\big{)}}_{Y_{2}}$
$\displaystyle\quad+\underbrace{\frac{1}{4}\bar{g}^{\nu}_{\mu}\gamma^{\theta}\gamma^{\varphi}\big{(}\bar{F_{\theta}}^{\sigma}-\frac{1}{2}\gamma^{5}\bar{H_{\theta}}^{\sigma}\big{)}\big{(}\bar{F}_{\sigma\varphi}-\frac{1}{2}\gamma^{5}\bar{H}_{\sigma\varphi}\big{)}}_{Y_{3}},$
(A.8c)
$\displaystyle{E^{\phi_{\nu}}}_{\psi_{\mu}}=-{E^{\psi_{\nu}}}_{\phi_{\mu}}$
$\displaystyle=\underbrace{-\frac{1}{4}\gamma^{\beta}\gamma^{\sigma}\gamma^{\nu}\big{(}D_{\sigma}\bar{F}_{\beta\mu}+\frac{1}{2}\gamma^{5}D_{\sigma}\bar{H_{\beta}}_{\mu}\big{)}}_{Y_{4}}$
$\displaystyle\quad+\underbrace{\frac{1}{4}\big{(}D_{\sigma}\bar{F^{\nu}}_{\beta}+\frac{1}{2}\gamma^{5}D_{\sigma}\bar{H^{\nu}}_{\beta}\big{)}\gamma_{\mu}\gamma^{\sigma}\gamma^{\beta}}_{Y_{5}}.$
(A.8d)

Then,

$\text{tr}(X_{1})=0,\thinspace\text{tr}(X_{2})=0,\thinspace\text{tr}(X_{3})=-8\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\thinspace\text{tr}(X_{4})=0,\thinspace\text{tr}(X_{5})=0.$

So,

$\displaystyle\begin{split}&{E^{\psi_{\mu}}}_{\psi_{\mu}}={E^{\phi_{\mu}}}_{\phi_{\mu}}=\sum_{i=1}^{3}\text{tr}(X_{i})=-8\bar{F}_{\mu\nu}\bar{F}^{\mu\nu},\\\
&{E^{\psi_{\mu}}}_{\phi_{\mu}}=-{E^{\phi_{\mu}}}_{\psi_{\mu}}=\sum_{i=4}^{5}\text{tr}(X_{i})=0.\end{split}$
(A.9)

Also,

$\begin{split}X_{1}Y_{1}&=2R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma},\\\
X_{2}Y_{2}&=16R_{\mu\nu}R^{\mu\nu},\\\
X_{3}Y_{3}&=8(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}\\\
&\enspace-4R_{\mu\nu}R^{\mu\nu},\end{split}\hskip
7.22743pt\begin{split}&X_{4}Y_{4}=0,\\\ &X_{5}Y_{5}=0,\\\
&X_{1}Y_{2}=X_{2}Y_{1}=0,\end{split}\hskip
7.22743pt\begin{split}X_{1}Y_{3}=X_{3}Y_{1}&=0,\\\
X_{2}Y_{3}=X_{3}Y_{2}&=0,\\\ X_{4}Y_{5}=X_{5}Y_{4}&=2R_{\mu\nu}R^{\mu\nu}\\\
&\enspace-2R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}.\end{split}$

Therefore,

$\displaystyle\begin{split}{E^{\psi_{\mu}}}_{\psi_{\nu}}{E^{\psi_{\nu}}}_{\psi_{\mu}}={E^{\phi_{\mu}}}_{\phi_{\nu}}{E^{\phi_{\nu}}}_{\phi_{\mu}}&=\sum_{i,j=1}^{3}X_{i}Y_{j}\\\
&=2R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+12R_{\mu\nu}R^{\mu\nu}+8(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2},\\\
{E^{\psi_{\mu}}}_{\phi_{\nu}}{E^{\phi_{\nu}}}_{\psi_{\mu}}={E^{\phi_{\mu}}}_{\psi_{\nu}}{E^{\psi_{\nu}}}_{\phi_{\mu}}&=\sum_{i,j=4}^{5}X_{i}Y_{j}\\\
&=4R_{\mu\nu}R^{\mu\nu}-4R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}.\end{split}$
(A.10)

#### Calculating $\text{tr}(\Omega_{\rho\sigma}\Omega^{\rho\sigma})$

$\displaystyle\begin{split}\text{tr}(\Omega_{\rho\sigma}\Omega^{\rho\sigma})&={(\Omega_{\rho\sigma})^{\psi_{\mu}}}_{\psi_{\nu}}{(\Omega^{\rho\sigma})^{\psi_{\nu}}}_{\psi_{\mu}}+{(\Omega_{\rho\sigma})^{\phi_{\mu}}}_{\phi_{\nu}}{(\Omega^{\rho\sigma})^{\phi_{\nu}}}_{\phi_{\mu}}\\\
&\quad+{(\Omega_{\rho\sigma})^{\psi_{\mu}}}_{\phi_{\nu}}{(\Omega^{\rho\sigma})^{\phi_{\nu}}}_{\psi_{\mu}}+{(\Omega_{\rho\sigma})^{\phi_{\mu}}}_{\psi_{\nu}}{(\Omega^{\rho\sigma})^{\psi_{\nu}}}_{\phi_{\mu}},\end{split}$
(A.11)

We have,

$\displaystyle{(\Omega_{\rho\sigma})^{\psi_{\mu}}}_{\psi_{\nu}}={(\Omega_{\rho\sigma})^{\phi_{\mu}}}_{\phi_{\nu}}=\mathbb{I}_{4}{R^{\mu}}_{\nu\rho\sigma}+\frac{1}{4}\bar{g}^{\mu}_{\nu}\gamma^{\theta}\gamma^{\varphi}R_{\rho\sigma\theta\varphi}+{[\omega_{\rho},\omega_{\sigma}]^{\psi_{\mu}}}_{\psi_{\nu}},$
(A.12a)
$\displaystyle{(\Omega_{\rho\sigma})^{\psi_{\mu}}}_{\phi_{\nu}}=-{(\Omega_{\rho\sigma})^{\phi_{\mu}}}_{\psi_{\nu}}={D_{[\rho}{\omega_{\sigma]}}^{\psi_{\mu}}}_{\phi_{\nu}},$
(A.12b)
$\displaystyle{(\Omega^{\rho\sigma})^{\psi_{\nu}}}_{\psi_{\mu}}={(\Omega^{\rho\sigma})^{\phi_{\nu}}}_{\phi_{\mu}}=\mathbb{I}_{4}{{R^{\nu}}_{\mu}}^{\rho\sigma}+\frac{1}{4}g^{\nu}_{\mu}\gamma^{\theta^{\prime}}\gamma^{\varphi^{\prime}}{R^{\rho\sigma}}_{\theta^{\prime}\varphi^{\prime}}+{[\omega^{\rho},\omega^{\sigma}]^{\psi_{\nu}}}_{\psi_{\mu}},$
(A.12c)
$\displaystyle{(\Omega^{\rho\sigma})^{\phi_{\nu}}}_{\psi_{\mu}}=-{(\Omega^{\rho\sigma})^{\psi_{\nu}}}_{\phi_{\mu}}={D^{[\rho}{\omega^{\sigma]}}^{\phi_{\nu}}}_{\psi_{\mu}}.$
(A.12d)

So,

$\displaystyle\begin{split}{(\Omega_{\rho\sigma})^{\psi_{\mu}}}_{\psi_{\nu}}{(\Omega^{\rho\sigma})^{\psi_{\nu}}}_{\psi_{\mu}}&=\big{(}\underbrace{\mathbb{I}_{4}{R^{\mu}}_{\nu\rho\sigma}+\frac{1}{4}\bar{g}^{\mu}_{\nu}\gamma^{\theta}\gamma^{\varphi}R_{\rho\sigma\theta\varphi}\big{)}\big{(}\mathbb{I}_{4}{{R^{\nu}}_{\mu}}^{\rho\sigma}+\frac{1}{4}\bar{g}^{\nu}_{\mu}\gamma^{\theta^{\prime}}\gamma^{\varphi^{\prime}}{R^{\rho\sigma}}_{\theta^{\prime}\varphi^{\prime}}}_{\textbf{Term1}}\big{)}\\\
&\qquad+\underbrace{2{R^{\mu}}_{\nu\rho\sigma}{[\omega^{\rho},\omega^{\sigma}]^{\psi_{\nu}}}_{\psi_{\mu}}}_{\textbf{Term2}}+\underbrace{\frac{1}{2}\gamma^{\theta}\gamma^{\varphi}R_{\rho\sigma\theta\varphi}{[\omega^{\rho},\omega^{\sigma}]^{\psi_{\mu}}}_{\psi_{\mu}}}_{\textbf{Term3}}\\\
&\qquad+\underbrace{{[\omega_{\rho},\omega_{\sigma}]^{\psi_{\mu}}}_{\psi_{\nu}}{[\omega^{\rho},\omega^{\sigma}]^{\psi_{\nu}}}_{\psi_{\mu}}}_{\textbf{Term4}}.\end{split}$
(A.13)

We compute,

  1. 1.

Term1
$\displaystyle=\big{(}\mathbb{I}_{4}{R^{\mu}}_{\nu\rho\sigma}+\frac{1}{4}\bar{g}^{\mu}_{\nu}\gamma^{\theta}\gamma^{\varphi}R_{\rho\sigma\theta\varphi}\big{)}\big{(}\mathbb{I}_{4}{{R^{\nu}}_{\mu}}^{\rho\sigma}+\frac{1}{4}\bar{g}^{\nu}_{\mu}\gamma^{\theta^{\prime}}\gamma^{\varphi^{\prime}}{R^{\rho\sigma}}_{\theta^{\prime}\varphi^{\prime}}\big{)}$
$\displaystyle=-6R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma},$ (A.14)

  2. 2.

Term2
$\displaystyle=2{R^{\mu}}_{\nu\rho\sigma}{[\omega^{\rho},\omega^{\sigma}]^{\psi_{\nu}}}_{\psi_{\mu}}$
$\displaystyle=4{R^{\mu}}_{\nu\rho\sigma}{(\omega^{\rho})^{\psi_{\nu}}}_{\phi_{\tau}}{(\omega^{\sigma})^{\phi_{\tau}}}_{\psi_{\mu}}$
$\displaystyle=4\underbrace{{R^{\mu}}_{\nu\rho\sigma}}_{\text{X}}\Big{(}\underbrace{\frac{1}{4}\gamma^{\alpha}\gamma^{\rho}\gamma^{\nu}({\bar{F}_{\alpha\tau}}+\frac{1}{2}\gamma^{5}{\bar{H}_{\alpha\tau}})}_{M_{1}}+\underbrace{\frac{1}{4}(\bar{F^{\nu}}_{\alpha}+\frac{1}{2}\gamma^{5}\bar{H^{\nu}}_{\alpha})\gamma_{\tau}\gamma^{\rho}\gamma^{\alpha}}_{M_{2}}\Big{)}$
$\displaystyle\enspace\times\Big{(}\underbrace{-\frac{1}{4}\gamma^{\beta}\gamma^{\sigma}\gamma^{\tau}(\bar{F}_{\beta\mu}+\frac{1}{2}\gamma^{5}\bar{H}_{\beta\mu})}_{N_{1}}\underbrace{-\frac{1}{4}(\bar{F^{\tau}}_{\beta}+\frac{1}{2}\gamma^{5}\bar{H^{\tau}}_{\beta})\gamma_{\mu}\gamma^{\sigma}\gamma^{\beta}}_{N_{2}}\Big{)}.$

Then,

$XM_{1}N_{1}=0,\thinspace XM_{2}N_{2}=0,\thinspace XM_{1}N_{2}=0,\thinspace
XM_{2}N_{1}=0.$

So,

$\textbf{Term2}=4\sum_{i,j=1}^{2}XM_{i}N_{j}=0,$ (A.15)

  3. 3.

Term3
$\displaystyle=\frac{1}{2}\gamma^{\theta}\gamma^{\varphi}R_{\rho\sigma\theta\varphi}{[\omega^{\rho},\omega^{\sigma}]^{\psi_{\mu}}}_{\psi_{\mu}}$
$\displaystyle=\gamma^{\theta}\gamma^{\varphi}R_{\rho\sigma\theta\varphi}{(\omega^{\rho})^{\psi_{\mu}}}_{\phi_{\tau}}{(\omega^{\sigma})^{\phi_{\tau}}}_{\psi_{\mu}}$
$\displaystyle=\underbrace{\gamma^{\theta}\gamma^{\varphi}R_{\rho\sigma\theta\varphi}}_{Y}\Big{(}\underbrace{\frac{1}{4}\gamma^{\alpha}\gamma^{\rho}\gamma^{\mu}({\bar{F}_{\alpha\tau}}+\frac{1}{2}\gamma^{5}{\bar{H}_{\alpha\tau}})}_{U_{1}}+\underbrace{\frac{1}{4}(\bar{F^{\mu}}_{\alpha}+\frac{1}{2}\gamma^{5}\bar{H^{\mu}}_{\alpha})\gamma_{\tau}\gamma^{\rho}\gamma^{\alpha}}_{U_{2}}\Big{)}$
$\displaystyle\enspace\times\Big{(}\underbrace{-\frac{1}{4}\gamma^{\beta}\gamma^{\sigma}\gamma^{\tau}(\bar{F}_{\beta\mu}+\frac{1}{2}\gamma^{5}\bar{H}_{\beta\mu})}_{V_{1}}\underbrace{-\frac{1}{4}(\bar{F^{\tau}}_{\beta}+\frac{1}{2}\gamma^{5}\bar{H^{\tau}}_{\beta})\gamma_{\mu}\gamma^{\sigma}\gamma^{\beta}}_{V_{2}}\Big{)}.$

Then,

$YU_{1}V_{1}=0,\thinspace YU_{2}V_{2}=0,\thinspace YU_{1}V_{2}=0,\thinspace
YU_{2}V_{1}=0.$

So,

$\textbf{Term3}=\sum_{i,j=1}^{2}YU_{i}V_{j}=0,$ (A.16)

  4. 4.

Term4
$\displaystyle={[\omega_{\rho},\omega_{\sigma}]^{\psi_{\mu}}}_{\psi_{\nu}}{[\omega^{\rho},\omega^{\sigma}]^{\psi_{\nu}}}_{\psi_{\mu}}$
$\displaystyle=2\underbrace{{(\omega_{\rho})^{\psi_{\mu}}}_{\phi_{\tau}}{(\omega_{\sigma})^{\phi_{\tau}}}_{\psi_{\nu}}}_{T_{1}}\Big{(}\underbrace{{(\omega^{\rho})^{\psi_{\nu}}}_{\phi_{\lambda}}{(\omega^{\sigma})^{\phi_{\lambda}}}_{\psi_{\mu}}}_{T_{2}}-\underbrace{{(\omega^{\sigma})^{\psi_{\nu}}}_{\phi_{\lambda}}{(\omega^{\rho})^{\phi_{\lambda}}}_{\psi_{\mu}}}_{T_{3}}\Big{)},$

where

$\displaystyle T_{1}$
$\displaystyle=\Big{(}\underbrace{\frac{1}{4}\gamma^{\alpha}\gamma_{\rho}\gamma^{\mu}({\bar{F}_{\alpha\tau}}+\frac{1}{2}\gamma^{5}{\bar{H}_{\alpha\tau}})}_{X_{1}}+\underbrace{\frac{1}{4}(\bar{F^{\mu}}_{\alpha}+\frac{1}{2}\gamma^{5}\bar{H^{\mu}}_{\alpha})\gamma_{\tau}\gamma_{\rho}\gamma^{\alpha}}_{X_{2}}\Big{)}$
$\displaystyle\enspace\times\Big{(}\underbrace{-\frac{1}{4}\gamma^{\beta}\gamma_{\sigma}\gamma^{\tau}(\bar{F}_{\beta\nu}+\frac{1}{2}\gamma^{5}\bar{H}_{\beta\nu})}_{Y_{1}}\underbrace{-\frac{1}{4}(\bar{F^{\tau}}_{\beta}+\frac{1}{2}\gamma^{5}\bar{H^{\tau}}_{\beta})\gamma_{\nu}\gamma_{\sigma}\gamma^{\beta}}_{Y_{2}}\Big{)},$
$\displaystyle T_{2}$
$\displaystyle=\Big{(}\underbrace{\frac{1}{4}\gamma^{\theta}\gamma^{\rho}\gamma^{\nu}({\bar{F}_{\theta\lambda}}+\frac{1}{2}\gamma^{5}{\bar{H}_{\theta\lambda}})}_{A_{1}}+\underbrace{\frac{1}{4}(\bar{F^{\nu}}_{\theta}+\frac{1}{2}\gamma^{5}\bar{H^{\nu}}_{\theta})\gamma_{\lambda}\gamma^{\rho}\gamma^{\theta}}_{A_{2}}\Big{)}$
$\displaystyle\enspace\times\Big{(}\underbrace{-\frac{1}{4}\gamma^{\varphi}\gamma^{\sigma}\gamma^{\lambda}(\bar{F}_{\varphi\mu}+\frac{1}{2}\gamma^{5}\bar{H}_{\varphi\mu})}_{B_{1}}\underbrace{-\frac{1}{4}(\bar{F^{\lambda}}_{\varphi}+\frac{1}{2}\gamma^{5}\bar{H^{\lambda}}_{\varphi})\gamma_{\mu}\gamma^{\sigma}\gamma^{\varphi}}_{B_{2}}\Big{)},$
$\displaystyle T_{3}$
$\displaystyle=\Big{(}\underbrace{\frac{1}{4}\gamma^{\theta}\gamma^{\sigma}\gamma^{\nu}({\bar{F}_{\theta\lambda}}+\frac{1}{2}\gamma^{5}{\bar{H}_{\theta\lambda}})}_{C_{1}}+\underbrace{\frac{1}{4}(\bar{F^{\nu}}_{\theta}+\frac{1}{2}\gamma^{5}\bar{H^{\nu}}_{\theta})\gamma_{\lambda}\gamma^{\sigma}\gamma^{\theta}}_{C_{2}}\Big{)}$
$\displaystyle\enspace\times\Big{(}\underbrace{-\frac{1}{4}\gamma^{\varphi}\gamma^{\rho}\gamma^{\lambda}(\bar{F}_{\varphi\mu}+\frac{1}{2}\gamma^{5}\bar{H}_{\varphi\mu})}_{D_{1}}\underbrace{-\frac{1}{4}(\bar{F^{\lambda}}_{\varphi}+\frac{1}{2}\gamma^{5}\bar{H^{\lambda}}_{\varphi})\gamma_{\mu}\gamma^{\rho}\gamma^{\varphi}}_{D_{2}}\Big{)}$

Then,

$\begin{split}&X_{1}Y_{1}A_{1}B_{1}=-2R_{\mu\nu}R^{\mu\nu},\\\
&X_{1}Y_{1}A_{2}B_{2}=-2R_{\mu\nu}R^{\mu\nu},\\\ &X_{1}Y_{1}A_{1}B_{2}=0,\\\
&X_{1}Y_{1}A_{2}B_{1}=0,\\\ &X_{2}Y_{2}A_{1}B_{1}=-2R_{\mu\nu}R^{\mu\nu},\\\
&X_{2}Y_{2}A_{2}B_{2}=-2R_{\mu\nu}R^{\mu\nu},\\\ \end{split}\hskip
7.22743pt\begin{split}X_{2}Y_{2}A_{1}B_{2}&=0,\\\ X_{2}Y_{2}A_{2}B_{1}&=0,\\\
X_{1}Y_{2}A_{1}B_{1}&=0,\\\ X_{1}Y_{2}A_{2}B_{2}&=0,\\\
X_{1}Y_{2}A_{1}B_{2}&=2R_{\mu\nu}R^{\mu\nu}\\\
&\quad-4(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2},\\\ \end{split}\hskip
7.22743pt\begin{split}X_{1}Y_{2}A_{2}B_{1}&=-2R_{\mu\nu}R^{\mu\nu},\\\
X_{2}Y_{1}A_{1}B_{1}&=0,\\\ X_{2}Y_{1}A_{2}B_{2}&=0,\\\
X_{2}Y_{1}A_{1}B_{2}&=-2R_{\mu\nu}R^{\mu\nu},\\\
X_{2}Y_{1}A_{2}B_{1}&=2R_{\mu\nu}R^{\mu\nu}\\\
&\quad-4(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}.\end{split}$

So,

$T_{1}T_{2}=\sum_{i,j,k,l=1}^{2}X_{i}Y_{j}A_{k}B_{l}=-8R_{\mu\nu}R^{\mu\nu}-8(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}.$

Also,

$\begin{split}&X_{1}Y_{1}C_{1}D_{1}=4R_{\mu\nu}R^{\mu\nu},\\\
&X_{1}Y_{1}C_{2}D_{2}=4R_{\mu\nu}R^{\mu\nu},\\\ &X_{1}Y_{1}C_{1}D_{2}=0,\\\
&X_{1}Y_{1}C_{2}D_{1}=0,\\\ &X_{2}Y_{2}C_{1}D_{1}=4R_{\mu\nu}R^{\mu\nu},\\\
&X_{2}Y_{2}C_{2}D_{2}=4R_{\mu\nu}R^{\mu\nu},\\\ \end{split}\hskip
7.22743pt\begin{split}X_{2}Y_{2}C_{1}D_{2}&=0,\\\ X_{2}Y_{2}C_{2}D_{1}&=0,\\\
X_{1}Y_{2}C_{1}D_{1}&=0,\\\ X_{1}Y_{2}C_{2}D_{2}&=0,\\\
X_{1}Y_{2}C_{1}D_{2}&=-4R_{\mu\nu}R^{\mu\nu}\\\
&\quad+8(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2},\\\ \end{split}\hskip
7.22743pt\begin{split}X_{1}Y_{2}C_{2}D_{1}&=4R_{\mu\nu}R^{\mu\nu},\\\
X_{2}Y_{1}C_{1}D_{1}&=0,\\\ X_{2}Y_{1}C_{2}D_{2}&=0,\\\
X_{2}Y_{1}C_{1}D_{2}&=4R_{\mu\nu}R^{\mu\nu},\\\
X_{2}Y_{1}C_{2}D_{1}&=-4R_{\mu\nu}R^{\mu\nu}\\\
&\quad+8(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}.\end{split}$

So,

$T_{1}T_{3}=\sum_{i,j,k,l=1}^{2}X_{i}Y_{j}C_{k}D_{l}=16R_{\mu\nu}R^{\mu\nu}+16(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}.$

Hence,

$\textbf{Term4}=2(T_{1}T_{2}-T_{1}T_{3})=-48R_{\mu\nu}R^{\mu\nu}-48(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}.$
(A.17)

Finally, we have

$\displaystyle{(\Omega_{\rho\sigma})^{\psi_{\mu}}}_{\psi_{\nu}}{(\Omega^{\rho\sigma})^{\psi_{\nu}}}_{\psi_{\mu}}$
$\displaystyle={(\Omega_{\rho\sigma})^{\phi_{\mu}}}_{\phi_{\nu}}{(\Omega^{\rho\sigma})^{\phi_{\nu}}}_{\phi_{\mu}}$
$\displaystyle=\textbf{Term1}+\textbf{Term2}+\textbf{Term3}+\textbf{Term4}$
$\displaystyle=-6R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-48R_{\mu\nu}R^{\mu\nu}-48(\bar{F}_{\mu\nu}\bar{F}^{\mu\nu})^{2}.$
(A.18)

Again,

$\displaystyle{(\Omega_{\rho\sigma})^{\psi_{\mu}}}_{\phi_{\nu}}{(\Omega^{\rho\sigma})^{\phi_{\nu}}}_{\psi_{\mu}}$
$\displaystyle={D_{[\rho}{\omega_{\sigma]}}^{\psi_{\mu}}}_{\phi_{\nu}}{D^{[\rho}{\omega^{\sigma]}}^{\phi_{\nu}}}_{\psi_{\mu}}$
$\displaystyle=2\underbrace{{{(D_{\rho}\omega_{\sigma})}^{\psi_{\mu}}}_{\phi_{\nu}}}_{T_{4}}\Big{(}\underbrace{{{(D^{\rho}\omega^{\sigma})}^{\phi_{\nu}}}_{\psi_{\mu}}}_{T_{5}}-\underbrace{{{(D^{\sigma}\omega^{\rho})}^{\phi_{\nu}}}_{\psi_{\mu}}}_{T_{6}}\Big{)},$
(A.19) $\displaystyle T_{4}$
$\displaystyle=\underbrace{\frac{1}{4}\gamma^{\alpha}\gamma_{\sigma}\gamma^{\mu}(D_{\rho}{\bar{F}_{\alpha\nu}}+\frac{1}{2}\gamma^{5}D_{\rho}{\bar{H}_{\alpha\nu}})}_{L_{1}}+\underbrace{\frac{1}{4}(D_{\rho}\bar{F^{\mu}}_{\alpha}+\frac{1}{2}\gamma^{5}D_{\rho}\bar{H^{\mu}}_{\alpha})\gamma_{\nu}\gamma_{\sigma}\gamma^{\alpha}}_{L_{2}},$
$\displaystyle T_{5}$
$\displaystyle=\underbrace{-\frac{1}{4}\gamma^{\beta}\gamma^{\sigma}\gamma^{\nu}(D^{\rho}\bar{F}_{\beta\mu}+\frac{1}{2}\gamma^{5}D^{\rho}\bar{H}_{\beta\mu})}_{P_{1}}\underbrace{-\frac{1}{4}(D^{\rho}\bar{F^{\nu}}_{\beta}+\frac{1}{2}\gamma^{5}D^{\rho}\bar{H^{\nu}}_{\beta})\gamma_{\mu}\gamma^{\sigma}\gamma^{\beta}}_{P_{2}},$
$\displaystyle T_{6}$
$\displaystyle=\underbrace{-\frac{1}{4}\gamma^{\beta}\gamma^{\rho}\gamma^{\nu}(D^{\sigma}\bar{F}_{\beta\mu}+\frac{1}{2}\gamma^{5}D^{\sigma}\bar{H}_{\beta\mu})}_{Q_{1}}\underbrace{-\frac{1}{4}(D^{\sigma}\bar{F^{\nu}}_{\beta}+\frac{1}{2}\gamma^{5}D^{\sigma}\bar{H^{\nu}}_{\beta})\gamma_{\mu}\gamma^{\rho}\gamma^{\beta}}_{Q_{2}}.$

Then,

$\begin{split}&L_{1}P_{1}=0,\\\ &L_{2}P_{2}=0,\\\
&L_{1}P_{2}=8R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-8R_{\mu\nu}R^{\mu\nu},\\\
&L_{2}P_{1}=8R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-8R_{\mu\nu}R^{\mu\nu},\end{split}\hskip
21.68121pt\begin{split}&L_{1}Q_{1}=0,\\\ &L_{2}Q_{2}=0,\\\
&L_{1}Q_{2}=2R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-2R_{\mu\nu}R^{\mu\nu},\\\
&L_{2}Q_{1}=2R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-2R_{\mu\nu}R^{\mu\nu}.\end{split}$

So,

$\displaystyle T_{4}T_{5}$
$\displaystyle=\sum_{i,j=1}^{2}L_{i}P_{j}=16R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-16R_{\mu\nu}R^{\mu\nu},$
$\displaystyle T_{4}T_{6}$
$\displaystyle=\sum_{i,j=1}^{2}L_{i}Q_{j}=4R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}.$

Hence, we have

$\displaystyle{(\Omega_{\rho\sigma})^{\psi_{\mu}}}_{\phi_{\nu}}{(\Omega^{\rho\sigma})^{\phi_{\nu}}}_{\psi_{\mu}}={(\Omega_{\rho\sigma})^{\phi_{\mu}}}_{\psi_{\nu}}{(\Omega^{\rho\sigma})^{\psi_{\nu}}}_{\phi_{\mu}}$
$\displaystyle=2(T_{4}T_{5}-T_{4}T_{6})$
$\displaystyle=24R_{\mu\nu\rho\sigma}\bar{F}^{\mu\nu}\bar{F}^{\rho\sigma}-24R_{\mu\nu}R^{\mu\nu}.$
(A.20)

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