Logarithmic correction to the entropy of extremal black holes in N = 1 Einstein-Maxwell supergravity

: We study one-loop covariant eﬀective action of “non-minimally coupled” N =1 , d = 4 Einstein-Maxwell supergravity theory by heat kernel tool. By ﬂuctuating the ﬁelds around the classical background, we study the functional determinant of Laplacian dif-ferential operator following Seeley-DeWitt technique of heat kernel expansion in proper time. We then compute the Seeley-DeWitt coeﬃcients obtained through the expansion. A particular Seeley-DeWitt coeﬃcient is used for determining the logarithmic correction to Bekenstein-Hawking entropy of extremal black holes using quantum entropy function formalism. We thus determine the logarithmic correction to the entropy of Kerr-Newman, Kerr and Reissner-Nordström black holes in “non-minimally coupled” N = 1 , d = 4 Einstein-Maxwell supergravity theory.


Introduction
Studying the spectrum of effective action is one of the exciting topics in quantum field theory at all times as it contains the relevant information of any theory.It provides the knowledge of physical amplitudes and quantum effects of the fields present in the theory.The analysis of effective action by perturbative expansion of fields in various loops is a powerful way to study a particular theory.Heat kernel proves to be an efficient tool for analysis of effective action at one-loop level [1,2].The basic principle of this heat kernel method is to study an asymptotic expansion for the spectrum of the differential operator of effective action.It helps us in the investigation of one-loop divergence, quantum anomalies, vacuum polarization, Casimir effect, ultraviolet divergence, spectral geometry, etc.Within heat kernel expansion method, there are various approaches  to analyze the effective action such as covariant perturbation expansion, higher derivative expansion, group theoretic approach, diagram technique, wave function expansion, Seeley-DeWitt expansion, etc. 1 However, among them, Seeley-DeWitt expansion [13] is better over other methods for various reasons stated here.The other methods possess either some model dependency or background geometry dependency.Whereas the Seeley-DeWitt expansion is more general and standard.This neither depends upon background geometry nor on supersymmetry and is applicable for all background field configuration. Tese properties make the Seeley-DeWitt approach more appropriate on those manifolds where other approaches fail to work.In the regime of black hole physics, the heat kernel expansion coefficients are found to be very useful for computing the logarithmic corrections to the entropy of black holes [29][30][31][32][33][34][35][36][37][38].
This paper is motivated by a list of reported works [29][30][31][32][33][34][35][36][37][38][39].In [29], Sen et al. determined the logarithmic corrections of extremal Kerr-Newman type black holes in non-supersymmetric 4D Einstein-Maxwell theory using the quantum entropy function formalism [40][41][42].Here the essential coefficient, a 4 , is calculated following the general Seeley-DeWitt approach [13].In our previous work [30], we have also followed the same approach as [29] and computed the logarithmic correction to the entropy of extremal black holes in minimal N = 2 Einstein-Maxwell supergravity theory (EMSGT) in 4D.The computation of the heat kernel coefficients for extended supersymmetric Einstein-Maxwell theory for N ≥ 2 is presented in [31,32] by Larsen et al..However, the authors have used the strategy of field redefinition approach within the Seeley-DeWitt technique to compute the required coefficients and obtained the logarithmic corrected entropy of various types of non-extremal black holes.In contrast, we are interested in the computation of logarithmic correction to the entropy of extremal black holes by using the general approach [13] for the computation of required coefficients.In [33][34][35][36], the heat kernel coefficients are computed via the wave-function expansion approach and then used for the determination of logarithmic corrections to the entropy of different extremal black holes in N = 2, N = 4, and N = 8 EMSGTs.Again, in [37,38], on-shell and off-shell approaches have been used to determine the required coefficients.In these wave function expansion, on-shell and off-shell methods, one should have explicit knowledge of symme-try properties of the background geometry as well as eigenvalues and eigenfunctions of Laplacian operator acting over the background.Whereas the Seeley-DeWitt expansion approach [13], followed in [29,30,39] as well as in the present work, is applicable for any arbitrary background geometry.
Following the above developments, the analysis of the effective action in N = 1, d = 4 EMSGT is highly inquisitive in nature.We prefer to do this study via the heat kernel tool following the Seeley-DeWitt expansion approach.The attractor mechanism for various black holes in N = 1 supergravity is presented in [43].In [44], the authors have studied local supersymmetry of N = 1 EMSGT in four dimensions.Moreover, it is also interesting to find the logarithmic correction to the black hole entropy of various extremal black hole solutions such as Reissner-Nordström, Kerr and Kerr-Newman black holes in this theory.It will give an intuitive picture about the microstates of black holes within the theory.In the context of quantum gravity, this can be treated as the infrared window into the microstates of the given black holes.
In the current paper, we consider a "non-minimal" N = 1 EMSGT2 in four dimensions on a compact Riemannian manifold without boundary.Here the gaugino field of the vector multiplet is non-minimally coupled to the gravitino field of the supergravity multiplet through the gauge field strength.We fluctuate the fields in the theory around an arbitrary background and construct the quadratic fluctuated (one-loop) action.We determine the functional determinant of one-loop action and then compute the first three Seeley-DeWitt coefficients for this theory.The coefficients are presented in (4.20).Then employing quantum entropy function formalism, the particular coefficient a 4 is used to determine the logarithmic correction to Bekenstein-Hawking entropy for Kerr-Newman, Kerr and Reissner-Nordström black holes in their extremal limit.The results are expressed in (5.25) to (5.27).The logarithmic correction to the extremal Reissner-Nordström black hole entropy in two different classes of N = 1 theory (minimal and truncated from N = 2) are predicted in [45].Our results are unique and essential from them in the sense that we are considering extremal Reissner-Nordström, Kerr and Kerr-Newman black holes in the general non-minimal N = 1 theory.It is also impossible to predict the Reissner-Nordström result of this paper from the work [45] and vice versa.
The present paper is arranged as follows.In Section 2, we have briefly revised the heat kernel technique for the analysis of effective action and introduced the methodology prescribed by Vassilevich et al. [13] for the computation of required Seeley-DeWitt coefficients.We described the four dimensional "non-minimal" N = 1 Einstein-Maxwell supergravity theory and the classical equations of motion for the action in Section 3. In Section 4, we carried out the heat kernel treatment to obtain the first three Seeley-DeWitt coefficients of the "non-minimal" N = 1, d = 4 EMSGT.The coefficients for bosonic fields are summarized from our earlier work [30], whereas the first three Seeley-DeWitt coefficients for fermionic fields are explicitly computed.In Section 5, we first discussed a general approach to determine the logarithmic correction to entropy of extremal black holes using Seeley-DeWitt coefficients via quantum entropy function formalism.Using our results for the Seeley-DeWitt coefficient, we then calculate logarithmic corrections to the entropy Kerr-Newman, Kerr, Reissner-Nordström extremal black holes in "nonminimal" N = 1, d = 4 EMSGT.We summarize our results in Section 6.Some details of our calculation are given in Appendix A.

Effective action analysis via heat kernel
In this section, we are going to discuss the one-loop effective action in 4D Euclidean spacetime using the Seeley-DeWitt expansion approach of heat kernel and review the basic calculation framework of this approach from [13].As initiated by Fock, Schwinger and DeWitt [15,[46][47][48], a Green function is introduced and studied by the heat kernel technique for this analysis.The integral of this Green function over the proper time coordinate is a convenient way to study the one-loop effective action.The proper time method defines the Green function in the neighborhood of light cone, which made it suitable for studying UV divergence and spectral geometry of effective action in one-loop approximation [49].

Loop expansion and effective action
We begin with a review of the heat kernel approach demonstrated in [13] for studying the one-loop effective action by computing the functional determinant of quantum fields fluctuated around the background.We also analyze the relationship between effective action and heat kernels. 3e consider a set of arbitrary fields ϕ m4 on four dimensional compact, smooth Riemannian manifold without boundary.Then, the generating functional for Green function with these fields ϕ m in a Euclidean path integral representation5 is expressed as where [Dϕ m ] denotes the functional integration over all the possible fields ϕ m present in the theory.S is the Euclidean action with lagrangian density L carrying all information of fields in the manifold defined by metric g µν .We fluctuate the fields around any background solution to study the effect of perturbation in the action, where φm and φm symbolize the background and fluctuated fields, respectively.We are actually interested in the quantum characteristics of the action around a classical background, which serves as a semi-classical system in an energy scale lower than the plank scale.For that, we consider the classical solutions φm of the theory as a background and the Taylor expansion of the action (2.2) around the classical background fields up to quadratic fluctuation fields yields, where S cl is the classical action that only depends on the background.S 2 denotes the quadratic interaction part of the quantum fields in action via the differential operator Λ.After imposing the quadratic fluctuation (2.4), the Gaussian integral of generating functional (2.1) gives the form of one-loop effective action W as [1, 28] where the sign of χ depicts the position of detΛ in the denominator or numerator for bosonic or fermionic field taken under consideration. 6W contains the detailed information of the fluctuated quantum fields at one-loop level.In order to analyze the spectrum of the differential operator Λ acted on the perturbed fields, we use the heat kernel technique by defining the heat kernel K as [1, 13] where K(x, y; s) is the Green function between the points x and y that satisfies the heat equation, with the boundary condition, Here the proper time s is known as the heat kernel parameter.The one-loop effective action W is related to the trace of heat kernel, i.e., heat trace D(s) via [13,30] where is the cut-off limit 7 of effective action (2.9) to counter the divergence in the lower limit regime and λ i 's are non zero eigenvalues of Λ which are discrete, but they may also be continuous.D(s) is associated with a perturbative expansion in proper time s as [13, 15-27] The proper time expansion of K(x, x; s) gives a series expansion where the coefficients a 2n , termed as Seeley-DeWitt coefficients [13,[15][16][17][18][19][20], are functions of local background invariants such as Ricci tensor, Riemann tensor, gauge field strengths as well as covariant derivative of these parameters.Our first task is to compute these coefficients a 2n8 for an arbitrary background on a manifold without boundary.We then emphasize the applicability of a particular Seeley-DeWitt coefficient for determining the logarithmic correction to the entropy of extremal black holes.

Heat kernel calculational framework and methodology
In this section, we try to highlight the generalized approach for the computation of Seeley-DeWitt coefficients.For the four dimensional compact, smooth Riemannian manifold without boundary9 under consideration, we will follow the procedure prescribed by Vassilevich in [13] to compute the Seeley-DeWitt coefficients.We aim to express the required coefficients in terms of local invariants. 10e start by considering an ansatz φm , a set of fields satisfying the classical equations of motion of a particular theory.We then fluctuate all the fields around the background as shown in (2.3) and the quadratic fluctuated action S 2 is defined as We study the spectrum of elliptic Laplacian type partial differential operator Λ mn on the fluctuated quantum fields on the manifold.Again, Λ mn should be a Hermitian operator for the heat kernel methodology to be applicable.Therefore, we have, φm Λ mn φn = ± [ φm (D µ D µ ) G mn φn + φm (N ρ D ρ ) mn φn + φm P mn φn ] , (2.12) where an overall +ve or -ve sign needs to be extracted out in the quadratic fluctuated form of bosons and fermions, respectively.G is effective metric in field space, i.e., for vector field G µν = ḡµν , for Rarita-Schwinger field G µν = ḡµν I 4 and for gaugino field G = I 4 .Here I 4 is the identity matrix associated with spinor indices of fermionic fields.P and N are arbitrary matrices, D µ is an ordinary covariant derivative acting on the quantum fields.Moreover, while dealing with a fermionic field (especially Dirac type) whose operator, say O is linear, one has to first make it Laplacian of the form (2.12) using the following relation [28,31,36], (2.13) Equation (2.12) can be rewritten in the form below φm Λ mn φn = ± ( φm (D µ D µ ) I mn φn + φm E mn φn ) , where E is endomorphism on the field bundle and D µ is a new effective covariant derivative defined with field connection ω µ , The new form of (2.The Seeley-DeWitt coefficients for a manifold without boundary can be determined from the knowledge of I, E and Ω αβ .In the following, we give expressions for the first three Seeley-DeWitt coefficients a 0 , a 2 , a 4 in terms of local background invariant parameters [13,26,27], where R, R µν and R µνρσ are the usual curvature tensors computed over the background metric.
There exist many studies about other higher-order coefficients, e.g., a 6 [27], a 8 [5,6], a 10 [7], and higher heat kernels [8,9].However, we will limit ourselves up to a 4 , which helps to determine the logarithmic divergence of the string theoretic black holes.Apart from this, the coefficient a 2 describes the quadratic divergence and renormalization of Newton constant and a 0 encodes the cosmological constant at one-loop approximation [31,32].

"Non-minimal" N = 1 Einstein-Maxwell supergravity theory
A locally supersymmetric Einstein-Maxwell theory in four dimensions is well studied by Ferrara et al. [44].From here, we structure a 4D N = 1 supergravity embedded Einstein-Maxwell theory where the supergravity multiplet is non-minimally coupled to the vector multiplet.The resultant theory, recognized as a "non-minimal" N = 1, d = 4 EMSGT, has the same on-shell bosonic and fermionic degrees of freedom.The bosonic sector is controlled by a 4D Einstein-Maxwell theory [29], while the fermionic sector casts a nonminimal exchange of graviphoton between the gravitino and gaugino species. 11We aim to calculate the Seeley-DeWitt coefficients in the mentioned theory for the fluctuation of its fields.We also summarize the classical equations of motion and identities for the theory.

Supergravity solution and classical equations of motion
The "non-minimal" N = 1 EMSGT in four dimensions is defined with the following field contents: a gravitational spin 2 field g µν (graviton) and an abelian spin 1 gauge field A µ (photon) along with their superpartners, massless spin 3/2 Rarita-Schwinger ψ µ (gravitino) and spin 1/2 λ (gaugino) fields.Note that the gaugino field (λ) interacts non-minimally with the gravitino field (ψ µ ) via non-vanishing gauge field strength.The theory is described by the following action [44] where, R is the Ricci scalar curvature and is the field strength of gauge field A µ .
Let us consider an arbitrary background solution denoted as (ḡ µν , Āµ ) satisfying the classical equations of motion of Einstein-Maxwell theory.The solution is then embedded in N = 1 supersymmetric theory in four dimensions.The geometry of action (3.1) is defined by the Einstein equation, where Fµν = ∂ [µ Āν] is the background gauge field strength.Equation (3.4) also defines background curvature invariants in terms of field strength tensor.For such a geometrical space-time, the trace of (3.4) yields R = 0 as a classical solution.Hence one can ignore the terms proportional to R in further calculations.The reduced equation of motion for the N = 1, d = 4 EMSGT is thus given as The theory also satisfies the Maxwell equations and Bianchi identities of the form, The equations of motion (3.5), (3.6) and the condition R = 0 play a crucial role in simplifying the Seeley-DeWitt coefficients for the N = 1, d = 4 EMSGT theory and express them in terms of local background invariants.
4 Heat kernels in "non-minimal" N = 1, d = 4 EMSGT In this section, we study the spectrum of one-loop action corresponding to the quantum fluctuations in the background fields for the "non-minimal" N = 1 EMSGT in a compact four dimensional Riemannian manifold without boundary.We will focus on the computation of the first three Seeley-DeWitt coefficients for the fluctuated bosonic and fermionic fields present in the concerned theory using the heat kernel tool, as described in Section 2. In the later sections, we throw light on the applicability of these computed coefficients from the perspective of logarithmic correction to the entropy of extremal black holes.

Bosonic sector
Here we are interested in the analysis of quadratic order fluctuated action S 2 and computation of the Seeley-DeWitt coefficients for non-supersymmetric 4D Einstein-Maxwell theory with only bosonic fields, i.e., g µν and A µ , described by the action (3.2).We consider the following fluctuations of the fields present in the theory The resulting quadratic order fluctuated action S 2 is then studied and the required coefficients were calculated following the general Seeley-DeWitt technique [13] for heat kernel.The first three Seeley-DeWitt coefficients for bosonic fields read as12

Fermionic sector
In this subsection, we deal with the superpartner fermionic fields of N = 1, d = 4 EMSGT, i.e., ψ µ and λ.As discussed in Section 3, these fermionic fields are non-minimally coupled.The quadratic fluctuated action for these fields is given in (3.1).Before proceeding further, it is necessary to gauge fix the action for gauge invariance.We achieve that by adding a gauge fixing term L gf to the lagrangian (3.3), This gauge fixing process cancels the gauge degrees of freedom and induces ghost fields to the theory described by the following lagrangian, In (4.4) b, c are Faddeev-Popov ghosts, and ẽ is a ghost associated with the unusual nature of gauge fixing [33].These ghost fields are bosonic ghost having spin half statistics in ghost lagrangian.The final quadratic fluctuated gauge fixed action for the fermionic sector, thus obtained along with unphysical ghost fields is expressed as where we have, Our primary task in this subsection is to compute the Seeley-DeWitt coefficients for action (4.5) and then add it to the coefficients obtained for the bosonic part (4.2).Let us begin with the computation of required coefficients for fermionic fields without ghost fields.The corresponding quadratic fluctuated action can be written as where O is the differential operator which can be expressed in Dirac form as 13φm Since we consider the Euclidean continuation of N = 1 supergravity in 4D Einstein-Maxwell theory, the gamma matrices are Hermitian.Hence, the differential operator O mn is linear, Dirac type and Hermitian.In order to compute the Seeley-DeWitt coefficients following the general technique described in Section 2.2, it is to be made Laplacian.Following (2.13) we have, (4.9) The form of Λ mn can further be simplified by using various identities and ignoring terms dependent on the Ricci scalar R, (4.10) The Laplacian operator Λ obtained in (4.10) fits in the essential format (2.12) required for the heat kernel analysis, thus providing the expressions of I, N ρ and P as φm I mn φn = ψµ I 4 g µν ψ ν + λI 4 λ, (4.11) One can extract ω ρ from N ρ by using (2.17), which on further simplification using gamma matrices properties, reads as The strategy is to express the kinetic differential operator Λ in the form that fits in the prescription (2.14) and find the required quantities.Determination of E and Ω αβ from P , N ρ and ω ρ is now straightforward by using (2.18), (2.19), (4.12) and (4.14). 14One can then calculate the required trace values from the expression of I, E and Ω αβ .The results are (4.15) In the above trace result of I, the values 16 and 4 correspond to the degrees of freedom associated with the gravitino and gaugino fields, respectively.So, the relevant Seeley-DeWitt coefficients for the gauged fermionic part, computed using (4.15) in (2.20) for the Majorana fermions, are Here a -ve sign corresponding to the value of χ from (2.20) for fermion spin-statistics and a factor of 1/2 for considering Majorana degree of freedom are imposed manually in determining the above coefficients.
Next, we aim to compute the Seeley-DeWitt coefficients for the ghost fields described by the lagrangian (4.4).The ghost fields b, c and ẽ are three minimally coupled Majorana spin 1/2 fermions.So, the contribution of Seeley-DeWitt coefficients from the ghost fields will be -3 times of the coefficients of a free Majorana spin 1/2 field, where a 1/2,f 2n (x) is the Seeley-DeWitt coefficients for a free Majorana spin 1/2 field calculated in [39].The Seeley-DeWitt coefficients for ghost sector are The net Seeley-DeWitt coefficients for the fermionic sector can be computed by summing up the results (4.16) and (4.18), (4π

Total Seeley-DeWitt coefficients
The total Seeley-DeWitt Coefficients obtained by adding the bosonic contribution (4.2) and the fermionic contribution (4.19) would be, The coefficient a B+F 4 obtained in (4.20) is independent of the terms R µνθφ F µν F θφ and ( F µν Fµν ) 2 , and only depends upon the background metric which predicts the rotational invariance of the result under electric and magnetic duality.

Logarithmic correction of extremal black holes in
"non-minimal" N = 1, d = 4 EMSGT In this section, we discuss the applicability of Seeley-DeWitt coefficients in the area of black hole physics.We use our results, particularly a 4 coefficient for the computation of logarithmic correction part of extremal black hole entropy in "non-minimal" N = 1, d = 4 EMSGT.

General methodology for computing logarithmic correction using Seeley-DeWitt coefficient
We review the particular procedure adopted in [29,30,[33][34][35][36] for determining the logarithmic correction of entropy of extremal black holes using quantum entropy function formalism [40][41][42].An extremal black hole is defined with near horizon geometry AdS 2 × K, where K is a compactified space fibered over AdS 2 space.The quantum corrected entropy (S BH ) of an extremal black hole is related to the near horizon partition function (Z AdS 2 ) via AdS 2 /CF T 1 correspondence [40] as where E 0 is ground state energy of the extremal black hole carrying charges and β is the boundary length of regularized AdS 2 .We denote coordinate of AdS 2 by (η, θ) and that of K by (y).If the volume of AdS 2 is made finite by inserting an IR cut-off η ≤ η 0 for regularization, then the one-loop correction to the partition function Z AdS 2 is presented as with the one-loop effective action, where G(y) = √ ḡ/ sinh η and ∆L eff is the lagrangian density corresponding to the oneloop effective action.The integration limit over (y) depends upon the geometry associated with K space.The above form of W can be equivalently interpreted as [35,36]  These one-loop corrections serve as logarithmic corrections to the entropy of extremal black holes if one considers only massless fluctuations in the one-loop.In the standard heat kernel analysis of the one-loop effective lagrangian ∆L eff , only the a 4 (x) Seeley-DeWitt coefficient provides a term proportional to logarithmic of horizon area A H [29,30], where K zm (x, x; 0) is the heat kernel contribution for the modes of Λ having vanishing eigenvalues and Y is the scaling dimension associated with zero mode corrections of massless fields under considerations.For extremal black holes defined in large charge and angular momentum limit, the present set up computes the logarithmic correction into two parts -a local part (∆S local ) controlled by the a 4 (x) coefficient and a zero mode part (∆S zm ) controlled by the zero modes of the massless fluctuations in the near-horizon, 15 ) M r and Y r are respectively the number of zero modes and scaling dimension for different boson (B) and fermion (F ) fluctuations.These quantities take different values depending on dimensionality and types of field present [31].In four dimensions we have, 3 + X : metric 1 : gauge field, 4 : gravitino field for BPS solution, 0 : gravitino field for non-BPS solution, (5.9) 2 : metric, 1 : gauge fields, 1/2 : spin 1/2 fields, 3/2 : spin 3/2 fields, (5.10)where X is the number of rotational isometries associated with the angular momentum J of black holes, (5.12) where b = J/M .The horizon radius is given as At the extremal limit M → b 2 + Q 2 , the classical entropy of the black hole in near horizon becomes and the extremal near horizon Kerr-Newman metric is given by [29] (5.15) For the purpose of using the a 4 (x) result (4.20) in the relation (5.7), we consider the background invariants [50,51], (5.16) In the extremal limit, one can find [29] hor dψdφG(ψ)R µνρσ R µνρσ = 8π e(e 2 + 1) The extremal black holes in N = 1, d = 4 Einstein-Maxwell supergravity theory are non-BPS in nature [43,45].Hence, the zero mode contribution to the correction in entropy of non-BPS extremal Kerr-Newman, Kerr and Reissner-Nordström black holes are computed by using (5.9) to (5.11)  e(e 2 + 1)

. 11 )
The local mode contribution (5.7) for the extremal solutions of the N = 1, d = 4 EMSGT can be computed from the coefficient a 4 (x) (4.20) in the near-horizon geometry.15Total logarithmic correction is obtained as ∆S BH = ∆S local + ∆S zm .5.2 Extremal black holes in "non-minimal" N = 1, d = 4 EMSGT and their logarithmic correctionsAn N = 1, d = 4 EMSGT can have Kerr-Newman, Kerr and Reissner-Nordström solutions.The metric ḡ of general Kerr-Newman black hole with mass M , charge Q and angular momentum J is given by