department of physics, indian institute of technology ... · gourav banerjee1, sudip karan2 and...

Logarithmic correction to the entropy ofextremal black holes in N = 1 Einstein-Maxwellsupergravity

Gourav Banerjee1 Sudip Karan2 and Binata Panda3

123 Department of PhysicsIndian Institute of Technology (Indian School of Mines)Dhanbad Jharkhand-826004 INDIA

Email 1gourav 9124apismacin 2sudipkaranapismacin 3binataiitismacin

Abstract

We study one-loop covariant effective action of ldquonon-minimally coupledrdquo N = 1d = 4 Einstein-Maxwell supergravity theory by heat kernel tool By fluctuatingthe fields around the classical background we study the functional determinantof Laplacian differential operator following Seeley-DeWitt technique of heat kernelexpansion in proper time We then compute the Seeley-DeWitt coefficients ob-tained through the expansion A particular Seeley-DeWitt coefficient is used fordetermining the logarithmic correction to Bekenstein-Hawking entropy of extremalblack holes using quantum entropy function formalism We thus determine the log-arithmic correction to the entropy of Kerr-Newman Kerr and Reissner-Nordstromblack holes in ldquonon-minimally coupledrdquo N = 1 d = 4 Einstein-Maxwell supergrav-ity theory

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21

Contents

1 Introduction 2

2 Effective action analysis via heat kernel 421 Loop expansion and effective action 522 Heat kernel calculational framework and methodology 7

3 ldquoNon-minimalrdquo N = 1 Einstein-Maxwell supergravity theory 831 Supergravity solution and classical equations of motion 9

4 Heat kernels in ldquonon-minimalrdquo N = 1 d = 4 EMSGT 1041 Bosonic sector 1042 Fermionic sector 1043 Total Seeley-DeWitt coefficients 14

5 Logarithmic correction of extremal black holes in ldquonon-minimalrdquo N =1 d = 4 EMSGT 1551 General methodology for computing logarithmic correction using Seeley-

DeWitt coefficient 1552 Extremal black holes in ldquonon-minimalrdquo N = 1 d = 4 EMSGT and their

logarithmic corrections 17

6 Concluding remarks 19

A Trace calculations 19

1

1 Introduction

Studying the spectrum of effective action is one of the exciting topics in quantum fieldtheory at all times as it contains the relevant information of any theory It provides theknowledge of physical amplitudes and quantum effects of the fields present in the theoryThe analysis of effective action by perturbative expansion of fields in various loops is apowerful way to study a particular theory Heat kernel proves to be an efficient tool foranalysis of effective action at one-loop level [1 2] The basic principle of this heat kernelmethod is to study an asymptotic expansion for the spectrum of the differential operatorof effective action It helps us in the investigation of one-loop divergence quantumanomalies vacuum polarization Casimir effect ultraviolet divergence spectral geometryetc Within heat kernel expansion method there are various approaches [1ndash28] to analyzethe effective action such as covariant perturbation expansion higher derivative expansiongroup theoretic approach diagram technique wave function expansion Seeley-DeWittexpansion etc1 However among them Seeley-DeWitt expansion [13] is better over othermethods for various reasons stated here The other methods possess either some modeldependency or background geometry dependency Whereas the Seeley-DeWitt expansionis more general and standard This neither depends upon background geometry nor onsupersymmetry and is applicable for all background field configuration These propertiesmake the Seeley-DeWitt approach more appropriate on those manifolds where otherapproaches fail to work In the regime of black hole physics the heat kernel expansioncoefficients are found to be very useful for computing the logarithmic corrections to theentropy of black holes [29ndash39] and the current paper will mainly focus on this direction

This paper is motivated by a list of reported works [29ndash40] In [29] Sen et aldetermined the logarithmic corrections of extremal Kerr-Newman type black holes innon-supersymmetric 4D Einstein-Maxwell theory using the quantum entropy functionformalism [41ndash43] Here the essential coefficient a4 is calculated following the generalSeeley-DeWitt approach [13] In our previous work [30] we have also followed the sameapproach as [29] and computed the logarithmic correction to the entropy of extremalblack holes in minimal N = 2 Einstein-Maxwell supergravity theory (EMSGT) in 4DThe computation of the heat kernel coefficients for extended supersymmetric Einstein-Maxwell theory for N ge 2 is presented in [31 32] by Larsen et al However the authorshave used the strategy of field redefinition approach within the Seeley-DeWitt techniqueto compute the required coefficients and obtained the logarithmic corrected entropy ofvarious types of non-extremal black holes In contrast we are interested in the computa-tion of logarithmic correction to the entropy of extremal black holes by using the generalapproach [13] for the computation of required coefficients In [33ndash36] the heat kernelcoefficients are computed via the wave-function expansion approach and then used forthe determination of logarithmic corrections to the entropy of different extremal blackholes in N = 2 N = 4 and N = 8 EMSGTs Again in [37 38] on-shell and off-shellapproaches have been used to determine the required coefficients In these wave functionexpansion on-shell and off-shell methods one should have explicit knowledge of symme-

1For a general review regarding several approaches of heat kernel expansion one can refer [13]

2

try properties of the background geometry as well as eigenvalues and eigenfunctions ofLaplacian operator acting over the background Whereas the Seeley-DeWitt expansionapproach [13] followed in [293040] as well as in the present work is applicable for anyarbitrary background geometry

Logarithmic corrections are special quantum corrections to the Bekenstein-Hawkingentropy of black holes that can be completely determined from the gravity side with-out being concerned about the UV completion of the theory [29ndash39] In the context ofquantum gravity logarithmic corrections can be treated as the lsquoinfrared windowrsquo into themicrostates of the given black holes Over the last few decades supergravities are themost intensively used platform to investigate logarithmic corrections within string theorySupergravity theories can be realized as the low energy gravity end of compactified stringtheories (mainly on Calabi-Yau manifold) and this makes logarithmic correction resultsin supergravities a more significant IR testing grounds for string theories2 Following thedevelopments discussed in the previous paragraph the analysis of the effective action inN = 1 d = 4 EMSGT is highly inquisitive for the determination of logarithmic correc-tion to the entropy of extremal black holes In the current paper we prefer to do thisstudy via the heat kernel tool following the Seeley-DeWitt expansion approach [13] anduse the results into the framework of the quantum entropy function formalism [41ndash43]The attractor mechanism for various black holes in N = 1 supergravity is presentedin [46] In [47] the authors have studied local supersymmetry of N = 1 EMSGT infour dimensions Moreover we aim to find the logarithmic correction to the black holeentropy of various extremal black hole solutions such as Reissner-Nordstrom Kerr andKerr-Newman black holes in a general class of N = 1 EMSGT in four dimensions Ourresults will provide a strong constraint to an ultraviolet completion of the theory if anindependent computation is done to determine the same entropy corrections from themicroscopic description

In the current paper we particularly consider a ldquonon-minimalrdquo N = 1 EMSGT3

in four dimensions on a compact Riemannian manifold without boundary Here thegaugino field of the vector multiplet is non-minimally coupled to the gravitino field of thesupergravity multiplet through the gauge field strength We fluctuate the fields in thetheory around an arbitrary background and construct the quadratic fluctuated (one-loop)action We determine the functional determinant of one-loop action and then computethe first three Seeley-DeWitt coefficients for this theory The coefficients are presented in(420) Then employing quantum entropy function formalism the particular coefficienta4 is used to determine the logarithmic correction to Bekenstein-Hawking entropy forKerr-Newman Kerr and Reissner-Nordstrom black holes in their extremal limit Theresults are expressed in (525) to (527)

Earlier the logarithmic correction to the extremal Reissner-Nordstrom black holeentropy in two different classes of N = 1 d = 4 EMSGT (truncated from N = 2 andminimal) are predicted in [48] In the ldquotruncatedrdquo N = 1 theory all the field couplings

2An excellent comparative review of logarithmic corrections and their microscopic consistency instring theory can be found in [4445]

3For details of this theory refer to Section 3

3

are considered in such a way that the N = 2 multiplets can be kinematically decomposedinto N = 1 multiplets For this truncation one needs to assume a list of approximations(see sec 41 of [48]) In the ldquominimalrdquoN = 1 theory the coupling term between gravitino(of supergravity multiplet) and gaugino (of vector multiplet) field is customized so thatthere exist minimal couplings between the supergravity multiplet and vector multiplets4

But in the present paper we will proceed with a third and more general class of N = 1theory ndash the ldquonon-minimalrdquo N = 1 EMSGT where no such approximations are assumedfor field couplings and the general non-minimal coupling between the supergravity andvector multiplet is also turned on All the three classes of N = 1 theory are distinct andunable to reproduce each otherrsquos results because of the different choice of field couplingsBoth the ldquotruncatedrdquo and ldquominimalrdquo N = 1 theories are approximated while the ldquonon-minimalrdquo N = 1 EMSGT is a general one In work [48] logarithmic corrections foronly extremal Reissner-Nordstrom are achieved using N = 2 data and a particular localsupersymmetrization process On the other hand our plan is to calculate logarithmiccorrections to the entropy of extremal Reissner-Nordstrom Kerr and Kerr-Newman blackholes by directly analyzing the quadratic fluctuated action of the general ldquonon-minimalrdquoN = 1 EMSGT As discussed our results are unique novel and impossible to predictfrom [48] and vice versa

The present paper is arranged as follows In Section 2 we have briefly revised theheat kernel technique for the analysis of effective action and introduced the methodol-ogy prescribed by Vassilevich et al [13] for the computation of required Seeley-DeWittcoefficients We described the four dimensional ldquonon-minimalrdquo N = 1 Einstein-Maxwellsupergravity theory and the classical equations of motion for the action in Section 3 InSection 4 we carried out the heat kernel treatment to obtain the first three Seeley-DeWittcoefficients of the ldquonon-minimalrdquoN = 1 d = 4 EMSGT The coefficients for bosonic fieldsare summarized from our earlier work [30] whereas the first three Seeley-DeWitt coef-ficients for fermionic fields are explicitly computed In Section 5 we first discussed ageneral approach to determine the logarithmic correction to entropy of extremal blackholes using Seeley-DeWitt coefficients via quantum entropy function formalism Usingour results for the Seeley-DeWitt coefficient we then calculate logarithmic correctionsto the entropy Kerr-Newman Kerr Reissner-Nordstrom extremal black holes in ldquonon-minimalrdquo N = 1 d = 4 EMSGT We summarize our results in Section 6 Some detailsof our calculation are given in Appendix A

2 Effective action analysis via heat kernel

In this section we are going to discuss the one-loop effective action in 4D Euclideanspacetime using the Seeley-DeWitt expansion approach of heat kernel and review thebasic calculation framework of this approach from [13] As initiated by Fock Schwinger

4In general there exist a non-minimal coupling term ψFλ between gravitino ψ (of supergravitymultiplet) and gaugino λ (of vector multiplets) via the vector field (of vector multiplets) strength F inan N = 1 theory [47] The ldquominimally coupledrdquo N = 1 class assumed a constant vector strength matrixF [48]

4

and DeWitt [15 49ndash51] a Green function is introduced and studied by the heat kerneltechnique for this analysis The integral of this Green function over the proper timecoordinate is a convenient way to study the one-loop effective action The proper timemethod defines the Green function in the neighborhood of light cone which made itsuitable for studying UV divergence and spectral geometry of effective action in one-loopapproximation [52]

21 Loop expansion and effective action

We begin with a review of the heat kernel approach demonstrated in [13] for studyingthe one-loop effective action by computing the functional determinant of quantum fieldsfluctuated around the background We also analyze the relationship between effectiveaction and heat kernels5

We consider a set of arbitrary fields ϕm6 on four dimensional compact smooth Rie-

mannian manifold without boundary Then the generating functional for Green functionwith these fields ϕm in a Euclidean path integral representation7 is expressed as

Z =

int[Dϕm] exp(minusS[ϕm]) (21)

S[ϕm] =

intd4xradicgL(ϕm partmicroϕm) (22)

where [Dϕm] denotes the functional integration over all the possible fields ϕm presentin the theory S is the Euclidean action with lagrangian density L carrying all informa-tion of fields in the manifold defined by metric gmicroν We fluctuate the fields around anybackground solution to study the effect of perturbation in the action

ϕm = ϕm + ϕm (23)

where ϕm and ϕm symbolize the background and fluctuated fields respectively Weare actually interested in the quantum characteristics of the action around a classicalbackground which serves as a semi-classical system in an energy scale lower than theplank scale For that we consider the classical solutions ϕm of the theory as a backgroundand the Taylor expansion of the action (22) around the classical background fields up toquadratic fluctuation fields yields

S asymp Scl + S2 = Scl +

intd4xradicgϕmΛmnϕn (24)

where Scl is the classical action that only depends on the background S2 denotes thequadratic interaction part of the quantum fields in action via the differential operator

5For an explicit elaboration one can refer to [30]6Note that this set also includes metric field7The Euclidean path integral is followed by Wick rotation Note that we have set ~ = c = 1 and

GN = 116π throughout the paper

5

Λ After imposing the quadratic fluctuation (24) the Gaussian integral of generatingfunctional (21) gives the form of one-loop effective action W as [1 28]

W =χ

2ln det(Λ) =

χ

2tr ln Λ forall

χ = +1 Bosons

χ = minus1 Fermions (25)

where the sign of χ depicts the position of detΛ in the denominator or numerator forbosonic or fermionic field taken under consideration8 W contains the detailed informa-tion of the fluctuated quantum fields at one-loop level In order to analyze the spectrum ofthe differential operator Λ acted on the perturbed fields we use the heat kernel techniqueby defining the heat kernel K as [1 13]

K(x y s) = 〈x| exp(minussΛ) |y〉 (26)

where K(x y s) is the Green function between the points x and y that satisfies the heatequation

(parts + Λx)K(x y s) = 0 (27)

with the boundary condition

K(x y 0) = δ(x y) (28)

Here the proper time s is known as the heat kernel parameter The one-loop effectiveaction W is related to the trace of heat kernel ie heat trace D(s) via [1330]

W = minus1

2

int infinε

1

sdsχD(s) D(s) = tr (eminussΛ) =

sumi

eminussλi (29)

where ε is the cut-off limit9 of effective action (29) to counter the divergence in the lowerlimit regime and λirsquos are non zero eigenvalues of Λ which are discrete but they mayalso be continuous D(s) is associated with a perturbative expansion in proper time sas [1315ndash27]

D(s) =

intd4xradicgK(x x s) =

intd4xradicg

infinsumn=0

snminus2a2n(x) (210)

The proper time expansion of K(x x s) gives a series expansion where the coefficientsa2n termed as Seeley-DeWitt coefficients [13 15ndash20] are functions of local backgroundinvariants such as Ricci tensor Riemann tensor gauge field strengths as well as covariantderivative of these parameters Our first task is to compute these coefficients a2n

10 for anarbitrary background on a manifold without boundary We then emphasize the applica-bility of a particular Seeley-DeWitt coefficient for determining the logarithmic correctionto the entropy of extremal black holes

8exp(minusW ) =int

[Dϕm] exp(minusintd4xradicgϕΛϕ) = detminusχ2 Λ [32]

9ε limit is controlled by the Planck parameter ε sim l2p sim GN sim 116π

10a2n+1 = 0 for a manifold without boundary

6

22 Heat kernel calculational framework and methodology

In this section we try to highlight the generalized approach for the computation ofSeeley-DeWitt coefficients For the four dimensional compact smooth Riemannian man-ifold without boundary11 under consideration we will follow the procedure prescribedby Vassilevich in [13] to compute the Seeley-DeWitt coefficients We aim to express therequired coefficients in terms of local invariants12

We start by considering an ansatz ϕm a set of fields satisfying the classical equationsof motion of a particular theory We then fluctuate all the fields around the backgroundas shown in (23) and the quadratic fluctuated action S2 is defined as

S2 =


We study the spectrum of elliptic Laplacian type partial differential operator Λmn on thefluctuated quantum fields on the manifold Again Λmn should be a Hermitian operatorfor the heat kernel methodology to be applicable Therefore we have

ϕmΛmnϕn = plusmn [ϕm (DmicroDmicro)Gmnϕn + ϕm (NρDρ)mn ϕn + ϕmP

mnϕn] (212)

where an overall +ve or -ve sign needs to be extracted out in the quadratic fluctuatedform of bosons and fermions respectively G is effective metric in field space ie forvector field Gmicroν = gmicroν for Rarita-Schwinger field Gmicroν = gmicroνI4 and for gaugino fieldG = I4 Here I4 is the identity matrix associated with spinor indices of fermionic fieldsP and N are arbitrary matrices Dmicro is an ordinary covariant derivative acting on thequantum 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 (212)using the following relation [283136]

ln det(Λmn) =1

2ln det(OmpOp

ndagger) (213)

Equation (212) can be rewritten in the form below

ϕmΛmnϕn = plusmn (ϕm (DmicroDmicro) Imnϕn + ϕmEmnϕn) (214)

where E is endomorphism on the field bundle andDmicro is a new effective covariant derivativedefined with field connection ωmicro

Dmicro = Dmicro + ωmicro (215)

The new form of (214) of Λ is also minimal Laplacian second-order pseudo-differentialtype as desired One can obtain the form of I ωρ and E by comparing (212) and (214)with help of the effective covariant derivative (215)

ϕm (I)mn ϕn =ϕmGmnϕn (216)

11The same methodology is also applicable for manifold with a boundary defined by some boundaryconditions (see section 5 of [13])

12Kindly refer our earlier works [3040] for details on this discussion

7

ϕm (ωρ)mn ϕn =ϕm

(1

2Nρ

)mnϕn (217)

ϕmEmnϕn = ϕmP

mnϕn minus ϕm(ωρ)mp(ωρ)pnϕn minus ϕm(Dρωρ)

mnϕn (218)

And the field strength Ωαβ associated with the curvature of Dmicro is obtained via

ϕm(Ωαβ)mnϕn equiv ϕm [DαDβ]mn ϕn (219)

The Seeley-DeWitt coefficients for a manifold without boundary can be determined fromthe knowledge of I E and Ωαβ In the following we give expressions for the first threeSeeley-DeWitt coefficients a0 a2 a4 in terms of local background invariant parameters[132627]

χ(4π)2a0(x) = tr (I)

χ(4π)2a2(x) =1

6tr (6E +RI)

χ(4π)2a4(x) =1

360tr

60RE + 180E2 + 30ΩmicroνΩmicroν + (5R2 minus 2RmicroνRmicroν + 2RmicroνρσRmicroνρσ)I

(220)

where R Rmicroν and Rmicroνρσ are the usual curvature tensors computed over the backgroundmetric

There exist many studies about other higher-order coefficients eg a6 [27] a8 [5 6]a10 [7] and higher heat kernels [89] However we will limit ourselves up to a4 which helpsto determine the logarithmic divergence of the string theoretic black holes Apart fromthis the coefficient a2 describes the quadratic divergence and renormalization of Newtonconstant and a0 encodes the cosmological constant at one-loop approximation [3132]

3 ldquoNon-minimalrdquo N = 1 Einstein-Maxwell supergrav-

ity theory

The gauge action of pure N = 1 supergravity containing spin 2 graviton with superpart-ner massless spin 32 Rarita-Schwinger fields was first constructed and the correspondingsupersymmetric transformations were studied by Ferrara et al [53] Later this work isextended in [47] by coupling an additional vector multiplet (matter coupling) to the pureN = 1 supergravity Here the vector multiplet is non-minimally coupled to the super-gauged action of N = 1 theory The resultant ldquonon-minimalrdquo N = 1 d = 4 EMSGT isthe object of interest of the current paper Two other special classes of N = 1 supergrav-ity theories namely ldquominimally coupledrdquo N = 1 supergravity theory and N = 1 theoryobtained from truncation of N = 2 theory were also studied by Ferrara et al in [4854]This ldquonon-minimalrdquo N = 1 d = 4 EMSGT has the same on-shell bosonic and fermionicdegrees of freedom The bosonic sector is controlled by a 4D Einstein-Maxwell the-ory [29] while the fermionic sector casts a non-minimal exchange of graviphoton between

8

the gravitino and gaugino species We aim to calculate the Seeley-DeWitt coefficients inthe mentioned theory for the fluctuation of its fields We also summarize the classicalequations of motion and identities for the theory

31 Supergravity solution and classical equations of motion

The ldquonon-minimalrdquo N = 1 EMSGT in four dimensions is defined with the followingfield contents a gravitational spin 2 field gmicroν (graviton) and an abelian spin 1 gaugefield Amicro (photon) along with their superpartners massless spin 32 Rarita-Schwinger ψmicro(gravitino) and spin 12 λ (gaugino) fields Note that the gaugino field (λ) interactsnon-minimally with the gravitino field (ψmicro) via non-vanishing gauge field strength

The theory is described by the following action [47]

S =

intd4xradicgLEM +

intd4xradicgLf (31)

whereLEM = Rminus F microνFmicroν (32)

Lf = minus1

2ψmicroγ

[microρν]Dρψν minus1

2λγρDρλ+

1

2radic

2ψmicroFαβγ

αγβγmicroλ (33)

R is the Ricci scalar curvature and Fmicroν = part[microAν] is the field strength of gauge field AmicroLet us consider an arbitrary background solution denoted as (gmicroν Amicro) satisfying the

classical equations of motion of Einstein-Maxwell theory The solution is then embeddedin N = 1 supersymmetric theory in four dimensions The geometry of action (31) isdefined by the Einstein equation

Rmicroν minus1

2gmicroνR = 8πGNTmicroν = 2FmicroρFν

ρ minus 1

2gmicroνF

ρσFρσ (34)

where Fmicroν = part[microAν] is the background gauge field strength Equation (34) also definesbackground curvature invariants in terms of field strength tensor For such a geometricalspace-time the trace of (34) yields R = 0 as a classical solution Hence one can ignorethe terms proportional to R in further calculations The reduced equation of motion forthe N = 1 d = 4 EMSGT is thus given as

Rmicroν = 2FmicroρFνρ minus 1

2gmicroνF

ρσFρσ (35)

The theory also satisfies the Maxwell equations and Bianchi identities of the form

DmicroFmicroν = 0 D[microFνρ] = 0 Rmicro[νθφ] = 0 (36)

The equations of motion (35) (36) and the condition R = 0 play a crucial role insimplifying the Seeley-DeWitt coefficients for the N = 1 d = 4 EMSGT theory andexpress them in terms of local background invariants

9

4 Heat kernels in ldquonon-minimalrdquo N = 1 d = 4 EMSGT

In this section we study the spectrum of one-loop action corresponding to the quantumfluctuations in the background fields for the ldquonon-minimalrdquo N = 1 EMSGT in a compactfour dimensional Riemannian manifold without boundary We will focus on the computa-tion of the first three Seeley-DeWitt coefficients for the fluctuated bosonic and fermionicfields present in the concerned theory using the heat kernel tool as described in Section2 In the later sections we throw light on the applicability of these computed coefficientsfrom the perspective of logarithmic correction to the entropy of extremal black holes

41 Bosonic sector

Here we are interested in the analysis of quadratic order fluctuated action S2 and com-putation of the Seeley-DeWitt coefficients for non-supersymmetric 4D Einstein-Maxwelltheory with only bosonic fields ie gmicroν and Amicro described by the action (32) Weconsider the following fluctuations of the fields present in the theory

gmicroν = gmicroν + gmicroν Amicro = Amicro + Amicro (41)

The resulting quadratic order fluctuated action S2 is then studied and the required coeffi-cients were calculated following the general Seeley-DeWitt technique [13] for heat kernelThe first three Seeley-DeWitt coefficients for bosonic fields read as13

(4π)2aB0 (x) = 4

(4π)2aB2 (x) = 6F microνFmicroν

(4π)2aB4 (x) =

1

180(199RmicroνρσR

microνρσ + 26RmicroνRmicroν)

(42)

42 Fermionic sector

In this subsection we deal with the superpartner fermionic fields of N = 1 d = 4EMSGT ie ψmicro and λ As discussed in Section 3 these fermionic fields are non-minimallycoupled The quadratic fluctuated action for these fields is given in (31) Before pro-ceeding further it is necessary to gauge fix the action for gauge invariance We achievethat by adding a gauge fixing term Lgf to the lagrangian (33)

Lgf =1

4ψmicroγ

microγρDργνψν (43)

This gauge fixing process cancels the gauge degrees of freedom and induces ghost fieldsto the theory described by the following lagrangian

Lghostf = macrbγρDρc+ macreγρDρe (44)

13One may go through our earlier work [30] for explicit computation of Seeley-DeWitt coefficients forthe bosonic gravity multiplet

10

In (44) b c are Faddeev-Popov ghosts and e is a ghost associated with the unusual natureof gauge fixing [33] These ghost fields are bosonic ghost having spin half statistics inghost lagrangian The final quadratic fluctuated gauge fixed action for the fermionicsector thus obtained along with unphysical ghost fields is expressed as

S2F =

intd4xradicg(Lf + Lgf + Lghostf

) (45)

where we have

Lf + Lgf =1

4ψmicroγ

νγργmicroDρψν minus1

2λγρDρλ

+1

4radic

2ψmicroFαβγ

αγβγmicroλminus 1

4radic

2λγνγαγβFαβψν

(46)

Our primary task in this subsection is to compute the Seeley-DeWitt coefficients foraction (45) and then add it to the coefficients obtained for the bosonic part (42)

Let us begin with the computation of required coefficients for fermionic fields withoutghost fields The corresponding quadratic fluctuated action can be written as

minus1

2

intd4xradicgϕmO

mnϕn (47)

where O is the differential operator which can be expressed in Dirac form as14

ϕmOmnϕn = minus i

2ψmicroγ

νγργmicroDρψν + iλγρDρλ

minus i

2radic

2ψmicroFαβγ

αγβγmicroλ+i

2radic


(48)

Since we consider the Euclidean continuation of N = 1 supergravity in 4D Einstein-Maxwell theory the gamma matrices are Hermitian Hence the differential operator Omn

is linear Dirac type and Hermitian In order to compute the Seeley-DeWitt coefficientsfollowing the general technique described in Section 22 it is to be made Laplacian

14The spinors in (46) satisfy the Majorana condition But in order to follow the heat kernel methodwe first compute the Seeley-DeWitt coefficients for Dirac spinors and then obtain the same for theMajorana spinors

11

Following (213) we have

ϕmΛmnϕn = ϕm

(O)mp (O)pndagger

ϕn

= ψmicro

minus1

4γτγργmicroγνγσγτDρDσ +

1

8γαγβγmicroγνγθγφFαβFθφ

ψν

+ λ

1

8γτγαγβγθγφγτ FαβFθφ minus γργσDρDσ

λ

+ ψmicro

minus 1

4radic

2γτγργmicroγαγβγτ (DρFαβ + FαβDρ) +

1

2radic

2γαγβγmicroγρFαβDρ

λ

+ λ

1

4radic

2γτγαγβγνγργτ FαβDρ minus

1

2radic

2γργνγαγβ(DρFαβ + FαβDρ)

ψν

(49)

The form of Λmn can further be simplified by using various identities and ignoring termsdependent on the Ricci scalar R

ϕmΛmnϕn = minusI4gmicroνψmicroD

ρDρψν minus I4λDρDρλ+ ψmicro

Rmicroν minus 1

2γνγθRmicro

θ

+1

2γmicroγθRν

θ minus1

2γργσRmicroν

ρσ +1


ψν

+1

4λ(

γφγαγβγθ + γθγβγαγφ)FαβFθφ

λ

minus 1

2radic

2λγργνγαγβ(DρFαβ)

ψν

minus 1

2radic

2ψmicro

(γβγργmicroγα + γαγmicroγργβ

)(DρFαβ)

λ

minus 1

2radic

2ψmicro

(γβγργmicroγα + γαγmicroγργβ minus γαγβγmicroγρ

)Fαβ

Dρλ

+1

2radic

2λ(

γργαγβγν + γνγβγαγρ minus γργνγαγβ)Fαβ

Dρψν

(410)

The Laplacian operator Λ obtained in (410) fits in the essential format (212) requiredfor the heat kernel analysis thus providing the expressions of I Nρ and P as

ϕmImnϕn = ψmicroI4gmicroνψν + λI4λ (411)

12

ϕmPmnϕn = ψmicro

minusRmicroνI4 +

1

2γνγαRmicro

α

minus 1

2γmicroγαRν

α +1

2γηγρRmicroν

ηρ minus1


ψν

minus 1

4λγφγαγβγθ + γθγβγαγφ

FαβFθφλ

+1

2radic

2ψmicro

(γβγργmicroγα + γαγmicroγργβ)DρFαβ

λ

+1

2radic

2λγργνγαγβDρFαβ

ψν

(412)

ϕm (Nρ)mn ϕn =1

2radic

2ψmicro


)Fαβ

λ

minus 1

2radic

2λ(


ψν

(413)

One can extract ωρ from Nρ by using (217) which on further simplification using gammamatrices properties reads as

ϕm(ωρ)mnϕn = ψmicro

minus 1

4radic

2γθγφγργmicroFθφ +

1radic2γτγρF micro

τ minus1radic2γτγmicroF ρ

τ

λ

+ λ

1

4radic

2γνγργθγφFθφ +

1radic2γργτ F ν

τ minus1radic2γνγτ F ρ

τ

ψν

(414)

The strategy is to express the kinetic differential operator Λ in the form that fits in theprescription (214) and find the required quantities Determination of E and Ωαβ fromP Nρ and ωρ is now straightforward by using (218) (219) (412) and (414)15 Onecan then calculate the required trace values from the expression of I E and Ωαβ Theresults are

tr (I) = 16 + 4 = 20

tr (E) = minus8F microνFmicroν

tr(E2)

= 10(F microνFmicroν)2 + 3RmicroνRmicroν + 2RmicroνθφRmicroνθφ minus 2RαβθφF

αβF θφ

tr(ΩαβΩαβ

)= minus13

2RmicroνθφRmicroνθφ + 12RαβθφF

αβF θφ minus 6RmicroνRmicroν

minus 60(F microνFmicroν)2

(415)

In the above trace result of I the values 16 and 4 correspond to the degrees of freedomassociated with the gravitino and gaugino fields respectively So the relevant Seeley-DeWitt coefficients for the gauged fermionic part computed using (415) in (220) for the

15The full expression of E and Ωαβ are given in (A2) and (A7)

13

Majorana fermions are

(4π)2af0(x) = minus10

(4π)2af2(x) = 4F microνFmicroν

(4π)2af4(x) = minus 1

144

(41RmicroνθφRmicroνθφ + 64RmicroνRmicroν

)

(416)

Here a -ve sign corresponding to the value of χ from (220) for fermion spin-statisticsand a factor of 12 for considering Majorana degree of freedom are imposed manually indetermining the above coefficients

Next we aim to compute the Seeley-DeWitt coefficients for the ghost fields describedby the lagrangian (44) The ghost fields b c and e are three minimally coupled Majoranaspin 12 fermions So the contribution of Seeley-DeWitt coefficients from the ghost fieldswill be -3 times of the coefficients of a free Majorana spin 12 field

aghostf2n (x) = minus3a

12f2n (x) (417)

where a12f2n (x) is the Seeley-DeWitt coefficients for a free Majorana spin 12 field calcu-

lated in [40] The Seeley-DeWitt coefficients for ghost sector are

(4π)2aghostf0 (x) = 6


(4π)2aghostf4 (x) = minus 1

240

(7RmicroνθφRmicroνθφ + 8RmicroνR

microν)

(418)

The net Seeley-DeWitt coefficients for the fermionic sector can be computed by summingup the results (416) and (418)

(4π)2aF0 (x) = minus4

(4π)2aF2 (x) = 4F microνFmicroν

(4π)2aF4 (x) = minus 1

360


)

(419)

43 Total Seeley-DeWitt coefficients

The total Seeley-DeWitt Coefficients obtained by adding the bosonic contribution (42)and the fermionic contribution (419) would be

(4π)2aB+F0 (x) = 0

(4π)2aB+F2 (x) = 10F microνFmicroν

(4π)2aB+F4 (x) =

1

24

(19RmicroνθφRmicroνθφ minus 8RmicroνRmicroν

)

(420)

The coefficient aB+F4 obtained in (420) is independent of the terms RmicroνθφF

microνF θφ and(F microνFmicroν)

2 and only depends upon the background metric which predicts the rotationalinvariance of the result under electric and magnetic duality

14

5 Logarithmic correction of extremal black holes in

ldquonon-minimalrdquo N = 1 d = 4 EMSGT

In this section we discuss the applicability of Seeley-DeWitt coefficients in the area ofblack hole physics We use our results particularly a4 coefficient for the computation oflogarithmic correction part of extremal black hole entropy in ldquonon-minimalrdquoN = 1 d = 4EMSGT

51 General methodology for computing logarithmic correctionusing Seeley-DeWitt coefficient

We review the particular procedure adopted in [29 30 33ndash36] for determining the loga-rithmic correction of entropy of extremal black holes using quantum entropy function for-malism [41ndash43] An extremal black hole is defined with near horizon geometry AdS2timesKwhere K is a compactified space fibered over AdS2 space The quantum corrected entropy(SBH) of an extremal black hole is related to the near horizon partition function (ZAdS2)via AdS2CFT1 correspondence [41] as

eSBHminusE0β = ZAdS2 (51)

where E0 is ground state energy of the extremal black hole carrying charges and β is theboundary length of regularized AdS2 We denote coordinate of AdS2 by (η θ) and thatof K by (y) If the volume of AdS2 is made finite by inserting an IR cut-off η le η0 forregularization then the one-loop correction to the partition function ZAdS2 is presentedas

Z1-loopAdS2

= eminusW (52)

with the one-loop effective action

W =

int η0

0

dη

int 2π

0

dθ

inthor

d2yradicg∆Leff

= 2π(cosh η0 minus 1)

inthor

d2yG(y)∆Leff

(53)

where G(y) =radicg sinh η and ∆Leff is the lagrangian density corresponding to the one-

loop effective action The integration limit over (y) depends upon the geometry associatedwith K space The above form of W can be equivalently interpreted as [3536]

W = minusβ∆E0 +O(βminus1)minus 2π

inthor

d2yG(y)∆Leff (54)

where β = 2π sinh η0 and the shift in the ground state energy ∆E0 = minusintd2yG(y)∆Leff

For the extremal black holes the W form (54) extracts the one-loop corrected entropy(∆SBH) from the β independent part of the relation (51) as

∆SBH = 2π

inthor

d2yG(y)∆Leff (55)

15

These one-loop corrections serve as logarithmic corrections to the entropy of extremalblack holes if one considers only massless fluctuations in the one-loop In the standardheat kernel analysis of the one-loop effective lagrangian ∆Leff only the a4(x) Seeley-DeWitt coefficient provides a term proportional to logarithmic of horizon area AH [2930]

∆Leff minusχ

2

(a4(x)minus (Y minus 1)Kzm(x x 0)

)lnAH (56)

where Kzm(x x 0) is the heat kernel contribution for the modes of Λ having vanishingeigenvalues and Y is the scaling dimension associated with zero mode corrections ofmassless fields under considerations For extremal black holes defined in large charge andangular momentum limit the present set up computes the logarithmic correction intotwo parts ndash a local part (∆Slocal) controlled by the a4(x) coefficient and a zero mode part(∆Szm) controlled by the zero modes of the massless fluctuations in the near-horizon16

∆Slocal = minusπint

hor

d2yG(y)χa4(x) lnAH (57)

∆Szm = minus1

2

(sumrisinB

(Yr minus 1)M r minussumrisinF

(2Yr minus 1)M r

)lnAH (58)

Mr and Yr are respectively the number of zero modes and scaling dimension for differentboson (B) and fermion (F) fluctuations These quantities take different values dependingon dimensionality and types of field present In four dimensions we have [2931ndash3739]

Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =

3 +X metric in extremal limit

0 gravitino field for non-BPS solutions(510)

where X is the number of rotational isometries associated with the angular momentumJ of black holes

X =

1 forallJ3 = constant J2 = arbitrary

3 forallJ3 = J2 = 0(511)

One should not be concerned about the Mr values of the gauge and gaugino fields becausetheir particular Yr values suggest that they have no zero-mode correction contributionin the formula (58) The local mode contribution (57) for the extremal solutions ofthe N = 1 d = 4 EMSGT can be computed from the coefficient a4(x) (420) in thenear-horizon geometry

16Total logarithmic correction is obtained as ∆SBH = ∆Slocal + ∆Szm

16

52 Extremal black holes in ldquonon-minimalrdquo N = 1 d = 4 EMSGTand their logarithmic corrections

An N = 1 d = 4 EMSGT can have Kerr-Newman Kerr and Reissner-Nordstrom so-lutions The metric g of general Kerr-Newman black hole with mass M charge Q andangular momentum J is given by

ds2 = minusr2 + b2 cos2 ψ minus 2Mr +Q2

r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ

r2 + b2 minus 2Mr +Q2dr2 + (r2 + b2 cos2 ψ)dψ2

+

((r2 + b2 cos2 ψ)(r2 + b2) + (2Mr minusQ2)b2 sin2 ψ

r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b

r2 + b2 cos2 ψsin2 ψdtdφ

(512)

where b = JM The horizon radius is given as

r = M +radicM2 minus (Q2 + b2) (513)

At the extremal limit M rarrradicb2 +Q2 the classical entropy of the black hole in near

horizon becomesScl = 16π2(2b2 +Q2) (514)

and the extremal near horizon Kerr-Newman metric is given by [29]

ds2 = (Q2 + b2 + b2 cos2 ψ)(dη2 + sinh2 ηdθ2 + dψ2)

+(Q2 + 2b2)2

(Q2 + b2 + b2 cos2 ψ)sin2 ψ

(dχminus i 2Mb

Q2 + 2b2(cosh η minus 1)dθ

)2

(515)

For the purpose of using the a4(x) result (420) in the relation (57) we consider thebackground invariants [55 56]

RmicroνρσRmicroνρσ =8

(r2 + b2 cos2 ψ)6

6M2(r6 minus 15b2r4 cos2 ψ + 15b4r2 cos4 ψ

minus b6 cos6 ψ)minus 12MQ2r(r4 minus 10r2b2 cos2 ψ + 5b4 cos4 ψ)

+Q4(7r4 minus 34r2b2 cos2 ψ + 7b4 cos4 ψ)

RmicroνRmicroν =4Q4

(r2 + b2 cos2 ψ)4

G(y)KN = G(ψ) = (Q2 + b2 + b2 cos2 ψ)(Q2 + 2b2) sinψ

(516)

In the extremal limit one can find [29]inthor

dψdφG(ψ)RmicroνρσRmicroνρσ =8π

e(e2 + 1)52(2e2 + 1)

3(2e2 + 1)2 tanminus1

(eradice2 + 1

)17

+ eradice2 + 1(minus8e6 minus 20e4 minus 8e2 + 1)

(517)int

hor

dψdφG(ψ)RmicroνRmicroν =2π

e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)

where e = bQ The local mode contribution in the logarithmic correction to the entropyof Kerr-Newman extremal black holes in N = 1 d = 4 EMSGT can be drawn by usingthe results (420) (517) and (518) in the relation (57)

∆SlocalKN =1

48

minus

51(2e2 + 1) tanminus1(

eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)

For Kerr black holes we set the limit e rarr infin in (519) which gives the local modecontribution

∆SlocalKerr =19

12lnAH (520)

And for Reissner-Nordstrom black holes we set the limit e rarr 0 in (519) yielding thefollowing result

∆SlocalRN = minus5

4lnAH (521)

The extremal black holes in N = 1 d = 4 Einstein-Maxwell supergravity theory are non-BPS in nature [4648] Hence the zero mode contribution to the correction in entropy ofnon-BPS extremal Kerr-Newman Kerr and Reissner-Nordstrom black holes are computedby using (510) to (511) in the relation (58) as

∆SzmKN = minus1

2(2minus 1)(3 + 1) lnAH = minus2 lnAH (522)

∆SzmKerr = minus1


∆SzmRN = minus1


Merging the above local and zero mode contributions the total logarithmic correction∆SBH (= ∆Slocal + ∆Szm) to the entropy of extremal Kerr-Newman Kerr and Reissner-Nordstrom black holes in ldquonon-minimalrdquo N = 1 d = 4 EMSGT are computed as

∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH minus 2 lnAH (525)

18

∆SBHKerr =19

12lnAH minus 2 lnAH = minus 5

12lnAH (526)

∆SBHRN = minus5

4lnAH minus 3 lnAH = minus17

4lnAH (527)

6 Concluding remarks

In summary we have analyzed the trace anomalies of quadratic fluctuated fields in theldquonon-minimalrdquo N = 1 d = 4 EMSGT and calculated the first three Seeley-DeWittcoefficients using the approach described in Section 22 An essential advantage of theSeeley-DeWitt coefficient results (420) is that these are not background-geometry specificand hence can be used for any arbitrary solution within the ldquonon-minimalrdquo N = 1 d = 4EMSGT

As an application we have used the third coefficient a4 in the quantum entropyfunction formalism and computed the local contribution of logarithmic correction to theBekenstein-Hawking entropy of extremal black holes in the ldquonon-minimalrdquoN = 1 d =4 EMSGT The zero mode contribution of the logarithmic correction to the entropyhas been computed separately by analyzing the number of zero modes and the scalingdimensions of the fields in the theory We therefore obtain the net logarithmic correctionto the Bekenstein-Hawking entropy of extremal Kerr-Newman (525) Kerr (526) andReissner-Nordstrom (527) non-BPS black holes in the ldquonon-minimalrdquo N = 1 d = 4EMSGT These results are new and significant In contrast the logarithmic correctionto the entropy of extremal Reissner-Nordstrom black holes in two other classes of N =1 d = 4 EMSGT are computed in [48] However our work deals with a more generalN = 1 d = 4 EMSGT theory where the vector multiplet is non-minimally coupled thesupergravity multiplet Due to the presence of non-minimal coupling it is impossible toreproduce the results of [48] from our work and vice-versa It will also be interesting toreproduce these results by other methods of the heat kernel The calculated results mayalso give some valuable inputs for developing insight regarding N = 1 Einstein-Maxwellsupergravity theory in four dimensions and can also provide an infrared window to anymicroscopic theory for the same

Acknowledgments

The authors would like to thank Ashoke Sen and Rajesh Kumar Gupta for valuablediscussions and useful comments on the work We also acknowledge IISER-Bhopal forthe hospitality during the course of this work

A Trace calculations

Here we will explicitly show the expression of E and Ωαβ and the computation of tracesof E E2 and ΩαβΩαβ for the fermionic sector of N = 1 d = 4 EMSGT The trace

19

results will be used to calculate a4 required in the logarithmic correction to the entropyof extremal black holes In trace calculations we have used the following identities [30]

(DρFmicroν)(DρF microν) = RmicroνρσF

microνF ρσ minusRmicroνRmicroν

(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF

microνF ρσ minusRmicroνRmicroν)(A1)

The expression of E by using ωρ (414) will be

ϕmEmnϕn =ψmicroE

ψmicroψνψν + λEλλλ+ ψmicroEψmicroλλ+ λEλψνψν

=ψmicro

minus 3

8γαγβγmicroγνγθγφFαβFθφ minus

1

4γθγφγαγmicroFθφF ν

α

minus 1

4γθγφγαγνFθφF micro

α +1

8gmicroνγαγβγθγφFαβFθφ minus

3

2γθγφF microνFθφ

minus 3γαγβF microαF ν

β +1

2γmicroγαRν

α minus γνγαRmicroα +

1

2γηγρRmicroν

ηρ

+3

4γmicroγνF θφFθφ minus

1

2gmicroνI4F

θφFθφ + I4Rmicroνψν + λ

1

2γαγβγθγφFαβFθφ

λ

+ ψmicro

minus 3

4radic

2γθγφγργmicroDρFθφ +

1radic2γφγρDρF micro

φ +1radic2γθγφDmicroFθφ

λ

+ λminus 3

4radic

2γνγργθγφDρFθφ minus

1radic2γργθDρF ν

θ +1radic2γθγφDνFθφ

ψν

(A2)

Trace of E

tr (E) = tr (Eψmicroψmicro + Eλ

λ + Eψmicroλ + Eλ

ψmicro) (A3)

From (A2) we have

tr (Eψmicroψmicro) = minus4F microνFmicroν

tr (Eλλ) = minus4F microνFmicroν

tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)

Equations (A3) and (A4) gives the tr (E) (415)

Trace of E2

tr (E2) = tr(Eψmicro

ψνEψνψmicro + Eλ

λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)

Using eq (A2) we have

tr (EψmicroψνE

ψνψmicro) = 5RmicroνRmicroν + 2(F microνFmicroν)

2 + 2RmicroνθφRmicroνθφ

tr (EλλE

λλ) = 8(F microνFmicroν)

2 minus 4RmicroνRmicroν

tr (EψmicroλE

λψmicro) = RmicroνRmicroν minusRmicroνθφF

microνF θφ

tr (EλψmicroE

ψmicroλ) = RmicroνRmicroν minusRmicroνθφF

microνF θφ

(A6)

Equations (A5) and (A6) yields the tr (E2) (415)

20

Trace of ΩαβΩαβ

The expression for Ωαβ using ωρ (414) is

ϕm(Ωαβ)mnϕn =ψmicro(Ωαβ)ψmicroψνψν + λ(Ωαβ)λλλ+ ψmicro(Ωαβ)ψmicroλλ+ λ(Ωαβ)λψνψν

=ψmicro

I4R

microναβ +

1

4gmicroνγξγκRαβξκ + [ωα ωβ]ψmicroψν

ψν

+ λ1

4γθγφRαβθφ + [ωα ωβ]λλ

λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)

One can calculate tr (ΩαβΩαβ) as

tr (ΩαβΩαβ) =tr

(Ωαβ)ψmicroψν (Ωαβ)ψν

ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ

+ (Ωαβ)ψmicroλ(Ωαβ)λ

ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro

νγξγκRαβξκ︸︷︷︸R2

+(

(ωα)ψmicroλ(ωβ)λ

ψν minus (ωβ)ψmicroλ(ωα)λ

ψν︸︷︷︸R3

)

timesI4Rν

microαβ︸︷︷︸S1

+1

4gνmicroγσγκRαβ

σκ︸︷︷︸S2

+(

(ωα)ψνλ(ωβ)λ

ψmicro minus (ωβ)ψνλ(ωα)λ

ψmicro︸︷︷︸S3

)

(A9)

(Ωαβ)λλ(Ωαβ)λ

λ= 1

4γθγφRαβθφ︸︷︷︸

R4

+ (ωα)λψmicro(ωβ)ψmicro

λ minus (ωβ)λψmicro(ωα)ψmicro

λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4

+ (ωα)λψν (ωβ)ψν

λ minus (ωβ)λψν

(ωα)ψνλ︸︷︷︸

S5

(A10)

(Ωαβ)ψmicroλ(Ωαβ)λ

ψmicro=(Dα(ωβ)ψmicro

λ minusDβ(ωα)ψmicroλ)

︸︷︷︸R6

times(Dα(ωβ)λ

ψmicro minusDβ(ωα)λψmicro)

︸︷︷︸S6

(A11)

and

(Ωαβ)λψν (Ωαβ)ψν

λ=(Dα(ωβ)λ

ψν minusDβ(ωα)λψν)

︸︷︷︸R7

times(Dα(ωβ)ψν

λ minusDβ(ωα)ψνλ)

︸︷︷︸S7

(A12)

21

Then

tr (R1S1) = minus4RmicroνθφRmicroνθφ


tr (R1S2) = 0

tr (R2S1) = 0

tr (R1S3) = minus4RmicroνθφFmicroνF θφ


tr (R2S3) = 2RmicroνRmicroν


tr (R3S3) = 6RmicroνRmicroν minus 36(F microνFmicroν)2

tr (R4S4) = minus1

2RmicroνθφRmicroνθφ

tr (R4S5) = minus2RmicroνRmicroν



tr (R6S6) = 10RmicroνθφFmicroνF θφ minus 10RmicroνRmicroν


(A13)

Equations (A9) to (A13) gives

tr ((Ωαβ)ψmicroψν (Ωαβ)ψν

ψmicro) = minus6RmicroνθφRmicroνθφ minus 8RmicroνθφF

microνF θφ

+ 10RmicroνRmicroν minus 36(F microνFmicroν)2

tr ((Ωαβ)λλ(Ωαβ)λ

λ) = minus1

2RmicroνθφRmicroνθφ + 4RmicroνRmicroν minus 24(F microνFmicroν)

2

tr ((Ωαβ)ψmicroλ(Ωαβ)λ

ψmicro) = 10RmicroνθφF

microνF θφ minus 10RmicroνRmicroν

tr ((Ωαβ)λψν (Ωαβ)ψν

λ) = 10RmicroνθφF


(A14)

Finally from (A8) and (A14) we have obtained tr (ΩαβΩαβ) (415)

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23

[22] S M Christensen and M J Duff New gravitational index theorems and supertheo-rems Nucl Phys B 154 (1979) 301 [inSPIRE]

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[36] A Sen Logarithmic Corrections to N=2 Black Hole Entropy An Infrared Windowinto the Microstates Gen Rel Grav 44 (2012) 1207 [arXiv11083842] [inSPIRE]

24

[37] C Keeler F Larsen and P Lisbao Logarithmic Corrections to N ge 2 Black HoleEntropy Phys Rev D 90 (2014) 043011 [arXiv14041379] [inSPIRE]

[38] F Larsen and P Lisbao Quantum Corrections to Supergravity on AdS2 times S2 PhysRev D 91 (2015) 084056 [arXiv14117423] [inSPIRE]

[39] A Sen Logarithmic Corrections to Schwarzschild and Other Non-extremal BlackHole Entropy in Different Dimensions JHEP 04 (2013) 156 [arXiv12050971] [in-SPIRE]

[40] S Karan S Kumar and B Panda General heat kernel coefficients for mass-less free spin-32 RaritandashSchwinger field Int J Mod Phys A 33 (2018) 1850063[arXiv170908063] [inSPIRE]

[41] A Sen Entropy Function and AdS(2) CFT(1) Correspondence JHEP 11 (2008)075 [arXiv08050095] [inSPIRE]

[42] A Sen Arithmetic of Quantum Entropy Function JHEP 08 (2009) 068[arXiv09031477] [inSPIRE]

[43] A Sen Quantum Entropy Function from AdS(2)CFT(1) Correspondence Int JMod Phys A 24 (2009) 4225 [arXiv08093304] [inSPIRE]

[44] I Mandal and A Sen Black Hole Microstate Counting and its Macroscopic Counter-part Class Quant Grav 27 (2010) 214003 [Nucl Phys B Proc Suppl 216 (2011)147] [arXiv10083801] [inSPIRE]

[45] A Sen Microscopic and Macroscopic Entropy of Extremal Black Holes in StringTheory Gen Rel Grav 46 (2014) 1711 [arXiv14020109] [inSPIRE]

[46] L Andrianopoli R DrsquoAuria S Ferrara and M Trigiante Black-hole attractors inN=1 supergravity JHEP 07 (2007) 019 [hep-th0703178] [inSPIRE]

[47] S Ferrara J Scherk and P van Nieuwenhuizen Locally Supersymmetric Maxwell-Einstein Theory Phys Rev Lett 37 (1976) 1035 [inSPIRE]

[48] S Ferrara and A Marrani Generalized Mirror Symmetry and Quantum Black HoleEntropy Phys Lett B 707 (2012) 173 [arXiv11090444] [inSPIRE]

[49] V Fock Proper time in classical and quantum mechanics Phys Z Sowjetunion 12(1937) 404 [inSPIRE]

[50] J S Schwinger On gauge invariance and vacuum polarization Phys Rev 82 (1951)664 [inSPIRE]

[51] B S DeWitt Quantum field theory in curved space-time Phys Rept 19 (1975) 295[inSPIRE]

25

[52] I G Avramidi Heat kernel and quantum gravity Springer Berlin Heidelberg Ger-many (2000)

[53] D Z Freedman P van Nieuwenhuizen and S Ferrara Progress Toward a Theory ofSupergravity Phys Rev D 13 (1976) 3214 [inSPIRE]

[54] L Andrianopoli R DrsquoAuria and S Ferrara Consistent reduction of N=2 rarr N=1four-dimensional supergravity coupled to matter Nucl Phys B 628 (2002) 387[hep-th0112192] [inSPIRE]

[55] R C Henry Kretschmann scalar for a kerr-newman black hole Astrophys J 535(2000) 350 [astro-ph9912320] [inSPIRE]

[56] C Cherubini D Bini S Capozziello and R Ruffini Second order scalar invariantsof the Riemann tensor Applications to black hole space-times Int J Mod Phys D11 (2002) 827 [gr-qc0302095] [inSPIRE]

26

1 Introduction




3 ``Non-minimal N=1 Einstein-Maxwell supergravity theory


4 Heat kernels in ``non-minimal N=1 d=4 EMSGT

41 Bosonic sector

42 Fermionic sector


5 Logarithmic correction of extremal black holes in ``non-minimal N=1d=4 EMSGT

51 General methodology for computing logarithmic correction using Seeley-DeWitt coefficient

52 Extremal black holes in ``non-minimal N=1d=4 EMSGT and their logarithmic corrections



Contents

1 Introduction 2

2 Effective action analysis via heat kernel 421 Loop expansion and effective action 522 Heat kernel calculational framework and methodology 7

3 ldquoNon-minimalrdquo N = 1 Einstein-Maxwell supergravity theory 831 Supergravity solution and classical equations of motion 9

4 Heat kernels in ldquonon-minimalrdquo N = 1 d = 4 EMSGT 1041 Bosonic sector 1042 Fermionic sector 1043 Total Seeley-DeWitt coefficients 14

5 Logarithmic correction of extremal black holes in ldquonon-minimalrdquo N =1 d = 4 EMSGT 1551 General methodology for computing logarithmic correction using Seeley-

DeWitt coefficient 1552 Extremal black holes in ldquonon-minimalrdquo N = 1 d = 4 EMSGT and their

logarithmic corrections 17

6 Concluding remarks 19

A Trace calculations 19

1

1 Introduction




2








3







4






Z =


S[ϕm] =



ϕm = ϕm + ϕm (23)







5


W =χ

2ln det(Λ) =

χ

2tr ln Λ forall

χ = +1 Bosons





(parts + Λx)K(x y s) = 0 (27)


K(x y 0) = δ(x y) (28)


W = minus1

2

int infinε

1


sumi

eminussλi (29)


D(s) =


intd4xradicg

infinsumn=0




8exp(minusW ) =int




6




S2 =




mnϕn] (212)


ln det(Λmn) =1

2ln det(OmpOp

ndagger) (213)









7


(1

2Nρ

)mnϕn (217)

ϕmEmnϕn = ϕmP


mnϕn (218)




χ(4π)2a0(x) = tr (I)

χ(4π)2a2(x) =1

6tr (6E +RI)

χ(4π)2a4(x) =1

360tr


(220)




ity theory


8





S =

intd4xradicgLEM +

intd4xradicgLf (31)


Lf = minus1

2ψmicroγ


2λγρDρλ+

1

2radic

2ψmicroFαβγ




Rmicroν minus1


ρ minus 1

2gmicroνF

ρσFρσ (34)



2gmicroνF

ρσFρσ (35)




9



41 Bosonic sector




(4π)2aB0 (x) = 4


(4π)2aB4 (x) =

1



(42)

42 Fermionic sector


Lgf =1

4ψmicroγ





10


S2F =


) (45)

where we have

Lf + Lgf =1

4ψmicroγ


2λγρDρλ

+1

4radic

2ψmicroFαβγ


4radic


(46)



minus1

2

intd4xradicgϕmO

mnϕn (47)


ϕmOmnϕn = minus i

2ψmicroγ


minus i

2radic

2ψmicroFαβγ

αγβγmicroλ+i

2radic


(48)




11


ϕmΛmnϕn = ϕm

(O)mp (O)pndagger

ϕn

= ψmicro

minus1


1


ψν

+ λ

1


λ

+ ψmicro

minus 1

4radic


1

2radic


λ

+ λ

1

4radic


1

2radic


ψν

(49)




Rmicroν minus 1

2γνγθRmicro

θ

+1

2γmicroγθRν

θ minus1

2γργσRmicroν

ρσ +1


ψν

+1

4λ(


λ

minus 1

2radic


ψν

minus 1

2radic

2ψmicro


)(DρFαβ)

λ

minus 1

2radic

2ψmicro


)Fαβ

Dρλ

+1

2radic

2λ(


Dρψν

(410)



12

ϕmPmnϕn = ψmicro

minusRmicroνI4 +

1

2γνγαRmicro

α

minus 1

2γmicroγαRν

α +1

2γηγρRmicroν

ηρ minus1


ψν

minus 1


FαβFθφλ

+1

2radic

2ψmicro


λ

+1

2radic


ψν

(412)

ϕm (Nρ)mn ϕn =1

2radic

2ψmicro


)Fαβ

λ

minus 1

2radic

2λ(


ψν

(413)



minus 1

4radic




τ

λ

+ λ

1

4radic




τ

ψν

(414)


tr (I) = 16 + 4 = 20


tr(E2)


αβF θφ

tr(ΩαβΩαβ

)= minus13




(415)



13





144


)

(416)




12f2n (x) (417)






240


microν)

(418)





360


)

(419)



(4π)2aB+F0 (x) = 0


(4π)2aB+F4 (x) =

1

24


)

(420)




14








Z1-loopAdS2

= eminusW (52)


W =

int η0

0

dη

int 2π

0

dθ

inthor

d2yradicg∆Leff


inthor

d2yG(y)∆Leff

(53)




inthor

d2yG(y)∆Leff (54)



∆SBH = 2π

inthor

d2yG(y)∆Leff (55)

15


∆Leff minusχ

2


)lnAH (56)



hor


∆Szm = minus1

2

(sumrisinB


(2Yr minus 1)M r

)lnAH (58)


Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =




X =


3 forallJ3 = J2 = 0(511)



16




r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector







1 Introduction




2








3







4






Z =


S[ϕm] =



ϕm = ϕm + ϕm (23)







5


W =χ

2ln det(Λ) =

χ

2tr ln Λ forall

χ = +1 Bosons





(parts + Λx)K(x y s) = 0 (27)


K(x y 0) = δ(x y) (28)


W = minus1

2

int infinε

1


sumi

eminussλi (29)


D(s) =


intd4xradicg

infinsumn=0




8exp(minusW ) =int




6




S2 =




mnϕn] (212)


ln det(Λmn) =1

2ln det(OmpOp

ndagger) (213)









7


(1

2Nρ

)mnϕn (217)

ϕmEmnϕn = ϕmP


mnϕn (218)




χ(4π)2a0(x) = tr (I)

χ(4π)2a2(x) =1

6tr (6E +RI)

χ(4π)2a4(x) =1

360tr


(220)




ity theory


8





S =

intd4xradicgLEM +

intd4xradicgLf (31)


Lf = minus1

2ψmicroγ


2λγρDρλ+

1

2radic

2ψmicroFαβγ




Rmicroν minus1


ρ minus 1

2gmicroνF

ρσFρσ (34)



2gmicroνF

ρσFρσ (35)




9



41 Bosonic sector




(4π)2aB0 (x) = 4


(4π)2aB4 (x) =

1



(42)

42 Fermionic sector


Lgf =1

4ψmicroγ





10


S2F =


) (45)

where we have

Lf + Lgf =1

4ψmicroγ


2λγρDρλ

+1

4radic

2ψmicroFαβγ


4radic


(46)



minus1

2

intd4xradicgϕmO

mnϕn (47)


ϕmOmnϕn = minus i

2ψmicroγ


minus i

2radic

2ψmicroFαβγ

αγβγmicroλ+i

2radic


(48)




11


ϕmΛmnϕn = ϕm

(O)mp (O)pndagger

ϕn

= ψmicro

minus1


1


ψν

+ λ

1


λ

+ ψmicro

minus 1

4radic


1

2radic


λ

+ λ

1

4radic


1

2radic


ψν

(49)




Rmicroν minus 1

2γνγθRmicro

θ

+1

2γmicroγθRν

θ minus1

2γργσRmicroν

ρσ +1


ψν

+1

4λ(


λ

minus 1

2radic


ψν

minus 1

2radic

2ψmicro


)(DρFαβ)

λ

minus 1

2radic

2ψmicro


)Fαβ

Dρλ

+1

2radic

2λ(


Dρψν

(410)



12

ϕmPmnϕn = ψmicro

minusRmicroνI4 +

1

2γνγαRmicro

α

minus 1

2γmicroγαRν

α +1

2γηγρRmicroν

ηρ minus1


ψν

minus 1


FαβFθφλ

+1

2radic

2ψmicro


λ

+1

2radic


ψν

(412)

ϕm (Nρ)mn ϕn =1

2radic

2ψmicro


)Fαβ

λ

minus 1

2radic

2λ(


ψν

(413)



minus 1

4radic




τ

λ

+ λ

1

4radic




τ

ψν

(414)


tr (I) = 16 + 4 = 20


tr(E2)


αβF θφ

tr(ΩαβΩαβ

)= minus13




(415)



13





144


)

(416)




12f2n (x) (417)






240


microν)

(418)





360


)

(419)



(4π)2aB+F0 (x) = 0


(4π)2aB+F4 (x) =

1

24


)

(420)




14








Z1-loopAdS2

= eminusW (52)


W =

int η0

0

dη

int 2π

0

dθ

inthor

d2yradicg∆Leff


inthor

d2yG(y)∆Leff

(53)




inthor

d2yG(y)∆Leff (54)



∆SBH = 2π

inthor

d2yG(y)∆Leff (55)

15


∆Leff minusχ

2


)lnAH (56)



hor


∆Szm = minus1

2

(sumrisinB


(2Yr minus 1)M r

)lnAH (58)


Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =




X =


3 forallJ3 = J2 = 0(511)



16




r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector














3







4






Z =


S[ϕm] =



ϕm = ϕm + ϕm (23)







5


W =χ

2ln det(Λ) =

χ

2tr ln Λ forall

χ = +1 Bosons





(parts + Λx)K(x y s) = 0 (27)


K(x y 0) = δ(x y) (28)


W = minus1

2

int infinε

1


sumi

eminussλi (29)


D(s) =


intd4xradicg

infinsumn=0




8exp(minusW ) =int




6




S2 =




mnϕn] (212)


ln det(Λmn) =1

2ln det(OmpOp

ndagger) (213)









7


(1

2Nρ

)mnϕn (217)

ϕmEmnϕn = ϕmP


mnϕn (218)




χ(4π)2a0(x) = tr (I)

χ(4π)2a2(x) =1

6tr (6E +RI)

χ(4π)2a4(x) =1

360tr


(220)




ity theory


8





S =

intd4xradicgLEM +

intd4xradicgLf (31)


Lf = minus1

2ψmicroγ


2λγρDρλ+

1

2radic

2ψmicroFαβγ




Rmicroν minus1


ρ minus 1

2gmicroνF

ρσFρσ (34)



2gmicroνF

ρσFρσ (35)




9



41 Bosonic sector




(4π)2aB0 (x) = 4


(4π)2aB4 (x) =

1



(42)

42 Fermionic sector


Lgf =1

4ψmicroγ





10


S2F =


) (45)

where we have

Lf + Lgf =1

4ψmicroγ


2λγρDρλ

+1

4radic

2ψmicroFαβγ


4radic


(46)



minus1

2

intd4xradicgϕmO

mnϕn (47)


ϕmOmnϕn = minus i

2ψmicroγ


minus i

2radic

2ψmicroFαβγ

αγβγmicroλ+i

2radic


(48)




11


ϕmΛmnϕn = ϕm

(O)mp (O)pndagger

ϕn

= ψmicro

minus1


1


ψν

+ λ

1


λ

+ ψmicro

minus 1

4radic


1

2radic


λ

+ λ

1

4radic


1

2radic


ψν

(49)




Rmicroν minus 1

2γνγθRmicro

θ

+1

2γmicroγθRν

θ minus1

2γργσRmicroν

ρσ +1


ψν

+1

4λ(


λ

minus 1

2radic


ψν

minus 1

2radic

2ψmicro


)(DρFαβ)

λ

minus 1

2radic

2ψmicro


)Fαβ

Dρλ

+1

2radic

2λ(


Dρψν

(410)



12

ϕmPmnϕn = ψmicro

minusRmicroνI4 +

1

2γνγαRmicro

α

minus 1

2γmicroγαRν

α +1

2γηγρRmicroν

ηρ minus1


ψν

minus 1


FαβFθφλ

+1

2radic

2ψmicro


λ

+1

2radic


ψν

(412)

ϕm (Nρ)mn ϕn =1

2radic

2ψmicro


)Fαβ

λ

minus 1

2radic

2λ(


ψν

(413)



minus 1

4radic




τ

λ

+ λ

1

4radic




τ

ψν

(414)


tr (I) = 16 + 4 = 20


tr(E2)


αβF θφ

tr(ΩαβΩαβ

)= minus13




(415)



13





144


)

(416)




12f2n (x) (417)






240


microν)

(418)





360


)

(419)



(4π)2aB+F0 (x) = 0


(4π)2aB+F4 (x) =

1

24


)

(420)




14








Z1-loopAdS2

= eminusW (52)


W =

int η0

0

dη

int 2π

0

dθ

inthor

d2yradicg∆Leff


inthor

d2yG(y)∆Leff

(53)




inthor

d2yG(y)∆Leff (54)



∆SBH = 2π

inthor

d2yG(y)∆Leff (55)

15


∆Leff minusχ

2


)lnAH (56)



hor


∆Szm = minus1

2

(sumrisinB


(2Yr minus 1)M r

)lnAH (58)


Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =




X =


3 forallJ3 = J2 = 0(511)



16




r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector













4






Z =


S[ϕm] =



ϕm = ϕm + ϕm (23)







5


W =χ

2ln det(Λ) =

χ

2tr ln Λ forall

χ = +1 Bosons





(parts + Λx)K(x y s) = 0 (27)


K(x y 0) = δ(x y) (28)


W = minus1

2

int infinε

1


sumi

eminussλi (29)


D(s) =


intd4xradicg

infinsumn=0




8exp(minusW ) =int




6




S2 =




mnϕn] (212)


ln det(Λmn) =1

2ln det(OmpOp

ndagger) (213)









7


(1

2Nρ

)mnϕn (217)

ϕmEmnϕn = ϕmP


mnϕn (218)




χ(4π)2a0(x) = tr (I)

χ(4π)2a2(x) =1

6tr (6E +RI)

χ(4π)2a4(x) =1

360tr


(220)




ity theory


8





S =

intd4xradicgLEM +

intd4xradicgLf (31)


Lf = minus1

2ψmicroγ


2λγρDρλ+

1

2radic

2ψmicroFαβγ




Rmicroν minus1


ρ minus 1

2gmicroνF

ρσFρσ (34)



2gmicroνF

ρσFρσ (35)




9



41 Bosonic sector




(4π)2aB0 (x) = 4


(4π)2aB4 (x) =

1



(42)

42 Fermionic sector


Lgf =1

4ψmicroγ





10


S2F =


) (45)

where we have

Lf + Lgf =1

4ψmicroγ


2λγρDρλ

+1

4radic

2ψmicroFαβγ


4radic


(46)



minus1

2

intd4xradicgϕmO

mnϕn (47)


ϕmOmnϕn = minus i

2ψmicroγ


minus i

2radic

2ψmicroFαβγ

αγβγmicroλ+i

2radic


(48)




11


ϕmΛmnϕn = ϕm

(O)mp (O)pndagger

ϕn

= ψmicro

minus1


1


ψν

+ λ

1


λ

+ ψmicro

minus 1

4radic


1

2radic


λ

+ λ

1

4radic


1

2radic


ψν

(49)




Rmicroν minus 1

2γνγθRmicro

θ

+1

2γmicroγθRν

θ minus1

2γργσRmicroν

ρσ +1


ψν

+1

4λ(


λ

minus 1

2radic


ψν

minus 1

2radic

2ψmicro


)(DρFαβ)

λ

minus 1

2radic

2ψmicro


)Fαβ

Dρλ

+1

2radic

2λ(


Dρψν

(410)



12

ϕmPmnϕn = ψmicro

minusRmicroνI4 +

1

2γνγαRmicro

α

minus 1

2γmicroγαRν

α +1

2γηγρRmicroν

ηρ minus1


ψν

minus 1


FαβFθφλ

+1

2radic

2ψmicro


λ

+1

2radic


ψν

(412)

ϕm (Nρ)mn ϕn =1

2radic

2ψmicro


)Fαβ

λ

minus 1

2radic

2λ(


ψν

(413)



minus 1

4radic




τ

λ

+ λ

1

4radic




τ

ψν

(414)


tr (I) = 16 + 4 = 20


tr(E2)


αβF θφ

tr(ΩαβΩαβ

)= minus13




(415)



13





144


)

(416)




12f2n (x) (417)






240


microν)

(418)





360


)

(419)



(4π)2aB+F0 (x) = 0


(4π)2aB+F4 (x) =

1

24


)

(420)




14








Z1-loopAdS2

= eminusW (52)


W =

int η0

0

dη

int 2π

0

dθ

inthor

d2yradicg∆Leff


inthor

d2yG(y)∆Leff

(53)




inthor

d2yG(y)∆Leff (54)



∆SBH = 2π

inthor

d2yG(y)∆Leff (55)

15


∆Leff minusχ

2


)lnAH (56)



hor


∆Szm = minus1

2

(sumrisinB


(2Yr minus 1)M r

)lnAH (58)


Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =




X =


3 forallJ3 = J2 = 0(511)



16




r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector












Z =


S[ϕm] =



ϕm = ϕm + ϕm (23)







5


W =χ

2ln det(Λ) =

χ

2tr ln Λ forall

χ = +1 Bosons





(parts + Λx)K(x y s) = 0 (27)


K(x y 0) = δ(x y) (28)


W = minus1

2

int infinε

1


sumi

eminussλi (29)


D(s) =


intd4xradicg

infinsumn=0




8exp(minusW ) =int




6




S2 =




mnϕn] (212)


ln det(Λmn) =1

2ln det(OmpOp

ndagger) (213)









7


(1

2Nρ

)mnϕn (217)

ϕmEmnϕn = ϕmP


mnϕn (218)




χ(4π)2a0(x) = tr (I)

χ(4π)2a2(x) =1

6tr (6E +RI)

χ(4π)2a4(x) =1

360tr


(220)




ity theory


8





S =

intd4xradicgLEM +

intd4xradicgLf (31)


Lf = minus1

2ψmicroγ


2λγρDρλ+

1

2radic

2ψmicroFαβγ




Rmicroν minus1


ρ minus 1

2gmicroνF

ρσFρσ (34)



2gmicroνF

ρσFρσ (35)




9



41 Bosonic sector




(4π)2aB0 (x) = 4


(4π)2aB4 (x) =

1



(42)

42 Fermionic sector


Lgf =1

4ψmicroγ





10


S2F =


) (45)

where we have

Lf + Lgf =1

4ψmicroγ


2λγρDρλ

+1

4radic

2ψmicroFαβγ


4radic


(46)



minus1

2

intd4xradicgϕmO

mnϕn (47)


ϕmOmnϕn = minus i

2ψmicroγ


minus i

2radic

2ψmicroFαβγ

αγβγmicroλ+i

2radic


(48)




11


ϕmΛmnϕn = ϕm

(O)mp (O)pndagger

ϕn

= ψmicro

minus1


1


ψν

+ λ

1


λ

+ ψmicro

minus 1

4radic


1

2radic


λ

+ λ

1

4radic


1

2radic


ψν

(49)




Rmicroν minus 1

2γνγθRmicro

θ

+1

2γmicroγθRν

θ minus1

2γργσRmicroν

ρσ +1


ψν

+1

4λ(


λ

minus 1

2radic


ψν

minus 1

2radic

2ψmicro


)(DρFαβ)

λ

minus 1

2radic

2ψmicro


)Fαβ

Dρλ

+1

2radic

2λ(


Dρψν

(410)



12

ϕmPmnϕn = ψmicro

minusRmicroνI4 +

1

2γνγαRmicro

α

minus 1

2γmicroγαRν

α +1

2γηγρRmicroν

ηρ minus1


ψν

minus 1


FαβFθφλ

+1

2radic

2ψmicro


λ

+1

2radic


ψν

(412)

ϕm (Nρ)mn ϕn =1

2radic

2ψmicro


)Fαβ

λ

minus 1

2radic

2λ(


ψν

(413)



minus 1

4radic




τ

λ

+ λ

1

4radic




τ

ψν

(414)


tr (I) = 16 + 4 = 20


tr(E2)


αβF θφ

tr(ΩαβΩαβ

)= minus13




(415)



13





144


)

(416)




12f2n (x) (417)






240


microν)

(418)





360


)

(419)



(4π)2aB+F0 (x) = 0


(4π)2aB+F4 (x) =

1

24


)

(420)




14








Z1-loopAdS2

= eminusW (52)


W =

int η0

0

dη

int 2π

0

dθ

inthor

d2yradicg∆Leff


inthor

d2yG(y)∆Leff

(53)




inthor

d2yG(y)∆Leff (54)



∆SBH = 2π

inthor

d2yG(y)∆Leff (55)

15


∆Leff minusχ

2


)lnAH (56)



hor


∆Szm = minus1

2

(sumrisinB


(2Yr minus 1)M r

)lnAH (58)


Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =




X =


3 forallJ3 = J2 = 0(511)



16




r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector








W =χ

2ln det(Λ) =

χ

2tr ln Λ forall

χ = +1 Bosons





(parts + Λx)K(x y s) = 0 (27)


K(x y 0) = δ(x y) (28)


W = minus1

2

int infinε

1


sumi

eminussλi (29)


D(s) =


intd4xradicg

infinsumn=0




8exp(minusW ) =int




6




S2 =




mnϕn] (212)


ln det(Λmn) =1

2ln det(OmpOp

ndagger) (213)









7


(1

2Nρ

)mnϕn (217)

ϕmEmnϕn = ϕmP


mnϕn (218)




χ(4π)2a0(x) = tr (I)

χ(4π)2a2(x) =1

6tr (6E +RI)

χ(4π)2a4(x) =1

360tr


(220)




ity theory


8





S =

intd4xradicgLEM +

intd4xradicgLf (31)


Lf = minus1

2ψmicroγ


2λγρDρλ+

1

2radic

2ψmicroFαβγ




Rmicroν minus1


ρ minus 1

2gmicroνF

ρσFρσ (34)



2gmicroνF

ρσFρσ (35)




9



41 Bosonic sector




(4π)2aB0 (x) = 4


(4π)2aB4 (x) =

1



(42)

42 Fermionic sector


Lgf =1

4ψmicroγ





10


S2F =


) (45)

where we have

Lf + Lgf =1

4ψmicroγ


2λγρDρλ

+1

4radic

2ψmicroFαβγ


4radic


(46)



minus1

2

intd4xradicgϕmO

mnϕn (47)


ϕmOmnϕn = minus i

2ψmicroγ


minus i

2radic

2ψmicroFαβγ

αγβγmicroλ+i

2radic


(48)




11


ϕmΛmnϕn = ϕm

(O)mp (O)pndagger

ϕn

= ψmicro

minus1


1


ψν

+ λ

1


λ

+ ψmicro

minus 1

4radic


1

2radic


λ

+ λ

1

4radic


1

2radic


ψν

(49)




Rmicroν minus 1

2γνγθRmicro

θ

+1

2γmicroγθRν

θ minus1

2γργσRmicroν

ρσ +1


ψν

+1

4λ(


λ

minus 1

2radic


ψν

minus 1

2radic

2ψmicro


)(DρFαβ)

λ

minus 1

2radic

2ψmicro


)Fαβ

Dρλ

+1

2radic

2λ(


Dρψν

(410)



12

ϕmPmnϕn = ψmicro

minusRmicroνI4 +

1

2γνγαRmicro

α

minus 1

2γmicroγαRν

α +1

2γηγρRmicroν

ηρ minus1


ψν

minus 1


FαβFθφλ

+1

2radic

2ψmicro


λ

+1

2radic


ψν

(412)

ϕm (Nρ)mn ϕn =1

2radic

2ψmicro


)Fαβ

λ

minus 1

2radic

2λ(


ψν

(413)



minus 1

4radic




τ

λ

+ λ

1

4radic




τ

ψν

(414)


tr (I) = 16 + 4 = 20


tr(E2)


αβF θφ

tr(ΩαβΩαβ

)= minus13




(415)



13





144


)

(416)




12f2n (x) (417)






240


microν)

(418)





360


)

(419)



(4π)2aB+F0 (x) = 0


(4π)2aB+F4 (x) =

1

24


)

(420)




14








Z1-loopAdS2

= eminusW (52)


W =

int η0

0

dη

int 2π

0

dθ

inthor

d2yradicg∆Leff


inthor

d2yG(y)∆Leff

(53)




inthor

d2yG(y)∆Leff (54)



∆SBH = 2π

inthor

d2yG(y)∆Leff (55)

15


∆Leff minusχ

2


)lnAH (56)



hor


∆Szm = minus1

2

(sumrisinB


(2Yr minus 1)M r

)lnAH (58)


Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =




X =


3 forallJ3 = J2 = 0(511)



16




r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector










S2 =




mnϕn] (212)


ln det(Λmn) =1

2ln det(OmpOp

ndagger) (213)









7


(1

2Nρ

)mnϕn (217)

ϕmEmnϕn = ϕmP


mnϕn (218)




χ(4π)2a0(x) = tr (I)

χ(4π)2a2(x) =1

6tr (6E +RI)

χ(4π)2a4(x) =1

360tr


(220)




ity theory


8





S =

intd4xradicgLEM +

intd4xradicgLf (31)


Lf = minus1

2ψmicroγ


2λγρDρλ+

1

2radic

2ψmicroFαβγ




Rmicroν minus1


ρ minus 1

2gmicroνF

ρσFρσ (34)



2gmicroνF

ρσFρσ (35)




9



41 Bosonic sector




(4π)2aB0 (x) = 4


(4π)2aB4 (x) =

1



(42)

42 Fermionic sector


Lgf =1

4ψmicroγ





10


S2F =


) (45)

where we have

Lf + Lgf =1

4ψmicroγ


2λγρDρλ

+1

4radic

2ψmicroFαβγ


4radic


(46)



minus1

2

intd4xradicgϕmO

mnϕn (47)


ϕmOmnϕn = minus i

2ψmicroγ


minus i

2radic

2ψmicroFαβγ

αγβγmicroλ+i

2radic


(48)




11


ϕmΛmnϕn = ϕm

(O)mp (O)pndagger

ϕn

= ψmicro

minus1


1


ψν

+ λ

1


λ

+ ψmicro

minus 1

4radic


1

2radic


λ

+ λ

1

4radic


1

2radic


ψν

(49)




Rmicroν minus 1

2γνγθRmicro

θ

+1

2γmicroγθRν

θ minus1

2γργσRmicroν

ρσ +1


ψν

+1

4λ(


λ

minus 1

2radic


ψν

minus 1

2radic

2ψmicro


)(DρFαβ)

λ

minus 1

2radic

2ψmicro


)Fαβ

Dρλ

+1

2radic

2λ(


Dρψν

(410)



12

ϕmPmnϕn = ψmicro

minusRmicroνI4 +

1

2γνγαRmicro

α

minus 1

2γmicroγαRν

α +1

2γηγρRmicroν

ηρ minus1


ψν

minus 1


FαβFθφλ

+1

2radic

2ψmicro


λ

+1

2radic


ψν

(412)

ϕm (Nρ)mn ϕn =1

2radic

2ψmicro


)Fαβ

λ

minus 1

2radic

2λ(


ψν

(413)



minus 1

4radic




τ

λ

+ λ

1

4radic




τ

ψν

(414)


tr (I) = 16 + 4 = 20


tr(E2)


αβF θφ

tr(ΩαβΩαβ

)= minus13




(415)



13





144


)

(416)




12f2n (x) (417)






240


microν)

(418)





360


)

(419)



(4π)2aB+F0 (x) = 0


(4π)2aB+F4 (x) =

1

24


)

(420)




14








Z1-loopAdS2

= eminusW (52)


W =

int η0

0

dη

int 2π

0

dθ

inthor

d2yradicg∆Leff


inthor

d2yG(y)∆Leff

(53)




inthor

d2yG(y)∆Leff (54)



∆SBH = 2π

inthor

d2yG(y)∆Leff (55)

15


∆Leff minusχ

2


)lnAH (56)



hor


∆Szm = minus1

2

(sumrisinB


(2Yr minus 1)M r

)lnAH (58)


Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =




X =


3 forallJ3 = J2 = 0(511)



16




r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector








(1

2Nρ

)mnϕn (217)

ϕmEmnϕn = ϕmP


mnϕn (218)




χ(4π)2a0(x) = tr (I)

χ(4π)2a2(x) =1

6tr (6E +RI)

χ(4π)2a4(x) =1

360tr


(220)




ity theory


8





S =

intd4xradicgLEM +

intd4xradicgLf (31)


Lf = minus1

2ψmicroγ


2λγρDρλ+

1

2radic

2ψmicroFαβγ




Rmicroν minus1


ρ minus 1

2gmicroνF

ρσFρσ (34)



2gmicroνF

ρσFρσ (35)




9



41 Bosonic sector




(4π)2aB0 (x) = 4


(4π)2aB4 (x) =

1



(42)

42 Fermionic sector


Lgf =1

4ψmicroγ





10


S2F =


) (45)

where we have

Lf + Lgf =1

4ψmicroγ


2λγρDρλ

+1

4radic

2ψmicroFαβγ


4radic


(46)



minus1

2

intd4xradicgϕmO

mnϕn (47)


ϕmOmnϕn = minus i

2ψmicroγ


minus i

2radic

2ψmicroFαβγ

αγβγmicroλ+i

2radic


(48)




11


ϕmΛmnϕn = ϕm

(O)mp (O)pndagger

ϕn

= ψmicro

minus1


1


ψν

+ λ

1


λ

+ ψmicro

minus 1

4radic


1

2radic


λ

+ λ

1

4radic


1

2radic


ψν

(49)




Rmicroν minus 1

2γνγθRmicro

θ

+1

2γmicroγθRν

θ minus1

2γργσRmicroν

ρσ +1


ψν

+1

4λ(


λ

minus 1

2radic


ψν

minus 1

2radic

2ψmicro


)(DρFαβ)

λ

minus 1

2radic

2ψmicro


)Fαβ

Dρλ

+1

2radic

2λ(


Dρψν

(410)



12

ϕmPmnϕn = ψmicro

minusRmicroνI4 +

1

2γνγαRmicro

α

minus 1

2γmicroγαRν

α +1

2γηγρRmicroν

ηρ minus1


ψν

minus 1


FαβFθφλ

+1

2radic

2ψmicro


λ

+1

2radic


ψν

(412)

ϕm (Nρ)mn ϕn =1

2radic

2ψmicro


)Fαβ

λ

minus 1

2radic

2λ(


ψν

(413)



minus 1

4radic




τ

λ

+ λ

1

4radic




τ

ψν

(414)


tr (I) = 16 + 4 = 20


tr(E2)


αβF θφ

tr(ΩαβΩαβ

)= minus13




(415)



13





144


)

(416)




12f2n (x) (417)






240


microν)

(418)





360


)

(419)



(4π)2aB+F0 (x) = 0


(4π)2aB+F4 (x) =

1

24


)

(420)




14








Z1-loopAdS2

= eminusW (52)


W =

int η0

0

dη

int 2π

0

dθ

inthor

d2yradicg∆Leff


inthor

d2yG(y)∆Leff

(53)




inthor

d2yG(y)∆Leff (54)



∆SBH = 2π

inthor

d2yG(y)∆Leff (55)

15


∆Leff minusχ

2


)lnAH (56)



hor


∆Szm = minus1

2

(sumrisinB


(2Yr minus 1)M r

)lnAH (58)


Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =




X =


3 forallJ3 = J2 = 0(511)



16




r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector











S =

intd4xradicgLEM +

intd4xradicgLf (31)


Lf = minus1

2ψmicroγ


2λγρDρλ+

1

2radic

2ψmicroFαβγ




Rmicroν minus1


ρ minus 1

2gmicroνF

ρσFρσ (34)



2gmicroνF

ρσFρσ (35)




9



41 Bosonic sector




(4π)2aB0 (x) = 4


(4π)2aB4 (x) =

1



(42)

42 Fermionic sector


Lgf =1

4ψmicroγ





10


S2F =


) (45)

where we have

Lf + Lgf =1

4ψmicroγ


2λγρDρλ

+1

4radic

2ψmicroFαβγ


4radic


(46)



minus1

2

intd4xradicgϕmO

mnϕn (47)


ϕmOmnϕn = minus i

2ψmicroγ


minus i

2radic

2ψmicroFαβγ

αγβγmicroλ+i

2radic


(48)




11


ϕmΛmnϕn = ϕm

(O)mp (O)pndagger

ϕn

= ψmicro

minus1


1


ψν

+ λ

1


λ

+ ψmicro

minus 1

4radic


1

2radic


λ

+ λ

1

4radic


1

2radic


ψν

(49)




Rmicroν minus 1

2γνγθRmicro

θ

+1

2γmicroγθRν

θ minus1

2γργσRmicroν

ρσ +1


ψν

+1

4λ(


λ

minus 1

2radic


ψν

minus 1

2radic

2ψmicro


)(DρFαβ)

λ

minus 1

2radic

2ψmicro


)Fαβ

Dρλ

+1

2radic

2λ(


Dρψν

(410)



12

ϕmPmnϕn = ψmicro

minusRmicroνI4 +

1

2γνγαRmicro

α

minus 1

2γmicroγαRν

α +1

2γηγρRmicroν

ηρ minus1


ψν

minus 1


FαβFθφλ

+1

2radic

2ψmicro


λ

+1

2radic


ψν

(412)

ϕm (Nρ)mn ϕn =1

2radic

2ψmicro


)Fαβ

λ

minus 1

2radic

2λ(


ψν

(413)



minus 1

4radic




τ

λ

+ λ

1

4radic




τ

ψν

(414)


tr (I) = 16 + 4 = 20


tr(E2)


αβF θφ

tr(ΩαβΩαβ

)= minus13




(415)



13





144


)

(416)




12f2n (x) (417)






240


microν)

(418)





360


)

(419)



(4π)2aB+F0 (x) = 0


(4π)2aB+F4 (x) =

1

24


)

(420)




14








Z1-loopAdS2

= eminusW (52)


W =

int η0

0

dη

int 2π

0

dθ

inthor

d2yradicg∆Leff


inthor

d2yG(y)∆Leff

(53)




inthor

d2yG(y)∆Leff (54)



∆SBH = 2π

inthor

d2yG(y)∆Leff (55)

15


∆Leff minusχ

2


)lnAH (56)



hor


∆Szm = minus1

2

(sumrisinB


(2Yr minus 1)M r

)lnAH (58)


Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =




X =


3 forallJ3 = J2 = 0(511)



16




r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector









41 Bosonic sector




(4π)2aB0 (x) = 4


(4π)2aB4 (x) =

1



(42)

42 Fermionic sector


Lgf =1

4ψmicroγ





10


S2F =


) (45)

where we have

Lf + Lgf =1

4ψmicroγ


2λγρDρλ

+1

4radic

2ψmicroFαβγ


4radic


(46)



minus1

2

intd4xradicgϕmO

mnϕn (47)


ϕmOmnϕn = minus i

2ψmicroγ


minus i

2radic

2ψmicroFαβγ

αγβγmicroλ+i

2radic


(48)




11


ϕmΛmnϕn = ϕm

(O)mp (O)pndagger

ϕn

= ψmicro

minus1


1


ψν

+ λ

1


λ

+ ψmicro

minus 1

4radic


1

2radic


λ

+ λ

1

4radic


1

2radic


ψν

(49)




Rmicroν minus 1

2γνγθRmicro

θ

+1

2γmicroγθRν

θ minus1

2γργσRmicroν

ρσ +1


ψν

+1

4λ(


λ

minus 1

2radic


ψν

minus 1

2radic

2ψmicro


)(DρFαβ)

λ

minus 1

2radic

2ψmicro


)Fαβ

Dρλ

+1

2radic

2λ(


Dρψν

(410)



12

ϕmPmnϕn = ψmicro

minusRmicroνI4 +

1

2γνγαRmicro

α

minus 1

2γmicroγαRν

α +1

2γηγρRmicroν

ηρ minus1


ψν

minus 1


FαβFθφλ

+1

2radic

2ψmicro


λ

+1

2radic


ψν

(412)

ϕm (Nρ)mn ϕn =1

2radic

2ψmicro


)Fαβ

λ

minus 1

2radic

2λ(


ψν

(413)



minus 1

4radic




τ

λ

+ λ

1

4radic




τ

ψν

(414)


tr (I) = 16 + 4 = 20


tr(E2)


αβF θφ

tr(ΩαβΩαβ

)= minus13




(415)



13





144


)

(416)




12f2n (x) (417)






240


microν)

(418)





360


)

(419)



(4π)2aB+F0 (x) = 0


(4π)2aB+F4 (x) =

1

24


)

(420)




14








Z1-loopAdS2

= eminusW (52)


W =

int η0

0

dη

int 2π

0

dθ

inthor

d2yradicg∆Leff


inthor

d2yG(y)∆Leff

(53)




inthor

d2yG(y)∆Leff (54)



∆SBH = 2π

inthor

d2yG(y)∆Leff (55)

15


∆Leff minusχ

2


)lnAH (56)



hor


∆Szm = minus1

2

(sumrisinB


(2Yr minus 1)M r

)lnAH (58)


Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =




X =


3 forallJ3 = J2 = 0(511)



16




r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector








S2F =


) (45)

where we have

Lf + Lgf =1

4ψmicroγ


2λγρDρλ

+1

4radic

2ψmicroFαβγ


4radic


(46)



minus1

2

intd4xradicgϕmO

mnϕn (47)


ϕmOmnϕn = minus i

2ψmicroγ


minus i

2radic

2ψmicroFαβγ

αγβγmicroλ+i

2radic


(48)




11


ϕmΛmnϕn = ϕm

(O)mp (O)pndagger

ϕn

= ψmicro

minus1


1


ψν

+ λ

1


λ

+ ψmicro

minus 1

4radic


1

2radic


λ

+ λ

1

4radic


1

2radic


ψν

(49)




Rmicroν minus 1

2γνγθRmicro

θ

+1

2γmicroγθRν

θ minus1

2γργσRmicroν

ρσ +1


ψν

+1

4λ(


λ

minus 1

2radic


ψν

minus 1

2radic

2ψmicro


)(DρFαβ)

λ

minus 1

2radic

2ψmicro


)Fαβ

Dρλ

+1

2radic

2λ(


Dρψν

(410)



12

ϕmPmnϕn = ψmicro

minusRmicroνI4 +

1

2γνγαRmicro

α

minus 1

2γmicroγαRν

α +1

2γηγρRmicroν

ηρ minus1


ψν

minus 1


FαβFθφλ

+1

2radic

2ψmicro


λ

+1

2radic


ψν

(412)

ϕm (Nρ)mn ϕn =1

2radic

2ψmicro


)Fαβ

λ

minus 1

2radic

2λ(


ψν

(413)



minus 1

4radic




τ

λ

+ λ

1

4radic




τ

ψν

(414)


tr (I) = 16 + 4 = 20


tr(E2)


αβF θφ

tr(ΩαβΩαβ

)= minus13




(415)



13





144


)

(416)




12f2n (x) (417)






240


microν)

(418)





360


)

(419)



(4π)2aB+F0 (x) = 0


(4π)2aB+F4 (x) =

1

24


)

(420)




14








Z1-loopAdS2

= eminusW (52)


W =

int η0

0

dη

int 2π

0

dθ

inthor

d2yradicg∆Leff


inthor

d2yG(y)∆Leff

(53)




inthor

d2yG(y)∆Leff (54)



∆SBH = 2π

inthor

d2yG(y)∆Leff (55)

15


∆Leff minusχ

2


)lnAH (56)



hor


∆Szm = minus1

2

(sumrisinB


(2Yr minus 1)M r

)lnAH (58)


Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =




X =


3 forallJ3 = J2 = 0(511)



16




r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector








ϕmΛmnϕn = ϕm

(O)mp (O)pndagger

ϕn

= ψmicro

minus1


1


ψν

+ λ

1


λ

+ ψmicro

minus 1

4radic


1

2radic


λ

+ λ

1

4radic


1

2radic


ψν

(49)




Rmicroν minus 1

2γνγθRmicro

θ

+1

2γmicroγθRν

θ minus1

2γργσRmicroν

ρσ +1


ψν

+1

4λ(


λ

minus 1

2radic


ψν

minus 1

2radic

2ψmicro


)(DρFαβ)

λ

minus 1

2radic

2ψmicro


)Fαβ

Dρλ

+1

2radic

2λ(


Dρψν

(410)



12

ϕmPmnϕn = ψmicro

minusRmicroνI4 +

1

2γνγαRmicro

α

minus 1

2γmicroγαRν

α +1

2γηγρRmicroν

ηρ minus1


ψν

minus 1


FαβFθφλ

+1

2radic

2ψmicro


λ

+1

2radic


ψν

(412)

ϕm (Nρ)mn ϕn =1

2radic

2ψmicro


)Fαβ

λ

minus 1

2radic

2λ(


ψν

(413)



minus 1

4radic




τ

λ

+ λ

1

4radic




τ

ψν

(414)


tr (I) = 16 + 4 = 20


tr(E2)


αβF θφ

tr(ΩαβΩαβ

)= minus13




(415)



13





144


)

(416)




12f2n (x) (417)






240


microν)

(418)





360


)

(419)



(4π)2aB+F0 (x) = 0


(4π)2aB+F4 (x) =

1

24


)

(420)




14








Z1-loopAdS2

= eminusW (52)


W =

int η0

0

dη

int 2π

0

dθ

inthor

d2yradicg∆Leff


inthor

d2yG(y)∆Leff

(53)




inthor

d2yG(y)∆Leff (54)



∆SBH = 2π

inthor

d2yG(y)∆Leff (55)

15


∆Leff minusχ

2


)lnAH (56)



hor


∆Szm = minus1

2

(sumrisinB


(2Yr minus 1)M r

)lnAH (58)


Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =




X =


3 forallJ3 = J2 = 0(511)



16




r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector







ϕmPmnϕn = ψmicro

minusRmicroνI4 +

1

2γνγαRmicro

α

minus 1

2γmicroγαRν

α +1

2γηγρRmicroν

ηρ minus1


ψν

minus 1


FαβFθφλ

+1

2radic

2ψmicro


λ

+1

2radic


ψν

(412)

ϕm (Nρ)mn ϕn =1

2radic

2ψmicro


)Fαβ

λ

minus 1

2radic

2λ(


ψν

(413)



minus 1

4radic




τ

λ

+ λ

1

4radic




τ

ψν

(414)


tr (I) = 16 + 4 = 20


tr(E2)


αβF θφ

tr(ΩαβΩαβ

)= minus13




(415)



13





144


)

(416)




12f2n (x) (417)






240


microν)

(418)





360


)

(419)



(4π)2aB+F0 (x) = 0


(4π)2aB+F4 (x) =

1

24


)

(420)




14








Z1-loopAdS2

= eminusW (52)


W =

int η0

0

dη

int 2π

0

dθ

inthor

d2yradicg∆Leff


inthor

d2yG(y)∆Leff

(53)




inthor

d2yG(y)∆Leff (54)



∆SBH = 2π

inthor

d2yG(y)∆Leff (55)

15


∆Leff minusχ

2


)lnAH (56)



hor


∆Szm = minus1

2

(sumrisinB


(2Yr minus 1)M r

)lnAH (58)


Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =




X =


3 forallJ3 = J2 = 0(511)



16




r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector











144


)

(416)




12f2n (x) (417)






240


microν)

(418)





360


)

(419)



(4π)2aB+F0 (x) = 0


(4π)2aB+F4 (x) =

1

24


)

(420)




14








Z1-loopAdS2

= eminusW (52)


W =

int η0

0

dη

int 2π

0

dθ

inthor

d2yradicg∆Leff


inthor

d2yG(y)∆Leff

(53)




inthor

d2yG(y)∆Leff (54)



∆SBH = 2π

inthor

d2yG(y)∆Leff (55)

15


∆Leff minusχ

2


)lnAH (56)



hor


∆Szm = minus1

2

(sumrisinB


(2Yr minus 1)M r

)lnAH (58)


Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =




X =


3 forallJ3 = J2 = 0(511)



16




r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector














Z1-loopAdS2

= eminusW (52)


W =

int η0

0

dη

int 2π

0

dθ

inthor

d2yradicg∆Leff


inthor

d2yG(y)∆Leff

(53)




inthor

d2yG(y)∆Leff (54)



∆SBH = 2π

inthor

d2yG(y)∆Leff (55)

15


∆Leff minusχ

2


)lnAH (56)



hor


∆Szm = minus1

2

(sumrisinB


(2Yr minus 1)M r

)lnAH (58)


Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =




X =


3 forallJ3 = J2 = 0(511)



16




r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector








∆Leff minusχ

2


)lnAH (56)



hor


∆Szm = minus1

2

(sumrisinB


(2Yr minus 1)M r

)lnAH (58)


Yr =

2 metric

1 gauge fields

12 spin 12 fields

32 gravitino fields

(59)

Mr =




X =


3 forallJ3 = J2 = 0(511)



16




r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector










r2 + b2 cos2 ψdt2

+r2 + b2 cos2 ψ


+


r2 + b2 cos2 ψ

)sin2 ψdφ2

+ 2(Q2 minus 2Mr)b


(512)







+(Q2 + 2b2)2


(dχminus i 2Mb


)2

(515)



(r2 + b2 cos2 ψ)6





(r2 + b2 cos2 ψ)4


(516)



e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)17


(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector








(517)int

hor


e(e2 + 1)52(2e2 + 1)


(eradice2 + 1

)+ eradice2 + 1(8e2 + 5)

(518)


∆SlocalKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)

lnAH

(519)


∆SlocalKerr =19

12lnAH (520)



4lnAH (521)


∆SzmKN = minus1


∆SzmKerr = minus1


∆SzmRN = minus1



∆SBHKN =1

48

minus


eradice2+1

)e(e2 + 1)52

+(152e6 + 380e4 + 168e2 minus 9)

(e2 + 1)2(2e2 + 1)


18

∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector







∆SBHKerr =19


12lnAH (526)

∆SBHRN = minus5


4lnAH (527)




Acknowledgments




19




(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector










(DmicroFρν)(DνF

ρmicro) =1

2(RmicroνρσF



ϕmEmnϕn =ψmicroE


=ψmicro

minus 3


1


α

minus 1


α +1


3



β +1

2γmicroγαRν


1

2γηγρRmicroν

ηρ

+3


1

2gmicroνI4F


1


λ

+ ψmicro

minus 3

4radic




λ

+ λminus 3

4radic




ψν

(A2)

Trace of E



ψmicro) (A3)

From (A2) we have



tr (Eψmicroλ) = 0

tr (Eλψmicro) = 0

(A4)


Trace of E2



λEλλ + Eψmicro

λEλψmicro + Eλ

ψmicroEψmicroλ

) (A5)


tr (EψmicroψνE



tr (EλλE



tr (EψmicroλE


microνF θφ

tr (EλψmicroE


microνF θφ

(A6)


20




=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector










=ψmicro

I4R

microναβ +

1


ψν

+ λ1


λ

+ ψmicro

D[αωβ]

ψmicroλλ+ λ

D[αωβ]

λψνψν

(A7)




ψmicro+ (Ωαβ)λ

λ(Ωαβ)λλ


ψmicro+ (Ωαβ)λ

ψν (Ωαβ)ψνλ

(A8)

where


ψmicro

=I4Rmicro

ναβ︸︷︷︸

R1

+1

4gmicro


+(



ψν︸︷︷︸R3

)

timesI4Rν


+1


σκ︸︷︷︸S2

+(




)

(A9)


λ= 1


R4



λ︸︷︷︸R5

times 1

4γργσRαβ

ρσ︸︷︷︸S4




S5

(A10)




︸︷︷︸R6

times(Dα(ωβ)λ


︸︷︷︸S6

(A11)

and


λ=(Dα(ωβ)λ


︸︷︷︸R7

times(Dα(ωβ)ψν


︸︷︷︸S7

(A12)

21

Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector







Then



tr (R1S2) = 0

tr (R2S1) = 0






tr (R4S4) = minus1







(A13)




microνF θφ



λ) = minus1


2







(A14)


References







22
















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector






















23
















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector






















24
















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector






















25






26

1 Introduction







41 Bosonic sector

42 Fermionic sector












26

1 Introduction







41 Bosonic sector

42 Fermionic sector







department of physics, indian institute of technology ... · gourav banerjee1, sudip karan2 and...

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