Generalized Einstein-Maxwell theory: Seeley-DeWitt coefficients and logarithmic corrections to the entropy of extremal and non-extremal black holes

We present a consolidated manual of Euclidean gravity approaches for finding the logarithmic corrections to the entropy of the full Kerr-Newman family of black holes in both extremal and non-extremal limits. Seeley-DeWitt coefficients for the quadratic fluctuations of a concern gravity theory appear to be the key ingredients in this manual. Following the manual, we calculate the first three Seeley-DeWitt coefficients and logarithmic corrections to the entropy of extremal and non-extremal black holes in a generalized Einstein-Maxwell theory minimally-coupled to additional massless scalar, vector, spin-1/2 Dirac and spin-3/2 Rarita-Schwinger fields. We finally employ the Seeley-DeWitt data to reproduce the logarithmic entropy corrections for extremal black holes in all $\mathcal{N} \geq 2$ Einstein-Maxwell supergravity via an alternative local supersymmetrization method.


Introduction
In any quantum gravity model, including string theory, it has been found that the leading quantum correction to the Bekenstein-Hawking entropy formula of black holes carrying large charges 1 is proportional to the logarithm of horizon area, called the logarithmic correction [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. These quantum corrections are special features of black hole entropy that can 1 In the large-charge limit, the charge, angular momentum, mass and other black hole parameters are scaled so that the black hole becomes large (i.e., the horizon area AH l 2 P , lP is the Planck length), keeping different dimensionless ratios unchanged.

An effective manual for logarithmic correction to black hole entropy
In this section, we present a consolidated manual of how Euclidean gravity approaches provide a standard path to evaluate logarithmic correction to the entropy of extremal and non-extremal Kerr-Newman family of black holes using the Seeley-DeWitt coefficients.

The working formula
Let us consider a four-dimensional Euclidean gravitational theory with the matter fields ξ and the metric g describing corresponding space-time geometry over a compact manifold. If we fluctuate g and ξ around an arbitrary classical background solution (ḡ,ξ) for small quantum fluctuationsξ m = {g,ξ}, g =ḡ +g, ξ =ξ +ξ, where Λ is the kinetic differential operator that characterizes the quadratic fluctuations. We can now introduce the heat kernel K(x, y; s) that encodes all the data about the spectrum of the operator Λ [20-22, 29, 47], Here {f i } are the eigenfunctions of the operator Λ with eigenvalues {λ i } and s is a proper time with units of (length) 2 , called the heat kernel time. By the proper time representation [39,40], the quantum corrected one-loop effective action W is then expressed in terms of the heat trace D(s) as [26] where χ = ±1 for bosonic and fermionic fluctuations, respectively; is a UV cutoff, restricted by ∼ l p 2 ∼ G N . 7 In order to evaluate K(x, x; s) and D(s), one can cast the Seeley-DeWitt expansion as s → 0, where the coefficients a 2n (x) of the perturbative expansion are known as the Seeley-DeWitt coefficients [41][42][43][44][45][46]. The s independent part of the expansion (2.7) allows us to find a logarithmic term from the integration (2.6) in the range s A H (A H is the black hole horizon area), ∞ ds s χD(s) = · · · + d 4 x detḡ a 4 (x) ln A H G N + · · · . (2.8) In any Euclidean gravity approach, this logarithmic term corrects the black hole entropy if one integrates out only massless modes in the one-loop effective action W [20][21][22][23][24][25][26][28][29][30][31][32][33][34]. Therefore, the logarithmic correction to black hole entropy is calculated by the general formula, 8 with the following local (C local ) and zero-mode (C zm ) contributions where n 0 ξm serves the zero-mode counts for a particular fluctuationξ m having scaling dimension βξ m . It is also reasonable to investigate whether the universal nature of black hole entropy sustains after incorporating the logarithmic corrections. In the describe Euclidean gravity framework, the logarithmic corrections ∆S BH are obtained as C ln A H G N where the prefactor C = 1 2 (C local + C zm ) is termed as the coefficient of logarithmic correction. The ln A H G N part is global in any generic gravity theory, while the coefficient of logarithmic correction C is generally "geometric" (i.e., depends on the black hole geometric characteristics like mass, charge, angular momentum, etc.). But C may also become "non-geometric", then the particular logarithmic correction result is fully universal.

Computation of the zero-mode part of logarithmic entropy correction
The contributions of various fields to the zero-mode correction C zm are not new; they have been computed and analyzed in many works [21,22,29,32]. A generalized and concise review can also be found in [33]. For both the scalar and spin-1/2 fields n 0 0 = n 0 1/2 = 0, hence they have no contribution in the C zm formula (2.9c). The four-dimensional vector fields have β 1 = 1, so they also contribute nothing to C zm via the formula (2.9c). For a spin-3/2 field β 3/2 = 3 in d = 4; n 0 3/2 = −4 for BPS solutions in N ≥ 2, d = 4 supergravities and 0 for all kinds of non-supersymmetric black holes. The four-dimensional metric has β 2 = 2 with n 0 2 = −3 − K for the extremal case and n 0 2 = −K for the non-extremal case. Here K (number of rotational isometries) is 3 for the non-rotating black holes and 1 for the rotating black holes. In summary, C zm receives a contribution from only the metric for non-supersymmetric black holes, while both the metric and gravitino field contribute for BPS black holes. And χ in the formula (2.9c) takes care of the bosonic and fermionic nature of the fields. Make sure to set χ = −1 for fermions and χ = +1 for bosons.

Computation of the local part of logarithmic entropy correction
We will now outline computation strategies of the local correction C local for both the extremal and non-extremal Kerr-Newman family of black holes. This analysis is twofolda standard computation approach of Seeley-DeWitt coefficients for any arbitrary gravity theory and the appropriate limits of necessary background invariants over the extremal and non-extremal Kerr-Newman black hole geometries.

A standard approach for computing Seeley-DeWitt coefficients
In order to compute a 4 (x), we will pursue a standard and efficient approach reviewed in [47]. The basic technical set-up of this approach is briefly depicted as follows. First, one needs to adjust (up to a total derivative 9 ) the quadratic-fluctuated action (2.3) so that the 9 In our choice of d = 4 compact manifold (without boundary), all the total derivative terms emerge as boundary terms in the integration (2.5) via the relation (2.7) and hence contribute nothing. This particular setting is useful in dealing with quantum fluctuations around the boundary of asymptotically flat black holes (e.g., Kerr-Newman family of black holes). kinetic operator Λ becomes Hermitian, Laplace-type and minimal of the following form ξ m Λξ mξnξ n = ±ξ m (D ρ D ρ )Iξ mξn + (N ρ D ρ )ξ mξn + Pξ mξn ξ n . (2.10) Here D ρ is a Christoffel-spin-connected covariant derivative. I is an arbitrary matrix induced from combinations of the background metricḡ and the identity operator in spin space. I acts as an effective metric that simultaneously contracts all the indices of any particular fluctuation. N ρ , P are also arbitrary matrices in terms of background fieldsξ and the background metricḡ. The standard form (2.10) can be generalized further so that it incorporates interactions between the fluctuations. The generalized operator form is prescribed asξ where the redefined covariant derivative D ρ , the gauge connection ω ρ , the commutator curvature Ω ρσ ≡ [D ρ , D σ ] and the matrix-valued potential E are defined as Note that all the matrices are labeled by "ξ mξn ", describing with which pair of fluctuations the matrices are contracted whereξ m includes particular fluctuations-types along with their tensor indices. Any fluctuation is considered as the "minimally-coupled" if it evaluates ω ρ = 0 (i.e., no other interactions except the one with background gravity via the √ detḡ term in its quadratic-fluctuated action). The commutations of covariant derivatives acting on the scalar φ, vector a µ , spin-1/2 Dirac λ, spin-3/2 Rarita-Schwinger ψ µ and metric h µν fluctuations have the following standard definitions With all these data, the formulae for the first three Seeley-DeWitt coefficients are listed as [47] a 0 (x) = χ 16π 2 tr(I), where χ = +1, −1 and -1/2 for the fluctuation of bosons, Dirac spinors and Majorana spinors, respectively. The above described approach naturally recognizes any fermionic fluctuation as a Dirac spinor [22,28,33]. Majorana spinors have half the degrees of freedom of Dirac fermions, and hence for casting them via the current approach, one needs to employ an additional 1/2 factor in the formulae (2.14). Weyl spinors are prohibited in this approach due to having both the right and left chiral states and must be redefined into Dirac or Majorana forms. The crucial benefit of the present Seeley-DeWitt computation approach is that after taking quadratic fluctuations around a classical background, we have direct formulae to calculate the Seeley-DeWitt coefficients in terms of background invariants like R µνρσ R µνρσ , R µν R µν , R 2 , R µνρσF µνF ρσ ,F µνF µν , (F µνF µν ) 2 , etc. Thus the results are global and not limited to any particular background of the theory. Apart from several useful applications, the Seeley-DeWitt coefficients have essential utility in one-loop quantum corrections. Since the logarithmic corrections are evaluated by a 4 (x), we find it sufficient to calculate the Seeley-DeWitt coefficients only up to this order.

Strategies for extremal and non-extremal Kerr-Newman black holes
Kerr-Newman black holes are the most general stationary solutions to the equations of motion of Einstein-Maxwell theory [48]. In standard spherical coordinates (t, r, ψ, φ), a Kerr-Newman black hole with charge Q, mass M and angular momentum J is characterized by the metric [30], with the following geometric invariants [58,59] R µν R µν = 4Q 4 (r 2 + b 2 cos 2 ψ) 4 , 16) where b = J/M . One can achieve the metric forms of Schwarzschild, Kerr and Reissner-Nordström black holes in appropriate limits of the Kerr-Newman metric (2.15). We are now going to discuss two separate strategies for calculating C local corrections to the entropy of extremal and non-extremal Kerr-Newman family of black holes. For both the strategies, all the charges of the Kerr-Newman black hole need to be scaled by a common large scale, say L, so that the angular momentum, charge and horizon area of the black hole are scaled as J ∼ L 2 , Q ∼ L and A H ∼ L 2 (the large-charge limits) [30,32].

Strategy A (for extremal black holes):
For the extremal Kerr-Newman family of black holes, we follow a strategy that casts the quantum entropy function formalism [35][36][37]. This Euclidean gravity approach is quite efficient in calculating the one-loop quantum corrections of extremal black holes by only using the near-horizon geometry data [20][21][22][23][26][27][28][29][30][31]. The near-horizon geometry of an extremal black hole is structured as AdS 2 × K (K is a compact space that includes the angular coordinates) and can be described by a Euclidean path integral partition function Z AdS 2 of various fields asymptotically approaching the classical near-horizon background. Z AdS 2 could be expressed in the form e −α × Z finite AdS 2 for some constant α and boundary length of the regulated AdS 2 . Then the principles of AdS 2 /CF T 1 correspondence allow the quantum entropy function formalism to identify the finite part Z finite AdS 2 as an alternative definition of the quantum degeneracy for extremal black holes from the macroscopic side. Therefore, the C local formula for extremal black holes becomes where the a 4 (x) coefficient needs to be integrated only over the extremal near-horizon geometry (of the form AdS 2 × K) by dropping all the terms proportional to the boundary of IR regulated AdS 2 [21,22,[27][28][29][30]. Now for the fluctuations of Einstein-Maxwell theories, a 4 (x) is always electro-magnetic dual invariant (i.e., the R µνρσF µνF ρσ and (F µνF µν ) 2 terms are absent in the final expression 10 ) and the theories also satisfy the R = 0 condition 11 for any background solution. Therefore, one needs only the "finite" near-horizon extremal limits of Ricci and Riemann tensor squares for calculating logarithmic corrections to the entropy of extremal Kerr-Newman black holes [30], .
The strategy of finding C local for the extremal Kerr-Newman family of black holes involves the following algorithm - 4. For extremal Kerr (Q = 0, J = 0) and Reissner-Nordström (Q = 0, J = 0) black holes, set b → ∞ and b → 0 respectively in the Kerr-Newman C local result. The undetermined value of b for Schwarzschild black holes (Q = 0, J = 0) justifies that the extremal-Schwarzschild limit is not possible.
Strategy B (for non-extremal black holes): In Strategy A, the quantum entropy function formalism exactly predicts the degeneracy of extremal black holes by alternatively defining a near-horizon partition function. But for the generic non-extremal Kerr-Newman family of black holes, we cast the Euclidean gravity approach developed in [32] where a special treatment 12 is used to extract out the particular black hole partition function by eliminating the thermal gas contribution of all particles (massless and massive) present in the theory. This treatment effectively leads to the logarithmic corrections for a particular choice of integration range of the heat kernel time s and writes the following C local formula for non-extremal black holes where a 4 (x) needs to be integrated over the full black hole geometry. Now the Seeley-DeWitt coefficient a 4 (x) encodes all the trace anomaly data related to the logarithmic corrections via (2.19) and can be written in the following form where the anomalies W µνρσ W µνρσ and E 4 are recognized as Weyl tensor square and 4D Euler-Gauss-Bonnet density, respectively, for the constant coefficients c and a (a.k.a. the central charges of corresponding conformal anomalies). For any arbitrary backgrounds, one can use the standard forms - (2.21) The standard definition of four-dimensional Euler characteristic suggests the integral of E 4 is a pure number for non-extremal black holes, i.e.,  On the other hand, the integral of W µνρσ W µνρσ over the full non-extremal Kerr-Newman geometry (2.15) can be evaluated in terms of different dimensionless ratios of black hole parameters as [32] full geometry (2.23b) Thus, we arrive at a modified working formula of C local for the non-extremal black holes, 13 (2.24) The strategy of finding C local for the non-extremal Kerr-Newman family of black holes involves the following algorithm -1. Calculate a 4 (x) following the method of section 2.3.1, and express them only into R µνρσ R µνρσ and R µν R µν invariants of the Einstein-Maxwell backgrounds.  We, therefore, have all the necessary ingredients for evaluating logarithmic entropy corrections for all the Kerr-Newman family of black holes. The "Strategy A" and "Strategy B" provide the local corrections (C local ) for the extremal and non-extremal cases, respectively. 14 The zero-mode corrections (C zm ) can be extracted using the inputs of section 2.2 13 At any point, one can also solve eqs. (2.22) and (2.23) for the integrals of RµνρσR µνρσ and Rµν R µν over the Kerr-Newman geometry and proceed with the primary formula (2.19).
14 In principle, one can use "Strategy B" for the extremal Kerr-Newman black holes. But it will be challenging to achieve the finite integration values of WµνρσW µνρσ and E4 (or RµνρσR µνρσ and Rµν R µν ) in the extremal limit M = Q 2 + b 2 . In contrast, the quantum entropy function formalism [35][36][37] used in "Strategy A" is a popular trick that easily determines the logarithmic entropy corrections for extremal black holes using finite near-horizon limits of the relevant background-geometric invariants.
in the general formula (2.9c). Finally, the central formula (2.9a) evaluates the necessary logarithmic correction results for the Kerr-Newman family of black holes. It is important to highlight that the whole framework 2.3 of calculating logarithmic corrections does not rely on supersymmetry and hence entirely appropriate for all extremal and non-extremal black holes in supergravity embedded Einstein-Maxwell theories.
3 Seeley-DeWitt coefficients and logarithmic entropy corrections in the "minimally-coupled" Einstein-Maxwell theory A pure or simple EMT casts a vector field A µ coupled minimally to metric g µν in four dimensions via the action (G N = 1/16π), where R is the Ricci scalar constructed from g µν and whereF µν = ∂ µĀν − ∂ νĀµ is the background field strength, R µν and R are background Ricci parameters induced fromḡ µν . We now turn to a generalization of the simple EMT: a massless scalar field φ, an additional massless vector field a µ , a massless spin-1/2 Dirac field λ and a massless spin-3/2 Rarita-Schwinger field ψ µ (Majorana form) are minimally coupled to the pure Einstein-Maxwell system (3.1). The generalized theory is structured such that all the additionally-coupled fields must fluctuate around the background of the pure Einstein-Maxwell system for the requirement of sharing the common Kerr-Newman family of solutions. The action describing the resultant d = 4 "minimally-coupled" Einstein-Maxwell theory, denoted as S EM(mc) , can be structured by coupling the free actions of the massless fields minimally to the pure Einstein-Maxwell action (3.1), and γ µρν is an antisymmetrized product 16 of Euclidean gamma matrices γ µ which follow the 4D Clifford algebra (with the identity matrix I 4 ), For investigating quadratic fluctuation of the content in the "minimally-coupled" EMT (3.3), we consider the following fluctuations - 15 The derivation of the Einstein equation (3.2) is provided in appendix B. 16 In our convention, the antisymmetrized products of gamma matrices are defined as γ α 1 α 2 ...αn = 1 n! P (−1) P γ α 1 γ α 2 · · · γ αn , where P stands for the type (i.e., even or odd) of permutations.
As a result, the "minimally-coupled" EMT (3.3) satisfies the same Einstein equation (3.2) and the other equations of motion as the simple EMT (3.1) for the common background solution (ḡ µν ,Ā µ ). We then execute the quadratic fluctuation of the action (3.3). The particular kind of couplings allows us to distribute the fluctuationsξ m = {h µν , a µ , φ, a µ , λ, ψ µ } into various sectors, where the quadratic-fluctuated Einstein-Maxwell sector δ 2 S EM and the additionally-coupled field sectors δ 2 S scalar , δ 2 S vector , δ 2 S Dirac , δ 2 S RS are expressed as well as analyzed in sections 3.1 to 3.5. Furthermore, the "minimally-coupled" EMT does not give rise to new black holes beyond the Kerr-Newman family of solutions, and hence the minimally-coupled fields provide additional contributions to the logarithmic correction results of the pure Einstein-Maxwell system. Our purpose is to compute all these logarithmic correction contributions for the Kerr-Newman family of black holes in both extremal and non-extremal limits. For that, we need to analyze the quadratic fluctuated action components and evaluate the Seeley-DeWitt coefficients. Following the prescription of section 2, we now pursue this direction further. Note that all the significant terms and data relevant to the pure Einstein-Maxwell sector and the additionally-coupled scalar field, vector field, spin-1/2 Dirac field, spin-3/2 Rarita-Schwinger field sectors are respectively labeled by "EM", "scalar", "vector", "Dirac" and "RS".

Contributions of the Einstein-Maxwell sector
Investigating the Einstein-Maxwell part of the action (3.3) via the Seeley-DeWitt approach is not an easy task; it involves a lengthy but systematic process. The initial challenge is preparing the quadratic order fluctuated action for the fluctuations (3.5) and expressing it into the prescribed Laplace-type form (2.10). Then one needs to encounter a mountain of tedious trace calculations. For the quadratic fluctuationsξ m = {h µν , a µ }, the Einstein-Maxwell sector in the action (3.3) can be decoupled into two separate subparts, where δ 2 S Ricci and δ 2 S Maxwell respectively denote the quadratic fluctuated Ricci scalar part and Maxwell part: The Ricci scalar part We begin by expressing the standard form of Christoffel symbol Γ ρ µν in terms of the fluctuated g µν and g µν , where the background Christoffel symbolΓ σµν and the covariant derivative D ρ operating on the metric fluctuation h µν are defined as (3.10) One can now execute the product (3.9) by eliminating all the terms higher than second order of the fluctuation h µν and express, which adjusts the expression of Ricci tensor R µν = g ρσ R σµρν up to quadratic order by avoiding all the total derivative terms as Here R µν is the background Ricci tensor induced fromΓ ρ µν . Further, contracting the R µν expression (3.12) by the fluctuated g µν form (3.6), one can achieve a simplified quadratic fluctuated Ricci scalar R form as where R =ḡ µν R µν is the background Ricci scalar. We, therefore, obtain, where only second-order product terms of the metric fluctuation h µν are considered. At this stage, we can choose a harmonic gauge which provides the following simplified, gauge-fixed and quadratic fluctuated form of the Ricci scalar part (3.8b) Derivation of the above quadratic fluctuated form involves the elimination of all total derivative terms, the use of the commutation relation (2.13e), and the condition R = 0 for the "minimally-coupled" EMT background.

The Maxwell part
With the help of fluctuations (3.6) and considering up to the second-order fluctuated terms for the fluctuationsξ m = {h µν , a µ }, we obtain where only second-order product terms of the metric and gauge field fluctuations are considered. Furthermore, up to a total derivative and with the help of a µ commutation relation (2.13b), one can also express, After substituting the forms (3.17) and (3.18), we gauge fix the action (3.8c) by choosing a Lorenz gauge D µ a µ = 0 in the form of the following gauge-fixing term 17 and obtain the simplified, gauge-fixed and quadratic fluctuated form of the Maxwell part (3.8c) as We now need to recombine δ 2 S Ricci (3.16) and δ 2 S Maxwell (3.20) parts in order to present the Einstein-Maxwell sector's contribution in the quadratic fluctuation of the action (3.3), We further perform multiple customizations over the above form to extract the necessary Laplace-type operator Λ. This includes bypassing the kinetic term h µ µ D ρ D ρ h ν ν via casting an effective metric, neglecting all total derivative terms, and considering all symmetric properties of the fluctuations and their all possible pairs. All these lead to the following Laplace-type structure for the Einstein-Maxwell fluctuationsξ m = {h µν , a µ }, where I, P and ω ρ (= 1 2 N ρ ) hold the forms, According to the formulae (2.12), the above data provide us the following results for E and Ω ρσ where the appropriate forms of (ω ρ ) hµν aα and (ω ρ ) aαhµν in the result (3.24b) can be arranged from eq. (3.23f). From here, our next challenge is to calculate the crucial trace values tr(I), tr(E), tr(E 2 ) and tr(Ω ρσ Ω ρσ ) as urged by the formulae (2.14) for finding the Seeley-DeWitt coefficients. Appendix C contains a detailed outline of these lengthy trace calculations. In terms of background invariants, the trace results are recorded as tr(I) = 14,  In addition, we must need to include appropriate ghost fields for countering the effect of gauge-fixing terms (3.15) and (3.19). All these ghost fields can be described via a combined action [20], which further yields the following results for E and Ω ρσ   Employing the Seeley-DeWitt and trace anomaly data into "Strategy A" and "Strategy B", we find the local corrections to the extremal and non-extremal Kerr-Newman black hole entropy due to the Einstein-Maxwell sector. The results are .

Contributions of the minimally-coupled scalar field
In the fluctuation of the action (3.3), the quadratic fluctuated part of the minimally-coupled massless scalar field φ is where we can adjust up to a total derivative, and reexpress into the following form of the Laplace-type operator Λ (3.36) Comparing it with the standard schematic (2.10), we read off P = N ρ = 0 and then write down the results for the matrices I, E, ω ρ and Ω ρσ , Inserting all the trace data in the formulae (2.14), we find the first three Seeley-DeWitt coefficients for the massless scalar field fluctuation as   Table 2: Logarithmic correction contributions (∆S BH ) of the massless scalar field to the entropy of Kerr-Newman family of black holes in the d = 4 "minimally-coupled" EMT.

Contributions of the minimally-coupled vector field
The quadratic fluctuated part of the additionally-coupled vector field a µ in the fluctuation of the action (3.3) is presented as where we have omitted all total derivative terms and also used the commutation relation (2.13b) for the vector field fluctuation a µ . We then gauge fix the action (3.41) by incorporating the gauge fixing term, The gauge-fixed action (without ghost) provides the desired Laplace-type operator Λ as yielding (N ρ ) a µ a ν = 0, P a µ a ν = −R µν in the schematic (2.10). With the aid of the formulae (2.12), one obtains results for the useful matrices,   The particular kind of gauge-fixing term (3.42) demands to account for the following ghost action [20] Table 3: Logarithmic correction contributions (∆S BH ) of the massless additionally-coupled vector field to the entropy of Kerr-Newman family of black holes in the d = 4 "minimallycoupled" EMT.

Contributions of the minimally-coupled spin-1/2 Dirac field
In the quadratic fluctuation of the action (3.3), the contribution of the minimally-coupled massless Dirac spinor λ is where The Seeley-DeWitt coefficients for the fluctuation of an elementary free spin-1/2 Dirac field around an arbitrary background have already been reported in our earlier work [60]. We find it worthy of reviewing the work [60] in order to obtain necessary results from the quadratic fluctuated action (3.51). Unlike bosons, the quadratic fluctuations of fermions are characterized by first-order operators. After correctly identifying the first order Diractype operator / D for the spin-1/2 fluctuation (3.51), we structure the required Laplace-type operator Λ as following where we have employed various gamma matrix identities, the spin-1/2 commutation relation (2.13c) and the R = 0 condition for the Einstein-Maxwell background. The particular form of Λ yields P = N ρ = 0, which further expresses the forms of matrices defined in eq. (2.12) as (3.54) We then compute the following trace results tr(I) = 4, tr(E) = 0, tr(E 2 ) = 0, and find the first three Seeley-DeWitt coefficients for the minimally-coupled spin-1/2 Dirac field fluctuation, Finally, the logarithmic correction contribution of the minimally-coupled spin-1/2 Dirac field to the entropy of extremal and non-extremal Kerr-Newman black family of holes are calculated as in table 4. In these corrections, the minimally-coupled spin-1/2 Dirac spinor contributes nothing to the C zm formula (2.9c).

Contributions of the minimally-coupled spin-3/2 Rarita-Schwinger field
The quadratic fluctuated part of the minimally-coupled massless Rarita-Schwinger field ψ µ (Majorana type) is expressed as where one can adjust, with the following gauge-fixing term for casting the gauge γ ρ ψ ρ = 0  Table 4: Logarithmic correction contributions (∆S BH ) of the massless spin-1/2 Dirac field to the entropy of Kerr-Newman family of black holes in the d = 4 "minimally-coupled" EMT.
contribution for the particular gauge-fixing term chosen. But we aim to calculate the complete (gauge-fixed (3.60) and ghost (3.65)) Seeley-DeWitt coefficient results for the quadratic fluctuated minimally-coupled spin-3/2 Majorana spinor (3.58). After combining (3.58) and (3.59), the gauge-fixed action extracts a first-order Dirac-type 19 operator / D that further structures the second-order Laplace-type operator Λ as following where various gamma identities, spin-3/2 commutation relation (2.13d), and R = 0 condition are employed to obtain the last line equality. As per the schematic (2.10), we read off N ρ and P , and then find the matrices I, ω ρ , E and Ω ρσ ,

(3.64)
It is also customary to include a ghost action that counters the gauge-fixing term (3.59), whereb,c, andẽ are three minimally-coupled bosonic ghosts (i.e., spin-1/2 Majorana spinors) [20]. The combined Seeley-DeWitt contribution of these ghosts is equivalently obtained as a RS,ghost where the minus sign is included for the reverse spin-statistics of ghosts, 3 serves the ghost multiplicity and 1/2 factor converts the single spin-1/2 Dirac results a Dirac 2n (recorded in eq. (3.56)) into the Majorana kind. After combining a RS,no-ghost 2n and a RS,ghost 2n contributions, the complete Seeley-DeWitt results for the minimally-coupled Rarita-Schwinger field fluctuation are  In the end, we utilize the a RS 4 (x) and trace anomaly data and achieve the logarithmic correction contributions of the minimally-coupled Rarita-Schwinger field to the entropy of extremal and non-extremal Kerr-Newman family of black holes. The results are presented in table 5. These corrections do not receive any zero-mode contribution from the nonsupersymmetric Rarita-Schwinger field, as discussed in section 2.2.

Discussions
This section generalizes the d = 4 "minimally-coupled" EMT further by coupling any arbitrary numbers of massless fields, which leads to a set of generalized Seeley-DeWitt coefficient and logarithmic correction formulae for all the extremal and non-extremal Kerr-Newman family of black holes. We then employ the generalized "minimally-coupled" data in successful derivation of the logarithmic corrections to the entropy of extremal black holes in N ≥ 2, d = 4 Einstein-Maxwell supergravity theories. Finally, we conclude by summarizing and discussing the results.

(4.2)
The data (4.1) and (4.2) provide the following local corrections to the entropy of extremal and non-extremal Kerr-Newman black holes On the other hand, the zero-mode correction only includes the contribution from metric in the formula (2.9c) and should be added to the Einstein-Maxwell sector. For Kerr-Newman black holes, The logarithmic corrections to the entropy of Kerr-Newman black holes in the generalized "minimally-coupled" EMT are × 13n EM − n 0 2 + 44n 1 + 2n 1/2 + n 3/2 + 1440 ln A H , (4.5a) 822n EM + 3n 0 + 36n 1 + 18n 1/2 − 231n 3/2 − 90 ln A H . (4.5b) In particular limits (refer to section 2.3.2) of the Kerr-Newman results, one can extract the logarithmic entropy corrections for Kerr and Reissner-Nordström black holes as below: For the extremal and non-extremal Kerr black holes, we need to set b → ∞ and Q = 0 in the C local results (4.3a) and (4.3b), respectively. This yields exactly the same extremal and non-extremal local corrections for the Kerr black hole, The corresponding zero-mode corrections are obtained as For the extremal and non-extremal Reissner-Nordström black holes, we need to set b → 0 and b → 0 in the C local results (4.3a) and (4.3b), respectively. This provides extremal and non-extremal local corrections for the Reissner-Nordström black holes, The corresponding zero-mode corrections are obtained as  424n EM + n 0 + 62n 1 + 11n 1/2 − 229 2 n 3/2 + 540 ln A H , For the Schwarzschild black holes, we need to set Q = 0 in eq. (4.3b), which yields the same C local as the non-extremal Kerr result (4.6). Also, Schwarzschild black hole is non-rotating and hence the zero-mode correction C zm is the same as the non-extremal Reissner-Nordström result (4.10b). Hence, the logarithmic correction to the entropy of Schwarzschild black hole in the generalized d = 4 "minimally-coupled" EMT is (4.12) The above results are our principal focus in this work. The corrections for extremal black holes (eqs. (4.5a), (4.8a) and (4.11a)) are novel reports to the literature. The nonextremal black hole results (eqs. (4.5b), (4.8b), (4.11b) and (4.12)) agree with [32], where the a 4 (x) formula 20 is managed from a set of secondary data provided in [52][53][54][55][56]. Also, the Schwarzschild corrections due to vector and spin-3/2 Rarita-Schwinger fields in eq. (4.12) are entirely consistent with [57], where results are obtained via the tunneling approach.

A local method for logarithmic correction to the extremal black hole entropy in N ≥ 2 Einstein-Maxwell supergravity
Logarithmic entropy corrections for the black holes in Einstein-Maxwell embedded supergravity theories are already reported in various works [21-28, 33, 61]. The basic technical approach is common -analysis of relevant quadratic fluctuated action via the heat kernel method. But supergravity actions are incredibly complicated and unwieldy. The direct calculations are hardly manageable and so far only accomplished for N = 1, 2, 4, 8 Einstein-Maxwell supergravity theories (EMSGTs). In [24,28,33,61], the results are estimated for all N ≥ 3, d = 4 EMSGTs by considering the N ≥ 3 → N = 2 decompositions. For the derivations of the minimally-coupled EMSGT sectors, the "minimally-coupled" EMT results 20 One can find a mismatch of the Rµν R µν coefficient for spin-3/2 fields with [32]. But this is justified because the coefficient of Rµν R µν for the spin-3/2 field is not absolute, rather gauge dependent (while the coefficient of RµνρσR µνρσ is gauge-independent) [53]. After choosing the gauge γ ρ ψρ = 0, the coefficient of Rµν R µν in 360(4π) 2 a4(x) will always be 4 for a spin-3/2 Majorana spinor, which is also consistent with [54].
of the previous section are found to have essential utility [28]. But, the quadratic fluctuated supergravity actions mostly include "non-minimal" coupling terms [21,22,26,28,33]. Consequently, one can not directly employ the "minimally-coupled" EMT data in the full reproduction of logarithmic entropy corrections in the EMSGTs. However, we find an alternative but indirect way of getting a supersymmetrized form of the a 4 (x) formula (4.1) for near-horizon backgrounds, which can be utilized for the full reproduction of logarithmic corrections to the entropy of extremal black holes in N ≥ 2, d = 4 EMSGTs. The entire approach is depicted as follows.
At first, we need to express the a 4 (x) formula (4.1) in terms of the Weyl tensor square W µνρσ W µνρσ and Euler density E 4 (using the trace anomaly form (2.20)), The supersymmetric completion of the above expression entirely depends on the invariants W µνρσ W µνρσ and E 4 . The Euler density E 4 is a topological invariant and hence selfsupersymmetric. On the other hand, it is well reported that the supersymmetrization of W µνρσ W µνρσ , evaluated on near-horizon black hole backgrounds [62][63][64][65][66], is surprisingly found to be the same as E 4 [67,68]. Again, the quantum entropy function formalism [35][36][37] used in "Strategy A" requires only the near-horizon details of extremal black holes.
All that mentioned suggests a simple and straightforward way to supersymmetrize any theory by analyzing only the Gauss-Bonnet term E 4 as well as bypassing the need for Weyl tensor square term W µνρσ W µνρσ and then achieve logarithmic entropy corrections for the extremal black holes in supergravity theories. As discussed, substituting W µνρσ W µνρσ = E 4 in (4.13) will lead to the supersymmetrized a 4 (x) formula for near-horizon backgrounds in N ≥ 2, d = 4 EMSGTs, (4.14) This necessarily yields a C local formula (via the "Strategy A") for the extremal Kerr-Newman black holes, where the equality is obtained after using the extremal near-horizon limits (2.18) in the form of E 4 . In addition, one obtains the following zero-mode corrections using the formula (2.9c) for Kerr-Newman, for Reissner-Nordström. (4.16) Note that extremal Reissner-Nordström black holes with near-horizon geometry AdS 2 × S 2 are the only possible BPS solutions in the four-dimensional N ≥ 2 EMSGTs. Summing all up, we write a combined logarithmic entropy correction formula that reads as follows By setting particular multiplicity values in the above formula, one can extract logarithmic corrections to the entropy of extremal Kerr-Newman, Kerr (b → ∞) and Reissner-Nordström (b → 0) black holes in all N ≥ 2, d = 4 EMSGTs. For a matter coupled N = 2, d = 4 EMSGT, with the supergravity multiplet (n EM = 1, n 3/2 = 2) coupled to n V vector multiplets (n 0 = 2, n 1 = 1, n 1/2 = 1) and n H hyper multiplets (n 0 = 4, n 1/2 = 1), one obtains Substituting the above relation into the formula (4.17), we get All these logarithmic entropy corrections (eqs. (4.18) and (4.20)) exhibit perfect matching with the available direct approach results in [21-24, 26, 28]. The whole process of estimating logarithmic entropy corrections for extremal Kerr-Newman, Kerr and Reissner-Nordström black holes in N ≥ 2, d = 4 EMSGTs is an indirect "local method". It is strictly limited to the analysis of extremal black holes via "Strategy A". As compared to the direct approaches, it is much simpler because one needs not required to deal with overly complicated quadratic fluctuated supergravity actions and execute a mountain of complex trace calculations.

Summary and conclusions
To sum up, we have provided a consolidated manual for investigating logarithmic correction to the entropy of Kerr-Newman family of black holes for both extremal and non-extremal limits. The whole framework is divided into two separate strategies ("Strategy A" and "Strategy B") based on the Euclidean gravity approaches [32,[35][36][37]. Seeley-DeWitt coefficients are found to be the crucial and common ingredients in these strategies, where a standard method [47] computes them in various background invariants. Following this manual, we have calculated the first three Seeley-DeWitt coefficients for the fluctuations of the generalized d = 4 "minimally-coupled" EMT and employ them in obtaining logarithmic correction to the entropy of extremal and non-extremal black holes in the theory. The investigation of a global platform for simultaneously calculating logarithmic corrections to the entropy of all the Kerr-Newman family (Kerr-Newman, Kerr, Reissner-Nordström and Schwarzschild) of black holes in both extremal and non-extremal limits is novel. The extremal black hole results are mostly new reports. The non-supersymmetric "minimallycoupled" data, generalized for arbitrary numbers of minimally-coupled fields, have essential utility in alternative derivations of the results for different N ≥ 2, d = 4 EMSGTs via both direct (e.g., [28]) and indirect local methods (see section 4.2).
This work reported the presence of both the geometric dependent as well as fully universal logarithmic entropy corrections ∆S BH for the black hole solutions in the d = 4 "minimally-coupled" EMT. Our analysis showed that the quantum corrections for the charged (Kerr-Newman and Reissner-Nordström) black holes are geometric via the parameters Q, b, b , β, r H , while ∆S BH for the uncharged (Kerr and Schwarzschild) black holes have no dependence on their parameters, i.e., universal in both extremal and non-extremal limits. But, we found an exception for the extremal Reissner-Nordström black holes that possess a universal form of logarithmic entropy corrections. Almost a similar outcome is exhibited by the extremal black holes in N ≥ 2, d = 4 EMSGTs. However, the extremal Kerr-Newman results show a slight deviation: universal only in N = 3 EMSGT but have geometric dependence in all other N ≥ 2 EMSGTs. Any progress in searching the reason behind the typical patterns of ∆S BH would be welcome. We have not witnessed any vanishing ∆S BH for black holes in the d = 4 "minimally-coupled" EMT, as reported for extremal Reissner-Nordström black holes in the N = 4, d = 4 EMSGT. One should not worry about the logarithmic correction contributions that are found to be negative. The total quantum corrected black hole entropy A H 4G N + ∆S BH is always positive due to the presence of the leading positive Bekenstein-Hawking term A H 4G N in the large charge limit. All the calculated ∆S BH results as well as their particular characteristics are significant and can be served as 'macroscopic experimental data' to understand the microstructure of the general black holes in any quantum theory of gravity in the future.
But for the case of zero modes, the functional integral (A.4) cannot sustain its Gaussian form, and hence one needs to remove the zero modes from Z 1-loop to carry out the heat kernel treatment of W. However, we can add their contribution back by substituting the zeromode part of Z 1-loop with ordinary volume integrals over different asymptotic symmetries that induce the zero modes [20][21][22]29]. As a result, the one-loop partition function Z 1-loop is disintegrated into a product of two separate parts, where det Λ is the determinant over non-zero modes of Λ and Z zero (L) is the zero-mode integral that scales non-trivially with an overall length scale L of the background metric. For this choice of scaling, the non-zero eigenvalues of the Laplace-type operator Λ scale as L −2 , which essentially sets a new rescaled heat kernel parameters = s/L 2 of the integration range /L 2 s 1 (or equivalently s L 2 ). Then, using the Seeley-DeWitt expansion (2.7) in the relation (2.5), we express the non-zero mode contribution to the one-loop effective action as (A.6) Here "· · · " represents all non-logarithmic terms containing the other Seeley-DeWitt coefficients. On the other hand, the zero mode contribution to the path integral from all fluctuationsξ m , each having n 0 ξm zero modes, can be represented as [20][21][22]29] Z zero (L) = L ξ m χβξ m n 0 ξm Z 0 , (A.7) where Z 0 does not scale with L. βξ m are numbers that depend on the type of fluctuation and space-time dimensions. In a d-dimensional theory, it is found that β 1 = d−2 2 for vector, β 2 = d 2 for metric, β 3/2 = d − 1 for gravitino, etc. Finally, substituting both (A.6) and (A.7) contributions in the relation (A.5) and using n zm = ξ m n 0 ξm , we write the following restructured one-loop effective action form where the first part containing a 4 (x) coefficient is termed as the "local" contribution and the second part controlled by the zero-mode parameters βξ m , n 0 ξm is recognized as the "zeromode" contribution. We refer the readers to [23,75] for more details regarding the above analysis.
If W is identified as the one-loop quantum effective action to the partition function describing the macroscopic horizon degeneracy of a black hole with horizon area A H , then the form (A.8) provides the particular logarithmic correction formula (2.9). One needs to consider only the terms proportional to ln A H in the relation ∆S BH = ln Z 1-loop = −W for the large-charge limit A H ∼ L 2 .
Note on the strategy for non-extremal black holes: In the Euclidean gravity approach [32], the non-extremal black holes are in equilibrium with thermal gas present in the theory. In order to identify only the particular piece of black hole partition function, one must subtract the thermal gas contributions. The special treatment is to consider two black hole solutions (same type but different scaling) in the same theory -System 1: black hole with the length parameter a confined in a box of size ζ, System 2: black hole with the length parameter a confined in a box of size ζa /a.
It is argued in [32] that the thermal gas contribution remains invariant in both systems and hence, the difference between corresponding one-loop effective actions becomes Now the eigenvalues of system 2 are scaled in terms of those in system 1 as which appropriately fits the ∆W into the form (A.6) for an upper integration limit = /L 2 with L = a /a. This finally leads us to the same effective action form (A.8) and logarithmic correction formula (2.9) for the non-extremal black hole after extracting only the terms proportional to ln a 2 . The above note is mostly based on [32,76]. 21
The above results produce the exact list (3.25) via the definitions (C.1).