Supersymmetric Localization for BPS Black Hole Entropy: 1-loop Partition Function from Vector Multiplets

We use the techniques of supersymmetric localization to compute the BPS black hole entropy in N=2 supergravity. We focus on the n_v+1 vector multiplets on the black hole near horizon background which is AdS_2 x S^2 space. We find the localizing saddle point of the vector multiplets by solving the localization equations, and compute the exact one loop partition function on the saddle point. Furthermore, we propose the appropriate functional integration measure. Through this measure, the one loop determinant is written in terms of the radius of the physical metric, which depends on the localizing saddle point value of the vector multiplets. The result for the one loop determinant is consistent with the logarithmic corrections to the BPS black hole entropy from vector multiplets.


Introduction
A consistent theory of quantum gravity should be able to provide the statistical interpretation of Bakenstein-Hawking entropy which is given by one quarter of the area of the horizon in Planck units [1,2]. String theory being a candidate for the quantum theory of gravity provides a natural framework to study classical and quantum properties of black holes. In last decade there has been tremendous progress in this direction in the large cases of supersymmetric extremal black hole after the work of Strominger and Vafa [3]. In particular, now we have a very good understanding of statistical degeneracy for a large class of supersymmetric extremal black hole in N = 4 and N = 8 string theory which in the thermodynamic limit reduces to Bakenstein-Hawking entropy [4][5][6][7][8][9][10][11][12][13][14]. In order to extend this comparison beyond thermodynamic limit, one needs to understand how to compute the corrections to Bakenstein-Hawking entropy in both microscopic and macroscopic level. In a quantum theory one would expect that both the microscopic and macroscopic entropy will receive corrections from perturbative and non perturbative effects. At the microscopic level understanding, these corrections involves computation of degeneracy to a greater accuracy and it's asymptotic expansion [15,16]. On the other hand at the macroscopic level, one needs a full quantum generalisation of the entropy formula. The area law is generalized to Wald entropy formula [17,18] to take into account the higher order derivative corrections which include the α -corrections in string theories. For the single centered extremal black case, the formula was further generalized by Sen [19,20] based on AdS 2 /CFT 1 . The extremal black hole has the AdS 2 factor in it's near horizon geometry, so it is of the from, AdS 2 × K, in 4 spacetime dimensions. Here, K becomes S 2 for the supersymmetric case because the supersymmetry requires the extremal black holes to be spherically symmetric. According to this proposal, the full quantum entropy associated with the horizon of an extremal black hole is given in terms of expectation value of Wilson loop at the boundary of the AdS 2 . The proposal takes the form, where < > f inite AdS 2 denotes the finite part of unnormalised Euclidean path integral and the quantum entropy associated with the horizon is given by S hor (p, q) = ln W (p, q). (1.2) Since the proposal involves the path integral over all fields including the metric, there is no notion of fixed background. But, as is denoted by the subscript in (1.1), the boundary condition is fixed by the attractor values of the black hole background, which is the AdS 2 geometry. The Wilson loop wraps the boundary of AdS 2 . The insertion of the Wilson line at the boundary means that we change the boundary condition from Dirichlet to Neumann condition for gauge field. Neumann condition fixes the electric fields at the boundary i.e. electric charges and hence the proposal computes the entropy in the microcanonical ensemble. Further, we need to extract finite part of the functional integral to see the physically meaningful quantity. Since there is an IR divergence due to the infinite volume of AdS 2 space, the IR divergence should be removed by regularization and the holographic renormalization. The classical limit of this partition function reduces to exponential of Wald entropy. Furthermore, one can use this proposal to compute the full quantum corrections. It includes not only α correction but also g s quantum correction as well as the non perturbative corrections to the entropy. To compute this, one has to integrate over all string fields on each saddle point. Since this integral over all string fields is quite difficult and challenging, the strategy we follow is to first integrate out all massive KK modes and stringy modes, and write down a Wilsonian effective action. This effective action will be given in terms of few massless supergravity fields and include all higher derivative corrections together with non perturbative corrections coming from world sheet instantons. Thus we are left with the path integral over massless fields with the above boundary conditions and we takes this as our starting point. By computing the path integration, the proposal of the quantum entropy function has been tested. Perturbative calculation on a classical saddle point and comparing it with the similar expansion on the microscopic side has led to perfect match of logarithmic correction in case of BPS black hole in N = 4 and N = 8 supergravities [21][22][23][24] in 4-dimensions and BMPV black hole in 5-dimensions [25].
The computation of the path integral can also be performed by using, so called, supersymmetric localization. It is a powerful method, making the exact computation possible in a supersymmetric theory. This method has been used quite successfully in the cases of supersymmteric gauge theories in various dimensions and on various compact manifolds [26][27][28][29][30][31]. The argument of the localization principle is so general that this principle can also be applied to the supergravity computation. The argument of the supersymmetric localization is following [32]. Let us suppose that Q be a fermonic symmetry which gives rise to a compact bosonic symmetry, We would like to compute an integral of some Q invariant function h and Q invariant action S, where we let the measure dµ is also invariant under the Q. We deform a partition function by adding the Q-exact function QV with parameter t, Z t = dµ h e −S−tQV . (1.5) where V is a fermionic function and invariant under the H-transformation. Since the action S, measure dµ and the localization action are invariant under the the supersymmetry Q, the modified partition function Z t is independent of the parameter t. d dt Z t = − dµ QV h e −S−tQV = − dµ Q V h e −S−tQV = 0 .
(1. 6) In the limit t → ∞, the semiclassical approximation with respect to 1/t is exact.
One left with the integration over the submanifold M Q where M Q is the manifold where QV = 0 and dµ Q is the induced measure on the submanifold M Q . For supergravity case, a rigid supersymmetry parameter can be chosen, where the Q 2 should preserve the asymptotic boundary conditions. In the case of the black hole entropy, we choose a Killing spinor of AdS 2 ×S 2 . The supersymmetric localization principle requires the off-shell closure of the supersymmetry algebra. The N = 2 supergravities coupled to vector multiplets in 4-dimensions has an off shell formulation in terms of conformal supergravity [33][34][35]. It is a gauge theory, where all the N = 2 superconformal symmetries are promoted to the local symmetries, which coupled to the matter fields, and gauge equivalent to the Poincare supergravities. The Weyl multiplet having off-shell degrees of freedom includes the gauge fields for all the local symmetries, where the graviton and gravitini are contained. To have the degrees of freedom for N = 2 Poincare supergravity, one needs to add additional matter multiplets which is called compensating multiplets. One of the advantages of this formulation is that the off-shell supersymmetry algebra does not depend on the choice of prepotential and as a result the solution for the localization equations and the computation of one loop partition function do not depend on the details of prepotential.
To utilize the advantage of the conformal supergravity, we use the freedom of a choice of the gauge condition. Note that the metic g µν in Weyl multiplet is not the physical metric and conformaly related to the metric in Einstein frame G µν , where K(X,X) is the Kähler potential that is function of the scalars in the vector multiplets. A conventional gauge for the scale symmetry is choosing the e K = 1, and it constrains the the degree of freedom of n v + 1 scalars. Instead of this gauge, we use another choice: the radius of the AdS 2 ×S 2 metric g µν to be constant, and all the n v + 1 scalars to be free to fluctuate. Throughout this paper, we follow this gauge choice. Note that the conformal mode of the physical metric G µν is encoded in the fluctuating scalars in vectormultiplets.
The application of the supersymmetric localization to quantum entropy function was initiated in [36][37][38]. In the work of [37], the authors consider 1 8 th BPS black hole in N = 8 supersymmetric string theory for which microscopic answer is known. After considering truncations of N = 8 supergravity to N = 2 supergravity with only vector multiplets, and assuming that the one loop determinant coming from localizing action is trivial, they find that the on shell action evaluated on the localization solutions together with proper path integral measure itself reproduces the modified Bessel function, which is the microscopic answer for 1 8 th BPS black hole in N = 8 theory. The agreement with microscopic answer is remarkable, however we still need to understand the assumptions taken in this process. The integration measure should be the result of the one loop determinant coming from all the multiplets including Weyl multiplets and gravitini multiplets.
In this paper, as a first step, we focus on the fluctuations of n v + 1 vector multiplets and compute the Z 1−loop while keeping the Weyl multiplet and all other multiplets to their classical near horizon background. One of the important issues which needs a very careful analysis, is the choice of reality properties for the fields in the Euclidean theory. Another issue in doing such a computations is the right choice of path integral measure. In the ref. [36,37,39] the authors made a choice of the analytic continuation for the fields in the vector multiplet which is consistent with the supersymmetry transformation. However in our paper, we will follow the analytic continuation motivated from the work of [26,31] and the principle of localization. At present it is not clear to us which one is the right choice of analytic continuation. Probably the complete analysis in presence of Weyl multiplet and the comparison with a microscopic answer might able to fix the right choice of analytic continuation and also the path integral measure.
We summarize our results here. We first find the solutions of the localization equations using our choice of reality properties and find that the solutions of localization equations are labelled by 2 real parameters for each vector multiplet. This is in contrast to the solutions obtained in [36,39] where the solutions are labelled by one real parameter for each vector multiplet. We then compute the determinant of the quadratic fluctuations of the Q-exact deformations about the localization solution. Since we are dealing with the abelian vector multiplets, the answer does not seem to depend on the parameters of the localization solutions. Also, since the off shell supersymmetry transformations for the fields involve unphysical metric which has dilatation weight −2, our answer of the one loop determinant with the ordinary measure is not scale invariant. However, given that our calculation is in conformal supergravity where all the symmetries are realised as gauge symmetry, one would expect that with the gauge invariant measure the one loop determinant should be scale invariant. We propose the scale invariant path integral measure involving vector multiplet field including ghost fields. With the proposed measure we find that the answer do depends on the localization solution through the physical metric which is scale invariant.
The organization of the paper are as follows. In section 2, we describe N = 2 vector multiplet. By taking the Euclidean continuation starting from Minkowskian supergravity, we describe the vector multiplet on Euclidean background and the reality properties of the fields in the vector multiplet. We then describe the supersymmetry algebra on AdS 2 × S 2 background and our choice of localization supercharge. In section 3, we describe the localization Lagrangian and the solution of localization equations. In section 4, we compute the one loop determinant about the localization background by computing the index using Atiyah-Bott fixed point formula. In this section, we assume the trivial functional integration measure and obtain our result in terms of unphysical metric. In the next section, we propose the form of the scale invariant path integral measure and redo the calculation of the one loop determinant, and therefore our main result is expressed in terms of physical variables. We end our paper by pointing out issues and open problems in the discussion section.
Note added: While this paper was being prepared for publication, we received communication from S. Murthy and V. Reys of a paper which contains overlapping results.

Euclidean continuation
In order to get off-shell N = 2 vector multiplets in Euclidean background, we start from N = 2 conformal supergravity coupled to n v + 1 vector multiplets by setting the Weyl multiplet as a background. Here we also translate the Lorenzian signature to Euclidean signature. For the details of the conformal supergravties, convention of gamma matrices, spinors and relation to those of Euclidean signature, we refer to the Appendix A and B.
Let us see how the fermionic fields are translated to those in Euclidean signature. Since the 4 dimensional Euclidean space does not allow the Majorana spinor representation, it is useful to redefine fields in such a way that they satisfy the symplectic Majorana condition. For the chiral and anti-chiral projection of the gaugino, poincare supersymmetry parameter and conformal supersymmetry parameter, we use following redefinition, where although we keep using four component notation, we use unbarred and barred notation to denote chiral and anti-chiral projected spinors. The symplectic Majorana condition in Minkowski space is where C − is the charge conjugation matrix. They satisfy Note that the chiral and anti-chiral projection is not compatible with the (symplectic) Majorana condition, so the condition (2.2) relates the chiral spinors and anti-chiral spinors.
After hiding † operation on all spinors in the theory using the symplectic Majorana condition (2.2), the action and the supersymmetry transformation rule do not distinguish whether they are of Minkowkian or Euclidean theory. So, we are free to go to the Euclidean theory by taking analytic continuation However, we note that the property of the fermions under the complex conjugation is different. In the Euclidean 4-dimensional space, we treat the chiral and antichiral spinors as independent fields, as they are no longer related by the complex conjugate. Instead, we can impose the following reality condition, i.e. symplectic Majorana condition, for each chiral and anti-chiral spinors, where the spinors with lower SU (2) index is defined as However, while we will choose the Killing spinors for the supersymmetric localization to satisfy this reality condition, spinor fields may not strictly follow this condition because we will further impose analytic continuation in such a way that the path integration is well defined. The killing spinor equations are obtained from the variation of the gravitino, Here T ab andT ab are self-dual and anti-self-dual auxiliary tensor in Weyl multiplet. 1 And the covariant derivative includes gauge fields of both SU (2) R and U (1) R . These equations determine η i in terms of killing spinors, We also read off the auxiliary equations from variation of the auxiliary fermionic fields, χ i and φ i µ , in the Weyl multiplet,

Vector multiplets in Euclidean theory and analytic continuation
In this section, we present the vector multiplets in Euclidean theory that is compatible with N = 2 supersymmetry, and then take the analytic continuation for the contour of the path integration. N = 2 vector multiplet consist of scalars X andX, one vector field W µ , SU (2) R triplet auxiliary field Y ij and SU (2) R doublet fermion λ i . For our purpose of extremal black hole, we will only consider abelian vector multiplets. The supersymmetry transformations of the vector multiplet fields are given by

10)
Qλ Ii = 2iγ a D aX where the covariant derivatives are and F µν is defined as The square of the supersymmetry transformations are give by 14) The square of the supersymmetry (2.13) is summarized into (2.16) Note that the reality condition in (2.5) is compatible with the supersymmetry transformation if the bosonic fields and the background Weyl multiplet satisfy That is to say, the reality condition of fermions in (2.5) and bosons in (2.17) is preserved under the supersymmetry transformation rules given in (2.11). In particular, the symmetry parameters appeared in the algebra, (2.36), satisfy the following reality conditions, (2.18) Therefore, the reality condition of all the fields is preserved. Here, the fact that parameter Θ is real reflects that the abelian factor of the R-symmetry group for the Euclidean space is SO(1, 1) R , whereas the U (1) R is for the Minkowskian space.
However, we take further analytic continuation as in [26,31], so that rather than (2.17) the actual path integral will be taken along the path with the condition 2 , This is for the positive definiteness of the original action. As the Eulclidean Lagrangian is of the form L E ∼ ∂ µX ∂ µ X + Y ij Y ij , the integration contour (2.19) makes the action positive and the path integration well defined. In this analytic continuation, the abelian R-symmetry is U (1) R as of the Minkowskian theory. This reality condition is not compatible with the supersymmetry transformation. For example, the square of Q gives rise to a complexified gauge symmetry transformation as the parameterΦ in the algebra (2.36) is complex valued. Nevertheless, the argument of localization still holds because the action is invariant under supersymmetry transformations [26]. The motivation for choosing the analytic continuation in (2.19) will also be clear for the purpose of the supersymmetric localization, which will be explained later in section 3.1.

Supersymmetry on AdS 2 ×S 2
In 4-dimensions, a supersymmetric extremal black hole has near horizon geometry of the form AdS 2 ×S 2 . Also all other field configurations at the near horizon are consistent with the isometry of the AdS 2 ×S 2 . In the quantum entropy function, this background serves as the boundary condition for fields in the path integral. In the Lorentzian signature, AdS 2 ×S 2 geometry implies the following ansatz, And by the attractor equations, the constant and X I * are fixed in terms of the electric field and magnetic charges, e I * and p I * , and the complex constant ω, Solving the above equations fixes the value of the scalar field X I * in terms of electric field and magnetic charge, Using the global U (1) R rotation from the superconformal Weyl multiplet, we will set ω =ω = 4/ . Thus with this choice of ω andω, the U (1) R symmetry is explicitly broken.
In the Euclidean AdS 2 ×S 2 case, the near horizon field configurations take following form In the above we have used the r = cosh η. With the above vielbein, the non vanishing component of the spin connections are The background value of (2.23) implies that the auxiliary Killing spinor equations (2.9) become It is solved in [41] and there are 8 Killing spinors. For the purpose of the supersymmetric localization, we will choose the following two Killing spinors among them. In terms of Dirac spinor notation, 27) and in the following gamma matrix representation, These Killing spinors satisfy the the symplectic Majorana condition Our choice of charges generates the killing vector field of the compact isometry transformation as where we denote L as the rotation of the AdS 2 and J as the rotation of the S 2 . We also note that Then the symmetry parameters (2.14) are given by Therefore, the parameters (2.16) in the supersymmetry algebra (2.15) arê Note here that v µ A µ = 0 and v µ V µ i j = 0 as the background value of them are zero. It still holds when we consider the Weyl multiplet as localization saddle point [39].

Localization Lagrangian
We deform the physical action by adding the following localization Lagrangian, Here, we take overall normalization factor (ξ j ξ j +ξ jξ j ) −1 such that we will get standard kinetic terms for scalars and fermions. Note that the localization Lagrangian is by construction positive definite as it involves the dagger operation. The dagger operation should be taken carefully because it relies on which contour of integration that we choose. For the positive definiteness of the Euclidean action, we gave up the the reality condition (2.5) for fermions and performed further analytic continuation, following the contours defined in (2.19).
Our deformation action (3.1) also motivates the choice of the integration contour given in (2.19) for the purpose of the supersymmetric localization. The reality condition for bosons (2.17) and the (2.5) for fermions does not make the V in (3.1) invariant under the abelian R-symmetry, SO(1, 1) R . If one allows the fluctuation of the Weyl multiplet to consider the localization in supergravity, then Q 2 V may not be zero as the SO(1, 1) R symmetry parameter appeared in the algebra in (2.16) isΘ = v µ A µ + Θ. This beaks the principle of the supersymmetric localization in supergravity. Explicitly, we can see that the localization action is not R-symmetry invariant by looking at the expression of the localization Lagrangian, Also, these are not the standard kinetic terms for the Lagrangian having non elliptic kinetic operator. So it may causes further problem at doing path integral. Under the condition (2.19), the localization Lagrangian would have standard kinetic terms for bosons and fermions, These are invariant under the U (1) R symmetry.

Localization saddle points
To look at the localization saddle point, let us consider the bosonic part of the localization Lagrangian. After some algebra, one can rewrite the bosonic part of the localization Lagrangian (3.1) as follows 3 where the symbol ± indicates the self-dual or anti-self-dual parts such that, • The localization solution : The condition 0 = L Q b gives 6 localization saddle point equations. Classical background is the trivial solution. On top of this we find the other off shell solutions. From the first and the last two lines in (3.4), one finds the unique solution for X I 1 and Y I ij [36,39], Similarly first three lines of (3.4) provides equations for F I ab and X I 2 . We find a nontrivial smooth solution which is given by However, we cannot prove that this is the unique smooth solution but we will provide evidence in support of it in the appendix D. One important feature of the above solution is that although there is a non trivial field strength along S 2 , the total flux however is zero. Thus the magnetic charge for this off shell solution is same as the attractor value.
•At north/south pole and the origin: At the fixed points η = 0 with ψ = 0 or ψ = π, it seems that further singular solutions can be enhanced because ξ i ξ i = 0 or ξ iξ i = 0 at this point and the number of localization equations are reduced. However, we will argue that there are no nontrivial solutions that are localized at the fixed points.
Consider the point η = 0 and ψ = 0. Using the fact that and after some algebra, one finds that the localization Lagrangian reduces to From the first line, we get X I 1 and X I 2 to be constant. Since e 1 θ = e 3 φ = 0 at the north pole, we get the anti-self dual equation from the first term of the second line, Similarly, at η = 0 and ψ = π, we get the self-dual equation, However, there is no U (1) instantons in 4-dimensions, so there is no localized nontrivial solutions.
•Discrete zero modes of the gauge field Apart from the zero mode in (3.6) and (3.7), AdS 2 has further normalisable zero mode of the gauge field [42], which not only makes the localisation action but also the original action vanish. The explicit form of the discrete zero mode is given by (3.12) These are not pure gauge mode as the scalar field Φ l are not normalisable.

1-loop partition function
In this section, we compute the 1-loop partition function by computing equivariant index. For this, we introduce BRST symmetry to fix the gauge and combine it with the localization supercharge. Through out this section, we assume the ordinary path integration measure. The correct measure will be taken into account in the next section.

BRST and combined cohomology
• Cohomological variables and supersymmetry complex: It is useful to present the supersymmetry in the cohomological form by changing the variables. Our fermionic variables are reorganized as Then the inverse relation is In terms of these variable, the supersymmetry transformations are HereΦ I contains the degree of freedom X I 1 as in (2.36), and B Iij contains the degree of freedom Y Iij as Note that all the bosonic variables are organized into (X I 2 , W I µ , B Iij ,Φ I ) and all the fermionic variables are into (Ψ I , Ψ I µ , Ξ Iij ). And the Q 2 acts as In general, Q could act as . However, we note from (2.36) thatL ab =Θ =Θ i j = 0.
• BRST complex: To treat the gauge fixing of the U (1) nv+1 Yang-Mills gauge symmetry, we introduce the ghost fields and use BRST quantization. The BRST complex is Here the c I ,c I and B I are the ghost, anti-ghost and the standard Lagrange multiplier, so that the gauge fixing is performed by adding the terms, , so the length dimension for the ghost multiple is set by Note that the AdS 2 ×S 2 space does not have normalizable zero mode of c I ,c I and B I . The boundary condition of the path integral does not allow the non-normalizable modes, so we do not need special treatment for freezing out these kind of zero modes. This differs from the case of S 4 space. We refer [26] as the S 4 example where there are constant zero modes so the additional constant fields are introduced to freeze out those modes. The gauge fixing Lagrangian is Q B -exact, so where we put constants factors −1 and −2 to set the length dimension −4 for the Lagrangian.
•Combined complex: Since the gauge fixing Lagrangian (4.8) is not Q invariant, we need to consider new complex and modify the gauge fixing Lagrangian. Combining the BRST symmetry with supersymmetry, we make the combined complex. For this we define the supersymmetry for the ghost and we introduce the combined operatorQ := Q + Q B . Then we get the following combinedQ-complex, QB I = L vc I ,Qc I = B I , In fact supersymmetry transformation for ghost (4.9) was defined such that theQ 2 acts asQ 2 = L v := H. (4.12) We now useQ-exact gauge fixing term, This is equivalent to the (4.8) as 14) and the terms, −ic I ∇ µ Ψ Iµ and − ξ 2c I L vc I , do not contribute to the determinant. It is becausec can be connected only to c but there are no vertices in those extra terms containing c. Now, the physical action isQ invariant since it is invariant under the Q and Q B symmetry. Also the gauge fixing Lagrangian isQ invariant. For the supersymmetric localization, we now deform the physical action by adding the followingQ exact terms, (4.15) SinceQλ iI = Qλ iI andQλ iI = Qλ iI , the localization equation obtained from (3.4) will not be changed. To express it in terms of the set of the cohomological variables in (4.12), we use the inverse relation (4.2). Then we find, where explicitly, we note that 17) and their conjugation are given by

1-loop determinant
To evaluate the 1-loop determinant, we consider the fluctuations of fields that are orthogonal to the zero modes for whichQV = 0. We denote such fluctuations, in terms of the new set of variables in (4.12), as (X ,QX , Ξ ,QΞ ). Also, since the 1loop gives the exact quantum contribution, it is enough to consider only the quadratic terms. One can formally write the quadratic terms of the localization Lagrangian aŝ Among the fluctuation modes of X and Ξ , some of them can be annihilated byQ 2 . Let us classify the set of the path integration variable into two parts, Then, since H commute withQ and D ij , the terms in the localization Lagrangian can be separated aŝ where the kinetic operators of bosons and fermions, K b and K f , are divided as  Note that the determinant of K 0 b and K 0 f cancels with each other since D 10 is non degenerate for the corresponding modes. Also note that So the 1-loop determinant is given, up to a sign, by This 1-loop determinant is encoded in the following quantity, which will be expressed as a polynomial of U (1) representation. Calculating this, we can read off the eigenvalues and the degeneracies and then obtain the ratio of the determinant as To compute the (4.25), we add and subtract the zero mode contributions to express the trace as the summation over the complete set of basis of fields, (4.27) Here, the trace, Tr 0 , includes the zero modes given in (3.7) as well as the discrete zero modes of the gauge field (3.12). 4 The contributions from X 0 and Ξ 0 cancel each other since operator D 10 maps the fields X to the dual of the fields Ξ and it is non-degenerate for those mode. The trace over the zero modes also does not have contribution. By defining q := e it/ , To see this, note that the D 10 maps the eigenmode of H on the bundle X to the eigen mode with the same eigenvalues on the Ξ, unless these modes are in kernel or cokernel of the operator D 10 .
To compute the index, (4.29), we will first show that the operator D 10 is transversally elliptic with respect to the U (1) action generated by H, i.e. elliptic in all directions transversal to the H-orbit. If the operator is transversally elliptic on compact manifold, it is guaranteed that each subspace with the same eigenvalue of H in kernal and cokernel is finite dimensional [46,47]. We will assume that it still holds for the AdS 2 ×S 2 and we will compute the index (4.29) using Atiyah-Bott fixed point formula.
To show the transversally ellipticity, we compute the symbol of the operator D 10 . The D 10 appears as ΞD 10 X in the expression of V in (4.19). We take the relevant terms, and use the explicit expression in (4.18) with neglectingQc I andQΞ Iij as they are not relevant for the operator D 10 . To explicitly write the symbol of D 10 operator, denoted as σ(D 10 ), we consider only the highest derivative terms and replace ∂ µ by ip µ . It is convenient to introduce orthonormal four unit vector fields u µ a as, (4.31) In particular, Then, the highest derivative terms of ΞD 10 X term are where we denoted c ψ = cos(ψ) , s ψ = sin(ψ) , ch η = cosh(η) , sh η = sinh(η) , α = ch 2 η − c 2 ψ .
we get Nontrivial contribution to the index arises from the upper-left 4 × 4 block of the matrix in the middle, (4.39) We note that the above matrix is not invertible at the equator cos ψ = 0 of the S 2 . This is because σσ T = (cosh 2 η(p 2 1 + p 2 2 + p 2 3 ) + cos 2 ψp 2 4 ) · I and is zero for p 1 = p 2 = p 3 = 0 and p 4 = 0. However, if we restrict the momentum to be orthogonal to the Killing vector v µ , then σ is invertible as long as (p 1 , p 2 , p 3 ) are not all zero. Therefore the operator D 10 is transversally elliptic with respect to the symmetry L v . Now, we use the Atiyah-Bott fixed point formula to compute the equivariant index (4.29). The Atiyah-Bott formula is reviewed in the Appendix E and the formula is give by where the γ is the transformation of the section induced by the f (x). The formula reduces the trace of the operator e tH into the summation over the fixed point of the operator H. In our case, there are two fixed points. One is the north pole of the S 2 together with the origin of the AdS 2 and the other is the south pole of the S 2 together with the origin of the AdS 2 . Near the fixed points the space is locally R 2 × R 2 so it is parametrized by the orthonormal coordinate (x 1 , x 2 , x 3 , x 4 ), where the (x 1 , x 2 ) are the local coordinate on the AdS 2 and (x 3 , x 4 ) are the local coordinate on the S 2 . Let us define the complexified coordinates Under the operator e tH they transform as z 1 → e it/ z 1 := qz 1 , z 2 → e −it/ z 2 :=qz 2 , w 1 → e it/ w 1 := qw 1 , w 2 → e it/ w 2 := qw 2 . (4.42) Note here that the operator H generates as the L−J rotation, where L and J are the rotation on the AdS 2 and S 2 . So the the coordinate z 2 rotates in the opposite way to the z 1 . Also, the coordinate w 2 rotates the opposite again to the z 2 coordinate as it is the coordinate at the south pole of the S 2 . Let us consider how the fields transform. The nontrivial part is for Ξ ij . One can explicitly see from (4.35) that the fermions (Ξ 1 , Ξ 2 , Ξ 3 ) defined in (4.34) are dual of self-dual field at the north pole and the dual of anti-self-dual field at south pole such that they are contracted with (F + 14 , F + 13 , F + 12 ) and (F − 14 , F − 13 , F − 12 ) respectively. 5 In terms of the complexified coordinates, the self-dual and anti-self-dual field strength have the following basis, (4.43) By the definition (4.34), we see that the fermions (Ξ 11 , Ξ 22 , Ξ 12 ) have the following basis at north pole, ∂ ∂z 2 , (4.44) and at the south pole, and similarly we get at the south pole Applying the fixed point formula (4.40) and using the (4.42), (4.46) and (4.47) , we obtain the following result for each vector multiplet, (4.48) 5 In the case we choose another Killing spinor which squares to L + J, we get opposite transformation rule for the z 2 and w 2 . But now (Ξ 1 , Ξ 2 , Ξ 3 ) are dual of (F − 14 , F − 13 , F − 12 ) at North pole and (F + 14 , F + 13 , F + 12 ) at South pole respectively, so we get same transformation rule as in (4.46) and (4.47), giving the same result (4.50). Notice that in the S 4 computation, for the similar killing spinor which squares to L + J, the fields (Ξ 1 , Ξ 2 , Ξ 3 ) are dual of (F + 14 , F + 13 , F + 12 ) at North pole and (F − 14 , F − 13 , F − 12 ) at South pole respectively. This is the main difference of the index computation in S 4 and AdS 2 × S 2 .
The first term is from origin of AdS 2 and north pole of S 2 and the second term is from origin of AdS 2 and south pole of S 2 . One obtains the degeneracies of eigen values of H by expanding this expression in power series of q. Here we follow the way of expansion as was done in [26,47]. The result for the determinant is in fact independent on the way of expansions for the index. By expanding we finally arrive at From the result (4.50), we read off the 1-loop partition function for n v + 1 vector multiplets, Using the ζ-function regularization, For given radius , we get the exact 1-loop partition function. However, we note that this radius is not the physical radius because it can be chosen to be an arbitrary constant value as the choice of D-gauge fixing. In the next section, we will show that, by appropriate integration measure, the 1-loop partition function is independent of the gauge choice and depends on the radius of the physical AdS 2 × S 2 metric.

Integration measure
In the previous section, we assumed that the integration measure is trivial. As a result, the one loop partition function is not independent of the choice of D-gauge. This implies that the trivial path integration measure is not scale invariant. In this section, we will properly define the path integration measure and show that the result of the 1-loop partition function is indeed gauge invariant. Further, the result depends on the solutions of localization equations through the radius of the physical AdS 2 ×S 2 metric. To define the measure we use the ultra locality arguments [40], as well as the condition that the result should be in terms of the physical quantities. Let us consider the kinetic terms in the action, The metric g µν is not a physical metric (it has dilatation weight -2) and is related to physical metric G µν which is the metric in Einstein frame by redefinition, Note that the radius of the AdS 2 × S 2 metric g µν is fixed to the constant and the physical radius P is not fixed but depends on the scalars, i.e. P = P (X ,X) as the Kähler potential K is the function of the scalars. In terms of the physical metric, we get standard Einstein-Hilbert action, and the kinetic term of the vector multiplets fields are 3) Looking at the factors in front of the each kinetic term, the definition of the norm for each field is defined as where we denote g 0µν as AdS 2 ×S 2 metric with unit radius. By following the normalization conditions,

5) the integration measure is determined as
Similarly, we determine the measure for the ghost multiplet by looking at the gauge fixing action in Einstein frame, The definition of norm 6 , and the normalization condition, determine the integration measure for the ghost multiplets The measure (5.7) and (5.11) will give the result in terms of the physical quantities. One can consistently see that the 1-loop determinant for each kinetic operator for each field will be given in terms of det G and det / ∂ G , not in terms of det g and det / ∂ g . It seems that a naive counting of the scale factor and radius factor and P seems completely cancel by bosonic measure and fermionic measure. However, they are infinite product. The regularized number of those factors should not be canceled each other and should cancel the scale factor appears in the 1-loop partition function (4.51) such that the result should be in terms of the physical radius P .
Let us reconsider the computation of the 1-loop partition function. By the supersymmetric localization, the measure depends only on the saddle point value, i.e. P = P ( C), where C parametrizes all saddle point of the scalar in the vector multiplets. We now redefine the cohomological variables by following field redefinition, We also redefine theQ operatorQ Since the P ( C) is function of the saddle points,Q does not act on the P ( C). Using this operator we define the other primed cohomological variables, In terms of these new variables and new supercharge,X ,QΞ ,QX ,Ξ, we rewrite the localization lagrangian as in (4.19), where theD ij are properly defined by multiplying diagonal matrices whose elements are composed of and P . We can follow the same analysis as below the (4.19) and

Discussion
Quantum entropy function as a expectation value of the Wilson loop at the boundary of AdS 2 computes the entropy of the extremal back holes for the fixed value of electric and magnetic charge. In this paper, we considered the n v + 1 vector multiplets on the AdS 2 ×S 2 background and used the supersymmetric localization to compute their exact contribution to the quantum entropy function. In this process, we deform the action byQ exact action and find the solutions of the localization equations. We then compute the one loop determinant coming from the quadratic fluctuation of thê QV action. In order to express the result in terms of physical variables, we propose the scale invariant path integral measure for vector multiplet fields.
Our work is motivated from the work in [36,37,39]. However, there is a difference between our analysis and the analysis presented in those works. Our path integral for (n v + 1) vector multiplets localizes to (2n v + 2) dimensional ordinary integral unlike in the case of [36,37,39] where the path integral localises to (n v + 1) ordinary integral. This difference arises because there is inherent subtly of choice of the analytic continuation for the Euclidean path integral. We choose the analytic continuation motivated by the work [26,31]. It will be very important to understand a priori principle for the choice of contour of integration in the Euclidean path integral. We find that the solution for X I 1 and Y I ij given by (3.6) is unique and same as in [36,37,39]. However, we find out that there are more solutions (3.7), but we could not able to prove that this is the unique solution. Although we provide evidence in support of this in the appendix, it will be interesting to prove it explicitly.
Another important aspect our work is the proposal for the path integral measure. This proposal is based on the fact that the path integral measure should be scale invariant and the results in the one loop determinant being dependent only on the physical quantities. As a result, the 1-loop determinant is scale invariant and depends on the physical metric. It is remarkable that although we are considering abelian vector multiplets, the 1-loop result depends on the continuous parameters C I of localizing solutions through the physical metric. However, not only for the scale symmetry but also for supersymmetry, the measure should be invariant for the purpose of the localization. We were not able to show this and assumed that supersymmetric invariance is satisfied.
Our result for the one loop partition function matches with the on shell computation of the logarithmic correction in N = 2 black hole entropy [43]. In the on shell computation the contribution of each vector multiplet to the logarithmic correction is − 1 12 ln A H where A H = 4π 2 P is the area of horizon. To completely reproduce this on shell result, we still need to do the integration over bosonic zero modes, the parameters C I 1 and C I 2 . Given that it matches with the one loop determinant, it seems plausible that the integration over bosonic zero modes do not give any further P dependence. We will report on this in future publication.
Our computation should be thought of as an important step towards the completing the exact computation of the black hole entropy initiated in [36,37,39]. For completing this story, we still have many open problems to solve. We have to incorporate the quantum fluctuation of all other multiplets, particularly Weyl multiplets. In particular, including the hypermultiplets would be one of the tricky issues since there is no off-shell formulation of the hypermultiplets with finite number of auxiliary fields in supergravity, but it should be very important to treat the general N = 2 supergravities because the hypermultiplets should be incorporated to complete the off shell conformal supergravity as a compensating multiplets. It will also help to have complete analysis of localization for the Weyl multiplets as mentioned in [39]. Furthermore, one needs to also consider gravitini multiplets. It is particularly necessary for theories with higher supersymmetries like N = 4 or N = 8. Here, we may have to understand similar issues that was appeared in our work on vector multiplets. For example, we may need to specify the analytic continuations of all the fields, properly treat the gauge fixing of all the gauge symmetries in the conformal supergravity and should properly define the gauge invariant functional measure for the path integral. We left these exercises for the future work.

A Gamma matrices and spinors
Our convention of gamma matrices and the reality properties follows the paper, [44].

A.1 (1, 3) dimensions
In Minkowskian four dimensions, there are two choices, C ± and B ± , such that (A.1) Two representations are related by C + = C − γ 5 . Chirality operator The choice of C + and B + allows us to set Majorana spinor, defined as or equivalently, ψ * = B + ψ , (A. 6) whereas, for the choice of C − and B − , the symplectic Majorana spinors can be defined, 7 These spinors are not compatible with Weyl representation, such that under the chiral decompsition, Two chirally projected Majorana spinors ψ i ± symplectic Majorana spinors λ i ± can be related by In Euclidean four dimensions we also have two choices such that, and compatible with Weyl condition, (A.14)

A.3 Fierz identities
It is useful to note the following gamma matrix algebra, For example Gamma matrices form a complete basis so it is followed by the Fierz arrangement, where C can be either C + or C − in (A.1). For example, with bosonic fermions η ± , ξ ± and λ ± with positive or negative chiralities, (A.20) A.4 Minkowskian theory to Euclidean theory for N = 2 While the Minkowskian space allows the Majorana representation, the Euclidean theory does not. They have different properties under the complex conjugation. So in order to relate Minkowskian and Euclidean theory, we have to hide the complex conjugate operation. We change the Dirac conjugation of spinors,ψ := ψ † A, into the charge conjugation ψ T C − γ 5 using the Majorana relation given in (A.5) and the relation between C + and C − in (A.1). Now, since the same C − can be used both Minkowskian and Euclidean spacetime, as in (A.1) and (A.11), Euclideanization is straightforward. In the case of N = 2 theory, it is convenient to use the symplectic Majorana spinor representation because it is allowed both in Minkowskian and Euclidean theory. Using the relation (A.10), we can redefine spinor fields to satisfy the symplectic Majorana condition. After hidding the † operation in the theory using the symmplectic Majorana conjugate, one cannot distinguish whether it is Minkowskian or Euclidean theory and we are free to move by analytic continuation t = −iθ.
Tensor density that can be used in self or anti-selfdual equation should also be modified.
In Euclidean space t = −iθ In both of Minkowskian and Euclidean space, the self or anti-selfdual condition is written in terms of the tensor density, e µνλρ , in (A.21) and (A. 22), where T µν± are self and anti-selfdual tensor respectively.
B Superconformal calculus for d = 4 and N = 2 SUGRA We review the superconformal calculus for d = 4 and N = 2 off-shell supergratives. We refer the reader to [45] for detailed review, and to [33][34][35] for the original development.

B.1 Weyl multiplet
The first step is to construct superconformal gauge theory by promoting all the N = 2 superconformal generator as local symmetries. By all the local superconformal transformation, the covariant derivative is defined as where the sum is for all superconformal generators except the translation generator [44] and the δ is gauge transformation with the gauge field, h µ (T ), as parameter. Later, we will introduce D µ as a covariant derivative with respect to M, D, A, V . The gauge fields h µ (T ) and the symmetry parameters for each symmetry generators are contained in the table 1, and the table 2 shows the charges of the gauge field and supersymmetry parameters. The contents of the Weyl multiplet is given by the following 24 + 24 off-shell degrees of freedom, where the ω ab µ , f a µ , φ i µ are not included because they are not independent fields but composite fields in terms of others. The constrained relations are presented in (B.7). The fields T ij ab , χ , D are the auxiliary tensor, spinor and scalar fields, and the auxiliary tenor satisfies antiselfdual condition Table 2. Weyl weight ω, U (1) R weight c and fermion chirality with respect to γ 5 for each the Weyl multiplet component field and supersymmetry parameters. whose complex conjugation gives selfdual tensor, Conventional notations are The SU (2) gauge fields V µ i j is anti-hermitian and traceless •Conventional constraints: In order to relate ω ab µ , φ i µ , f i µ with other fields, we impose the following constraints, Here, the modified field strengths are and the dual tensors are defined as, Under the conventional constraints, (B.7), the composite fields are expressed in terms of Weyl multiplet, (B.11) •The transformation law and the superconformal algebra Q − S − K− transformation rules for the Weyl multiplet fields are, where The sum over T is for all superconformal transformation except the general coordinate transformation, and the parameters are 8 16) and the δ gauge in general includes additional abelian, non-abelian or central charge gauge transformations.

B.2 Vector multiplets
Consider N υ + 1 vector multiplet, restricting ourselves to the case of abelian gauge symmetries, Complex scalar X I , a vector gauge field W I µ , SU (2) triplet auxiliary scalars Y I ij where Y ijI := (Y I ij ) * . Table 3. Weyl weight ω, U (1) R weight c and fermion chirality with respect to γ 5 for each vector multiplet component field One linear combination of the abelian gauge symmetries corresponds to the gauged central charge transformation, and the corresponding field strength belongs to the graviphoton. Note that we must have at least one vector multiplet in the theory in order to make contact with N = 2 Poincaré supergravity, because the Weyl multiplet does not account for the graviphoton.
Here the covariant field strength is which satisfies the Bianchi identity where, The covariant derivatives are (B.26) and the SUSY transformation of them are The algebra includes central charge gauge symmetry, Prepotential, F (X) is a holomorphic function, which is homogeneous of secondegree, i.e., for any complex parameter λ. Some identities are Lagrangian(Bosonic): (B.33) Conventional gauge fixing conditions:

B.3 Chiral notation
In the Minkowskian N = 2 supereravities, we adopt the so-called chiral notation, which is to keep track of spinor chiralities through writing the SU (2) R index as an upper or lower index. For instance, consider two Majorana spinors, ψ i . The chiral projection of them are Depending on the spinor an upper index might be associated with left or with right chirality. The assignments are listed in various tables 1, 2, 3. This chiral decomposition is not compatible with Majorana condition as we also see in (A.1), Since we can take B + = 1, the complex conjugation can be thought as raising and lowering the SU (2) R indices. Dirac conjugation is defined as which is followed byΨ i 1 γ a 1 ···an Ψ j 2 = 0 , for odd n Ψ i 1 γ a 1 ···an Ψ 2j = 0 , for even n . (B.39) We have further useful relations,

C Bispinors
We presents explicit values of some bispinors, which are useful for detailed calculation.
If we multiply the above differential equation with sinh η sin ψ (sinh 2 η + sin 2 ψ) 2 and use the variable r = cosh η and x = cos ψ, we obtain the following equation First let us restrict to solutions that are independent of θ and φ. In this case, the asymptotic behavior is Y ,0 (ψ, φ)/r = P (x)/r , where P is Legendre polynomial. Since all the coefficient of the differential operators are polynomial of r and x, we assume that the solutions can be written as c n,p x p r n .

(D.3)
In case where = 1, the solution X 2 = x/r is a solution that is presented in (3.7). In case where = 2, the simplest solution is and it diverges at r = 1. We can rewrite it as follows X 2 = 1 4r 6r + (r 2 − x 2 )log r − 1 r + 1 . (D.5) Although there are other solutions with = 2, if we fix the coefficient of the highest power and subtract the above solution from other solutions, the highest power of r becomes less than −2. So we can consider them as solutions with > 2. In case where = 3, the simplest solution is which diverges at r = 1 again. It can be rewritten as We conjecture that for any positive integer , there is a unique solution of the following form where e is 0 or 1 when is even or odd respectively and the floor function 2 denotes the largest integer not greater than 2 . The series expansion of r continues infinitely for ≥ 2.
Next let us consider solutions whose asymptotic behaviors are e imθ Y , (ψ, φ)/r . We propose that are the solutions. They also diverge at η = 0.

E Fixed point formula
In this appendix we will show a proof of Atiyah-Bott fixed point formula [46,47], which we used in the section 4.2. Let E 0 → E 1 be an complex of vector bundles over a manifold X and D 10 : Γ(E 0 ) → Γ(E 1 ) is a differential operator. For a given map f : X → X, we can define f * E i , which is a pullback of E i by f . Moreover the map f induces a map γ : E i f (p) → E i p , where p ∈ X. Let us define a map T := γ • f * . If the fixed points on X under f are isolated, we have the following formula Tr Ker D 10 T − Tr Coker D 10 T = x∈fixed point set if the left-hand side is well-defined. This is called Atiyah-Bott formula.
Since we focus on cases where D 10 commutes with T here, Tr Ker D 10 T −Tr Coker D 10 T = Tr Γ(E 0 ) T − Tr Γ(E 1 ) T . Let us take an example where E 0 is an cotangent bundle and prove the following part in the formula 9 Tr Γ(E 0 ) T " = " x∈fixed point set in that case. The above equality holds if the left-hand side is well-defined. Since Γ(E 0 ) is infinite dimensional, one should be careful about it. An element of Γ(E 0 ) can be written as µ A µ (x)dx µ . If x goes to y = f (x) by f , A µ (x)dx µ is mapped to A µ (y)dy µ by f * and it is further mapped to A ν (y)(dy ν /dx µ )dx µ by γ. Generally if a given operator Q maps u µ (x)dx µ to v µ (x)dx µ =: (Q ν µ u ν )dx µ , we can define the kernel K Q ν µ for the operator Q such that dy K Q ν µ (x, y)u ν (y) = v µ (x). In our case, the kernel for T is For a general operator Q, the trace of Q over Γ(E 0 ) can be rewritten as follows if the left-hand side is well-defined. Let us derive the above relation. Fist we choose a complete set of orthonormal basis {|A {p} } {p} in Γ(E 0 ) . Each component A µ of one-form field µ A µ dx µ can take different field configuration labeled by p µ and {p} is the set {p µ } µ of labels for all components. We can rewrite the trace in the left-hand side of the above relation as where we used the definition of the kernel K Q ν µ . By applying the following completeness condition to the right-hand side of (E.5), we can derive (E.4). Let us apply the relation (E.4) to the operator T . By using (E.3) and replacing the integration variable x with z := x − f (x), we can derive the formula (E.2) as follows For general E i , we can also derive the formula (E.2) in the same way. Although Tr Γ(E i ) T itself may not be well-defined, Tr Γ(E 0 ) T − Tr Γ(E 1 ) T is well-defined in the following cases. If T commutes with D 10 , Tr Γ(E 0 ) T − Tr Γ(E 1 ) T = Tr Ker D 10 T − Tr Coker D 10 T . Let us decompose the spaces Ker D 10 and Coker D 10 into subspaces such that each subspace has different eigenvalue for T . If each subspace is finite dimensional, Tr Ker D 10 T − Tr Coker D 10 T is well-defined and the following formula holds Tr Ker D 10 T − Tr Coker D 10 T = x∈fixed point set