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Register a new userSupersymmetric Localization for BPS Black Hole Entropy: 1-loop Partition Function from Vector Multiplets
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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.
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We study supersymmetric index of 4d $SU(N)$ $mathcal{N}=4$ super Yang-Mills theory on $S^1 times M_3$. We compute asymptotic behavior of the index in the limit of shrinking $S^1$ for arbitrary $N$ by a refinement of supersymmetric Cardy formula. The asymptotic behavior for the superconformal index case ($M_3 =S^3$) at large $N$ agrees with the Bekenstein-Hawking entropy of rotating electrically charged BPS black hole in $AdS_5$ via a Legendre transformation as recently shown in literature. We also find that the agreement formally persists for finite $N$ if we slightly modify the AdS/CFT dictionary between Newton constant and $N$. This implies an existence of non-renormalization property of the quantum black hole entropy. We also study the cases with other gauge groups and additional matters, and the orbifold $mathcal{N}=4$ super Yang-Mills theory. It turns out that the entropies of all the CFT examples in this paper are given by $2pi sqrt{Q_1 Q_2 +Q_1 Q_3 +Q_2 Q_3 -2c(J_1 +J_2 )} $ with charges $Q_{1,2,3}$, angular momenta $J_{1,2}$ and central charge $c$. The results for other $M_3$ make predictions to the gravity side.
We evaluate the mixed partition function for dyonic BPS black holes using the recently proposed degeneracy formula for the STU model. The result factorizes into the OSV mixed partition function times a proportionality factor. The latter is in agreement with the measure factor that was recently conjectured for a class of N=2 black holes that contains the STU model.
We use localization to compute the partition function of a four dimensional, supersymmetric, abelian gauge theory on a hemisphere coupled to charged matter on the boundary. Our theory has eight real supercharges in the bulk of which four are broken by the presence of the boundary. The main result is that the partition function is identical to that of ${mathcal N}=2$ abelian Chern-Simons theory on a three-sphere coupled to chiral multiplets, but where the quantized Chern-Simons level is replaced by an arbitrary complexified gauge coupling $tau$. The localization reduces the path integral to a single ordinary integral over a real variable. This integral in turn allows us to calculate the scaling dimensions of certain protected operators and two-point functions of abelian symmetry currents at arbitrary values of $tau$. Because the underlying theory has conformal symmetry, the current two-point functions tell us the zero temperature conductivity of the Lorentzi
The type IIA superstring partition function Z_IIA on the euclidean attractor geometry AdS_2 x S^2 x CY_3, computes the modified elliptic genus Z_BH of the associated black hole. The hybrid formalism of superstrings defined as a conformally invariant sigma model on the coset supermanifold PSU(1,1|2)/U(1)xU(1), together with Calabi-Yau and chiral boson CFTs, is used to calculate Z_IIA. The sigma model action on AdS_2 x S^2 is explicitly written in U(1)xU(1) invariant variables. The N=2 generators of AdS_2 x S^2 x CY_3 are enlarged and embedded in an N=4 topological algebra. The world sheet superconformal invariance is then used to construct a nilpotent BRST operator, in contrast to the kappa symmetry analysis used by Beasely et. al. in hep-th/0608021. The sigma model action is explicitly shown to be closed under this BRST operator. Localization arguments are then used to deform the world sheet path integral with the addition of a BRST exact term, where, contributions arise only from the center of AdS_2 and, the north and south poles of S^2. This leads to the OSV result Z_BH = Z_IIA = |Z_top|^2, where |Z_top|^2 is the square of the topological string partition function.
BPS black hole degeneracies can be expressed in terms of an inverse Laplace transform of a partition function based on a mixed electric/magnetic ensemble, which involves a non-trivial integration measure. This measure has been evaluated for black holes with various degrees of supersymmetry and for N=4 supersymmetric black holes all results agree. It generally receives contributions from non-holomorphic corrections. An explicit evaluation of these corrections in the context of the effective action of the FHSV model reveals that these are related to, but quantitatively different from, the non-holomorphic corrections to the topological string, indicating that the relation between the twisted partition functions of the latter and the effective action is more subtle than has so far been envisaged. The effective action result leads to a duality invariant BPS free energy and arguments are presented for the existence of consistent non-holomorphic deformations of special geometry that can account for these effects. A prediction is given for the measure based on semiclassical arguments for a class of N=2 black holes. Furthermore an attempt is made to confront some of the results of this paper to a recent proposal for the microstate degeneracies of the STU model.
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