Black hole entropy function, attractors and precision counting of microstates

Abstract

In these lecture notes we describe recent progress in our understanding of attractor mechanism and entropy of extremal black holes based on the entropy function formalism. We also describe precise computation of the microscopic degeneracy of a class of quarter BPS dyons in N = 4 supersymmetric string theories, and compare the statistical entropy of these dyons, expanded in inverse powers of electric and magnetic charges, with a similar expansion of the corresponding black hole entropy. This comparison is extended to include the contribution to the entropy from multi-centred black holes as well.

Heat kernels on the AdS2 cone and logarithmic corrections to extremal black hole entropy

“…In that case, the integral in (2.1) receives contributions from the deformed contour and from the poles of the integrand which were crossed during this deformation. It can further be shown [11] that the dominant contribution to d (Q, P ) arises from the the poles (2.2) of the integrand, while the contribution from the deformed contour is subleading. Then using the residue theorem one first reduces the integration variables in (2.1) from three to two by performing the integration overv.…”

“…We refer the reader to [11] for an indepth account and a more extensive set of references. In particular, we will study the degeneracy of 1 4 -BPS…”

“…For more details, we refer the reader to [10,11,15]. The poles of the integrand are the zeros of Φ 10 which are located at…”

“…The strategy one follows in evaluating the integral in (2.1) has been explicated in [11]. The procedure is to deform the contour C to values of (ρ 2 ,σ 2 ,v 2 ) of the order of 1/charge.…”

“…This program has been very explicitly carried out for a class of supersymmetric black holes and a detailed matching of the string answer has been done in the large charge limit, where the string answer explicitly matches with the Bekenstein-Hawking, or more generally, the Wald entropy [2] of the black hole in question [3][4][5][6][7][8][9][10]. We refer the reader to [11] for a review and a more exhaustive set of references. 1 However, the string answer is the full quantum answer for the entropy of the black hole and therefore also contains corrections to the Wald formula and it is interesting to ask if a direct physical interpretation of these corrections is possible, at least in some cases.…”