Supersymmetric Index from Black Hole Entropy

For BPS black holes with at least four unbroken supercharges, we describe how the macroscopic entropy can be used to compute an appropriate index, which can be then compared with the same index computed in the microscopic description. We obtain exact results incorporating all higher order quantum corrections in the limit when only one of the charges, representing momentum along an internal direction, approaches infinity keeping all other charges fixed at arbitrary finite values. In this limit, we find that the microscopic index is controlled by certain anomaly coefficients whereas the macroscopic index is controlled by the coefficients of certain Chern-Simons terms in the effective action. The equality between the macroscopic and the microscopic index then follows as a consequence of anomaly inflow. In contrast, the absolute degeneracy does not have any such simple expression in terms of the anomaly coefficients or coefficients of Chern-Simons terms. We apply our analysis to several examples of spinning black holes in five dimensions and non-spinning black holes in four dimensions to compute the index exactly in the limit when only one of the charges becomes large, and find perfect agreement with the result of exact microscopic counting. Our analysis resolves a puzzle involving M5-branes wrapped on a 5-cycle in K3 x T^3.