Supersymmetric Index from Black Hole Entropy

TL;DR

The paper defines and computes a space-time supersymmetric index for BPS black holes with at least four supercharges, showing that in a Cardy-like limit where a single internal momentum charge becomes large the macroscopic index—encoded in Chern-Simons coefficients via anomaly inflow—precisely matches the microscopic index across diverse compactifications (K3×S^1, K3×T^2, T^4, T^6, and M-theory realizations). It develops a macroscopic computation that isolates the bulk AdS$_3$ contributions and the exterior hair modes, and demonstrates that the exterior sector cancels constant one-loop shifts, yielding a simple CS-driven expression for the index. The results are supported by exact microscopic calculations using modular forms and Cardy-type asymptotics in several duality frames, with perfect agreement in all examined cases, thereby resolving prior puzzles about M5-brane configurations and their Cardy limits. A general, anomaly-based argument then explains the universal agreement between microscopic and macroscopic indices, clarifying the roles of regular versus irregular (hair) modes and the necessity of focusing on the index rather than the absolute degeneracy. Overall, the work strengthens the AdS$_3$/CFT$_2$ perspective on black hole microstates and provides robust tools for exact index matching in string theory.

Abstract

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.