Single-centered black hole microstate degeneracies from instantons in supergravity

TL;DR

This paper derives holographic constraints on black hole microstate degeneracies by computing the quantum entropy in supergravity, incorporating instanton effects, and comparing results with known microscopic BPS counting functions involving mock modular forms.

Contribution

It introduces a systematic method to connect supergravity instanton sums with microscopic degeneracies, revealing the influence of mock modular structures.

Findings

Agreement with microscopic degeneracies for the first seven magnetic charges

Instanton sums produce Bessel function expansions similar to Rademacher series

Discrepancies near the order where mock modular behavior becomes significant

Abstract

We obtain holographic constraints on the microscopic degeneracies of black holes by computing the exact macroscopic quantum entropy using localization, including the effects of string worldsheet instantons in the supergravity effective action. For $\frac14$-BPS black holes in type II string theory on $K3 \times T^{2}$, the constraints can be explicitly checked against expressions for the microscopic BPS counting functions that are known in terms of certain mock modular forms. We find that the effect of including the infinite sum over instantons in the holomorphic prepotential of the supergravity leads to a sum over Bessel functions with successively sub-leading arguments as in the Rademacher expansion of Jacobi forms -- but begins to disagree with such a structure near an order where the mock modular nature becomes relevant. This leads to a systematic method to recover the polar terms of the microscopic degeneracies from the degeneracy of instantons (the Gromov-Witten invariants). We check explicitly that our formula agrees with the known microscopic answer for the first seven values of the magnetic charge invariant.