Mock modularity from black hole scattering states

TL;DR

This paper explains the non-holomorphic modularity of black hole degeneracy generating functions by computing an index for two-centered black holes, revealing contributions from bound states and scattering states.

Contribution

It provides a physical interpretation for the non-holomorphic parts of the modular forms via supersymmetric index calculations of black hole scattering states.

Findings

The continuum contribution matches the non-holomorphic completion term.

The index includes both bound state and scattering state contributions.

The approach clarifies the modular properties of black hole degeneracies.

Abstract

The exact degeneracies of quarter-BPS dyons in Type II string theory on $K3 \times T^2$ are given by Fourier coefficients of the inverse of the Igusa cusp form. For a fixed magnetic charge invariant $m$, the generating function of these degeneracies naturally decomposes as a sum of two parts, which are supposed to account for single-centered black holes, and two-centered black hole bound states, respectively. The decomposition is such that each part is separately modular covariant but neither is holomorphic, calling for a physical interpretation of the non-holomorphy. We resolve this puzzle by computing the supersymmetric index of the quantum mechanics of two-centered half-BPS black-holes, which we model by geodesic motion on Taub-NUT space subject to a certain potential. We compute a suitable index using localization methods, and find that it includes both a temperature-independent contribution from BPS bound states, as well as a temperature-dependent contribution due to a spectral asymmetry in the continuum of scattering states. The continuum contribution agrees precisely with the non-holomorphic completion term required for the modularity of the generating function of two-centered black hole bound states.