Black Hole Quantum Mechanics and Generalized Error Functions

TL;DR

This paper derives the non-holomorphic completion of mock modular forms related to BPS black hole microstates in string theory, using localization techniques in supersymmetric quantum mechanics.

Contribution

It generalizes the error function completion for multiple centers, extending previous results to arbitrary numbers of BPS black hole constituents.

Findings

Derived the non-holomorphic completion for any number of centers.

Connected the completion to integrals over phase space and generalized error functions.

Validated the approach by explicit localization calculations.

Abstract

In Type II Calabi-Yau string compactifications, S-duality predicts that suitable generating series of BPS indices counting microstates of D4-D2-D0 black holes are in general mock modular forms of higher depth. The non-holomorphic contributions needed to cancel the anomaly under modular transformations involve certain indefinite theta series with kernels constructed from generalized error functions. Physically, these contributions are expected to arise from a spectral asymmetry in the continuum of scattering states of $n$ BPS dyons with mutually non-local charges. For $n=2$, the (standard, depth one) error function completion was derived long ago by explicitly computing the bosonic and fermionic density of states in the two-body supersymmetric quantum mechanics. Here we derive the general non-holomorphic completion for an arbitrary number of centers by evaluating the refined Witten index of the supersymmetric quantum mechanics using localization. In a nutshell, the index reduces to an integral over $\mathbb{R}^{3n-3}$ (the relative location of the centers), and splits into an integral over the $2n-2$ dimensional phase space of BPS ground states times an integral over $n-1$ transverse directions, which ultimately produces the expected generalized error functions.