Mock modularity of Calabi-Yau threefolds

TL;DR

This paper studies the mock modular properties of generating functions related to BPS indices in string theory, providing a method to solve their modular anomaly equations for arbitrary charges using indefinite theta series.

Contribution

It introduces a new method to solve modular anomaly equations for generating functions of BPS indices in Calabi-Yau compactifications, applicable to arbitrary D4-brane charges.

Findings

Derived explicit solutions for generating functions using indefinite theta series.

Established a systematic approach to determine these functions from finite Fourier data.

Enhanced understanding of the modular structure of BPS state counting functions.

Abstract

Generating functions $h_r(\tau)$ of D4-D2-D0 BPS indices, appearing in Calabi-Yau compactifications of type IIA string theory and identical to rank 0 Donaldson-Thomas invariants, are known to be higher depth mock modular forms satisfying a specific modular anomaly equation, with depth determined by the D4-brane charge $r$. We develop a method to solve the anomaly equation for arbitrary charges, in terms of indefinite theta series. This allows us to find the generating functions up to modular forms that can be fixed by computing just a finite number of Fourier coefficients of $h_r$.