Modular anomaly of BPS black holes

TL;DR

This paper simplifies the modular anomaly in generating functions of BPS indices for D4-D2-D0 black holes, which are mock modular forms with complex anomalies, making it easier to analyze their properties.

Contribution

It introduces a method to significantly simplify the modular anomaly of these generating functions, facilitating their study and understanding.

Findings

Simplified the expression of the modular anomaly

Reduced complexity in analyzing mock modular forms

Enhanced understanding of BPS black hole state counting

Abstract

Generating functions of BPS indices, counting states of D4-D2-D0 black holes in Calabi-Yau compactifications of type IIA string theory and identified with rank 0 Donaldson- Thomas invariants, are examples of mock modular forms. They have a quite complicated modular anomaly expressed as a sum over three different types of trees weighted by generalized error functions and their derivatives. We show that this anomaly can be significantly simplified, which in turn simplfies finding the corresponding mock modular generating functions.