Black holes and higher depth mock modular forms

TL;DR

This paper explores the modular properties of black hole degeneracy generating functions in string theory, revealing their nature as mock modular forms of higher depth due to divisor sum structures.

Contribution

It introduces a novel analysis of the modular anomaly in BPS degeneracy functions when the divisor is reducible, showing they form higher depth mock modular forms.

Findings

Derived modular properties from S-duality invariance.

Identified modular anomaly for reducible divisors.

Expressed the anomaly via non-holomorphic completions with error functions.

Abstract

By enforcing invariance under S-duality in type IIB string theory compactified on a Calabi-Yau threefold, we derive modular properties of the generating function of BPS degeneracies of D4-D2-D0 black holes in type IIA string theory compactified on the same space. Mathematically, these BPS degeneracies are the generalized Donaldson-Thomas invariants counting coherent sheaves with support on a divisor $\cal D$, at the large volume attractor point. For $\cal D$ irreducible, this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping M5-brane on $\cal D$ and is therefore known to be modular. Instead, when $\cal D$ is the sum of $n$ irreducible divisors ${\cal D}_i$, we show that the generating function acquires a modular anomaly. We characterize this anomaly for arbitrary $n$ by providing an explicit expression for a non-holomorphic modular completion in terms of generalized error functions. As a result, the generating function turns out to be a (mixed) mock modular form of depth $n-1$.