Mock modularity at work, or black holes in a forest

TL;DR

This paper reviews how mock modular forms, originating from Ramanujan, are used to understand black hole microstates in string theory, linking advanced mathematics with physical theories and topological invariants.

Contribution

It synthesizes recent developments connecting mock modular forms to black hole microstate counting and introduces new mathematical structures on Calabi-Yau moduli spaces.

Findings

Mock modular forms encode black hole microstate generating functions.

Applications include computing topological invariants like Donaldson-Thomas and Vafa-Witten.

New non-commutative structures on Calabi-Yau moduli spaces are constructed.

Abstract

Mock modular forms, first invented by Ramanujan, provide a beautiful generalization of the usual modular forms. In recent years, it was found that they capture generating functions of the number of microstates of BPS black holes appearing in compactifications of string theory with 8 and 16 supercharges. This review describes these results and their applications which range from the actual computation of these generating functions for both compact and non-compact compactification manifolds (encoding, respectively, Donaldson-Thomas and Vafa-Witten topological invariants) to the construction of new non-commutative structures on moduli spaces of Calabi-Yau threefolds.