Mock modularity and refinement: from BPS black holes to Vafa-Witten theory

The generating functions of degeneracies of D4-D2-D0 black holes in Type II string compactifications on Calabi-Yau threefolds are examples of (higher depth) mock modular forms. I'll explain how S-duality can be used to derive an explicit form for their modular completions, which becomes particularly simple in the presence of a refinement. This result turns out to have many applications going beyond the original context. In particular, I'll show that it can be used i) to reproduce and generalize in an easy way the known results on modular properties of the generating functions of BPS dyons in N=4 string compactifications; ii) to find Vafa-Witten invariants of arbitrary(!) rank on CP^2, Hirzebruch and del Pezzo surfaces; iii) to obtain holomorphic anomaly equations for BPS partition functions; iv) to reveal a non-commutative structure induced by the refinement on the moduli space of compactified theory.