TH String Theory Seminar

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

by Sergey Alexandrov (Montpellier University)

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The generating functions of degeneracies of D4-D2-D0 black holes in Type II string compactifications on Calabi-Yau threefolds are example 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 in a very 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.


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