Abstract. I'll present recent results on modular properties of generating functions of degeneracies of BPS black holes appearing in string theory compactifications on Calabi-Yau threefolds. These generating functions turn out to be examples of (higher depth) mock modular forms and therefore have modular anomalies. I'll present the explicit form for their modular completions which becomes particularly simple in the presence of a refinement. In this case the construction suggests that the refinement induces a quantization of the corresponding moduli space. I'll also discuss implications of these results for Vafa-Witten theory and holomorphic anomaly equations for BPS partition functions.
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