(Mock-)modularity of BPS black holes

I'll show that the generating function of degeneracies of black holes appearing in superstring theory compactified on a Calabi-Yau manifold possesses remarkable modular properties. These properties are a non-trivial consequence of string dualities and imply that the generating function has a modular anomaly of a specific type. The anomaly can be removed by constructing a modular completion of the generating function which will be presented explicitly. These results appear to be at the intersection of several branches of physics and mathematics such that string theory compactifications, BPS state counting and wall-crossing, mock modular functions and indefinite theta-series, quaternionic manifolds and their twistor spaces. I'll try to give at least a minimal introduction to each subject which should provide their basic understanding.