Résumé
BPS indices encoding the entropy of supersymmetric black holes in compactifications of Type II string theory on Calabi-Yau threefolds, known in mathematics as generalized Donaldson-Thomas invariants, possess remarkable (mock) modular properties. I'll explain the physical origin of mock modularity and show, for a set of one-parameter threefolds, how it can be used, together with wall-crossing and direct integration of topological string, to compute the BPS indices and other topological invariants. As a result, one obtains explicit (mock) modular functions encoding infinite sets of D4-D2-D0 BPS indices as well as new boundary conditions for the holomorphic anomaly equation of the topological string partition function allowing to overcome the limitations of the direct integration method. In the end, I'll present prelimenary results on the asymptotic growth of DT invariants hinting for the existence of some phase transitions.