Résumé
This thesis addresses three problems arising in type II string theory compactified on a Calabi-Yau manifold. In the first one we study the hypermultiplet moduli space (HM), by working on its twistor space. Using data derived via mirror symmetry and S-duality, we compute NS5-instanton corrections to the HM metric in the one-instanton approximation.
These corrections are weighted by D4-D2-D0 BPS indices, which coincide with rank 0 Donaldson-Thomas invariants and count the (signed) number of BPS black hole microstates. These invariants exhibit wall-crossing behavior and induce a Riemann-Hilbert problem. This problem can describe many setups, including the D-instanton corrected twistor space of the HM in type II string theory and is of independent mathematical interest. We consider a quantum deformation of the RH problem, induced by the refined BPS indices. Using a formulation of the problem in terms of a non-commutative Moyal star product, we provide a perturbative solution to it. From the adjoint form of this solution, we identify a generating function for coordinates on the still mysterious quantum analog of the twistor space.
Finally, we study the modular properties of the D4-D2-D0 BPS indices, more precisely of their generating functions. It was previously argued, using S-duality, that the generating functions are higher depth mock modular forms. Moreover, they satisfy a modular completion equation, which fixes their shadow in terms of other (lower rank) generating functions. We start by bringing about a significant simplification to these equations and recovering subtle contributions that were overlooked. Then, we provide (a recipe for) solutions to these modular completion equations, up to all the holomorphic modular ambiguities that need to be fixed independently. For this, we use indefinite generalized theta series and Jacobi-like forms to write the solutions.