Résumé:

I'll review the progress achieved in constructing the non-perturbative hypermultiplet moduli space using twistorial methods. The metric on this space determines the low-energy effective action of Type II string theory compactified on a Calabi-Yau threefold. It receives instanton corrections due to D-branes and NS5-branes and I'll show how all of them can be nicely incorporated at the level of the twistor space. I'll argue that the inclusion of NS5-brane instantons might be closely related to quantization of a certain integrable structure. If the time permits, I'll present also a conjecture for the Witten index of 4d N=2 gauge theories inspired by the twistorial description of the hypermultiplet moduli space.