Duality in hypermultiplet moduli spaces

The hypermultiplet moduli space in type II string theory compactified on a Calabi-Yau 3-folds provides a framework for a far-reaching generalization of classical and homological mirror symmetry, as well as a convenient packaging of BPS black hole degeneracies consistent with wall-crossing. In addition to the usual action of the monodromy group and discrete Peccei-Quinn symmetries, it should also be invariant under S-duality, which mixes the usual D-brane instantons (or objects in the derived/Fukaya category) with a new type of instantons (NS5-branes, or Kaluza-Klein monopoles). For rigid Calabi-Yau three-folds, the group degenerated by these generators can be identified as a Picard modular subgroup of SU(2,1). For non-rigid ones, it probably includes SL(3,Z) or even larger arithmetic groups. I will use these symmetries to obtain the contributions of NS5-branes at the semi-classical level.