  We investigate quantum corrections to the hypermultiplet moduli space M in
Calabi-Yau compactifications of type II string theories, with particular
emphasis on instanton effects from Euclidean NS5-branes. Based on the
consistency of D- and NS5-instanton corrections, we determine the topology of
the hypermultiplet moduli space at fixed string coupling, as previewed in
arXiv:1009.3026. On the type IIB side, we compute corrections from
(p,k)-fivebrane instantons to the metric on M (specifically, the correction to
the complex contact structure on its twistor space Z) by applying S-duality to
the D-instanton sum. For fixed fivebrane charge k, the corrections can be
written as a non-Gaussian theta series, whose summand for k=1 reduces to the
topological A-model amplitude. By mirror symmetry, instanton corrections
induced from the chiral type IIA NS5-brane are similarly governed by the wave
function of the topological B-model. In the course of this investigation we
clarify charge quantization for coherent sheaves and find hitherto unnoticed
corrections to the Heisenberg, monodromy and S-duality actions on M, as well as
to the mirror map for Ramond-Ramond fields and D-brane charges.
