Conifold singularities, resumming instantons and non-perturbative mirror symmetry

We determine the instanton corrected hypermultiplet moduli space in type IIB compactifications near a Calabi-Yau conifold point where the size of a two-cycle shrinks to zero. We show that D1-instantons resolve the conifold singularity caused by worldsheet instantons. Furthermore, by resumming the instanton series, we reproduce exactly the results obtained by Ooguri and Vafa on the type IIA side, where membrane instantons correct the hypermultiplet moduli space. Our calculations therefore establish that mirror symmetry holds non-perturbatively in the string coupling.