S-duality in twistor space

A bstract In type IIB string compactifications on a Calabi-Yau threefold, the hypermultiplet moduli space MH must carry an isometric action of the modular group SL(2 , Z ), inherited from the S-duality symmetry of type IIB string theory in ten dimensions. We investigate how this modular symmetry is realized at the level of the twistor space of MH , and construct a general class of SL(2 , Z )-invariant quaternion-Kähler metrics with two commuting isometries, parametrized by a suitably covariant family of holomorphic transition functions. This family should include MH corrected by D3-D1-D(-1)-instantons (with five-brane corrections ignored) and, after taking a suitable rigid limit, the Coulomb branch of five-dimensional N = 2 gauge theories compactified on a torus, including monopole string instantons. These results allow us to considerably simplify the derivation of the mirror map between type IIA and IIB fields in the sector where only D1-D(-1)-instantons are retained.