S-duality in Twistor Space

TL;DR

This paper explores how S-duality symmetry is realized in the twistor space of the hypermultiplet moduli space in type IIB string theory, constructing invariant metrics and simplifying the mirror map derivation.

Contribution

It constructs SL(2,Z)-invariant quaternion-Kahler metrics with two isometries, applicable to corrected moduli spaces and gauge theory Coulomb branches.

Findings

Constructed SL(2,Z)-invariant metrics with two isometries.

Provided a simplified derivation of the mirror map for D1-D(-1)-instantons.

Included instanton corrections in the moduli space framework.

Abstract

In type IIB string compactifications on a Calabi-Yau threefold, the hypermultiplet moduli space $M_H$ 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 $M_H$, and construct a general class of SL(2,Z)-invariant quaternion-Kahler metrics with two commuting isometries, parametrized by a suitably covariant family of holomorphic transition functions. This family should include $M_H$ corrected by D3-D1-D(-1)-instantons (with fivebrane 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.