Linear perturbations of quaternionic metrics

TL;DR

This paper develops a twistor-based framework for analyzing linear deformations of quaternionic-Kahler manifolds, providing new tools to study their moduli spaces, with applications to string theory hypermultiplet moduli spaces.

Contribution

It introduces a twistor approach to encode deformations of quaternionic-Kahler manifolds directly via their twistor space, bypassing the Swann bundle, and applies this to cases with isometries in string theory.

Findings

Deformations are encoded in variations of complex contact transformations on twistor space.

The method applies to quaternionic-Kahler metrics with commuting isometries.

Illustrations include hypermultiplet moduli spaces in string theory.

Abstract

We extend the twistor methods developed in our earlier work on linear deformations of hyperkahler manifolds [arXiv:0806.4620] to the case of quaternionic-Kahler manifolds. Via Swann's construction, deformations of a 4d-dimensional quaternionic-Kahler manifold $M$ are in one-to-one correspondence with deformations of its $4d+4$-dimensional hyperkahler cone $S$. The latter can be encoded in variations of the complex symplectomorphisms which relate different locally flat patches of the twistor space $Z_S$, with a suitable homogeneity condition that ensures that the hyperkahler cone property is preserved. Equivalently, we show that the deformations of $M$ can be encoded in variations of the complex contact transformations which relate different locally flat patches of the twistor space $Z_M$ of $M$, by-passing the Swann bundle and its twistor space. We specialize these general results to the case of quaternionic-Kahler metrics with $d+1$ commuting isometries, obtainable by the Legendre transform method, and linear deformations thereof. We illustrate our methods for the hypermultiplet moduli space in string theory compactifications at tree- and one-loop level.