Twistor spaces for HKT manifolds

TL;DR

This paper constructs a twistor space for HKT manifolds, revealing its complex structure and how it encodes the metric and torsion, thus advancing the geometric understanding of supersymmetric sigma models.

Contribution

It introduces a twistor space for HKT manifolds with a natural complex structure and provides a reconstruction theorem linking the manifold's data to the twistor space.

Findings

Twistor space has a natural complex structure.

HKT metric and torsion can be reconstructed from the twistor space.

Sigma model superfields are described as holomorphic maps into the twistor space.

Abstract

We construct the twistor space associated with an HKT manifold, that is, a hyper-K\"ahler manifold with torsion, a type of geometry that arises as the target space geometry in two-dimensional sigma models with (4,0) supersymmetry. We show that this twistor space has a natural complex structure and is a holomorphic fibre bundle over the complex projective line with fibre the associated HKT manifold. We also show how the metric and torsion of the HKT manifold can be determined from data on the twistor space by a reconstruction theorem. We give a geometric description of the sigma model (4,0) superfields as holomorphic maps (suitably understood) from a twistorial extension of (4,0) superspace (harmonic superspace) into the twistor space of the sigma model target manifold and write an action for the sigma model in terms of these (4,0) superfields.