Twistor Spaces for QKT Manifolds

TL;DR

This paper explores the geometric structures of QKT manifolds arising in supersymmetric sigma models, constructing their twistor spaces and establishing links to hyper-K"ahler manifolds, with implications for theoretical physics and differential geometry.

Contribution

It introduces the construction of twistor spaces for QKT manifolds and demonstrates their properties, including conditions under which they are K"ahler and their relation to hyper-K"ahler manifolds.

Findings

QKT manifolds serve as target spaces in (4,0) supersymmetric sigma models.

Constructed twistor spaces are K"ahler with a complex contact structure under certain conditions.

Established a correspondence between QKT and hyper-K"ahler manifolds in specific dimensions.

Abstract

We find that the target space of two-dimensional (4,0) supersymmetric sigma models with torsion coupled to (4,0) supergravity is a QKT manifold, that is, a quaternionic K\"ahler manifold with torsion. We give four examples of geodesically complete QKT manifolds one of which is a generalisation of the LeBrun geometry. We then construct the twistor space associated with a QKT manifold and show that under certain conditions it is a K\"ahler manifold with a complex contact structure. We also show that, for every 4k-dimensional QKT manifold, there is an associated 4(k+1)-dimensional hyper-K\"ahler one.