Quaternionic K\"ahler and hyperK\"ahler manifolds with torsion and twistor spaces

TL;DR

This paper explores the geometric structures of quaternionic Kähler and hyperKähler manifolds with torsion, linking their properties to twistor spaces and supersymmetric sigma models with Wess-Zumino terms.

Contribution

It establishes a novel connection between the geometry of QKT and HKT manifolds and the structures of their twistor spaces, especially regarding the Swann bundle and almost hermitian structures.

Findings

Swann bundle of QKT admits HKT structure with symmetry

Twistor space admits an almost hermitian structure with skew-symmetric Nijenhuis tensor

Connects geometric structures to supersymmetric quantum field theories

Abstract

The target space of a (4,0) supersymmetric two-dimensional sigma model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy contained in Sp(n)Sp(1) (resp. Sp(n)), QKT (resp. HKT)-spaces. We study the geometry of QKT, HKT manifold and their twistor spaces. We show that the Swann bundle of a QKT manifold admits a HKT structure with special symmetry if and only if the twistor space of the QKT manifold admits an almost hermitian structure with totally skew-symmetric Nijenhuis tensor, thus connecting two structures arising from quantum field theories and supersymmetric sigma models with Wess-Zumino term.