The Generalised Complex Geometry of $(p,q)$ Hermitian Geometries

TL;DR

This paper extends the framework of generalized complex geometry to encompass various $(p,q)$ Hermitian geometries arising from different supersymmetric sigma models, unifying them under a common geometric formulation.

Contribution

It introduces a generalized complex geometry formulation for $(p,q)$ Hermitian geometries, broadening Gualtieri's approach to include new cases like $(4,2)$ and $(4,1)$.

Findings

Unified geometric description of $(p,q)$ Hermitian geometries.

Explicit formulas for the map to generalized geometry.

Extension of Gualtieri's formulation to new supersymmetric cases.

Abstract

We define $(p,q)$ hermitian geometry as the target space geometry of the two dimensional $(p,q)$ supersymmetric sigma model. This includes generalised K\"{a}hler geometry for $(2,2)$, generalised hyperk\"{a}hler geometry for $(4,2)$, strong K\"{a}hler with torsion geometry for $(2,1)$ and strong hyperk\"{a}hler with torsion geometry for $(4,1)$. We provide a generalised complex geometry formulation of hermitian geometry, generalising Gualtieri's formulation of the $(2,2)$ case. Our formulation involves a chiral version of generalised complex structure and we provide explicit formulae for the map to generalised geometry.