Generalized Kahler geometry

TL;DR

This paper explores the fundamental aspects of generalized Kahler geometry, its equivalence with bi-Hermitian structures, and its connections to holomorphic Dirac geometry, deformation theory, and gerbes, extending classical geometric concepts.

Contribution

It provides a comprehensive analysis of generalized Kahler geometry, establishing its relations to bi-Hermitian structures, holomorphic Dirac geometry, and deformation theory, and discusses analogies with pre-quantum line bundles.

Findings

Equivalence with bi-Hermitian geometry on 2D sigma model targets

Connection to holomorphic Dirac geometry and derived deformation theory

Analogy between pre-quantum line bundles and gerbes in this context

Abstract

Generalized Kahler geometry is the natural analogue of Kahler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure. We explore the fundamental aspects of this geometry, including its equivalence with the bi-Hermitian geometry on the target of a 2-dimensional sigma model with (2,2) supersymmetry, as well as the relation to holomorphic Dirac geometry and the resulting derived deformation theory. We also explore the analogy between pre-quantum line bundles and gerbes in the context of generalized Kahler geometry.