A Hitchin Connection for a large class of families of K\"ahler Structures

TL;DR

This paper constructs a generalized Hitchin connection for a broad class of compatible complex structures on symplectic manifolds, extending previous work and applicable to various geometric settings.

Contribution

It introduces a new Hitchin connection applicable to arbitrary prequantizable symplectic manifolds satisfying a Fano type condition, broadening the scope of previous constructions.

Findings

Hitchin connection constructed for large class of Kähler structures

Connection is well-defined in neighborhoods of natural complex structures

Applicable to flat space, tori, and moduli spaces of flat connections

Abstract

In this paper we construct a Hitchin connection in a setting, which significantly generalizes the setting covered by the first author previously, which in turn was a generalisation of the moduli space case covered by Hitchin in his original work on the Hitchin connection. In fact, our construction provides a Hitchin connection, which is a partial connection on the space of all compatible complex structures on an arbitrary, but fixed prequantizable symplectic manifold, which satisfies a certain Fano type condition. The subspace of the tangent space to the space of compatible complex structures on which the constructed Hitchin connection is defined, is in fact of finite co-dimension, if the symplectic manifold is compact. In a number of examples, including flat symplectic space, symplectic tori and moduli spaces of flat connections for a compact Lie group, we prove that our Hitchin connection is defined in a neighbourhood of the natural families of complex structures compatible with the given symplectic form, which these spaces admits.