Genus-one complex quantum Chern--Simons theory

TL;DR

This paper explores the geometric quantisation of genus-one complex Chern--Simons theory, introducing a complexified Hitchin connection and relating it to Witten's connection via a Bargmann transform, advancing understanding of moduli space quantisation.

Contribution

It introduces a complexified Hitchin connection in Kähler quantisation for genus-one surfaces and links it to Witten's connection through a Bargmann transform, expanding the geometric quantisation framework.

Findings

Defined a natural complexified Hitchin connection for genus-one surfaces.

Identified Witten's connection with the complexified Hitchin connection.

Connected polarised sections over moduli spaces via a Bargmann transform.

Abstract

We consider the geometric quantisation of Chern--Simons theory for closed genus-one surfaces and semisimple complex groups. First we introduce the natural complexified analogue of the Hitchin connection in K\"{a}hler quantisation, with polarisations coming from the nonabelian Hodge hyper-K\"{a}hler geometry of the moduli spaces of flat connections, thereby complementing the real-polarised approach of Witten. Then we consider the connection of Witten, and we identify it with the complexified Hitchin connection using a version of the Bargmann transform on polarised sections over the moduli spaces.