Geometric quantization of the moduli space of the Self-duality equations on a Riemann surface

TL;DR

This paper explores the geometric quantization of the moduli space of solutions to self-duality equations on a Riemann surface, clarifying its symplectic structure and constructing a prequantum line bundle using Quillen's method.

Contribution

It provides a detailed analysis of the symplectic structure of the moduli space and constructs a prequantum line bundle, extending Hitchin's work with new explicit details and methods.

Findings

Clarified the symplectic structure of the moduli space

Constructed a prequantum line bundle using Quillen's method

Proved the Quillen curvature is proportional to the symplectic form

Abstract

The self-duality equations on a Riemann surface arise as dimensional reduction of self-dual Yang-Mills equations. Hitchin had showed that the moduli space ${\mathcal M}$ of solutions of the self-duality equations on a compact Riemann surface of genus $g >1$ has a hyperK\"{a}hler structure. In particular ${\mathcal M}$ is a symplectic manifold. In this paper we elaborate on one of the symplectic structures, the details of which is missing in Hitchin's paper. Next we apply Quillen's determinant line bundle construction to show that ${\mathcal M}$ admits a prequantum line bundle. The Quillen curvature is shown to be proportional to the symplectic form mentioned above. We do it in two ways, one of them is a bit unnatural (published in R.O.M.P.) and a second way which is more natural.