Geometric Quantization of the Hitchin System

TL;DR

This paper develops a geometric quantization framework for the Hitchin moduli space by constructing a prequantum line bundle using the Quillen bundle and Higgs fields, enabling a holomorphic Hilbert space formulation.

Contribution

It introduces a novel quantization method for the Hitchin system by modifying the Quillen metric to match the Kahler form, establishing a holomorphic Hilbert space structure.

Findings

The Kahler form is shown to be integral.

The Quillen bundle descends as a prequantum line bundle.

Holomorphic square integrable sections form the Hilbert space.

Abstract

This paper is about geometric quantization of the Hitchin system. We quantize a Kahler form on the Hitchin moduli space (which is half the first Kahler form defined by Hitchin) by considering the Quillen bundle as the prequantum line bundle and modifying the Quillen metric using the Higgs field so that the curvature is proportional to the Kahler form. We show that this Kahler form is integral and the Quillen bundle descends as a prequantum line bundle on the moduli space. It is holomorphic and hence one can take holomorphic square integrable sections as the Hilbert space of quantization of the Hitchin moduli space.