HyperK\"{a}hler prequantization of the Hitchin system and Chern-Simons theory with complex gauge group

TL;DR

This paper explicitly constructs the hyperK"ahler structure on the Hitchin moduli space, demonstrates the existence of three prequantum line bundles with curvatures matching the symplectic forms, and relates these to Chern-Simons theory with complex gauge group.

Contribution

It provides a detailed hyperK"ahler structure construction and links prequantum line bundles to Chern-Simons theory, filling gaps in Hitchin's original work.

Findings

Explicit hyperK"ahler structure on the Hitchin moduli space.

Construction of three prequantum line bundles with compatible curvatures.

Connection of prequantum line bundles to Chern-Simons gauge theory.

Abstract

Hitchin has shown that the moduli space ${\mathcal M}$ of the dimensionally reduced self-dual Yang-Mills equations has a hyperK\"{a}hler structure. In this paper we first explicitly show the hyperK\"{a}hler structure, the details of which is missing in Hitchin's paper. We show here that ${\mathcal M}$ admits three pre-quantum line bundles, corresponding to the three symplectic forms. We use Quillen's determinant line bundle construction and show that the Quillen curvatures of these prequantum line bundles are proportional to each of the symplectic forms mentioned above. The prequantum line bundles are holomorphic with respect to their respective complex structures. We show how these prequantum line bundles can be derived from cocycle line bundles of Chern-Simons gauge theory with complex gauge group in the case when the moduli space is smooth.