Quillen bundle and Geometric Prequantization of Non-Abelian Vortices on a Riemann surface

TL;DR

This paper advances the geometric prequantization of non-abelian vortex moduli spaces on Riemann surfaces by explicitly constructing a Quillen determinant line bundle with a modified metric, linking symplectic geometry and quantum line bundles.

Contribution

It explicitly calculates the symplectic form and constructs a Quillen determinant line bundle with a modified metric for non-abelian vortices, extending known abelian results.

Findings

Explicit symplectic form from the $L^2$ metric

Construction of a Quillen line bundle with a modified metric

Family of symplectic forms parametrized by a section $ ext{Ψ}_0$

Abstract

In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising from the $L^2$ metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum line bundle turns out to be Quillen's determinant line bundle with a modified Quillen metric. Next, as in the case of abelian vortices, we construct Quillen line bundles over the moduli space whose curvatures form a family of symplectic forms which are parametrised by $\Psi_0$, a section of a certain bundle.