Geometric Prequantization of the Moduli Space of the Vortex equations on a Riemann surface

TL;DR

This paper explicitly constructs a family of symplectic and prequantum line bundle structures on the moduli space of vortex solutions on a Riemann surface, enriching the geometric quantization framework.

Contribution

It introduces a family of symplectic structures parametrized by sections of line bundles and constructs corresponding prequantum line bundles, advancing geometric quantization of vortex moduli spaces.

Findings

Explicit symplectic structures on vortex moduli space

Family of prequantum line bundles with curvature proportional to symplectic forms

Enhanced understanding of geometric quantization for vortex equations

Abstract

The moduli space of solutions to the vortex equations on a Riemann surface are well known to have a symplectic (in fact K\"{a}hler) structure. We show this symplectic structure explictly and proceed to show a family of symplectic (in fact, K\"{a}hler) structures $\Omega_{\Psi_0}$ on the moduli space, parametrised by $\Psi_0$, a section of a line bundle on the Riemann surface. Next we show that corresponding to these there is a family of prequantum line bundles ${\mathcal P}_{\Psi_0} $on the moduli space whose curvature is proportional to the symplectic forms $\Omega_{\Psi_0}$.