Quillen-type bundle and geometric prequantization on moduli space of the Seiberg-Witten equations on product of Riemann surfaces

TL;DR

This paper establishes a symplectic structure on the moduli space of Seiberg-Witten equations on a product of Riemann surfaces and constructs a Quillen-type line bundle for geometric prequantization.

Contribution

It demonstrates the existence of a symplectic form and constructs a Quillen-type determinant line bundle on the moduli space, enabling geometric prequantization.

Findings

Symplectic structure on the moduli space is proven to exist.

A Quillen-type determinant line bundle is constructed.

Curvature of the line bundle is proportional to the symplectic form.

Abstract

We show the existence of a symplectic structure on the moduli space of the Seiberg-Witten equations on $\Sigma \times \Sigma$ where $\Sigma$ is a compact oriented Riemann surface. To prequantize the moduli space, we construct a Quillen-type determinant line bundle on it and show its curvature is proportional to the symplectic form.