Vortices and Jacobian varieties

TL;DR

This paper explores the geometric structure of vortex moduli spaces on Riemann surfaces, revealing how their metrics relate to the Jacobian variety and how they degenerate in the dissolving limit, impacting vortex dynamics.

Contribution

It provides a detailed analysis of the moduli space geometry for N-vortices with N less than the genus, linking vortex metrics to the Jacobian and describing metric degeneration phenomena.

Findings

For N=1, the moduli space metric converges to a Bergman metric.

For N>1, the vortex metric degenerates on the Abel-Jacobi critical locus.

The degeneracy influences multivortex dynamic behavior.

Abstract

We investigate the geometry of the moduli space of N-vortices on line bundles over a closed Riemann surface of genus g > 1, in the little explored situation where 1 =< N < g. In the regime where the area of the surface is just large enough to accommodate N vortices (which we call the dissolving limit), we describe the relation between the geometry of the moduli space and the complex geometry of the Jacobian variety of the surface. For N = 1, we show that the metric on the moduli space converges to a natural Bergman metric on the Riemann surface. When N > 1, the vortex metric typically degenerates as the dissolving limit is approached, the degeneration occurring precisely on the critical locus of the Abel-Jacobi map at degree N. We describe consequences of this phenomenon from the point of view of multivortex dynamics.