Vortices and Jacobian varieties

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.Comment: 36 pages, 2 figure