Vortices and Jacobian varieties

Abstract

We investigate the geometry of the moduli space of N vortices on line bundles over a closed Riemann surface Σ 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 Σ . For N = 1 , we show that the metric on the moduli space converges to a natural Bergman metric on Σ . 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 of Σ at degree N . We describe consequences of this phenomenon from the point of view of multivortex dynamics.