Vortices and Jacobian varieties

We investigate the geometry of the moduli space of N$N$ vortices on line bundles over a closed Riemann surface Σ$\Sigma$ of genus g>1$g>1$, in the little explored situation where 1≤N<g$1\le N<g$. In the regime where the area of the surface is just large enough to accommodate N$N$ 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$\Sigma$. For N=1$N=1$, we show that the metric on the moduli space converges to a natural Bergman metric on Σ$\Sigma$. When N>1$N>1$, 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$\Sigma$ at degree N$N$. We describe consequences of this phenomenon from the point of view of multivortex dynamics.