Vortices and Jacobian varieties

Details Vortices and Jacobian varieties Vortices and Jacobian varieties

2010-10-04

arxiv.org/abs/1010.0644v1

J.Geom.Phys.61:1135-1155, 2011

Physics

High Energy Physics

High Energy Physics - Theory

36 pages, 2 figures

Scientific paper

10.1016/j.geomphys.2011.02.017

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.

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