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arXiv · 1010.0644

Vortices and Jacobian varieties

Nicholas S. Manton·Nuno M. Romão
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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.

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Nicholas S. Manton, Nuno M. Romão. 2010-10-04. Vortices and Jacobian varieties. https://doi.org/10.1016/j.geomphys.2011.02.017

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