Vortices and Jacobian varieties

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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. © 2011 Elsevier B.V.

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Manton, N. S., & Romão, N. M. (2011). Vortices and Jacobian varieties. Journal of Geometry and Physics, 61(6), 1135–1155. https://doi.org/10.1016/j.geomphys.2011.02.017

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