Vortices and Jacobian varieties

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.

36 pages, 2 figures

References in corpus (3)

• On the L^2-metric of vortex moduli spaces
• One-vortex moduli space and Ricci flow
• On the curvature of vortex moduli spaces

Cited by in corpus (13)

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• Vortex Motion on Surfaces of Small Curvature
• On the curvature of vortex moduli spaces
• Moduli of vortices and Grassmann manifolds
• The Volume of the Quiver Vortex Moduli Space
• Higgs and Coulomb Branch Descriptions of the Volume of the Vortex Moduli Space
• Vortex counting and the quantum Hall effect
• On the Kaehler metrics over ${mathrm{Sym}^{d}(X)$
• Kähler quantization of vortex moduli
• Vortices and the Abel-Jacobi map
• Pairs of pants, Pochhammer curves and $L^2$-invariants
• The ground state energy of a charged particle on a Riemann surface
• Non-abelian vortices on CP^1 and Grassmannians