2010/10/04 by N. S. Manton, Nicholas S. Manton, Nuno M. Romão
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Applied mathematics #Geography #Jacobian matrix and determinant #Mathematics #Meteorology #Nonlinear Waves and Solitons #Pure mathematics #Quantum chaos and dynamical systems #Vortex #hep-th #math.AG
paper · pdf · doi:10.1016/j.geomphys.2011.02.017
published as J.Geom.Phys.61:1135-1155, 2011 · 36 pages, 2 figures
arxiv created 2010/10/04 · openalex publication_date 2011/02/22 · arxiv updated 2015/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.