Vortices and the Abel–Jacobi map

Abstract

The abelian Higgs model on a compact Riemann surface Σ supports vortex solutions for any positive vortex number d∈Z. Moreover, the vortex moduli space for fixed d has long been known to be the symmetrized d-th power of Σ, in symbols, Symd(Σ). This moduli space is Kähler with respect to the physically motivated metric whose geodesics describe slow vortex motion.In this paper we appeal to classical properties of Symd(Σ) to obtain new results for the moduli space metric. Our main tool is the Abel–Jacobi map, which maps Symd(Σ) into the Jacobian of Σ. Fibres of the Abel–Jacobi map are complex projective spaces, and the first theorem we prove states that near the Bradlow limit the moduli space metric restricted to these fibres is a multiple of the Fubini–Study metric. Additional significance is given to the fibres of the Abel–Jacobi map by our second result: we show that if Σ is a hyperelliptic surface, there exist two special fibres which are geodesic submanifolds of the moduli space. Even more is true: the Abel–Jacobi map has a number of fibres which contain complex projective subspaces that are geodesic.