Periodic Solutions of the Abelian Higgs Model and Rigid Rotation of Vortices

Abstract.

The metric and potential energy on the reduced moduli space of selfdual vortices in the Abelian Higgs model on $S^2$ are computed in a certain limit, first identified by Bradlow. In this limit it is proved that the Higgs field is asymptotic to a standard holomorphic section. These results are then used to prove a theorem asserting the existence of time-periodic solutions of the Abelian Higgs model on ${\Bbb R} \times S^2$ which represent two vortices in rigid rotation about one another. The theorem answers affirmatively the question, raised by Jaffe and Taubes, of whether a balance between the inter-vortex attraction and the centrifugal repulsion provides for the existence of such solutions (as it does in the classical two body problem for point particles.) The starting point of the analysis is the adiabatic limit system, i.e. the Hamiltonian system defined by restricting the Abelian Higgs model to the moduli space of self-dual vortices. The Hamiltonian consists of a potential energy term and kinetic energy term which is given by the metric on the moduli space induced from $L^2$. It is shown under two assumptions on the metric and potential energy that the adiabatic limit system admits periodic solutions of the required type. Periodic solutions to the full system are then obtained by an application of the implicit function theorem. Explicit examples where the assumptions on the adiabatic limit system hold are provided by the computations of the metric and potential in the Bradlow limit.