Speaker
Description
In this talk, I will discuss a subclass of gauged nonlinear sigma models with gauge group U(1) and a general Kaehler target. At critical coupling, such models admit topological soliton solutions called BPS vortices, which solve a system of first order PDEs called the Bogomolny equations. They attain a topological lower energy bound characterized by the degree of a holomorphic map. The most famous example is when the target is the complex plane, describing the Abelian-Higgs model.
I will present a geometric formalism to compute the second variation of the energy functional around a general BPS vortex (whose existence we establish for a compact target). On a non-compact domain, such as R^2, it is a nontrivial question whether the associated Jacobi operator has any L^2 normalisable eigenfunctions. Such eigenfunctions are called shape modes, and, if present, they have a strong effect on the low energy scattering behaviour of vortices. We prove (on R^2) that all vortices have at least one shape mode, and that such shape modes can be constructed by solving a simple scalar Schroedinger PDE. This is based on joint work with Derek Harland and Martin Speight.