The abelian Yang-Mills-Higgs equations in two space dimensions admit topological vortex solutions. I will discuss the asymptotic stability of the degree-one vortex at self-dual coupling, for small equivariant perturbations. An important feature of the problem is the presence of an internal mode in the spectral gap of the linearized operator. The interaction between this discrete mode and the continuous spectrum leads to a damping mechanism, quantified by a nonlinear Fermi Golden rule. The main difficulties in the proof of stability are 1) The slow decay of the internal mode. 2) Proving decay for the projection of the perturbation onto the continuous spectrum, which satisfies a Klein-Gordon equation with a matrix potential. The latter is achieved by combining the spacetime resonance method for the flat Klein-Gordon equation with local energy decay for the linearized operator.

This is joint work with José Palacios, Fabio Pusateri, Jonas Lührmann, and Wilhelm Schlag.