Vortices and the real time dynamics of the Abelian Higgs model in 2D

I will discuss the role of topological fluctuations (vortices) in the dynamics of the Abelian Higgs model in two spatial dimensions. I will describe how long lived clusters of like sign vortices appear in the cooling of type II materials (when the Higgs is heavier than the gauge field), and predict some of their properties. In type I (when the Higgs is light) I will show that vortices are intimatelly connected with the nucleation of the low temperature superconducting phase and that they undergo a clustering instability analogous to that of gravitational systems. The Abelian Higgs model describes in some circumstances the long wave length behavior of superconductors. Some experimental signals of our results are discussed in this context.