Date/Time
Date(s) - 08/03/2024
1:30 pm - 2:30 pm

Location: HH 410

Speaker: Amirmasoud Geevechi (University of Toronto)

Title: Slow Motion of Vortex filaments in the Abelian Higgs Model

Abstract:
Abelian Higgs model is a system of partial differential equations for describing the interaction of the Higgs field and the electromagnetic field. The equations enjoy some local symmetry or also called gauge symmetry. The solutions to the time-independent and 2D version of the equations have been constructed by Jaffe and Taubes in 1980. These are called vortex configurations. In this talk, I will mention the main result of my thesis with Prof. Robert Jerrard about how one can glue the 2D solutions and add perturbations to them in order to construct time-dependent solutions in (3+1)D. The final result is that one can construct solutions in (3+1)D arbitrary close to wave maps to the moduli space of vortex configurations, for long time. This is a generalization of a result by David Stuart in 1994 where the dynamics in (2+1)D has been approximated by a Hamiltonian dynamics on the moduli space. We will see that suitable gauge
conditions are crucial in various steps of the construction in order to make the equations massive and stable. This is the so-called Higgs mechanism. Also, I will mention that one main difference between (3 + 1)D and (2 + 1)D is that the gauge flexibility cannot treat all of the variables in the same way and one has to use more explicit structure of the equations.