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Differential Geometry Seminar
Adiabatic approximation of Abelian Higgs model in dimension 1+3
Adiabatic approximation of Abelian Higgs model in dimension 1+3
Organizers
Speaker
Time
Tuesday, December 10, 2024 3:00 PM - 4:00 PM
Venue
A7-101
Online
Zoom 638 227 8222
(BIMSA)
Abstract
In this talk, I will present our result about the approximation of the Abelian Yang-Mills-Higgs equations by wave maps. These equations describe the interaction of the Higgs field and the electromagnetic field. They have $U(1)$ gauge symmetries and are hyperbolic. At the critical coupling, the finite energy and static solutions in dimension $2$ have been classified by Jaffe and Taubes in 1980, the so called (anti)-vortex configurations and they form a moduli space which is a smooth Riemannian manifold. In this talk I will explain that one can find adiabatic solutions in dimension $(1+3)$ whose dynamics are approximated by wave maps to the moduli space, a generalization of the geodesic approximation in dimension $(1+2)$ which has been proven by Stuart in 1994. We use the gluing method to construct solutions and take advantage of the so-called Higgs mechanism, the existence of a spectral gap for the linearization of the equations, under specific gauge conditions. This is a joint work with Robert Jerrard.