The Yang-Mills-Higgs functional on complex line bundles: $\Gamma$-convergence and the London equation

TL;DR

This paper studies the Abelian Yang-Mills-Higgs functional on complex line bundles over manifolds, proving convergence results that connect minimiser energies to minimal surfaces and solutions of the London equation.

Contribution

It establishes a $\Gamma$-convergence result for the functional in the non-self dual scaling, linking minimiser behavior to geometric and physical equations.

Findings

Minimiser energy concentrates on area-minimising surfaces.

Curvature converges to solutions of the London equation.

Results extend Ginzburg-Landau models to curved settings.

Abstract

We consider the Abelian Yang-Mills-Higgs functional, in the non-self dual scaling, on a complex line bundle over a closed Riemannian manifold of dimension $n\geq 3$. This functional is the natural generalisation of the Ginzburg-Landau model for superconductivity to the non-Euclidean setting. We prove a $\Gamma$-convergence result, in the strongly repulsive limit, on the functional rescaled by the logarithm of the coupling parameter. As a corollary, we prove that the energy of minimisers concentrates on an area-minimising surface of dimension $n-2$, while the curvature of minimisers converges to a solution of the London equation.