The Yang–Mills–Higgs Functional on Complex Line Bundles: \(\Gamma \)-Convergence and the London Equation

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 \(n\ge 3\). 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.