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

Abstract

AbstractWe 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 $$n\ge 3$$ n ≥ 3 . This functional is the natural generalisation of the Ginzburg–Landau model for superconductivity to the non-Euclidean setting. We prove a $$\Gamma $$ Γ -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 $$n-2$$ n - 2 , while the curvature of minimisers converges to a solution of the London equation.