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

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