Convergence of the self‐dual U(1)‐Yang–Mills–Higgs energies to the (n−2)$(n-2)$‐area functional

Abstract

AbstractGiven a hermitian line bundle on a closed Riemannian manifold , the self‐dual Yang–Mills–Higgs energies are a natural family of functionals defined for couples consisting of a section  and a hermitian connection ∇ with curvature . While the critical points of these functionals have been well‐studied in dimension two by the gauge theory community, it was shown in [52] that critical points in higher dimension converge as (in an appropriate sense) to minimal submanifolds of codimension two, with strong parallels to the correspondence between the Allen–Cahn equations and minimal hypersurfaces. In this paper, we complement this idea by showing the Γ‐convergence of to (2π times) the codimension two area: more precisely, given a family of couples with , we prove that a suitable gauge invariant Jacobian converges to an integral ‐cycle Γ, in the homology class dual to the Euler class , with mass . We also obtain a recovery sequence, for any integral cycle in this homology class. Finally, we apply these techniques to compare min‐max values for the ‐area from the Almgren–Pitts theory with those obtained from the Yang–Mills–Higgs framework, showing that the former values always provide a lower bound for the latter. As an ingredient, we also establish a Huisken‐type monotonicity result along the gradient flow of .