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

2021/03/26 by Parise, Davide, Pigati, Alessandro, Stern, Daniel · 1 citation
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2103.14615

Abstract

Given a hermitian line bundle L→ M on a closed Riemannian manifold (Mⁿ,g), the self-dual Yang-Mills-Higgs energies are a natural family of functionals amp;E_ε(u,∇):=∫_M(|∇ u|²+ε²|F_∇|²+((1-|u|²)²)/(4ε²)) defined for couples (u,∇) consisting of a section u∈Γ(L) and a hermitian connection ∇ with curvature F_∇. While the critical points of these functionals have been well-studied in dimension two by the gauge theory community, it was shown in previous work of the second- and third-named authors that critical points in higher dimension converge as ε→ 0 (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 E_ε to (2π times) the codimension two area: more precisely, given a family of couples (u_ε,∇_ε) with sup_εE_ε(u_ε,∇_ε)

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