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

TL;DR

This paper proves that self-dual $U(1)$-Yang-Mills-Higgs energies on a closed manifold converge to the $(n-2)$-area functional, establishing a link between gauge theory and minimal submanifolds through $ ext{Gamma}$-convergence and homology cycles.

Contribution

It demonstrates the $ ext{Gamma}$-convergence of the energies to the $(n-2)$-area functional and compares min-max values from geometric measure theory with gauge theory results.

Findings

Convergence of Jacobians to integral cycles dual to the Euler class.

Establishment of a lower bound for min-max $(n-2)$-area values via Yang-Mills-Higgs energies.

A Huisken-type monotonicity formula along the gradient flow of the energy.

Abstract

Given a hermitian line bundle $L\to M$ on a closed Riemannian manifold $(M^n,g)$, the self-dual Yang-Mills-Higgs energies are a natural family of functionals \begin{align*} &E_\epsilon(u,\nabla):=\int_M\Big(|\nabla u|^2+\epsilon^2|F_\nabla|^2+\frac{(1-|u|^2)^2}{4\epsilon^2}\Big) \end{align*} defined for couples $(u,\nabla)$ consisting of a section $u\in\Gamma(L)$ and a hermitian connection $\nabla$ with curvature $F_\nabla$. 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 $\epsilon\to 0$ (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 $\Gamma$-convergence of $E_\epsilon$ to ($2\pi$ times) the codimension two area: more precisely, given a family of couples $(u_\epsilon,\nabla_\epsilon)$ with $\sup_\epsilon E_\epsilon(u_\epsilon,\nabla_\epsilon)<\infty$, we prove that a suitable gauge invariant Jacobian $J(u_\epsilon,\nabla_\epsilon)$ converges to an integral $(n-2)$-cycle $\Gamma$, in the homology class dual to the Euler class $c_1(L)$, with mass $2 \pi \mathbb{M}(\Gamma)\le\liminf_{\epsilon \rightarrow 0}E_\epsilon(u_\epsilon,\nabla_\epsilon)$. 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 $(n-2)$-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 $E_{\epsilon}$.