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

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_ε(u,\nabla):=\int_M\Big(|\nabla u|^2+ε^2|F_\nabla|^2+\frac{(1-|u|^2)^2}{4ε^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 $2\pi \mathbb{M}(\Gamma )\le {\mathrm{liminf}}_{\epsilon \to 0}{E}_{\epsilon }(u_{\epsilon },{\nabla }_{\epsilon })$. 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 $E_{\epsilon }$.

52 pages; comments welcome