Nonabelian Yang-Mills-Higgs and Plateau's problem in codimension three

Abstract

We investigate the asymptotic behavior of the $\mathrm{SU}(2)$-Yang-Mills-Higgs energy $E(\Phi ,A)={\int }_M|d_A\Phi {|}^2+|F_A{|}^2$ in the large mass limit, proving convergence to the codimension-three area functional in the sense of De Giorgi's $\Gamma$-convergence. More precisely, for a compact manifold with boundary $M$ and any family of pairs ${\Phi }_m\in {\Omega }^0(M;\mathfrak{su}(2))$ and $A_m\in {\Omega }^1(M;\mathfrak{su}(2))$ indexed by a mass parameter $m\to \infty$, satisfying $$E({\Phi }_m,A_m)\le Cm\text{and}\underset{m\to \infty }{\lim }\frac{1}{m}{\int }_M(m-|{\Phi }_m|{)}^2=0,$$ we prove that the $(n-3)$-currents dual to $\frac{1}{2\pi m}\mathrm{tr}(d_{A_m}{\Phi }_m\wedge F_{A_m})$ converge subsequentially to a relative integral $(n-3)$-cycle $T$ of mass \begin{equation} \mathbb{M}(T)\leq \liminf_{m\to\infty}\frac{1}{4πm}E(Φ_m,A_m), \end{equation} and show conversely that any integral $(n-3)$-current $T$ with $[T]=0\in H_{n-3}(M,\partial M;\mathbb{Z})$ admits such an approximation, with equality in the above inequality. In the special case of pairs $({\Phi }_m,A_m)$ satisfying the generalized monopole equation $\ast d_{A_m}{\Phi }_m=F_{A_m}\wedge \Theta$ for a calibration form $\Theta \in {\Omega }^{n-3}(M)$, we deduce that the limit $\nu ={\lim }_{m\to \infty }\frac{1}{2\pi m}|d_{A_m}{\Phi }_m{|}^2$ of the Dirichlet energy measures satisfies $\nu \le |T|$, with equality if and only if $T$ is calibrated by $\Theta$, giving evidence for predictions of Donaldson-Segal in the settings of $G_2$-manifolds and Calabi-Yau $3$-folds.

arXiv comment37 pages