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

Davide Parise, Alessandro Pigati, Daniel Stern

TL;DR

This work establishes a robust bridge between gauge theory and codimension-three minimal geometry by proving that the SU(2) Yang–Mills–Higgs energy, in a large-mass (or small-\epsilon) regime, Γ-converges to the codimension-three area functional. The authors construct a current-based description via $Z(\Phi,A)$ and show that, under natural bounded-energy and Higgs-field concentration conditions, the rescaled energy concentrates on integral $(n-3)$-currents $T$, with a sharp mass inequality and a recovery sequence for any such current. In the monopole regime with a calibrating form $Θ$, the limiting energy measure is controlled by the current’s mass and is calibrated when the current is calibrated, providing gauge-theoretic evidence for predictions in $G_2$- and Calabi–Yau settings (Donaldson–Segal program). Moreover, the paper demonstrates that codimension-three Plateau problems can be solved variationally via SU(2)–Higgs–Yang–Mills theory: minimizers (or minimizing sequences) of the energy converge to area-minimizing $(n-3)$-currents, and the framework extends to boundary-value problems and boundary-gluing constructions. Overall, the results offer a concrete adiabatic-disintegration dictionary between gauge theory and calibrated geometry in higher codimension, with potential impacts on enumerative invariants and calibrated submanifold theory.

Abstract

We investigate the asymptotic behavior of the $\mathrm{SU}(2)$-Yang-Mills-Higgs energy $E(Φ,A)=\int_M|d_AΦ|^2+|F_A|^2$ in the large mass limit, proving convergence to the codimension-three area functional in the sense of De Giorgi's $Γ$-convergence. More precisely, for a compact manifold with boundary $M$ and any family of pairs $Φ_m\inΩ^0(M;\mathfrak{su}(2))$ and $A_m\in Ω^1(M;\mathfrak{su}(2))$ indexed by a mass parameter $m\to\infty$, satisfying $$E(Φ_m,A_m)\leq Cm\quad\text{and}\quad\lim_{m\to\infty}\frac{1}{m}\int_M(m-|Φ_m|)^2=0,$$ we prove that the $(n-3)$-currents dual to $\frac{1}{2πm}\mathrm{tr}(d_{A_m}Φ_m\wedge F_{A_m})$ 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 $(Φ_m,A_m)$ satisfying the generalized monopole equation $*d_{A_m}Φ_m=F_{A_m}\wedge Θ$ for a calibration form $Θ\in Ω^{n-3}(M)$, we deduce that the limit $ν=\lim_{m\to\infty}\frac{1}{2πm}|d_{A_m}Φ_m|^2$ of the Dirichlet energy measures satisfies $ν\leq |T|$, with equality if and only if $T$ is calibrated by $Θ$, giving evidence for predictions of Donaldson-Segal in the settings of $G_2$-manifolds and Calabi-Yau $3$-folds.

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

TL;DR

This work establishes a robust bridge between gauge theory and codimension-three minimal geometry by proving that the SU(2) Yang–Mills–Higgs energy, in a large-mass (or small-\epsilon) regime, Γ-converges to the codimension-three area functional. The authors construct a current-based description via and show that, under natural bounded-energy and Higgs-field concentration conditions, the rescaled energy concentrates on integral -currents , with a sharp mass inequality and a recovery sequence for any such current. In the monopole regime with a calibrating form , the limiting energy measure is controlled by the current’s mass and is calibrated when the current is calibrated, providing gauge-theoretic evidence for predictions in - and Calabi–Yau settings (Donaldson–Segal program). Moreover, the paper demonstrates that codimension-three Plateau problems can be solved variationally via SU(2)–Higgs–Yang–Mills theory: minimizers (or minimizing sequences) of the energy converge to area-minimizing -currents, and the framework extends to boundary-value problems and boundary-gluing constructions. Overall, the results offer a concrete adiabatic-disintegration dictionary between gauge theory and calibrated geometry in higher codimension, with potential impacts on enumerative invariants and calibrated submanifold theory.

Abstract

We investigate the asymptotic behavior of the -Yang-Mills-Higgs energy in the large mass limit, proving convergence to the codimension-three area functional in the sense of De Giorgi's -convergence. More precisely, for a compact manifold with boundary and any family of pairs and indexed by a mass parameter , satisfying we prove that the -currents dual to converge subsequentially to a relative integral -cycle 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 -current with admits such an approximation, with equality in the above inequality. In the special case of pairs satisfying the generalized monopole equation for a calibration form , we deduce that the limit of the Dirichlet energy measures satisfies , with equality if and only if is calibrated by , giving evidence for predictions of Donaldson-Segal in the settings of -manifolds and Calabi-Yau -folds.

Paper Structure

This paper contains 12 sections, 19 theorems, 291 equations.

Table of Contents

  1. Introduction
  2. toc$\Gamma$-convergence of the rescaled Yang--Mills--Higgs energiestoc$\bm{\Gamma}$-convergence of the rescaled Yang--Mills--Higgs energies
  3. Large mass limits of generalized monopoles and general critical points
  4. Preliminaries, notation, and some key identities
  5. tocConventions for $\mathfrak{su}(2)$-valued and $\mathbb{H}$-valued formstocConventions for $\bm{\mathfrak{su}(2)}$-valued and $\bm{\mathbb{H}}$-valued forms
  6. tocThe Yang--Mills--Higgs functionals and $Z(\Phi,A)$tocThe Yang--Mills--Higgs functionals and $\bm{Z(\Phi,A)}$
  7. Liminf Inequality
  8. Reduction to dimension 3
  9. Liminf inequality: proof in dimension three
  10. Limsup inequality
  11. Proof of Theorem \ref{['thm: recovery']}: constructing a recovery sequence
  12. Approximating the Plateau problem

Key Result

Theorem 1.2

Given a sequence $\epsilon_j\to 0$, a sequence of smooth $\mathop{\mathrm{SU}}\nolimits(2)$-connections $A_j$ and sections $\Phi_j\in \Gamma(\mathcal{E})$ such that and there exist an $(n-3)$-current $T$, restricting to an integral cycle in the interior of $M$, and a measure $\mu\in C^0(M)^*$ such that, along a subsequence, as $(n-3)$-currents, in $C^0(M)^*$, and the weight measure $|T|=\theta

Theorems & Definitions (36)

  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.8
  • proof
  • Remark 1.11
  • Remark 2.1
  • Proposition 2.2
  • ...and 26 more