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

2025/02/11 by Parise, Davide, Pigati, Alessandro, Stern, Daniel · 2 citations
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2502.07756

Abstract

We investigate the asymptotic behavior of the SU(2)-Yang-Mills-Higgs energy E(Φ,A)=∫_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∈Ω⁰(M;\mathfraksu(2)) and A_m∈ Ω¹(M;\mathfraksu(2)) indexed by a mass parameter m→∞, satisfying E(Φ_m,A_m)≤ Cm\quadand lim_m→∞(1)/(m)∫_M(m-|Φ_m|)²=0, we prove that the (n-3)-currents dual to (1)/(2πm)tr(d_AmΦ_m\wedge F_Am) converge subsequentially to a relative integral (n-3)-cycle T of mass \mathbbM(T)≤ \liminf_m→∞(1)/(4πm)E(Φ_m,A_m), and show conversely that any integral (n-3)-current T with [T]=0∈ H_n-3(M,∂ M;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_AmΦ_m=F_Am\wedge Θ for a calibration form Θ∈ Ω^n-3(M), we deduce that the limit ν=lim_m→∞(1)/(2πm)|d_AmΦ_m|² of the Dirichlet energy measures satisfies ν≤ |T|, with equality if and only if T is calibrated by Θ, giving evidence for predictions of Donaldson-Segal in the settings of G₂-manifolds and Calabi-Yau 3-folds.

Cited by

• Anisotropic min-max via phase transitions
• The large mass limit of monopoles: abelian limits and Dirac singularities

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