Large mass limits of ${\mathrm{G}}_2$ and Calabi--Yau monopoles: calibrated concentration, Higgs zeros, and abelianization

Abstract

We study large mass monopoles with structure group $\mathrm{SU}(2)$ or $\mathrm{SO}(3)$ on asymptotically conical ${\mathrm{G}}_2$-manifolds and Calabi--Yau $3$-folds, with fixed asymptotic class. After placing the AC asymptotic theory, the variational compactness theory of Parise--Pigati--Stern, and Li's singular abelian compactness theory in a common $\Theta$-monopole framework, we prove that the mass-renormalized Yang--Mills--Higgs and intermediate energy measures converge to $8\pi \Vert T\Vert$ for a compactly supported calibrated integral codimension-three cycle $T$. This identifies the two limiting currents and shows that the variational calibration inequalities are saturated. Using this common limit as the starting point for a finer analysis, if $\mathcal{S}$ is the calibrated support, $\mathcal{Z}$ the Kuratowski upper limit of the Higgs zero sets, and $\mathcal{C}$ the limiting nonabelian locus, defined as the Kuratowski upper limit of Li's curvature concentration loci, then $\mathcal{S}\subset \mathcal{Z}\subset \mathcal{C}=\mathcal{S}\cup \mathcal{O}$, where $\mathcal{O}$ is precisely the obstruction to effective codimension-three monotonicity. For the cohomogeneity-one large mass families on the Bryant--Salamon ${\mathrm{G}}_2$-manifolds and the Stenzel Calabi--Yau $3$-fold, we prove that $\mathcal{O}=\varnothing$. On $X\setminus \mathcal{C}$ the sequence abelianizes; corrected longitudinal curvatures converge smoothly, and the remaining compactness alternatives are governed by $L^2$-harmonic $2$-forms.

148 pages, no figures. Comments are welcome