G2与卡拉比-丘单极子的大质量极限：校准集中、希格斯零点与阿贝尔化
Large mass limits of $\mathrm{G}_2$ and Calabi--Yau monopoles: calibrated concentration, Higgs zeros, and abelianization
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中文总结 AI 辅助
本文研究渐近锥形G2流形与卡拉比-丘三维流形上的大质量单极子，建立统一的Θ-单极子框架，推导其能量测度收敛结果，明确校准支撑、希格斯零点集与非阿贝尔轨迹的关系，并验证特定流形族的阻碍为空。
中文摘要 AI 辅助
我们研究在渐近锥形G2流形与卡拉比-丘三维流形上、具有固定渐近类的结构群为SU(2)或SO(3)的大质量单极子。在将AC渐近理论、Parise-Pigati-Stern的变分紧性理论以及Li的奇异阿贝尔紧性理论纳入统一的Θ-单极子框架后，我们证明质量重整化的杨-米尔斯-希格斯能量与中间能量测度收敛于8π∥T∥，其中T为具有紧支撑的校准积分余维3闭链。这确定了两个极限流，并表明变分校准不等式达到饱和。以该共同极限为起点进行更精细的分析，若S为校准支撑，Z为希格斯零点集的Kuratowski上极限，C为Li的曲率集中轨迹的Kuratowski上极限所定义的极限非阿贝尔轨迹，则S⊂Z⊂C=S∪O，其中O恰好是有效余维3单调性的阻碍。对于Bryant-Salamon G2流形与Stenzel卡拉比-丘三维流形上的等变大质量族，我们证明O=∅。在X\backslash C上，序列阿贝尔化；校正后的纵向曲率光滑收敛，剩余的紧性替代由L²调和2-形式控制。
英文摘要
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 $Θ$-monopole framework, we prove that the mass-renormalized Yang--Mills--Higgs and intermediate energy measures converge to $8π\|T\|$ 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.
