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The large mass limit of $G_2$ and Calabi-Yau monopoles

Yang Li

TL;DR

This work analyzes the large mass limit of $SU(2)$ G$_2$-monopoles and Calabi–Yau monopoles on asymptotically conical manifolds. By decomposing the curvature into parallel and perpendicular parts and establishing a suite of sharp estimates (monotonicity, $\epsilon$-regularity, Higgs-field decay, and exponential decay of $F^{\perp}$), the authors show curvature concentrates along a calibrated cycle $Q$, while the generic region collapses to an abelian $U(1)$ monopole. A mollified Chern form and a Whitney/deformation framework yield a weak limit data $(F_{\infty},\Phi_{\infty},Q)$ with $Q$ an integral coassociative (resp. SLag) current; the limit charges a singular abelian monopole on $M\setminus\mathrm{supp}(Q)$ and an energy identity equates the limiting energy to contributions from monopole bubbles along normal fibers. The results provide a rigorous gauge-theoretic realization of the Donaldson–Segal program, linking large-mass monopoles to calibrated geometry and offering a nonperturbative mechanism for the appearance of coassociative and special Lagrangian cycles with integral multiplicities. The framework also highlights intricate bubbling phenomena and multiple open problems, including multivalued Fueter sections and extensions to general Lie groups.

Abstract

We develop a structure theory for the limit of $SU(2)$ $G_2$-monopoles (resp. Calabi-Yau monopoles) on a principal $SU(2)$-bundle over an asymptotically conical $G_2$-manifolds (resp. Calabi-Yau 3-folds) as the mass parameter tends to infinity, while the topologial data for the bundle stays fixed. We show how to extract a singular abelian $G_2$-monopole (resp. Calabi-Yau monopole) with Dirac singularity along a calibrated cycle in the large mass limit, and we prove an energy identity for monopole bubbles.

The large mass limit of $G_2$ and Calabi-Yau monopoles

TL;DR

This work analyzes the large mass limit of G-monopoles and Calabi–Yau monopoles on asymptotically conical manifolds. By decomposing the curvature into parallel and perpendicular parts and establishing a suite of sharp estimates (monotonicity, -regularity, Higgs-field decay, and exponential decay of ), the authors show curvature concentrates along a calibrated cycle , while the generic region collapses to an abelian monopole. A mollified Chern form and a Whitney/deformation framework yield a weak limit data with an integral coassociative (resp. SLag) current; the limit charges a singular abelian monopole on and an energy identity equates the limiting energy to contributions from monopole bubbles along normal fibers. The results provide a rigorous gauge-theoretic realization of the Donaldson–Segal program, linking large-mass monopoles to calibrated geometry and offering a nonperturbative mechanism for the appearance of coassociative and special Lagrangian cycles with integral multiplicities. The framework also highlights intricate bubbling phenomena and multiple open problems, including multivalued Fueter sections and extensions to general Lie groups.

Abstract

We develop a structure theory for the limit of -monopoles (resp. Calabi-Yau monopoles) on a principal -bundle over an asymptotically conical -manifolds (resp. Calabi-Yau 3-folds) as the mass parameter tends to infinity, while the topologial data for the bundle stays fixed. We show how to extract a singular abelian -monopole (resp. Calabi-Yau monopole) with Dirac singularity along a calibrated cycle in the large mass limit, and we prove an energy identity for monopole bubbles.

Paper Structure

This paper contains 36 sections, 48 theorems, 273 equations.

Table of Contents

  1. Introduction and background
  2. Curvature concentration for $G_2$/Calabi-Yau monopoles
  3. Asymptotic geometry of $G_2$/Calabi-Yau monopoles
  4. Large mass limit and calibrated cycles
  5. Some estimates on $G_2$/Calabi-Yau monopoles
  6. Yang-Mills-Higgs equation and monotonicity formula
  7. $\epsilon$-regularity for Yang-Mills-Higgs equation
  8. Higgs field estimate
  9. An integral Hessian estimate
  10. Differential inequalities and exponential decay
  11. Exponential decay of perpendicular curvature
  12. Curvature concentration locus
  13. Parallel curvature component
  14. $L^1$ type estimate on the parallel curvature component
  15. Chern form of the singular $U(1)$ connection
  16. ...and 21 more sections

Key Result

Theorem 1.12

There exists some $L^2$-harmonic 2-form $\sigma\in L^2\mathcal{H}^2(M)\simeq H^2_c(M)$ depending on $(A, \Phi)$ with $\left\lVert\sigma\right\rVert_{L^2}\leq Cm$, such that the following convergence results hold:

Theorems & Definitions (117)

  • Example 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.7
  • Example 1.9
  • Definition 1.10
  • Definition 1.11
  • ...and 107 more