The large mass limit of monopoles: abelian limits and Dirac singularities

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Original abstract

Let $(A_i,Φ_i)$ be finite energy $\mathrm{SU}(2)$ monopoles of charge $k>0$ on an asymptotically conical $3$-manifold with one end, with masses $m_i\to\infty$. After passing to a subsequence, the mass-renormalized energy measures concentrate at finitely many points $x_a$ with concentration weights $4πK_a$, where $K_a$ is the total charge of the complete finite cluster of mass-one Euclidean monopoles lying over $x_a$. We prove that, on the complement $M$ of these points, the fields abelianize exponentially. After translating the Higgs fields by their masses along the unit Higgs directions and applying gauge transformations, the translated pairs converge smoothly locally to a reducible monopole $(A_\infty,Φ_\infty)$ of the form \[ Φ_\infty=-uΨ_\infty, \qquad F_{A_\infty}=-*du\,Ψ_\infty, \qquad u=4π\sum_aK_aG(\,\cdot\,,x_a), \] where $Ψ_\infty$ is a parallel unit section and $G$ is the minimal positive Green function. Consequently, $x_a$ is a Dirac singularity of charge $K_a$. The singular part of the residual limit is determined by the weighted $0$-cycle of concentration points and total cluster charges, and does not retain the individual Euclidean profiles or their separation hierarchy. We also show that $k-\sum_aK_a$ is exactly the charge escaping through the asymptotically conical end, and describe the residual flat abelian ambiguity.

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• Official / arXiv page
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