Removable singularities for the Yang-Mills-Higgs equations in two dimensions

Abstract

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Energy gap for Yang–Mills connections, II: Arbitrary closed Riemannian manifolds

“…Proof. We adapt the proof of [53,Theorem 4.1]. Noting that * F A ∈ Ω 0 (B r ; g) when d = 2, the Kato Inequality [21,Equation (6.20)] and the Yang-Mills equation for A (see Section 2) imply that (A.4) |d|F A || = |d| * F A || ≤ |d A * F A | = |d * A F A | = 0 on B r .…”

The coupled Yang-Mills-Dirac equations for differential forms

“…In dimension 2 the Dirac equation reduces to the eigenvalue equation for a twisted Cauchy-Riemann operator. Point singularities in this dimension can be removed by an argument similar to [9], provided F A e L ι and φ e H 12 at the puncture and provided that a holonomy condition is assumed. All these arguments require techniques developed by K. Uhlenbeck for the pure Yang-Mills equations in dimension 4 ([10], [11]).…”

“…Then the pair (g(A), g{φ)), g e G, is also a solution and we identify (A, φ) with (g(A), g{φ)). There is a g 0 £ G such that (go{A) 9 gp{φ)) is a solution of an elliptic system of partial differential equations, although the system satisfied by (A, φ) may not be elliptic.…”

Removable point singularities for the Yang Mills Dirac equations in two dimensions

“…

In [12], we proved a removable point singularity theorem for the coupled Yang-Mills Higgs Equations over a two dimensional base manifold. Here, we prove a similar theorem for the coupled Yang Mills Dirac Equations.

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