Some boundedness properties of solutions to the Vafa-Witten equations on closed four-manifolds

Abstract

We consider a set of gauge-theoretic equations on closed oriented four-manifolds, which was introduced by Vafa and Witten. The equations involve a triple consisting of a connection and extra fields associated to a principal bundle over a closed oriented four-manifold. They are similar to Hitchin's equations over compact Riemann surfaces, and as part of the resemblance, there is no $L^2$-bound on the curvature without an $L^2$-bound on the extra fields. In this article, however, we observe that under the particular circumstance where the curvature does not become concentrated and the limiting connection is not locally reducible, one obtains an $L^2$-bound on the extra fields.

33 pages, final version, to appear in The Quarterly Journal of Mathematics

References in corpus (1)

• On the behavior of sequences of solutions to U(1) Seiberg-Witten systems in dimension 4

Cited by in corpus (9)

• On orientations for gauge-theoretic moduli spaces
• On the singular sets of solutions to the Kapustin-Witten equations and the Vafa-Witten ones on compact Kähler surfaces
• The behavior of sequences of solutions to the Vafa-Witten equations
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• A lower bound on the solutions of Kapustin-Witten equations
• A perturbation and generic smoothness of the Vafa-Witten moduli spaces on closed symplectic four-manifolds
• Stable sheaves with twisted sections and the Vafa-Witten equations on smooth projective surfaces
• A compactness theorem for stable flat $SL(2,\mathbb{C})$ connections on $3$-folds
• On a topology property for the moduli space of Kapustin-Witten equations