The Vafa-Witten equations on an oriented Riemannian 4-manifold are first order, non-linear equations for a pair of connection on a principal SO3 bundle over a 4-manifold and a self-dual 2-form with values in the associated Lie algebra bundle. This talk will describe a theorem about the behaviour of
sequences of solutions to the Vafa-Witten equations which have no convergent subsequence. This theorem says in part that a renormalization of a subsequence of the self-dual 2-form components of any given solution sequence converges on the complement of a closed set with Hausdorff dimension at most 2; and the limit defines a harmonic 2-form with values in a real line bundle. This behaviour generalizes Karen Uhlenbeck’s compactness theorem for the self-dual Yang-Mills equations; it is similar to what happens in other first order generalizations of the Seiberg-Witten/self-duality equations.
This video is part of Harvard University‘s conference JDG 2017.
