Research Seminar: Geometric Analysis by Johan Klemmensen

Speaker: Johan Klemmensen

Title: On Classifying Solutions of The Kapustin-Witten Equations via Higgs Bundles

Abstract: The Kapustin-Witten (KW) equations arise from a topological twist of $\mathscr{N}=4$ super Yang-Mills theory. This talk develops a Kobayashi-Hitchin type correspondence between Nahm pole solutions to the Kapustin-Witten on $S^1\times \Sigma \times \R_+$ and stable $\SLR$ Higgs bundles over $\Sigma$, as developed by Witten, Mazzeo, and He. A description of $SL(2,\C)$ Higgs bundles focusing on $SL(2,\R)$ Higgs bundles is given, providing a classification by solutions of the Hitchin equations. A short treatment of b-calculus is presented, emphasizing the real blowup of points and polyhomogeneous conormal functions. The Kapustin-Witten (KW) equations are then introduced as gauge-theoretical differential equations on a Riemannian four-manifold. The Nahm pole boundary condition with and without knot singularities is studied by embedding knots along the boundary. Then, a dimensional reduction of these equations reveals the extended Bogomolny equations (EBE). The moduli space of solutions of the EBE on $\Sigma \times \R_+$ with Nahm pole boundary condition and converging to a flat $SL(2,\R)$ connection as $y\to \infty$ is completely classified via $SL(2,\R)$ Higgs bundles, providing the Kobayashi-Hitchin type correspondence. The classification finally lifts back to the Kapustin-Witten equations. The dimension of the moduli space with knot singularities is thought to have applications in computing the Jones polynomial.