Panagiotis Dimakis, Université du Québec à Montréal, CIRGET
Since their introduction in 2006, the Kapustin-Witten (KW) equations have become the subject of a number of conjectures. Given a knot $K$ embedded in a closed $3$-manifold $Y$, the most prominent conjecture predicts that the number of solutions to the KW equations on $Y\times\mathbb{R}_+$ with boundary conditions determined by the embedding and with fixed topological charge, is a topological invariant of the knot. A major obstacle with this conjecture is the difficulty of constructing solutions satisfying these boundary conditions. In this talk we assume $Y\cong \Sigma\times\mathbb{R}_+$ and study solutions to the dimensionally reduced KW equations with the required boundary conditions. We prove that the moduli spaces are diffeomorphic to certain holomorphic lagrangian sub-manifolds inside the moduli of Higgs bundles associated to $\Sigma$. Time permitting, we explain how one could use this result to construct knot invariants.
MC 5417