UC Davis

We study the topological A-twist of 3d N = 4 Yang-Mills gauge theories, with an eye towards geometric representation theory and knot theory. We present an explicit, geometric category describing 1/2-BPS vortex line operators in these theories, as well as collisions of local operators bound to them in terms of convolution techniques generalizing the work of Braverman-Finkelberg-Nakajima on Coulomb branches of vacua. Given a suitable Dirichlet boundary, we show that local operators bound to these vortex line operators can be represented as linear operators between the Borel-Moore homologies of generalized affine Springer fibers, vastly generalizing classical work on affine Springer representations of Hecke algebras and affine Weyl groups. We end with an application to knot theory. We apply 3d mirror symmetry to a recent construction in B-twisted 3d N = 4 gauge theory of HOMFLY-PT knot homology due to Oblomkov-Rozansky to obtain a mirror construction in the A-twist. The mirror construction exactly reproduces a different realization of HOMFLY-PT homology for positive algebraic links due to Oblomkov-Rasmussen-Shende, providing a robust check of our proposed mirror construction.