Symplectic duality, as defined by Braden-Licata-Proudfoot-Webster, is an equivalence between categories of deformation quantization modules on certain pairs of holomorphic symplectic manifolds. Surprisingly all known symplectic dual pairs arise as Higgs and Coulomb branches of 3d N=4 supersymmetric quantum field theories.
In this talk I would like to argue that symplectic duality, as formulated by BLPW, is just a shadow of a more fundamental equivalence between certain factorization categories. More precisely each 3d N=4 theory gives rise two topological field theories: the 3d A-model and the 3d B-model. The factorization categories of line operators in these theories can be computed by compactifying on a circle where they look like ordinary 2d A/B-models on loop spaces. Then symplectic duality is a corollary of the equivalence between A-model line operators in a theory and the B-model line operators in the mirror theory.
When a 3d theory is equipped with an action of a reductive group G the category of A-type line operators carries an action of D-mod(G((t))) and the B-type line operators carry an action of QCoh(Loc_G(D^*)). This allows us to make a link between symplectic duality and the local geometric Langlands program.
This work is joint with Philsang Yoo, Tudor Dimofte, and Davide Gaiotto.