Aganagic's invariant is Khovanov homology

Abstract

On the Coulomb branch of a quiver gauge theory, there is a family of functions parameterized by choices of points in the punctured plane. Aganagic has predicted that Khovanov homology can be recovered from the braid group action on Fukaya-Seidel categories arising from monodromy in said space of potentials. These categories have since been rigorously studied, and shown to contain a certain (combinatorially defined) category on which Webster had previously constructed a (combinatorially defined) braid group action from which the Khovanov homology can be recovered. Here we show, by a direct calculation, that the aforementioned containment intertwines said combinatorially defined braid group action with the braid group action arising naturally from monodromy. This provides a mathematical verification that Aganagic's proposal gives a symplectic construction of Khovanov homology -- with both gradings, and over the integers.