Aganagic's invariant is Khovanov homology

TL;DR

This paper verifies Aganagic's prediction that Khovanov homology can be obtained from braid group actions on Fukaya-Seidel categories related to quiver gauge theories, providing a symplectic construction with full gradings over integers.

Contribution

It provides a direct calculation confirming that the braid group action from monodromy matches the combinatorial action used to define Khovanov homology, validating Aganagic's proposal.

Findings

Confirmed the intertwining of braid group actions from monodromy and combinatorial definitions.

Established a symplectic construction of Khovanov homology with both gradings.

Validated Aganagic's prediction for a geometric realization of 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.