Aganagic's invariant is Khovanov homology

TL;DR

The paper proves that Aganagic's Coulomb-branch (monodromy) construction of knot invariants and Webster's diagrammatic braid action are canonically intertwined for the case of the Dynkin diagram vertex, thereby giving a rigorous symplectic realization of Khovanov homology over the integers. By embedding Webster's weighty combinatorial category into the Fukaya-Seidel category of the multiplicative Coulomb branch and establishing a natural intertwining between the braid actions, the authors show Kh(\overline{\beta}) is recovered as a Hom-space in Fukaya-Seidel theory, with the Jones-graded and quantum gradings matched via the J, u, and \hbar gradings. The core steps involve (i) a detailed calculation of the braid action on simple KLRW objects and their cones, (ii) an explicit geometric realization of these objects as Lagrangians in Fukaya-Seidel categories with a cylindrical model for disk counts, and (iii) a precise comparison of caps and cups across the two formalisms, culminating in an intertwining theorem for braiding and cup/cap operations. The results provide a rigorous, integral verification of Aganagic's proposal and connect diagrammatic categorifications with symplectic constructions in a concrete, computable setting. This sharpens the link between combinatorial categorifications and symplectic/topological field theory perspectives on knot homology, with potential to extend to broader Lie types and superalgebras via the same monodromy framework.

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.

Figures (20)

• Figure 1: Nontrivial KLRW relations. Exchanging $i$ and $j$ in diagrams (b) and (d), i.e. if we have an arrow $(j) \gets (i)$, the right-hand-side gets an extra (-1) factor.
• Figure 2:
• Figure 3: Example of objects in $Fuk_{| | |}(\mathcal{M}^\times(\Gamma, \vec{d}), \mathcal{W}(a))$
• Figure 4: The disk count yielding $p_1 \cdot p_2 = p_3$ where $p_1 = \vcenter{}$, $p_2 = \vcenter{}$, and $p_3 = \vcenter{}$
• Figure 5: The image of the morphism $\vcenter{}$ under the embedding of Theorem \ref{['five authors theorem']}

Theorems & Definitions (52)

• Theorem 1.1
• Remark 2.1
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Remark 2.4
• Remark 2.5
• Lemma 3.1
• proof