Natural transformations between braiding functors in the Fukaya category

TL;DR

This paper analyzes the space of natural transformations between braiding functors in the Fukaya category of a Coulomb branch, computing their cohomological classes and Hochschild cohomology to understand higher $A_ abla$-structure.

Contribution

It provides explicit computations of natural transformations and Hochschild cohomology in the Fukaya category related to Coulomb branches, advancing categorical braid theory.

Findings

Computed all cohomologically distinct natural transformations between key functors.

Calculated the Hochschild cohomology of the Fukaya category explicitly.

Classified higher $A_ abla$-structure data for braiding functors.

Abstract

We study the space of $A_\infty$-natural transformations between braiding functors acting on the Fukaya category associated to the Coulomb branch $\mathcal{M}(\bullet,1)$ of the $\mathfrak{sl}_2$ quiver gauge theory. We compute all cohomologically distinct $A_\infty$-natural transformations $\mathrm{Nat}(\mathrm{id}, \mathrm{id})$ and $\mathrm{Nat}(\mathrm{id}, \beta_i^-)$, where $\beta_i^-$ denotes the negative braiding functor. Our computation is carried out in a diagrammatic framework compatible with the established embedding of the KLRW category into this Fukaya category. We then compute the Hochschild cohomology of the Fukaya category using an explicit projective resolution of the diagonal bimodule obtained via the Chouhy-Solotar reduction system, and use this to classify all cohomologically distinct natural transformations. These results determine the higher $A_\infty$-data encoded in the braiding functors and their natural transformations, and provide the first step toward a categorical formulation of braid cobordism actions on Fukaya categories.