Natural transformations between braiding functors in the Fukaya category

TL;DR

This work advances a categorical understanding of braid actions in symplectic geometry by analyzing $A_\infty$ natural transformations between braiding functors on the Fukaya category attached to the Coulomb branch $\mathcal{M}(\bullet,1)$ for the $\mathfrak{sl}_2$ quiver theory. It develops a diagrammatic embedding of the KLRW category into the Fukaya framework, enabling explicit computation of Hochschild cohomology and the full set of cohomologically distinct natural transformations $\mathrm{Nat}(\mathrm{id},\mathrm{id})$ and $\mathrm{Nat}(\mathrm{id},\beta_i^-)$. The results reveal the higher $A_\infty$-data governing braiding functors and their interrelations, providing a concrete step toward a categorical braid cobordism action on Fukaya categories. The methodology leverages the Chouhy–Solotar resolution for a tractable diagonal-bimodule projective resolution, yielding explicit representatives and connecting KLRW diagrammatics to multiplicative Coulomb-branch Fukaya categories. Overall, the paper lays groundwork for integrating braid cobordisms into Fukaya-categorical contexts and enriching the algebraic understanding of braiding phenomena in symplectic geometry.

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 $\mathrm{Nat}(\mathrm{id}, β_i^-)$, where $β_i^-$ 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.

Figures (7)

• Figure 1: A degree-2 element in $\mathop{\mathrm{Hom}}\nolimits(I_{i-1}, \beta_{i^-} I_{i-1})$
• Figure 2: The generators $T_i$, the red dots and stars denote the stops and punctures
• Figure 3: Morphisms between two generating $T_i$ objects.
• Figure 4: Proof of $\mu^2(s_1,a_{12})=a_{12}s$.
• Figure 5: Proof of $\mu^2(p_2,q_1)=s_2$.

Theorems & Definitions (12)

• Definition 1.1
• Remark 1.2
• Remark 2.1
• Definition 2.2
• Remark 2.3
• Theorem 2.4
• Definition 2.5
• Theorem 2.6
• proof
• Definition 2.7