Natural transformations between braiding functors in the Fukaya category

2025/11/13 by Tong, Yujin
#FOS: Mathematics #Quantum Algebra (math.QA) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2511.10462

Abstract

We study the space of A_∞-natural transformations between braiding functors acting on the Fukaya category associated to the Coulomb branch M(\bullet,1) of the \mathfraksl₂ quiver gauge theory. We compute all cohomologically distinct A_∞-natural transformations Nat(id, id) and Nat(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_∞-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.

Citations

• Hamiltonian flow between standard module Lagrangians
• Aganagic's invariant is Khovanov homology
• Quiver Hecke algebras from Floer homology in Couloumb branches
• Knot Categorification from Mirror Symmetry, Part II: Lagrangians
• Deformations of path algebras of quivers with relations
• Projective resolutions of associative algebras and ambiguities
• 2-Kac-Moody algebras

Related

• Braided Deformations of Monoidal Categories and Vassiliev Invariants
• Aganagic's invariant is Khovanov homology
• Hochschild cohomology of the Fukaya category via Floer cohomology with coefficients
• The Fukaya category pairs with Lagrangian cobordisms exactly
• Lie algebra actions on module categories for truncated shifted Yangians