Fukaya category of coulomb branch

Abstract[Background & Objective] This project aims to investigate the profound connection between the symplectic topology of the non-abelian Coulomb branch $\mathcal{M}_C$ and geometric representation theory on the algebraic side, within the context of 3d $\mathcal{N}=4$ gauge theories (specifically for $G=SU(3)$ with the adjoint fundamental representation $\mathbb{C}^3$). Our core objective is to provide a rigorous, geometric dictionary based on Lagrangian Floer theory between the category of perfect modules of $\mathcal{M}_C$ and the Category $\mathcal{O}$ of flavoured KLRW algebras at the classical limit $(\eta=0, h=0)$. [Methodology: Covering Spaces and Equivariant Categories] Given the complexity of the singularities in the non-abelian Coulomb branch, this project approaches the study through its abelianized covering space $\widetilde{\mathcal{M}_C}$. We utilize hyperplane arrangements on the real slice of the complexified Cartan subalgebra $\mathfrak{t}_{\mathbb{C}}$—comprising matter walls determined by mass parameters and root/Coxeter walls generated by the Weyl group—to provide a localized, combinatorial description of $\widetilde{\mathcal{M}_C}$. Building upon this, we leverage the group action of the Weyl group $W = S_3$ to precisely recover the geometric information of the actual Coulomb branch $\mathcal{M}_C$ by constructing the equivariant Fukaya category $\mathcal{W}(\widetilde{\mathcal{M}_C})^{S_3}$. [Core Mechanism: Geometric Degeneration and Algebraic Generation] The core focus of this research lies in the isomorphic mechanism between the space of equivariant morphisms on the symplectic geometry side and the Skew Group Ring structure on the algebraic side. Specifically, we examine the geometric behavior when crossing the root hyperplanes: as the equivariant parameter reaches the wall, the dimension of the corresponding fixed locus increases, leading to a geometric degeneration. We demonstrate that by calculating the intersections and counting holomorphic polygons (Floer differentials) of Lagrangian submanifolds and their $S_3$ orbits on these degenerated fixed locus walls, we can precisely induce the generators of the nil-Hecke algebra on the algebraic level. [Expected Outcomes] Through the aforementioned mechanisms, this project will explicitly reveal the topological origins of the crossings between red strands (matter) and black strands (roots) in the KLRW diagrammatic calculus. This not only explains the emergence of KLRW algebraic relations at the level of symplectic geometry but also further validates the strict equivalence between $\mathcal{W}(\mathcal{M}_C) \simeq \mathcal{W}(\widetilde{\mathcal{M}_C})^{S_3}$ and the Category $\mathcal{O}$ of KLRW algebras, providing an intuitive and highly computable hyperplane arrangement model for higher categorification.