2025/11/20 by Er, Arif, Tan, Meng-Chwan
Mathematics · #Advanced Operator Algebra Research #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Geometric and Algebraic Topology #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2511.15953
openalex publication_date 2025/11/20 · openalex created_date 2025/11/23 · openalex updated_date 2026/07/28
We explain why on certain five-manifolds, topological 5d N = 2 gauge theory of Haydys-Witten twist with gauge group G, is dual to that of Geyer-Mülsch twist with gauge group LG, where G is a real, compact Lie group with Langlands dual LG. In turn, via their 2d and 3d gauged A/B-twisted Landau-Ginzburg model interpretations, we can show that (i) a Fukaya-Seidel-type A_∞-1-category of an HW4-instanton Floer homology of three-manifolds and (ii) a Fueter-type A_∞-2-category of an HW3-instanton Floer homology of two-manifolds, are dual to (i) an Orlov-type A_∞-1-category of a novel holomorphic LGℍ-flat Floer homology of three-manifolds and (ii) a Rozansky-Witten-type A_∞-2-category of a novel holomorphic LG_\mathbbO-flat Floer homology of two-manifolds, respectively. We also derive their Atiyah-Floer-type correspondences to symplectic categories. Our work, which demonstrates a mirror symmetry and Langlands duality of (higher) A_∞-categories of Floer homologies, therefore furnishes purely physical proofs and gauge-theoretic generalizations of the mathematical conjectures by Bousseau [1] and Doan-Rezchikov [2], and more.