Topological 5d $\mathcal{N}=2$ Gauge Theories: Mirror Symmetry and Langlands Duality of $A_{\mathrm{\infty }}$-categories of Floer Homologies

We explain why on certain five-manifolds, topological 5d $\mathcal{N}=2$ gauge theory of Haydys-Witten twist with gauge group $G$, is dual to that of Geyer-Mülsch twist with gauge group ${}^{L}G$, where $G$ is a real, compact Lie group with Langlands dual ${}^{L}G$. In turn, via their 2d and 3d gauged A/B-twisted Landau-Ginzburg model interpretations, we can show that (i) a Fukaya-Seidel-type $A_{\mathrm{\infty }}$-1-category of an HW${}_4$-instanton Floer homology of three-manifolds and (ii) a Fueter-type $A_{\mathrm{\infty }}$-2-category of an HW${}_3$-instanton Floer homology of two-manifolds, are dual to (i) an Orlov-type $A_{\mathrm{\infty }}$-1-category of a novel holomorphic ${}^L{G}_{\mathbb{H}}$-flat Floer homology of three-manifolds and (ii) a Rozansky-Witten-type $A_{\mathrm{\infty }}$-2-category of a novel holomorphic ${}^L{G}_{\mathbb{O}}$-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) $A_{\mathrm{\infty }}$-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.