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

TL;DR

This work builds a unified physical framework in which five-dimensional Haydys–Witten and Geyer–Mülsch twists realize dual gauge theories linked by Langlands duality. Through systematic dimensional reductions and dualities (including BJSV reductions, S-duality, and HMS concepts), the authors construct and relate higher A_infty-categories of Floer homologies: Orlov-type, RW-type, Fueter-type, and Fukaya-Seidel-type, across four-, three-, and two-manifolds. They provide physical realizations of novel Floer invariants, derive Atiyah–Floer-type correspondences, and establish a web of Langlands dual relations among these categories, including a physical proof of Doan–Rezchikov’s conjecture on correspondences between KRS-type and Orlov-type categories. The results yield a gauge-theoretic generalization of HMS for Landau–Ginzburg models, grounded in 4d N=4 S-duality and its 5d descendants, with broad implications for the geometric Langlands program and higher-categorical mirror symmetry. Overall, the paper synthesizes topological gauge theories, sigma-models, LG models, and brane categories to reveal deep dualities among Floer-theoretic invariants in a physically rigorous setting.

Abstract

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 $^LG$, 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 $A_\infty$-1-category of an HW$_4$-instanton Floer homology of three-manifolds and (ii) a Fueter-type $A_\infty$-2-category of an HW$_3$-instanton Floer homology of two-manifolds, are dual to (i) an Orlov-type $A_\infty$-1-category of a novel $^LG_{\mathbb{H}}$-flat Floer homology of three-manifolds and (ii) a Rozansky-Witten-type $A_\infty$-2-category of a novel $^LG_{\mathbb{O}}$-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) $A_\infty$-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.

Figures (12)

• Figure 1: Heegaard splits of $M_3$ and $M_5$.
• Figure 2: Union of 2d A-models along their common boundary $\mathscr{R}$.
• Figure 3: BPS worldsheet with strings $\gamma^{IJ}_\pm(t, \theta, \mathfrak{B}_3)$ and boundaries corresponding to $\mathfrak{C}^I_{\text{BF}_{\mathbb{H}}}(\theta)$ and $\mathfrak{C}^J_{\text{BF}_{\mathbb{H}}}(\theta)$.
• Figure 4: Tree-level scattering BPS worldsheet of incoming ($-$) and outgoing ($+$) LG $\mathfrak{B}_3^{\theta}$-strings.
• Figure 5: 3d RW model on $I \times \mathbb{R}_t \times \mathbb{R}_{\tau}$ physically realizing (I) a derived category of coherent sheaves with (a) 1-objects being branes $\mathscr{P}_{\{0, 1\}}(\theta)$, and (b) 1-morphisms being $\mathscr{P}^{\theta}$-strings ending on the 1-objects; as well as (II) an RW 2-category with (i) 2-objects being complex-Lagrangian branes $\widehat{L}_{\{0, 1\}}^{*}(\theta)$, (ii) 2-morphisms being $\mathscr{P}^{\theta}$-strings with vertices being the 2-objects, (iii) 1-objects being $\widehat{L}^{\theta}$-strings, and (iv) 1-morphisms being $\mathscr{P}^{\theta}$-strings with edges being the 1-objects.