Topological Gauge Theories with Sixteen Supercharges: Higher $A_\infty$-categorification of Floer Homologies

TL;DR

This work builds a comprehensive gauge-theoretic framework to physically realize higher A_infty-categories that categorify Floer homologies across dimensions, starting from HW and Spin(7) theories with sixteen supercharges. By repeatedly performing Kaluza-Klein reductions to 3d/2d LG models and 1d SQMs, the authors produce Fueter-type A_infty-2/3-categories and Fukaya-Seidel-type A_infty-categories of solitons, membranes, and sheets whose endpoints encode BF/holomorphic DT data on manifolds such as M_2, CY_2, and CY_2 × S^1. They demonstrate equivalences between Fueter-type 2-categories and FS-type categories, and provide physical proofs and generalizations of conjectures by Bousseau, Doan-Rezchikov, and Cao, thereby unifying higher categorifications of HW and holomorphic DT Floer homologies in a gauge-theoretic setting. The results collectively advance a gauge-theoretic generalization of Atiyah-Floer-type correspondences and support Cao’s vision of a Cauchy-Riemann-Fueter framework linking multi-layered A_infty-structures with holomorphic invariants across dimensions.

Abstract

This work is a sequel to [arXiv:2410.18575], and a third and final installment of the program initiated in [arXiv:2311.18302]. We show how, via a 3d gauged Landau-Ginzburg model interpretation of certain topologically-twisted 5d $\mathcal{N} = 2$ and 8d $\mathcal{N} = 1$ gauge theories, one can derive novel Fueter type $A_{\infty}$-2-categories that 2-categorify the 3d-Haydys-Witten, Haydys-Witten, and holomorphic Donaldson-Thomas Floer homology of two, four, and five-manifolds, respectively. Via a 2d gauged Landau-Ginzburg model interpretation of the aforementioned twisted gauge theories, these Fueter type $A_{\infty}$-2-categories can be shown to be equivalent to corresponding Fukaya-Seidel type $A_{\infty}$-categories. In the 8d case, one can also derive higher $A_{\infty}$-categories, such as a novel Cauchy-Riemann-Fueter type $A_{\infty}$-3-category that 3-categorifies the Haydys-Witten Floer homology of four-manifolds via a 4d gauged Landau-Ginzburg model interpretation of the theory. Together with previous results from [arXiv:2410.18575] and [arXiv:2311.18302], our work furnishes purely physical proofs and generalizations of the mathematical conjectures by Bousseau [3], Doan-Rezchikov [4], and Cao [5].

Figures (7)

• Figure 1: BPS worldsheet with $\mathcal{M}^{\theta}(\mathbb{R}_{\tau}, \mathfrak{A}_2)$-solitons $\sigma^{IJ, KL}_\pm(t, \theta, \mathfrak{A}_2)$ and boundaries labeled by $\gamma^{IJ}(\theta, \mathfrak{A}_2)$ and $\gamma^{KL}(\theta, \mathfrak{A}_2)$.
• Figure 2: $\mathfrak{A}_2^{\theta}$-sheet $\Sigma^{IJ, KL}(t, \tau, \theta, \mathfrak{A}_2)$ with edges being $\mathfrak{A}_2^{\theta}$-solitons $\Gamma^{IJ}(\tau, \theta, \mathfrak{A}_2)$ and $\Gamma^{KL}(\tau, \theta, \mathfrak{A}_2)$, and vertices corresponding to $\mathcal{E}^I_{\text{BF}}(\theta)$, $\mathcal{E}^J_{\text{BF}}(\theta)$, $\mathcal{E}^K_{\text{BF}}(\theta)$, and $\mathcal{E}^L_{\text{BF}}(\theta)$.
• Figure 3: BPS worldvolume with $\mathfrak{A}_2^{\theta}$-sheets $\Sigma^{IJ, KL}_\pm(t, \tau, \theta, \mathfrak{A}_2)$, boundaries labeled by $\mathfrak{A}_2^{\theta}$-solitons $\Gamma^{IJ}(\tau, \theta, \mathfrak{A}_2)$ and $\Gamma^{KL}(\tau, \theta, \mathfrak{A}_2)$, and edges labeled by $\mathcal{E}^I_{\text{BF}}(\theta)$, $\mathcal{E}^J_{\text{BF}}(\theta)$, $\mathcal{E}^K_{\text{BF}}(\theta)$, and $\mathcal{E}^L_{\text{BF}}(\theta)$.
• Figure 4: Tree-level scattering BPS worldsheet of incoming ($-$) and outgoing ($+$) LG $\mathcal{M}^{\theta}(\mathbb{R}_{\tau}, \mathfrak{A}_2)$-soliton strings.
• Figure 5: Tree-level scattering BPS worldvolume of incoming ($-$) and outgoing ($+$) LG $\mathfrak{A}_2^{\theta}$-soliton membranes.