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

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].