Topological 8d mathcalN=1 gauge theory: Novel Floer homologies, and A₈₍₅ₓₘ-categories of six, five, and four-manifolds

This work is a continuation of the program initiated in 1. We show how one can define novel gauge-theoretic (holomorphic) Floer homologies of seven, six, and five-manifolds, from the physics of a topologically-twisted 8d N=1 gauge theory on a Spin (7) -manifold via its supersymmetric quantum mechanics interpretation. They are associated with G₂ instanton, Donaldson-Thomas, and Haydys-Witten configurations on the seven, six, and five-manifolds, respectively. We also show how one can define hyperkähler Floer homologies specified by hypercontact three-manifolds, and symplectic Floer homologies of instanton moduli spaces. In turn, this will allow us to derive Atiyah-Floer type dualities between the various gauge-theoretic Floer homologies and symplectic intersection Floer homologies of instanton moduli spaces. Via a 2d gauged Landau-Ginzburg model interpretation of the 8d theory, one can derive novel Fukaya-Seidel type A_-categories that categorify Donaldson-Thomas, Haydys-Witten, and Vafa-Witten configurations on six, five, and four-manifolds, respectively – thereby categorifying the aforementioned Floer homologies of six and five-manifolds, and the Floer homology of four-manifolds from 1 – where an Atiyah-Floer type correspondence for the Donaldson-Thomas case can be established. Last but not least, topological invariance of the theory suggests a relation amongst these Floer homologies and Fukaya-Seidel type A_-categories for certain Spin (7) -manifolds. Our work therefore furnishes purely physical proofs and generalizations of the conjectures by Donaldson-Thomas 2, Donaldson-Segal 3, Cherkis 4, Hohloch-Noetzel-Salamon 5, Salamon 6, Haydys 7, and Bousseau 8, and more.