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We show how one can define novel gauge-theoretic Floer homologies of four,three and two-manifolds from the physics of a certain topologically-twisted 5d${\cal N}=2$ gauge theory via its supersymmetric quantum mechanicsinterpretation. They are associated with Vafa-Witten, Hitchin and complexifiedBF configurations on the four, three and two-manifolds, respectively. We alsoshow how one can define novel symplectic Floer homologies of Hitchin spaces,which in turn will allow us to derive novel Atiyah-Floer correspondences thatrelate our gauge-theoretic Floer homologies to symplectic intersection Floerhomologies of Higgs bundles. Furthermore, topological invariance and 5d"S-duality" suggest a web of relations and a Langlands duality amongst thesenovel Floer homologies and their loop/toroidal group generalizations. Last butnot least, via a 2d gauged Landau-Ginzburg model interpretation of the 5dtheory, we derive, from the soliton string theory that it defines and the 5dpartition function, a Fukaya-Seidel type $A_\infty$-category of Hitchinconfigurations on three-manifolds and its Atiyah-Floer correspondence. Our worktherefore furnishes purely physical realizations and generalizations of themathematical conjectures and constructions of Haydys [1], Wang [2] andAbouzaid-Manolescu [3], and more.

Topological 5d $\mathcal {N} = 2$ Gauge Theory: Novel Floer Homologies, their Dualities, and an $A_\infty$-category of Three-Manifolds