| Abstract: |
We shall show how one can define novel gauge-theoretic Floer homologies of four, three and two-manifolds that are associated with Vafa-Witten, Hitchin and complexified BF configurations, respectively, from the physics of a certain topologically-twisted 5d N=2 gauge theory. Via topological invariance and a 5d “S-duality”, we shall derive novel Atiyah-Floer correspondences of these gauge-theoretic Floer homologies which relate them to symplectic intersection Floer homologies of Higgs bundles, and a web of relations involving their loop/toroidal group generalizations and their Langlands dual. Lastly, through a soliton string theory interpretation of the 5d theory, we shall derive a Fukaya-Seidel type A-infinity category of the aforementioned Floer homology of three-manifolds, and its Atiyah-Floer correspondence. We therefore furnish purely physical proofs and generalizations of the mathematical conjectures by Haydys [1], Abouzaid-Manolescu [2], and Bousseau [3], and more. |
