Week 18.12.2023 – 24.12.2023

This talk is about our latest work in [arXiv:2311.18302]. 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 Hitchin configurations on three-manifolds and its Atiyah-Floer correspondence. We therefore furnish purely physical realizations and generalizations of the mathematical conjectures and constructions of Haydys [1], Wang [2] and Abouzaid-Manolescu [3], and more.