Vafa-Witten theory: invariants, Floer homologies, Higgs bundles, a geometric Langlands correspondence, and categorification

Abstract

We revisit Vafa–Witten theory in the more general setting whereby the underlying moduli space is not that of instantons, but of the full Vafa–Witten equations. We physically derive (i) a novel Vafa–Witten four-manifold invariant associated with this moduli space, (ii) their relation to Gromov–Witten invariants, (iii) a novel Vafa–Witten Floer homology assigned to three-manifold boundaries, (iv) a novel Vafa–Witten Atiyah–Floer correspondence, (v) a proof and generalization of a conjecture by Abouzaid–Manolescu in $\href{https://doi.org/10.4171/jems/994}{[2]}$ about the hypercohomology of a perverse sheaf of vanishing cycles, (vi) a Langlands duality of these invariants, Floer homologies and hypercohomology, and (vii) a quantum geometric Langlands correspondence with purely imaginary parameter that specializes to the classical correspondence in the zero-coupling limit, where Higgs bundles feature in (ii), (iv), (vi) and (vii). We also explain how these invariants and homologies will be categorified in the process, and discuss their higher categorification. We thereby relate differential and enumerative geometry, topology and geometric representation theory in mathematics, via a maximally-supersymmetric topological quantum field theory with electric-magnetic duality in physics.