Vafa-Witten Invariants and S-Duality Conjecture on Four-Manifolds with b+ > 1

This monograph explores the mathematical structure of Vafa-Witten invariants arising from the topological twisting of N=4 supersymmetric Yang-Mills theory on compact four-manifolds. We focus specifically on the regime where the self-dual Betti number satisfies b+ > 1, a condition that ensures the vanishing of wall-crossing phenomena and stabilizes the invariants against metric perturbations. We rigorously define the Vafa-Witten moduli space as the solution set to a system of non-linear partial differential equations involving a connection and a self-dual two-form. The central theme is the S-duality conjecture, which posits that the generating function of these invariants transforms as a modular form under the action of the duality group SL(2, Z) on the complexified coupling constant. We review recent advances in the algebro-geometric formulation by Tanaka and Thomas, as well as analytic results by Mares and Taubes, providing a comprehensive synthesis of the current status of the conjecture.