Vafa-Witten Invariants and Instanton Moduli Spaces on Compact Four-Manifolds

Mathematical Applications of Science Fiction We present a transfinite and rigorous construction of Vafa–Witten invariants for compact, oriented Riemannian four-manifolds with positive definite second Betti number greater than one. By analyzing the twisted N equals 4 supersymmetric Yang–Mills theory, we define the moduli space of solutions to the Vafa–Witten equations as a derived stack equipped with a virtual fundamental class. We establish the compactness of the moduli space under suitable stability conditions and prove several technical results concerning the Fredholm properties of the linearized deformation complex. Furthermore, we derive the modular transformation properties of the associated partition function under the S-duality group. This approach unifies the algebro-geometric theory of stable sheaves with the analytic framework of gauge theory, providing a rigorous formulation of the Euler characteristics of instanton moduli spaces.