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Abstract
Given an oriented Riemannian four-manifold equipped with a principal bundle, we investigate the moduli space MVW of solutions to the Vafa-Witten equations. These equations arise from a twist of N = 4 supersymmetric Yang-Mills theory. Physicists believe that this theory has a well-defined partition function, which depends on a single complex parameter. On one hand, the S-duality conjecture predicts that this partition function is a modular form. On the other hand, the Fourier coefficients of the partition function are supposed to be the "Euler characteristics" of various moduli spaces M̄ASD of compactified anti-self-dual instantons. For several algebraic surfaces, these Euler characteristics were verified to be modular forms.
Except in certain special cases, it's unclear how to precisely define the partition function. If there is a mathematically sensible definition of the partition function, we expect it to arise as a gauge-theoretic invariant of the moduli spaces MVW. The aim of this thesis is to initiate the analysis necessary to define such invariants. We establish various properties, computations, and estimates for the Vafa-Witten equations. In particular, we give a partial Uhlenbeck compactification of the moduli space.