Modern aspects of topological gauge theories - Polynomial invariants and mock modular forms

Abstract

In this dissertation we present new results in the field of topologically twisted gauge theories evaluated on compact four-manifolds without boundary. We focus on the Donaldson-Witten theory, that is the N = 2 topologically twisted super Yang-Mills theory, We revisit and study the contribution on the Coulomb branch of the path integral of the low energy effective theory which is non-vanishing only for four-manifolds with b+2 ≤ 1. We establish new ways to evaluate path integrals of this theory using mock modular forms. Furthermore we propose a new regularization and renormalization of the path integral required for conservation of the BRST symmetry of the topological theory. We conclude by generalizing these considerations to the Donaldson-Witten theory in the presence of supersymmetric surface defects.