Equivariant Localization Techniques in Topological Quantum Field Theories

We present how TQFTs of cohomological type can be described in the context of equivariant cohomology. In this setup, we show how appropriate equivariant localization formulas can be applied to the correlation functions of these theories, yielding to a simpler description of the path integrals involved. In particular, for the topological Yang-Mills theory in four dimensions, we derive (at least formally) Witten's conjecture relating Donaldson and Seiberg-Witten invariants. We conclude by presenting several ideas on how to replace the formal aspects of these techniques with rigorous arguments.