Duality in Topological Quantum Field Theories

In this talk, we will study Topological Quantum Field Theories (TQFT for short) as powerful tools for understanding and computing topological invariants. We will review some of the categorical machinery needed, like monoidal categories, and their physical inspiration as models for path integrals. Focusing on the case of surfaces, we will characterize 2d TQFT in purely algebraic terms as Frobenius algebras.

As these low dimensional cases show, the monoidality of these constructions imposes very strong duality restrictions on the theory, which lie in the germ of Baez and Dolan’s cobordism hypothesis. Finally, we will explain how these difficulties can be overcome by considering lax monoidal TQFT and almost TQFT and how they can be used for explicit computations of topological invariants.