250121 VO Introduction to Topological Quantum Field Theory (2014S)

The functorial approach to topological quantum field theory goes back to Atiyah and Segal. After a motivational discussion of the Feynman path integral and a brief, self-contained introduction to monoidal categories, the lecture is roughly divided into three parts. (1) We will first study general properties of "closed d-dimensional TQFTs", and work out specific details for d=1,2,3. Interesting examples in the two-dimensional case are topological Yang-Mills theory, sigma models and Landau-Ginzburg models. (2) We then move to "open/closed 2d TQFT" a la Moore-Segal and Lazaroiu, which "live" on surfaces that may have non-trivial boundary conditions. We will see that such TQFTs are equivalently described by an interplay of commutative Frobenius algebras and Calabi-Yau categories. (3) Finally we add further structure to the surfaces by embedding certain one-dimensional submanifolds. This leads to "2d TQFT with defects". A minimal generators-and-relations description analogous to (1) and (2) is not known in this case, but we will explain that 2-categories are the natural language here. We shall end the lecture course with an outlook on connections to rational conformal field theory, knot homologies, and categorical quantum mechanics.