Séance Séminaire

Séminaire des Doctorant·e·s

Smooth Topological Field Theories

In my talk I will give an introduction to (smooth) TFTs in the categorical setting. In the 1980s Atiyah and Segal proposed an axiomatization of topological quantum field theories, known as functorial topological field theories (TFTs). Physically speaking, the Atiyah-Segal axioms associate to a given n-dimensional space manifold X a Hilbert space, which is thought of as the space of states of a physical system living on X. The time evolution of the physical system is topologically encoded in an (n+1)-dimensional spacetime manifold, to which the axioms assign a time evolution operator, i.e. a linear map from the Hilbert space of initial states to the Hilbert space of final states. I will explain how this assignment can be stated in a categorical setting, leading to de definition of functorial TFTs. Smooth field theories over a fixed manifold were introduced by Segal. Intuitively, a smooth field theory is a family of ordinary field theories parametrized by a manifold X, which is realized by equipping bordisms with smooth maps to X. They play a key role in the Stolz Teichner program, which aims at relating smooth field theories and generalized cohomology theories.