Current trimester: TQFT and Knot Theory
Kickoff at IHP: 22.04.2026. Inscription : https://indico.math.cnrs.fr/event/16226/
Introduction to non semi-simple TQFT
byChristian Blanchet
Abstract:A Topological Quantum Field Theory in dimension 2+1 assigns to a surface a vector space (states), together with an action of mapping classes and more generally cobordisms. This includes invariants of 3-dimensional manifolds (correlation functions) possibly decorated with links or colored graphs. The starting point was the interpretation by Witten in the late 1980’ of the Jones polynomial invariant of knots in terms of Chern-Simons theory and the mathematical construction by Reshetikhin-Turaev of the expected model, now called WRT theory. WRT is based on simple modules over the quantum group sl(2) at root of unity, which means that an important part of the representation theory is neglected. Starting with a motivating example we will introduce to TQFTs which do use those neglectons. The non-simple TQFTs have two flavors. The graded version, first constructed by B- Costantino-Geer-Patureau involves manifolds with cohomology class (or abelian flat connection). The non graded version further developped by Marco de Renzi and collaborators recovers Kerler-Lyubashenko representations of mapping class groups on tensor powers of the adjoint representation. We will describe the constructions and the resulting TQFT spaces.