Introduction to non semi-simple TQFT

• Christian BLANCHET

Room: Amphithéâtre Gaston Darboux 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.

TQFTs for physics

• Giovanni RIZI (IHES)

Room: Amphithéâtre Gaston Darboux In this talk I will discuss some instances of how Topological Quantum Field Theories (TQFTs) appear and are used in high-energy physics contexts. TQFTs describe gapped phases of Quantum Field Theories (QFTs) and their relation with generalized symmetries makes them a useful classification tool for such phases. I will also highlight instances in which TQFT tools are useful beyond classification purposes and allow to describe more dynamical phenomena, such as modified crossing in 1+1 dimensional scattering processes. Beyond describing gapped phases, TQFTs are a useful method to describe the symmetry structure of general QFTs via the Symmetry Topological Field Theory (SymTFT) construction. Time permitting I will also discuss some connections between TQFTs with gravity and holographic dualities.

Towards a homological reconstruction of TQFTs

• Jules MARTEL (Université de Cergy-Pontoise, Pontoise, France)

Room: Amphithéâtre Gaston Darboux TQFTs are the idea of Witten that we can study quantum field theories from the point of view of the topology of state spaces and of topological transitions. Mathematically, it was formalised by Atiyah as a linearization of a cobordism category. It was concretely realized by Reshetikhin--Turaev (RT) coupled with Blanchet--Habbeger--Masbaum--Vogel universal construction. The RT philosophy relies on a diagrammatic model for cobordisms (based on knot diagrams) and uses modules on quantum groups (or more generally monoidal categories) to model linearly these diagrams. This has more recently been extended so to permit the utilization of non semisimple monoidal categories as input of the construction giving rise to a new generation of TQFTs, for which a nice example is the Kerler--Lyubashenko (KL) construction. These constructions need abstract algebra tools applied on diagrammatic representations of manifolds, and we will try to avoid this by using homology theories, allowing a more global definition of TQFTs. In this talk I'll introduce a new philosophy to build TQFTs based on homology of configuration spaces. Representations of mapping class groups constitute an important byproduct of TQFTs, while more natural ones can be built out of twisted homologies of configuration spaces of surfaces. We will show that from this latter new framework we can recover step by step the properties of KL TQFTs associated with quantum groups and even a unifying framework. I'll stay introductory most of the talk but the constructions and their motivations will be the occasion to review joint works with: S. Bigelow, M. De Renzi, R. Detcherry or Q. Faes (depending on the time).