Quantum Maths Seminar, external speaker:
Speaker: Johannes Berger (University of Hamburg)
Title: Invariants and TFT from non-semisimple modular categories: Review & examples.
Abstract: In 2019 De Renzi, Gainutdinov, Geer, Patureau-Mirand, and Runkel constructed invariants of framed links and three-manifolds from the input datum of a (not necessarily semisimple) modular tensor category. In fact, their invariants are parameterized by so-called modified traces on tensor ideal in our category. Using the universal construction of Blanchet-Habegger-Masbaum-Vogel, the invariants based on a specific modified trace were then extended to what could be called a TFT: the result is a symmetric monoidal functor from some category of 3-dimensional bordisms (i.e. with some extra conditions) to the category of vector spaces.
I will review the construction, and intersperse the presentation with some results from my thesis and joint work with Gainutdinov and Runkel. These results mainly regard
1. the two main ingredients in the construction (the modified trace and the non-semisimple Kirby color), in the case that the modular tensor category is given as modules over a Hopf algebra (+ adjectives).
2. examples of the resulting invariants. We will see e.g. how invariants based on the same category but different ideals distinguish or fail to distinguish some framed links.
One hour talk, followed by informal discussion.
All are welcome. Please contact the seminar organizer to be added to the talk.