# Lecture and seminar series on modified traces in algebra and topology

What?

Monoidal categories with duals have a notion of the trace of a morphism, generalizing the classical traces of matrices. These categorical traces are a well-studied key tool in low-dimensional topology, representation theory and other fields. However, it is often the case that categories of interest are not semisimple and the categorical traces vanish, becoming a trivial concept. The the past 10 years it became clear there exist non-trivial replacements for traces on non-semisimple categories called modified traces. The study of these traces leads to new, interesting quantum invariants of links and 3-manifolds as well as applications in the study of representation theory.

The aim of these two lecture series is to give a reasonably self-contained introduction to modified traces, with an eye on its origin in representation theory and low-dimensional topology.

# Current talks

**These talks are hosted jointly by HIM and the University of Hamburg, see here**.

**Tuesday, December 8th, 2020
**

**13:30-14:45**

**Thorsten Heidersdorf: "Generalized negligible morphisms''**

I will introduce generalized negligible morphisms and their tensor ideals. In many cases these admit modified trace functions. I will discuss the occurrence of these tensor ideals for tilting modules of quantum groups and algebraic groups and for several Deligne categories.

**Tuesday, December 8th, 2020
**

**15:00-16:15**

**Ingo Runkel: "3d TQFT from not necessarily semisimple modular tensor categories''**

Given a modular tensor category C, not necessarily semisimple, one can construct a 3d TQFT that operates on bordisms with C-colored ribbon graphs. The construction proceeds in three steps: 1) define a functor on so-called bichrome graphs; 2) combine this with a modified trace on C to obtain a surgery invariant of three-manifolds; 3) apply the universal construction to get a 3d TQFT. In the case that C is semisimple, this reproduces the Reshetikhin-Turaev TQFT. In this seminar, I will go through the above steps and discuss some properties of the resulting TQFT. This is joint work with Marco De Renzi, Azat Gainutdinov, Nathan Geer and Bertrand Patureau-Mirand.

**Tuesday, December 15th, 2020
**

**13:30-14:45**

**David Reutter: From non-unital skein theory to modified traces and non-semisimple topological field theories**

This talk is a progress report on ongoing joint work with Kevin Walker.

The three-dimensional Reshetikhin-Turaev (RT) topological field theory associated to a semisimple modular tensor category M can be understood as a boundary theory for a certain fully extended oriented 4-dimensional TQFT, originally introduced by Crane and Yetter. To a 3-manifold M, this field theory assigns a `skein space’ of embedded ribbon diagrams in M, and the extension to 4-dimensions is completely determined by specifying a `trace’ on the modular category; a morphism from the skein space of the 3-sphere to the ground field fulfilling certain non-degeneracy conditions.

In this talk, I will describe an analogous oriented 4-dimensional theory associated to a non-semisimple ribbon category C. On oriented 3-manifolds M, this theory may be expressed as a `non-unital skein theory’ of certain non-empty embedded ribbon graphs in M. Extensions of this field theory to oriented 4-bordisms are in bijective correspondence with non-degenerate modified traces on C.

In the modular case, a normalized variant of the Lyubashenko-Kerler invariant of 3-manifolds may be recovered from this 4D theory analogously to how one recovers RT invariants from Crane-Yetter theory. More generally, we expect to recover the 3D De Renzi—Gainutdinov—Geer-Patureau-Mirand—Runkel TQFT as a certain not-everywhere-defined boundary theory.

**Tuesday, December 15th, 2020
**

**15:00-16:15**

**Christoph Schweigert: Relative Serre functors, Frobenius algebras and some applications to conformal field theory**

In this talk, we explain some aspects of pivotal module categories over pivotal finite tensor categories. We show that they lead to module traces in the sense of Schaumann and are a rich source of Frobenius algebras, even if the categories involved are not semisimple. The theory is nicely compatible with the central monad and also yields Frobenius algebras in the Drinfeld center. This leads to conjectures for the field content of two-dimensional conformal field theories which pass several consistency checks.

**Lecture series: Speakers, titles and abstracts**

Jonathan Kujawa (University of Oklahoma) - An introduction to modified traces

The trace of a map and the dimension of a representation are fundamental invariants in representation theory. They are useful both for proving results in representation theory and for applications in other areas (e.g., low-dimensional topology). Unfortunately, in the non-semi-simple setting these invariants are often trivial. For example, this happens in modular representation theory, graded representation theory, and representations of quantum groups at roots of unity. Over the last decade, it became clear that many of these settings admit replacement invariants which can serve much the same role. However, it is still mysterious when exactly they exist.

In this lecture series I will give a friendly introduction to the theory of these invariants with an emphasis on examples, basic properties, and some applications within representation theory.

Marco de Renzi (Universität Zürich) – TQFTs from non-semisimple modular categories and modified traces

Topological Quantum Field Theories (TQFTs for short) provide very sophisticated tools for the study of topology in dimension 2 and 3: they contain invariants of 3-manifolds that can be computed by cut-and-paste methods, and their extremely rich structure induces representations of mapping class groups of surfaces. In recent years, so-called non-semisimple constructions have deeply generalized the standard semisimple approach of Reshetikhin and Turaev to the theory, producing powerful topological invariants and representations of mapping class groups with remarkable new properties. In this series of talks, we will explain how to construct a TQFT starting from an algebraic ingredient called (non-semisimple) modular category, and using the theory of modified traces. We will also describe some features of the invariants and representations issued by this construction.

Based on joint works with A. Gainutdinov, N. Geer, B. Patureau, and I. Runkel.

**Schedule**

All lectures will take place 19:00 local time (that is 5pm UTC), and always Mondays, Tuesdays and Thursdays; the first part of this mini lecture series is scheduled for 12.Oct.2020, 13.Oct.2020 and 15.Oct.2020, the second for 19.Oct. 2020, 20.Oct.2020 and 22.Oct.2020.

Talks are given online and zoom links will be sent to the registered participants in due time. Recordings of the talks will be posted online.

Each talk will be 25 + 5 (break) + 25 minutes plus time for questions.

**Registration**

If you are interested in attending the lecture series please register online here.

**Questions?**

If you have any questions, please feel free to contact any of the organizers:

Nicolle Gonzales, Amit Hazi, Louise Sutton, Daniel Tubbenhauer, Paul Wedrich, Jieru Zhu

**Slides/Videos**

**Jonathan Kujawa (University of Oklahoma) - An introduction to modified traces**

Lecture I

Lecture II

Lecture III

**Marco de Renzi (Universität Zürich) – TQFTs from non-semisimple modular categories and modified traces**

Lecture I

Lecture II

Lecture III