# Recent seminars

## 25/01/2023, Wednesday, 17:00–18:00 Europe/Lisbon Room P3.10, Mathematics Building — Online

In 2010, Mazorchuk and Miemietz laid the foundations of a systematic theory of finitary 2-representations of finitary 2-categories, which are the categorical analog of finite-dimensional representations of finite-dimensional algebras. In the last couple of years, this theory has been much further developed and has led to interesting classification results for e.g. certain finitary 2-representations of Soergel bimodules of finite Coxeter type, which form an important class of examples.

Together with Miemietz and Vaz, I've recently started to look at 2-representations of Soergel bimodules of affine type A, which form a 2-category that is no longer finitary but only locally wide finitary, a generalization which was introduced and studied by Macpherson. This has major consequences for their 2-representations, e.g. they now come in 3 different flavors: finitary, wide finitary and triangulated.

In my talk, I will first very briefly review finitary 2-representation theory of finitary 2-categories and recall the example of Soergel bimodules of finite Coxeter type. After that, I will zoom in on Soergel bimodules of affine type A and their three types of 2-representations. I will try to sketch some general features, but the talk will nevertheless be very example-based, since our research is still in its early stages.

Local participants are welcome to attend the live presentation in room 3.10.

## 09/12/2022, Friday, 12:00–13:00 Europe/Lisbon Room P3.10, Mathematics Building — Online

Zoran Petric, Mathematical Institute SANU, Belgrade, Serbia
A diagrammatic presentation of the category 3Cob

The category 3Cob has closed oriented surfaces as objects and 3-dimensional cobordisms, i.e. 3-dimensional compact oriented manifolds (possibly with boundary) with canonical orientation preserving (reversing) identification of the incoming (outgoing) boundary. The composition is defined in terms of gluing. We present this category using a diagrammatic language similar to the language of standard surgery presentation of closed, orientable, connected 3-manifolds, save that besides framed links we use wedges of circles in our diagrams.

We will explain how to interpret such a diagram as an arrow of 3Cob and give an outline of the composition calculus for diagrams. This is a joint work with Jovana Nikolic and Mladen Zekic.

zoran-petric-slides

NOTE: Please note the unusual day and time.  Local participants are invited to join the speaker and ourselves in room 3.10 (3rd floor, Mathematics Department, Instituto Superior Técnico).

## 07/12/2022, Wednesday, 17:00–18:00 Europe/Lisbon Room P3.10, Mathematics Building — Online

Konstantin Eder, University of Erlangen–Nürnberg
Super Cartan geometry and (loop) quantum supergravity

In this talk, a mathematically rigorous approach toward geometric supergravity will be discussed which, in the physical literature, is usually known as the Castellani-D'Auria-Fré approach. To this end, using tools from supergeometry, the notion of a super Cartan geometry will be introduced. Interestingly, in order to consistently incorporate the anticommutative nature of fermionic fields, the ordinary category of supermanifolds needs to be generalized in a physically consistent way leading to the notion of so-called enriched supermanifolds. We then apply this formalism to discuss a geometric formulation of (generalized) pure Anti-de Sitter supergravity with N=1,2 supersymmetry in D=4 modified by an additional Holst term. In this context, we will also talk about so-called picture changing operators (PCO) and how they can be implemented in a mathematically rigorous way. Finally, an outlook will be given for applications of this formalism to (loop) quantum supergravity and the description of quantum supersymmetric black holes.

Local participants are invited to join us in room 3.10 (3rd floor, Mathematics Department, Instituto Superior Técnico).

## 30/11/2022, Wednesday, 17:00–18:00 Europe/Lisbon Room P3.10, Mathematics Building — Online

A representation of the string 2-group

The string 2-group is supposed to play the role of the spin group, but in string theory instead of quantum mechanics. Several aspects of this analogy are by now well understood. In this talk I will talk about joint work with Matthias Ludewig and Peter Kristel on a further aspect, namely the representation theory of the string 2-group. This was an open problem for a long time. Our solution combines higher-categorical topology with operator algebras, and allows a neat definition of Stolz-Teichner's "stringor bundle" as an associated 2-vector bundle.

Local participants are invited to join us in room 3.10 (3rd floor, Mathematics Department, Instituto Superior Técnico).

## 16/11/2022, Wednesday, 17:00–18:00 Europe/Lisbon Room P3.10, Mathematics Building — Online

Fiona Torzewska, University of Leeds
Topological quantum field theories and homotopy cobordisms

I will begin by explaining the construction of a category $\operatorname{CofCos}$, whose objects are topological spaces and whose morphisms are cofibrant cospans. Here the identity cospan is chosen to be of the form $X\to X\times [0,1] \rightarrow X$, in contrast with the usual identity in the bicategory $\operatorname{Cosp}(V)$ of cospans over a category $V$. The category $\operatorname{CofCos}$ has a subcategory $\operatorname{HomCob}$ in which all spaces are homotopically $1$-finitely generated. There exist functors into $\operatorname{HomCob}$ from a number of categorical constructions which are potentially of use for modelling particle trajectories in topological phases of matter: embedded cobordism categories and motion groupoids for example. Thus, functors from $\operatorname{HomCob}$ into $\operatorname{Vect}$ give representations of the aforementioned categories.

I will also construct a family of functors $Z_G : \operatorname{HomCob} \to \operatorname{Vect}$, one for each finite group $G$, showing that topological quantum field theories previously constructed by Yetter, and an untwisted version of Dijkgraaf-Witten, generalise to functors from $\operatorname{HomCob}$. I will construct this functor in such a way that it is clear the images are finite dimensional vector spaces, and the functor is explicitly calculable.