Structures on Tensor Categories and Topological Field Theories

Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg

Erlangen, March 4-6 2014

Photograph by Johanna Kulzer   This is a three day learning event on recent developments in (extended) topological field theories. The focus is the interplay of structures on tensor categories and topological field theories, including structures related to non-semisimple categories. The idea is not to have yet another workshop or conference but to study these topics in depth, with a schedule more like the one of an Oberwolfach workshop, with a lecture series and plenty of time for discussions. It will include a lecture series by Christopher Schommer-Pries on topological field theories and the underlying categorical structures

Speakers

Speakers include:

• Bruce Bartlett, Oxford University
• Alain Bruguières, Université Montpellier II
• Thomas Kerler, Ohio State University
• Christopher Schommer-Pries, Max Planck Institute for Mathematics, Bonn
• Noah Snyder, Columbia University

Programme

Schedule

Talks

• • Bruce Bartlett: Three-dimensional bordism representations from generators-and-relations I will give an overview of joint work establishing a simple generators-and-relations presentation of the 3-dimensional oriented bordism bicategory, and also its „signature“ central extension. This allows for an elementary proof that a representation of this bicategory (i.e a 123 TQFT) corresponds in a 2-1 fashion to a modular category, which must be anomaly-free in the oriented case. J/w Chris Douglas, Jamie Vicary, Chris Schommer-Pries. Slides

• • Alain Bruguières: Hopf Structures and Tensor Categories Slides

• • Thomas Kerler: Half Projective TQFTs The notion of a half-projective TQFT refers to a modification of Atiyah’s axioms as a functor on cobordism categories that introduces a connectivity anomalie. The latter is with respect to a particular element x in the ground ring R of the target category of the TQFT, and is redundant if x is a unit in R. We will discuss several examples of half-projective TQFTs in which x is not a unit and the extension is non-trivial. These include non-semisimple TQFTs for which x=0 and invariants vanish on circle products, as well as integral TQFTs for which x generates a non-trivial ideal in R. We will present open questions on connecting this formalism to recent developments in gauge and Morse theoretically constructed TQFTs, as well as on generalizations of the half-projective axiom to extended TQFTs Slides, Slides Print Version

• • Chris Schommer-Pries: Structures on tensor categories inspired by local topological field theory One philosophical interpretation of the cobordism hypothesis is that, although it at first appears geometric, the cobordism category has a rich algebraic structure and is in fact the universal example of this structure. This means that anywhere this algebraic structure appears we obtain a powerful connection with the topology of manifolds. Each cobordism becomes an equation, and topological identities become powerful algebraic identities. The goal of these lectures will be to explore this phenomenon in detail in the particular case of tensor categories. We hope to show how many aspects of the theory of tensor categories have a natural „cobordism theoretical“ interpretation.

• • Christoph Schweigert: Invariants for mapping class group actions from ribbon Hopf algebra automorphisms Slides
  • Noah Snyder: Radford’s theorem for finite tensor categories In this talk I will explain how the techniques from Chris’s talk can be adapted to finite but non-semisimple tensor categories. Instead of a 3-dimensional topological field theory we get a 2d TFT with a little bit of extra 3-dimensional information. In particular, there is enough 3-dimensional topology in order to show that the topological belt trick proves Radford’s theorem on quadruple duals for finite tensor categories.

Lunch Breaks

Lunch will be provided at the university restaurant in the same building. There will be a choice of different dishes, including meat or fish, vegetarian options and a salad bar.

Welcome

On Tuesday, March 4, there will be an informal welcome with beer and pretzels directly after the questions and discussion session. It will take place in the conference building.

Conference Dinner

The conference dinner will take place Wednesday, March 5, 7:00 pm at the restaurant Schwarzer Bär, Innere Brucker Straße 19, 91054 Erlangen. The restaurant is in Erlangen city centre in walking distance from all hotels (Map).

Organisers

• • Catherine Meusburger, Department Mathematik, Friedrich-Alexander Universität Erlangen-Nürnberg, catherine.meusburger@math.uni-erlangen.de

• • Christoph Schweigert, Fachbereich Mathematik, Universität Hamburg, christoph.schweigert@uni-hamburg.de

Conference Secretary

• Johanna Kulzer Department Mathematik FAU Erlangen-Nürnberg Cauerstraße 11 91058 Erlangen GermanyTel: 0049 9131 85 67035 Fax: 0049 9131 85 67036 Johanna.Kulzer@math.uni-erlangen.de

Travel and Location

The workshop will take place in Erlangen, which is a university town in the south-east of Germany in the region of Franconia. It is easy to reach by a combination of air and train travel. The conference venue is the Department of Mathematics, University of Erlangen-Nürnberg, Cauerstraße 11, 91058 Erlangen. (Map, Campus Map ) The talks will take place in the lecture theatre Hörsaal 13 on the first floor.

Support

This workshop is an activity of the Emerging Fields Project Quantum Geometry and funded by the University of Erlangen-Nuremberg via its Emerging Fields Initiative.