Workshop 2014
Structures on Tensor Categories and Topological Field Theories
Department Mathematik, FriedrichAlexanderUniversität ErlangenNürnberg
Erlangen, March 46 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 nonsemisimple 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 SchommerPries 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 SchommerPries, Max Planck Institute for Mathematics, Bonn
 Noah Snyder, Columbia University
Programme
Schedule
Time  Tuesday March 4  Wednesday March 5  Thursday March 6 

9:30  Registration  
10:00  Chris SchommerPries  Chris SchommerPries  Chris SchommerPries 
11:00  Coffee break  Coffee break  Coffee break 
11:30  Chris SchommerPries  Chris SchommerPries  Chris SchommerPries 
12:30  Lunch break  Lunch break  Lunch break 
14:30  Alain Bruguières Slides  Thomas Kerler Slides  Noah Snyder 
15:30  Coffee Break  Coffee Break  Coffee Break 
16:00  Bruce Bartlett Slides  Christoph Schweigert Slides  Questions & Discussion 
17:00  Questions & Discussion  Questions & Discussion  
17:30  Welcome  
19:00  Conference Dinner 
Talks

 Bruce Bartlett: Threedimensional bordism representations from generatorsandrelations I will give an overview of joint work establishing a simple generatorsandrelations presentation of the 3dimensional 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 21 fashion to a modular category, which must be anomalyfree in the oriented case. J/w Chris Douglas, Jamie Vicary, Chris SchommerPries. Slides

 Alain Bruguières: Hopf Structures and Tensor Categories Slides

 Thomas Kerler: Half Projective TQFTs The notion of a halfprojective 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 halfprojective TQFTs in which x is not a unit and the extension is nontrivial. These include nonsemisimple TQFTs for which x=0 and invariants vanish on circle products, as well as integral TQFTs for which x generates a nontrivial 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 halfprojective axiom to extended TQFTs Slides, Slides Print Version

 Chris SchommerPries: 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 nonsemisimple tensor categories. Instead of a 3dimensional topological field theory we get a 2d TFT with a little bit of extra 3dimensional information. In particular, there is enough 3dimensional 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, FriedrichAlexander Universität ErlangenNürnberg, catherine.meusburger@math.unierlangen.de

 Christoph Schweigert, Fachbereich Mathematik, Universität Hamburg, christoph.schweigert@unihamburg.de
Conference Secretary
 Johanna Kulzer Department Mathematik FAU ErlangenNürnberg Cauerstraße 11 91058 Erlangen GermanyTel: 0049 9131 85 67035 Fax: 0049 9131 85 67036 Johanna.Kulzer@math.unierlangen.de
Travel and Location
The workshop will take place in Erlangen, which is a university town in the southeast 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 ErlangenNürnberg, 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 ErlangenNuremberg via its Emerging Fields Initiative.