Workshop 2011
Workshop on Representation Theoretical and Categorical Structures in Quantum Geometry and Conformal Field Theory
Department Mathematik, FriedrichAlexanderUniversität ErlangenNürnberg
Erlangen, October 31 to November 2 2011
Photograph by Martina Lanini
Scientific Objective
Manifold invariants constructed from the representation categories of Hopf algebras and from more general tensor categories play an important role in quantum geometry, conformal field theory, representation theory and topological field theory. The workshop aims to bring together scientists from these disciplines to encourage interaction and to develop a common language across the boundaries of these subjects. Within these three main subjects, the thematic focus will be on the following research topics:

 Quantum geometry and quantum gravity: state sum models, quantum group representations in quantum geometry, models involving (higher) categories, 2groups in quantum geometry, inclusion of matter and observables.
 (Higher) categories and (extended) TFTs: discrete models involving higher categories, relation between TuraevViro and ReshetikhinTuraev invariants, categorical structures and higher TFTs.
 Representation theoretical structures inspired by rational CFT: module categories, categorical center, chiral theories as a threecategory.
Speakers
 Benjamin Balsam, Stony Brook
 John Barrett, University of Nottingham
 Christoher Douglas, University of Oxford
 Jürgen Fuchs, Karlstad University
 Winston Fairbairn, FAU ErlangenNürnberg
 Krzysztof Gawędzki, ENS Lyon
 Liang Kong, Tsinghua University
 Marco Mackaay, Algarve University
 Shahn Majid, Queen Mary University
 Jeffrey Morton, IST, Lisbon
 Michael Müger, University of Nijmegen
 Thomas Nikolaus, University of Regensburg
 Karim Noui, Université de Tours
 Victor Ostrik, University of Oregon
 Hendryk Pfeiffer, UBC
 Roger Picken, IST, Lisbon
 Ingo Runkel, Hamburg University
 Urs Schreiber, Utrecht University
 Rafal Suszek, University of Warsaw
 Alexis Virelizier, Université Montpellier 2
 Derek Wise, FAU ErlangenNürnberg
Programme
For each of the main topics there will be two introductory talks (1h each) which give an indepth presentation of the subject as well as several talks covering specific aspects of the topic (1 h) and shorter contributions (30 min). All talks will be aimed at a mixed audience of mathematicians and physicists and emphasize the connections to the other fields.
Schedule
Time  Monday Oct. 31  Tuesday Nov. 1  Wednesday Nov. 2 

8:30 a.m.  Registration  Virelizier I  
9 a.m.  Barrett I  Ostrik II  
9:30 a.m.  Coffee Break  
10.a.m.  Coffee break  Ostrik I  Coffee break 
10:30am  Barrett II  Virelizier II  
10:45 a.m.  Kong  
11:15 am  Douglas  Wise  
11:45 am  Pfeiffer  Lunch Break  
12:15 pm  Mackaay  
12:45 pm  Lunch break  
1:15 pm  Lunch break  
2 pm  Müger  Fairbairn  
2:30 p.m.  Gawędzki  
3 pm  Majid  Balsam  
3:30 p.m.  Picken  
4 p.m.  Suszek  Coffee break  
4:30 p.m.  Coffee break  Schreiber  Morton 
5 p.m.  Runkel  Fuchs  
5:30 p.m.  Nikolaus  Noui  
6 p.m.  Welcome  
6:30 p.m.  
7 p.m.  
7:30 p.m.  Conference Dinner 
Lunch Breaks
Lunch will be provided at the conference venue (Kollegienhaus, Universitätsstraße 15, 91054 Erlangen). If you have any dietary restrictions or special requirements, please indicate so in the registration form.
Welcome
On Monday, October 31, there will be an informal welcome with local drinks and snacks directly after the last talk. It will take place in the conference venue (Kollegienhaus, Universitätsstraße 15, 91054 Erlangen).
Conference Dinner
The conference dinner will take place Tuesday, November 1 2011, 7:30 pm at the restaurant Schwarzer Bär, Innere Brucker Straße 19, 91054 Erlangen. The restaurant is a 5 minute walk from the conference venue (map with directions).
Talks

 Benjamin Balsam, Stony Brook: Turaev Viro theory and Extended TQFTs In the last 20 years, TuraevViro theory has been discovered and described by at least three independent research groups in the contexts of topology (TuraevViro), condensed matter physics (LevinWen) and coding theory (Kitaev). First, I’ll give a brief introduction to all three descriptions and explain why they are equivalent. Using the statesum description of TuraevViro theory, we prove the following theorem: Let C be a spherical fusion category, Z(C) its Drinfeld Center. TuraevViro TQFT determined by a spherical fusion category C is equivalent to the ReshetikhinTuraev TQFT determined by Z(C). Finally, we consider TV theory at the (0,1,2) levels and show how the TQFT behaves as a fully extended theory. This is joint work with Sasha Kirillov.

 John Barrett, University of Nottingham: Quantum gravity and category theory The talks will summarise attempts to construct theories of quantum gravity using methods of category theory; the focus will be on the physical ideas and the development of the subject. Some possible new directions will be sketched.

 Christoher Douglas, University of Oxford: Tensor categories and topology I’ll discuss joint work with Chris SchommerPries and Noah Snyder in which we show that fusion categories are fullydualizable objects of a symmetric monoidal 3category of tensor categories. This provides a local, that is fully extended, 3dimensional 3framed topological field theory for each fusion category. We investigate the correspondence between algebraic properties of the fusion category and topological properties of the associated field theory. In particular, we explain how the sphericality of a fusion category corresponds to the spinindependence of the associated field theory. We also describe 2dimensional field theories associated to finite tensor and rigid tensor categories, and give a transparent topological proof of the fact that the quadruple dual functor on a fusion category is trivial.

 Winston Fairbairn, FAU ErlangenNürnberg: State sum models in 3d quantum gravity I will review the role played by state sum models in threedimensional quantum gravity. I will firstly discuss classical aspects of gravity in three spacetime dimensions and show how to introduce particles into the framework. I will then introduce the PonzanoRegge and TuraevViro state sum models and discuss their relation to (Euclidean) threedimensional quantum gravity. Finally, I will show how to generalise the PonzanoRegge model as a mean to incorporate particles.

 Jürgen Fuchs, Karlstad University: A categorical construction of invariants of mapping class groups To any finitedimensional factorizable ribbon Hopf algebra H we construct an Hbimodule F that has a natural structure of a commutative symmetric Frobenius algebra. We use the structural morphisms of F to construct, for any genusg Riemann surface S with m punctures, an element Z in the space of bimodule morphisms from K^g to F^m, where K is Lyubashenko’s Hopf algebra object for the category of Hbimodules. We show that for g=1 the morphism Z is invariant under a natural action (found by Lyubashenko) of the mapping class group of S, and conjecture that this extends to all other values of the genus. There are analogous results for the situation that H is supplemented by a ribbon automorphism of H. All these structures are expected to have analogues for a more general class of monoidal categories that share properties of the categories Hmod.

 Krzysztof Gawędzki, ENS Lyon: Global gauge anomalies in 2D I shall discuss the invariance under gauge transformations nonhomotopic to identity of gauged 2D sigma models with WessZumino terms. On mathematical side, such invariance is guaranteed by the existence of equivariant structures on appropriate gerbes and gerbe modules. Applications to WZW and coset models of CFT will be briefly covered.

 Liang Kong, Tsinghua University: Topological orders and tensor categories Topological order is an important subject in condensed matter physics. It may have applications to quantum computing. LevinWen models describe a large class of nonchiral topological orders. In my talk, I will show how to use the representation theory of tensor categories to classify all gapped boundaries and defects of codimension 1,2, and 3 in LevinWen models. In particular, I will show that a boundary theory determines the bulk theory uniquely, but a bulk theory only determine boundary theories up to Morita equivalence. I will also discuss its connection to extended TuraevViro TQFT. This is a joint work with Alexei Kitaev.

 Marco Mackaay, Algarve University: The Schur 2category and Chuang and Rouquier’s colored braid complex In my talk I will explain how a quotient of Khovanov and Lauda’s categorification of quantum sl(n) categorifies the qSchur algebra. This Schur category contains Soergel’s category of bimodules, which categorifies the Hecke algebra. In the second part of my talk I will recall Rouquier’s categorification of the projection of the braid group onto the Hecke algebra. To each braid Rouquier associated a complex of Soergel bimodules, which he showed to be invariant up to homotopy under the braidlike Reidemeister moves. I will show how this can be generalized for colored braids, using complexes in the Schur category. The results I will present have appeared or will appear in joint papers with (various subsets of) Khovanov, Lauda, Stosic and Vaz.

 Shahn Majid, Queen Mary University: Bar categories and noncommutative geometry on braided algebras The talk covers the categorical formulation of `complex conjugation‘ in noncommutative geometry using the notion of a bar category and how it applies to quantum groups at roots of unity. I will also explain how it is the noncommutative geometry of U_q(g), not of the quantum group coordinate algebras, that emerges from 3D quantum gravity and TQFT. More precisely, it is the noncommutative geometry of the braidedquantum groups B_q(G) associated to the representation category in the canonical construction of Lyubashenko and myself. I construct a canonical differential exterior algebra on these

 Jeffrey Morton, IST, Lisbon: Extended TQFT in a Bimodule 2Category I will describe an extended (2categorical) topological QFT with target 2category consisting of C*algebras and bimodules. The construction is explained as factorizable into a classical field theory valued in groupoids, and a quantization functor, as in the program of FreedHopkinsLurieTeleman. I will explain the Lagrangian action functional in terms of cohomological twisting of the groupoids in the classical part of the theory, and describe how this is incorporated into the quantization functor. This project is joint work with Derek Wise.

 Michael Müger, University of Nijmegen: Orbifolds of rational CFTs and braided crossed Gcategories I review my work on the role of braided crossed Gcategories (due to Turaev and others) in the description of the representation category of conformal orbifold models.

 Thomas Nikolaus, University of Regensburg: Equivariant DijkgraafWitten theory For a finite group G there is a well known Quantum field theory called DijkgraafWitten theory. This can be described as an extended 3d TFT. From that theory one can extract an interesting tensor category which can also be described as the representation category of a Quantum group D(G) (the Drinfel’d double of G). We present an equivariant extension of DijkgraafWitten theory. This leads us to equivariant generalizations of D(G) and its represenation categories. We furthermore discuss the issue of modularity and the orbifold theory.

 Karim Noui, Université de Tours: ChernSimons theory and Black Holes

 Victor Ostrik, University of Oregon: Tensor categories and conformal field theory I will survey the role played by the theory of tensor categories in conformal field theory.

 Hendryk Pfeiffer, UBC: Combinatorial characterization of fusion categories Every multifusion category C can be characterized as the category of finitedimensional comodules of a quotient of the path algebra of a quiver GxG. Here G is a finite directed graph that depends on the fusion rules and on the choice of a generator of C. The path algebra k(GxG) is a Weak Bialgebra, and the quotient is modulo two types of relations. The first enforce that the tensor powers of the generator have the appropriate endomorphism algebras, thus providing a SchurWeyl dual picture. The second type of relations removes suitable grouplikes in order obtain a finitedimensional Weak Hopf Algebra whose category of comodules is the desired fusion category with all the additional structure. As an example, I show the modular categories associated with U_q(sl_2) for suitable roots of unity in this picture. If there is time, I sketch how to obtain canonical bases for their objects.

 Roger Picken, IST, Lisbon: Surface parallel transport based on a crossed module, and beyond I will describe an approach to nonabelian parallel transport along surfaces, taking values in a crossed module of groups, using a cubical, as opposed to simplicial, framework. I also plan to discuss some interesting potential applications and generalizations, such as Wilson surfaces, 3D quantum gravity, and nonabelian 3D parallel transport. This is largely based on work with João Faria Martins.

 Ingo Runkel, Hamburg University: Low dimensional field theories with domain walls and related categorical structures I will discuss domain walls and other defects in twodimensional topological and conformal field theory, and briefly also in threedimensional topological field theory. The focus will be on the categorical notions that enter their mathematical description.

 Urs Schreiber, Utrecht University: Higher WZW terms Using homotopy theory of smooth higher groupoids, I show how to construct the ChernSimons 2gerbe over the moduli stack of Gbundles with connection and how it loops to the WZW gerbe on G itself. The whole construction works also for higher smooth groups and produces higher ChernSimons theories with associated higher WZW gerbes. I briefly indicate an example from 5brane physics. This is joint work with Hisham Sati and Domenico Fiorenza.

 Rafal Suszek, University of Warsaw: A CFTinspired fusion of Gspaces The Verlinde fusion ring R_k(G) of a simple compact connected Lie group G at level k is a representationtheoretic concept that plays an important role in the description of the RCFT of the levelk WessZuminoWitten type with G as a target space. In particular, it models the fusion algebra of the socalled maximally symmetric conformal defects. Using basic tools of the 2category of bundle gerbes with connection associated with the RCFT, I shall introduce a bimonoidal category with antiinvolution closely related to the category of quasiHamiltonian Gspaces of Alekseev, Malkin and Meinrenken, and subsequently present the existing (partial) evidence in favour of the conjecture that the category gives a geometric model of R_k(G).

 Alexis Virelizier, Université Montpellier 2: 3dimensional TQFTs Two fundamental constructions of 3dimensional topological quantum field theories (TQFTs) are due to ReshetikhinTuraev and TuraevViro. The ReshetikhinTuraev construction, based on surgery presentations of 3manifolds, is widely viewed as a mathematical realization of Witten’s ChernSimons TQFT. The TuraevViro construction, based on triangulations of 3manifolds, is closely related to the PonzanoRegge statesum model for 3dimensional quantum gravity. I will explain how these two constructions are related via the DrinfeldJoyalStreet center of monoidal categories. The key points are first to define the TuraevViro state sum invariant not only on triangulations but on skeletons (for example, any CWdecomposition of a 3manifold is a skeleton), and second to use a Hopf monadic description of the DrinfeldJoyalStreet center. This is a joint work with Vladimir Turaev.

 Derek Wise, FAU ErlangenNürnberg: Higher gauge theory and teleparallel gravity I will describe a relationship between higher gauge theory—the generalization of gauge theory from groups to 2groups—and the geometry of flat linear connections. Since general relativity can be reformulated as a „teleparallel“ theory, involving a flat connection with torsion, this leads to an unexpected way of understanding gravity as a higher gauge theory. This will be a report on work in progress with John Baez. A draft paper is available at: http://math.ucr.edu/home/baez/teleparallel.pdf.
Conference Venue
The workshop location is the Kollegienhaus, Universitätsstraße 15, 91054 Erlangen of the University of ErlangenNürnberg (Map). It is located in the city centre of Erlangen, close to the train station and borders on the historical castle garden.
Support
This workshop is funded by the ESF Research Networking Programme „Quantum Geometry and Quantum Gravity“ and by the Priority Programme 1388: Representation Theory of the German Research Foundation (DFG). We express our gratitude for their support.
Organisers
Catherine Meusburger  Christoph Schweigert 
Department Mathematik FAU ErlangenNürnberg Cauerstraße 11 91058 Erlangen Germany  Fachbereich Mathematik Universität Hamburg Bundesstraße 55 (Geomatikum) 20146 Hamburg Germany 
Tel: 0049 9131 85 67034  Tel: 0049 40 42838 5170 
Fax: 0049 9131 85 67036  Fax: 0049 40 42838 5190 
Catherine.Meusburger@math.unierlangen.de  Christoph.Schweigert@unihamburg.de 