Workshop 2011

Workshop on Representation Theoretical and Categorical Structures in Quantum Geometry and Conformal Field Theory

Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg

Erlangen, October 31 to November 2 2011

Photo Workshop

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, 2-groups in quantum geometry, inclusion of matter and observables.
    • (Higher) categories and (extended) TFTs: discrete models involving higher categories, relation between Turaev-Viro and Reshetikhin-Turaev invariants, categorical structures and higher TFTs.
    • Representation theoretical structures inspired by rational CFT: module categories, categorical center, chiral theories as a three-category.

Speakers

  • Benjamin Balsam, Stony Brook
  • John Barrett, University of Nottingham
  • Christoher Douglas, University of Oxford
  • Jürgen Fuchs, Karlstad University
  • Winston Fairbairn, FAU Erlangen-Nürnberg
  • 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 Erlangen-Nürnberg

Programme

For each of the main topics there will be two introductory talks (1h each) which give an in-depth 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, Turaev-Viro theory has been discovered and described by at least three independent research groups in the contexts of topology (Turaev-Viro), condensed matter physics (Levin-Wen) and coding theory (Kitaev). First, I’ll give a brief introduction to all three descriptions and explain why they are equivalent. Using the state-sum description of Turaev-Viro theory, we prove the following theorem: Let C be a spherical fusion category, Z(C) its Drinfeld Center. Turaev-Viro TQFT determined by a spherical fusion category C is equivalent to the Reshetikhin-Turaev 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 Schommer-Pries and Noah Snyder in which we show that fusion categories are fully-dualizable objects of a symmetric monoidal 3-category of tensor categories. This provides a local, that is fully extended, 3-dimensional 3-framed 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 spin-independence of the associated field theory. We also describe 2-dimensional 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 Erlangen-Nürnberg: State sum models in 3d quantum gravity I will review the role played by state sum models in three-dimensional quantum gravity. I will firstly discuss classical aspects of gravity in three space-time dimensions and show how to introduce particles into the framework. I will then introduce the Ponzano-Regge and Turaev-Viro state sum models and discuss their relation to (Euclidean) three-dimensional quantum gravity. Finally, I will show how to generalise the Ponzano-Regge model as a mean to incorporate particles.

 

    • Jürgen Fuchs, Karlstad University: A categorical construction of invariants of mapping class groups To any finite-dimensional factorizable ribbon Hopf algebra H we construct an H-bimodule F that has a natural structure of a commutative symmetric Frobenius algebra. We use the structural morphisms of F to construct, for any genus-g 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 H-bimodules. 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 H-mod.

 

    • Krzysztof Gawędzki, ENS Lyon: Global gauge anomalies in 2D I shall discuss the invariance under gauge transformations non-homotopic to identity of gauged 2D sigma models with Wess-Zumino 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. Levin-Wen models describe a large class of non-chiral 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 Levin-Wen 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 Turaev-Viro TQFT. This is a joint work with Alexei Kitaev.

 

    • Marco Mackaay, Algarve University: The Schur 2-category 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 q-Schur 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 braided-quantum 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 2-Category I will describe an extended (2-categorical) topological QFT with target 2-category 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 Freed-Hopkins-Lurie-Teleman. 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 G-categories I review my work on the role of braided crossed G-categories (due to Turaev and others) in the description of the representation category of conformal orbifold models.

 

    • Thomas Nikolaus, University of Regensburg: Equivariant Dijkgraaf-Witten theory For a finite group G there is a well known Quantum field theory called Dijkgraaf-Witten 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 Dijkgraaf-Witten 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: Chern-Simons 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 multi-fusion category C can be characterized as the category of finite-dimensional 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 Schur-Weyl dual picture. The second type of relations removes suitable group-likes in order obtain a finite-dimensional 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 non-abelian 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 non-abelian 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 two-dimensional topological and conformal field theory, and briefly also in three-dimensional 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 Chern-Simons 2-gerbe over the moduli stack of G-bundles 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 Chern-Simons theories with associated higher WZW gerbes. I briefly indicate an example from 5-brane physics. This is joint work with Hisham Sati and Domenico Fiorenza.

 

    • Rafal Suszek, University of Warsaw: A CFT-inspired fusion of G-spaces The Verlinde fusion ring R_k(G) of a simple compact connected Lie group G at level k is a representation-theoretic concept that plays an important role in the description of the RCFT of the level-k Wess-Zumino-Witten type with G as a target space. In particular, it models the fusion algebra of the so-called maximally symmetric conformal defects. Using basic tools of the 2-category of bundle gerbes with connection associated with the RCFT, I shall introduce a bimonoidal category with anti-involution closely related to the category of quasi-Hamiltonian G-spaces 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: 3-dimensional TQFTs Two fundamental constructions of 3-dimensional topological quantum field theories (TQFTs) are due to Reshetikhin-Turaev and Turaev-Viro. The Reshetikhin-Turaev construction, based on surgery presentations of 3-manifolds, is widely viewed as a mathematical realization of Witten’s Chern-Simons TQFT. The Turaev-Viro construction, based on triangulations of 3-manifolds, is closely related to the Ponzano-Regge state-sum model for 3-dimensional quantum gravity. I will explain how these two constructions are related via the Drinfeld-Joyal-Street center of monoidal categories. The key points are first to define the Turaev-Viro state sum invariant not only on triangulations but on skeletons (for example, any CW-decomposition of a 3-manifold is a skeleton), and second to use a Hopf monadic description of the Drinfeld-Joyal-Street center. This is a joint work with Vladimir Turaev.

 

    • Derek Wise, FAU Erlangen-Nürnberg: Higher gauge theory and teleparallel gravity I will describe a relationship between higher gauge theory—the generalization of gauge theory from groups to 2-groups—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 Erlangen-Nürnberg (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 Erlangen-Nürnberg 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.uni-erlangen.de Christoph.Schweigert@uni-hamburg.de