# Arbeitsgemeinschaft: Conformal Field Theory

### Christoph Schweigert

Universität Hamburg, Germany### Yasuyuki Kawahigashi

University of Tokyo, Japan### Victor Ostrik

University of Oregon, Eugene, United States

## Abstract

The Arbeitsgemeinschaft mit aktuellem Thema “Algebraic structures in conformal field theories”, organized by Y. Kawahigashi (University of Tokyo), V. Ostrik (University of Oregon) and C. Schweigert (University of Hamburg), was held from April 1 to 7, 2007.

Two-dimensional conformal field theory plays a fundamental role in the theory of two-dimensional critical systems of classical statistical mechanics, in quasi one-dimensional condensed matter physics and in string theory. The study of defects in systems of condensed matter physics, of percolation probabilities and of (open) string perturbation theory in the background of certain solitonic solutions of string theory, the so-called D-branes, forces one to analyze conformal field theories on surfaces that may have boundaries and/or can be non-orientable. This study has recently led to deeper insight into the mathematical structure of conformal field theory.

Many mathematical disciplines have contributed to a better understanding of conformal field theory and have received stimulating input from questions arising in conformal field theories. There are two major approaches to chiral conformal field theory: one that is based on operator algebras and one based on vertex algebras. In both approaches, chiral conformal field theory is described by a certain infinite dimensional algebraic structure. In the first approach, conformal field theory is described by a net of von Neumann algebras, where each von Neumann algebra consists of bounded linear operators and is generated by local observables. This approach was initiated by Haag and Kastler more than 40 years ago (and applies also to quantum field theories in dimensions other than two). The latter is based on algebraic axiomatization of quantum fields in chiral conformal field theory and was initiated by Frenkel, Lepowsky, Meurman and Borcherds in the 1980s.

Both algebraic structures lead to representation categories which are tensor categories and, in the case of rational chiral conformal field theories, more specifically modular tensor categories. In the operator algebraic approach, the representation category consists of representations of a net of von Neumann algebras on a Hilbert space; while its original form is due to Doplicher, Haag and Roberts, for chiral conformal field theory certain adaptations have to be made. The representation category of vertex algebras consists of modules over (conformal) vertex algebras. In this Arbeitsgemeinschaft, we have studied algebraic structures related to tensor categories arising in conformal field theory.

These tensor categories also encode the monodromy representations of the vector bundles of conformal blocks for rational vertex algebras, objects that are of interest for algebraic geometry. Moreover, modular tensor categories are a crucial ingredient in the construction of three-dimensional topological field theories.

While chiral conformal field theories have certain physical applications in the description of quantum Hall systems, full local conformal field theories are relevant for the physical applications referred to in the first paragraph. Recently, it has been understood that the construction of a full local conformal field theory is best described using the structure of a module category over the tensor category that describes the chiral data.

In view of the above background, we started the Arbeitsgemeinschaft with a general introduction (Svegstrup) to tensor categories and module categories to provide an oecumenic language for both approaches. Then we had three talks on how tensor categories naturally arise in various approaches to conformal field theory: through operator algebras (Bartels), loop groups (Bunke), and Frobenius algebras (Grossman).

The notion of a fusion category provides an abstract framework for various kinds of representation categories. The quantum double construction, originally due to Drinfeld, is a very important method to produce a modular tensor category. It applies to a finite group and also to a more general tensor category. Whenever a finite group acts on a conformal field theory by symmetries, one can pass to a new theory fixed by such a symmetry. This new theory is called an orbifold, and constructions of this type are studied in many different approaches. Doubles of finite groups, their representation categories and orbifold theories in the operator algebraic approach were introduced in the talks by Müller and Gray.

Certain module categories can be classified and the $A$-$D$-$E$ classification for the case of the quantum $SL(2)$ due to Kirillov and Ostrik is a basic example. Another type of classification for fusion categories is the one for given fusion rules or a given number of simple objects. Various such classification results have been obtained by Ostrik and collaborators; they have been the subject of two talks (Phung, Cuntz). The operator algebraic approach based on theory of subfactors by Jones has produced a few exceptional tensor categories (the topic of Peters' talk) which have not been obtained by other approaches such as theory of quantum groups. Our understanding of such structures in the framework of conformal field theory is still very poor and further developments are expected.

There have been various concrete constructions of $(2+1)$-dimensional topological quantum field theory in the sense of Atiyah. Two of such constructions, due to Reshetikhin–Turaev and Turaev–Viro, are particularly closely related to tensor categories. In these constructions, a three-dimensional closed manifold is realized through Dehn surgery and triangulation, respectively, and combinatorial data arising from a tensor category produce a number for each closed manifold which is a topological invariant of the manifold. This was explained in a first talk by Suszek; in a second talk (Schommer-Pries), the construction of three-dimensional topological field theories from subfactors was presented.

Conformal blocks play a fundamental role in conformal field theory. They were defined in the concrete context of Wess–Zumino–Witten models in Graziano's talk and the Knizhnik–Zamolodchikov connection on them was introduced in Nieper–Wißkirchen's talk.

Correlation functions, symmetries and dualities are studied in the categorical framework of conformal field theory. Techniques from topological field theory provide tools to study such objects. This was the topic of two talks (Lehn, Zito). In this context, a tensor functor from a tensor category to a category of bimodules called $α$-induction plays a prominent role; this was the topic of a Asaeda's talk. A study of boundary conformal field theory in the operator algebraic approach, based on recent results of Longo–Rehren was presented by Bahns. It gives a concrete realization of the general abstract structure.

We had 17 talks by the participants and two sessions where 12 participants gave 10-minute presentations on their work. We also had two short supplemental presentations by the organizers. We had a sufficient amount of time for free discussions among the participants. The meeting had 54 participants from various countries in Europe and the U.S., Canada, Japan and India.

## Cite this article

Christoph Schweigert, Yasuyuki Kawahigashi, Victor Ostrik, Arbeitsgemeinschaft: Conformal Field Theory. Oberwolfach Rep. 4 (2007), no. 2, pp. 945–994

DOI 10.4171/OWR/2007/17