arXiv:hep-th/0309269v1 30 Sep 2003 hep-th/0309269 HU-EP-03/68 September 2003 Algebras in tensor categories and coset conformal field theories J. Fr¨ ohlich, 1 J. Fuchs, 2 I. Runkel, 3 C. Schweigert 4 § 1 Institut f¨ ur Theoretische Physik, ETH Z¨ urich, CH – 8093 Z¨ urich, Switzerland 2 Institutionen f¨ or fysik, Karlstads Universitet, S – 651 88 Karlstad, Sweden 3 Institut f¨ ur Physik, Humboldt-Universit¨ at, D – 12 489 Berlin, Germany 4 Fachbereich Mathematik, Universit¨ at Hamburg, D – 20 146 Hamburg, Germany Abstract: The coset construction is the most important tool to construct rational conformal field theories with known chiral data. For some cosets at small level, so-called maverick cosets, the familiar analysis using selection and identification rules breaks down. Intriguingly, this phenomenon is linked to the existence of exceptional modular invariants. Recent progress in CFT, based on studying algebras in tensor categories, allows for a universal construction of the chiral data of coset theories which in particular also applies to maverick cosets. 1 Coset conformal field theories The coset construction is among the oldest [1] tools for obtaining rational two-dimensional conformal field theories and has been very successful. It has been used to construct prominent classes of models, such as (super-)Virasoro minimal models and Kazama-Suzuki models. Still, it presents a number of mysteries, even in the case of unitary conformal field theories, to which we will restrict ourselves in this contribution. The coset construction is based on the following data: A (finite-dimensional, complex, reductive) Lie algebra g together with a choice k of levels, i.e. a positive integer for each simple ideal of g, and a Lie subalgebra g ′ of g. The embedding of g ′ into g determines the levels k ′ of g ′ . The aim of the coset construction is to obtain conformal field theories whose chiral data – like conformal weights, fusion rules, braiding and fusing matrices – are completely known, and moreover can be expressed entirely in terms of the chiral data for (g,k) and (g ′ ,k ′ ). This goal can indeed be achieved. However, as will become evident below, the way this aim is reached is quite a bit more subtle than one might anticipate. At first sight, understanding coset theories proceeds according to the following well- known pattern: The g/g ′ coset theory has a description in terms of a gauged WZW sigma model [2] with target space the Lie group G (the compact simply connected covering group associated to g), in which the action of the subgroup G ′ of G is gauged. For constructing the space of states, this immediately suggests to start with positive energy representations § corresponding author : schweigert@math.uni-hamburg.de