Categorical structures in (the representation theory of) conformal field theory
by
Michael Müger
Korteweg de Vries Institute for Mathematics, University of Amsterdam

We review recent work by the author and others concerning categorical structures in (rigorous) conformal field theory.

1. (Very brief) The representation category of a (suitably defined) rational chiral conformal field theory (RCFT) is a modular category, i.e. a semisimple C-linear ribbon category with trivial center.

2. Finite local extensions of a RCQT A are classified by certain commutative algebras in the (braided) category Rep A, and for an extension B corresponding to the monoid (A, m, e), Rep B is equivalent to the category of dyslectic (A, m, e) modules in Rep A (in the sense of Pareigis).

3. If a RCFT A comes with an action of a finite group G then Rep A is the grade zero subcategory of a braided crossed G-category (Turaev) G-Rep(B). In the simplest case, where Rep B is trivial, the possible categories G-Rep(B) are classified by Ospel's `quasiabelian cohomology' H³_qa (G, C^*). (This is in analogy to the well known classification of group categories by H³(G, k^*) and of braided group categories by H³_ab(G, k^*), where G must be abelian.)

4. The classification of so-called modular invariants leads to not necessarily commutative monoids in Rep A. We explain the current understanding.

Date received: May 16, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cajf-15.