On the Quantum Symmetry of Rational Field Theories

The aim of this talk is to describe a possible understanding of the quantum symmetry of two-dimensional (D = 2) rational quantum field theories (or D = 1 chiral rational conformal field theories). We start by briefly sketching the operator-algebraic approach to relativistic quantum field theory (for a review, see for example [1,2]) and in particular the Doplicher--Haag--Roberts program for the description of the superselection sectors ([3] for D > 2 and [4] for D = 2). The category CA of localized endomorphisms of the observable algebra is introduced. This is a strict monoidal, rigid category which is symmetric in D > 2, i.e. one has permutation statistics, but only braided in D = 2, i.e., in D = 2 one has generically braid group statistics. We restrict our attention to D = 2 and to the rational case, i.e., when CA has a finite number of simple objects. In the case of chiral conformal field theories the corresponding category has been described in [5]. Doplicher and Roberts [6] have completed the DHR program in D > 2, showing that CA is equivalent to the category of finite-dimensional representations of some compact Lie group – the group of “internal” symmetries of the theory. The fact that for D = 2 the category CA is a braided one has lead various people to argue that the internal symmetries are given by quantum groups. 1 Considering rational theories one has to restrict oneself to quantum groups at roots of unity. For generic values of the deformation parameter the quantum groups of Drinfeld and Jimbo and Faddeev--Reshetikhin--Takhtadzhan are indeed deformations of the group algebra or universal enveloping algebra of ordinary simple Lie groups or algebras, and in fact the representation theory remains unchanged. On the other hand, at q a root of unity much of the similarity with the undeformed case breaks down. Quantum groups at roots of unity are not semisimple, and as a consequence their category of representations contains indecomposable (i.e., reducible, but not fully reducible) representations. Though this is well known, a number of papers simply ignored this fact. The careful analysis shows that