Field identifications in coset conformal theories from projection matrices.

We demonstrate the usefulness of projection matrices for finite subalgebras h c 3 and their affine counterparts h c i in finding field identifications (and selection rules) in coset conformal field theories. Submitted to Phys. Lett. B. * Work supported by the U.S. Department of Energy under Contract DE-AC0376SF00515. Address after September 1, 1989: Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 14853. t Work supported in part by NSERC of Canada. Address after September 1, 1989: Departement de physique, Universitk Laval, Laval, Quebec, Canada GlK 7P4. Coset conformal field theories[ 11 may include all two-dimensional rational conformal field theories[2]. For every finite Lie subalgebra h C 3, one can construct a two-dimensional conformal field theoryi. If G, H are the (covering) Lie groups whose algebras are S, h, the embedding h c s will quite generally specify a relation between the centres B(G), B(H) of the two groups. We will . explain how these relations may be identified. One of their consequences in the coset conformal field theory is a selection rule saying that certain primary fields do not occur[3,4]. Let i, k denote the Kac-Moody algebras that are the central extensions of the loop algebras of 3, h, respectively. Then the finite subalgebra h c s with index of embedding e induces an affine subalgebra kek c i’“, where the superscripts are the levels (see, for example, Reference [5]3. There exist automorphisms of G (k) which are not themselves elements of 4 (L) and are therefore called outer automorphisms[6]. The outer automorphisms of 4 permute the fundamental weights &‘(p = 0, 1, . . . , R; R = rank(g)) in -such a way as to leave the Dynkin diagram of ij invariant. Similarly, outer automorphisms of & permute the fundamental weights wO( cy = O,l, . . . ,r; r = rank(h) ) of A. The group of outer automorphisms of i, O(G), is isomorphic to the centre B(G). Relations between the centres B(G) and B(H) are therefore accompanied 1 We assume h is a maximal subalgebra of ij, otherwise the coset theory factors into (a/i) @ (k/h) theories, where k C ij is maximal.