A matrix S for all simple current extensions Original Research Article

A formula is presented for the modular transformation matrix S for any simple current extension of the chiral algebra of a conformal field theory. This provides in particular an algorithm for resolving arbitrary simple current fixed points, in such a way that the matrix S we obtain is unitary and symmetric and furnishes a modular group representation. The formalism works in principle for any conformal field theory. A crucial ingredient is a set of matrices SJab, where J is a simple current and a and b are fixed points of J. We expect that these input matrices realize the modular group for the torus one-point functions of the simple currents. In the case of WZW models these matrices can be identified with the S-matrices of the orbit Lie algebras that were introduced recently [J. Fuchs et al., preprint hep-th/9506135, Commun. Math. Phys., in press]. As a special case of our conjecture we obtain the modular matrix S for WZW theories based on group manifolds that are not simply connected, as well as for most coset models.