The action of outer automorphisms on bundles of chiral blocks

On the bundles of WZW chiral blocks over the moduli space of a punctured rational curve we construct isomorphisms that implement the action of outer automorphisms of the underlying affine Lie algebra. These bundle-isomorphisms respect the Knizhnik-Zamolodchikov connection and have finite order. When all primary fields are fixed points, the isomorphisms are endomorphisms; in this case, the bundle of chiral blocks is shown to be a reducible vector bundle. A conjecture for the trace of such endomorphisms is presented, both for genus zero and for higher genera; the proposed relation generalizes the Verlinde formula and is compatible with factorization. Our results have applications to conformal field theories based on non-simply connected groups and to the classification of boundary conditions in such theories.