The action of outer automorphisms on bundles of chiral blocks

TL;DR

This paper constructs isomorphisms on bundles of WZW chiral blocks that implement outer automorphisms, respecting the Knizhnik-Zamolodchikov connection, with implications for conformal field theories and boundary condition classification.

Contribution

It introduces bundle-isomorphisms corresponding to outer automorphisms that preserve the connection and proposes a trace conjecture generalizing the Verlinde formula.

Findings

Isomorphisms respect the Knizhnik-Zamolodchikov connection.

Conjecture for the trace of endomorphisms generalizes the Verlinde formula.

Applications to non-simply connected groups and boundary conditions.

Abstract

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 typically a reducible vector bundle. A conjecture for the trace of such endomorphisms is presented; the proposed relation generalizes the Verlinde formula. Our results have applications to conformal field theories based on non-simply connected groups and to the classification of boundary conditions in such theories.