On the level-dependence of Wess–Zumino–Witten three-point functions

Abstract

Three-point functions of Wess-Zumino-Witten models are investigated. In particular, we study the level-dependence of three-point functions in the models based on algebras su(3) and su(4). We find a correspondence with Berenstein-Zelevinsky triangles. Using previous work connecting those triangles to the fusion multiplicities, and the Gepner-Witten depth rule, we explain how to construct the full three-point functions. We show how their level-dependence is similar to that of the related fusion multiplicity. For example, the concept of threshold level plays a prominent role, as it does for fusion.

Fusion rules and the Patera-Sharp generating-function method

“…This was done in [20]. Three-point functions were calculated that can be regarded as generating functions for tensor product couplings, and a very simple method was found for adapting the results to fusion couplings.…”

Affine su(3) and su(4) fusion multiplicities as polytope volumes

“…still provided (17). The set of threshold levels for the T λ,µ,ν distinct couplings is easily read off:…”

“…Since a generalisation of the BZ triangles to other Lie algebras is presently not known, our belief is mainly based on our alternative approach to the computation of fusion multiplicities. It relies on the depth rule and the relation to three-point functions in Wess-Zumino-Witten conformal field theory [16,17]. In that work BZ triangles only appear as guidelines, while the basic building blocks are certain polynomials.…”

PURELY AFFINE ELEMENTARY su(N) FUSIONS

“…In [6], we showed how the polynomial description above is ideally suited to the implementation of the Gepner-Witten depth rule [2] of affine fusion. The rule encodes the level-dependence.…”

“…The extra EFs will be called purely affine EFs. The only known example is in su(4) [5,6], and it reads…”