Group-level duality of WZW fusion coefficients and Chern-Simons link observables

We present the duality of the fusion rule multiplicities, modular tranformation matrices and conformal weights of the Wess-Zumino-Witten models based on any of the classical affine Lie algebras SU(N) K,Sp(N) Kand SO(N) K under the exchange of N and K. We also exhibit the transformation properties of these quantities under the symmetries of the extended Dynkin diagrams, which play a central role in this group-level duality. We interpret these results in Chern-Simons theory, and derive identities between link observables. Several surprising results involving spinor representations of SO(N) are proven