Level-rank duality of the U(N) WZW model, Chern-Simons theory, and 2d qYM theory

We study the WZW, Chern-Simons, and 2d qYM theories with gauge group U(N). The U(N) WZW model is only well-defined for odd level K, and this model is shown to exhibit level-rank duality in a much simpler form than that for SU(N). The U(N) Chern-Simons theory on Seifert manifolds exhibits a similar duality, distinct from the level-rank duality of SU(N) Chern-Simons theory on S-3. When q = e(2 pi i/(N+K)), the observables of the 2d U(N) qYM theory can be expressed as a sum over a finite subset of U(N) representations. When N and K are odd, the qYM theory exhibits N <-> K duality, provided q = e(2 pi i/(N+K)) and theta =0 mod 2 pi /(N + K).