  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).

High Energy Physics - Theory