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

TL;DR

The paper shows that level-rank duality for the U(N) WZW model extends coherently to three related theories: Chern-Simons theory on Seifert manifolds and 2d q-deformed Yang-Mills theory. It establishes a clean duality between the U(N) and U(K) WZW models at odd N and K via a one-to-one map of primaries and a simple relation between modular data, which then propagates to CS observables on Seifert manifolds and to the truncated qYM partition function at roots of unity q = e^{2π i/(N+K)}. Under these conditions, the partition function and Wilson line observables satisfy an N↔K duality up to known phases or complex conjugation, with the duality intimately tied to the level-rank structure of the underlying WZW theory. The work clarifies how these dualities differ across manifolds and theories and highlights potential implications for large-N dualities in topological field theories, while noting the q-real case relevant to black hole counting does not directly realize the same duality. Overall, the paper unifies several manifestations of level-rank duality in low-dimensional gauge theories.

Abstract

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).