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

TL;DR

This paper explores the level-rank duality in U(N) WZW, Chern-Simons, and 2d qYM theories, revealing simplified duality relations and connections between these models for odd levels and specific parameters.

Contribution

It demonstrates a simpler form of level-rank duality for U(N) WZW and Chern-Simons theories and establishes duality conditions for 2d qYM theory with odd N and K.

Findings

U(N) WZW model exhibits simpler level-rank duality for odd K.

U(N) Chern-Simons theory on Seifert manifolds shows a similar duality.

2d U(N) qYM theory exhibits N <--> K duality under specific conditions.

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).