Quantisation of the SU(N) WZW Model at Level $k$

TL;DR

This paper analyzes the quantization of the SU(N) WZW model at level k, deriving explicit braiding matrices, relating them to Racah matrices, and providing constructions for the k=1 case, advancing understanding of its algebraic structure.

Contribution

It explicitly computes the braiding matrix for SU(N) at level k, relates it to Racah matrices, and constructs chiral vertex operators for the k=1 case.

Findings

Braiding matrix explicitly derived for fundamental representation.

Deformation parameter t determined as exp(iπ/(k+N)).

Explicit free field realization for k=1 case provided.

Abstract

The quantisation of the Wess-Zumino-Witten model on a circle is discussed in the case of $SU(N)$ at level $k$. The quantum commutation of the chiral vertex operators is described by an exchange relation with a braiding matrix, $Q$. Using quantum consistency conditions, the braiding matrix is found explicitly in the fundamental representation. This matrix is shown to be related to the Racah matrix for $U_t(SL(N))$. From calculating the four-point functions with the Knizhnik-Zamolodchikov equations, the deformation parameter $t$ is shown to be $t=\exp({i\pi /(k+N)})$ when the level $k\ge 2$. For $k=1$, there are two possible types of braiding, $t=\exp({i\pi /(1+N)})$ or $t=\exp(i\pi)$. In the latter case, the chiral vertex operators are constructed explicitly by extending the free field realisation a la Frenkel-Kac and Segal. This construction gives an explicit description of how to chirally factorise the $SU(N)_{k=1}$ WZW model.