Quantisation of the SU (N) WZW model at level k

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 operator, Q. Using quantum consistency conditions, the braiding operator is found explicitly in the fundamental representation. Its matrix elements are identified as the Racah coefficients for Ut(SL(N)). From studying the monodromy of the four-point functions by solving Knizhnik-Zamolodchikov equations, the deformation parameter t is shown to be t = exp(iπ/ (k + N)) when the level k ⩾ 2. For k = 1, there are two possible types of braiding, t = exp(iπ/(1 + N)) or t = exp(iπ). In the latter case, the chiral vertex operators are constructed explicitly by extending the free field realisation à la Frenkel-Kac and Segal. This construction gives an explicit description of how to chirally factorise the SU(N)k=1 WZW model.