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 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.

DAMTP-94-42, 21 pages

References in corpus (4)

• Lattice Wess-Zumino-Witten Model and Quantum Groups
• Quantisation of a particle moving on a group manifold
• On quantum group symmetries of conformal field theories
• Braiding in Conformal Field Theory and Solvable Lattice Models

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