标题： SU(N) WZW 模型在第$k$层的量子化 显示英文标题

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

摘要： 讨论了在圆周上量子化Wess-Zumino-Witten模型在$SU(N)$水平为$k$的情况。 手征顶点算符的量子交换由带有编织矩阵$Q$的交换关系描述。 利用量子一致性条件，在基本表示下明确找到了编织矩阵。 此矩阵被证明与$U_t(SL(N))$的 Racah 矩阵有关。 通过使用Knizhnik-Zamolodchikov方程计算四点函数，当水平为$k\ge 2$时，变形参数$t$被证明为$t=\exp({i\pi /(k+N)})$。 对于$k=1$，存在两种可能的编织类型，$t=\exp({i\pi /(1+N)})$或$t=\exp(i\pi)$。 在后一种情况下，通过扩展弗伦克尔-卡奇和塞加尔的自由场实现，明确构造了手征顶点算子。 这种构造给出了如何对手征地分解$SU(N)_{k=1}$WZW 模型的一个明确描述。 显示英文摘要

摘要： 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.