论文标题
重新研究$ gl（1 | 1）$ WZW模型的免费现场表示形式
The free field representation for the $GL(1|1)$ WZW model revisited
论文作者
论文摘要
在使用$ bc $系统的免费字段实现中的$ gl（1 | 1）$ WZW模型将重新审视。通过实现$ bc $系统，我们描述了Neveu--Schwarz和Ramond部门模块$ \ MATHCAL V^{\ text {ns}} _ {en} = \ bigoplus_ {l \ in \ Mathbb Z} v^{\ text {r}} _ {en} = \ bigoplus_ {l \ in \ mathbb z+{1 \ over2}} \ mathcal v^l_ {en} $在给定的fermion数字$ l $的子空间中。我们表明，有两个相互本地运营商的扇区，每个部门由所有Neveu-Schwarz运营商和带有整数或半级旋转的Ramond操作员组成。对于对应于空间的最高权重向量的操作员，找到了共形块和结构常数。考虑了交叉和编织矩阵，六角形和五边形方程被证明满足了典型的模块。考虑中间状态，具有非典型（对数）模块的共形块的简化病例。在退化情况下，相关函数的已知共形块分解显示与交叉和编织关系中的退化分裂有关。讨论了非典型模块中的标量产品。解释了该标量产物中统一函数中统一函数中统一的分解。
The $GL(1|1)$ WZW model in the free field realization that uses the $bc$ system is revisited. By bosonizing the $bc$ system we describe the Neveu--Schwarz and Ramond sector modules $\mathcal V^{\text{NS}}_{en}=\bigoplus_{l\in\mathbb Z}\mathcal V^l_{en}$ and $\mathcal V^{\text{R}}_{en}=\bigoplus_{l\in\mathbb Z+{1\over2}}\mathcal V^l_{en}$ in terms of the subspaces of a given fermion number $l$. We show that there are two sectors of mutually local operators, each consists of all Neveu--Schwarz operators and of Ramond operators with either integer or half-integer spins. Conformal blocks and structure constants are found for operators that correspond the highest weight vectors of the spaces $\mathcal V^l_{en}$. The crossing and braiding matrices are considered and the hexagon and pentagon equations are shown to be satisfied for typical modules. The degenerate case of conformal blocks with atypical (logarithmic) modules as intermediate states is considered. The known conformal block decomposition of correlation functions in the degenerate case is shown to be related to the degeneration splitting in the crossing and braiding relations. The scalar product in atypical modules is discussed. The decomposition of unity in the full correlation functions in the degenerate case in terms of this scalar product is explained.
