The free field representation for the $GL(1|1)$ WZW model revisited

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.