The free field representation for the GL(1∣1) WZW model revisited
Abstract
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 NeveuSchwarz and Ramond sector modules ${{ \mathcal V }}_{{en}}^{\mathrm{NS}}={\displaystyle \bigoplus }_{l\in {\mathbb{Z}}}{{ \mathcal V }}_{{en}}^{l}$ and ${{ \mathcal V }}_{{en}}^{{\rm{R}}}={\displaystyle \bigoplus }_{l\in {\mathbb{Z}}+\displaystyle \frac{1}{2}}{{ \mathcal V }}_{{en}}^{l}$ 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 NeveuSchwarz operators and of Ramond operators with either integer or halfinteger spins. Conformal blocks and structure constants are found for operators that correspond the highest weight vectors of the spaces ${{ \mathcal V }}_{{en}}^{l}$ . 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.
 Publication:

Physica Scripta
 Pub Date:
 January 2023
 DOI:
 10.1088/14024896/aca5c4
 arXiv:
 arXiv:2205.00557
 Bibcode:
 2023PhyS...98a5025L
 Keywords:

 MooreSeiberg equations;
 free field realization;
 GL(1∣1) supersymmetric WZW model;
 logarithmic conformal filed theory;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 31 pages