Conformal blocks of Wess-Zumino-Witten model from its free-field representation

Abstract

A bstract A powerful approach to the celebrated Wess-Zumino-Witten (WZW) model is provided by its free-field realization. However, explicit calculations of conformal blocks are not described in the literature in full detail. We begin this study with the simplest cases of the $$ \hat{sl}{(2)}_k $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>sl</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:msub> <mml:mfenced> <mml:mn>2</mml:mn> </mml:mfenced> <mml:mi>k</mml:mi> </mml:msub> </mml:math> and $$ \hat{sl}{(3)}_k $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>sl</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:msub> <mml:mfenced> <mml:mn>3</mml:mn> </mml:mfenced> <mml:mi>k</mml:mi> </mml:msub> </mml:math> WZW models, with special emphasis on their global sl (2) and sl (3) symmetries of the resulting correlators, which are not explicit in this formalism. Also non-trivial is the verification of the Knizhnik-Zamolodchikov equations in the $$ \hat{sl}{(3)}_k $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>sl</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:msub> <mml:mfenced> <mml:mn>3</mml:mn> </mml:mfenced> <mml:mi>k</mml:mi> </mml:msub> </mml:math> case, where the answers take the form of double integrals over screening charge positions and do not look like ordinary hypergeometric functions.