On the gl(1|1) Wess-Zumino-Witten Model

TL;DR

This paper advances the understanding of the gl(1|1) Wess-Zumino-Witten model by analyzing its correlation functions, solving the Knizhnik-Zamolodchikov equations, and establishing crossing symmetry, with connections to symplectic fermions.

Contribution

It provides explicit solutions to the KZ equations for multiple point functions and demonstrates crossing symmetry, extending the link between the gl(1|1) model and symplectic fermions.

Findings

Explicit four-point function for mixed representations

Solutions to KZ equations via hypergeometric functions

Proof of crossing symmetry at generic momenta

Abstract

We continue the study of the gl(1|1) Wess-Zumino-Witten model. The Knizhnik-Zamolodchikov equations for the one, two, three and four point functions are analyzed, for vertex operators corresponding to typical and projective representations. We illustrate their interplay with the logarithmic global conformal Ward identities. We compute the four point function for one projective and three typical representations. Three coupled first order Knizhnik-Zamolodchikov equations are integrated consecutively in terms of generalized hypergeometric functions, and we assemble the solutions into a local correlator. Moreover, we prove crossing symmetry of the four point function of four typical representations at generic momenta. Throughout, the map between the gl(1|1) Wess-Zumino-Witten model and symplectic fermions is exploited and extended.