Conformal Blocks for admissible representations in SL(2) current algebra

TL;DR

This paper develops a method to compute conformal blocks for admissible representations in SL(2) current algebra using free field constructions, extending to non-integrable cases and verifying key properties.

Contribution

It introduces a novel approach to handle fractional powers of free fields in screening charges, enabling explicit integral representations for a broader class of representations.

Findings

Derived explicit conformal block expressions for admissible representations.

Verified consistency with Knizhnik-Zamolodchikov equations.

Analyzed fusion rules and compared with existing literature.

Abstract

We show how to deal with screening charges involving fractional powers of free fields. This enables us to use the free field Wakimoto construction to obtain complete expressions for integral representations of conformal blocks for N-point functions on the sphere, also in the case of non-integrable representations, in particular for admissible representations. We verify several formal properties including the Knizhnik-Zamolodchikov equations. We discuss the fusion rules which result from our treatment, and compare with the literature.