Conformal blocks for admissible representations of $SL(2)$ current algebra

TL;DR

This paper reviews explicit integral formulas for conformal blocks in admissible representations of the SL(2) current algebra, utilizing Wakimoto free field construction and addressing fractional powers of free fields.

Contribution

It introduces a second screening charge depending on fractional powers and develops techniques to handle these complexities for conformal blocks.

Findings

Derived explicit integral representations for conformal blocks on the sphere.

Verified conformal blocks satisfy Knizhnik-Zamolodchikov equations.

Discussed fusion rules for admissible representations.

Abstract

A review is presented of the recently obtained expressions for conformal blocks for {\it admissible} representations in $SL(2)$ current algebra based on the Wakimoto free field construction. In this realization one needs to introduce a second screening charge, one which depends on fractional powers of free fields. The techniques necessary to deal with these complications are developed, and explicit general integral representations for conformal blocks on the sphere are provided. The fusion rules are discussed and as a check it is verified that the conformal blocks satisfy the Knizhnik-Zamolodchikov equations. (Talk presented by J. Rasmussen at the Leuven workshop, July 10-14 1995, to appear in the proceedings)