Free field realization of SL(2) correlators for admissible representations, and hamiltonian reduction for correlators

TL;DR

This paper develops a free field realization for SL(2) WZW theories with admissible representations, introduces fractional calculus for screening charges, and derives integral representations of conformal blocks satisfying KZ equations, relating them to minimal models via hamiltonian reduction.

Contribution

It introduces fractional calculus techniques to handle fractional powers in screening charges and explicitly relates SL(2) correlators to minimal models through hamiltonian reduction.

Findings

Derived explicit integral representations of conformal blocks.

Showed conformal blocks satisfy Knizhnik-Zamolodchikov equations.

Established connection between SL(2) correlators and minimal models.

Abstract

Talk presented by J.L. Petersen at the 29th Symposium Ahrenshop, Buckow August 29-September 2, 1995. A presentation is given of the free field realization relevant to SL(2) WZW theories with a Hilbert space based on admissible representations. It is known that this implies the presence of two screening charges, one involving a fractional power of a free field. We develop the use of fractional calculus for treating in general such cases. We derive explicit integral representations of $N$-point conformal blocks. We show that they satisfy the Knizhnik-Zamolodchikov equations and we prove how they are related to minimal conformal blocks via a formulation of hamiltonian reduction advocated by Furlan, Ganchev, Paunov and Petkova.