Affine Jordan cells, logarithmic correlators, and hamiltonian reduction

TL;DR

This paper develops a logarithmic extension of SL(2,R) Wess-Zumino-Witten models using affine Jordan cells, solving associated Ward identities, and extending Hamiltonian reduction to logarithmic correlators, revealing hierarchical structures in the chiral blocks.

Contribution

It introduces affine Jordan cells into SL(2,R) models, solves conformal and SL(2,R) Ward identities for logarithmic correlators, and extends Hamiltonian reduction to these new structures.

Findings

Logarithmic terms appear in two- and three-point chiral blocks.

Chiral blocks satisfy modified Knizhnik-Zamolodchikov equations.

Hamiltonian reduction applies straightforwardly to logarithmic correlators.

Abstract

We study a particular type of logarithmic extension of SL(2,R) Wess-Zumino-Witten models. It is based on the introduction of affine Jordan cells constructed as multiplets of quasi-primary fields organized in indecomposable representations of the Lie algebra sl(2). We solve the simultaneously imposed set of conformal and SL(2,R) Ward identities for two- and three-point chiral blocks. These correlators will in general involve logarithmic terms and may be represented compactly by considering spins with nilpotent parts. The chiral blocks are found to exhibit hierarchical structures revealed by computing derivatives with respect to the spins. We modify the Knizhnik-Zamolodchikov equations to cover affine Jordan cells and show that our chiral blocks satisfy these equations. It is also demonstrated that a simple and well-established prescription for hamiltonian reduction at the level of ordinary correlators extends straightforwardly to the logarithmic correlators as the latter then reduce to the known results for two- and three-point conformal blocks in logarithmic conformal field theory.