Singular Vectors in Logarithmic Conformal Field Theories

TL;DR

This paper extends the concept of null vectors to indecomposable representations in logarithmic conformal field theories, providing a formalism that aids in their classification and understanding.

Contribution

It develops a compact formalism for null vectors in logarithmic CFTs, enabling the representation of the stress energy tensor on Jordan cells as linear differential operators.

Findings

Null vectors impose strong constraints on logarithmic CFTs

The formalism facilitates classification of these theories

Representation of stress energy tensor on Jordan cells is achieved

Abstract

Null vectors are generalized to the case of indecomposable representations which are one of the main features of logarithmic conformal field theories. This is done by developing a compact formalism with the particular advantage that the stress energy tensor acting on Jordan cells of primary fields and their logarithmic partners can still be represented in form of linear differential operators. Since the existence of singular vectors is subject to much stronger constraints than in regular conformal field theory, they also provide a powerful tool for the classification of logarithmic conformal field theories.