On Staggered Indecomposable Virasoro Modules

TL;DR

This paper develops a general theory for staggered indecomposable Virasoro modules with non-diagonalisable L_0, focusing on identification and existence proofs, using elementary techniques and relevant examples from logarithmic conformal field theory.

Contribution

It provides the first comprehensive framework for understanding staggered Virasoro modules, including methods for their identification and construction.

Findings

Established criteria for identifying staggered modules

Proved existence of certain staggered modules

Illustrated concepts with physically relevant examples

Abstract

In this article, certain indecomposable Virasoro modules are studied. Specifically, the Virasoro mode L_0 is assumed to be non-diagonalisable, possessing Jordan blocks of rank two. Moreover, the module is further assumed to have a highest weight submodule, the "left module", and that the quotient by this submodule yields another highest weight module, the "right module". Such modules, which have been called staggered, have appeared repeatedly in the logarithmic conformal field theory literature, but their theory has not been explored in full generality. Here, such a theory is developed for the Virasoro algebra using rather elementary techniques. The focus centres on two different but related questions typically encountered in practical studies: How can one identify a given staggered module, and how can one demonstrate the existence of a proposed staggered module. The text is liberally peppered throughout with examples illustrating the general concepts. These have been carefully chosen for their physical relevance or for the novel features they exhibit.