What the characters of irreducible subrepresentations of Jordan cells can tell us about LCFT

TL;DR

This paper explores how characters of irreducible subrepresentations in Jordan cells reveal key features of logarithmic conformal field theories, especially in the context of triplet algebras and minimal models.

Contribution

It demonstrates that characters of irreducible submodules encode significant information about LCFTs and fit into the ADET-classification, highlighting their importance in classifying rational CFTs.

Findings

Characters form a basis for torus amplitudes

Characters are modular forms of inhomogeneous weight

LCFT characters fit into ADET-classification

Abstract

In this article, we review some aspects of logarithmic conformal field theories which can be inferred from the characters of irreducible submodules of indecomposable modules. We will mainly consider the W(2,2p-1,2p-1,2p-1) series of triplet algebras and a bit logarithmic extensions of the minimal Virasoro models. Since in all known examples of logarithmic conformal field theories the vacuum representation of the maximally extended chiral symmetry algebra is an irreducible submodule of a larger, indecomposable module, its character provides a lot of non-trivial information about the theory such as a set of functions which spans the space of all torus amplitudes. Despite such characters being modular forms of inhomogeneous weight, they fit in the ADET-classification of fermionic sum representations. Thus, they show that logarithmic conformal field theories naturally have to be taken into account when attempting to classify rational conformal field theories.