On Reducible but Indecomposable Representations of the Virasoro Algebra

TL;DR

This paper explores reducible yet indecomposable representations of the Virasoro algebra, focusing on Jordan block structures and their implications for logarithmic conformal field theories, especially at specific central charges.

Contribution

It introduces the concept of Jordan lowest weight modules and classifies the moduli space of staggered modules, advancing understanding of logarithmic representations in conformal field theory.

Findings

Classification of Jordan lowest weight modules

Determination of the moduli space of staggered modules

Analysis of representations at central charge c=-2

Abstract

Motivated by the necessity to include so-called logarithmic operators in conformal field theories (Gurarie, 1993) at values of the central charge belonging to the logarithmic series c_{1,p}=1-6(p-1)^2/p, reducible but indecomposable representations of the Virasoro algebra are investigated, where L_0 possesses a nontrivial Jordan decomposition. After studying `Jordan lowest weight modules', where L_0 acts as a Jordan block on the lowest weight space (we focus on the rank two case), we turn to the more general case of extensions of a lowest weight module by another one, where again L_0 cannot be diagonalized. The moduli space of such `staggered' modules is determined. Using the structure of the moduli space, very restrictive conditions on submodules of `Jordan Verma modules' (the generalization of the usual Verma modules) are derived. Furthermore, for any given lowest weight of a Jordan Verma module its `maximal preserving submodule' (the maximal submodule, such that the quotient module still is a Jordan lowest weight module) is determined. Finally, the representations of the W-algebra W(2,3^3) at central charge c=-2 are investigated yielding a rational logarithmic model.