On Reducible but Indecomposable Representations of the Virasoro Algebra

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.