On staggered indecomposable Virasoro modules

Abstract

In this article, certain indecomposable Virasoro modules are studied. Specifically, the Virasoro mode L0 is assumed to be nondiagonalizable, possessing Jordan blocks of rank 2. 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 centers 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. Given just the values of the highest weights of the left and right modules, themselves subject to simple necessary conditions, invariants are defined which together with the knowledge of the left and right modules uniquely identify a staggered module. The possible values of these invariants form a vector space of dimension 0, 1, or 2, and the structures of the left and right modules limit the isomorphism classes of the corresponding staggered modules to an affine subspace (possibly empty). The number of invariants and affine restrictions is purely determined by the structures of the left and right modules. Moreover, in order to facilitate applications, the expressions for the invariants and restrictions are given by formulas as explicit as possible (they generally rely on expressions for Virasoro singular vectors). Finally, 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.