Pathspace Decompositions for the Virasoro Algebra and its Verma Modules

TL;DR

This paper introduces a new approach to analyze Virasoro algebra modules using pathspace decompositions, providing explicit algorithms and insights into their structure, especially for nonunitary models.

Contribution

It develops admissible pathspace representations for Virasoro modules, offering explicit construction algorithms and a novel decomposition of the algebra's generators.

Findings

Decomposition of Virasoro modules into positive and negative definite parts.

Explicit algorithms for constructing admissible representations.

Analysis of Virasoro generators considering nonunitarity.

Abstract

Starting from a detailed analysis of the structure of pathspaces of the ${\cal A}$-fusion graphs and the corresponding irreducible Virasoro algebra quotients $V(c,h)$ for the ($2,q$ odd) models, we introduce the notion of an admissible pathspace representation. The pathspaces ${\cal P_A}$ over the ${\cal A}$-Graphs are isomorphic to the pathspaces over Coxeter $A$-graphs that appear in FB models. We give explicit construction algorithms for admissible representations. From the finitedimensional results of these algorithms we derive a decomposition of $V(c,h)$ into its positive and negative definite subspaces w.r.t. the Shapovalov form and the corresponding signature characters. Finally, we treat the Virasoro operation on the lattice induced by admissible representations adopting a particle point of view. We use this analysis to decompose the Virasoro algebra generators themselves. This decomposition also takes the nonunitarity of the ($2,q$) models into account.