Application of vertex algebras to the structure theory of certain representations over the Virasoro algebra

TL;DR

This paper explores the structure and irreducibility of tensor products of Virasoro algebra modules, introducing new criteria and discovering novel irreducible modules, with implications for vertex operator algebra theory.

Contribution

It generalizes Zhang's irreducibility criterion, introduces new irreducible Virasoro modules, and links module reducibility to intertwining operators in vertex operator algebras.

Findings

Irreducibility depends on roots of a polynomial from singular vectors

Discovery of a new type of irreducible Vir-module with infinite-dimensional weight spaces

Complete structure analysis for tensor products in minimal models

Abstract

In this paper we discuss the structure of the tensor product V'_{\alpha,\beta}\otimes L(c,h) of irreducible module from intermediate series and irreducible highest weight module over the Virasoro algebra. We generalize Zhang's irreducibility criterion, and show that irreducibility depends on the existence of integral roots of a certain polynomial, induced by a singular vector in the Verma module V(c,h). A new type of irreducible Vir-module with infinite-dimensional weight subspaces is found. We show how the existence of intertwining operator for modules over vertex operator algebra yields reducibility of V'_{\alpha ,\beta}\otimes L(c,h) which is a completely new point of view to this problem. As an example, the complete structure of the tensor product with minimal models c=-22/5 and c=1/2 is presented.