Quasi root systems and vertex operator realizations of the Virasoro algebra

TL;DR

This paper introduces quasi root systems as a generalization of root systems in Lie algebra theory and constructs the Virasoro algebra using vertex operators derived from these systems.

Contribution

It defines quasi root systems, explores their properties, and provides explicit solutions for the Virasoro algebra realization in terms of these systems.

Findings

Introduction of quasi root systems as a new mathematical concept

Explicit solutions for the Virasoro algebra using quasi root systems

A comprehensive list of examples of quasi root systems

Abstract

A construction of the Virasoro algebra in terms of free massless two-dimensional boson fields is studied. The ansatz for the Virasoro field contains the most general unitary scaling dimension 2 expression built from vertex operators. The ansatz leads in a natural way to a concept of a quasi root system. This is a new notion generalizing the notion of a root system in the theory of Lie algebras. We introduce a definition of a quasi root system and provide an extensive list of examples. Explicit solutions of the ansatz are presented for a range of quasi root systems.