Classification of screening systems for lattice vertex operator algebras

TL;DR

This paper classifies and analyzes systems of screening operators in lattice vertex operator algebras, identifying four main types and providing a comprehensive classification for rank two and certain higher-rank lattices.

Contribution

It introduces a classification of screening pairs and systems in lattice vertex operator algebras, including a detailed analysis for rank two and specific higher-rank lattices.

Findings

Classified screening pairs into four types.

Identified important role of one screening type in W-algebras.

Classified all screening systems for rank two lattices.

Abstract

We study and classify systems of certain screening operators arising in a generalized vertex operator algebra, or more generally an abelian intertwining algebra with an associated vertex operator (super)algebra. Screening pairs arising from weight one primary vectors acting commutatively on a lattice vertex operator algebra (the vacuum module) are classified into four general types, one type of which has been shown to play an important role in the construction and study of certain important families of $\mathcal{W}$-vertex algebras. These types of screening pairs we go on to study in detail through the notion of a system of screeners, which are lattice elements or `screening momenta' which give rise to screening pairs. We classify screening systems for all positive definite integral lattices of rank two, and for all positive definite even lattices of arbitrary rank when these lattices are generated by a screening system.