Classification of screening systems for lattice vertex operator algebras

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.