Screening Operators for Lattice Vertex Operator Algebras and Resulting Constructions

We give a rigorous treatment of the notion of screening pairs of screening operators $(\ ilde{Q},Q)$ for a rank $d$ lattice vertex operator superalgebra $V_L$. Certain such screening pairs proved to be useful machinery in the study of the internal structure of the $\mathcal{W}\mbox{-algebra}$ $\mathcal{W}(p)=\mbox{ker }\ ilde{Q}$ by Adamovi\'c and Milas and in proving the $C_2$-cofinite property in the rank 1 case. We analyze and classify when screening pairs can arise for $L$ of arbitrary rank, and then give a classification of when configurations of multiple screening pairs can occur for lattice vertex operator algebras with lattices of rank $2$ and ADE-type root lattices. We then show how to construct subalgebras of $V_L$ by considering the kernel of a screening operator for $V_L$ when $L$ is of rank $2$, and how one can take the intersection of the kernels of certain commuting screening operators to obtain interesting vertex operator algebras that can be analyzed using this intersection of kernels structure. The subalgebras studied here share features similar to the $\mathcal{W}(p)$-algebra in the rank 1 setting, in that they are simple, $C_2$-cofinite, and irrational. We classify the irreducible modules for these vertex operator superalgebras.