Vertex Algebras at the Corner

Abstract

We introduce a class of Vertex Operator Algebras which arise at junctions of supersymmetric interfaces in ${\cal N}=4$ Super Yang Mills gauge theory. These vertex algebras satisfy non-trivial duality relations inherited from S-duality of the four-dimensional gauge theory. The gauge theory construction equips the vertex algebras with collections of modules labelled by supersymmetric interface line defects. We discuss in detail the simplest class of algebras $Y_{L,M,N}$, which generalizes $W_N$ algebras. We uncover tantalizing relations between $Y_{L,M,N}$, the topological vertex and the $W_{1+\infty}$ algebra.