Large $N$ Vertex Algebras via Deligne Category

In this paper, we propose a new construction of vertex algebras using the Deligne category. This approach provides a rigorous framework for defining the so-called large $N$ vertex algebra, which has appeared in recent physics literatures. We first define the notion of a vertex algebra in a symmetric monoidal category and extend familiar constructions in ordinary vertex algebras to this broader categorical context. As an application, we consider a $\beta\gamma$ vertex algebra in the Deligne category and construct the large N vertex algebra from it. We study some simple properties of this vertex algebra and analyze a certain vertex Poisson algebra limit.