Large $N$ Vertex Algebras via Deligne Category
Keyou Zeng
TL;DR
This work provides a rigorous framework for vertex algebras inside symmetric monoidal (and Karoubian) categories, with a detailed construction of vertex algebras in Deligne categories and their ind-completions. The authors define a VA object in such categories, develop its basic structure (vacuum, translation, locality, OPE) and functorial properties, and then realize a concrete large $N$ vertex algebra via BRST reduction of a $eta ext{-}oldsymbol{eta}$ system in $ ext{Rep}( ext{GL}_N)$. The main achievement is a systematic categorical realization of Costello–Gaiotto’s large $N$ VA as $H^ullet(oldsymbol{A}_N,Q)$, including an $ ext{N}=4$ super-Virasoro symmetry and two compatible vertex Poisson structures encoding planar and non-planar deformations. This framework not only gives a solid mathematical footing for large $N$ limits but also connects to 4d/2d dualities and planar algebraic limits via deformation quantization. Overall, the paper provides a versatile categorical toolkit for constructing and studying large $N$ vertex algebras and their semiclassical/Poisson limits.
Abstract
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 $βγ$ 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.