Limits of Vertex Algebras and Large N Factorization

TL;DR

This paper explores the limits of sequences of vertex algebras, establishing conditions for their direct limits to form vertex algebras and analyzing large N limits in permutation orbifolds relevant to holographic CFTs.

Contribution

It provides a framework for understanding when the direct limit of vertex algebras is a vertex algebra and characterizes large N limits in permutation orbifolds.

Findings

Large N limits exist for nested oligomorphic permutation orbifolds.

Necessary and sufficient conditions for factorization of large N limits.

Clarifies potential vertex operator algebras for holographic conformal field theories.

Abstract

We investigate the limit of sequences of vertex algebras. We discuss under what condition the vector space direct limit of such a sequence is again a vertex algebra. We then apply this framework to permutation orbifolds of vertex operator algebras and their large N limit. We establish that for any nested oligomorphic permutation orbifold such a large N limit exists, and we give a necessary and sufficient condition for that limit to factorize. This helps clarify the question of what VOAs are candidates for holographic conformal field theories in physics.