The Large $N$ Limit of Orbifold Vertex Operator Algebras

TL;DR

This paper studies the behavior of permutation orbifolds of vertex operator algebras as the permutation group size grows large, introducing nested oligomorphic orbifolds and conditions for their convergence to a vertex algebra.

Contribution

It introduces the concept of nested oligomorphic permutation orbifolds and analyzes conditions for their fixed point VOAs to converge in the large N limit.

Findings

Limit, if it exists, has a vertex algebra structure.

Provided an example with $ ext{GL}(N,q)$ showing convergence.

Established conditions for convergence of fixed point VOAs.

Abstract

We investigate the large $N$ limit of permutation orbifolds of vertex operator algebras. To this end, we introduce the notion of nested oligomorphic permutation orbifolds and discuss under which conditions their fixed point VOAs converge. We show that if this limit exists, then it has the structure of a vertex algebra. Finally, we give an example based on $\mathrm{GL}(N,q)$ for which the fixed point VOA limit is also the limit of the full permutation orbifold VOA.