Permutation Orbifolds in the large N Limit

TL;DR

This work maps out the space of permutation orbifolds at large central charge and identifies precise conditions under which such theories can be holographically dual to weakly coupled AdS gravity. By introducing oligomorphic permutation groups and explicit constructions like $S_{\sqrt{N}}\wr S_{\sqrt{N}}$ and $S_{\sqrt{N}}\times S_{\sqrt{N}}$, it shows how to obtain a finite low-energy spectrum and analyzes growth in both untwisted and twisted sectors. It demonstrates that symmetric orbifold correlation functions factorize at large $N$ and extends the factorization result to a broad class of oligomorphic groups, including democratic ones, with explicit counting. The findings illuminate when permutation-constructed CFTs resemble generalized free fields in AdS and highlight how the structure of the permutation group controls the bulk-like behavior and phase structure of the dual theory.

Abstract

The space of permutation orbifolds is a simple landscape of two dimensional CFTs, generalizing the well-known symmetric orbifolds. We consider constraints which a permutation orbifold with large central charge must obey in order to be holographically dual to a weakly coupled (but possibly stringy) theory of gravity in AdS. We then construct explicit examples of permutation orbifolds which obey these constraints. In our constructions the spectrum remains finite at large N, but differs qualitatively from that of symmetric orbifolds. We also discuss under what conditions the correlation functions factorize at large N and thus reduce to those of a generalized free field in AdS. We show that this happens not just for symmetric orbifolds, but also for permutation groups which act "democratically" in a sense which we define.

Figures (1)

• Figure 1: The blue dots are the $f_K$ for $S_{\sqrt{N}}\times S_{\sqrt{N}}$. The blue line gives an exponential fit. For reference, the purple squares are the $f_K$ for $S_{\sqrt{N}}\wr S_{\sqrt{N}}$.