Permutation orbifolds and holography

TL;DR

This paper constructs a class of two-dimensional conformal field theories using permutation orbifold techniques, identifying conditions on permutation groups that ensure a holographic dual with finite state degeneracy and sparse spectrum.

Contribution

It introduces the use of oligomorphic permutation groups to ensure holographic properties in permutation orbifold CFTs and analyzes their spectral and geometric features.

Findings

Permutation orbifolds with oligomorphic groups have finite state degeneracy.

Sparse low-lying spectrum constrains the permutation group's orbits.

Holographic spectral properties relate to the geometry of covering spaces.

Abstract

Two dimensional conformal field theories with large central charge and a sparse low-lying spectrum are expected to admit a classical string holographic dual. We construct a large class of such theories employing permutation orbifold technology. In particular, we describe the group theoretic constraints on permutation groups to ensure a (stringy) holographic CFT. The primary result we uncover is that in order for the degeneracy of states to be finite in the large central charge limit, the groups of interest are the so-called oligomorphic permutation groups. Further requiring that the low-lying spectrum be sparse enough puts a bound on the number of orbits of these groups (on finite element subsets). Along the way we also study familiar cyclic and symmetric orbifolds to build intuition. We also demonstrate how holographic spectral properties are tied to the geometry of covering spaces for permutation orbifolds.