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Permutation orbifolds and holography

Felix M. Haehl, Mukund Rangamani

TL;DR

This paper introduces a systematic, group-theoretic framework to construct two-dimensional CFTs with large central charge and sparse low-lying spectra via permutation orbifolds. By analyzing the large-$N$ limit through cycle-index (Pólya) counting, the authors identify oligomorphic permutation groups as the structural criterion ensuring finite degeneracy of low-lying states, with an additional Hartman bound to guarantee holographic viability. They provide explicit treatments of cyclic and symmetric orbifolds, establish universal low-energy behavior and growth bounds, and extend the construction to wreath-product orbifolds to broaden the holographic landscape. The approach clarifies how spectral properties tie to covering-space geometry and opens avenues for higher-genus studies and entanglement entropy analyses in holographic contexts.

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.

Permutation orbifolds and holography

TL;DR

This paper introduces a systematic, group-theoretic framework to construct two-dimensional CFTs with large central charge and sparse low-lying spectra via permutation orbifolds. By analyzing the large- limit through cycle-index (Pólya) counting, the authors identify oligomorphic permutation groups as the structural criterion ensuring finite degeneracy of low-lying states, with an additional Hartman bound to guarantee holographic viability. They provide explicit treatments of cyclic and symmetric orbifolds, establish universal low-energy behavior and growth bounds, and extend the construction to wreath-product orbifolds to broaden the holographic landscape. The approach clarifies how spectral properties tie to covering-space geometry and opens avenues for higher-genus studies and entanglement entropy analyses in holographic contexts.

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.

Paper Structure

This paper contains 32 sections, 6 theorems, 94 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A review of permutation orbifolds
  3. Cyclic orbifolds
  4. Example:
  5. Symmetric orbifolds
  6. Example:
  7. Spectral properties of cyclic and symmetric orbifolds
  8. Example: The free boson orbifold
  9. Asymptotic behaviour of cyclic and symmetric orbifolds
  10. Oligomorphy and holography
  11. Pólya counting for the partition sum
  12. Cycle index for $\mathbb{Z}_N$ orbifolds:
  13. Cycle index for $S_N$ orbifolds:
  14. Cycle index for general orbifolds:
  15. Bound on growth rate of holographic CFT spectra
  16. ...and 17 more sections

Key Result

Theorem 1

At large large $N$, the geometries which are dominant in the torus partition function of $\mathbb{Z}_N$ and $S_N$ orbifold CFTs are obtained as follows. Take $N$ numbered copies of the torus with modular parameter $\tau$ and arrange them in any order in any number of groups which is consistent with

Figures (4)

  • Figure 1: Plot of the vacuum subtracted rescaled free energies defined in \ref{['eq:deltaF']} of orbifold CFTs ${\cal C}_{N,\Omega}$ with respect to a cyclic group $\mathbb{Z}_N$ (blue line) and a symmetric group $S_N$ (red line) for $N=103$. ${\cal C}$ is taken to be the free boson CFT at unit radius. The dashed (orange) line shows the limit of the symmetric orbifold as $N\rightarrow \infty$. Any other CFT with a holographic dual has the same universal large central charge asymptotics. The vertical line is $T =\frac{1}{2\pi}$, drawn to guide the eye for the critical temperature, and corresponds to a square torus.
  • Figure 2: Illustration of the content of (\ref{['eq:Z4covers']}). Every graph consists of 4 boxes each representing one sheet of a $4$-fold cover of the torus with modular parameter $\tau$. Depending on how the sheets are sewn together, we obtain different covering spaces which are all tori with different modular parameters (edges without arrows are glued to the opposite edge). The partition function $Z(\tau,{\bar{\tau}})$ of a given parent CFT ${\cal C}$ has to be evaluated on all these covers in order to get the partition function of the orbifold theory ${\cal C}^{\otimes 4}/\mathbb{Z}_4$. This can be seen from the fact that each of the covers has an automorphism group of sheet permutations that is generated by a set of elements of $\mathbb{Z}_4$.
  • Figure 3: Box diagrams of all eleven $4$-sheeted unbranched covers of a torus whose automorphism groups of sheet permutations are not generated by elements of $\mathbb{Z}_4$. Together with the geometries of Fig. \ref{['fig:Z4covers']} these form the complete set of all $4$-sheeted unbranched covers of the torus. All of the these $21$ covers are relevant for computing the symmetric orbifold partition function.
  • Figure 4: Illustration of the $S_p \wr \mathbb{Z}_q$ wreath product action on $pq$-sheeted covers of a torus. The cyclic symmetry $\mathbb{Z}_q$ acts on a set of $q$ base sheets of the parent torus. This set of $q$ tori is fibred where each fiber consists of a $p$-sheeted unbranched cover. The symmetric group $S_p$ acts independently on each fiber. The imprimitive wreath product action preserves the fibration, i.e., it preserves the presence in each column of one copy of each orbifold sheet.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6