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Narain to Narnia

Nathan Benjamin, Christoph A. Keller, Hirosi Ooguri, Ida G. Zadeh

TL;DR

The article advances the program of holography with ensemble averages by showing the Narain moduli-space average for c free bosons is captured by a bulk abelian Chern-Simons theory coupled to topological gravity, and that this correspondence extends nontrivially to toroidal ℤ_N orbifolds. It provides a concrete bulk interpretation of twist-field correlators as sums over discrete-gauge vortices organized by rational tangles, with the Poincaré sum reproducing the moduli-averaged CFT data. However, the authors find that this ensemble picture does not generically extend to K3 or Calabi–Yau moduli spaces, nor does it universally apply to minimal models, suggesting a central-charge criterion c=c_{crit} governs when such dualities hold. The work thus clarifies where ensemble holography with bulk geometry sums can be realized and identifies topological and spectral obstructions in more intricate CFTs, offering insight into the role of bulk topology, discrete gauge structure, and modular data in holographic averaging.

Abstract

We generalize the holographic correspondence between topological gravity coupled to an abelian Chern-Simons theory in three dimensions and an ensemble average of Narain's family of massless free bosons in two dimensions, discovered by Afkhami-Jeddi et al. and by Maloney and Witten. We find that the correspondence also works for toroidal orbifolds but not for K3 or Calabi-Yau sigma-models and not always for the minimal models. We conjecture that the correspondence requires that the central charge is equal to the critical central charge defined by the asymptotic density of states of the chiral algebra. For toroidal orbifolds, we extend the holographic correspondence to correlation functions of twist operators by using topological properties of rational tangles in the three-dimensional ball, which represent configurations of vortices associated to a discrete gauge symmetry.

Narain to Narnia

TL;DR

The article advances the program of holography with ensemble averages by showing the Narain moduli-space average for c free bosons is captured by a bulk abelian Chern-Simons theory coupled to topological gravity, and that this correspondence extends nontrivially to toroidal ℤ_N orbifolds. It provides a concrete bulk interpretation of twist-field correlators as sums over discrete-gauge vortices organized by rational tangles, with the Poincaré sum reproducing the moduli-averaged CFT data. However, the authors find that this ensemble picture does not generically extend to K3 or Calabi–Yau moduli spaces, nor does it universally apply to minimal models, suggesting a central-charge criterion c=c_{crit} governs when such dualities hold. The work thus clarifies where ensemble holography with bulk geometry sums can be realized and identifies topological and spectral obstructions in more intricate CFTs, offering insight into the role of bulk topology, discrete gauge structure, and modular data in holographic averaging.

Abstract

We generalize the holographic correspondence between topological gravity coupled to an abelian Chern-Simons theory in three dimensions and an ensemble average of Narain's family of massless free bosons in two dimensions, discovered by Afkhami-Jeddi et al. and by Maloney and Witten. We find that the correspondence also works for toroidal orbifolds but not for K3 or Calabi-Yau sigma-models and not always for the minimal models. We conjecture that the correspondence requires that the central charge is equal to the critical central charge defined by the asymptotic density of states of the chiral algebra. For toroidal orbifolds, we extend the holographic correspondence to correlation functions of twist operators by using topological properties of rational tangles in the three-dimensional ball, which represent configurations of vortices associated to a discrete gauge symmetry.

Paper Structure

This paper contains 36 sections, 174 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Review of Narain averaging duality
  3. Averaging over $T^c/\mathbb Z_2$
  4. Boundary CFT
  5. Bulk Chern-Simons Theory
  6. Averaging over $T^{2d}/\mathbb{Z}_N$
  7. Average over Kähler Moduli Space
  8. The Poincaré Series
  9. Bulk Chern-Simons Theory
  10. Averaging correlation functions of $\mathbb{Z}_2$ orbifolds
  11. Ensemble Average of CFT Correlation Functions
  12. $W_{\bf\epsilon_0,\epsilon_1}(\tau)$
  13. Eigenfunctions of the Laplacian
  14. Fixing the cusps
  15. Bulk Interpretation
  16. ...and 21 more sections

Figures (4)

  • Figure 1: Two rational tangles related by a monodromy transformation.
  • Figure 2: Consider a minimal surface bounded by each vortex and the boundary of the 3 ball, and cut open the 3-ball along the two minimal surfaces as shown in the diagram. The pink region is the boundary, the black lines are the two-tangles, and the orange surfaces the branch sheets (or in the language of Thurston, the orange surfaces are the doorways to Narnia). Gluing two copies of 3-balls across these minimal surfaces gives a solid torus of modulus $\tau$.
  • Figure 3: (a) The contractible cycle. (b) The non-contractible cycle
  • Figure 4: (a) an example of a rational tangle. (b) an example of an irrational tangle which cannot be generated by a finite sequence of exchanges of points on the boundary.