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Free partition functions and an averaged holographic duality

Nima Afkhami-Jeddi, Henry Cohn, Thomas Hartman, Amirhossein Tajdini

TL;DR

The paper studies ensembles of two-dimensional free boson CFTs defined by Narain moduli, showing that averaging over lattices yields an exactly computable torus partition function that reorganizes as a sum over three-dimensional topologies, suggesting a holographic dual in terms of a U(1) gravity theory.Using spinning modular bootstrap, Siegel averaging, and circle-method techniques, it derives both upper bounds on the spectral gap and the ensemble-averaged density of states, establishing Δ_1∼c/(2π e) as c→∞ and identifying exact cases (e.g., c=1 self-dual boson) where optimality holds analytically.The bulk calculation reproduces the same spectrum via a sum over SL(2,Z) images of the vacuum character, and the Siegel-Weil formula provides a principled link between the averaged CFT data and a gravitational path integral, supporting a holographic interpretation of ensemble-averaged Narain CFTs.Overall, the work connects modular invariance, lattice averaging, and holography in a tractable setting, offering a concrete framework for ensemble-averaged AdS_3 duals and insights into spectral gaps and sphere-packing-like structure in high dimensions.

Abstract

We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with $U(1)^c \times U(1)^c$ symmetry and a composite boundary graviton. Additionally, for small central charge $c$, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

Free partition functions and an averaged holographic duality

TL;DR

The paper studies ensembles of two-dimensional free boson CFTs defined by Narain moduli, showing that averaging over lattices yields an exactly computable torus partition function that reorganizes as a sum over three-dimensional topologies, suggesting a holographic dual in terms of a U(1) gravity theory.Using spinning modular bootstrap, Siegel averaging, and circle-method techniques, it derives both upper bounds on the spectral gap and the ensemble-averaged density of states, establishing Δ_1∼c/(2π e) as c→∞ and identifying exact cases (e.g., c=1 self-dual boson) where optimality holds analytically.The bulk calculation reproduces the same spectrum via a sum over SL(2,Z) images of the vacuum character, and the Siegel-Weil formula provides a principled link between the averaged CFT data and a gravitational path integral, supporting a holographic interpretation of ensemble-averaged Narain CFTs.Overall, the work connects modular invariance, lattice averaging, and holography in a tractable setting, offering a concrete framework for ensemble-averaged AdS_3 duals and insights into spectral gaps and sphere-packing-like structure in high dimensions.

Abstract

We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with symmetry and a composite boundary graviton. Additionally, for small central charge , we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

Paper Structure

This paper contains 24 sections, 5 theorems, 155 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partition functions
  4. Spinning modular bootstrap
  5. Narain compactifications
  6. Upper bounds on the spectral gap
  7. Numerical bootstrap bounds
  8. Analytic functional for the self-dual boson
  9. Seeking optimal Narain lattices
  10. Averaging over Narain lattices
  11. Averaging over lattices
  12. Geometry of Narain lattices
  13. Counting states
  14. Modular invariance
  15. Spectral gap
  16. ...and 9 more sections

Key Result

Theorem 4.1

If $c>2$, then the density of non-vacuum primary states of spin $\ell$ and scaling dimension $\Delta$ in a random Narain CFT of signature $(c,c)$ is given by for $\Delta \ge |\ell|$ and $0$ otherwise. In other words, for each measurable subset $A$ of $[|\ell|,\infty)$, the expected number of non-vacuum primary states in a random Narain CFT with spin $\ell$ and scaling dimension $\Delta \in A$ is

Figures (7)

  • Figure 1: Upper bound on $\Delta_1$ from the spinning modular bootstrap, at truncation order $K =25$.
  • Figure 2: Comparison of the upper bound to the piecewise linear function $\min\left(\frac{c+2}{6}, \frac{c+4}{8}\right)$.
  • Figure 3: Comparison of the spinning bootstrap to the spinless bootstrap bound $\Delta^\textup{LP}(c)$.
  • Figure 4: The square $2R$ that attains a sharp bound, together with the unit circle and the hyperbolas $x^2 - \bar{x}^2 = \pm 2$.
  • Figure 5: An octagon $2S_\varepsilon$ such that $f(0,0)>0$, together with the unit circle and the hyperbolas $x^2 - \bar{x}^2 = \pm 2$.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Theorem 4.1: Siegel
  • Lemma B.1
  • proof
  • Proposition B.2
  • Lemma B.3
  • proof : Proof of Proposition \ref{['prop:narainequiv']}
  • proof : Proof of Lemma \ref{['lemma:iwasawa']}
  • Proposition B.4
  • proof