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Random Statistics of OPE Coefficients and Euclidean Wormholes

Alexandre Belin, Jan de Boer

TL;DR

The paper proposes the OPE Randomness Hypothesis, modeling heavy-operator OPE coefficients as Gaussian random variables to generalize ETH in chaotic CFTs. It derives explicit CFT expressions for genus-2 observables and shows that contractions among OPE coefficients generate a non-factorizing, connected piece that matches a Euclidean wormhole contribution in AdS$_3$ gravity. This provides a physical interpretation of wormholes as manifestations of OPE statistics and explains why low-energy gravity cannot resolve microscopic OPE details. The work discusses extensions to other regimes, higher dimensions, and chaotic versus integrable theories, outlining implications for factorization and holographic correlations.

Abstract

We propose an ansatz for OPE coefficients in chaotic conformal field theories which generalizes the Eigenstate Thermalization Hypothesis and describes any OPE coefficient involving heavy operators as a random variable with a Gaussian distribution. In two dimensions this ansatz enables us to compute higher moments of the OPE coefficients and analyse two and four-point functions of OPE coefficients, which we relate to genus-2 partition functions and their squares. We compare the results of our ansatz to solutions of Einstein gravity in AdS$_3$, including a Euclidean wormhole that connects two genus-2 surfaces. Our ansatz reproduces the non-perturbative correction of the wormhole, giving it a physical interpretation in terms of OPE statistics. We propose that calculations performed within the semi-classical low-energy gravitational theory are only sensitive to the random nature of OPE coefficients, which explains the apparent lack of factorization in products of partition functions.

Random Statistics of OPE Coefficients and Euclidean Wormholes

TL;DR

The paper proposes the OPE Randomness Hypothesis, modeling heavy-operator OPE coefficients as Gaussian random variables to generalize ETH in chaotic CFTs. It derives explicit CFT expressions for genus-2 observables and shows that contractions among OPE coefficients generate a non-factorizing, connected piece that matches a Euclidean wormhole contribution in AdS gravity. This provides a physical interpretation of wormholes as manifestations of OPE statistics and explains why low-energy gravity cannot resolve microscopic OPE details. The work discusses extensions to other regimes, higher dimensions, and chaotic versus integrable theories, outlining implications for factorization and holographic correlations.

Abstract

We propose an ansatz for OPE coefficients in chaotic conformal field theories which generalizes the Eigenstate Thermalization Hypothesis and describes any OPE coefficient involving heavy operators as a random variable with a Gaussian distribution. In two dimensions this ansatz enables us to compute higher moments of the OPE coefficients and analyse two and four-point functions of OPE coefficients, which we relate to genus-2 partition functions and their squares. We compare the results of our ansatz to solutions of Einstein gravity in AdS, including a Euclidean wormhole that connects two genus-2 surfaces. Our ansatz reproduces the non-perturbative correction of the wormhole, giving it a physical interpretation in terms of OPE statistics. We propose that calculations performed within the semi-classical low-energy gravitational theory are only sensitive to the random nature of OPE coefficients, which explains the apparent lack of factorization in products of partition functions.

Paper Structure

This paper contains 13 sections, 32 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Gravity is a theory of random variables
  3. Statistics of OPE coefficients
  4. Four OPE coefficients and the variance of $C_{ijk}\bar{C}_{ijk}$.
  5. Genus-2 partition function from the CFT
  6. Square of the partition function
  7. Genus-2 partition function in AdS$_3$ gravity
  8. Other OPE channels and handlebodies
  9. Discussion
  10. Beyond the single large energy limit
  11. Higher dimensions
  12. Chaotic versus integrable theories
  13. Torus correlation functions and spectral form factor

Figures (3)

  • Figure 1: A genus-2 surface with 3 cycles highlighted. One handlebody solution corresponds to a solid filing of the genus-2 surface such that the three red cycles are made contractible in the bulk.
  • Figure 2: The euclidean wormhole (\ref{['wormholemetric']}) connecting through the bulk two genus-2 surfaces that were originally disconnected.
  • Figure 3: A handlebody where the three blue cycles are made contractible in the bulk. This saddle is equivalent to decomposing the genus two partition function such that the three blue cycles are cut open. This yields a sum over OPE coefficients $\sum_{i,j,k}C_{iij}C_{jkk}$, corresponding to a product of torus one-point functions.