Non-Gaussianities in the Statistical Distribution of Heavy OPE Coefficients and Wormholes

TL;DR

This work uses Virasoro crossing kernels to derive high-genus asymptotics for heavy OPE coefficients in chaotic 2d CFTs, revealing non-Gaussian corrections that are exponentially suppressed in entropy much like ETH. By analyzing skyline and comb channels across genus, the authors obtain universal and light-data–dependent formulas for higher moments of heavy OPE configurations, highlighting how non-Gaussianities arise and how they influence products of partition functions. They connect these CFT results to gravity, showing that non-Gaussianities modify Euclidean wormhole contributions and even suggest new gravitational saddles when light operators are present; they also propose a generating function framework based on typicality to organize these statistics. The findings illuminate the interplay between conformal bootstrap constraints, chaotic dynamics in CFTs, and semiclassical gravity, with implications for factorization and the wormhole program in AdS$_3$/CFT$_2$.

Abstract

The Eigenstate Thermalization Hypothesis makes a prediction for the statistical distribution of matrix elements of simple operators in energy eigenstates of chaotic quantum systems. As a leading approximation, off-diagonal matrix elements are described by Gaussian random variables but higher-point correlation functions enforce non-Gaussian corrections which are further exponentially suppressed in the entropy. In this paper, we investigate non-Gaussian corrections to the statistical distribution of heavy-heavy-heavy OPE coefficients in chaotic two-dimensional conformal field theories. Using the Virasoro crossing kernels, we provide asymptotic formulas involving arbitrary numbers of OPE coefficients from modular invariance on genus-$g$ surfaces. We find that the non-Gaussianities are further exponentially suppressed in the entropy, much like the ETH. We discuss the implication of these results for products of CFT partition functions in gravity and Euclidean wormholes. Our results suggest that there are new connected wormhole geometries that dominate over the genus-two wormhole.

Figures (13)

• Figure 1: On the left the trivalent graph associated with the configuration of OPE coefficients in the skyline channel. On the right, the comb channel.
• Figure 2: The two conformal block decompositions of the four-point function with four different primary operators. The fusion kernel is the change-of-basis matrix relating the two bases of conformal blocks in the s- and t- channel.
• Figure 3: The modular kernel relating the two different conformal block decompositions of the torus one-point function.
• Figure 4: The crossing kernels relating the different pair-of-pants decompositions or channels of the genus-three Riemann surface. From left to right the names of these channels are: skyline, sunset, necklace, and comb. The trivalent graphs associated to each decomposition are shown below each surface. Note that the fusion kernel may change the topology of the graph while the modular kernel always leaves the diagram invariant.
• Figure 5: The sequence of transformations we use to solve for the statistics of the OPE coefficients associated with the skyline channel.