Generalized Spectral Form Factors and the Statistics of Heavy Operators

TL;DR

This work introduces generalized spectral form factors that incorporate heavy OPE data from conformal field theories, enabling chaos diagnostics beyond energy-level statistics. By formulating an EFT of quantum chaos on a tripled Hilbert space, the authors derive Haar-average statistics for heavy-heavy-heavy OPE coefficients in line with the ORH, and predict a genus-2 spectral form factor with a ramp and plateau akin to standard SFFs. They show that, at late times, the genus-2 SFF is governed by spectral correlations (sine-kernel-like behavior) with corrections from operator contractions, and discuss gravitational interpretations via genus-2 wormholes. The results suggest that ramp-plateau behavior is a generic feature of generalized spectral form factors and establish a framework to probe OPE randomness in chaotic CFTs. Open questions remain about symmetry classes, descendants, and the precise Thouless-time scaling in higher-genus probes.

Abstract

The spectral form factor is a powerful probe of quantum chaos that diagnoses the statistics of energy levels, but is blind to other features of a theory such as matrix elements of operators or OPE coefficients in conformal field theories. In this paper, we introduce generalized spectral form factors: new probes of quantum chaos sensitive to the dynamical data of a theory. These quantities can be studied using an effective theory of quantum chaos. We focus our attention on a particular combination of heavy-heavy-heavy OPE coefficients that generalizes the genus-2 partition function of two-dimensional CFTs, for which we define a spectral form factor. We probe heavy-heavy-heavy OPE coefficients and find statistical correlations that agree with the OPE Randomness Hypothesis: these coefficients have a random matrix component in the ergodic regime. The EFT of quantum chaos predicts that the genus-2 spectral form factor displays a ramp and a plateau. Our results suggest that this is a common property of generalized spectral form factors.

Figures (3)

• Figure 1: Genus-2 surface with three different cycles related to the three moduli $\tau_i$.
• Figure 2: The genus-2 spectral form factor as a function of time. After an initial decay, the signal displays a ramp and plateau at sufficiently late times. The ramp and the plateau are the average of the noisy signal. The $\sigma$-model captures the moments of this noisy signal.
• Figure 3: Sunset diagrams are the diagrammatic representation of the contractions depicted in \ref{['eq.sunset']}. Each vertex of the diagram represents an insertion of a OPE coefficient. The ones corresponding to the insertions of $c^*$s are marked with a star, while the unmarked vertices are the $c$ insertions. Each edge in the diagram corresponds to the index contractions between OPE coefficients.