Symmetries and spectral statistics in chaotic conformal field theories II: Maass cusp forms and arithmetic chaos

TL;DR

This paper investigates the spectral properties of Maass cusp forms in chaotic conformal field theories, revealing how their statistical features relate to random matrix theory and gravitational duals, advancing understanding of arithmetic chaos.

Contribution

It introduces a novel spectral bootstrap approach for cusp forms, linking their statistical distributions to universal random matrix behavior in conformal field theories.

Findings

Spectral form factor sensitive to average cusp form statistics

Universal linear ramp indicates eigenvalue repulsion

Cusp form correlations match AdS3 gravity wormhole amplitudes

Abstract

We continue the study of random matrix universality in two-dimensional conformal field theories. This is facilitated by expanding the spectral form factor in a basis of modular invariant eigenfunctions of the Laplacian on the fundamental domain. The focus of this paper is on the discrete part of the spectrum, which consists of the Maass cusp forms. Both their eigenvalues and Fourier coefficients are sporadic discrete numbers with interesting statistical properties and relations to analytic number theory; this is referred to as `arithmetic chaos'. We show that the near-extremal spectral form factor at late times is only sensitive to a statistical average over these erratic features. Nevertheless, complete information about their statistical distributions is encoded in the spectral form factor if all its spin sectors exhibit universal random matrix eigenvalue repulsion (a `linear ramp'). We `bootstrap' the spectral correlations between the cusp form basis functions that correspond to a universal linear ramp and show that they are unique up to theory-dependent subleading corrections. The statistical treatment of cusp forms provides a natural avenue to fix the subleading corrections in a minimal way, which we observe leads to the same correlations as those described by the [torus]$\times$[interval] wormhole amplitude in AdS${}_3$ gravity.