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

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.

Figures (12)

• Figure 1: A depiction of the statistical approximation to cusp form data. The "spectral staircase" and the erratically distributed Fourier coefficients are replaced with their average values. We will justify this 'statistical' coarse-graining in the time regime that is relevant for random matrix universality. It should be distinguished from 'microcanonical' coarse-graining, which is always required to discuss correlations in the CFT spectrum.
• Figure 2: Numerical verification of the encoding of a linear ramp in correlations of even (left) and odd (right) Maass cusp forms, according to \ref{['eq:discSumRes']}, for $y_1=y_2\equiv y$. The summation over $n$ is performed up to some cutoff such that convergence is achieved within the displayable accuracy. The plots show that the sum converges to the ramp linear $y/(2\pi)$ up to an $m$-dependent constant that is subleading as $y\rightarrow \infty$.
• Figure 3: We compute the Maass cusp form sum using the variance of the Fourier coefficients instead of their exact values in \ref{['eq:z-m-R-result-disc-zn3']}. For large $y$ increasingly many Fourier coefficients contribute to the sum over $n$, which means that their square can be increasingly well approximated by their variance. We therefore reproduce the linear ramp asymptotically (up to a subleading constant shift), c.f. figure \ref{['fig:rampCusp']}. The left (right) shows the case of even (odd) parity cusp forms. In the odd case the ramps for different spins lie almost on top of each other. Insets show larger values of $y$.
• Figure 4: We quantify how the sum over cusp forms indexed by $n$ depends on the terms with small $n$ (and hence small $R_n^\pm$). We plot the ramp in the spectral form factor computed using only values of $n$ for which $R_n^\pm > R_\text{min}$ and normalize it by the complete result. Asymptotically as $y \rightarrow \infty$ this ratio converges to $1$, no matter how many low-lying values of $R_n^\pm$ we exclude. We show the cases of even (left) and odd (right) cusp forms separately (they are almost indistinguishable). Solid lines correspond to spin $m=1$, dashed lines to $m=2$.
• Figure 5: Comparison of the exact counting function of discrete eigenvalues of the Laplacian with the Weyl law approximation as given in \ref{['eq:muRapproxapp']}.

Theorems & Definitions (7)

• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 1
• Lemma 4
• Theorem 2
• Lemma 5