An integrable road to a perturbative plateau

TL;DR

The paper demonstrates that the late-time plateau of the spectral form factor in 2D dilaton gravity emerges from a perturbative sum over wormhole geometries in a tau-scaling regime, with genus-$g$ contributions growing as $T^{2g+1}$ and the series converging to the microcanonical plateau. This universal behavior is grounded in the integrable KdV hierarchy, open-closed duality, and the intersection-number structure of moduli spaces, which cause precise cancellations in Weil-Petersson volumes and their ribbon-graph representations. By mapping tau_k to cusp defects in dilaton gravity and presenting a matrix-integral dual, the authors unify topological gravity, JT/dilaton gravity, and random-matrix theory, showing how the plateau persists across a broad class of models. They further propose a Lorentzian topology-change perspective to physically motivate the time-power scaling, indicating a deep, universal mechanism behind late-time gravity dynamics.

Abstract

As has been known since the 90s, there is an integrable structure underlying two-dimensional gravity theories. Recently, two-dimensional gravity theories have regained an enormous amount of attention, but now in relation with quantum chaos - superficially nothing like integrability. In this paper, we return to the roots and exploit the integrable structure underlying dilaton gravity theories to study a late time, large $e^{S_\text{BH}}$ double scaled limit of the spectral form factor. In this limit, a novel cancellation due to the integrable structure ensures that at each genus $g$ the spectral form factor grows like $T^{2g+1}$, and that the sum over genera converges, realising a perturbative approach to the late-time plateau. Along the way, we clarify various aspects of this integrable structure. In particular, we explain the central role played by ribbon graphs, we discuss intersection theory, and we explain what the relations with dilaton gravity and matrix models are from a more modern holographic perspective.

Figures (7)

• Figure 1: The double scaling limit of the sum over genus $g$ wormholes in the Airy model, up to $g=g_\text{max}$ (numbers shown) with $e^{\textsf{S}_0}=10$ and $\beta=1/2$.
• Figure 2: The torus with one puncture $V_{1,1}(b)$ becomes a ribbon graph, when the hole $b$ becomes large. In this limit the moduli space is spanned by the lengths of the ribbons with the constraint $b=2\ell_1+2\ell_2+2\ell_3$, and we integrate with the flat measure to recover $V_{1,1}(b)=b^2/48$.
• Figure 3: Open-closed duality in a nutshell. Here we have in mind computing $V_{1,1}(b)$ in the theory with an exponential of $\tau_k$ inserted in the action, see \ref{['69']} below. Either we use the complicated action \ref{['69']} (left); or we expand out the deformations (right), in which case we have a sum over cusp-defects but we just use the simple JT gravity action at the cost of computing a more complicated observable with many boundaries (or operators). Alternatively, in the context of this section, we could think of the left picture as JT gravity and the right picture as Airy with an exponential of \ref{['520']} inserted. The idea is always the same.
• Figure 4: The $\tau$-scaling limit of the sum over genus $g$ wormholes in the Airy model \ref{['airyexpa']} up to $g=g_\text{max}$ (numbers shown) with $e^{\textsf{S}_0}=10$ and $\beta=1/2$. This series converges to the exact expression \ref{['airyexact']} for all $T$ (infinite radius of convergence) and in particular it converges to the plateau.
• Figure 5: The $\tau$-scaling limit of the sum over genus $g$ wormholes in JT gravity \ref{['JTexact']} with $e^{\textsf{S}_0}=10$ and $\beta=1/2$. We have not shown individual terms in the series, because it converges only when $Te^{-\textsf{S}_0}<1/2\pi$, which is a very short time on this plot. Importantly though, the sum over all genus uniquely gives the exact answer \ref{['JTexact']}. The JT plateau is much higher and hence reached much later than the Airy plateau, because of the exponentially larger spectral density.