Quantum chaos in 2D gravity

TL;DR

This work constructs a fully non-perturbative framework for quantum chaos in two-dimensional gravity by linking JT gravity to Kodaira-Spencer universe field theory and a flavor matrix theory (fMT). It shows that late-time spectral correlations are governed by a nonlinear σ-model on the coset $\ ext{AIII}_{n|n}$, arising from a stationary-phase analysis of a $\mathrm{GL}(n|n)$ flavor matrix integral derived from brane correlators. The authors interpret this via open-closed string duality and a geometric transition between brane-based flavor descriptions and the closed KS theory, thereby unifying JT gravity, KS theory, and the SYK universality class at the spectral edge. The framework provides a quantitative description of the ergodic phase in 2D gravity and points to extensions to higher dimensions and to fully non-perturbative holographic completions.

Abstract

We present a quantitative and fully non-perturbative description of the ergodic phase of quantum chaos in the setting of two-dimensional gravity. To this end we describe the doubly non-perturbative completion of semiclassical 2D gravity in terms of its associated universe field theory. The guiding principle of our analysis is a flavor-matrix theory (fMT) description of the ergodic phase of holographic gravity, which exhibits $\mathrm{U}(n|n)$ causal symmetry breaking and restoration. JT gravity and its 2D-gravity cousins alone do not realize an action principle with causal symmetry, however we demonstrate that their {\it universe field theory}, the Kodaira-Spencer (KS) theory of gravity, does. After directly deriving the fMT from brane-antibrane correlators in KS theory, we show that causal symmetry breaking and restoration can be understood geometrically in terms of different (topological) D-brane vacua. We interpret our results in terms of an open-closed string duality between holomorphic Chern-Simons theory and its closed-string equivalent, the KS theory of gravity. Emphasis will be put on relating these geometric principles to the characteristic spectral correlations of the quantum ergodic phase.

Figures (9)

• Figure 1: Diagram of the related theories discussed in this article. (Left) JT gravity, defined perturbatively as a sum over topologies weighted by $(e^{-S_0})^{2g-2+n}$. It may be completed non-perturbatively by a large $L$ color matrix theory (middle), double-scaled to the spectral edge. This is dual to a flavor matrix integral, which can be derived exactly from JT universe field theory (top) or from the brane worldvolume theory (bottom). A saddle point approximation of fMT then leads to the $\sigma$-model of quantum chaos.
• Figure 2: Spectral density of a chaotic quantum system. A total number of $L$ microstates is contained in a compact spectral support defined by a non-vanishing average spectral density $\rho(E)$. In the gravitational context, one is often interested in energies 'double scaled' to the ground state, $E=0$, inset left. Different from the energy levels of a generic system (right inset), levels of chaotic systems are almost uniformly spaced (middle) and cannot 'touch'.
• Figure 3: A schematic picture of the different types of D-branes in KS theory. Color branes (red) and flavor branes (blue) each have open string degrees of freedom associated to the branes themselves (indicated by the color matrix $H$ and flavor matrix $A$ resp.) and open string degrees of freedom $\Psi, \overline{\Psi}$ connecting both types of branes.
• Figure 4: Left: The setup of a stack of $L$ color branes and $n$ flavor (anti-)branes. The open string degrees of freedom that stretch between the two types of branes can be described by a field $\Psi^a_{\mu}$ in the bi-fundamental representation of $U(L)\otimes \mathrm{U}(n|n)$. Right: After taking $L\to \infty$ the color branes dissolve into a flux for the holomorphic three-form, which is represented by a branch cut (dashed line) in the $x$-plane. The closed string description is that of JT gravity on the world-sheet. The flavor branes (blue) in the closed string background, on which JT universes with fixed energy boundaries can end, correspond to probes for the color degrees of freedom.
• Figure 5: Real part of the Airy potential $\frac{y^3}{3}$. The blue shaded areas are regions where the real part is negative. Left, black striped: integration contour $\mathcal{C}$ for the brane insertions. Right, red and yellow striped: two distinct choices of $\mathcal{C}'_\pm$ for the anti-brane insertion.