Echoes of chaos from string theory black holes

TL;DR

This work studies the weak-coupling (integrable) D1-D5 CFT as a microscopic black-hole model and investigates whether chaos-like late-time structure can emerge under temporal coarse-graining. By employing progressive time-averaging inspired by random matrix theory, the authors show that the light-probe two-point function around Ramond ground states exhibits a universal early-time decay followed by a dip, ramp, and plateau, reminiscent of the spectral form factor in chaotic systems. They derive analytic expressions for the late-time ramp and plateau, finding that the plateau height scales as $(\log S)/S$ with entropy $S$, while the ramp grows logarithmically and the dip scales with $\sqrt{S}$, underscoring qualitative chaos-like features despite integrability. These results illuminate how coarse-graining can reveal chaotic-like dynamics in black-hole microstate models and outline a path to explore chaotic behavior by perturbing away from the integrable limit. The findings bridge D1-D5 CFT behavior with RMT/SYK phenomenology and suggest extensions to finite coupling where chaotic dynamics are expected to emerge.

Abstract

The strongly coupled D1-D5 conformal field theory is a microscopic model of black holes which is expected to have chaotic dynamics. Here, we study the weak coupling limit of the theory where it is integrable rather than chaotic. In this limit, the operators creating microstates of the lowest mass black hole are known exactly. We consider the time-ordered two-point function of light probes in these microstates, normalized by the same two-point function in vacuum. These correlators display a universal early-time decay followed by late-time sporadic behavior. To find a prescription for temporal coarse-graining of these late fluctuations we appeal to random matrix theory, where we show that a progressive time-average smooths the spectral form factor (a proxy for the 2-point function) in a typical draw of a random matrix. This coarse-grained quantity reproduces the matrix ensemble average to a good approximation. Employing this coarse-graining in the D1-D5 system, we find that the early-time decay is followed by a dip, a ramp and a plateau, in remarkable qualitative agreement with recent studies of the Sachdev-Ye-Kitaev (SYK) model. We study the timescales involved, comment on similarities and differences between our integrable model and the chaotic SYK model, and suggest ways to extend our results away from the integrable limit.

Figures (9)

• Figure 1: The regularized two-point function (\ref{['gencorr']}).
• Figure 2: Log-log plot of the spectral form factor \ref{['eq:formfactor']} with $\beta=1$ for a single 1024$\times$1024 matrix drawn from the Gaussian Unitary Ensemble (GUE). The early part is self-averaging but the late part is superseded by noise.
• Figure 3: Log-log plot of the average spectral form factor with $\beta=1$ for five hundred 1024$\times$1024 matrices drawn from the Gaussian unitary ensemble (GUE) (black), and the sliding window average \ref{['eq:slidingwindow']} with fixed time windows $\Delta t =10,60,110,160$ (color) for a single instance of a random matrix. Notice that for averaging with a fixed time window there is tension between preserving the dip and having a sufficiently smooth ramp and plateau.
• Figure 4: Log-log plot of the average spectral form factor with $\beta=1$ for five hundred 1024$\times$1024 matrices drawn from the Gaussian unitary ensemble (GUE) (purple), and the sliding window average \ref{['eq:slidingwindow']} for a progressive time window $\Delta t=0.8 t$ (blue) for a single instance of a random matrix. The progressive window captures the behavior of the ensemble average, in particular the dip, the ramp and the plateau.
• Figure 5: Left: The continuous orange line represents the regularized two-point function (\ref{['eq:reg2pt']}). The blue dotted line is its progressive time-average. Right: The progressive time-average of (\ref{['eq:reg2pt']}) for $\eta=0.05+0.025 j, \;\; j=0,...,10$. Smaller values of $\eta$ correspond to larger $N$ and smaller plateau height.