Poincare Recurrences and Topological Diversity

TL;DR

The paper tests whether finite-entropy holographic CFTs exhibit Poincaré recurrences by computing bulk $AdS_3$ correlators through a modular sum over $SL(2,\mathbb{Z})$ images of BTZ. It finds that, although the ensemble includes thermal AdS and various black-hole saddles, perturbative corrections become large at a universal critical time $t_0 \sim \pi k/(2\Delta)$ and the total correlator fails to be quasi-periodic, even after summing many geometries. Toy models and a winding-string example illustrate that exact periodicity would require a complete nonperturbative resummation and that semiclassical approaches around a single saddle are insufficient. The results imply that bulk gravity alone is unlikely to reproduce finite-entropy recurrences, reinforcing the view that unitarity and information recovery in black-hole evaporation demand nonperturbative, possibly non-geometric, aspects of quantum gravity. This work thus clarifies the limitations of semiclassical AdS/CFT in capturing long-time unitary dynamics and motivates further study of the gravitational path integral and Stokes-related phenomena in the quest to address the information paradox.

Abstract

Finite entropy thermal systems undergo Poincare recurrences. In the context of field theory, this implies that at finite temperature, timelike two-point functions will be quasi-periodic. In this note we attempt to reproduce this behavior using the AdS/CFT correspondence by studying the correlator of a massive scalar field in the bulk. We evaluate the correlator by summing over all the SL(2,Z) images of the BTZ spacetime. We show that all the terms in this sum receive large corrections after at certain critical time, and that the result, even if convergent, is not quasi-periodic. We present several arguments indicating that the periodicity will be very difficult to recover without an exact re-summation, and discuss several toy models which illustrate this. Finally, we consider the consequences for the information paradox.

Figures (5)

• Figure 1: The extended conformal diagram of the BTZ black hole. The regions I and II are outside the black hole, each with its own asymptotic AdS boundary. Regions III and IV are inside the horizon. The two boundaries are connected by spacelike trajectories passing through the horizon.
• Figure 2: On the right, qualitative behavior of $f(t)$ computed by summing the BTZ black hole and thermal AdS, in the high-temperature regime where the black hole dominates. On the left, a typical quasi-periodic function with the same long time average.
• Figure 3: Two possible slicings of a solid torus as $t=0$ data for Lorenztian space times: on the left the 'donut slicing', giving two copies of AdS in an entangled state; on the right the 'bagel slicing', giving the BTZ black hole.
• Figure 4: Black holes contributing to our problem are represented as dots in the $-1/\tau$ plane. At high temperature the action of each of these black holes approaches zero as we come close to the real axis.
• Figure 5: Two point correlators as functions of Lorentzian time $t$ for different geometries. We have taken $\beta = 0.95$ and $k=1$ (larger $k$ would give an exponential weight to the non-rotating BTZ). The upper left figure is the non-rotating BTZ black hole, which is the dominant contribution at high temperature. On the upper right is thermal AdS, which gives an oscillating function with period $2 \pi$ in AdS units. On the lower left is the $a=1$, $b=10$ rotating black hole. There are peaks which become relevant at long times, but the overall contribution is decaying. The lower right figure is a detail of the lower left.