2D CFT Partition Functions at Late Times

TL;DR

This work analyzes the late-time behavior of the analytically continued partition function $Z(\beta+it)Z(\beta-it)$ in holographic 2d CFTs via the spectral form factor $g(\beta,t)$ to probe information loss and spectrum discreteness. It shows that each Virasoro character decays at late times, implying information is not restored by Virasoro symmetry alone; a universal late-time contribution, together with random matrix theory behavior, governs chaotic theories up to times exponential in the central charge. The authors identify a dip time $t_d$ signaling a crossover to RMT-like ramp and plateau, with $t_d$ scaling as $\exp\left( \frac{12\pi^2 k}{(1+s)\beta} \right)$ depending on vacuum versus light contributions, and argue there is a parametrically long period of late-time growth afterward. In integrable theories, information is restored by an infinite sum over modular images or non-perturbative saddles, hinting at a bulk mechanism where the full spectrum emerges from a discrete set of saddles, analogous to a Rademacher expansion. Overall, the paper provides a coherent picture in which information loss is evaded dynamically via the interplay of infinitely many Virasoro characters or bulk saddles, with clear implications for the bulk interpretation of black hole information in AdS$_3$/CFT$_2$.

Abstract

We consider the late time behavior of the analytically continued partition function $Z(β+ it) Z(β- it)$ in holographic $2d$ CFTs. This is a probe of information loss in such theories and in their holographic duals. We show that each Virasoro character decays in time, and so information is not restored at the level of individual characters. We identify a universal decaying contribution at late times, and conjecture that it describes the behavior of generic chaotic $2d$ CFTs out to times that are exponentially large in the central charge. It was recently suggested that at sufficiently late times one expects a crossover to random matrix behavior. We estimate an upper bound on the crossover time, which suggests that the decay is followed by a parametrically long period of late time growth. Finally, we discuss integrable theories and show how information is restored at late times by a series of characters. This hints at a possible bulk mechanism, where information is restored by an infinite sum over non-perturbative saddles.

Figures (7)

• Figure 1: The spectral form factor $g(\beta,t)$ for the GUE ensemble of random matrices, using 50 matrices of rank 2,000 and computed with $\beta=5$. At late times we have a period of close to linear growth we call the ramp, folowed by a plateau.
• Figure 2: The spectral form factor $g(\beta,t)$ for a single matrix, using the same parameters as Figure \ref{['fig:GUEg']}. The late time ramp and plateau are barely visible.
• Figure 3: The spectral form factors corresponding to the BTZ black hole contribution $g_{\rm BTZ}(\beta,t)$ (blue) and corresponding to the dominant image of the vacuum $g_{n}(\beta,t)$ (red). Here, for $t\neq2\pi n$, we interpolate by taking $n={\rm integer\, part}(t/2\pi)$. This accounts for the discontinuities in the red, dashed line. The peaks of this contribution are attained at discrete times $t_n$ (purple dots). Going to the late dominant frame does not avoid late time decay, violating the late time bound \ref{['gbound']} (black, dotted). Inset: The dominant contribution at $t_n$ (purple) with a fit to a $t^{-3}$ power law (black).
• Figure 4: Here, we show the upper half-plane tiled by fundamental domains of $SL(2;\mathbb{Z})$. As we increase the temperature, which corresponds to lowering the red line, we cross more and more fundamental domains.
• Figure 5: Here, in the top line, we display the behavior of our universal contribution, $g_{\star}(\beta,t)$ at various temperatures. On the bottom line, for comparison, we display the spectral form factor for a sample modular invariant function $\psi_{2}(\tau)$. As we increase temperature both are controlled by more and more saddles.