On the Late-Time Behavior of Virasoro Blocks and a Classification of Semiclassical Saddles

TL;DR

The paper classifies all semiclassical saddles contributing to Virasoro blocks at large $c$ and analyzes their late-time behavior in heavy-light AdS$_3$/CFT$_2$ setups. Using the monodromy method, it shows leading saddles decay exponentially with a universal rate independent of the exchanged dimension, while two infinite families of subleading saddles exist, including one that does not decay at late times; a transseries/relis resurgence perspective clarifies the nonperturbative structure. An explicit algebraic treatment for degenerate external states provides a cross-check and connects to the monodromy framework, validating the saddle classification across kinematic regimes including intermediate times. The results imply that semiclassical Virasoro blocks alone do not resolve information loss, highlighting the importance of nonperturbative $e^{-c}$ effects and suggesting avenues toward a bulk path-integral understanding of Virasoro blocks in AdS$_3$ gravity.

Abstract

Recent work has demonstrated that black hole thermodynamics and information loss/restoration in AdS$_3$/CFT$_2$ can be derived almost entirely from the behavior of the Virasoro conformal blocks at large central charge, with relatively little dependence on the precise details of the CFT spectrum or OPE coefficients. Here, we elaborate on the non-perturbative behavior of Virasoro blocks by classifying all `saddles' that can contribute for arbitrary values of external and internal operator dimensions in the semiclassical large central charge limit. The leading saddles, which determine the naive semiclassical behavior of the Virasoro blocks, all decay exponentially at late times, and at a rate that is independent of internal operator dimensions. Consequently, the semiclassical contribution of high-energy states does not resolve a well-known version of the information loss problem in AdS$_3$. However, we identify two infinite classes of sub-leading saddles, and one of these classes does not decay at late times.

Figures (17)

• Figure 1: This figure depicts a generic configuration of the Lorentzian heavy-light correlator, with dashed lines drawn in to indicate past and future lightcones emanating from the operator ${\cal O}_L(0)$. The lightcones appear as branch cuts in CFT correlators.
• Figure 2: This figure shows the time-dependence of leading semiclassical saddles contributing to $\frac{1}{c} \log {\cal V}$, with different $\alpha_I = 1, 3/5, i/2, 5i/4$ (black, solid; gray, dot-dashed; red, dashed; and blue, dotted, respectively) and fixed $\alpha_L =0.99$ and $T_H = 2 \pi$. The solid black line corresopnds to $\alpha_I = 1$, which is the vacuum Virasoro block. For ease of comparison we have made an overall constant shift in each $f$ to emphasize that the late-time exponential decay is completely independent of the intermediate operator dimension. See fig. \ref{['fig:FollowLeadingNonVac']} for more details.
• Figure 3: The path along which we must translate $\psi(y)$, the solutions to equation (\ref{['eq:MonodromyEquation']}), in order to define a $2 \times 2$ monodromy matrix. The path must encircle $y=0$ and $y=z$, but not $y=1$.
• Figure 4: A surplus angle geometry from the insertion of a negative weight state in AdS$_3$.
• Figure 5: This figure depicts a generic configuration of the Lorentzian heavy-light correlator, with dashed lines drawn in to indicate past and future lightcones emanating from the operator ${\cal O}_L(0)$. Due to the cylindrical geometry, as $t$ increases the operator ${\cal O}_L(t)$ must pass through the future lightcone of ${\cal O}_L(0)$ at regular intervals. From the point of view of the conventional $z$-plane, depicted on the right, the multi-sheeted CFT correlator transitions to a different sheet for each $2\pi$ increment of $t$.