A Numerical Approach to Virasoro Blocks and the Information Paradox

Abstract

We chart the breakdown of semiclassical gravity by analyzing the Virasoro conformal blocks to high numerical precision, focusing on the heavy-light limit corresponding to a light probe propagating in a BTZ black hole background. In the Lorentzian regime, we find empirically that the initial exponential time-dependence of the blocks transitions to a universal $t^{-\frac{3}{2}}$ power-law decay. For the vacuum block the transition occurs at $t \approx \frac{πc}{6 h_L}$, confirming analytic predictions. In the Euclidean regime, due to Stokes phenomena the naive semiclassical approximation fails completely in a finite region enclosing the `forbidden singularities'. We emphasize that limitations on the reconstruction of a local bulk should ultimately stem from distinctions between semiclassical and exact correlators.

Figures (23)

• Figure 1: This figure suggests the analytic continuations necessary to obtain a heavy-light correlator with increasing (Lorentzian) time separation between the light operators. We take $r \lesssim 1$ to avoid singularities on the lightcones displayed on the left; one can also use $r$ as a proxy for a Euclidean time separation between the light operators.
• Figure 2: This figure shows the Penrose diagram for an energy eigenstate black hole in AdS, suggesting the role of ingoing and outgoing modes behind the horizon and their relationship with local CFT operators. Analytic continuation provides a painfully naive but instrumentally effective method for studying correlators behind the horizon.
• Figure 3: The $q(z)$ map takes the universal cover of the $z$-plane (the sphere with punctures at $0,1,\infty$) to $|q|<1$. This figure suggests the relationship between the $z$ plane, the unit $\rho$ disk, and the unit $q$ disk, with branch cuts indicated with colored lines Maldacena:2015iua. The relations between these variables are $q=e^{-\pi\frac{K(1-z)}{K(z)}}$ and $z = \frac{4 \rho}{(1+\rho)^2}$, and the inverse transformations are $z=\left(\frac{\theta_2(q)}{\theta_3(q)}\right)^4$ and $\rho=\frac{z}{(1+\sqrt{1-z})^2}$. The Virasoro blocks converge throughout $|q| < 1$, with OPE limits occurring on the $q$ unit circle.
• Figure 4: This figure displays contours of constant $|q|$ inside the $\rho$ unit circle, which corresponds to the entire $z$-plane via $z = \frac{4 \rho}{(1+\rho)^2}$. Since this is only the first sheet of the $z$-plane, it corresponds to the region in the $q$-disk enclosed by the two blue lines connecting $\pm1$ in figure \ref{['fig:zrhoqBranchCuts']}. The correlator can have singularities in the OPE limits $\rho \to -1, 1$ and these correspond to $q \to -1, 1$ as well. Away from these limits $|q| < |\rho |$ and the $q$-expansion converges much more rapidly than the $\rho$ expansion.
• Figure 5: These plots display the maximum $|q|$ where the $q$-expansion converges for various choices of parameters. Convergence improves when $h_L$ and $h_H$ move closer to $c/24$ and when $c$ decreases. The intermediate primary dimension $h$ seems to have little effect on convergence. These plots define 'convergence' as $\left|\left|\frac{{\cal V}_{0.95N}(q)}{{\cal V}_N(q)}\right|-1\right|<10^{-5}$, where ${\cal V}_M$ includes an expansion up to order $q^M$.