Spinning constraints on chaotic large $c$ CFTs

TL;DR

This work analyzes out-of-time-order correlators in large-$c$ 2d CFTs to understand chaotic dynamics beyond the vacuum block. It shows that while the vacuum Virasoro block yields the maximal Lyapunov exponent $\lambda_L=\frac{2\pi}{\beta}$, non-vacuum blocks with large spin and finite twist can dominate at scrambling times, prompting bounds and partial constraints on the OPE data via pillow-coordinatized Virasoro blocks and their density of states. The authors perform numerical checks using Zamolodchikov recursion, derive analytic estimates for heavy, light, and large-spin sectors, and connect these results to bulk AdS$_3$ gravity by matching non-vacuum Virasoro blocks to worldline actions of massive and spinning particles in BTZ shockwave backgrounds. The conclusions indicate a delicate conspiracy among intermediate states required to preserve chaos bounds, and lay groundwork for refining holographic criteria through Virasoro mean field theory and bulk interpretation of spinning exchanges.

Abstract

We study out-of-time ordered four-point functions in two dimensional conformal field theories by suitably analytically continuing the Euclidean correlator. For large central charge theories with a sparse spectrum, chaotic dynamics is revealed in an exponential decay; this is seen directly in the contribution of the vacuum block to the correlation function. However, contributions from individual non-vacuum blocks with large spin and small twist dominate over the vacuum block. We argue, based on holographic intuition, that suitable summations over such intermediate states in the block decomposition of the correlator should be sub-dominant, and attempt to use this criterion to constrain the OPE data with partial success. Along the way we also discuss the relation between the spinning Virasoro blocks and the on-shell worldline action of spinning particles in an asymptotically AdS spacetime.

Figures (5)

• Figure 1: The image of the map $q(z)$ given in Eq. \ref{['eqn:qzmap']} which takes us from the $\mathbb{C}$ parametrized by $z$ to the unit disc in $q$-space. The shaded domain is the entire $z$ plane and the red trajectory is a path along which we analytically continue the Euclidean four-point function to the desired OTO correlator plotted here for $\epsilon_1=0,\,\epsilon_2=0.2,\,\epsilon_3=0.1,\,\epsilon_4=0.3,\,\beta=x=1$ and $t$ from 0 to $\infty$.
• Figure 2: The Virasoro vacuum blocks $G_0(x,t)$ in \ref{['eqn:GExp']} at $x=0$ as a function of $t$ with $\beta=1$ and $(\epsilon_1,\epsilon_2,\epsilon_3,\epsilon_4)=(-{1\over 4},{1\over 4},0,{1\over 2})$. Left: Increasing central charges $c$ are shown from green to red. Right: A comparison between the numerically computed vacuum block \ref{['eqn:VBafterA']} (shown in red) and the semiclassical heavy-light vacuum block \ref{['eqn:G0Gh']} (shown in blue).
• Figure 3: The logarithm of the ratio of the non-vacuum and vacuum blocks $G_{h,\bar{h}}(x,t)$ in \ref{['eqn:GExp']} at $x=0$ as a function of $t$ with $\beta=1$ and $(\epsilon_1,\epsilon_2,\epsilon_3,\epsilon_4)=(0,{1\over 2},-{1\over 4},{1\over 4})$. Increasing dimensions $h$ of the exchanged operators are shown from green to red.
• Figure 4: A sketch of the massive particle worldlines used for computing the chaos correlator from the bulk AdS geometry, depicted on the shockwave Penrose diagram.
• Figure 5: A rough characterization of the results for the chaos correlator (temporal dependence alone) in various domains analyzed in the paper.