A Quantum Correction To Chaos

TL;DR

The paper investigates quantum corrections to chaotic dynamics in 2d CFTs at large central charge by computing Virasoro vacuum-block contributions in AdS$_3$/CFT$_2$. It identifies a $1/c$ quantum correction to the Lyapunov exponent, yielding $oxed{ \\lambda_L = rac{2\\pi}{\\beta} (1 + 12/c) }$, while also revealing additional $1/c$-suppressed pieces in OTOCs that do not reduce to a simple exponent. The authors reconcile these corrections with the MSS chaos bound by carefully analyzing the regime and showing that, near scrambling time, subleading effects can preserve causality and analyticity constraints. They further connect chaos growth to bulk locality and scattering, discussing double-trace contributions, bulk point singularities, and causality conditions in both flat space and AdS/CFT. The work highlights how quantum gravitational effects modify scrambling in holographic theories and outlines future directions for higher-order corrections and broader dimensionality.

Abstract

We use results on Virasoro conformal blocks to study chaotic dynamics in CFT$_2$ at large central charge c. The Lyapunov exponent $λ_L$, which is a diagnostic for the early onset of chaos, receives $1/c$ corrections that may be interpreted as $λ_L = \frac{2 π}β \left( 1 + \frac{12}{c} \right)$. However, out of time order correlators receive other equally important $1/c$ suppressed contributions that do not have such a simple interpretation. We revisit the proof of a bound on $λ_L$ that emerges at large $c$, focusing on CFT$_2$ and explaining why our results do not conflict with the analysis leading to the bound. We also comment on relationships between chaos, scattering, causality, and bulk locality.

Figures (6)

• Figure 1: This figure depicts the various regimes of the out of time order correlator $F_\beta$ of equation (\ref{['eq:OutOfOrderCorr']}). At early times $t \sim \beta$ we expect $1-F_\beta \lesssim \frac{1}{c}$; this quantity subsequently grows exponentially with Lyapunov exponent $\lambda_L$ until a scrambling time of order $t_* \approx \frac{\beta}{2 \pi} \log c$.
• Figure 2: The figure on the left shows a choice of locations for $V$ and $W$Roberts:2014ifa in the Euclidean plane at Lorentzian $t=0$. The distance between the two $V$ and the two $W$ is small, regulated by $\epsilon_i - \epsilon_j$. On the right we have a cartoon of the configuration at finite $x, t$ from the perspective of AdS$_3$/CFT$_2$. This configuration has the physically intuitive advantage that all operators are on the same side of a single black brane in the Poincaré patch.
• Figure 3:
• Figure 4: CFT 4-point correlators can have branch cuts between their OPE singularities at $0, 1,$ and $\infty$. This figure shows the analytic continuation in the $z$ plane necessary to obtain the out of time order correlator of equation (\ref{['eq:OutOfOrderCorr']}). While $z$ pases through the branch cut extending from $1$ to $\infty$, the $\bar{z}$ variable remains on the original sheet.
• Figure 5: This figure shows diagrams in AdS$_3$ corresponding to various contributions Fitzpatrick:2015dlt to the Virasoro vacuum block. The diagram at left is 1-graviton exchange, and is proportional to $h_W h_V/c$. The central diagram, proportional to $h_W h_V^2/c^2$ is a semi-classical correction incorporating gravitational back-reaction. The diagram at right is a true quantum correction proportional to $h_W h_V/c^2$ which is responsible for $1/c$ corrections to the Lyapunov exponent.