Bounding the Space of Holographic CFTs with Chaos

TL;DR

This paper uses chaos bounds in quantum systems to constrain the landscape of holographic conformal field theories and their dual higher spin gravity theories, ruling out certain models and exploring the properties of others.

Contribution

It establishes constraints on 2D holographic CFTs with higher spin currents and analyzes the chaos properties of theories with $W_{}[lambda]$ symmetry, connecting them to higher spin gravity duals.

Findings

Unitary, sparse 2D CFTs with large central charge and bounded higher spin currents are inconsistent.

The chaos bound is saturated in certain 2D holographic CFTs, linking Lyapunov exponent to OPE properties.

Theories with classical $W_{}[lambda]$ symmetry do not exhibit chaos and violate unitarity for $||>2$.

Abstract

Thermal states of quantum systems with many degrees of freedom are subject to a bound on the rate of onset of chaos, including a bound on the Lyapunov exponent, $\lambda_L\leq 2\pi /\beta$. We harness this bound to constrain the space of putative holographic CFTs and their would-be dual theories of AdS gravity. First, by studying out-of-time-order four-point functions, we discuss how $\lambda_L=2\pi/\beta$ in ordinary two-dimensional holographic CFTs is related to properties of the OPE at strong coupling. We then rule out the existence of unitary, sparse two-dimensional CFTs with large central charge and a set of higher spin currents of bounded spin; this implies the inconsistency of weakly coupled AdS$_3$ higher spin gravities without infinite towers of gauge fields, such as the $SL(N)$ theories. This fits naturally with the structure of higher-dimensional gravity, where finite towers of higher spin fields lead to acausality. On the other hand, unitary CFTs with classical $W_{\infty}[\lambda]$ symmetry, dual to 3D Vasiliev or hs[$\lambda$] higher spin gravities, do not violate the chaos bound, instead exhibiting no chaos: $\lambda_L=0$. Independently, we show that such theories violate unitarity for $|\lambda|>2$. These results encourage a tensionless string theory interpretation of the 3D Vasiliev theory. We also perform some CFT calculations of chaos in Rindler space in various dimensions.