Bounding the Space of Holographic CFTs with Chaos

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, $λ_L\leq 2π/β$. 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 $λ_L=2π/β$ 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 $W_{\infty}[λ]$ symmetry, dual to 3D Vasiliev or hs[$λ$] higher spin gravities, do not violate the chaos bound, instead exhibiting no chaos: $λ_L=0$. Independently, we show that such theories violate unitarity for $|λ|>2$. 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.