Bounding the space of holographic CFTs with chaos

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≤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 AdS₃ 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_∞[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.