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, lambda(L) <= 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 in finite 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-infinity[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 vertical bar lambda vertical bar > 2. These results encourage a tensionless string theory interpretation of the 3D Vasiliev theory.