Reparametrization mode Ward Identities and chaos in higher-pt. correlators in CFT2
Abstract
Recently introduced reparametrization mode operators in CFTs have been shown to govern stress tensor interactions via the shadow operator formalism and seem to govern the effective dynamics of chaotic systems. We initiate a study of Ward identities of reparametrization mode operators i.e. how two dimensional CFT Ward identities govern the behaviour of insertions of reparametrization modes ε in correlation functions: εεφφ. We find that in the semi-classical limit of large c they dictate the leading O(c-1) behaviour. While for the 4pt function this reproduces the same computation as done by Heahl, Reeves \& Rozali in Haehl:2019eae, in the case of 6pt function of pair-wise equal operators this provides an alternative way of computing the Virasoro block in stress-tensor comb channel. We compute a maximally out of time ordered correlation function in a thermal background and find the expected behaviour of an exponential growth governed by Lyapunov index λL=2π/β lasting for twice the scrambling time of the system t*=β2π\,c for the maximally braided type of out-of-time-ordering. However when only the internal operators of the comb channel are out-of-time-ordered, the correlator sees no exponential behaviour despite the inclusion of the Virasoro contribution. From a bulk perspective for the out-of-time-ordered 4pt function we find that the Casimir equation for the stress tensor block reproduces the linearised back reaction in the bulk.