Abstract
We study out-of-time ordered four-point functions in two dimensional conformal field theories by suitably analytically continuing the Euclidean correlator. For large central charge theories with a sparse spectrum, chaotic dynamics is revealed in an exponential decay; this is seen directly in the contribution of the vacuum block to the correlation function. However, contributions from individual non-vacuum blocks with large spin and small twist dominate over the vacuum block. We argue, based on holographic intuition, that suitable summations over such intermediate states in the block decomposition of the correlator should be sub-dominant, and attempt to use this criterion to constrain the OPE data with partial success. Along the way we also discuss the relation between the spinning Virasoro blocks and the on-shell worldline action of spinning particles in an asymptotically AdS spacetime.
Highlights
It is empirically clear that field theories with a large number of degrees of freedom and a sparse spectrum of low-lying operators satisfy necessary criteria to have a dual description in terms of gravitational dynamics in AdS
Making use of our assumption that hv hw), in which case mv(hv) c ≈ 2 hv, Assuming the operator product expansion (OPE) coefficients saturate this bound in the regime h < c ≈ mv c with 1, we can estimate the contribution of a Virasoro block of conformal dimension (h, h) to the correlator as2hw CV2 V (h, h) Fh(z)Fh (z)
A simple way to see that we aren’t quite seeing the physics we are interested in is that this limiting behavior is independent of the exchanged operator dimension h, whereas we expect there to be a delicate interplay among blocks near the scrambling time to be compatible with the chaos bound
Summary
It is empirically clear that field theories with a large number of degrees of freedom (measured e.g., by the central charge) and a sparse spectrum of low-lying operators satisfy necessary criteria to have a dual description in terms of gravitational dynamics in AdS. By truncating the correlator to include only the identity operator O0 = 1 and its Virasoro descendants, which assumes the low-lying spectrum is sparse, they were able to show that upon analytic continuation to the appropriate Lorentzian out-of-time-order, one obtains the desired form of the chaos correlator with maximal Lyapunov exponent This is due to a delicate cancellation between the contribution from Virasoro descendants of different spins, since each spin-s global primary operator contributes to the OTO correlator an exponential factor e (s−1)t, which for s > 2 violates the chaos bound. We set-up the problem of putting bounds on OPE data, and will show that the contributions from very heavy primaries and large spin primaries in the intermediate states are innocuous This is done by explicit evaluation of the conformal blocks to get a good estimate, and uses some recently derived bounds on the growth of OPE coefficients [23, 24]. The appendices A and B are devoted to providing details of the spinning particle analysis, while appendix C computes the Euclidean block for completeness
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