Generalized conformal structure, dilaton gravity and SYK

TL;DR

This work shows that SYK and related non-conformal holographic theories possess a generalized conformal structure driven by dimensionful couplings. It demonstrates that such theories admit a parent AdS3 description via dimensional reductions, constraining bulk interactions and Ward identities, and explains why chaos exhibits maximal Lyapunov growth independent of the running-scale L. By exploring single- and multi-scalar bulk realizations, black-hole thermodynamics, and reductions of higher-dimensional CFTs, the paper unifies SYK holography under a common AdS3-based framework and clarifies how chaos and two-point functions behave in generalized-conformal settings. The results provide a coherent prescription for incorporating multiple bulk scalars and for computing higher-point functions while preserving generalized conformal symmetry.

Abstract

A theory admits generalized conformal structure if the only scale in the quantum theory is set by a dimensionful coupling. SYK is an example of a theory with generalized conformal structure and in this paper we investigate the consequences of this structure for correlation functions and for the holographic realization of SYK. The Ward identities associated with the generalized conformal structure of SYK are implemented holographically in gravity/multiple scalar theories, which always have a parent AdS$_3$ origin. For questions involving only the graviton/running scalar sector, one can always describe the bulk running in terms of a single scalar but multiple running scalars are in general needed once one includes the bulk fields corresponding to all SYK operators. We then explore chaos in holographic theories with generalized conformal structure. The four point function explored by Maldacena, Shenker and Stanford exhibits exactly the same chaotic behaviour in any such theory as in holographic realizations of conformal theories i.e. the dimensionful coupling scale does not affect the chaotic exponential growth.