# Generalized conformal structure, dilaton gravity and SYK

## TL;DR

This paper explores the implications of generalized conformal structure in SYK and related holographic theories, analyzing correlation functions, Ward identities, and chaos, revealing that the scale does not influence chaotic behavior.

## Contribution

It demonstrates how generalized conformal structure affects holographic correlation functions and chaos in SYK-like theories, using gravity/multiple scalar models with AdS$_3$ origins.

## Key findings

- Ward identities are holographically implemented in gravity/scalar theories.
- Multiple scalar fields are needed for full SYK operator correspondence.
- Chaotic behavior remains unaffected by the scale in generalized conformal theories.

## 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.