All point correlation functions in SYK

TL;DR

This paper reveals that in large N melonic theories like SYK, all correlation functions can be derived from the three-point functions of bilinear operators, simplifying analysis of higher-point functions and revealing universal behaviors.

Contribution

It introduces a method to determine all correlation functions from the bilinear three-point function, applicable beyond melonic theories, and explicitly computes these functions for q-body SYK.

Findings

All correlation functions are determined by the bilinear three-point function.

Explicit calculation of the bilinear three-point function for q-body SYK.

Universality of correlators for operators with large dimensions.

Abstract

Large $N$ melonic theories are characterized by two-point function Feynman diagrams built exclusively out of melons. This leads to conformal invariance at strong coupling, four-point function diagrams that are exclusively ladders, and higher-point functions that are built out of four-point functions joined together. We uncover an incredibly useful property of these theories: the six-point function, or equivalently, the three-point function of the primary $O(N)$ invariant bilinears, regarded as an analytic function of the operator dimensions, fully determines all correlation functions, to leading nontrivial order in $1/N$, through simple Feynman-like rules. The result is applicable to any theory, not necessarily melonic, in which higher-point correlators are built out of four-point functions. We explicitly calculate the bilinear three-point function for $q$-body SYK, at any $q$. This leads to the bilinear four-point function, as well as all higher-point functions, expressed in terms of higher-point conformal blocks, which we discuss. We find universality of correlators of operators of large dimension, which we simplify through a saddle point analysis. We comment on the implications for the AdS dual of SYK.