All point correlation functions in SYK

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.