All point correlation functions in SYK

TL;DR

In SYK-like large-$N$ theories the six-point function, equivalently the three-point function of bilinear $O(N)$ singlets, determines all correlators at leading order in $1/N$ via contour-integral representations over exchanged dimensions and conformal blocks. The authors compute the bilinear three-point function $c_{123}$ (including contact and planar pieces) for general $q$, then express the bilinear four-point and higher-point functions in terms of higher-point conformal blocks, establishing a universal, analytic structure in operator dimensions. They further interpret these results in a bulk AdS$_2$ framework, with a tower of dual scalars $oldsymbol{ ext{φ}}_n$ of mass $m_n^2=h_n(h_n-1)$ and cubic couplings determined by the boundary OPE data, and analyze exchange and contact Witten diagrams as the bulk counterpart to the boundary ladder diagrams. The findings reveal a remarkably universal large-dimension sector, provide a concrete route to reconstruct bulk interactions from boundary data, and hint at a string-like bulk interpretation for 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.

Figures (18)

• Figure 1: The connected fermion six-point function, to leading nontrivial order in $1/N$, is given by a sum of Feynman diagrams, of the kind shown on the right. This consists of three fermion four-point functions, ladders, that are glued together. There are two classes of diagrams, as shown on the right; only the second is planar. This figure, as well as all others, is for $q=4$ SYK, and the lines denote the full propagators: they should be dressed with melons.
• Figure 2: The fermion eight-point function is composed of Feynman diagrams such as the one shown. It is built out of two six-point functions; the shaded circle is defined by Fig. \ref{['FigIntro1']}.
• Figure 3: The fermion four-point function, at order $1/N$, is a sum of ladder diagrams. There are also crossed diagrams, which are not shown.
• Figure 4: The contour of integration $\mathcal{C}$ in the complex $h$-plane.
• Figure 5: A pictorial representation of the four-point function, split into a product of two three-point functions $\langle \chi \chi \mathcal{O}\rangle$, see GR1, using the shadow formalism. See Eq. \ref{['PsiIntr']}.