The Bulk Dual of SYK: Cubic Couplings

TL;DR

This work initiates a concrete program to construct the bulk dual of the SYK model by exploiting the 1/N expansion to extract bulk data from fermionic correlators. By analyzing the fermion 2-, 4-, and 6-point functions, the authors identify an infinite tower of singlet bilinear operators O_n with dimensions h_n, map them to bulk scalars φ_n with masses m_n^2 = h_n(h_n−1), and determine the cubic couplings λ_nmk from the fermion six-point function. The resulting bulk couplings are organized into two contributions—the contact and planar diagrams—with distinct large-q behavior; notably, planar contributions dominate at large indices and resemble cubic couplings of a generalized free field theory. The analysis provides explicit formulas for the large-q limits and finite-q continuations, offering a route to the full bulk dynamics of SYK and guiding future explorations of higher-point interactions and potential nonlocal structures in the bulk.

Abstract

The SYK model, a quantum mechanical model of $N \gg 1$ Majorana fermions $χ_i$, with a $q$-body, random interaction, is a novel realization of holography. It is known that the AdS$_2$ dual contains a tower of massive particles, yet there is at present no proposal for the bulk theory. As SYK is solvable in the $1/N$ expansion, one can systematically derive the bulk. We initiate such a program, by analyzing the fermion two, four and six-point functions, from which we extract the tower of singlet, large $N$ dominant, operators, their dimensions, and their three-point correlation functions. These determine the masses of the bulk fields and their cubic couplings. We present these couplings, analyze their structure and discuss the simplifications that arise for large $q$.

Figures (10)

• Figure 1: The two classes of Feynman diagrams relevant for the fermion six-point function.
• Figure 2: A cubic coupling of the bulk fields $\phi_n$ gives rise to the tree level interaction of the bulk fields shown in this Feynman-Witten diagram. Extrapolating the bulk points to the boundary gives, via the AdS/CFT dictionary, the CFT three-point function of the composites $\mathcal{O}_n$.
• Figure 3: The fermion two-point function, to leading order in $1/N$, is a sum of melon diagrams. This figure, as well as all figures, are for $q=4$ SYK. In addition, we have suppressed the dashed lines commonly drawn to indicate $\langle J_{i_1\ldots i_q} J_{i_1 \ldots i_q}\rangle$ contractions.
• Figure 4: (a) The fermion four-point function, at order $1/N$, is a sum of ladder diagrams. (b) It can be represented as a sum of conformal blocks, involving the $\mathcal{O}_n$.
• Figure 5: The first set of diagrams ("contact" diagrams) contributing to the six-point function at order $1/N^2$. (a). These diagrams can be compactly drawn as three four-point functions (shaded squares) glued together as in (b).