The SYK model consists of N ≫ 1 fermions in 0 + 1 dimensions with a random, all-to-all quartic interaction. Recently, Kitaev has found that the SYK model is maximally chaotic and has proposed it as a model of holography. We solve the Schwinger-Dyson equation and compute the spectrum of two-particle states in SYK, finding both a continuous and discrete tower. The four-point function is expressed as a sum over the spectrum. The sum over the discrete tower is evaluated.The Sachdev-Ye-Kitaev model (SYK) [1, 2] is a 0 + 1 dimensional model of N ≫ 1 fermions with an all-to-all random quartic interaction. SYK has three notable features:Solvable at strong coupling. At large N one can sum all Feynman diagrams, and thereby compute correlation functions at strong coupling.Maximally chaotic. Chaos is quantified by the Lyapunov exponent, which is defined by an out-of-time-order four-point function [3,4]. The Lyapunov exponent of a black hole in Einstein gravity is 2π β [4-6], where β is the inverse temperature. This is the maximal allowed Lyapunov exponent [7], and SYK saturates the bound [1].Emergent conformal symmetry. In the context of the two-point function, there is emergent conformal symmetry at low energies [1,[8][9][10].where χ j are Majorana fermions {χ i , χ j } = δ ij , and the model has quenched disorder with the couplings J ijkl drawn from the distribution,leading to a disorder average of,The expressions for the correlation functions that will follow will always be the result after the disorder average has been performed. The Lagrangian trivially follows from

The SYK model: fermions with a q-body, Gaussian-random, all-to-all interaction, is the first of a fascinating new class of solvable large N models. We generalize SYK to include f flavors of fermions, each occupying N a sites and appearing with a q a order in the interaction. Like SYK, this entire class of models generically has an infrared fixed point. We compute the infrared dimensions of the fermions, and the spectrum of singlet bilinear operators. We show that there is always a dimension-two operator in the spectrum, which implies that, like in SYK, there is breaking of conformal invariance and maximal chaos in the infrared four-point function of the generalized model. After a disorder average, the generalized model has a global O(N 1 ) × O(N 2 ) × . . . × O(N f ) symmetry: a subgroup of the O(N ) symmetry of SYK;thereby giving a richer spectrum. We also elucidate aspects of the large q limit and the OPE, and solve q = 2 SYK at finite N .(1.1)The model has qualitatively similar properties for any choice of even q ≥ 4. The couplingsq are independently chosen from a Gaussian, O(N ) invariant, distribution with zero mean and a variance proportional to J 2 N 1−q . When evaluating observables, say correlation functions, a disorder average is performed at the end of the calculation. For the purposes of correlation functions, at large N , the model is self-averaging for q > 2: randomly chosen, but fixed, J i 1 ,...,i q give the same results as disorder averaged J i 1 ,...,i q . One can alternatively think of the J i 1 ,...,i q as nearly static free bosonic fields; at leading order in 1/N , this gives the same connected correlation functions [16], and furthermore, allows one to gauge the O(N ) symmetry [17]. To leading order in 1/N the fermions are non-interacting, and the two-point function of the fermions satisfies a simple integral equation which can be explicitly solved near the infrared fixed point. The fermions start with dimension 0 in the UV, and flow to dimension ∆ = 1/q in the IR. After the disorder average, the dynamics is invariant under an O(N ) global symmetry, χ i → O ij χ j , with OO T = 1, much like a vector model. The bilinear, primary, fermion operators, singlets under O(N ), are schematically N i=1 χ i ∂ 2n+1 τ χ i . In the UV, these operators have dimension 2n + 1. In the IR, the dimensions receive an order-one shift for small n, and approach 2∆ + 2n + 1 asymptotically for large n. The standard AdS/CFT dictionary relates the dimensions of CFT single-trace operators for matrix theories, or bilinear singlet operators for vector models, to the masses of particles in the bulk dual. This would imply that the SYK dual has a tower of particles in the bulk, with masses, in units of the AdS radius, roughly spaced by two. This spectrum differs from N = 4/AdS 5 × S 5 duality where for large 't Hooft coupling only a small number of massless modes survive, or vector model/Vasiliev duality, where a tower of massless modes appears in the bulk. In [14] it was noted that the bulk dual of SYK might be a string theo...

We study the 6j symbol for the conformal group, and its appearance in three seemingly unrelated contexts: the SYK model, conformal representation theory, and perturbative amplitudes in AdS. The contribution of the planar Feynman diagrams to the three-point function of the bilinear singlets in SYK is shown to be a 6j symbol. We generalize the computation of these and other Feynman diagrams to d dimensions. The 6j symbol can be viewed as the crossing kernel for conformal partial waves, which may be computed using the Lorentzian inversion formula. We provide closed-form expressions for 6j symbols in d = 1, 2, 4. In AdS, we show that the 6j symbol is the Lorentzian inversion of a crossing-symmetric tree-level exchange amplitude, thus efficiently packaging the double-trace OPE data. Finally, we consider one-loop diagrams in AdS with internal scalars and external spinning operators, and show that the triangle diagram is a 6j symbol, while one-loop n-gon diagrams are built out of 6j symbols.

The SYK model, a quantum mechanical model of N 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.