d-dimensional SYK, AdS loops, and 6j symbols

Abstract

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 doubletrace 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.