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.
Highlights
Clebsch-Gordan coefficients are familiar quantities in quantum mechanics, encoding the addition of two angular momenta
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 6j symbol can be viewed as the crossing kernel for conformal partial waves, which may be computed using the Lorentzian inversion formula
Summary
Clebsch-Gordan coefficients are familiar quantities in quantum mechanics, encoding the addition of two angular momenta. We reveal two other contexts in which these 6j symbols appear: Feynman diagrams of the SYK model and their generalization to higher dimensions, and Witten diagrams for tree-level and one-loop scattering amplitudes in AdS, dual to large N CFT correlators at leading and subleading orders in 1/N. There is a simple way to see that the planar three-point diagram is a 6j symbol: by taking the inner product with a threepoint function of the shadow operators, one obtains a tetrahedron, i.e. a 6j symbol This argument establishes an intriguing connection: the overlap of two conformal partial waves — a group-theoretic quantity — and the planar Feynman diagrams in an SYK correlation function — a dynamical quantity — are just two different ways of splitting a tetrahedron. In appendix D, we present derivations of some shadow transforms of three-point functions
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