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arxiv 1808.00612 v3 pith:EGNZADWL submitted 2018-08-02 hep-th cond-mat.str-elmath-phmath.MP

d-dimensional SYK, AdS Loops, and 6j Symbols

classification hep-th cond-mat.str-elmath-phmath.MP
keywords symboldiagramsconformalsymbolsfeynmaninversionlorentzianone-loop
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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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.

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