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

DOI10.1007/JHEP03(2019)052zbMATH Open1414.81211arXiv1808.00612OpenAlexW2885591200WikidataQ128256658 ScholiaQ128256658MaRDI QIDQ2421070

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Publication date: 8 June 2019

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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 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.

Full work available at URL: https://arxiv.org/abs/1808.00612

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