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Home > Journal of High Energy Physics (Springer/SISSA) > d -dimensional SYK, AdS loops, and 6 j symbols |

Liu, Junyu (0000000107068890, Walter Burke Institute for Theoretical Physics, Caltech, E 1200 California Blvd, Pasadena, CA, 91125, U.S.A.) (0000000107068890, Institute for Quantum Information and Matter, Caltech, E 1200 California Blvd, Pasadena, CA, 91125, U.S.A.) ; Perlmutter, Eric (0000000107068890, Walter Burke Institute for Theoretical Physics, Caltech, E 1200 California Blvd, Pasadena, CA, 91125, U.S.A.) ; Rosenhaus, Vladimir (0000 0004 1936 9676, Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA, 93106, U.S.A.) (0000 0001 2160 7918, School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ, 08540, U.S.A.) ; Simmons-Duffin, David (0000000107068890, Walter Burke Institute for Theoretical Physics, Caltech, E 1200 California Blvd, Pasadena, CA, 91125, U.S.A.)

14 March 2019

**Abstract: **We study the 6 j 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 6 j symbol. We generalize the computation of these and other Feynman diagrams to d dimensions. The 6 j 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 6 j symbols in d = 1, 2, 4. In AdS, we show that the 6 j 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 6 j symbol, while one-loop n -gon diagrams are built out of 6 j symbols.

**Published in: ****JHEP 1903 (2019) 052**
**Published by: **Springer/SISSA

**DOI: **10.1007/JHEP03(2019)052

**arXiv: **1808.00612

**License: **CC-BY-4.0