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

Liu, Junyu and Perlmutter, Eric and Rosenhaus, Vladimir and Simmons-Duffin, David (2019) d-dimensional SYK, AdS loops, and 6j symbols. Journal of High Energy Physics, 2019 (3). Art. No. 52. ISSN 1029-8479.

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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 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.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Liu, Junyu0000-0003-1669-8039
Simmons-Duffin, David0000-0002-2937-9515
Additional Information:© The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Received: December 12, 2018; Accepted: March 5, 2019; Published: March 11, 2019. We thank Bartek Czech, Abhijit Gadde, David Gross, Denis Karateev, Petr Kravchuk, Murat Kologlu, Shota Komatsu, Grisha Korchemsky, Douglas Stanford, Herman Verlinde, and Ellis Ye Yuan for discussions. VR thanks H. Verlinde for valuable early discussions on the possible connection between SYK and the 6j symbol. We also thank the December 2017 Southern California Strings Seminar at UCLA, where this work was initiated, and the Bootstrap 2018 workshop at Caltech, where this work was nearly completed, for hospitality. EP thanks the Yukawa Institute for Theoretical Physics and National Taiwan University for hospitality. VR thanks the Nordita program "Correlation functions in solvable models" for hospitality. JL is supported in part by the Institute for Quantum Information and Matter (IQIM), an NSF Physics Frontiers Center (NSF Grant PHY-1125565) with support from the Gordon and Betty Moore Foundation (GBMF-2644), and by the Walter Burke Institute for Theoretical Physics. DSD and EP are supported by Simons Foundation grant 488657, and by the Walter Burke Institute for Theoretical Physics. VR is support from NSF grant PHY-1125915 and PHY-1606531.
Group:IQIM, Institute for Quantum Information and Matter, Walter Burke Institute for Theoretical Physics
Funding AgencyGrant Number
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
Gordon and Betty Moore FoundationGBMF-2644
Walter Burke Institute for Theoretical Physics, CaltechUNSPECIFIED
Simons Foundation488657
Subject Keywords:AdS-CFT Correspondence; Black Holes; Conformal Field Theory
Record Number:CaltechAUTHORS:20190314-092752423
Persistent URL:
Official Citation:Liu, J., Perlmutter, E., Rosenhaus, V. et al. J. High Energ. Phys. (2019) 2019: 52.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:93798
Deposited By: Tony Diaz
Deposited On:14 Mar 2019 17:04
Last Modified:14 Mar 2019 17:04

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