Journal Article

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

Simmons-Duffin, David, Liu, Junyu, Perlmutter, Eric, Rosenhaus, Vladimir

2019

Overview

A bstract 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.

• Publisher: Springer Berlin Heidelberg,Springer Nature B.V,SpringerOpen
• Subject: AdS-CFT Correspondence / Amplitudes / Black Holes / Classical and Quantum Gravitation / Conformal Field Theory / Elementary Particles / Feynman diagrams / High energy physics / Operators (mathematics) / Physics / Physics and Astronomy / Quantum Field Theories / Quantum Field Theory / Quantum Physics / Regular Article - Theoretical Physics / Relativity Theory / Scalars / String Theory / Symbols