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

TL;DR

This paper explores the $6j$ symbol's role across SYK models, conformal theory, and AdS amplitudes, providing new formulas and insights into their interrelations and applications in higher dimensions and loop diagrams.

Contribution

It introduces closed-form expressions for $6j$ symbols in multiple dimensions and links them to crossing kernels, conformal partial waves, and AdS loop diagrams, revealing their unifying role.

Findings

$6j$ symbols are the crossing kernels for conformal partial waves.

Closed-form $6j$ symbols are derived for $d=1,2,4$.

One-loop AdS diagrams with scalars and spinning operators are expressed as $6j$ symbols.

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.