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

TL;DR

The paper reveals a unifying role for the conformal $6j$ symbol as a crossing kernel that links three seemingly disparate arenas: the planar diagrams of the SYK model, conformal representation theory, and AdS perturbation theory. By employing the Lorentzian inversion formula and shadow transforms, the authors derive closed-form $6j$ symbols in $d=1,2,4$ and demonstrate how these symbols encode double-trace OPE data and anomalous dimensions, both in CFTs and in AdS loop amplitudes. They further show that AdS tree-level exchanges can be recast as sums/inversions of $6j$ symbols, while one-loop and polygonal diagrams decompose into spectral integrals over these symbols, effectively packaging complex amplitudes into a compact, factorized algebraic structure. This framework provides practical tools for computing higher-point SYK correlators, extracting CFT data, and organizing AdS amplitudes, with several open questions guiding future exploration in odd dimensions, Virasoro extensions, and higher-loop generalizations.

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.

Figures (10)

• Figure 1: The inner product of an $s$-channel conformal partial wave and a $t$-channel conformal partial wave is a $6j$ symbol, represented by a tetrahedron. Each line represents a coordinate, and each vertex is a conformal three-point function.
• Figure 2: (a) The sum of the planar Feynman diagram contribution to the SYK three-point function of bilinears is given by three conformal three-point functions glued together. The figure on the right is not a Feynman diagram; each vertex is a three-point function. (b) The inner product with a bare three-point function of shadow operators extracts the structure constant, and is equal to a $6j$ symbol.
• Figure 3: The four-point function is a sum of ladder diagrams.
• Figure 4: A contribution to the bilinear four-point function. The shaded blob refers to the six-point function of fundamentals; for SYK, the six-point function is given by a sum of the planar and contact diagrams, though this is not relevant for our calculation, which relies only on the conformal properties of the ingredients.
• Figure 5: The SYK bilinear four-point planar diagram with no exchanged melons.