d-dimensional SYK, AdS loops, and 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.

Unitarity methods in AdS/CFT

“…However, to set the stage, suppose one does. An efficient algorithm for this purpose that manifests the simplifying role of conformal symmetry was developed in [27]. First, by manipulating AdS bulk-to-bulk propagators, one can recast Witten diagrams as spectral integrals over conformally-invariant gluings of CFT three-point functions.…”

“…10 In Euclidean signature the integration is over all of space, but in Lorentzian signature we can consider more general pairings of operators and domains of integration [42,56]. 11 Note that the 6j symbol is known explicitly in closed form only in d = 1, 2, 4 [27,57]. See also [58,59].…”

“…The main point will be to introduce the AdS unitarity cut, which factorizes tree-level exchanges into products of three-point functions, and its manifest relation to dDisc. The computation follows [27] and employs the technical toolkit of conformal integrals reviewed in Section 2 (see e.g. [27,[41][42][43]60]).…”

“…• We give a simple and elegant dictionary between the diagrammatic bulk unitarity method, which involves cutting and gluing AdS graphs, and the algebraic holographic unitarity method of [1], which involves summing over products of OPE data. Using CFT tools [27], we show that the two methods are equivalent. In doing so we recast the method of [1] in the language of the dDisc.…”

“…The loop expansion in AdS has been further studied along these lines in [8][9][10][11][12][13][14][15][16][17]. Other subsequent work on AdS loops from diverse perspectives includes [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. 1 The current paper aims to provide a more systematic approach to unitarity methods for correlators in AdS/CFT, cast in both bulk and boundary language.…”