A natural language for AdS/CFT correlators

Abstract

We provide dramatic evidence that 'Mellin space' is the natural home for correlation functions in CFTs with weakly coupled bulk duals. In Mellin space, CFT correlators have poles corresponding to an OPE decomposition into 'left' and 'right' sub-correlators, in direct analogy with the factorization channels of scattering amplitudes. In the regime where these correlators can be computed by tree level Witten diagrams in AdS, we derive an explicit formula for the residues of Mellin amplitudes at the corresponding factorization poles, and we use the conformal Casimir to show that these amplitudes obey algebraic finite difference equations. By analyzing the recursive structure of our factorization formula we obtain simple diagrammatic rules for the construction of Mellin amplitudes corresponding to tree-level Witten diagrams in any bulk scalar theory. We prove the diagrammatic rules using our finite difference equations. Finally, we show that our factorization formula and our diagrammatic rules morph into the flat space S-Matrix of the bulk theory, reproducing the usual Feynman rules, when we take the flat space limit of AdS/CFT. Throughout we emphasize a deep analogy with the properties of flat space scattering amplitudes in momentum space, which suggests that the Mellin amplitude may provide a holographic definition of the flat space S-Matrix.

Factorization of Mellin amplitudes

“…As explained in [3] (section 2.1), for each primary operator O p , with dimension ∆ and spin J, that appears in the OPE (1.1), the Mellin amplitude has an infinite sequence of poles…”

“…However, before that we must discuss the generalization of the Mellin representation (1.4) for correlation functions involving tensor operators (section 3). In order to make the analogy with scattering amplitudes more explicit, we start by reviewing their factorization properties 3 These formulas were first derived in [3] from the result of [3]. This mismatch follows from a different choice of normalization for the operators O.…”

Computing three-point functions for short operators

“…For related references, see e.g. [39][40][41][42][43]. It can be hoped that the methods used here will also be useful for higher-point correlators.…”

Deconstructing conformal blocks in 4D CFT

“…[1], there have been many developments, both analytical [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and numerical [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] in the 4D bootstrap. All numerical studies are still based on identical scalar correlators, unless supersymmetry or global symmetries are present.…”