A natural language for AdS/CFT correlators

By A. Fitzpatrick, Jared Kaplan, J. Penedones, S. Raju, B. C. Rees. Journal of High Energy Physics, 2011

330 citations recorded by the source corpus.

A bstractWe 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.

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Citations in the corpus

Spinning AdS propagators (2014, methodology): “This fact has been explored in many places in the literature to simplify computations, including computations of correlation functions of higher spin fields [9, 10, 11], of conformal blocks for external operators with spin [12, 13], and of Witten diagrams to derive Feynman rules in Mellin space [14,”
The analytic bootstrap and AdS superhorizon locality (2012, background): “This approach has been fruitful, especially when interpreted in Mellin space [19, 23–28], but it is an essentially perturbative approach analogous [29] to the use of dispersion relations for the study of perturbative scattering amplitudes.”
Witten diagrams revisited: the AdS geometry of conformal blocks (2015, background): “(9)For certain non-generic operator dimensions, the sum over poles actually truncates [21,23].”
Spinning conformal correlators (2011, methodology): “In the case of polynomial scattering amplitudes, we can use AdS/CFT to provide an explicit map from scattering amplitudes in Md+1 to CFTd correlators.”
Spinning conformal blocks (2011, background): “Some steps towards this goal were taken in [12, 13, 37]; we hope that the present formalism will allow for additional progress to be made.”
Looking for a bulk point (2015, unknown): “Mellin space looks like the best tool to study these issues, since the Mellin amplitudes have analytic properties similar to string tree-level amplitudes [12,66,67].”
Bootstrapping mixed correlators in the 3D Ising model (2014, background): “2) Other recent studies of global conformal blocks can be found in [11, 41, 42, 45, 46, 49–55], with connections to Mellin amplitudes in [56–65].”
Loops in AdS from conformal field theory (2016, background): “For d = 4,∆ = 2, upon matching to (143), we require 2R̂0 = 12α (2), which defines the α(2) normalization of [57].”
From correlation functions to event shapes (2013, background): “presentation of the correlation functions [37, 38, 39]. The Mellin representation is a natural language for the description of CFT correlators and came into widespread use in recent years (see, e.g., [40, 41, 42, 43]). For the connected part of the four-point Euclidean correlation function16 it read”
Unitarity and the holographic S-Matrix (2011, methodology): “An elementary feature of the AdS/CFT correspondence is that AdS global time (1)We reviewed aspects of CFT physics and the Mellin amplitude in our recent companion paper [9], and we discussed them in detail in [7], so we urge interested readers to consult these references for a review.”