A Natural Language for AdS/CFT Correlators (1107.1499v2)

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 intoleft' 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.

• The paper demonstrates that Mellin space provides a natural and efficient framework to express AdS/CFT correlators in the large N expansion.
• It derives a factorization formula for tree-level Witten diagrams using pole structures that mirror scattering amplitudes.
• The work bridges AdS/CFT duality with the flat space S-matrix, offering fresh insights into non-perturbative quantum gravity.

An Analytical Approach to AdS/CFT Correlators in Mellin Space

The paper "A Natural Language for AdS/CFT Correlators" offers a comprehensive paper on the application of Mellin space to correlation functions within conformal field theories (CFTs) that are dual to weakly coupled bulk theories in Anti-de Sitter (AdS) space. It explores the potential of Mellin space as a structured and efficient arena for such correlators, highlighting its similarities to momentum space in the context of scattering amplitudes in flat spacetime.

The authors begin by establishing Mellin space as the natural framework for expressing CFT correlators, particularly within the large N expansion. In this formalism, correlators are presented as meromorphic functions with pole structures that align with factorization channels familiar from scattering theory. Specifically, Mellin amplitudes in this context exhibit poles corresponding to the exchange of operators, analogous to physical resonances in scattering processes. The residues of these poles reflect the contributions from lower-point correlators, establishing a recursive relationship that simplifies computations of higher-point functions.

The authors derive a specific factorization formula applicable to tree-level Witten diagrams in any theory of scalar fields. This formula specifies that Mellin amplitudes exhibit poles at positions determined by the dimensions of the involved operators, with residues given by simple products of Mellin amplitudes from smaller diagrams. The strength of these results is illustrated by applying the factorization principle to compute explicit 5-point and 6-point correlation functions in a theory with cubic interactions among scalar fields.

Furthermore, the paper delineates a set of universal diagrammatic rules in Mellin space. These rules offer an intuitive method to construct Mellin amplitudes for any tree-level diagram by associating polynomial factors with derivative interactions and maintaining consistency across different recursive applications of the factorization formula.

A critical advance presented in this work is the holographic reproduction of the flat space S-matrix from Mellin space amplitudes. By examining the limit where AdS curvature vanishes, it becomes evident that the structure of the Mellin amplitude morphs into the familiar flat space scattering amplitudes. This is a testament to the robustness of the AdS/CFT duality and provides a significant bridge to understand scattering processes in curved backgrounds from their flat space counterparts.

The implications of this research extend to simplifying the computation of complex AdS/CFT correlators and, potentially, to provide new insights into non-perturbative aspects of quantum gravity and field theory. The factorization properties and diagrammatic rules might be adapted to other complex theories, including those involving tensor fields or loop-level processes. Furthermore, the results underscore Mellin space's utility in non-perturbative descriptions of holographic dualities, promoting the possibility of expressing these theories in terms of a generalized S-matrix formalism.

In conclusion, the paper "A Natural Language for AdS/CFT Correlators" demonstrates significant progress in applying the Mellin representation to AdS/CFT, providing new computational tools and insights that enhance the understanding of CFT correlators. This work also opens new pathways for future investigations into the intricacies of holographic dualities and the fundamental structure of quantum field theories.