On the differential representation and color-kinematics duality of AdS boundary correlators

Abstract

The AdS boundary correlators and their dual correlation functions of boundary operators have been the main dynamic observables of the holographic duality relating a bulk AdS theory and a boundary conformal field theory. We show that tree-level AdS boundary correlators for generic states can be expressed as nonlocal differential operators of a certain structure acting on contact Witten diagrams. We further write the boundary correlators in a form that is very similar to flat space amplitudes, with Mandelstam variables replaced by certain combinations of single-state conformal generators, prove that all tree-level AdS boundary correlators have a differential representation, and detail the conversion of such differential expressions to position space. We illustrate the construction through the computation of the boundary correlators of scalars coupled to gluons and gravitons; when converted to position space, they reproduce known results. Color-kinematics duality and BCJ relations can be defined in analogy with their flat space counterparts, and are respected by the scalar correlators with a gluon exchange. We also discuss potential approaches to the double copy and find that its direct generalization may require nontrivial extensions.

Analytical evaluation of AdS4 Witten diagrams as flat space multi-loop Feynman integrals

“…A further potentially fruitful way to proceed is to use different new techniques to calculate Witten diagrams, like the differential representation [14,84,85] and unitarity methods based on the split representation of the propagator [17,30,37] in combination with our method.…”

Snowmass White Paper: the Double Copy and its Applications

“…Here, the proposed double copy map is constructed by "squaring" gluon amplitudes in the same AdS space-time, based on observations of a double copy structure between the gluon and gravity boundary correlators in AdS momentum space [324][325][326][327] and Mellin space [328,329]. Additionally, there has been progress in realizing that gluon, NLSM and biadjoint scalar scattering in AdS manifests colorkinematics duality, with amplitudes obeying the BCJ relations [279][280][281][330][331][332]. These observations build on a variety of methods, ranging from Mellin representations of the AdS amplitudes [333,334] to geometric formulations of the kinematics in terms of isometry generators [145,330,331].…”

“…Additionally, there has been progress in realizing that gluon, NLSM and biadjoint scalar scattering in AdS manifests colorkinematics duality, with amplitudes obeying the BCJ relations [279][280][281][330][331][332]. These observations build on a variety of methods, ranging from Mellin representations of the AdS amplitudes [333,334] to geometric formulations of the kinematics in terms of isometry generators [145,330,331]. A key test of these methods in the coming years will be whether or not they can be used to compute gravitational scattering amplitudes in AdS beyond the state-of-the-art set with AdS/CFT techniques.…”

Snowmass White Paper: Gravitational Waves and Scattering Amplitudes

“…While still in its infancy, see e.g. [373][374][375][376][377][378][379][380][381][382][383] and the Snowmass White Paper [384], the generalization of the amplitudes program to curved space together with the gauge theory realization of classical solutions of general relativity [64,[385][386][387][388][389][390] may offer a new perspective on the ringdown phase, as well as, new means to explore analytically the correlation between initial parameters and intrinsic properties of the remnant, including its tidal deformability, etc. Clearly, a systematic development of these methods will have wider applications, establishing connections to other areas of physics, such as gauge/string duality.…”