On the Differential Representation and Color-Kinematics Duality of AdS Boundary Correlators

TL;DR

This work develops and evidences a differential representation for tree-level AdS boundary correlators, expressing correlators as nonlocal differential operators acting on a single scalar contact diagram and mapping flat-space kinematics to AdS via conformal generators $ ext{D}_{ij}$. It establishes a Berends–Giele recursion to separate exchange nonlocalities from a local factor and demonstrates explicit gluon- and graviton-mediated four-point scalar correlators, revealing color-kinematics duality and BCJ-like relations in AdS. The approach recovers flat-space intuitions in the large-$d$ limit and frames a potential AdS double-copy with necessary Casimir corrections, while providing a practical strategy to convert the differential form into sums of $D$-functions. The results point to deep structural parallels between AdS boundary physics and flat-space amplitudes, suggesting new recursions and factorization insights for AdS/CFT and possibly dS extensions.

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.