On-shell Correlators and Color-Kinematics Duality in Curved Symmetric Spacetimes

TL;DR

This work extends flat-space on-shell kinematics to curved symmetric spacetimes by introducing isometric momenta generated by spacetime isometries and using the Killing Casimir as the propagator. The authors define the on-shell correlator, develop its isometric representation, and show that locality enforces a natural separation into commuting propagators and noncommuting numerators, enabling a curved-space color-kinematics duality between BAS and the NLSM. They compute tree-level correlators in BAS and NLSM, demonstrate that kinematic Jacobi identities hold, and prove the fundamental BCJ relation via a null color replacement argument. The results yield a unified, geometry-agnostic framework for on-shell observables in symmetric spacetimes and open avenues toward loops, spin, and curved-space double-copy constructions with potential AdS/dS applications.

Abstract

We define a perturbatively calculable quantity--the on-shell correlator--which furnishes a unified description of particle dynamics in curved spacetime. Specializing to the case of flat and anti-de Sitter space, on-shell correlators coincide precisely with on-shell scattering amplitudes and boundary correlators, respectively. Remarkably, we find that symmetric manifolds admit a generalization of on-shell kinematics in which the corresponding momenta are literally the isometry generators of the spacetime acting on the external kinematic data. These isometric momenta are intrinsically non-commutative but exhibit on-shell conditions that are identical to those of flat space, thus providing a common language for computing and representing on-shell correlators which is agnostic about the underlying geometry. Afterwards, we compute tree-level on-shell correlators for biadjoint scalar (BAS) theory and the nonlinear sigma model (NLSM) and learn that color-kinematics duality is manifested at the level of fields under a mapping of the color algebra to the algebra of gauged isometries on the spacetime manifold. Last but not least, we present a field theoretic derivation of the fundamental BCJ relations for on-shell correlators following from the existence of certain conserved currents in BAS theory and the NLSM.