On-shell correlators and color-kinematics duality in curved symmetric spacetimes

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.

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.…”

“…Arguments that color-kinematics duality should persist in any curved spacetime have been put forth in [145,335], and it was shown in [330] that it does hold for NLSMs and the biadjoint scalar theory on symmetric spaces. Some explicit calculations have been possible for plane wave spacetimes [193,336]; these admit an S-matrix [337], extend to solutions of string theory [338][339][340][341] and can be viewed as local approximations to any spacetime [342].…”

“…Note that a complementary approach dubbed the Cosmological Bootstrap approaches this problem from a different perspective using unitarity, locality, and symmetries; see the dedicated White Paper[382] for a thorough discussion on this topic 20. Progress made in an AdS/CFT context[279-281, 327, 328, 331, 332] and maximally symmetric spacetimes[330,335] can provide lessons for a dS double copy.…”

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.…”