[Paper Review] On-shell Correlators and Color-Kinematics Duality in Curved Symmetric Spacetimes

Clifford Cheung, Julio Parra-Martinez+1 more|arXiv (Cornell University)|Jan 13, 2022

Black Holes and Theoretical Physics95 references55 citations

TL;DR

This paper introduces on-shell correlators as a unified framework for describing particle dynamics in curved symmetric spacetimes, generalizing flat-space scattering amplitudes and anti-de Sitter boundary correlators. By formulating kinematics using isometry generators as non-commutative momenta, it establishes a common language for on-shell conditions across geometries and demonstrates that color-kinematics duality emerges from a mapping of the color algebra to the algebra of gauged isometries in biadjoint scalar and nonlinear sigma model theories.

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.

Motivation & Objective

• To unify the description of particle dynamics in curved spacetimes by introducing a perturbatively calculable observable: the on-shell correlator.
• To generalize on-shell kinematics to symmetric manifolds by representing momenta as isometry generators (Killing vectors), which are non-commutative but satisfy identical on-shell conditions as in flat space.
• To demonstrate that color-kinematics duality in biadjoint scalar and nonlinear sigma model theories arises from a mapping of the color algebra to the algebra of gauged isometries on the spacetime manifold.
• To provide a field-theoretic derivation of the fundamental BCJ relations via conserved currents in biadjoint scalar and nonlinear sigma model theories.
• To extend the applicability of on-shell methods beyond flat space, including to anti-de Sitter and de Sitter spacetimes, and to explore implications for higher-point correlators and bootstrap methods.

Proposed method

• Define on-shell correlators as classical solutions to equations of motion that asymptote to superpositions of on-shell wavefunctions, with external propagators amputated and wavefunctions dressed.
• Introduce an isometric frame using Killing vectors $ K_A = K^\mu_A \partial_\mu $, where isometric momenta $ D_A $ are Lie derivatives along these vectors, forming a non-commutative algebra $[D_A, D_B] = F^C_{AB} D_C$.
• Construct on-shell wavefunctions as zero eigenfunctions of the Killing Casimir $ D^2 = D_A D^A $, generalizing flat-space on-shell conditions.
• Express tree-level on-shell correlators as differential operators acting on integrated products of wavefunctions, with propagators $ 1/D^2 $ and vertices as functions of isometric momenta $ D_A $, preserving momentum ordering.
• Derive the color-kinematics duality by mapping the color algebra in biadjoint scalar theory to the algebra of gauged isometries in the spacetime manifold.
• Derive the fundamental BCJ relations from the existence of conserved currents in biadjoint scalar and nonlinear sigma model theories using the isometric formalism.

Experimental results

Research questions

• RQ1 Can a unified framework for on-shell dynamics be constructed in curved symmetric spacetimes that generalizes both flat-space scattering amplitudes and anti-de Sitter boundary correlators?
• RQ2 How can on-shell kinematics be generalized in curved spacetimes using the isometry algebra, and what role does non-commutativity play in this generalization?
• RQ3 Does color-kinematics duality persist in curved spacetime, and if so, how is it realized through a mapping between color algebra and spacetime isometry algebra?
• RQ4 Can the fundamental BCJ relations be derived from field-theoretic currents in biadjoint scalar and nonlinear sigma model theories using the isometric formalism?
• RQ5 To what extent can this formalism be extended to include spin, loop diagrams, off-shell correlators, and de Sitter spacetime?

Key findings

• On-shell correlators in curved symmetric spacetimes reduce to standard on-shell scattering amplitudes in flat space and boundary correlators in anti-de Sitter space by construction.
• Isometric momenta $ D_A $, defined as Lie derivatives along Killing vectors, satisfy the same on-shell conditions as in flat space but are intrinsically non-commutative, with commutator $[D_A, D_B] = F^C_{AB} D_C$.
• The differential operator representation of on-shell correlators preserves the relative ordering of isometric momenta, generalizing flat-space Feynman diagram computations to curved spacetimes.
• Color-kinematics duality in biadjoint scalar and nonlinear sigma model theories is realized via a mapping of the color algebra to the algebra of gauged isometries on the spacetime manifold.
• The fundamental BCJ relations for on-shell correlators are derived from the existence of conserved currents in biadjoint scalar and nonlinear sigma model theories using the isometric formalism.
• The framework is applicable to a broad class of symmetric spacetimes, including de Sitter space, where on-shell correlators correspond to coefficients of the late-time wavefunction of the universe.