Double-Copy-Like Decomposition Explained

• Double-copy-like decomposition is a framework that maps physical and mathematical structures between gauge and gravity theories via exact, perturbative, or algebraic correspondences.
• It unifies diverse constructs—from Kerr–Schild metrics and color-kinematics duality to topological observables and paradoxical decompositions—across various domains.
• The methodology extends its impact to curved spacetimes, higher-spin models, and even measure theory, offering practical insights for both theoretical developments and computational applications.

Double-copy-like decomposition refers to a hierarchy of structural correspondences—exact, perturbative, or algebraic—by which one can decompose physical or mathematical objects in a manner analogous to, or inspired by, the double copy relations between gauge theory and gravity. At its core, this principle manifests in a broad range of contexts: classical solutions, quantum amplitudes, Lagrangian actions, topological and geometric data, and even solution spaces of PDEs. Central frameworks include the Kerr–Schild/Weyl classical double copy, color-kinematics factorization at amplitude and BRST levels, convolutional constructions on backgrounds, and their topological and higher-spin generalizations.

1. Foundational Structures: Kerr–Schild, Weyl, and Classical Double Copy

The canonical setting for double-copy-like decomposition is the classical solution space of general relativity and gauge theory, most notably via the Kerr–Schild ansatz. Metrics of the form

$$g_{\mu\nu} = \eta_{\mu\nu} + \phi(x)\,k_\mu(x)\,k_\nu(x)$$

with $k_\mu$ null and geodesic, linearize the Einstein field equations and admit a direct "single copy" to the gauge field $A_\mu = \phi(x)\,k_\mu(x)$, which solves linearized Yang–Mills or Maxwell equations. Stripping off $k_\mu$ in turn yields the "zeroth copy" biadjoint scalar $\phi(x)$, satisfying a harmonic or conformally coupled equation (Luna et al., 2016, Easson et al., 2023, Didenko et al., 2022).

The Weyl double copy generalizes this structure: the (anti-)self-dual Weyl curvature spinor or tensor can be written as the symmetric product of gauge curvature tensors modulated by a scalar, schematically

$C_{ABCD} = \frac{1}{S}\,f_{(AB} f_{CD)}$

where $S$ is the zeroth copy and $f_{AB}$ is the field strength (Alawadhi, 2023, Godazgar et al., 2021).

This structure extends to double Kerr–Schild metrics (e.g., Taub–NUT), which decompose as two such products and whose single copy corresponds to nontrivial dyonic solutions in gauge theory (Luna et al., 2015, Bahjat-Abbas et al., 2020). Non-abelian monopoles and global topological data (e.g., patching of Dirac strings) are also faithfully transported between gauge and gravity sectors by a double-copy-like map at the level of Wilson lines and holonomies (Alfonsi et al., 2020).

2. Color-Kinematics Duality and Algebraic Decomposition

One of the most far-reaching realizations of double-copy-like decomposition is the color-kinematics duality of scattering amplitudes. At tree and loop level, gauge-theory amplitudes $A_n$ admit representations

$A_n = \sum_{i} \frac{c_i n_i}{D_i}$

with color factors $c_i$, kinematic numerators $n_i$, and propagator structure $D_i$. The double copy replaces $c_i$ by a second copy of $n_i$ (or a numerically related set) to produce gravity amplitudes

$$\mathcal{M}_n = \sum_{i} \frac{n_i \tilde{n}_i}{D_i}$$

subject to the requirement that $n_i$ satisfy the same Jacobi-type algebra as $c_i$ (Luna et al., 2016, Ilderton et al., 2024). At the Lagrangian level, this is mirrored in prescriptions for "squaring" gauge-invariant structures to obtain the quadratic and cubic actions of double field theory (Diaz-Jaramillo et al., 2021), and is elevated to a homotopy-algebraic identity at the level of the BRST/BV complex (Borsten et al., 2021). In particular, L∞ and A∞ homotopy structures for Yang–Mills can be factorized and squared, directly yielding the corresponding objects for gravity or non-linear sigma models.

In 2D, these ideas are realized off-shell and non-perturbatively: biadjoint scalars (BAS) double-copy via Zakharov–Mikhailov (ZM) replacements to special Galileons (SG), at both Lagrangian and amplitudes levels, including integrable Lax data and infinite towers of conserved currents (Cheung et al., 2022).

3. Global and Topological Double-Copy-Like Correspondence

The double-copy paradigm extends to topological data and non-local observables. The mapping between non-abelian monopole gauge bundles (transition functions classified by $\pi_1(G)$, patching matrices, Wilson loops) and the discrete NUT charges in gravity encodes a double-copy-like relation for global structures (Alfonsi et al., 2020). At the operator level, the Wilson line of gauge theory is mapped (via a color-to-kinematics prescription) to a gravitational Wilson line, with memory and asymptotic charges following suit (Alawadhi, 2023, Godazgar et al., 2021).

Boundary data, such as BMS superrotations at null infinity, correspond to large gauge transformations on the gauge side; this is manifest in the asymptotic Weyl double copy, where gravitational radiation and memory map to electromagnetic soft modes and charges (Godazgar et al., 2021).

4. Extensions to Curved Backgrounds, Higher Spins, and Cosmology

Double-copy-like decompositions are now established in arbitrary backgrounds, higher-spin theories, and cosmological contexts:

• Curved backgrounds: Classical and convolutional double copy constructions have been shown to work for axio-dilatonic gravity in general backgrounds. The convolution kernel (biadjoint scalar propagator) mediates the pairing between background gauge fields and gravitational metrics, with explicit all-order results in symmetric (FRW) cosmologies (Ilderton et al., 2024). The prescription is recursion-based, fixing higher-order gravitational perturbations by quadratic maps from the gauge sector.
• Higher Spins: The generalized Kerr–Schild ansatz applies to arbitrary symmetric tensors in (A)dS backgrounds. Powers of null geodesic vectors multiplied by a common scalar profile yield solutions to Fronsdal equations, with "double-copy-like" decomposition tying together scalars, Maxwell fields, gravitons, and all higher spins into a hierarchy determined by background curvature (Didenko et al., 2022).
• Cosmological correlators: In de Sitter (inflationary) backgrounds, tree-level n-point functions factorize into a mode-function (time-dependent) part and a tensorial (polarization-momentum) part, where the latter is always a double copy (product or KLT sum) of gauge-theory building blocks (Fazio, 2019).

5. Generalizations: Homotopy Algebras, Massive Theories, and Low Dimensions

Mathematically, double-copy-like decompositions admit rigorous formulation in the language of homotopy algebras. The gauge BRST/BV complex, often organized as an L∞ algebra, can be strictified (by auxiliary fields) so that all interactions are cubic and the entire structure (including ghosts and anti-fields) factorizes. Gravity—and related theories such as special Galileon, DBI, conformal gravity—then arises as a "square" in this algebraic sense (Borsten et al., 2021).

In lower dimensions:

• The 3D Cotton double copy, involving topologically massive theories, marks the dimensional reduction and algebraic reinterpretation of the Weyl double copy, preserving the essential quadratic structure of curvature spinors in terms of gauge-theory field strengths (Emond et al., 2022).
• Anyon solutions in 2 + 1 dimensions double copy to gravitational anyons, capturing Aharonov–Bohm phases and topological mixing with dilatons (Burger et al., 2021).

6. Paradoxical Decompositions: Mathematical Analogues

Distinctly, double-copy-like decomposition also appears outside physical field theory as a paradigm for constructing paradoxical decompositions in measure theory. The decomposition of the real line into $2k$ (or countably many) pieces, which reassemble via measure-preserving (piecewise rigid) bijections into multiple copies of $\mathbb{R}$, furnishes a group-theoretic and combinatorial double copy (Kandola et al., 2015). The underlying structure is the free group action and partitioning of group elements, mirroring the multiplicities familiar in BCJ-like constructions.

Context/Domain	Double-Copy-Like Structure	Defining Features or Mechanisms
Classical field theory	Kerr–Schild, Weyl double copy	Null congruence, harmonic scalar
Amplitudes (QFT)	Color–kinematics duality (BCJ, KLT)	Algebraic replacement $c_i\to n_i$
Topology/global geometry	Wilson lines, holonomies	Patching, winding, $\pi_1$, A–B phase
Lagrangian algebra	Homotopy algebra squaring	L∞ factorization, strictification
PDEs/integrable models	Non-perturbative, two-step replacements	Fourier mode expansions, Moyal algebra
Measure theory	Paradoxical decomposition (Banach–Tarski)	Free group action, reassembly maps

7. Outlook and Open Questions

While double-copy-like decomposition is firmly established in classical, quantum, and even purely algebraic contexts, there remain significant open questions regarding uniqueness (non-abelian and abelian gauge fields mapping to the same gravity solutions), full extension to arbitrary backgrounds (robustness of color-kinematics duality), and the scope of convolutional constructions at loop level and for all possible UV completions (Ilderton et al., 2024, Borsten et al., 2021). The emergence of multicopy hierarchies in higher-spin/AdS theories (Didenko et al., 2022), the status of the double copy under dualities and large (topological) transformations, and the mathematical formalization within homotopy or category theory are active directions.

The breadth and depth of double-copy-like decomposition continue to illuminate profound unity across gauge, gravity, and more abstract mathematical structures. Its further development is anticipated to inform both fundamental theory and applied calculations in high-energy, mathematical, and mathematical-physics research.