3D gravity and double copy theory

Abstract: We introduce a novel reformulation of three-dimensional gravity in terms of divergenceless vector frames, inspired by the double copy for Chern-Simons theory. This formulation is on-shell equivalent to conventional 3D gravity and provides a transparent geometric interpretation of the double-copy construction. We relate the resulting theory to a Chern-Simons-like action, propose a higher-dimensional origin, and explore extensions that give rise to $AdS_3$

• The paper introduces an on-shell reformulation of 3D gravity using divergenceless vector frames derived from a double copy of Chern-Simons gauge theory.
• It derives corresponding actions and extended gauge structures, including non-local deformations that accommodate both flat and AdS3 geometries.
• It establishes a six-dimensional perspective via Poisson geometry reduction, offering insights into gauge-gravity dualities and potential holographic applications.

Reformulating 3D Gravity via Double Copy Theory

Overview

This work provides a new on-shell reformulation of three-dimensional (3D) gravity in terms of divergenceless vector frames, with the construction inspired by the double copy of Chern-Simons (CS) gauge theory. The formulation maintains complete equivalence with conventional 3D gravity while offering a manifest geometric realization of the double copy framework. The analysis establishes a direct connection between 3D gravity and Chern-Simons-like actions, explores a higher-dimensional (6D) perspective that elucidates the structure and symmetries of the theory, and introduces non-local extensions that accommodate $AdS_3$ geometries.

Reformulation of 3D Gravity

The starting point is the well-known feature that 3D gravity, formulated via the Einstein-Hilbert or first-order (vielbein plus spin connection) action, has no local propagating degrees of freedom, and is classically equivalent to a Chern-Simons gauge theory. The standard first-order action, upon imposition of zero torsion and curvature, reduces the field equations to the flatness of the spin connection and the nilpotency of the exterior derivative acting on the vielbein.

By passing to vector frame variables—fields $E_a$ that are inverse to the vielbein one-forms and subject to a divergence-free constraint with respect to a prescribed volume form—the equations of motion are recast as the simultaneous vanishing of Lie brackets between frames and the divergencelessness constraint. The physical content is preserved thanks to the global structure (constant frame rotations on overlaps), which matches the information retained in the original vielbein description.

The action constructed for these frame variables,

$$S_{DC}(E) = \frac{1}{6}\int \rho^2 \epsilon^{abc} \epsilon_{\mu\nu\rho} E^\mu_a E^\nu_b E^\rho_c \, d^3x,$$

is subject to $\partial_\mu (\rho E^\mu_a)=0$ and $\det E \neq 0$. Its variation yields precisely the required algebraic (commuting frame) equations, identifying it as a dynamical on-shell equivalent to conventional 3D gravity, but re-expressed in double copy-motivated variables.

Double Copy Action and Gauge Structure

Expanding around the trivial frame ($E_a^\mu = \delta_a^\mu + A_a^\mu$), the action yields, to quadratic and cubic order, a functional of divergenceless vector perturbations $A$,

$$S_{DC}[A]=\int \left[ \frac{1}{2} A_a^\mu \frac{\partial_b \partial^\nu}{\Box} A_c^\rho + \frac{1}{6}A_a^\mu A_b^\nu A_c^\rho \right] \epsilon^{abc} \epsilon_{\mu\nu\rho} d^3x,$$

with $\partial_\mu A^\mu_a = 0$. This is directly identified with the action found in the double copy construction for 3D Chern-Simons theory, exhibiting non-local quadratic terms that reproduce the gauge-invariant content.

The double copy field content is generalized to include, beyond the frame $E_a^\mu$, a symmetric traceless tensor $E_a$0 and an antisymmetric $E_a$1. These satisfy analogous divergenceless constraints and transform under an extended gauge symmetry parametrized by volume-preserving vector fields and a $E_a$2 parameter, encapsulated in the explicit transformation rules. The cubic vertex structure and gauge algebra align closely with those encountered in deformation theory and the BV formalism applied to CS-like models.

Six-Dimensional Origin and Poisson Geometry

A salient feature of the formulation is its interpretation as a dimensional reduction from a six-dimensional theory of divergenceless bivector fields with a unimodular Poisson structure. Specifically, the 6D action,

$E_a$3

evaluated on a mixed-geometry ansatz ($E_a$4; the bivector $E_a$5 decomposed into components built from $E_a$6, $E_a$7, $E_a$8), directly reduces to the aforementioned 3D actions. The construction leverages the Cartan calculus for multivector fields and the Schouten-Nijenhuis bracket, providing an algebraic structure that underlies both the on-shell dynamics and gauge symmetries. This places the double copy framework for 3D gravity as a real-analytic analogue of Kodaira-Spencer gravity, with a six-dimensional geometric origin.

Non-Local Extensions and $E_a$9 Geometries

The paper identifies a distinguished non-local quadratic pairing that can be incorporated into the action, parametrized by $$S_{DC}(E) = \frac{1}{6}\int \rho^2 \epsilon^{abc} \epsilon_{\mu\nu\rho} E^\mu_a E^\nu_b E^\rho_c \, d^3x,$$0,

$$S_{DC}(E) = \frac{1}{6}\int \rho^2 \epsilon^{abc} \epsilon_{\mu\nu\rho} E^\mu_a E^\nu_b E^\rho_c \, d^3x,$$1

This term is essential to produce the Maurer-Cartan equations for frames satisfying

$$S_{DC}(E) = \frac{1}{6}\int \rho^2 \epsilon^{abc} \epsilon_{\mu\nu\rho} E^\mu_a E^\nu_b E^\rho_c \, d^3x,$$2

which corresponds to the vector-frame structure of $$S_{DC}(E) = \frac{1}{6}\int \rho^2 \epsilon^{abc} \epsilon_{\mu\nu\rho} E^\mu_a E^\nu_b E^\rho_c \, d^3x,$$3 gravity. Through this non-local deformation, the extended formalism now captures negative cosmological constant solutions, integrating $$S_{DC}(E) = \frac{1}{6}\int \rho^2 \epsilon^{abc} \epsilon_{\mu\nu\rho} E^\mu_a E^\nu_b E^\rho_c \, d^3x,$$4 as a natural vacuum.

Implications and Prospects

The reformulation provides a transparent geometric (frame-based) picture of the double copy mechanism in lower-dimensional gravity, highlighting a precise match with deformations of topological field theories and BV structures. The link to six-dimensional Poisson geometry suggests new avenues for understanding background independence, generalized geometry, and spacetime gauge structures in the context of the double copy.

On a practical level, this approach offers a novel computational framework for analyzing perturbations and symmetries in 3D gravity, potentially facilitating new results in bulk/boundary correspondence, holography, and topological gravity quantization schemes. The 6D reduction hints at generalizations for topological gravities in higher dimensions, possibly pointing toward underlying string or sigma model structures.

Speculatively, the equivalence between the real double copy and sections of Kodaira-Spencer/Poisson sigma models positions the approach as an analytic tool for connecting gauge and gravity integrands at the level of Lagrangian field theory, not just scattering amplitudes. Future directions may include systematization of higher-dimensional analogues, detailed study of the structure of the gauge algebra, and implications for non-topological or dynamical gravities.

Conclusion

The work establishes an explicit, on-shell equivalent reformulation of 3D gravity inspired by double copy theory, recasting gravitational dynamics in terms of divergenceless vector frames and connecting the construction to topological Poisson structures in six dimensions. The formalism recovers conventional gravity (both flat and $$S_{DC}(E) = \frac{1}{6}\int \rho^2 \epsilon^{abc} \epsilon_{\mu\nu\rho} E^\mu_a E^\nu_b E^\rho_c \, d^3x,$$5 backgrounds) and naturally accommodates extended gauge structures. The identification of a higher-dimensional origin and correspondence with double-copy actions strengthens the geometric, algebraic, and physical links between gauge and gravity theories beyond the amplitude level, suggesting further potential for both theoretical development and practical applications in the study of gravitational field structures.