Generalize the colour-algebraic construction beyond an SU(2) subalgebra

Classify constant colour tensors and spacetime matrices that reduce the Yang–Mills and biadjoint-scalar equations to the same scalar equation without restricting the fields to an SU(2) subalgebra, thereby determining whether the three-dimensional Levi–Civita symbol and the O(3) colour matrix are accidental or manifestations of a broader algebraic mechanism.

Background

The correspondence constructed in the paper relies on an SU(2) × SU(2) colour structure, whose three-dimensional Levi–Civita symbols and constant O(3) matrix allow the quartic biadjoint-scalar equation to reduce to the same nonlinear scalar equation as the selected SU(2) Yang–Mills and conformally flat gravitational sectors.

Embedding the construction into an SU(2) subgroup of SU(3) or SU(N) is described as immediate but does not produce genuinely new colour dynamics. A substantive extension would require identifying more general constant colour tensors and spacetime matrices that preserve the common scalar reduction while allowing genuinely non-Abelian SU(3) or SU(N) dynamics.

References

A nontrivial extension would require constant colour tensors and spacetime matrices whose contractions reduce the Yang--Mills and biadjoint equations to the same scalar equation without confining the fields to an $SU(2)$ subalgebra. Classifying such tensors would show whether the role played here by the three-dimensional Levi--Civita symbol and the $O(3)$ matrix $U{aa'}$ is accidental or part of a broader algebraic mechanism.

— Towards a Non-Perturbative Classical Double Copy  (2608.14512 - Armstrong-Williams, 14 Aug 2026) in Section Discussion and Further Work