Generalized color relations for symmetric color structures

Establish whether analogues of the generalized Jacobi relations for totally antisymmetric n-index structure constants exist for symmetric color objects such as d^{abc} and the generators T^a_{ij}.

Background

The paper generalizes the Jacobi identity to totally antisymmetric structure constants with arbitrarily many indices and constructs several color-dual theories based on those relations. Its appendix also exhibits a two-dimensional Zakharov–Mikhailov extension involving the symmetric tensor d{abc}.

This motivates the unresolved question of whether comparable algebraic relations can be formulated for other color constituents, including symmetric d-tensors and fundamental generators, potentially enlarging the space of color–kinematics-dual theories.

References

One natural question regarding the color relations in eq:colorrel is whether there are any analogous generalizations for $d{abc}$ or $T_{ij}a$.

— Color Relations and Off-Shell Double-Copy for Towers of Theories  (2609.36604 - Mangan et al., 29 Sep 2026) in Section 4, Conclusions and future directions

This appendix has focused on off-shell color-dual theories that include a three-index symmetric color object, but it seems plausible that there might be an $n$-index generalization that obeys relations like eq:colorrel. We leave this to future work.

— Color Relations and Off-Shell Double-Copy for Towers of Theories  (2609.36604 - Mangan et al., 29 Sep 2026) in Appendix C, Color-dual Zakharov-Mikhailov theory with d^{abc}