We analyze theories with color-kinematics duality from an algebraic perspective and find that any such theory has an underlying ${\mathrm{BV}}^{\blacksquare }$-algebra, extending the ideas of Reiterer [A homotopy BV algebra for Yang–Mills and color–kinematics, arXiv:1912.03110.]. Conversely, we show that any theory with a ${\mathrm{BV}}^{\blacksquare }$-algebra features a kinematic Lie algebra that controls interaction vertices, both on shell and off shell. We explain that the archetypal example of a theory with a ${\mathrm{BV}}^{\blacksquare }$-algebra is Chern-Simons theory, for which the resulting kinematic Lie algebra is isomorphic to the Schouten-Nijenhuis algebra on multivector fields. The ${\mathrm{BV}}^{\blacksquare }$-algebra implies the known color-kinematics duality of Chern-Simons theory. Similarly, we show that holomorphic and Cauchy-Riemann Chern-Simons theories come with ${\mathrm{BV}}^{\blacksquare }$-algebras and that, on the appropriate twistor spaces, these theories organize and identify kinematic Lie algebras for self-dual and full Yang-Mills theories, as well as the currents of any field theory with a twistorial description. We show that this result extends to the loop level under certain assumptions.

• Received 22 December 2022
• Accepted 7 June 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.041603

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Published by the American Physical Society