### Speaker

### Description

We shall review recent work on when, how and in what sense colour-kinematics duality and the double copy can be realised at the level of off-shell fields and Lagrangians. In the case of pure Yang–Mills theory we argue that colour-kinematics duality can be made manifest in the Batalin–Vilkovisky action, but only up to Jacobian counterterms that must enter at the loop level. The latter implies a departure from what is normally understood by colour-kinematics duality in that the counterterms will generically break it. However, this notion of CK duality is quite natural, as a symmetry of the action itself, and implies the validity of the

double copy. We then describe the generalisation to super Yang–Mills theory, where Sen’s formalism for self-dual field strengths emerges automatically. We conclude by discussing the mathematical underpinnings of these observations in terms of homotopy algebras, where Chern–Simons theory provides the paradigmatic example. Figuratively, colour–kinematics duality is a symmetry of Yang–Mills theory in the same sense that a mug is a donut.