Title: On colour-kinematics duality and the double copy

Abstract: We begin by reviewing the colour-kinematics duality of (super) Yang-Mills theory and its double copy into (super)gravity. We then show that off-shell colour-kinematics duality can be made manifest in the Yang—Mills Batalin—Vilkovisky action, up to Jacobian counterterms. The latter implies a departure from what is normally understood by colour-kinematics duality in that the counterterms generically break it. However, this notion of CK duality is very natural and, most importantly, implies the validity of the double copy to all orders in perturbations theory. Perturbatively, at least, gravity is the square of Yang—Mills! We then describe generalisations to the non-linear sigma model and 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. Figuratively, colour-kinematics duality is a symmetry of Yang—Mills in the same sense that a mug is a donut.

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