Homotopy double copy of noncommutative gauge theories

We discuss the double copy formulation of Moyal-Weyl type noncommutative gauge theories from the homotopy algebraic perspective of factorisations of $L_\infty$-algebras. We define new noncommutative scalar field theories with rigid colour symmetries taking the role of the zeroth copy, where the deformed colour algebra plays the role of a kinematic algebra; some of these theories have a trivial classical limit but exhibit colour-kinematics duality, from which we construct the double copy theory explicitly. We show that noncommutative gauge theories exhibit a twisted form of colour-kinematics duality, which we use to show that their double copies match with the commutative case. We illustrate this explicitly for Chern-Simons theory, and also for Yang-Mills theory where we obtain a modified Kawai-Lewellen-Tye relation whose momentum kernel is linked to a binoncommutative biadjoint scalar theory. We reinterpret rank one noncommutative gauge theories as double copy theories, and discuss how our findings tie in with recent discussions of Moyal-Weyl deformations of self-dual Yang-Mills theory and gravity.