Vertex operators for the kinematic algebra of Yang-Mills theory

The kinematic algebra of Yang-Mills theory can be understood in the framework of homotopy algebras: the $L_{\infty}$ algebra of Yang-Mills theory is the tensor product of the color Lie algebra and a kinematic space that carries a $C_{\infty}$ algebra. There are also hidden structures that generalize Batalin-Vilkovisky algebras, which explain color-kinematics duality and the double copy but are only partially understood. We show that there is a representation of the $C_{\infty}$ algebra, in terms of vertex operators, on the Hilbert space of a first-quantized worldline theory. To this end we introduce $A_{\infty}$ morphisms, which define the vertex operators and which inject the $C_{\infty}$ algebra into the strictly associative algebra of operators on the Hilbert space. We also take first steps to represent the hidden structures on the same space.