Yang-Mills theory from the worldline

We construct off-shell vertex operators for the bosonic spinning particle. Using the language of homotopy algebras, we show that the full nonlinear structure of Yang-Mills theory, including its gauge transformations, is encoded in the commutator algebra of the worldline vertex operators. To do so, we deform the worldline BRST operator by coupling it to a background gauge field and show that the coupling is consistent on a suitable truncation of the Hilbert space. On this subspace, the square of the BRST operator is proportional to the Yang-Mills field equations, which we interpret as an operator Maurer-Cartan equation for the background. This allows us to define further vertex operators in different ghost numbers, which correspond to the entire $L_\infty$ algebra of Yang-Mills theory. Besides providing a precise map of a fully nonlinear field theory into a worldline model, we expect these results will be valuable to investigate the kinematic algebra of Yang-Mills, which is central to the double copy program.