  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.
