Homological perspective on edge modes in linear Yang–Mills and Chern–Simons theory

We provide an elegant homological construction of the extended phase space for linear Yang–Mills theory on an oriented and time-oriented Lorentzian manifold M with a time-like boundary $$\partial M$$ that was proposed by Donnelly and Freidel (JHEP 1609:102, 2016). This explains and formalizes many of the rather ad hoc constructions for edge modes appearing in the theoretical physics literature. Our construction also applies to linear Chern–Simons theory, in which case we obtain the extended phase space introduced by Geiller (Nucl Phys B 924:312, 2017).