Scattering amplitudes in YM and GR as minimal model brackets and their recursive characterization

Attached to both Yang-Mills and General Relativity about Minkowski spacetime are distinguished gauge independent objects known as the on-shell tree scattering amplitudes. We reinterpret and rigorously construct them as $L_\infty$ minimal model brackets. This is based on formulating YM and GR as differential graded Lie algebras. Their minimal model brackets are then given by a sum of trivalent (cubic) Feynman tree graphs. The amplitudes are gauge independent when all internal lines are off-shell, not merely up to $L_\infty$ isomorphism, and we include a homological algebra proof of this fact. Using the homological perturbation lemma, we construct homotopies (propagators) that are optimal in bringing out the factorization of the residues of the amplitudes. Using a variant of Hartogs extension for singular varieties, we give a rigorous account of a recursive characterization of the amplitudes via their residues independent of their original definition in terms of Feynman graphs (this does neither involve so-called BCFW shifts nor conditions at infinity under such shifts). Roughly, the amplitude with $N$ legs is the unique section of a sheaf on a variety of $N$ complex momenta whose residues along a finite list of irreducible codimension one subvarieties (prime divisors) factor into amplitudes with less than $N$ legs. The sheaf is a direct sum of rank one sheaves labeled by helicity signs. To emphasize that amplitudes are robust objects, we give a succinct list of properties that suffice for a dgLa so as to produce the YM and GR amplitudes respectively.