Full S -matrices and witten diagrams with relative L ∞ -algebras

The L∞-algebra approach to scattering amplitudes elegantly describes the non-trivial part of the S-matrix but fails to take into account the trivial part. We argue that the trivial contribution to the S-matrix should be accounted for by another, complementary L∞-algebra, such that a perturbative field theory is described by a cyclic relative L∞-algebra. We further demonstrate that this construction reproduces Witten diagrams that arise in AdS/CFT including, in particular, the trivial Witten diagrams corresponding to CFT two-point functions. We also discuss Chern-Simons theory and Yang-Mills theory on manifolds with boundaries using this approach.