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

A bstract 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.