Full S-matrices and Witten diagrams with (relative) L-infinity algebras

The $L_{\mathrm{\infty }}$-algebra approach to scattering amplitudes elegantly describes the nontrivial 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 $L_{\mathrm{\infty }}$-algebra, such that a perturbative field theory is described by a cyclic relative $L_{\mathrm{\infty }}$-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.