Homotopy transfer and effective field theory I: tree-level

Abstract

We use the dictionary between general field theories and strongly homotopyalgebras to provide an algebraic formulation of the procedure of integratingout of degrees of freedom in terms of homotopy transfer. This includes moregeneral effective theories in which some massive modes are kept while othermodes of a comparable mass scale are integrated out, as first explored by Senin the context of closed string field theory. We treatL∞-algebras both in termsof a nilpotent coderivation and, on the dual space, in terms of a nilpotentderivation (corresponding to the BRST charge of the field theory) and provideexplicit formulas for homotopy transfer. These are then shown to govern theintegrating out of degrees of freedom at tree level, while the generalization toloop level will be explored in a sequel to this paper.