Homotopy transfer for massive Kaluza-Klein modes

We develop techniques to treat massive Kaluza-Klein modes to arbitrary order in perturbation theory. The Higgs mechanism that renders the higher Kaluza-Klein modes massive is displayed. To this end we give an algorithm in perturbation theory that yields new fields with the following characteristics: they are gauge invariant under all higher-mode gauge transformations, which are broken, but they transform covariantly under the zero-mode gauge transformations, which are unbroken. We employ the formulation of field theory in terms of $L_{\infty}$ algebras together with their homotopy transfer, which here maps the gauge redundant fields of gravity to gauge invariant fields. We illustrate these results, as a proof of concept, for Kaluza-Klein theory on a torus. In an accompanying paper these results will be applied to a large class of generalized Scherk-Schwarz backgrounds in exceptional field theory.