Homotopy transfer for massive Kaluza-Klein modes

TL;DR

This paper introduces a perturbative method using $L_{ abla}$ algebras and homotopy transfer to systematically handle massive Kaluza-Klein modes and gauge invariance in higher-dimensional theories.

Contribution

It develops an algorithm for constructing gauge-invariant fields for massive Kaluza-Klein modes using homotopy transfer in $L_{ abla}$ algebras, applicable to complex backgrounds.

Findings

Provides a gauge-invariant formulation for massive modes.

Demonstrates the method on Kaluza-Klein theory on a torus.

Lays groundwork for applications in exceptional field theory.

Abstract

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.