Homotopy transfer for massive Kaluza-Klein modes

TL;DR

This work develops a perturbative, algebraic framework for handling massive Kaluza–Klein modes to arbitrary order by employing $L_{\\infty}$ algebras and homotopy transfer. It explicitly constructs gauge-invariant KK fields that remain invariant under non-zero-mode diffeomorphisms while transforming covariantly under zero-mode diffeomorphisms, thereby realizing a Higgs-like mechanism that renders higher KK modes massive. A torus compactification on $T^d$ serves as a concrete proof of concept, with a systematic procedure to compute the KK spectrum via the transported $L_{\\infty}$ structure and its cohomology, ensuring the correct degrees of freedom and masses. The results establish that the non-linear theory can be equivalently formulated in terms of gauge-invariant fields, via a transported action $S_{\\rm KK}[\\widehat{\\Phi}]$, and set the stage for applying these methods to generalized Scherk–Schwarz backgrounds in exceptional field theory and AdS/CFT contexts. Overall, the work provides a robust, gauge-consistent reorganization of higher-dimensional gravity KK towers that can underpin future holographic and EFT analyses.

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.

Figures (2)

• Figure 1: Homotopy transfer from $X$ to $\mathring{X}$.
• Figure 2: Proca chain complex