The dimensional reduction of linearized spin-2 theories invariant under transverse diffeomorphisms

TL;DR

This paper explores the dimensional reduction of linearized spin-2 theories invariant under transverse diffeomorphisms, analyzing their gauge invariances, unitarity, and behavior on curved backgrounds, revealing connections to Diff symmetry and massive scalar tensor models.

Contribution

It provides a detailed analysis of the Kaluza-Klein reduction of TDiff-invariant theories, showing how gauge invariances and unitarity are preserved and connecting massless and massive models.

Findings

Unitarity is preserved in all reduced models.

A smooth massless limit exists for the models.

Theories admit extensions to curved backgrounds with Einstein spaces.

Abstract

Here we perform the Kaluza-Klein dimensional reduction from $D+1$ to $D$ dimensions of massless Lagrangians described by a symmetric rank-2 tensor and invariant under transverse differmorphisms (TDiff). They include the linearized Einstein-Hilbert theory, linearized unimodular gravity and scalar tensor models. We obtain simple expressions in terms of gauge invariant field combinations and show that unitarity is preserved in all cases. After fixing a gauge, the reduced model becomes a massive scalar tensor theory. We show that the diffeomorphism (Diff) symmetry, instead of TDiff, is a general feature of the massless sector of consistent massive scalar tensor models. We discuss some subtleties when eliminating St\"uckelberg fields directly at action level as gauge conditions. We also show that the reduced models all have a smooth massless limit. A non local connection between the massless sector of the scalar tensor theory and the pure tensor TDiff model leads to a parametrization of the non conserved source which naturally separates spin-0 and spin-2 contributions in the pure tensor theory. The case of curved backgrounds is also investigated. If we truncate the non minimal couplings to linear terms in the curvature, vector and scalar constraints require Einstein spaces as in the Diff and WTDiff (Weyl plus Diff) cases. We prove that our linearized massive scalar tensor models admit those curved background extensions.