Gauged diffeomorphisms and hidden symmetries in Kaluza-Klein theories

TL;DR

This paper explores the symmetries of massive Kaluza-Klein modes in higher-dimensional theories, revealing an infinite-dimensional extension of diffeomorphisms and reformulating gravity using gauge theories and sigma models.

Contribution

It constructs the unbroken phase with massless modes, identifies global and local symmetries, and shows how gauging certain subgroups leads to a Chern-Simons formulation of gravity.

Findings

Infinite-dimensional extension of 3D diffeomorphisms.

Global SL(D-2,R) current algebra.

Gravity reformulated via Chern-Simons theory.

Abstract

We analyze the symmetries that are realized on the massive Kaluza-Klein modes in generic D-dimensional backgrounds with three non-compact directions. For this we construct the unbroken phase given by the decompactification limit, in which the higher Kaluza-Klein modes are massless. The latter admits an infinite-dimensional extension of the three-dimensional diffeomorphism group as local symmetry and, moreover, a current algebra associated to SL(D-2,R) together with the diffeomorphism algebra of the internal manifold as global symmetries. It is shown that the `broken phase' can be reconstructed by gauging a certain subgroup of the global symmetries. This deforms the three-dimensional diffeomorphisms to a gauged version, and it is shown that they can be governed by a Chern-Simons theory, which unifies the spin-2 modes with the Kaluza-Klein vectors. This provides a reformulation of D-dimensional Einstein gravity, in which the physical degrees of freedom are described by the scalars of a gauged non-linear sigma model based on SL(D-2,R)/SO(D-2), while the metric appears in a purely topological Chern-Simons form.