On the infinite-dimensional spin-2 symmetries in Kaluza-Klein theories

TL;DR

This paper explores the infinite-dimensional spin-2 symmetries in Kaluza-Klein theories, revealing their structure as broken phases of a massless theory with rich algebraic symmetries, including Kac-Moody and Virasoro algebras.

Contribution

It demonstrates how Kaluza-Klein spin-2 fields form a broken phase of a theory with infinite-dimensional symmetries, interpreted via Chern-Simons and algebra-valued geometry.

Findings

Gravity/spin-2 system in D=3 is a Chern-Simons theory of Kac-Moody algebra.

Global symmetry algebra includes Virasoro and affine Ehlers groups.

Broken phase involves gauging a subgroup, leading to an extended algebra with central extension.

Abstract

We consider the couplings of an infinite number of spin-2 fields to gravity appearing in Kaluza-Klein theories. They are constructed as the broken phase of a massless theory possessing an infinite-dimensional spin-2 symmetry. Focusing on a circle compactification of four-dimensional gravity we show that the resulting gravity/spin-2 system in D=3 has in its unbroken phase an interpretation as a Chern-Simons theory of the Kac-Moody algebra associated to the Poincare group and also fits into the geometrical framework of algebra-valued differential geometry developed by Wald. Assigning all degrees of freedom to scalar fields, the matter couplings in the unbroken phase are determined, and it is shown that their global symmetry algebra contains the Virasoro algebra together with an enhancement of the Ehlers group to its affine extension. The broken phase is then constructed by gauging a subgroup of the global symmetries. It is shown that metric, spin-2 fields and Kaluza-Klein vectors combine into a Chern-Simons theory for an extended algebra, in which the affine Poincare subalgebra acquires a central extension.