A variational principle for Kaluza-Klein type theories

TL;DR

This paper develops a variational principle for theories resembling Kaluza-Klein models, showing that solutions induce a geometric structure on a lower-dimensional manifold satisfying Einstein-Yang-Mills equations.

Contribution

It introduces a new variational framework for Kaluza-Klein type theories that naturally leads to Einstein-Yang--Mills solutions via symmetry breaking.

Findings

Solutions induce a principal bundle structure on the manifold.

The induced metric and connection solve Einstein-Yang--Mills equations.

The framework applies to any positive integer n and Lie group G.

Abstract

For any positive integer $n$ and any Lie group $\mathfrak{G}$, given a definite symmetric bilinear form on $\mathbb{R}^n$ and an $\hbox{Ad}$-invariant scalar product on the Lie algebra of $\mathfrak{G}$, we construct a variational problem on fields defined on an arbitrary oriented $(n+\hbox{dim}\mathfrak{G})$-dimensional manifold $\mathcal{Y}$. We show that, if $\mathfrak{G}$ is compact and simply connected, any global solution of the Euler--Lagrange equations leads, through a spontaneous symmetry breaking, to identify $\mathcal{Y}$ with the total space of a principal bundle over an $n$-dimensional manifold $\mathcal{X}$. Moreover $\mathcal{X}$ is then endowed with a (pseudo-)Riemannian metric and a connection which are solutions of the Einstein--Yang--Mills system of equations with a cosmological constant.