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# Advances in Theoretical and Mathematical Physics

## Volume 24 (2020)

### Number 2

### A variational principle for Kaluza–Klein types theories

Pages: 305 – 326

DOI: https://dx.doi.org/10.4310/ATMP.2020.v24.n2.a3

#### Author

#### Abstract

For any positive integer $n$ and any Lie group $\mathfrak{G}$, given a definite symmetric bilinear form on $\mathbb{R}^n$ and an Ad‑invariant scalar product on the Lie algebra of $\mathfrak{G}$, we construct a variational problem on fields defined on an arbitrary oriented $(n + \operatorname{dim} \mathfrak{G})$-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.

Published 10 July 2020