Advances in Theoretical and Mathematical Physics

Volume 24 (2020)

Number 2

A variational principle for Kaluza–Klein types theories

Pages: 305 – 326



Frédéric Hélein (UFR de Mathématiques, Université de Paris, France)


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