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arXiv · 1809.03375

A variational principle for Kaluza-Klein type theories

Frédéric Hélein·Frédéric FrÂ\'
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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.

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Frédéric Hélein, Frédéric FrÂ\'. 2018-09-10. A variational principle for Kaluza-Klein type theories. https://arxiv.org/abs/1809.03375

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