Canonical Higgs fields from higher-dimensional gravity

We consider the dimensional reduction of a gravitational field gg in a multidimensional universe endowed with a simple action of a compact Lie group. It is known that when the group preserves gg, this dimensional reduction leads in particular to scalar fields that correspond to an invariant metric on each orbit. We show that the action functionals of those fields (obtained from the reduction of Einstein’s action) exhibit, in the hyperbolic case, polynomials of several variables having a degree less or equal to 6.