Kaluza-Klein monopole with scalar hair

We construct a new family of rotating black holes with scalar hair and a regular horizon of spherical topology, within five dimensional ($d=5$) Einstein's gravity minimally coupled to a complex, massive scalar field doublet. These solutions represent generalizations of the Kaluza-Klein monopole found by Gross, Perry and Sorkin, with a twisted $S^1$ bundle over a four dimensional Minkowski spacetime being approached in the far field. The black holes are described by their mass, angular momentum, tension and a conserved Noether charge measuring the hairiness of the configurations. They are supported by rotation and have no static limit, while for vanishing horizon size, they reduce to boson stars. When performing a Kaluza-Klein reduction, the $d=5$ solutions yield a family of $d=4$ spherically symmetric dyonic black holes with gauged scalar hair. This provides a link between two seemingly unrelated mechanisms to endow a black hole with scalar hair: the $d=5$ synchronization condition between the scalar field frequency and the event horizon angular velocity results in the $d=4$ resonance condition between the scalar field frequency and the electrostatic chemical potential.