Black holes and boson stars with one Killing field in arbitrary odd dimensions
We extend the recent $D=5$ results of Dias, Horowitz, and Santos by finding asymptotically anti\char21{}de Sitter rotating black hole and boson star solutions with scalar hair in arbitrary odd spacetime dimension. Both the black holes and the boson stars are invariant under a single Killing vector field which corotates with the scalar field and, in the black hole case, is tangent to the generator of the horizon. Furthermore, we explicitly construct boson star and small black hole (${r}_{+}\ensuremath{\ll}\ensuremath{\ell}$) solutions perturbatively assuming a small amplitude for the scalar field, resulting in solutions valid for low energies and angular momenta. We find that just as in $D=5$, the angular momentum is primarily carried by the scalar field in $Dg5$, whereas unlike $D=5$ the energy is also primarily carried by the scalar field in $Dg5$; the thermodynamics in $D=5$ are governed by both the black hole and scalar field whereas in $Dg5$ they are governed primarily by the scalar field alone. We focus on cataloging these solutions for the spacetime dimensions of interest in string theory, namely $D=5,7,9,11$.