Photonic black resonators and photon stars in AdS$_5$

Rapidly rotating Myers-Perry-AdS$_5$ (MPAdS$_5$) black holes are shown to be unstable against rotational superradiance of a Maxwell field. From the onset of the instability, time-periodic neutral black hole solutions equipped with a nontrivial electromagnetic wave are obtained, which we call {\it photonic black resonators}. In the horizonless limit, they reduce to geon solutions which may be called {\it photon stars}. Specifically, we introduce a cohomogeneity-1 ansatz for the metric and Maxwell field and construct such solutions with an $R\times SU(2)$ isometry group. We compute thermodynamic quantities and obtain phase diagrams. It turns out that a photonic black resonator has a higher entropy than a MPAdS$_5$ black hole, while it also has a smaller entropy than a black resonator without the Maxwell field. This suggests what is expected for nonlinear dynamics following the Maxwell superradiant instability with the $SU(2)$ isometry.