Compact, charged boson-stars, -shells in the $\mathbb{C}P^N$ gravitating nonlinear sigma model

We study $U(1)$ gauged gravitating compact $Q$-ball, $Q$-shell solutions in a nonlinear sigma model with the target space $\mathbb{C}P^N$. The models with odd integer $N$ and a special potential can be parameterized by $N$-th complex scalar fields and they support compact solutions. Implementing the $U(1)$ gauge field in the model, the behavior of the solutions become complicated than the global model. Especially, they exhibit branch, i.e., two independent solutions with same shooting parameter. The energy of the solutions in the first branch behaves as $E\sim Q^{5/6}$ for small $Q$, where $Q$ stands for the $U(1)$ Noether charge. For the large $Q$, it gradually deviates from the scaling $E\sim Q^{5/6}$ and, for the $Q$-shells it is $E\sim Q^{7/6}$, which forms the second branch. A coupling with gravity allows for harboring of the Schwarzschild black holes for the $Q$-shell solutions, forming the charged boson shells. The space-time then consist of a charged black hole in the interior of the shell, surrounded by a $Q$-shell, and the outside becomes a Reissner-Nordstr\"om space-time. These solutions inherit the scaling behavior of the flat space-time.