Phase diagrams of the boson stars and shells of the $U(1)$ gauged $\mathbb{C}{P}^N$ nonlinear sigma model are studied. The solutions of the model exhibit both the ball- and the shell-shaped charge density depending on $N$. There appear four independent regions of the solutions which are essentially caused from the coexistence of electromagnetism and gravity. We examine several phase diagrams of the boson stars and the shells and discuss what and how the regions are emerged. A coupling with gravity allows for harboring of the charged black holes for the $Q$-shell solutions. Some solutions are strongly affected by the presence of the black holes and they allow to be smoothly connected. As a result, the regions are integrated by the harboring black holes.

中文翻译：

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在 CPN 非线性 sigma 模型中对包含黑洞的致密带电玻色子星和壳进行相位分析

玻色子星和壳的相图 $你(1)$ 测量的 $\mathbb{C}{磷}^N$非线性sigma模型进行了研究。该模型的解同时显示出球状和壳状电荷密度，具体取决于$N$. 出现了四个独立的解区域，它们本质上是由电磁和重力共存引起的。我们检查了玻色子星和壳的几个相图，并讨论了这些区域是什么以及如何出现的。与重力的耦合允许为$问$-shell 解决方案。一些解受到黑洞存在的强烈影响，它们允许平滑连接。结果，这些区域被窝藏的黑洞整合在一起。