# Gravitating compact $Q$-ball and $Q$-shell solutions in the $\mathbb{C}P^N$ nonlinear sigma model

## TL;DR

This paper explores gravitating $Q$-ball and $Q$-shell solutions within the $ ext{CP}^N$ sigma model, revealing how gravity influences their properties, including the formation of black holes and bounds on charge.

## Contribution

It introduces compact gravitating $Q$-ball and $Q$-shell solutions in the $ ext{CP}^N$ model with gravity, highlighting the impact of gravitational coupling on solution bounds and black hole formation.

## Key findings

- Energy scales as $|Q|^{5/6}$ for solutions.
- An upper bound on $|Q|$ appears with stronger gravity coupling.
- Maximal $|Q|$ decreases as gravitational coupling increases.

## Abstract

We study compact gravitating $Q$-ball, $Q$-shell solutions in a sigma model with the target space $\mathbb{C}P^N$. Models with odd integer $N$ and suitable potential can be parameterized by $N$-th complex scalar fields and they support compact solutions. A coupling with gravity allows for harboring of the Schwarzschild black holes for the $Q$-shell solutions. The energy of the solutions behaves as $E\sim |Q|^{5/6}$, where $Q$ stands for the $U(1)$ Noether charge, for both the gravitating and the black hole solutions.Notable difference from the solutions of the flat space is that upper bound of $|Q|$ appears when the coupling with gravity is stronger. The maximal value of $|Q|$ quickly reduces for larger coupling constant. It may give us a useful hint of how a star forms its shape with a certain finite number of particles.