Does a boson star rotate?

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Abstract
We investigate the possibility of slow rotation for a certain type of boson star. It is shown that a boson star made by a complex scalar field alone cannot rotate at least perturbatively. We have performed an extensive numerical study for the boson star made by a complex scalar field coupled with a U(1) gauge field to obtain negative results. It is argued that vortex lines will be generated for rapid rotation of the boson star.