  In this work we study the possibility that primordial magnetic fields
observed in galaxies could be produced by a dark matter halo made of charged
ultra-light bosons. In the model, we assume that ultra-light bosons arise as
excitations of a complex scalar field described by the Klein-Gordon equation
with local $U(1)$ symmetry which introduces electromagnetic fields that
minimally couple to the complex scalar current. We use classical solutions of
the Klein-Gordon-Maxwell system to describe the density profile of dark matter
and magnetic fields in the galaxies. We consider two cases assuming spherical
and dipolar spatial symmetries respectively. For the particular case of the LSB
spherical galaxy F563, we test the sensitivity of the predicted rotation curves
in the charged scalar field dark matter (cSFDM) model to variations of the
electromagnetic coupling and, by using the Fisher matrix error estimator, we
set a constraint over that coupling by requiring that theoretical rotation
curves lay inside the $1\sigma$ confidence region of observational data . We
find that cSFDM haloes are able to generate magnetic fields of the order of
$\mu G$ and reproduce the observed rotation curves of F563-V2 at the same time
if the ultra-light boson has a charge lower than $\sim 10^{-13}e$ for the
monopole-like density profile and lower than $10^{-14}e$ for the dipole-like
one.
