  We study Mott phases and superfluid-insulator (SI) transitions of ultracold
bosonic atoms in a two-dimensional square optical lattice at commensurate
filling and in the presence of a synthetic periodic vector potential
characterized by a strength $p$ and a period $l=qa$, where $q$ is an integer
and $a$ is the lattice spacing. We show that the Schr\"odinger equation for the
non-interacting bosons in the presence of such a periodic vector potential can
be reduced to an one-dimensional Harper-like equation which yields $q$ energy
bands. The lowest of these bands have either single or double minima whose
position within the magnetic Brillouin zone can be tuned by varying $p$ for a
given $q$. Using these energies and a strong-coupling expansion technique, we
compute the phase diagram of these bosons in the presence of a deep optical
lattice. We chart out the $p$ and $q$ dependence of the momentum distribution
of the bosons in the Mott phases near the SI transitions and demonstrate that
the bosons exhibit several re-entrant field-induced SI transitions for any
fixed period $q$. We also predict that the superfluid density of the resultant
superfluid state near such a SI transition has a periodicity $q$ ($q/2$) in
real space for odd (even) $q$ and suggest experiments to test our theory.
