  We systematically investigate the ground state and elementary excitations of
a Bose-Einstein Condensate with a synthetic vector potential, which is induced
by the many-body effects and atom-light coupling. For a sufficiently strong
inter-atom interaction, we find the condensate undergoes a Stoner-type
ferromagnetic transition through the self-consistent coupling with the vector
potential. For a weak interaction, the critical velocity of a supercurrent is
found anisotropic due to the density fluctuations affecting the gauge field. We
further analytically demonstrate the topological ground state with a coreless
vortex ring in a 3D harmonic trap and a coreless vortex-antivortex pair in a 2D
trap. The circulating persistent current is measurable in the time-of-flight
experiment or in the dipolar oscillation through the violation of Kohn theorem.
