String solutions in Chern-Simons-Higgs model coupled to an axion

We study a $d=2+1$ dimensional Chern-Simons gauge theory coupled to a Higgs scalar and an axion field, finding the form of the potential that allows the existence of self-dual equations and the corresponding Bogomolny bound for the energy of static configurations. We show that the same conditions allow for the $N=2$ supersymmetric extension of the model, reobtaining the BPS equations from the supersymmetry requirement. Explicit electrically charged vortexlike solutions to these equations are presented.