BPS vortices, Q-balls, and Q-vortices in N = 6 Chern-Simons matter theory

We investigate the vortex-type BPS equations in the ABJM theory without and with mass-deformation. We systematically classify the BPS equations in terms of the number of supersymmetry and the R-symmetries of the undeformed and mass-deformed ABJM theories. For the undeformed case, we analyze the N = 2 BPS equations for U(2)xU(2) gauge symmetry and obtain a coupled differential equation which can be reduced to either Liouville- or Sinh-Gordon-type vortex equations according to the choice of scalar functions. For the mass-deformed case with U(N)×U(N) gauge symmetry, we obtain some number of pairs of coupled differential equations from the N = 1, 2 BPS equations, which can be reduced to the vortex equations in Maxwell-Higgs theory or Chern-Simons matter theories as special cases. We discuss the solutions. In N = 3 vortex equations Chern-Simons-type vortex equation is not allowed. We also show that N = 5/2, 3/2, 1,2 BPS equations are equivalent to those with higher integer supersymmetries.