BPS Vortices, $Q$-balls, and $Q$-vortices in ${\cal N}=6$} Chern-Simons Matter Theory

TL;DR

This paper classifies and analyzes BPS vortex equations in ABJM theory with and without mass deformation, revealing their reduction to known vortex equations and discussing solution properties across different supersymmetry levels.

Contribution

It systematically classifies BPS vortex equations in ABJM theory, linking them to known vortex equations and exploring their solutions across various supersymmetry configurations.

Findings

Reduction of BPS equations to Liouville- or Sinh-Gordon-type equations.

Identification of vortex equations as special cases of Maxwell-Higgs or Chern-Simons theories.

Demonstration that lower supersymmetry BPS equations are equivalent to higher supersymmetry cases.

Abstract

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 ${\cal N}=2$ BPS equations for U(2)$\times$U(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($N$)$\times$U($N$) gauge symmetry, we obtain some number of pairs of coupled differential equations from the ${\cal N}=1,2$ 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 ${\cal N}=3$ vortex equations Chern-Simons-type vortex equation is not allowed. We also show that ${\cal N}=\frac52, \frac32, \frac12$ BPS equations are equivalent to those with higher integer supersymmetries.