Abelian Projections of the Mass-deformed ABJM theory and Weakly Curved Dual Geometry

We construct ${\cal N}=2,4$ supersymmetric abelian projections of the ${\cal N}=6$ mass-deformed ABJM theory. There are well-defined dual background geometries for the ${\cal N}=2$ abelian theory, while those geometries are unclear for the ${\cal N}=4$ abelian theory. The ${\cal N}=2$ theory is built on the supersymmetric vacua of the mass-deformed ABJM theory, which are proven to have one-to-one correspondence with the ${\mathbb Z}_k$ quotient of Lin-Lunin-Maldacena geometries. We select one special vacuum of the mass-deformed ABJM theory and show that the corresponding geometry is weakly curved at every point of the entire space transverse to the M2-branes in the large $N$ limit.