On 3d N=8 Lorentzian BLG theory as a scaling limit of 3d superconformal N=6 ABJM theory

We elaborate on the suggestion made in arXiv:0806.3498 that the 3d \N=8 superconformal SU(N) Chern-Simons-matter theory of Lorentzian Bagger-Lambert-Gustavson type (L-BLG) can be obtained by a scaling limit (involving sending the level k to infinity and redefining the fields) from the \N=6 superconformal U(N)xU(N) Chern-Simons-matter theory of Aharony, Bergman, Jafferis and Maldacena (ABJM). We show that to implement such a limit in a consistent way one is to extend the ABJM theory by an abelian "ghost" multiplet. The corresponding limit at the 3-algebra level also requires extending the non-antisymmetric Bagger-Lambert 3-algebra underlying the ABJM theory by a negative-norm generator. We draw analogy with similar scaling limits discussed previously for bosonic Chern-Simons theory and comment on some implications of this relation between the ABJM and L-BLG theories.