Scaling limit ofN=6superconformal Chern-Simons theories and Lorentzian Bagger-Lambert theories

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We show that the $\mathcal{N}=8$ superconformal Bagger-Lambert theory based on the Lorentzian 3-algebra can be derived by taking a certain scaling limit of the recently proposed $\mathcal{N}=6$ superconformal $U(N)\ifmmode\times\else\texttimes\fi{}U(N)$ Chern-Simons-matter theories at level $(k,\ensuremath{-}k)$. The scaling limit (and In\"on\"u-Wigner contraction) is to scale the trace part of the bifundamental fields as ${X}_{0}\ensuremath{\rightarrow}{\ensuremath{\lambda}}^{\ensuremath{-}1}{X}_{0}$ and an axial combination of the two gauge fields as ${B}_{\ensuremath{\mu}}\ensuremath{\rightarrow}\ensuremath{\lambda}{B}_{\ensuremath{\mu}}$. Simultaneously, we scale the level as $k\ensuremath{\rightarrow}{\ensuremath{\lambda}}^{\ensuremath{-}1}k$ and then take $\ensuremath{\lambda}\ensuremath{\rightarrow}0$ limit. Interestingly, the same constraint equation ${\ensuremath{\partial}}^{2}{X}_{0}=0$ is derived by imposing finiteness of the action. In this scaling limit, M2 branes are located far from the origin of ${\mathbf{C}}^{4}/{\mathbf{Z}}_{\mathbf{k}}$ compared to their fluctuations and ${\mathbf{Z}}_{\mathbf{k}}$ identification becomes a circle identification. Hence, the scaled theory describes $\mathcal{N}=8$ supersymmetric theory of 2-branes with dynamical coupling. The coupling constant is promoted to a space-time dependent $SO(8)$ vector ${X}_{0}^{I}$ and we show that the scaled theory has a generalized conformal symmetry as well as manifest $SO(8)$ with the transformation of the background fields ${X}_{0}^{I}$.