We construct three dimensional Chern-Simons-matter theories with gauge groups U(N)×U(N) and SU(N)×SU(N) which have explicit N = 6 superconformal symmetry. Using brane constructions we argue that the U(N) × U(N) theory at level k describes the low energy limit of N M2-branes probing a C 4 /Zk singularity. At large N the theory is then dual to M theory on AdS4 × S 7 /Zk. The theory also has a ’t Hooft limit (of large N with a fixed ratio N/k) which is dual to type IIA string theory on AdS4 × CP 3. For k = 1 the theory is conjectured to describe N M2-branes in flat space, although our construction realizes explicitly only six of the eight supersymmetries. We give some evidence for this conjecture, which is similar to the evidence for mirror symmetry in d = 3 gauge theories. When the gauge group is SU(2) × SU(2) our theory has extra symmetries and becomes identical to the Bagger-Lambert theory. Jun
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