ABJ(M) and Fractional M2's with Fractional M2 Charge

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Recently Aharony, Bergman and Jafferis (ABJ) have argued that a 3d U(N+M)xU(N) Chern-Simons gauge theory at level (k,-k) may have a vacuum with N=6 supersymmetry only if M<k+1 and if a certain period of the B-field in a IIA background is quantized. We use a braneology argument to argue that N=3 supersymmetry may be preserved under the weaker condition that 2Nk>M(M-k)-1 with no restriction on the B-field. IIB brane cartoons and 11d supergravity solutions corresponding to N=3 vacua that do not preserve N=6 supersymmetry are argued to represent cascading gauge theories, generalizing the N=2 Seiberg duality conjectured by Giveon and Kutasov. While as usual the M2-brane charge runs as a result of the twisted Bianchi identity for *G₄, the M5-brane charge running relies on the fact that it wraps a torsion homology cycle.