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We extend the hermitian three-algebra formulation of ABJM theory to include$U(1)$ factors. With attention payed to extra $U(1)$ factors, we refine theclassification of $\N=6$ ABJM theories. We argue that essentially the onlyallowed gauge groups are $SU(N)\times SU(N)$, $U(N)\times U(M)$ and$Sp(N)\times U(1)$ and that we have only one independent Chern-Simons level inall these cases. Our argument is based on integrality of the $U(1)$Chern-Simons levels and supersymmetry. A relation between monopole operatorsand Wilson lines in Chern-Simons theory suggests certain gauge representationsof the monopole operators. From this we classify cases where we can not expectenhanced $\N=8$ supersymmetry. We also show that there are two equivalentformulations of $\N=5$ ABJM theories, based on hermitian three-algebra andquaternionic three-algebra respectively. We suggest properties of monopoles in$\N=5$ theories and show how these monopoles may enhance supersymmetry from$\N=5$ to $\N=6$.

Monopoles, three-algebras and ABJM theories with $\N=5,6,8$ supersymmetry