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 the
classification of N=6 ABJM theories. We argue that essentially the only
allowed gauge groups are SU(N)×SU(N), U(N)×U(M) and
Sp(N)×U(1) and that we have only one independent Chern-Simons level in
all these cases. Our argument is based on integrality of the U(1)
Chern-Simons levels and supersymmetry. A relation between monopole operators
and Wilson lines in Chern-Simons theory suggests certain gauge representations
of the monopole operators. From this we classify cases where we can not expect
enhanced N=8 supersymmetry. We also show that there are two equivalent
formulations of N=5 ABJM theories, based on hermitian three-algebra and
quaternionic 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.Comment: 52 page