3d superconformal indices and isomorphisms of M2-brane theories

We test several expected isomorphisms between the U(N) x U(N) ABJM theory and (SU(N) x SU(N))/Z(N), theory including the BLG theory by comparing their superconformal indices. From moduli space analysis, it is expected that this equivalence can hold if and only if the rank N and Chern-Simons level k are coprime. We also calculate the index of the ABJ theory and investigate whether some theories with identical moduli spaces are isomorphic or not.