A monopole index for N=4 Chern-Simons theories

We compute a certain index for an N=4 Chern-Simons theory with gauge group U(N)r in the large N limit with taking account of monopole contribution, and compare it to the corresponding multi-particle index for M-theory in the dual geometry AdS4xX7. The internal space X7 has non-trivial two-cycles, and M2-branes wrapped on them contribute to the multi-particle index. We establish one-to-one map between r-1 independent magnetic charges on the gauge theory side and the same number of charges on the gravity side: the M-momentum and r-2 (=b2(X7)) wrapping numbers. With a certain assumption for the wrapped M2-brane contribution, we confirm the agreement of the indices for many sectors specified by the r-1 charges by using analytic and numerical methods.