A Monopole Index for N=4 Chern-Simons Theories

TL;DR

This work computes a monopole-inclusive index for ${ m N}=4$ Chern–Simons theories with gauge group ${ m U}(N)^r$ in the large-$N$ limit and demonstrates a precise match with the gravity multi-particle index on ${ m AdS}_4 imes X_7$, where $X_7$ contains nontrivial two-cycles. A one-to-one map is established between the $r-1$ independent magnetic charges and gravity charges: the M-momentum $P_M$ plus $r-2$ wrapping numbers, with twisted-sector contributions arising from wrapped M2-branes on the two-cycles. Analytically, the neutral sector of the indices is shown to agree; numerically, the charged sector matches across several ${ m N}=4$ theories with different untwisted/twisted hypermultiplet content, supported by a detailed selection-rule analysis linking gauge charges to gravity data. The results reinforce the interpretation of monopole operators as embodying the M-direction and wrapped M2-branes, and they illuminate the role of fivebrane linking numbers and Wilson lines in the gauge/gravity correspondence.

Abstract

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 AdS_4 x X_7. The internal space X_7 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 (=b_2(X_7)) 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.