The complete superconformal index for N=6 Chern-Simons theory

TL;DR

This work computes the complete superconformal index for the 3d N=6 Chern-Simons-matter theory with gauge group U(N)_k × U(N)_{-k} using localization on S^2 × S^1, including sectors with magnetic flux (monopole operators) that encode KK momentum along the M-theory circle. The index is assembled from a sum over flux distributions and holonomies, yielding a factorized large-N form I_N∞ = I^{(0)} I^{(+)} I^{(-)}, where I^{(±)} enumerate positive/negative KK-momentum sectors and I^{(0)} corresponds to zero momentum; analytic and numerical tests in 1–3 KK-momentum sectors show perfect agreement with the index of supersymmetric gravitons in AdS_4 × S^7/ℤ_k. The results highlight the crucial role of monopole operators in nontrivial representations and provide a nontrivial check of the AdS4/CFT3 correspondence beyond the 't Hooft limit, with potential extensions to other CS theories. Overall, the paper demonstrates a precise match between gauge theory and gravity indices in the large-N, finite-k regime, reinforcing the holographic duality and guiding future explorations of topological sectors in CS-matter theories.

Abstract

We calculate the superconformal index for N=6 Chern-Simons-matter theory with gauge group U(N) X U(N) at arbitrary allowed value of the Chern-Simons level k. The calculation is based on localization of the path integral for the index. Our index counts supersymmetric gauge invariant operators containing inclusions of magnetic monopole operators, where latter operators create magnetic fluxes on 2-sphere. Through analytic and numerical calculations in various sectors, we show that our result perfectly agrees with the index over supersymmetric gravitons in AdS_4 X S^7/Z_k in the large N limit. Monopole operators in nontrivial representations of U(N) X U(N) play important roles. We also comment on possible applications of our methods to other superconformal Chern-Simons theories.