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We construct monopole operators for 3d Yang-Mills-matter theories andChern-Simons-matter theories in canonical formalism. In this framework,monopole operators, although as the disorder operators, could be written interms of the fundamental fields of the theory, and thus could be treated in thesame way as the ordinary operators. We study the properties of the constructedmonopole operators. In Chern-Simons-matter theories, monopole operatorstransform as the local operators with the classical conformal dimension $ 0 $under the action of the dilation and are also covariantly constant. Insupersymmetric Chern-Simons-matter theories like the ABJM model, monopoleoperators commute with all of the supercharges, and thus are SUSY invariant.ABJM model with level $ k=1,2 $ is expected to have enhanced $SO(8) $R-symmetry due to the existence of the conserved extra R-symmetry currents $j^{AB}_{\mu} $ involving monopoles. With the explicit form of the monopoleoperators given, we prove the current conservation equation $\partial^{\mu}j^{AB}_{\mu} =0$ using the equations of motion. We also compute the extra $\mathcal{N}=2 $ supercharges, derive the extra $ \mathcal{N}=2 $ SUSYtransformation rules, and verify the closure of the $ \mathcal{N}=8 $supersymmetry.

Monopole operators and symmetry enhancement in ABJM theory revisited