Monopole operators and symmetry enhancement in ABJM theory revisited

TL;DR

This paper constructs monopole operators in 3d Yang-Mills and Chern-Simons-matter theories, demonstrating their properties and role in symmetry enhancement in ABJM theory, especially for levels 1 and 2.

Contribution

It provides an explicit construction of monopole operators in these theories and proves their role in symmetry enhancement and supersymmetry closure in ABJM models.

Findings

Monopole operators transform as local operators with zero conformal dimension.

Monopole operators commute with supercharges, ensuring SUSY invariance.

Proof of conserved R-symmetry currents involving monopoles in ABJM at levels 1 and 2.

Abstract

We construct monopole operators for 3d Yang-Mills-matter theories and Chern-Simons-matter theories in canonical formalism. In this framework, monopole operators, although as the disorder operators, could be written in terms of the fundamental fields of the theory, and thus could be treated in the same way as the ordinary operators. We study the properties of the constructed monopole operators. In Chern-Simons-matter theories, monopole operators transform as the local operators with the classical conformal dimension $ 0 $ under the action of the dilation and are also covariantly constant. In supersymmetric Chern-Simons-matter theories like the ABJM model, monopole operators 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 monopole operators 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 $ SUSY transformation rules, and verify the closure of the $ \mathcal{N}=8 $ supersymmetry.