I've been studying IR-dualities in 2+1 dimensions. I encountered monopole operators in the following papers:
1. "Time-Reversal Symmetry, Anomalies, and Dualities in (2+1)d" https://arxiv.org/abs/1712.08639
On page 10, starting from QED3 with Nf fermions of charge 1, the monopole operator is defined in the following way.
Let {ai,a†j}=δij be the annihilation and creation operators for the zero-modes of the Dirac fermion in the monopole background. Let |0⟩ be the bare monopole state. Then the monopole operator is defined to be associated with the state.
|Mi1i2⋯il⟩=ai1ai2⋯ail|0⟩
This state transforms in totally anti-symmetric representation of
SU(Nf), with
l indices, and is bosonic.
2. "A Duality Web in 2+1 Dimensions and Condensed Matter Physics" https://arxiv.org/abs/1606.01989
On page 13, it says that for the Dirac fermion coupled to a background spinc connection A,
iˉΨ⧸DAΨ
Both ˉΨ and Ψ have zero-modes leading after quantization to two different states differing by a factor of Ψ (or ˉΨ). These two states have spin zero and their electric charges differ by 1. To determine the charges, one has to add a bulk term
18πA∧dA.
Then the theory is time-reversal and time-reversal+charge conjugation invariant. This determines the charges to be
±12.
On page 14 and 15, it says from the duality
iˉΨ⧸DAΨ↔|Dbϕ|2−|ϕ|4+14πb∧db+12πb∧dA
The monopole operator Mb on the right hand side carries U(1)b charge 1 and U(1)A charge 1. It means that ϕ†Mb is U(1)b-gauge invariant. It has spin-12 because of the relative angular momentum between the electrically charged ϕ† and magnetically charged Mb. We have the identification
Ψ=ϕ†Mb.
Could anyone please help me understand the above definition of the monopole operator and the statements in the second paper?