# Question about Monopole Operator

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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 $QED_{3}$ with $N_{f}$ fermions of charge 1, the monopole operator is defined in the following way.
Let $\left\{a_{i},a_{j}^{\dagger}\right\}=\delta_{ij}$ be the annihilation and creation operators for the zero-modes of the Dirac fermion in the monopole background. Let $\left|0\right>$ be the bare monopole state. Then the monopole operator is defined to be associated with the state.
$$\left|\mathfrak{M}_{i_{1} i_{2}\cdots i_{l}}\right>=a_{i_{1}}a_{i_{2}}\cdots a_{i_{l}}\left|0\right>$$
This state transforms in totally anti-symmetric representation of $SU(N_{f})$, 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 $spin_{c}$ connection $\mathcal{A}$,
$$i\bar{\Psi}\displaystyle{\not} D _{\mathcal{A}}\Psi$$

Both $\bar{\Psi}$ and $\Psi$ have zero-modes leading after quantization to two different states differing by a factor of $\Psi$ (or $\bar{\Psi}$). These two states have spin zero and their electric charges differ by 1. To determine the charges, one has to add a bulk term
$$\frac{1}{8\pi}\mathcal{A}\wedge d\mathcal{A}.$$
Then the theory is time-reversal and time-reversal+charge conjugation invariant. This determines the charges to be $\pm\frac{1}{2}$.

On page 14 and 15, it says from the duality

$$i\bar{\Psi}\displaystyle{\not} D _{A}\Psi\leftrightarrow |D_{b}\phi|^{2}-|\phi|^{4}+\frac{1}{4\pi}b\wedge db+\frac{1}{2\pi}b\wedge dA$$

The monopole operator $\mathfrak{M}_{b}$ on the right hand side carries $U(1)_{b}$ charge 1 and $U(1)_{A}$ charge 1. It means that $\phi^{\dagger}\mathfrak{M}_{b}$ is $U(1)_{b}$-gauge invariant. It has spin-$\frac{1}{2}$ because of the relative angular momentum between the electrically charged $\phi^{\dagger}$ and magnetically charged $\mathfrak{M}_{b}$. We have the identification
$$\Psi=\phi^{\dagger}\mathfrak{M}_{b}.$$

Could anyone please help me understand the above definition of the monopole operator and the statements in the second paper?

Sorry I don't understand the effect of the CS term. Also, I am not familiar with $spin_{\mathbb{C}}$ connection. I don't understand the definition of the monopole operator either. Also, I believe that the monopole carries $U(1)_{b}$ charge $-1$, instead of $1$.
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