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  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?  


 

asked Feb 25 in Theoretical Physics by New Student (140 points) [ revision history ]
recategorized Mar 14 by Dilaton

I guess you understand the effect of Chern-Simons term. So what's exactly the part you don't get? 

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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