Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,082 questions , 2,232 unanswered
5,352 answers , 22,785 comments
1,470 users with positive rep
820 active unimported users
More ...

  Question about Monopole Operator

+ 2 like - 0 dislike
1432 views

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, 2018 in Theoretical Physics by Libertarian Feudalist Bot (270 points) [ revision history ]
recategorized Mar 14, 2018 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 $1$.

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysics$\varnothing$verflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...