# Why does a monopole operator break the global symmetry with topological current?

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I am currently reading the paper "A Duality Web in 2+ 1 Dimensions and Condensed Matter Physics" by Seiberg et al, and on page 22 they add to the Lagrangian a monopole operator of the form $\phi^\dagger \mathcal{M}_{\hat b}$. Firstly, is it perhaps a typo that the $\phi$ is unhatted? Should it be hatted so that it is charged under U(1)$_{\hat b}$ ? Secondly, how exactly does this operator break the global symmetry whose current is the topological current $d\hat b$? I have been trying to understand this under the light of "Generalized Global Symmetries", and if I understand correctly, this would constitute a 1-form global symmetry. However, I could not find in that paper a section which would explain why a monopole of this form would break the symmetry. I would be very grateful if someone could shed a little bit of light on this for me. Thank you!

I find that Nathan Seiberg explaining exactly the same is easier to follow here ( slides 7 - 11 )

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1. There is no typo. Physical observables must be (gauged-) charge neutral. The monopole operator $\mathcal{M}_{\hat{b}}$ and $\phi^{\dagger}$ carry the opposite gauged charged so their product is neutral.

2. Monopole operator breaks the conservation of the topological current because in the presence of a Dirac magnetic monopole, the gauge field is not globally well-defined anymore. i.e. $d\hat{f}=0$ does not implies that $\hat{f}=d\hat{b}$, where $\hat{f}$ is the field strength.

3. I don't think that the second paper "Generalized Global Symmetries" is really related with your questions.

answered Jul 7, 2018 by (260 points)

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