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

Question

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!

Comments

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

Answer 1

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.