Global $U(1)$ symmetry of 2+1d Abelian-Higgs Model

Question

In the Abelian-Higgs model, 

$$S=\int d^{3}x\left\{-\frac{1}{4g^{2}}F_{\mu\nu}F^{\mu\nu}+|D\phi|^{2}-a|\phi|^{2}-b|\phi|^{4}\right\}$$

there is a $U(1)$ gauge symmetry. In David Tongs' lecture notes 
http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf
on page 169, he says that there is also a less obvious global symmetry, with the current

$$\star j=\frac{1}{2\pi}db$$.

I understand that the current is conserved for an obvious reason. But why is the flux corresponding to a global $U(1)$ symmetry? What is this global $U(1)$ symmetry?

Answer 1

If we go to the dual version, i.e. Wilson-Fisher without gauge field but just global symmetry, then U(1) symmetry simply corresponds to boson number conservation.

Now go back to Abelian-Higgs, you can ask what's the dual version of the previous global U(1) symmetry: you would then notice that the role of previous U(1) symmetry is played by the flux conservation now. This means your U(1) gauge field is non-compact, and monopoles operators are forbidden (at least in this simple duality of continuous theory).

Just remember here monopole operator is dual to previous boson operator, then other things are clear.

Comments

I think the topological current is conserved automatically by its definition. Noether current is conserved on-shell, but here it is conserved automatically.

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Thank you. Could you explain what you mean by non-compact gauge field?

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"Non-campact"... OK... Are you from high energy?

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I am just a new student in theoretical physics. I know what compact means in topology but I don't know what non-compact gauge field is.

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I see. You might find Page 8 and the footnote 4 helpful in this paper: arXiv 1703.02426.

I noticed you're trying to learn duality, Subir's notes always contains a lot of details. For example, the following one has a lattice version of boson-vertex duality:

https://canvas.harvard.edu/courses/39684/files/folder/Lectures?preview=5387008

which explains lots of details, and importantly, the origin of non-compactness in duality.

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Thank you very much!!!

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Note that the meaning of noncompact Kite is using is different from how high energy theorists use it and is different from the fact that $U(1)$ is compact. Whether you have monopoles in the theory or not has nothing to do with the fact that the gauge group is $U(1)$ or $R$. $U(1)$ vs. $R$ is the difference between having quantized electric charges or not. In theories with rotational symmetry (and therefore quantized angular momentum since SO(n) is compact), having a magnetic monopoles implies electric charge quantization, but that's the only relation.