lagrangian formalism - Global $U(1)$ symmetry of 2+1D Abelian-Higgs Model

In the Abelian-Higgs model,

$$S=\int d^{3}x\left\{-\frac{1}{4g^{2}}F_{\mu\nu}F^{\mu\nu}+|D\phi|^{2}-a|\phi|^{2}-b|\phi|^{4}\right\}\tag{5.34}$$

there is a $U(1)$ gauge symmetry. In David Tongs' lecture notes The Quantum Hall Effect, chapter 5, on page 169, he says that there is also a less obvious global symmetry, with the current

$$\star j=\frac{1}{2\pi}db.\tag{5.35}$$

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

Any abelian gauge theory has a $\mathrm{U(1)}$ global symmetry with current $j = \star F$ by virtue of the Bianchi identity,

$$\mathrm{d} \star j = \mathrm{d} F = 0.$$

First suppose the theory is 4-dimensional, in which case this symmetry is a little more familiar. In this case $j$ is a 2-form. The associated charge

$$Q=\int_{S^2}\star j = \int_{S^2} F$$

measures the magnetic flux of a line operator $H(C)$ (the "$C$ which links the $S^2$. It corresponds to the worldline of a probe magnetic monopole, and $Q$ measures the magnetic flux of the monopole in the same way that $\int_{S^2} \star F$ measures the electric flux on the worldline of an electric charge. These are called 1-form global symmetries, because the charged operators are supported on lines.

The same story goes through in any dimension $d>2$. We obtain a $(d-3)$-form global symmetry, meaning the charged operators are supported on $(d-3)$-manifolds which link a 2-sphere over which we measure the charge $\int_{S^2} F$.

In 3 dimensions, $j=\star F$ is a 1-form, so this is an ordinary global symmetry. The 't Hooft operators are pointlike magnetic monopole operators, whose charge is again the magnetic flux.