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?

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 "`t Hooft line operator") which is supported on a line $C$ 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.

• We are considering a U(1) gauge theory, which means there is a gauge invariant local operator called $F$ in the spectrum, and it satisfies $\mathrm{d} F = 0$. There's nothing to check, it is part of the definition, so I'm not sure what you mean – Elliot Schneider Mar 10 '18 at 1:01
• @AccidentalFourierTransform A is not coupled to j, j is dA, and is conserved for topological reasons. Something else couples to j, say B, so the coupling looks like BdA. – Ryan Thorngren Mar 10 '18 at 1:23
• @AccidentalFourierTransform By definition, the quantum theory contains a local operator $F$ which is closed. Therefore the spectrum includes a conserved current $\star F$. If you like you can then couple the current to a background gauge field, but your comment seems to put the logical order backwards – Elliot Schneider Mar 10 '18 at 1:25
• That's wrong, because j is a function of A, which is dynamical, not of the background gauge field, which is not integrated over. You are confusing the Bianchi identity for the dynamical gauge field (which is guaranteed by the definition of the gauge theory) and the Bianchi identity for the background gauge field (which is equivalent to current conservation). – Ryan Thorngren Mar 10 '18 at 2:12
• U(1) is because the integrals of $db/2\pi$ (the charges) are integers on closed surfaces. – Ryan Thorngren Mar 10 '18 at 3:02

Conserved interger-period 2-forms correspond to $U(1)$ global symmetries by Noether's theorem. This symmetry acts on the instanton operators but not on the fields. If you apply particle-vortex duality, this is the shift symmetry of the dual $U(1)$ scalar.

• Thank you. Could you please add more details? I can only see the Noether current $-i(D\phi)^{\dagger}\phi+i\phi^{\dagger}D\phi$. – Libertarian Monarchist Bot Mar 10 '18 at 2:30
• Your $db/2\pi$ is the other Noether current. I wrote a paper about this with Zohar Komargodski, Adar Sharon, and Xinan Zhou arxiv.org/abs/1705.04786 . Check it out. – Ryan Thorngren Mar 10 '18 at 3:01
• But Noether current is conserved on-shell. This topological current is conserved automatically. Why is that related with $U(1)$ group? – Libertarian Monarchist Bot Mar 10 '18 at 3:30
• Could you please explain to me what $\int A\cup w_{3}$ means? Are there any maths textbooks explain this mathematics? – Libertarian Monarchist Bot Mar 10 '18 at 16:09
• This is explained in any textbook that discusses simplicial cohomology, Hatcher's Algebraic Topology, for instance, is available online math.cornell.edu/~hatcher/AT/ATpage.html . These expressions can also be understood in a continuum setting, which we explained here arxiv.org/abs/1404.3230 on page 21. – Ryan Thorngren Mar 10 '18 at 19:19