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!


Gauge theories are often equipped with higher form "electric" and "magnetic" global symmetries. Coupling such theories to electrically or magnetically charged sources explicitly breaks the corresponding symmetry (either partially or completely).

Let me first review how this works for the more familiar case of $\mathrm{U}(1)$ gauge theories in 4d. So consider a 1-form abelian gauge field $A$ with field strength $F=\mathrm{d}A$ on a 4-manifold $M$. Start with the pure gauge theory, with action

$$S \propto \int_M F \wedge \star F.$$

(We don't actually need to talk about an action, but it can make things more transparent in simple examples.) The equation of motion and Bianchi identity say that

$$\mathrm{d} \star F = 0 \quad\text{and}\quad \mathrm{d} F = 0.$$

In the quantum theory, these are operator equations. That means we have two different 2-form conserved currents $j= F$ and $\tilde j = \star F$, which satisfy $\mathrm{d} \star j = \mathrm{d} \star \tilde j=0$. Each of these is a $\mathrm{U}(1)$ 1-form global symmetry, often called "electric" ($\mathrm{U(1)_E}$) and "magnetic" ($\mathrm{U(1)_M}$), respectively. They are called 1-form symmetries because the charged objects, Wilson lines and 't Hooft lines, are supported on 1-manifolds. (Whereas for ordinary 0-form symmetries the charged objects are local operators). You can think of them as the worldlines of probe electric charges and magnetic monopoles. The $\mathrm{U(1)_E}$ and $\mathrm{U(1)_M}$ charges themselves are obtained by integrating $\star j$ and $\star \tilde j$ over 2-spheres that link these lines, $Q = \oint_{S^2} \star F$ and $\tilde Q = \oint_{S^2} F$. Of course, these just measure the electric and magnetic charges of the particle whose worldline they surround.

In these variables, a Wilson line simply takes the form $W_q(C) = e^{iq\oint_C A}$. The electric 1-form symmetry corresponds to the invariance of the action under the shift $A \to A + \lambda$, where $\lambda$ is a flat gauge field $(\mathrm{d}\lambda = 0),$ and clearly $W_q(C)$ transforms under this symmetry. The Wilson line inserts a source in the equation of motion, $\mathrm{d}\star F = q \delta_C$. The 't Hooft line is similarly the holonomy of the dual gauge field $\hat A$, and the magnetic 1-form symmetry would likewise correspond to shifting $\hat A$ by a flat gauge field if we wrote the theory in the dual variables. In terms of the original variables, the 't Hooft operator $H_m(C)$ corresponds to a prescription to delete $C$ from $M$ and demand that $\oint_{S^2} F = 2\pi m$, where $S^2$ is a sphere linking $C$. Equivalently, the 't Hooft operator inserts a source in the Bianchi identity, $\mathrm{d} F = 2\pi m \delta_C$.

So far we have discussed the pure gauge theory. Suppose now we couple it to electrically charged matter (say to a field with charge 1). The charged matter enters the equation of motion as a source, $\mathrm{d} \star F = \star j_E$, and explicitly breaks the electric 1-form symmetry. Similarly, coupling the theory to a monopole operator explicitly breaks the magnetic 1-form symmetry.

Hopefully now you can see the answer to your original question. The authors are considering a 3d theory with abelian gauge fields $b$ and $\hat b$. As ever, in the absence of magnetically charged matter, the Bianchi identities imply the existence of conserved currents $\star \mathrm{d}b$ and $\star \mathrm{d} \hat b$. Note that since we are in three dimensions these are ordinary 0-form global symmetries. So the theory has two $\mathrm{U(1)}$ ordinary global symmetries (both "magnetic" in the language above). Now they couple the theory to a monopole operator for $\hat b$. This introduces a magnetic source in the Bianchi identity for $\mathrm{d}\hat b$, and explicitly breaks the corresponding $\mathrm{U(1)}$ symmetry.

(I don't think $\phi^\dagger \mathcal{M}_{\hat b}$ is a typo. They say that $\mathcal{M}_{\hat b}$ carries charge one under the $b$ gauge symmetry and therefore multiply by $\phi^\dagger$ to make a gauge invariant operator.)


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