In the paper A Duality Web in 2+1 Dimensions and Condensed Matter Physics by Seiberg, Senthil, Wang, and Witten, they studied the particle-vortex dualities in $2+1$ dimensions.
On page 20, section 3.1, they considered phase transitions of the $2+1$ dimensional theory
$$\mathcal{L}=-\frac{1}{4e^{2}}f_{\mu\nu}f^{\mu\nu}+|D_{b}\phi|^{2}-\frac{1}{4e^{2}}\hat{f}_{\mu\nu}\hat{f}^{\mu\nu}+|D_{\hat{b}}\hat{\phi}|^{2}-V(|\phi|,|\hat{\phi}|)+\frac{1}{2\pi}\epsilon^{\mu\nu\rho}b_{\mu}\partial_{\nu}\hat{b}_{\rho},$$
where $(D_{b})_{\mu}=\partial_{\mu}+ib_{\mu}$, and $(D_{\hat{b}})_{\mu}=\partial_{\mu}+i\hat{b}_{\mu}$, and $f_{\mu\nu}=\partial_{\mu}b_{\nu}-\partial_{\nu}b_{\mu}$, and $\hat{f}_{\mu\nu}=\partial_{\mu}\hat{b}_{\nu}-\partial_{\nu}\hat{b}_{\mu}$.
This theory has two gauge redundancies $U(1)_{b}$ and $U(1)_{\hat{b}}$:
$$U(1)_{b}:\,\,\,b_{\mu}(x)\rightarrow b_{\mu}(x)-\partial_{\mu}\lambda(x),\quad \phi(x)\rightarrow e^{-i\lambda(x)}\phi(x)$$
$$U(1)_{\hat{b}}:\,\,\,\hat{b}_{\mu}(x)\rightarrow\hat{b}_{\mu}(x)-\partial_{\mu}\hat{\lambda}(x),\quad \hat{\phi}(x)\rightarrow e^{-i\hat{\lambda}(x)}\hat{\phi}(x)$$
(the BF-coupling $b\wedge d\hat{b}$ is invariant up to a total derivative under the above gauge transformations)
In addition, there are two generalized global symmetries (introduced by Gaiotto, Kapustin, Seiberg, and Willett) $U(1)_{f}$ and $U(1)_{\hat{f}}$ associated with the conservation of the topological currencies
$$j=\ast f,\quad\mathrm{and}\quad\hat{j}=\ast\hat{f},$$
where $f=db$, and $\hat{f}=d\hat{b}$ are the field strength. The conservation of these two topological currents follows trivially from the Bianchi identity.
On page 22, the author studied the consequence by adding a Dirac monopole operator $\mathcal{M}_{\hat{b}}(x)$ of gauge field $\hat{b}$ into the action. To be more specific, such an operator would break the conservation of $\hat{j}=\ast d\hat{b}$, and insert a Dirac monopole at the point $x$, which results in
$$d\ast\hat{j}=d\hat{f}=2\pi\delta(x)$$
In such a monopole configuration, the gauge field $\hat{b}$ is not globally defined, and the field strength $\hat{f}$ belongs to a non-trivial first Chern class. i.e.
$$\int_{S^{2}}\frac{\hat{f}}{2\pi}=1.$$
The authors claimed that adding such a monopole operator into the Lagrangian explicitly breaks the generalized global symmetry $U(1)_{\hat{f}}$. I will explain such an explicit symmetry breaking in the following simpler example.
Let's consider the free Maxwell theory in $2+1$ dimensions
$$S[A]=-\frac{1}{2}\int F\wedge\ast F$$
where $F=dA$, and $A$ is a $U(1)$-gauge field.
- This theory has two generalized global symmetries $U(1)_{e}$ and $U(1)_{m}$ associated with the topological currents
$$J_{e}=F,\quad\mathrm{and}\quad J_{m}=\ast F.$$
Their conservation follows directly from the EOM and the Bianchi identity
$$d\ast J_{e}=d\ast F=0,\quad d\ast J_{m}=dF=0.$$
The Lagrangian can be converted into the dual photon description. First, one imposes the Bianchi identity by hand into the path integral
$$\mathcal{Z}=\int\mathcal{D}F\int\mathcal{D}\sigma \exp\left\{i\int\left(-\frac{1}{2}F\wedge\ast F+\sigma dF\right)\right\}$$
where $\sigma$ is an auxiliary field, whose integral produces the Bianchi identity $dF=0$. Integrating out the gauge invariant variable $F$, one obtains the dual theory
$$\mathcal{Z}=\int\mathcal{D}\sigma\exp\left\{i\int\frac{1}{2}d\sigma\wedge\ast d\sigma\right\}$$
This theory should be equivalent to the original one, and its only Abelian symmetry is the shift
$$U(1):\sigma\rightarrow\sigma+\alpha$$
where $\alpha\in\mathbb{R}$. The corresponding Noether current is
$$J=d\sigma.$$
This symmetry should be identified with the global symmetry $U(1)_{e}$ or $U(1)_{m}$, depending on which one of $F$ or $\ast F$ is dualized.
The vaccua manifold of this theory can be identified with $\mathbb{R}$. Picking out one of its vacuum, the global symmetry is spontaneously broken.
- Next, one can add a Dirac monopole operator $\mathcal{M}(x)$ into the above theory. This can be achieved by imposing
$$dF=2\pi\delta(x)$$
into the path-integral. One has
$$\mathcal{Z}=\int\mathcal{D}F\int\mathcal{D}\sigma \exp\left\{i\int\left(-\frac{1}{2}F\wedge\ast F+\sigma(x)(dF-2\pi\delta(x))\right)\right\}$$
Integrating out $F$, one obtains
$$\mathcal{Z}=\int\mathcal{D}\sigma\exp\left\{i\int\left(\frac{1}{2}d\sigma\wedge\ast d\sigma-2\pi\sigma(x)\delta(x)\right)\right\}=\int\mathcal{D}\sigma e^{-2\pi i\sigma(0)}e^{\frac{i}{2}\int d\sigma\wedge\ast d\sigma}.$$
Therefore, one can define the monopole operator in the dual photon description by
$$\mathcal{M}(x)=e^{-2\pi i\sigma(x)}$$
and insert it into the path integral, and write
$$\mathcal{Z}=\int\mathcal{D}\sigma\mathcal{M}(0)e^{\frac{i}{2}\int d\sigma\wedge\ast d\sigma}.$$
Under the global $U(1)$ transformation, one has
$$\mathcal{M}(x)\rightarrow e^{-2\pi i\alpha}\mathcal{M}(x)$$
where $\alpha\in S^{1}$. Therefore the global $U(1)$ symmetry is broken to $\mathbb{Z}$.
On the other hand, I found something strange from David Tong's Lecture Notes on Gauge Theory. In section 8.2 page 377, he claimed that for the Abelian-Higgs model,
$$S=\int d^{3}x\left(-\frac{1}{4e^{2}}F_{\mu\nu}F^{\mu\nu}+|D_{\mu}\phi|^{2}-m^{2}|\phi|^{2}-\lambda|\phi|^{4}\right)$$
where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, the topological symmetry associated with $J=\ast F$ is unbroken in the Higgs phase when the Dirac monopole is present.
Can anybody help me understand why the generalized global symmetry in this case in unbroken even when the monopole is present?
