I'm reading this paragraph from Tong's gauge theory lecture note.
This is the previous phrase of my question:
This is my understanding of getting $\mathbb{Z}_N$ gauge symmetry from $\mathrm{U}(1)$ gauge symmetry: If the charge of the complex scalar is $N$, we could choose the vev of the scalar field to be $t \, e^{i N \alpha}$ where t is a real number, $\alpha=\frac{x}{N}\pi$ and x takes values $x = 0, 1, ... N-1$ such that the Lagrangian will be in the form where Higgs mechanism is transparent.
The picture I had in mind was
- I have a complex scalar field $\Phi(x)$. I could expand it as \begin{equation} \Phi(x)=(v+\rho(x))e^{i\phi(x)} \end{equation} if it has non-zero expectation value $\langle\Phi(x)\rangle=v$
- I have a $U(1)$ guage symmetry to utilize \begin{equation} \phi(x)\rightarrow \phi(x)+ N \alpha(x) \end{equation}
- I choose $\alpha(x)=-\frac{\phi(x)}{N} +\beta\pi$ where $\beta $ is a constant that could take values in $0,\frac{1}{N}, \cdots ,\frac{N-1}{N}$.
- Therefore, there is an extra $Z_N$ gauge symmetry left because all choices of $\beta$ give the same result.
Later Tong made a comment in his lecture note:
the $\mathrm{U}(1)$ gauge theory for A is actually Higgsed down to $\mathbb{Z}_N$, a fact which is clear in our initial formulation in $L_1$.
Is this just simply saying that the scalar field is of charge $N$ in $\mathrm{U}(1)$ symmetry which is transparent from the kinetic term in $L_1$? ($L_1$ is defined in the quote below)
Here we take the $U(1)$ gauge theory as our starting point. We can focus on the phase, $\phi\in[0,2\pi)$ of the scalar field. We have a gauge symmetry $\phi\to\phi + Nα$ where $α ∼ α + 2π$ is also periodic. In the Higgs phase, the scalar kinetic term is $$\mathcal{L}_1 = t^2 (d\phi − N A) ∧ \star (d\phi − N A)$$ for some $t ∈ \mathbf{R}$ which is set by the expectation value of the scalar. In the low-energy limit, $t^2 → ∞$ and we have $A = \frac{1}{N} d\phi$ which tells us that the connection must be flat. However, something remains because the holonomy around any non-contractible loop can be 2π $\frac{1}{2\pi}\oint A ∈ \frac{1}{N} \mathbf{Z}$.
Another question is why emphasizing the flat connection. Is this because he wants to emphasize the holonomy around any loop can be calculated?