I am following the book Topological Solitons by Manton and Sutcliffe and I am struggling to understand a boundary condition they choose to find the radial solutions of gauged vortices with finite energy.
The energy of the vortices is given by
\begin{equation} \label{gauged_static_energy_functional} E = \int_{}^{} \left(\frac{1}{2}\overline{D_i\phi}D_i\phi + \frac{1}{2}B^2 +U(\overline{\phi}\phi)\right)\,d^2x \ \end{equation}
It then assumes the ansatz:
\begin{equation} \begin{aligned} \phi(\rho, \theta) &= \widetilde{\phi}(\rho)e^{iN\theta} \\ A_{\rho}(\rho, \theta) &= A_{\rho}(\rho) \\ A_{\theta}(\rho, \theta) &= A_{\theta}(\rho) \end{aligned} \end{equation}
And uses a gauge transformation to set $A_\rho=0$. The energy for fields of this form is then evaluated:
\begin{equation} \label{polar_gauged_static_energy_functional} E = \pi \int_{0}^{\infty} \left[\frac{1}{\rho^2}\left(\frac{dA_{\theta}}{d\rho}\right)^2+\left(\frac{d\widetilde{\phi}}{d\rho}\right)^2+\frac{1}{\rho^2}(N-A_{\theta})^2\widetilde{\phi}^2+\frac{\lambda}{4}(\widetilde{\phi}^2-v^2)^2\right]\,\rho d\rho \ \end{equation}
Now it says that for this quantity to be finite, it requires $\lim_{\rho \to \infty} A_{\theta}(\rho) = N$ and $\lim_{\rho \to \infty} \widetilde{\phi}(\rho) = v$. I understand these are required so the energy density is $0$ at infinity. Then it requires $\widetilde{\phi}(0)=0$ which I understand is necessary so that the third term in the last expression doesn't diverge at the origin. However, I really don't know where the last condition, $A_{\theta}(0) = 0$, comes from.
