Boundary conditions for radial solution of gauged topological vortices

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_iphi}D_iphi + 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^{iNtheta}
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}}{drho}right)^2+left(frac{dwidetilde{phi}}{drho}right)^2+frac{1}{rho^2}(N-A_{theta})^2widetilde{phi}^2+frac{lambda}{4}(widetilde{phi}^2-v^2)^2right],rho drho
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.