Vector potential $A$ on a 2-sphere $S^2$ of radius $R$ with some points removed I am preparing myself for an exam and I got stuck with the following problem.
If I wanted to calculate the vector potential $A$ on a sphere (not off or in), where some points are removed, how would I have to tackle this problem effectively?
If I remove a point on the $2$-plane the standard solution is given by $A \propto \frac{1}{r^2}(-y,x,0)$, which is closed but not exact (due to the removed point at the origin). Can I lift this to $S^2$ and argue similarly?
 A: This vector potential can be written in every point on the plane except the origin as:
$$ A_x = -\frac{\partial  \psi}{\partial y}$$
$$ A_y = \frac{\partial \psi}{\partial x} $$
with
$$\psi = \frac{1}{2}\mathrm{log}(x^2+y^2)$$
$A$ is not exact, because $\psi$ is singular at the origin. But this means that the magnetic field is zero at every point except the origin. At the origin itself, the magnetic field must be infinite, because the flux through an arbitrary small loop is nonvanishing:
$$\Phi = \int A = \int_0^{2\pi} d\phi = 2 \pi$$
Such a magnetic field can be generated by a infinite solenoid whose radius is shrunk to zero while keeping the flux constant. 
In the hydrodynamical terminology, the function $\psi$ is called the stream function, it satisfies the Laplace's equation (harmonic function) except at the origin. This specific stream function describes a vortex
(The vector potential describes the velocity field of the vortex). The streamlines of this velocity field are circles around the origin and its magnitude is inversely proportional to the radius.
In order to see that the stream function is harmonic except at the singularity, and generalize the construction to the case of the sphere, we can use complex coordinates on the plane:
$$ z = x+iy$$
In this representation we have:
$$\psi = \frac{1}{2}\mathrm{log}(\bar{z}z)$$
Applying the Laplace operator, we get
$$\nabla^2 \psi=\partial_{\bar{z}} \partial_z \psi = \delta^2_L(z)$$
Where we have used 
$$ \frac{\partial}{\partial \bar{z}}\frac{1}{z} = \delta^{(2)}_L(z)$$
is the complex coordinate on the plane. $\delta^{(2)}_L$ is the two dimensional Dirac delta function with respect to the Lebesgue measure. i.e., 
$$\int_{\mathbb{C}} f(z) \delta^{(2)}_L(z-z_0) \mathrm{dRe}(z) \mathrm{dIm}(z) = f(z_0) $$
The vector potential in the complex representation has the form:
$$ A_z = \frac{1}{i}\frac{\partial  \psi}{\partial \bar{z}}$$
$$ A_{\bar{z}}= -\frac{1}{i}\frac{\partial \psi}{\partial z} $$
Explicitly:
$$A = \frac{1}{2i}\frac{zd\bar{z}-\bar{z}dz}{\bar{z}z}$$
This fact describes another physical interpretation of this vector potential as follows:
In two dimensions a function satisfying the Laplace's equation (harmonic function) (except at the point singularities) qualifies to be a stream function whose anti-symmetrized gradient (which is the vector potential in our problem) describes the velocity field of a vortex.
Please observe that that this velocity field is invariant under rotations about the origi n and its magnitude is inversely proportional to the distance from the origin. In this interpretation, the line integral of the vector potential is the vorticity. 
We can change the location of the singularity (flux line) to any other point in the plane, say $(x_0, y_0)$
$$A = \frac{(x-x_0)dy - (y-y_0)dx}{(x-x_0)^2+(y-y_0)^2}  = \frac{1}{2i}\frac{(z-z_0)d\bar{z}-(\bar{z}- \bar{z_0})dz}{(\bar{z}- \bar{z_0})(z-z_0)} $$
In this case it is not hard to verify that this vector potential can be derived from the stream function :
$$\psi = \frac{1}{2}\mathrm{log}((\bar{z}-\bar{z}_0)(z-z_0)) \equiv \mathrm{log}(|z-z_0|)$$
We can add several stream functions centered at various points on the plane with different vorticities to obtain a general solution representing fluxes at these points:
$$\psi =  \sum_k \Gamma_k \mathrm{log}(|z-z_k|)$$
The constant $\Gamma_k $ expresses the fluxes around the $k$-th center (or the vorticity in the hydrodynamical terminology).
One can easily verify that the single centered vector potential (and also the corresponding stream function) are invariants under the metric
preserving automorphisms  of the plane consisting of translations 
and rotations: (which can be compactly written in the complex notations
as:)
$$z \rightarrow e^{i\alpha} z + v$$
$$z_0  \rightarrow  e^{i\alpha} z_0 + v$$
From the  expression of the single centered stream function, one observes that the denominator is the geodesic distance on the plane, thus a candidate generalization to the sphere ($S^2$) would be the replacement of this by geodesic distance on the sphere:
$$|z-z_0|^2 \rightarrow \frac{|z-z_0|^2}{(1+\bar{z}z)(1+\bar{z}_0z_0)}$$
Where $z$ is the stereographic projection coordinate on the sphere:
$$ z = \mathrm{tan}{\frac{\theta}{2}}e^{i \phi}$$
($\theta$ and $\phi$ are the spherical surface coordinates).
Thus the candidate solution on the sphere is:
$$\psi= \mathrm{log}(|z-z_0| - \frac{1}{2} \mathrm{log} (1+\bar{z}z)- \frac{1}{2} \mathrm{log} (1+\bar{z}_0z_0) )$$
This solution is invariant under the metric preserving automorphisms of
the sphere:
$$ z\rightarrow \frac{\alpha z + \beta}{-\bar{\beta} z +\bar{\alpha} }$$,
with $|\alpha|^2+|\beta|^2 = 1$
The matrix: 
$$\begin{pmatrix}
\alpha & \beta\\ 
 -\bar{\beta}&\bar{\alpha}& 
 \end{pmatrix} \in SU(2)$$
which is the automorphism group of the round metric
Thus the candidate vector potential corresponding to this solution is
obtained by applying the gradient operator in the sphere's curvlinear
coordinates:
$$ A_z = \frac{1}{i}(1+\bar{z}z)^2 \frac{\partial  \psi}{\partial \bar{z}}$$
$$ A_{\bar{z}}= -\frac{1}{i}(1+\bar{z}z)^2 \frac{\partial \psi}{\partial z} $$
Explicitly
$$A = \frac{1}{2i} (1+\bar{z}z)  (1+\bar{z}_0z_0) \frac{(z-z_0)(1+\bar{z}_0 z)d\bar{z}-(\bar{z}- \bar{z_0}) (1+\bar{z}_0 z) dz}{(\bar{z}- \bar{z_0})(z-z_0)}$$
The Laplacian on the sphere is given by:
$$\nabla^2 = (1+\bar{z}z)^2 \frac{\partial}{\partial z}\frac{\partial}{\partial\bar{z}}$$
Applying the Laplacian operator to the candidate stream function, we obtain:
$$\nabla^2 \psi=  (1+\bar{z}z)^2  \delta^{(2)}_L(z-z_0)  +1 = \delta^{(2)}_S(z-z_0)  +1$$ 
Where $ \delta^{(2)}_S$ is the Dirac delta function corresponding to the spherical measure:
$$\int_{\mathbb{S^2}} f(z) \delta^{(2)}_S(z-z_0) \frac{ \mathrm{dRe}(z) \mathrm{dIm}(z) }{ (1+\bar{z}z) ^2}= f(z_0) $$
The additional constant term in the Laplacian constitutes a problem because it means that this stream function is not harmonic outside the singularities. The solution to this problem on the sphere is to add up several solutions with a vanishing total flux (vorticity). 
$$\sum_k \Gamma_k = 0$$
In this case the constant contributions from all centers will cancel.
