I have been taking a course on non-perturbative physics and currently the teacher is away so I cannot ask him.

In the lectures, he made the claim that a pair of vortices in the abelian-Higgs model should scatter perpendicular to each other.

The argument appears to be the following:

1). In the moduli space for a pair of vortices there are 2x2=4 coordinates. Take one pair of coordinates to describe the COM of the pair. The remaining pair of coordinates describe their vector separation.

2). Owing to the symmetry of the space under interchange of coordinates, the moduli space is $\mathbb{R^2}/\mathbb{Z_2}$.

3). As the physical displacement vector is rotated through $180^{\circ}$, then in the physics space, because of the $/\mathbb{Z_2}$, the physical angle is $180^{\circ}/2=90^{\circ}$.

I have two problems with this:

Firstly, I just do not understand what the third point means.

Secondly, this kind of argument seems too general and surely we could conclude that any pair of identical objects scatter at right-angles to their initial motion (not just the vortices of the abelian-Higgs model). In which case - why the fancy argument?



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