Interaction energy between vortices of vector fields on Riemannian surfaces

We study a variational Ginzburg-Landau type model depending on a small parameter $\epsilon>0$ for (tangent) vector fields on a $2$-dimensional Riemannian surface. As $\epsilon\to 0$, the vector fields tend to be of unit length and will have singular points of a (non-zero) index, called vortices. Our main result determines the interaction energy between these vortices as a $\Gamma$-limit (at the second order) as $\epsilon\to 0$.