Index-Like Theorems from Line Defect Vevs

In this paper we investigate the relation between complexified Fenchel-Nielsen coordinates and spectral network coordinates on Seiberg-Witten moduli space. The main technique is the comparison of exact expressions for the expectation value of 't Hooft defects in certain 4D $SU(2)$ $\mathcal{N}=2$ gauge theories. We derive an index-like theorem for a class of Dirac operators on singular monopole moduli spaces. Our expression determines the indices of Dirac operators on singular monopole moduli spaces in terms of characteristic numbers for vector bundles over certain Kronheimer-Nakajima quiver varieties.