Index theorem on magnetized blow-up manifold of $T^2\mathrm{/}{\mathbb{Z}}_N$

We investigate blow-up manifolds of $T^2\mathrm{/}{\mathbb{Z}}_N(N=2,3,4,6)$ orbifolds with magnetic flux $M$. Since the blow-up manifolds have no singularities, we can apply the Atiyah-Singer index theorem to them. Then, we establish the zero-mode counting formula $n_{+}-n_{-}=(M-V_{+})\mathrm{/}N+1$, where $V_{+}$ denotes the sum of winding numbers at fixed points on the $T^2\mathrm{/}{\mathbb{Z}}_N$ orbifolds, as the Atiyah-Singer index theorem on the orbifolds, and clarify physical and geometrical meanings of the formula.