  We investigate chiral zero modes and winding numbers at fixed points on
$T^2/\mathbb{Z}_N$ orbifolds. It is shown that the Atiyah-Singer index theorem
for the chiral zero modes leads to a formula $n_+-n_-=(-V_++V_-)/2N$, where
$n_{\pm}$ are the numbers of the $\pm$ chiral zero modes and $V_{\pm}$ are the
sums of the winding numbers at the fixed points on $T^2/\mathbb{Z}_N$. This
formula is complementary to our zero-mode counting formula on the magnetized
orbifolds with non-zero flux background $M \neq 0$, consistently with
substituting $M = 0$ for the counting formula $n_+ - n_- = (2M - V_+ +
V_-)/2N$.
