Zero-mode counting formula and zeros in orbifold compactifications

We thoroughly analyze the number of independent zero modes and their zero points on the toroidal orbifold $T^2/\mathbb{Z}_N$ ($N = 2, 3, 4, 6$) with magnetic flux background, inspired by the Atiyah-Singer index theorem. We first show a complete list for the number $n_{\eta}$ of orbifold zero modes belonging to $\mathbb{Z}_{N}$ eigenvalue $\eta$. Since it turns out that $n_{\eta}$ quite complicatedly depends on the flux quanta $M$, the Scherk-Schwarz twist phase $(\alpha_1, \alpha_2)$, and the $\mathbb{Z}_{N}$ eigenvalue $\eta$, it seems hard that $n_{\eta}$ can be universally explained in a simple formula. We, however, succeed in finding a single zero-mode counting formula $n_{\eta} = (M-V_{\eta})/N + 1$, where $V_{\eta}$ denotes the sum of winding numbers at the fixed points on the orbifold $T^2/\mathbb{Z}_N$. The formula is shown to hold for any pattern.