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 $T^2\mathrm{/}{\mathbb{Z}}_N$ ($N=2,3,4,6$) 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 $T^2\mathrm{/}{\mathbb{Z}}_N$. The formula is shown to hold for any pattern.