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/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 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 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/Z(N). The formula is shown to hold for any pattern.