Index theorem on T2/ZN orbifolds

We investigate chiral zero modes and winding numbers at fixed points on T2/ZN 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± are the numbers of the ± chiral zero modes and V± are the sums of the winding numbers at the fixed points on T2/ZN. This formula is complementary to our zero-mode counting formula on the magnetized orbifolds with nonzero flux background M≠0, consistently with substituting M=0 for the counting formula n+−n−=(2M−V++V−)/2N.