We analyze the number of independent chiral zero modes and the winding numbers at the fixed points on $T^2/{ \mathbb{ Z}}_N$ ($N=2,3,4,6$) orbifolds with magnetic flux. In the case of $N=2$, we derive the index formula $n_{ +}-n_{ -}=M/2+(-V_{ +}+V_{ -})/4=M/2-V_{ +}/2+1$ by using the trace formula, 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 $T^2/{ \mathbb{ Z}}_2$. We also obtain the formula $n_{ +}-n_{ -}=M/N+(-V_{ +}+V_{ -})/(2N)=M/N-V_{ +}/N+1$ for $N=3,4,6$ under an assumption.
Categories: hep-th
