Index and winding numbers on $T^2\mathrm{/}{\mathbb{Z}}_N$ orbifolds with magnetic flux

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