Index theorem on $T^2/{\mathbb{Z}}_N$ orbifolds

Abstract

We investigate chiral zero modes and winding numbers at fixed points on $T^2/{\mathbb{Z}}_N$ 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 $n_{\pm }$ are the numbers of the $\pm$ chiral zero modes and $V_{\pm }$ are the sums of the winding numbers at the fixed points on $T^2/{\mathbb{Z}}_N$. This formula is complementary to our zero-mode counting formula on the magnetized orbifolds with non-zero flux background $M\ne 0$, consistently with substituting $M=0$ for the counting formula $n_{+}-n_{-}=(2M-V_{+}+V_{-})/2N$.

33 pages, 3 figures

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Cited by in corpus (1)

• Classification of three generation models by orbifolding magnetized $T^2\times T^2$