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arXiv · 2010.14214

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

Makoto Sakamoto·Maki Takeuchi·Yoshiyuki Tatsuta
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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 $M \neq 0$, consistently with substituting $M = 0$ for the counting formula $n_+ - n_- = (2M - V_+ + V_-)/2N$.

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Makoto Sakamoto, Maki Takeuchi, Yoshiyuki Tatsuta. 2020-10-27. Index theorem on $T^2/\mathbb{Z}_N$ orbifolds. https://doi.org/10.1103/physrevd.103.025009

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