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arXiv · 2211.04595

Index theorem on magnetized blow-up manifold of $T^2/\mathbb{Z}_N$

Tatsuo Kobayashi·Hajime Otsuka·Makoto Sakamoto·Maki Takeuchi·Yoshiyuki Tatsuta·Hikaru Uchida
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Abstract

We investigate blow-up manifolds of $T^2/{\mathbb{Z}}_N\,(N=2,3,4,6)$ orbifolds with magnetic flux $M$. Since the blow-up manifolds have no singularities, we can apply the Atiyah-Singer index theorem to them. Then, we establish the zero-mode counting formula $n_{+}-n_{-}=(M-V_{+})/N+1$, where $V_{+}$ denotes the sum of winding numbers at fixed points on the $T^2/{\mathbb{Z}}_N$ orbifolds, as the Atiyah-Singer index theorem on the orbifolds, and clarify physical and geometrical meanings of the formula.

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Tatsuo Kobayashi, Hajime Otsuka, Makoto Sakamoto, Maki Takeuchi, Yoshiyuki Tatsuta, Hikaru Uchida. 2022-11-08. Index theorem on magnetized blow-up manifold of $T^2/\mathbb{Z}_N$. https://doi.org/10.1103/physrevd.107.075032

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