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

TL;DR

This paper derives a formula relating chiral zero modes and winding numbers on $T^2/\mathbb{Z}_N$ orbifolds, extending the index theorem to orbifold fixed points and connecting it with flux background scenarios.

Contribution

It presents a new index theorem formula for chiral zero modes on orbifolds, linking winding numbers at fixed points to zero mode counts, and complements existing flux-based counting methods.

Findings

Derived the index theorem formula for $T^2/\mathbb{Z}_N$ orbifolds.

Established the relationship between winding numbers and chiral zero modes.

Connected the zero-mode counting with flux background scenarios.

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 $n_+ - n_- = (2M - V_+ + V_-)/2N$.