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

TL;DR

This paper investigates the relationship between magnetic flux, zero modes, and winding numbers on orbifolds of the torus with specific Z_N symmetries, deriving index formulas for different N values.

Contribution

It derives explicit index formulas linking magnetic flux, winding numbers, and zero mode counts on T^2/Z_N orbifolds for N=2,3,4,6.

Findings

Derived index formula for N=2 using trace formula.

Generalized index formulas for N=3,4,6 under assumptions.

Established connection between winding numbers and zero mode differences.

Abstract

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