Zero-modes on orbifolds : magnetized orbifold models by modular transformation

TL;DR

This paper presents a systematic, modular transformation-based method to analyze zero-modes in magnetized orbifold models, providing a more direct and analytical approach than previous methods.

Contribution

It introduces a new analytical framework for counting zero-modes on orbifolds with magnetic fluxes using modular transformations, improving upon prior approaches.

Findings

Zero-mode number on $T^2/Z_N$ is generally $loor{M/N} + 1$.

The method aligns with previous results but is more direct.

The approach is consistent with the index theorem for Dirac operators.

Abstract

We study $T^2/Z_N$ orbifold models with magnetic fluxes. We propose a systematic way to analyze the number of zero-modes and their wavefunctions by use of modular transformation. Our results are consistent with the previous results, and our approach is more direct and analytical than the previous ones. The index theorem implies that the zero-mode number of the Dirac operator on $T^2$ is equal to the index $M$, which corresponds to the magnetic flux in a certain unit. Our results show that the zero-mode number of the Dirac operator on $T^2/Z_N$ is equal to $\lfloor M/N \rfloor +1$ except one case on the $T^2/Z_3$ orbifold.