Revisiting modular symmetry in magnetized torus and orbifold compactifications

TL;DR

This paper investigates the modular symmetry properties of wavefunctions in magnetized torus and orbifold compactifications, revealing their transformation as modular forms and group representations, with implications for string theory models.

Contribution

It provides a detailed analysis of how zero-mode wavefunctions transform under modular groups in magnetized torus and orbifold compactifications, including the effects of twists and shifts.

Findings

Wavefunctions behave as modular forms of weight 1/2

Zero-modes form representations of quotient groups of modular groups

Decomposition of wavefunctions on orbifolds into smaller representations

Abstract

We study the modular symmetry in $T^2$ and orbifold comfactifications with magnetic fluxes. There are $|M|$ zero-modes on $T^2$ with the magnetic flux $M$. Their wavefunctions as well as massive modes behave as modular forms of weight $1/2$ and represent the double covering group of $\Gamma \equiv SL(2,\mathbb{Z})$, $\widetilde{\Gamma} \equiv \widetilde{SL}(2,\mathbb{Z})$. Each wavefunction on $T^2$ with the magnetic flux $M$ transforms under $\widetilde{\Gamma}(2|M|)$, which is the normal subgroup of $\widetilde{SL}(2,\mathbb{Z})$. Then, $|M|$ zero-modes are representations of the quotient group $\widetilde{\Gamma}'_{2|M|} \equiv \widetilde{\Gamma}/\widetilde{\Gamma}(2|M|)$. We also study the modular symmetry on twisted and shifted orbifolds $T^2/\mathbb{Z}_N$. Wavefunctions are decomposed into smaller representations by eigenvalues of twist and shift. They provide us with reduction of reducible representations on $T^2$.