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

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.

30 pages

References in corpus (32)

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Cited by in corpus (38)

• Neutrino mixing from finite modular groups
• Modular $A_4$ invariance and neutrino mixing
• CP violation of quarks in $A_4$ modular invariance
• Finite modular subgroups for fermion mass matrices and baryon/lepton number violation
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• Modular forms of finite modular subgroups from magnetized D-brane models
• Modular $A_4$ invariance and leptogenesis
• $A_4$ lepton flavor model and modulus stabilization from $S_4$ modular symmetry
• Modular Flavor Symmetry on Magnetized Torus
• Modular $S_3$ invariant flavor model in SU(5) GUT
• Revisiting modular symmetry in magnetized torus and orbifold compactifications
• Type II seesaw models with modular $A_4$ symmetry
• Modular invariant flavor model of ${\mathrm{A}}_4$ and hierarchical structures at nearby fixed points
• $CP$ violation in modular invariant flavor models
• Modular flavor symmetries of three-generation modes on magnetized toroidal orbifolds
• Classification of discrete modular symmetries in Type IIB flux vacua
• Modular symmetry anomaly in magnetic flux compactification
• Towards unification of quark and lepton flavors in $A_4$ modular invariance
• Broken Scaling Neutrino Mass Matrix and Leptogenesis based on A${}_4$ Modular invariance
• Loop Fayet-Iliopoulos terms in $T^2/Z_2$ models: instability and moduli stabilization
• Quark and lepton hierarchies from $S_4^{\prime }$ modular flavor symmetry
• Quark and lepton flavors with common modulus $\tau$ in $A_4$ modular symmetry
• Non-invertible flavor symmetries in magnetized extra dimensions
• Classification of three generation models by orbifolding magnetized $T^2\times T^2$
• Zero-mode counting formula and zeros in orbifold compactifications
• Finite modular axion and radiative moduli stabilization
• Index theorem on magnetized blow-up manifold of $T^2/{\mathbb{Z}}_N$
• Classifications of magnetized $T^4$ and $T^4/Z_2$ orbifold models
• Index and winding numbers on $T^2/{\mathbb{Z}}_N$ orbifolds with magnetic flux
• Number of zero-modes on magnetized $T^4/Z_N$ orbifolds analyzed by modular transformation
• Modular symmetry anomaly and non-perturbative neutrino mass terms in magnetized orbifold models
• Zero-modes in magnetized $T^6/{\mathbb{Z}}_N$ orbifold models through $Sp(6,\mathbb{Z})$ modular symmetry
• Quark mass matrices in magnetized orbifold models with localized Fayet-Iliopoulos terms
• Majorana neutrino masses by D-brane instanton effects in magnetized orbifold models
• Zero-mode product expansions and higher order couplings in gauge backgrounds
• Trace conservation laws in $T^2/Z_m$ orbifold gauge theories
• Yukawa couplings from magnetized D-brane models on non-factorisable tori
• Yukawa textures in modular symmetric vacuum of magnetized orbifold models