Zero-mode counting formula and zeros in orbifold compactifications

Abstract

We thoroughly analyze the number of independent zero modes and their zero points on the toroidal orbifold $T^2/{\mathbb{Z}}_N$ ($N=2,3,4,6$) with magnetic flux background, inspired by the Atiyah-Singer index theorem. We first show a complete list for the number $n_{\eta }$ of orbifold zero modes belonging to ${\mathbb{Z}}_N$ eigenvalue $\eta$. Since it turns out that $n_{\eta }$ quite complicatedly depends on the flux quanta $M$, the Scherk-Schwarz twist phase $({\alpha }_1,{\alpha }_2)$, and the ${\mathbb{Z}}_N$ eigenvalue $\eta$, it seems hard that $n_{\eta }$ can be universally explained in a simple formula. We, however, succeed in finding a single zero-mode counting formula $n_{\eta }=(M-V_{\eta })/N+1$, where $V_{\eta }$ denotes the sum of winding numbers at the fixed points on the orbifold $T^2/{\mathbb{Z}}_N$. The formula is shown to hold for any pattern.

40 pages, 3 figures

References in corpus (22)

• Four-dimensional String Compactifications with D-Branes, Orientifolds and Fluxes
• Toward Realistic Intersecting D-Brane Models
• Open Strings
• Type-I strings on magnetised orbifolds and brane transmutation
• Noncommutative Compactifications of Type I Strings on Tori with Magnetic Background Flux
• Computing Yukawa Couplings from Magnetized Extra Dimensions
• A WAY TO BREAK SUPERSYMMETRY
• Three generation magnetized orbifold models
• $Z_N$ twisted orbifold models with magnetic flux
• Classification of three-generation models on magnetized orbifolds
• Zero-modes on orbifolds : magnetized orbifold models by modular transformation
• Gaussian Froggatt-Nielsen mechanism on magnetized orbifolds
• Operator analysis of physical states on magnetized $T^2/Z_N$ orbifolds
• Comprehensive Analysis of Yukawa Hierarchies on $T^2/Z_N$ with Magnetic Fluxes
• CP-violating phase on magnetized toroidal orbifolds
• Flavour mixings in flux compactifications
• Split symmetries
• Inflation from periodic extra dimensions
• Flavour physics without flavour symmetries
• Chiral fermions and anomaly cancellation on orbifolds with Wilson lines and flux
• Magnetized orbifolds and localized flux
• Proton decay in flux compactifications

Cited by in corpus (10)

• Classification of three generation models by orbifolding magnetized $T^2\times T^2$
• Index theorem on magnetized blow-up manifold of $T^2/{\mathbb{Z}}_N$
• Index theorem on $T^2/{\mathbb{Z}}_N$ orbifolds
• Modular symmetry and zeros in magnetic compactifications
• 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
• Mass spectrum in a six-dimensional $SU(n)$ gauge theory on a magnetized torus
• Three-generation solutions of equations of motion in heterotic supergravity