Index theorem on $T^2/{\mathbb{Z}}_N$ orbifolds

Abstract

We investigate chiral zero modes and winding numbers at fixed points on $T^2/{\mathbb{Z}}_N$ orbifolds. It is shown that the Atiyah-Singer index theorem for the chiral zero modes leads to a formula $n_{+}-n_{-}=(-V_{+}+V_{-})/2N$, 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 $T^2/{\mathbb{Z}}_N$. This formula is complementary to our zero-mode counting formula on the magnetized orbifolds with non-zero flux background $M\ne 0$, consistently with substituting $M=0$ for the counting formula $n_{+}-n_{-}=(2M-V_{+}+V_{-})/2N$.

arXiv comment33 pages, 3 figures

References in corpus (56)

• Evidence for F-Theory
• Bounds on Universal Extra Dimensions
• Bulk Fields and Supersymmetry in a Slice of AdS
• Hierarchies without Symmetries from Extra Dimensions
• Chiral Four-Dimensional N=1 Supersymmetric Type IIA Orientifolds from Intersecting D6-Branes
• Triplet-doublet Splitting, Proton Stability and Extra Dimension
• Fermion Masses, Mixings and Proton Decay in a Randall-Sundrum Model
• Type-I strings on magnetised orbifolds and brane transmutation
• Noncommutative Compactifications of Type I Strings on Tori with Magnetic Background Flux
• A Mini-Landscape of Exact MSSM Spectra in Heterotic Orbifolds
• Getting just the Standard Model at Intersecting Branes
• Searching for realistic 4d string models with a Pati-Salam symmetry -- Orbifold grand unified theories from heterotic string compactification on a Z6 orbifold
• A Minimal S^1/(Z_2 x Z_2') Orbifold GUT
• Computing Yukawa Couplings from Magnetized Extra Dimensions
• Yukawa couplings in intersecting D-brane models
• Supersymmetric Standard Model from the Heterotic String
• A WAY TO BREAK SUPERSYMMETRY
• The Heterotic Road to the MSSM with R parity
• Gauge Unification In Six Dimensions
• Building MSSM Flux Vacua
• D=4,N=1, Type IIB Orientifolds
• New Phases of String Theory and 6d RG Fixed Points via Branes at Orbifold Singularities
• Flavor Hierarchy From F-theory
• Anomalies on Orbifolds
• Supersymmetric Standard Model from the Heterotic String (II)
• Families as Neighbors in Extra Dimension
• MSSM vacua from Flux Compactifications
• Magnetized orbifold models
• Anomalies in orbifold field theories
• Chiral Compactification on a Square
• Supersymmetry Breaking, Fermion Masses and a Small Extra Dimension
• Three generation magnetized orbifold models
• Three fermionic generations on a topological defect in extra dimensions
• Six-Dimensional Gauge Theory on the Chiral Square
• $Z_N$ twisted orbifold models with magnetic flux
• Three generations on a local vortex in extra dimensions
• Fermion Generations from "Apple-Shaped" Extra Dimensions
• Classification of three-generation models on magnetized orbifolds
• Realistic Yukawa Structures from Orientifold Compactifications
• Spacetime as a topological insulator: Mechanism for the origin of the fermion generations
• Gaussian Froggatt-Nielsen mechanism on magnetized orbifolds
• Operator analysis of physical states on magnetized $T^2/Z_N$ orbifolds
• CP-violating phase on magnetized toroidal orbifolds
• Fermion masses and quantum numbers from extra dimensions
• Fermion generations, masses and mixings in a 6D brane model
• Quark mass hierarchy and mixing via geometry of extra dimension with point interactions
• $CP$ phase from twisted Higgs vacuum expectation value in extra dimension
• Flux Stabilization in 6 Dimensions: D-terms and Loop Corrections
• Chiral fermions and anomaly cancellation on orbifolds with Wilson lines and flux
• Fermion Families from Two Layer Warped Extra Dimensions
• Zero-mode counting formula and zeros in orbifold compactifications
• Dynamical generation of fermion mass hierarchy in an extra dimension
• 6d Dirac fermion on a rectangle; scrutinizing boundary conditions, mode functions and spectrum
• Magnetized orbifolds and localized flux
• Supersymmetry in 6d Dirac Action
• Transitions of Orbifold Vacua

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• 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 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 and zeros in magnetic compactifications
• Mass spectrum in a six-dimensional $SU(n)$ gauge theory on a magnetized torus