6d Dirac fermion on a rectangle; scrutinizing boundary conditions, mode functions and spectrum

Abstract

We classify possible boundary conditions of a 6d Dirac fermion $\Psi$ on a rectangle under the requirement that the 4d Lorentz structure is maintained, and derive the profiles and spectrum of the zero modes and nonzero KK modes under the two specific boundary conditions, (i) 4d-chirality positive components being zero at the boundaries and (ii) 2d-chirality positive components being zero at the boundaries. In the case of (i), twofold degenerated chiral zero modes appear which are localized towards specific directions of the rectangle pointed by an angle parameter $\theta$. This leads to an implication for a new direction of pursuing the origin of three generations in the matter fields of the standard model, even though triple-degenerated zero modes are not realized in the six dimensions. The emergence of the angle parameter $\theta$ originates from a rotational symmetry in the degenerated chiral zero modes on the rectangle extra dimensions since they do not feel the boundaries. In the case of (ii), this rotational symmetry is promoted to the two-dimensional conformal symmetry though no chiral massless zero mode appears. We also discuss the correspondence between our model on a rectangle and orbifold models in some details.

26 pages, 2 tables (v1); 39 pages, 3 tables, accepted to Nuclear Physics B (v2)

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• Index theorem on $T^2/{\mathbb{Z}}_N$ orbifolds
• 5d Dirac fermion on quantum graph
• Extended supersymmetry with central charges in Dirac action with curved extra dimensions
• Extended supersymmetry with central charges in higher dimensional Dirac action
• Quark and Lepton Mass and Mixing with non-universal $Z^{\prime }$ from a 5d Standard Model with gauged $SO(3)$
• Extended supersymmetry in Dirac action with extra dimensions