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

TL;DR

This paper classifies boundary conditions for a 6d Dirac fermion on a rectangle, analyzing resulting mode functions and spectra, with implications for understanding the origin of three generations in the standard model.

Contribution

It provides a detailed classification of boundary conditions maintaining 4d Lorentz symmetry and explores their effects on zero modes and spectra, introducing an angle parameter related to rotational symmetry.

Findings

Degenerated chiral zero modes localized by an angle parameter.

Emergence of a conformal symmetry in certain boundary conditions.

Insights into the origin of three generations in the standard model.

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.