Supersymmetry in 6d Dirac Action

TL;DR

This paper explores a 6d Dirac fermion model on a rectangle, revealing supersymmetry structures that classify boundary conditions and allow for massless chiral fermions, potentially addressing fermion generation issues.

Contribution

It demonstrates the role of supersymmetry in classifying boundary conditions and expanding the 6d Dirac field in Kaluza-Klein modes, even with bulk mass.

Findings

4d spectrum governed by N=2 supersymmetric quantum mechanics

Boundary conditions determine positive-energy solutions

Massless chiral fermions can arise despite bulk mass

Abstract

We investigate a 6d Dirac fermion on a rectangle. It is found that the 4d spectrum is governed by $N=2$ supersymmetric quantum mechanics. Then we demonstrate that the supersymmetry is very useful for classifying all the allowed boundary conditions and to expand the 6d Dirac field in Kaluza-Klein modes. A striking feature of the model is that even though the 6d Dirac fermion has non-vanishing bulk mass, the 4d mass spectrum can contain degenerate massless chiral fermions, which may provide a hint to solve the problem of the generation of quarks and leptons. It is pointed out that zero-energy solutions are not affected by the presence of the boundaries, while the boundary conditions work well for determining the positive-energy solutions. We also provide a brief discussion on possible boundary conditions in the general case, especially those on polygons.