Particular boundary condition ensures that a fermion in d=1+5, compactified on a finite disk, manifests in d=1+3 as massless spinor with a charge 1/2, mass protected and chirally coupled to the gauge field

TL;DR

This paper introduces a boundary condition in a 6D Kaluza-Klein model that guarantees the existence of a massless, chiral fermion with a specific charge in 4D after compactification on a finite disk, even with curved boundaries.

Contribution

It proposes a novel boundary condition for spinors in a 6D model that ensures a massless, chiral fermion with charge 1/2 in 4D, extending previous work to curved disks.

Findings

Massless chiral fermion with charge 1/2 exists in the model.

Boundary conditions can be chosen to control the fermion spectrum.

The operator of momentum is Hermitean on the boundary.

Abstract

The genuine Kaluza-Klein-like theories--with no fields in addition to gravity--have difficulties with the existence of massless spinors after the compactification of some space dimensions \cite{witten}. We proposed in previous paper a boundary condition for spinors in d=(1+5) compactified on a flat disk that ensures masslessness of spinors (with all positive half integer charges) in d=(1+3) as well as their chiral coupling to the corresponding background gauge gravitational field. In this paper we study the same toy model, proposing a boundary condition allowing a massless spinor of one handedness and only one charge (1/2) and infinitely many massive spinors of the same charge, allowing disc to be curved. We define the operator of momentum to be Hermitean on the vector space of spinor states--the solutions on a disc with the boundary.