Fermion Generations from "Apple-Shaped" Extra Dimensions

TL;DR

This paper explores how a specific 'apple-shaped' extra-dimensional surface with conical singularities can naturally produce three distinct massless fermion modes in four dimensions, corresponding to the three fermion generations.

Contribution

It demonstrates that a conical extra-dimensional manifold can generate three fermion generations distinguished by angular momentum, offering a geometric origin for fermion families.

Findings

Three massless fermion modes are localized in four dimensions.

Different modes have distinct angular momentum values.

The setup provides a geometric explanation for fermion generations.

Abstract

We examine the behavior of fermions in the presence of an internal compact 2-manifold which in one of the spherical angles exhibits a conical character with an obtuse angle. The extra manifold can be pictured as an apple-like surface i.e. a sphere with an extra "wedge" insert. Such a surface has conical singularities at north and south poles. It is shown that for this setup one can obtain, in four dimensions, three trapped massless fermion modes which differ from each other by having different values of angular momentum with respect to the internal 2-manifold. The extra angular momentum acts as the family label and these three massless modes are interpreted as the three generations of fundamental fermions.