Orbifold resolutions and fermion localization

TL;DR

This paper investigates how fermions behave on a regularized orbifold space by replacing singularities with spherical caps, revealing constraints on fermion configurations and their localization in the orbifold limit.

Contribution

It introduces a method to resolve orbifold singularities with spherical caps and analyzes the resulting constraints on fermion chiralities and charges.

Findings

Localized and bulk fermions emerge as the resolved space approaches the orbifold limit.

Not all fermion configurations on T^2/Z_2 admit simple resolutions.

Resolution imposes constraints on fermion chiralities and U(1) charges.

Abstract

We study the Dirac equation of chiral fermions on a regularized version of the two-dimensional T^2/Z_2 orbifold, where the conical singularities are replaced by suitable spherical caps with constant curvature. This study shows how localized and bulk fermions arise in the orbifold as the resolved space approaches the orbifold limit. Our analysis also shows that not all possible fermion configurations on T^2/Z_2 admit such a simple resolution. We focus our study to a fermion coupled to a U(1) gauge field. It is explicitly shown how a resolution of the orbifold puts severe constraints on the allowed chiralities and U(1) charges of the massless four dimensional fermions, localized or not, that can be present in the orbifold. The limit in which T^2/Z_2 (and its corresponding resolved space) collapses to S^1/Z_2 is also studied in detail.