Fermionic zero modes in gauge and gravity backgrounds On $T^2$

TL;DR

This paper investigates the existence of normalizable fermionic zero modes in gauge and gravity backgrounds on a two-dimensional torus, revealing conditions under which massless Dirac fermions can exist in such compactified spaces.

Contribution

It demonstrates the presence of normalizable fermionic zero modes on $T^2$ backgrounds under broad assumptions, with explicit examples for gauge backgrounds.

Findings

Massless Dirac fermions with zero modes exist on $T^2$ backgrounds.

Explicit solutions for certain gauge backgrounds are provided.

Zero modes are normalizable under general conditions.

Abstract

In this note we study fermionic zero modes in gauge and gravity backgrounds taking a two dimensional compact manifold $T^2$ as extra dimensions. The result is that there exist massless Dirac fermions which have normalizable zero modes under quite general assumptions about these backgrounds on the bulk. Several special cases of gauge background on the torus are discussed and some simple fermionic zero modes are obtained.