Dirac Operator Zero-modes on a Torus

TL;DR

This paper analyzes zero-modes of the Dirac operator on a torus with uniform gauge fields, relating wave functions to holomorphic functions and confirming their properties align with the index theorem.

Contribution

It constructs explicit zero-mode wave functions on a torus with twisted periodic conditions, linking them to holomorphic functions and clarifying their degeneracy and chirality.

Findings

Zero-modes are related to holomorphic functions on the complex torus.

Degeneracy and chirality are determined by the gauge background.

Results are consistent with the index theorem.

Abstract

We study Dirac operator zero-modes on a torus for gauge background with uniform field strengths. Under the basic translations of the torus coordinates the wave functions are subject to twisted periodic conditions. In a suitable torus coordinates the zero-mode wave functions can be related to holomorphic functions of the complex torus coordinates. We construct the zero-mode wave functions that satisfy the twisted periodic conditions. The chirality and the degeneracy of the zero-modes are uniquely determined by the gauge background and are consistent with the index theorem.