Charge quantization conditions based on the Atiyah--Singer index theorem

TL;DR

This paper derives charge quantization conditions using the Atiyah-Singer index theorem by analyzing the zero-modes of the Dirac operator on a sphere with a magnetic monopole background.

Contribution

It introduces a novel geometric approach to derive Dirac and Schwinger charge quantization conditions via the Atiyah-Singer index theorem.

Findings

Derivation of Dirac's quantization condition from index theorem

Derivation of Schwinger's quantization condition from index theorem

Solution of the massless Dirac equation on a sphere with monopole background

Abstract

Dirac's quantization condition, $eg=n/2$ ($n \in \Bbb Z$), and Schwinger's quantization condition, $eg=n$ ($n \in \Bbb Z$), for an electric charge $e$ and a magnetic charge $g$ are derived by utilizing the Atiyah-Singer index theorem in two dimensions. The massless Dirac equation on a sphere with a magnetic-monopole background is solved in order to count the number of zero-modes of the Dirac operator.