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

Abstract

Dirac's quantization condition, $eg=n/2$ ($n\in \mathbb{Z}$), and Schwinger's quantization condition, $eg=n$ ($n\in \mathbb{Z}$), 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.

16 pages, no figures; minor corrections, references added, published version