2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/182903Dirac'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.16 pages, no figures; minor corrections, references added, published versionHigh Energy Physics - TheoryMathematical Physicstext