Topological characterization of charge quantization

Abstract

Weinberg has written a paper showing how to calculate gauge coupling constants in (4 +N)-dimensional models withN dimensions forming a compact manifold. Each coupling constant is related to the inverse of an appropriate rootmean-square circumference of the manifold. We extend this work by showing that this charge is quantized, in the sense of a tower of particles each carrying a charge which is an integer multiple of a basic unit, if and only if Π1(I) =ZZ where Π1 is the first homotopy group,I is the isometry group of the compact manifold, andZZ is the additive group of integers.