To have arbitrarily small charges requires that the U (1) is not the circle of unit complex numbers under multiplication, but the real numbers ℝ under addition, which is only different in the global topology. Such a U (1) group is non-compact. The field θ transforms as an affine representation of the gauge group. Among the allowed gauge groups, only non-compact U (1) admits affine representations, and the U (1) of electromagnetism is experimentally known to be compact, since charge quantization holds to extremely high accuracy.