We compute the spectrum of scaling dimensions of Coulomb branch operators in
4d rank-2 N=2 superconformal field theories. Only a finite
rational set of scaling dimensions is allowed. It is determined by using
information about the global topology of the locus of metric singularities on
the Coulomb branch, the special K\"ahler geometry near those singularities, and
electric-magnetic duality monodromies along orbits of the U(1)R
symmetry. A set of novel topological and geometric results are developed which
promise to be useful for the study and classification of Coulomb branch
geometries at all ranks.Comment: 2 references added, submitted to JHE