The Singularity Structure of Scale-Invariant Rank-2 Coulomb Branches

We compute the spectrum of scaling dimensions of Coulomb branch operators in 4d rank-2 $\mathcal{N}{=}2$ 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 $\rm\, U(1)_R$ 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.