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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 Kahler 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.
We construct 4d superconformal field theories (SCFTs) whose Coulomb branches have singular complex structures. This implies, in particular, that their Coulomb branch coordinate rings are not freely generated. Our construction also gives examples of d
We study the classification of 2-dimensional scale-invariant rigid special Kahler (RSK) geometries, which potentially describe the Coulomb branches of N=2 supersymmetric field theories in four dimensions. We show that this classification is equivalen
We continue the classification of 2-dimensional scale-invariant rigid special Kahler (RSK) geometries. This classification was begun in [hep-th/0504070] where singularities corresponding to curves of the form y^2=x^6 with a fixed canonical basis of h
By studying Rozansky-Witten theory with non-compact target spaces we find new connections with knot invariants whose physical interpretation was not known. This opens up several new avenues, which include a new formulation of $q$-series invariants of
We derive closed formulae for the first examples of non-algebraic, elliptic `leading singularities in planar, maximally supersymmetric Yang-Mills theory and show that they are Yangian-invariant.