We initiate a systematic study of four dimensional N = 2 superconformal field theories (SCFTs) based on the analysis of their Coulomb branch geometries. Because these SCFTs are not uniquely characterized by their scale-invariant Coulomb branch geometries we also need information on their deformations. We construct all inequivalent such deformations preserving N = 2 supersymmetry and additional physical consistency conditions in the rank 1 case. These not only include all the ones previously predicted by S-duality, but also 16 additional deformations satisfying all the known N = 2 low energy consistency conditions. All but two of these additonal deformations have recently been identified with new rank 1 SCFTs; these identifications are briefly reviewed.Some novel ingredients which are important for this study include: a discussion of RGflows in the presence of a moduli space of vacua; a classification of local N = 2 supersymmetrypreserving deformations of unitary N = 2 SCFTs; and an analysis of charge normalizations and the Dirac quantization condition on Coulomb branches.This paper is the first in a series of three. The second paper [1] gives the details of the explicit construction of the Coulomb branch geometries discussed here, while the third [2] discusses the computation of central charges of the associated SCFTs.

We refine our previous proposal [1][2][3] for systematically classifying 4d rank-1 N = 2 SCFTs by constructing their possible Coulomb branch geometries. Four new recently discussed rank-1 theories [4,5], including novel N = 3 SCFTs, sit beautifully in our refined classification framework. By arguing for the consistency of their RG flows we can make a strong case for the existence of at least four additional rank-1 SCFTs, nearly doubling the number of known rank-1 SCFTs.The refinement consists of relaxing the assumption that the flavor symmetries of the SCFTs have no discrete factors. This results in an enlarged (but finite) set of possible rank-1 SCFTs. Their existence can be further constrained using consistency of their central charges and RG flows.

This is the third in a series of three papers on the systematic analysis of rank 1 four dimensional N = 2 SCFTs. In the first two papers [1, 2] we developed and carried out a strategy for classifying and constructing physical planar rank-1 Coulomb branch geometries of N = 2 SCFTs. Here we describe general features of the Higgs and mixed branch geometries of the moduli space of these SCFTs, and use this, along with their Coulomb branch geometry, to compute their conformal and flavor central charges. We conclude with a summary of the state of the art for rank-1 N = 2 SCFTs.

This is the second in a series of three papers on systematic analysis of rank 1 Coulomb branch geometries of four dimensional N =2 SCFTs. In [1] we developed a strategy for classifying physical rank-1 CB geometries of N =2 SCFTs. Here we show how to carry out this strategy computationally to construct the Seiberg-Witten curves and one-forms for all the rank-1 SCFTs. Explicit expressions are given for all 28 cases, with the exception of the N f =4 su(2) gauge theory and the E n SCFTs which were constructed in [2,3] and [4, 5]. A.2 Deformations of the III * singularity 62 A.2.1 {I 5 1 , I 4 } with sp(6) ⊕ su(2) flavor symmetry 62 A.2.2 {I 3 1 , I * 0 } with so(7) flavor symmetry 64 A.2.3 {I 2 1 , I * 1 } with su(2) ⊕ su(2) flavor symmetry 66 A.2.4 {I 2 , I * 1 } and {I 1 , I * 2 } 67 A.2.5 {I 1 , IV * } with su(2) flavor symmetry 68 A.3 Deformations of the IV * singularity 68 A.3.1 {I 4 1 , I 4 } with sp(4) ⊕ u(1) flavor symmetry 69 A.3.2 {I 2 1 , I * 0 } with su(3) flavor symmetry 70 A.3.3 {I 1 , I * 1 } with u(1) flavor symmetry 71 A.4 Deformations of the I * 0 singularity 72 A.4.1 {I 1 2 , I 4 } with sp(2) flavor symmetry 72 A.4.2 {I 2 3 } with sp(2) flavor symmetry 73 A.5 Deformation of the IV singularity 74 A.5.1 {I 4 1 } with su(3) flavor symmetry 74 A.6 Deformation of the III singularity 75 A.6.1 {I 3 1 } with su(2) flavor symmetry 75 A.7 Deformation of the II singularity 76 A.7.1 {I 2 1 } with no flavor symmetry 76

We initiate a systematic study of four dimensional N = 2 superconformal field theories (SCFTs) based on the analysis of their Coulomb branch geometries. Because these SCFTs are not uniquely characterized by their scale-invariant Coulomb branch geometries we also need information on their deformations. We construct all inequivalent such deformations preserving N = 2 supersymmetry and additional physical consistency conditions in the rank 1 case. These not only include all the ones previously predicted by S-duality, but also 16 additional deformations satisfying all the known N = 2 low energy consistency conditions. All but two of these additonal deformations have recently been identified with new rank 1 SCFTs; these identifications are briefly reviewed.Some novel ingredients which are important for this study include: a discussion of RG-flows in the presence of a moduli space of vacua; a classification of local N = 2 supersymmetry-preserving deformations of unitary N = 2 SCFTs; and an analysis of charge normalizations and the Dirac quantization condition on Coulomb branches. This paper is the first in a series of three. The second paper [1] gives the details of the explicit construction of the Coulomb branch geometries discussed here, while the third [2] discusses the computation of central charges of the associated SCFTs.