Construction and classification of Coulomb branch geometries
Philip Argyres, Mario Martone
TL;DR
This work outlines a program to classify four-dimensional ${\mathcal N}=2$ SCFTs by studying their Coulomb branch geometries ${\mathcal C}$, a complex space of dimension $r$ with a singular locus ${\mathcal V}$ where charged states become massless. It develops a framework based on the rigid special Kähler structure of ${\mathcal C}_{reg}$, electromagnetic duality monodromies in ${\rm Sp}(2r,\mathbb{Z})$, and a scale-invariant ${\mathbb C}^*$ action to organize theories by rank and mass deformations that deform but do not lift the CB. The paper reports substantive results, including a complete rank-1 CB classification and structural insights into rank-2 geometries, distinguishing metric and complex CB singularities and constraining CB operator dimensions ${\Delta_i}$ via monodromy; it also clarifies how deformations, RG flows, and central charges fit into this geometric picture. It further discusses finite versus infinite sets of geometries, CB stratification, and the role of polarization in connecting to ${\mathcal N}=4$ theories. Overall, the work provides a principled geometric route toward a finite catalog of ${\mathcal N}=2$ SCFTs at fixed rank and lays groundwork for extending these ideas to higher rank and non-principal polarizations.
Abstract
We give a non-technical summary of the classification program, very dear to the hearts of both authors, of four dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) based on the study of their Coulomb branch geometries. We outline the main ideas behind this program, review the most important results thus far obtained, and the prospects for future results. This contribution will appear in the volume "the Pollica perspective on the (super)-conformal world" but we decided to also make it available separately in the hope that it could be useful to those who are interested in obtaining a quick grasp of this rapidly developing program.