Towards a classification of rank r $$ \mathcal{N} $$ = 2 SCFTs. Part II. Special Kahler stratification of the Coulomb branch

Abstract

We study the stratification of the singular locus of four dimensional $$ \mathcal{N} $$ N = 2 Coulomb branches. We present a set of self-consistency conditions on this stratification which can be used to extend the classification of scale-invariant rank 1 Coulomb branch geometries to two complex dimensions, and beyond. The calculational simplicity of the arguments presented here stems from the fact that the main ingredients needed — the rank 1 deformation patterns and the pattern of inclusions of rank 2 strata — are discrete topological data which satisfy strong self-consistency conditions through their relationship to the central charges of the SCFT. This relationship of the stratification data to the central charges is used here, but is derived and explained in a companion paper [1] by one of the authors. We illustrate the use of these conditions by re-analyzing many previously-known examples of rank 2 SCFTs, and also by finding examples of new theories. The power of these conditions stems from the fact that for Coulomb branch stratifications a conjecturally complete list of physically allowed “elementary slices” is known. By contrast, constraining the possible elementary slices of symplectic singularities relevant for Higgs branch stratifications remains an open problem.

Towards the classification of rank-r $$ \mathcal{N} $$ = 2 SCFTs. Part I. Twisted partition function and central charge formulae

“…These formulae are valid for SCFTs of arbitrary rank and extend and generalize the work of Shapere and Tachikawa [10]. The main input which allowed this generalization is our improved understanding of the structure of the stratification of the singular locus of the Coulomb branch [35].…”

“…The work described here as well as in a companion paper [35], is motivated by the classification program of four dimensional N = 2 superconformal field theories [12] based on the systematic analysis of their Coulomb branch geometries. Particularly with the goal of extending the success story of rank-1 [34,[36][37][38] to arbitrary rank.…”

“…The calculation here is for pedagogical purposes only since we won't learn anything new on these very well studied lagrangian theories. We will leverage the power of our central charge formulae in [35] where we compute many new properties of N = 2 SCFTs of rank-2 theories and their moduli space. Let's start reminding the reader of the properties of these theories:…”

Coulomb and Higgs branches from canonical singularities. Part 0

“…There has been a lot of recent progress in studying the 4d N = 2 Coulomb branchfor a limited list of references see [53][54][55][56] -as well as the Higgs branch -see e.g. [57][58][59][60][61][62][63][64][65][66][67] and much of the SCFT/Vertex operator algebra literature [68], as in e.g.…”

S-fold magnetic quivers

“…More specifically, Hasse diagrams allow one to study the following moduli spaces: Higgs branches of theories in dimension 3−6; Coulomb branches of 3d theories; and even the full moduli space in 3d, which contains mixed branches. Hasse diagrams were also used to study the Coulomb branch and the full moduli space of 4d N = 2 theories in [43]. A feature of all magnetic quivers coming from S-folds studied in the following is the appearance of non-simply laced edges, where the order of the non-simply laced edges corresponds to the order of the S-fold (the folding parameter).…”