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

TL;DR

The paper builds a systematic framework to classify four-dimensional ${ m N}=2$ SCFTs by the special Kahler stratification of their Coulomb branches. It demonstrates how rank-2 geometries are tightly constrained by discrete rank-1 data and central charges, organizing the moduli space with Hasse diagrams and transverse slices that correspond to rank-1 SCFTs on each edge. Through a broad suite of examples—Lagrangian theories, Argyres–Douglas theories, F-theory constructions, and S-folds—it shows that the UV data of a rank-2 theory can be reconstructed from rank-1 building blocks and the central-charge relations developed in companion work. The work also reveals new irregular geometries, discrete gauging phenomena, and new predictions for rank-2 theories, while outlining open questions and avenues to extend the program to higher rank and more intricate moduli spaces.

Abstract

We study the stratification of the singular locus of four dimensional $\mathcal{N}=2$ 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 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.

Figures (42)

• Figure 1: Cartoon of a 3-dimensional scale-invariant Coulomb branch, where each real dimension in the figure represents 1 complex dimension. The cones and lines are meant to extend to infinity; they are truncated here due to lack of space. Figure (a) shows a space where ${\mathcal{S}}^{(d)}_i$ are the strata of dimension $d$ and $i$ is a unique label. ${\mathcal{S}}^{(3)}_1$ is the manifold of non-singular points of the Coulomb branch, ${\mathcal{S}}^{(0)}_0$ is the unique superconformal vacuum, and the other strata are manifolds of metric and/or complex structure singularities. The partial ordering by inclusion under closure among the strata is given in the Hasse diagram shown in figure (b). The union of the strata enclosed by the dashed line is the component ${\overline{\mathcal{S}}}^{(2)}_b$ associated to the stratum ${\mathcal{S}}^{(2)}_b$.
• Figure 2: Figure (a) shows a 3-dimensional scale-invariant Coulomb branch, where each real dimension in the figure represents 1 complex dimension and where (part of) a 2-dimensional component ${\overline{\mathcal{S}}}^{(2)}$ self-intersects along a 1-dimensional stratum ${\mathcal{S}}^{(1)}$. Also shown is a neighborhood (orange disk) in a transverse slice in ${\overline{\mathcal{S}}}^{(3)}$ through a point $p\in{\mathcal{S}}^{(1)}$ (orange dot). Figure (b) depicts the intersection of this neighborhood with ${\overline{\mathcal{S}}}^{(2)}$, where now each real dimension in the figure represents 1 real dimension. The intersection with ${\mathcal{S}}^{(2)}$ is the disjoint union of two punctured disks, ${\Delta}^*_1 \coprod {\Delta}^*_2$, pictured as cones with the point $p$ as their common vertex.
• Figure 3: Hasse diagram of a 3-dimensional scale-invariant Coulomb branch. Figure (a) shows the nodes labelled by the strata ${\mathcal{S}}^{(d)}_i$ of dimension $d$ and $i$ is a unique label. ${\mathcal{S}}^{(3)}_1$ is the manifold of non-singular points of the Coulomb branch, ${\mathcal{S}}^{(0)}_0$ is the unique superconformal vacuum. The union of the strata enclosed by the dashed line is the component ${\overline{\mathcal{S}}}^{(2)}_b$ associated to the stratum ${\mathcal{S}}^{(2)}_b$. Figure (b) labels the nodes by the theories ${\mathcal{T}}^{(r)}_i$ supported on the strata where $r$ is the rank of the theory and $i$ is the same unique stratum label. ${\mathcal{T}}^{(3)}_0$ is the whole theory, ${\mathcal{T}}^{(0)}_1$ is the empty theory, and the nodes enclosed by the dashed line form the Hasse diagram of the Coulomb branch of the ${\mathcal{T}}^{(2)}_x$ theory. A few edges have also been labelled by their elementary slices.
• Figure 4: Depiction of an $L_{(1,6)}(1,1,1)$ link consisting of the blue, red, and green circles. The solid gray torus is there for visualization purposes.
• Figure 5: The Hasse diagram for the Coulomb branch of the ${\mathcal{N}}=4$$\mathfrak{su}(3)$ theory.