Classification of all $\mathcal{N}\geq 3$ moduli space orbifold geometries at rank 2

TL;DR

The study advances the classification of moduli-space geometries for four-dimensional ${\mathcal N}\ge 3$ SCFTs at rank ${\le 2}$ by exploiting triple special Kähler (TSK) structure and finite symplectic group actions ${\Gamma}\subset {\rm Sp}(2r,\mathbb{Z})$, yielding ${\mathcal M}_{\Gamma} \cong \mathbb{C}^{3r}/{\Gamma}$. It systematically derives fixed-point data on Siegel space, computes CB slices ${\mathcal C}_{\Gamma}=\mathbb{C}^r/{\mu}_{\tau}({\Gamma})$, and uses refined Hilbert series and Molien-PLog analysis to filter physically viable geometries. The authors obtain 53 rank-2 geometries, of which 31 pass Hilbert-series constraints, with 23 matching known theories (including ${\mathcal N}=4$ and S-folds) and 8 representing genuinely new rank-2 ${\mathcal N}=3$ theories, three of which have non-freely generated CB chiral rings. The work highlights potential new ${\mathcal N}=3$ physics beyond discrete gaugings, and points to rich avenues for future checks via VOAs, mass deformations, and non-principal Dirac pairings, contributing a concrete, geometry-driven map of the rank-2 ${\mathcal N}\ge 3$ landscape.

Abstract

We classify orbifold geometries which can be interpreted as moduli spaces of four-dimensional $\mathcal{N}\geq 3$ superconformal field theories up to rank 2 (complex dimension 6). The large majority of the geometries we find correspond to moduli spaces of known theories or discretely gauged version of them. Remarkably, we find 6 geometries which are not realized by any known theory, of which 3 have an $\mathcal{N}=2$ Coulomb branch slice with a non-freely generated coordinate ring, suggesting the existence of new, exotic $\mathcal{N}=3$ theories.

Figures (2)

• Figure 1: ${\rm\, SU}(3)$ weight lattice, with the vector showing our choice of the weight of the supercharge, ${\overline Q}^{(0,1;1)}$ defining the chiral ring. (a) Red dots are the weights of the $(0,1)$ representation that correspond to $A_2\overline{B}_1[0;0]^{(0,1;4)}_2$. The product $(0,1) \otimes (0,1)$ decomposes in $(0,2)$ (in orange and yellow) which contains the null states, and $(1,0)$ (represented in yellow) which are non-null states. The components of a chiral multiplet in the $\overline{\bf3}$ annihilated by ${\overline Q}^{(0,1;1)}$ lie on the dashed line. (b) Blue dots are the $(1,1)$ weights corresponding to $B_1\overline{B}_1[0;0]^{(1,1;0)}_2$. The product $(1,1) \otimes (0,1)$ decomposes into the null states $(1,2)$ (all green dots, dark and light) and the non-null states $(2,0)$ and $(0,1)$ (lighter shades of green). The components of a chiral multiplet in the $(1,1)$ annihilated by ${\overline Q}^{(0,1;1)}$ lie on the dashed line. The light blue arrows show the choice of simple roots with respect to which our Dynkin labels are defined.
• Figure 2: We use the same conventions as in Figure \ref{['su3fig1']}. (a) The product of the $(2,0)$ (in red) with $(0,1)$ gives the null states in the $(2,1)$ (in pink and yellow) and the non-null states $(1,0)$ (in yellow). (b) The product of the $(0,2)$ (in blue) with $(0,1)$ gives the null states in the $(0,3)$ (in green) and the non-null states in the $(1,1)$ (in light green). In both cases the component of the chiral multiplets annihilated by ${\overline Q}^{(0,1;1)}$ lie on the dashed line.