4d N=2 SCFT and singularity theory Part III: Rigid singularity

TL;DR

The paper classifies 3-fold isolated quotient Gorenstein singularities $\mathbb{C}^3/G$ with $G\subset SL(3,\mathbb{C})$ by showing they are linearly equivalent to diagonal cyclic subgroups given by $\langle \zeta(1/n),\zeta(p/n),\zeta(q/n)\rangle$ with $1+p+q=n$, and constructs the invariant ring $S^G$ with minimal generators $g_1=x^n,g_2=y^n,g_3=z^n,g_4=xyz$ plus sets $A_{xy},A_{xz},A_{yz}$ and relations $R_{xy},R_{xz},R_{yz}$; the minimal embedding dimension is $d=4+l(n/(n-p))+l(n/(p+1))+l(n/(n-r))\ge 10$, consistent with a toric lattice-triangle description. The results establish rigidity (no nontrivial deformations) of these singularities, hence the corresponding 4d $\mathcal{N}=2$ SCFTs have no Coulomb branch, while nontrivial crepant resolutions provide Higgs-branch structure read from the Mori cone and imply no flavor symmetry due to trivial local class group. The toric viewpoint corroborates the classification via lattice triangles with no boundary lattice points, and the work illustrates how geometric smoothing/resolution data constrain moduli and dynamics of 4d SCFTs.

Abstract

We classify three fold isolated quotient Gorenstein singularity $C^3/G$. These singularities are rigid, i.e. there is no non-trivial deformation, and we conjecture that they define 4d $\mathcal{N}=2$ SCFTs which do not have a Coulomb branch.

Figures (1)

• Figure 1: Isolated toric Gorenstein singularity with trivial class group is defined by a lattice triangle with no lattice points on the boundary.

Theorems & Definitions (18)

• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Corollary 3.1
• Remark 3.1
• proof : Proof of Theorem \ref{['the1']}
• Theorem 3.4
• Remark 3.2
• Example 1
• Example 2