1-Form Symmetry, Isolated N=2 SCFTs, and Calabi-Yau Threefolds

TL;DR

The paper investigates 4D $\mathcal{N}=2$ SCFTs realized via type IIB string theory on isolated hypersurface singularities (IHS) and proves a central claim: isolated IHS-derived theories have trivial 1-form symmetry. By leveraging the Closset–Del Zotto–Giacomelli formalism and Milnor-ring data, the authors connect exactly marginal deformations to weight-1 monomials and show that, for these isolated theories, no nontrivial 1-form symmetry arises from gauge sectors with vanishing beta functions. They extend the result from the well-studied $(\mathfrak{g},\mathfrak{g}')$ theories to broader Type I constructions, discuss UV/IR symmetry matching across RG flows, and derive corollaries that constrain central charges $a$ and $c$ in terms of Coulomb-branch data. They further conjecture a general principle: interacting 4D $\mathcal{N}=2$ SCFTs with all Coulomb-branch chiral ring generators of dimension $<2$ possess no 1-form symmetry, a statement supported by evidence from class-S and certain $\mathcal{N}=3$ theories. The work thus links the geometric structure of singularities with global symmetry constraints, offering a route to classification via 1-form symmetry considerations.

Abstract

We systematically study 4D $\mathcal{N}=2$ superconformal field theories (SCFTs) that can be constructed via type IIB string theory on isolated hypersurface singularities (IHSs) embedded in $\mathbb{C}^4$. We show that if a theory in this class has no $\mathcal{N}=2$-preserving exactly marginal deformation (i.e., the theory is isolated as an $\mathcal{N}=2$ SCFT), then it has no 1-form symmetry. This situation is somewhat reminiscent of 1-form symmetry and decomposition in 2D quantum field theory. Moreover, our result suggests that, for theories arising from IHSs, 1-form symmetries originate from gauge groups (with vanishing beta functions). One corollary of our discussion is that there is no 1-form symmetry in IHS theories that have all Coulomb branch chiral ring generators of scaling dimension less than two. In terms of the $a$ and $c$ central charges, this condition implies that IHS theories satisfying $a<{1\over24}(15r+2f)$ and $c<{1\over6}(3r+f)$ (where $r$ is the complex dimension of the Coulomb branch, and $f$ is the rank of the continuous 0-form flavor symmetry) have no 1-form symmetry. After reviewing the 1-form symmetries of other classes of theories, we are motivated to conjecture that general interacting 4D $\mathcal{N}=2$ SCFTs with all Coulomb branch chiral ring generators of dimension less than two have no 1-form symmetry.