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

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.