Higher Form Symmetries of Argyres-Douglas Theories

TL;DR

This work determines the 1-form symmetry structure of four-dimensional N=2 theories engineered by IIB string theory on isolated hypersurface singularities, including Argyres-Douglas theories. It develops a framework based on noncommutative RR flux at infinity and a BPS quiver reformulation to extract the associated defect groups, linking geometric data to field-theoretic higher-form symmetries. For the Cecotti–Neitzke–Vafa AD theories it provides explicit defect-group results via Milnor-Orlik and Alexander polynomial data, and demonstrates how multiple global structures arise from these geometries. The findings supply nontrivial consistency checks for proposed N=1 Lagrangian flows and offer a general method to determine global structure in non-Lagrangian 4d N=2 theories, with potential implications for partition functions on nontrivial four-manifolds and lens-space indices.

Abstract

We determine the structure of 1-form symmetries for all 4d $\mathcal{N} = 2$ theories that have a geometric engineering in terms of type IIB string theory on isolated hypersurface singularities. This is a large class of models, that includes Argyres-Douglas theories and many others. Despite the lack of known gauge theory descriptions for most such theories, we find that the spectrum of 1-form symmetries can be obtained via a careful analysis of the non-commutative behaviour of RR fluxes at infinity in the IIB setup. The final result admits a very compact field theoretical reformulation in terms of the BPS quiver. We illustrate our methods in detail in the case of the $(\mathfrak{g}, \mathfrak{g}')$ Argyres-Douglas theories found by Cecotti-Neitzke-Vafa. In those cases where $\mathcal{N} = 1$ gauge theory descriptions have been proposed for theories within this class, we find agreement between the 1-form symmetries of such $\mathcal{N} = 1$ Lagrangian flows and those of the actual Argyres-Douglas fixed points, thus giving a consistency check for these proposals.

Figures (1)

• Figure 1: Second page for the Atiyah-Hirzebruch spectral sequence for the reduced K-homology of the horizon manifold on an isolated hypersurface singularity. We have denoted $H_2\coloneqq H_2(Y^5)$, $b_2\coloneqq \mathop{\mathrm{rk}}\nolimits (H_2\otimes \mathbb{Q})$, and shown the only differential that might potentially be non-vanishing. The entries shaded in blue are those contributing to $K_0(Y_5)$, and those in pink are those contributing to $K_1(Y_5)$.