On the Orbifold origin of Higher Form Symmetries in Geometric Engineering

TL;DR

The paper establishes a concrete link between orbifold singularities and higher-form symmetries in 5d SCFTs engineered via M-theory on Calabi–Yau threefolds, showing that the defect group of the theory coincides with a quantum symmetry of the associated BPS quiver. By performing (un-)orbifold operations at the quiver level (via skew-group algebras and related machinery), the authors construct minimal theories without higher-form symmetries and generate new theories with prescribed higher-form symmetries, validating the construction on toric and non-toric, non-complete-intersection CY$_3$ geometries. The approach bypasses explicit geometric resolutions by exploiting the McKay correspondence and quiver data, and it yields a versatile framework for predicting the presence or absence of higher-form symmetries across a broad class of 5d SCFTs. The results deepen the understanding of how geometry encodes extended operator spectra and offer a practical toolkit for exploring the landscape of 5d theories with intricate symmetry structures.

Abstract

In this work we explore the relation between orbifold singularities and higher form symmetries. Using the geometric engineering dictionary, we argue that the discrete higher symmetries of 5d SCFTs constructed from M-theory on a non-compact Calabi-Yau threefold can be related to a quantum symmetry of the associated BPS quiver. Through un-orbifolding the quantum symmetry we obtain a new theory without higher form symmetry, providing a notion of ''minimality'' for a theory. This procedure is carried out via algebraic manipulations of the BPS/McKay quiver describing the crepant resolution of the singular geometry. This technique can also be reverted and thus, starting from any ''minimal'' theory, one can orbifold it and generate new theories with the desired higher form symmetries. We test our technology on classes of 5d SCFTs that arise from M-theory geometric engineering on Calabi-Yau threefolds that are non-toric non-complete intersections, which have historically been challenging to tackle.

Figures (19)

• Figure 1: BPS quiver for $\mathbb{C}^2/\mathbb{Z}_3\times \mathbb{C}$.
• Figure 2: BPS quiver for $\mathbb{C}^3/\mathbb{Z}_9$ with action $(1,2,6)$.
• Figure 3: The rotation action identifying nodes 147, 258 and 369.
• Figure 4: The rotation action identifying nodes 147, 258 and 369.
• Figure 5: Unorbifolding procedure that produces a theory with no defect group. The orbifold action identifies the nodes on the left-hand side. E.g. the nodes $(1,4,7)$ are identified by the action, and appear as a single node on the right-hand side. The same goes for arrows: we denote as $A_{ij}$ an arrow from node $i$ to node $j$ in the quiver on the left. We explicitly show some identified arrows in the quiver on the right.

Theorems & Definitions (1)

• Remark B.1