2-Group Symmetries and M-Theory

TL;DR

The paper develops a boundary-geometric method to diagnose 2-group symmetries in M-theory engineered QFTs, translating field-theoretic data on 0-form and 1-form backgrounds into the topology of the singularity’s link. It provides a concrete toric-CY recipe, equating the 2-group data to H_1 of the link with a singular neighborhood excised, and demonstrates equivalence with intersection-theoretic approaches via Lefschetz duality and cokernels of pairing maps. Applying this framework to 5d SCFTs—including toric, brane-web, and non-Lagrangian cases—the authors compute 1-form symmetries and their nontrivial extensions, yielding explicit 2-group structures and confirming known results while extending them. The work delivers a practical, geometry-driven toolkit for identifying and understanding 2-groups in geometric engineering, with broad applicability to toric and generalized toric models and their brane-web duals.

Abstract

Quantum Field Theories engineered in M-theory can have 2-group symmetries, mixing 0-form and 1-form symmetry backgrounds in non-trivial ways. In this paper we develop methods for determining the 2-group structure from the boundary geometry of the M-theory background. We illustrate these methods in the case of 5d theories arising from M-theory on ordinary and generalised toric Calabi-Yau cones, including cases in which the resulting theory is non-Lagrangian. Our results confirm and elucidate previous results on 2-groups from geometric engineering.

Figures (7)

• Figure 1: The geometric description of a line changing operator $\mathcal{O}_{12}$ connecting line operators $L_1$ and $L_2$ as a chain (shown in yellow) connecting homologous cycles on the boundary.
• Figure 2: A line operator associated to a homologically trivial curve $\gamma$ on $\mathbf{L}^d$ ending on a point operator. From the point of view of the internal geometry $\mathbf{L}^d$ the point operator at the end corresponds to the M2-brane worldvolume wrapping a chain $C_\gamma$ with boundary $\gamma$.
• Figure 3: Schematic topology of $\mathcal{C}_{\mathbf{L}_{\mathcal{X}}}^3$Garcia-Etxebarria:2016bpb -- see also Albertini:2020mdx for the case $\mathcal{X}$ non-isolated. In this figure we have 4 vertices along an external edge corresponding to a $\mathbb C^2/\mathbb Z_3$ singularity. We denote as $\mathcal{S}$ the neighborhood of the singular locus we need excise when considering the 2-group structure -- clearly after the excision the Lens spaces $\mathbf{L}_1$ and $\mathbf{L}_4$ in the above figure become contratible and do not contibute to $H_1(\mathbf{L}^5_{\mathcal{X}} - \mathcal{S})$.
• Figure 4: Duality between geometry and $(p,q)$ web. The figure shows the combination of both brane-web and geometry. The location of the $(p,q)$ branes (in red) is dual to loci where $T^2$ cycles degenerate. This gives rise to the topologies we draw above. The excision locus, corresponding to the shaded region inside the wedge, is denoted by $\mathcal{S}$. Removing $\mathcal{S}$ is dual to deleting the corresponding parallel $(p,q)$ branes.
• Figure 5: The toric diagram from $SU(N)_N$ shown here on the left for $SU(6)_6$, which has $\mathbb{Z}_6$ 1-form symmetry. The orange lines indicate the triangles which determine the order of the $S^3/\mathbb{Z}_k$ quotient. The right figure shows the toric diagram after we excise the $A_1$ singularity (shown in green), which is associate to the $\mathcal{F}=SO(3)$ flavor symmetry group. The only remaining lens space is the $S^3/\mathbb{Z}_{2N}$. For even $N$ this forms a non-trivial extension with the 1-form symmetry $\mathbb{Z}_N$ and thereby a 2-group.