Higher-Form Symmetries in 5d

TL;DR

The paper develops a comprehensive framework for 1-form (higher-form) symmetries in 5d theories, linking gauge-theory criteria to a geometric criterion in M-theory on non-compact Calabi–Yau threefolds. It provides an explicit, intersection-based formula Γ ≃ H2^∂(M6)/im f2 (via the Smith normal form of the relevant intersection matrices) to determine the 1-form symmetry directly from the geometry, and verifies this across a wide class of examples, including toric CY3s and intersecting-surface constructions. The authors demonstrate the robustness of these symmetries under UV dualities and flop transitions, and connect 5d 1-form symmetries to the 6d defect group and to various dimensional reductions and general M-theory compactifications, including G2 spaces. The work also outlines extensions to higher-dimensional CYs, asymptotic flux perspectives, and potential observables sensitive to 1-form symmetry, laying groundwork for a systematic understanding of higher-form symmetries in string/M-theory contexts and their physical consequences.

Abstract

We study higher-form symmetries in 5d quantum field theories, whose charged operators include extended operators such as Wilson line and 't Hooft operators. We outline criteria for the existence of higher-form symmetries both from a field theory point of view as well as from the geometric realization in M-theory on non-compact Calabi-Yau threefolds. A geometric criterion for determining the higher-form symmetry from the intersection data of the Calabi-Yau is provided, and we test it in a multitude of examples, including toric geometries. We further check that the higher-form symmetry is consistent with dualities and is invariant under flop transitions, which relate theories with the same UV-fixed point. We explore extensions to higher-form symmetries in other compactifications of M-theory, such as $G_2$-holonomy manifolds, which give rise to 4d $\mathcal{N}=1$ theories.

Figures (4)

• Figure 1: Toric diagrams for $SU(2)_0$, $SU(2)_\pi$ and $\mathbb{P}^2$ theory (which is obtained from $SU(2)_\pi$ by an external flop). The internal vertex (blue) indicates the Cartan of the rank one gauge group, the external vertices correspond to non-compact divisors. Whenever there is a ruling, i.e. a partial singularization to an $A_1$ singularity, this is indicated by a green line.
• Figure 2: The toric diagrams for $SU(N)_k$ and $SU(N)_{\frac{N}{2} + k_1} \times SU(N)_{\frac{N}{2} + k_2}$ linear quiver. On the left the case of $k=0$ for $N=6$ is shown, on the right hand side $k_1= 1=k_2$. In both diagrams we show a specific full triangulation, i.e., the geometry in question is the fully resolved one, i.e., the theory on the Coulomb branch. The vertical green lines indicate the rulings that give the $SU(N)$ gauge groups.
• Figure 3: (a) Fully triangulated toric fan for $T_4$: the internal vertices are the compact divisors, associated to the Cartans of the gauge group. We marked the external vertices by the self-intersection numbers of the non-compact divisors with the compact divisors. These form the combined fiber diagram (CFD): $(-2)$ are the marked vertices, which form the subgraph that is the Dynkin diagram of the flavor symmetry $G_F= SU(N)^3$. $(-1)$ are the curves with self-intersection number $-1$, which correspond to matter hypermultiplets in bifundamentals in pair-wise combinations of the $SU(N)^3$ flavor symmetry. (b) the CFD for $T_N$.
• Figure 4: The non-Lagrangian theories determined in Eckhard:2020jyr obtained from $T_N$ after decoupling all hypermultiplets, as well as their flavor symmetries $G_F$.