0-Form, 1-Form and 2-Group Symmetries via Cutting and Gluing of Orbifolds

TL;DR

The paper develops a geometric pipeline to read off global symmetry data of supersymmetric quantum field theories engineered from localized orbifold singularities. By cutting and gluing along the boundary ∂X and exploiting Mayer-Vietoris sequences and orbifold (co)homology, the authors extract the 0-form flavor group, 1-form, and potential 2-group structures, including their Postnikov data, directly from topology. They apply the method to a broad set of examples—5D SCFTs from C^3/Γ, 5D theories from elliptically fibered CYs, and 4D SQCD-like theories from M-theory on local G2 spaces—demonstrating consistency with known field-theoretic expectations and providing explicit geometric predictions for the global form of flavor symmetries and higher-group structures. The framework unifies previous separate approaches and offers a computational, topology-driven tool for identifying intricate symmetry interplays in string-constructed QFTs, with potential extensions to more exotic higher-group structures and compact geometries.

Abstract

Orbifold singularities of M-theory constitute the building blocks of a broad class of supersymmetric quantum field theories (SQFTs). In this paper we show how the local data of these geometries determines global data on the resulting higher symmetries of these systems. In particular, via a process of cutting and gluing, we show how local orbifold singularities encode the 0-form, 1-form and 2-group symmetries of the resulting SQFTs. Geometrically, this is obtained from the possible singularities which extend to the boundary of the non-compact geometry. The resulting category of boundary conditions then captures these symmetries, and is equivalently specified by the orbifold homology of the boundary geometry. We illustrate these general points in the context of a number of examples, including 5D superconformal field theories engineered via orbifold singularities, 5D gauge theories engineered via singular elliptically fibered Calabi-Yau threefolds, as well as 4D SQCD-like theories engineered via M-theory on non-compact $G_2$ spaces.

Figures (9)

• Figure 1: Depiction of an SQFT realized at a localized region of a non-compact geometry $X$, with boundary $\partial X$. Flavor branes can extend out to infinity and intersect the boundary along a subspace $K$. Our procedure for extracting the global form of the flavor symmetry, 1-form symmetry and possible 2-group structures involves working with $\partial X^{\circ} = \partial X \backslash T(K)$, where $T(K)$ is a tubular neighborhood of $K$. The Mayer-Vietoris exact sequence then yields the relevant physical structures directly from geometry.
• Figure 2: On the left, we depict $\partial X$ with various orbifold singularities (stars) and an element in $\mathcal{C}^\vee$ (black loop). The gray disk bounds an element of $\mathcal{C}^\vee$, and shows that this element is topologically trivial under the inclusion of the orbifold loci back into $\partial X^\circ$. This motivates the identification $\textnormal{Tor\,ker}\,(j_1)=\mathcal{C}^\vee$. On the right, we depict an element in $\mathcal{A}$ with a black loop, and illustrate a gray disk to highlight that the topological equivalence relation inherent in $H_1(\partial X)$ is the same as that of $\mathcal{A}^\vee$ when wrapping $M2$-branes.
• Figure 3: Sketch of the base $\Delta$ of the torus fibration $\pi:\partial X\rightarrow \Delta$.
• Figure 4: Sketches of the deformation retracts for the base $\Delta$. In the first and second and third configuration the orbifold locus projects to $P_3$ and $P_1, P_2$ and $P_1, P_2 , P_3$ respectively. The $T^3/\:\!\Gamma$ fibration then deformation retracts to a fibration over the graphs $\mathfrak{G}$ marked red. We depict a decomposition of $\mathfrak{G}$ into intervals $I_*$ with one endpoint on the boundary of $\Delta$.
• Figure 5: Sketch of the IIA setup of type 1) with "vector-like" flavor symmetry. The total geometry is $T^*S^3$ with $N_c$ color D6-branes (black) wrapped on the compact $S^3$ and $N_f$ flavor D6-branes (red) wrapped on $T^2\times \mathbb{R}_+$ intersecting the $S^3$ in a circle and the boundary in $T^2$.