Junctions, Edge Modes, and $G_2$-Holonomy Orbifolds

TL;DR

This work develops a geometric template for engineering 4D edge modes coupled to strongly interacting 5D bulk theories using M-theory on singular $G_2$-holonomy orbifolds. By analyzing quotients of the Bryant–Salamon ASD bundle $X=oldsymbol{ ext Λ^2_{ ext{ASD}}(S^4)}$ and its variants, the authors show how position-dependent bulk couplings arise, yielding 4D quasi-SCFTs with rich symmetry structures inherited from 5D/7D bulk sectors. The paper catalogues Abelian quotient constructions (including Quadrions and related trinion-like configurations), studies generalized (higher-form) symmetries via defect groups, and demonstrates symmetry inheritance and breaking through explicit fixed-point analyses and Mayer–Vietoris mappings. These results illuminate how bulk dynamics control edge mode physics, enabling a wide class of novel 4D theories with intricate flavor and higher-form symmetry structures. The framework suggests further avenues, including anomaly inflow, non-Abelian quotients, and extensions to other special holonomy spaces.

Abstract

One of the general strategies for realizing a wide class of interacting QFTs is via junctions and intersections of higher-dimensional bulk theories. In the context of string/M-theory, this includes many $D > 4$ superconformal field theories (SCFTs) coupled to an IR free bulk. Gauging the flavor symmetries of these theories and allowing position dependent gauge couplings provides a general strategy for realizing novel higher-dimensional junctions of theories coupled to localized edge modes. Here, we show that M-theory on singular, asymptotically conical $G_2$-holonomy orbifolds provides a general template for realizing strongly coupled 5D bulk theories with 4D $\mathcal{N} = 1$ edge modes. This geometric approach also shows how bulk generalized symmetries are inherited in the boundary system.

Figures (18)

• Figure 1: Example of a 4D $\mathcal{N}=1$ quasi-SCFT obtained at a junction of four 7D theories, labeled A, B, C, and D above.
• Figure 2: 6D conformal matter as an example of an interface.
• Figure 3: Toric diagram of the $X^6_{p,q}$ singularity.
• Figure 4: Three sketches of the 5D bulk theory for different sizes of the zero section Vol$(S^4/\Gamma)\sim a$. Square (circle) nodes denote flavor (gauge) symmetries in 5D. The cone over gauge nodes denotes a gauging which breaks 5D Poincaré symmetry via a gauge coupling which depends on one 5D direction (the radial direction in $X_a$). In the 4D transverse directions along which the coupling does not vary 4D $\mathcal{N}=1$ supersymmetry is preserved. $a=\infty:$ two decoupled 5D $\mathcal{T}_N$ theories, formally at the north/south pole of an infinite volume $S^4/\Gamma$. $0<a<\infty:$ flavor loci are compactified transverse to the 5D SCFT locus and the volume of such transverse slices depend on the radial shell of $X_a$ they are contained in. $a=0:$ the $S^4/\Gamma$ collapses and the geometry is conical. The $\mathcal{T}_N$ theories supported on $\mathbb{R}$ over the north and south pole decompose into $\mathcal{T}_N^{\pm}$ supported on two half lines $\mathbb{R}^\pm$. 5D gauge symmetry loci also contain $\mathbb{R}$ and decompose similarly. No such splitting occurs for 7D SYM theory factors which appear as 5D flavor symmetries.
• Figure 5: General 5D theory with $\Gamma=\mathbb{Z}_K$. Two pairs of flavor symmetry loci between two 5D SCFT sectors are gauged, circular nodes, with one pair of flavor symmetries remaining, square nodes. Clearly, $\Gamma=\Gamma_N=\Gamma_S$ with the subscripts distinguish the group actions at the North and South pole. The dashed line denotes a massive mode resulting from an M2-brane wrapped on a possible 2-cycle of $S^4/\Gamma$. The cones of the gauge nodes denote a 5D gauging with gauge coupling depending on the radial coordinate $r$ of the $G_2$-holonomy space such that 4D $\mathcal{N}=1$ supersymmetry is preserved.