Compactification of M-/ string theory on manifolds with G 2 structure yields a wide variety of 4D and 3D physical theories. We analyze the local geometry of such compactifications as captured by a gauge theory obtained from a three-manifold of ADE singularities. Generic gauge theory solutions include a non-trivial gauge field flux as well as normal deformations to the three-manifold captured by non-commuting matrix coordinates, a signal of T-brane phenomena. Solutions of the 3D gauge theory on a three-manifold are given by a deformation of the Hitchin system on a marked Riemann surface which is fibered over an interval. We present explicit examples of such backgrounds as well as the profile of the corresponding zero modes for localized chiral matter. We also provide a purely algebraic prescription for characterizing localized matter for such T-brane configurations. The geometric interpretation of this gauge theory description provides a generalization of twisted connected sums with codimension seven singularities at localized regions of the geometry. It also indicates that geometric codimension six singularities can sometimes support 4D chiral matter due to physical structure "hidden" in T-branes.Manifolds of special holonomy are of great importance in connecting the higher-dimensional spacetime predicted by string theory to lower-dimensional physical phenomena. This is because such manifolds admit covariantly constant spinors, thus allowing the macroscopic dimensions to preserve some amount of supersymmetry.Historically, the most widely studied class of examples has centered on type II and heterotic strings compactified on Calabi-Yau threefolds [1]. This leads to 4D vacua with eight and four real supercharges, respectively. Such threefolds also play a prominent role in the study of F-theory and M-theory backgrounds, leading respectively to 6D and 5D vacua with eight real supercharges. Compactifications on Calabi-Yau spaces of other dimensions lead to a rich class of geometries, and correspondingly many novel physical systems in the macroscopic dimensions. In all these cases, the holomorphic geometry of the Calabi-Yau allows techniques from algebraic geometry to be used.There are, however, other manifolds of special holonomy, most notably those with G 2 and Spin(7) structure. For example, compactification of M-theory on G 2 and Spin (7) spaces provides a method for generating a broad class of 4D and 3D N = 1 vacua, respectively. 1 Despite these attractive features, it has also proven notoriously difficult to generate singular compact geometries of direct relevance for physics. 2 In the case of M-theory on a G 2 background, realizing a non-abelian ADE gauge group requires a three-manifold of ADE singularities (i.e., codimension four), and realizing 4D chiral matter requires codimension seven singularities. While there are now some techniques available to realize G 2 backgrounds with codimension four singularities, it is not entirely clear whether a smooth compact G 2 can be continuously deformed to such singular g...

The modern approach to m-form global symmetries in a d-dimensional quantum field theory (QFT) entails specifying dimension d − m − 1 topological generalized symmetry operators which non-trivially link with m-dimensional defect operators. In QFTs engineered via string constructions on a non-compact geometry X, these defects descend from branes wrapped on non-compact cycles which extend from a localized source / singularity to the boundary 𝝏X. The generalized symmetry operators which link with these defects arise from magnetic dual branes wrapped on cycles in 𝝏X. This provides a systematic way to read off various properties of such topological operators, including their worldvolume topological field theories, and the resulting fusion rules. We illustrate these general features in the context of 6D superconformal field theories, where we use the F-theory realization of these theories to read off the worldvolume theory on the generalized symmetry operators. Defects of dimension 3 which are charged under a suitable 3-form symmetry detect a non-invertible fusion rule for these operators. We also sketch how similar considerations hold for related systems.

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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.

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