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We use Higgs bundles to study the 3d $\mathcal{N} = 1$ vacua obtained fromM-theory compactified on a local $Spin(7)$ space given as a four-manifold $M_4$of ADE singularities with further generic enhancements in the singularity typealong one-dimensional subspaces. There can be strong quantum corrections to themassless degrees of freedom in the low energy effective field theory, buttopologically robust quantities such as "parity" anomalies are stillcalculable. We show how geometric reflections of the compactification spacedescend to 3d reflections and discrete symmetries. The "parity" anomalies ofthe effective field theory descend from topological data of thecompactification. The geometric perspective also allows us to track variousperturbative and non-perturbative corrections to the 3d effective field theory.We also provide some explicit constructions of well-known 3d theories,including those which arise as edge modes of 4d topological insulators, and 3d$\mathcal{N} = 1$ analogs of grand unified theories. An additional result ofour analysis is that we are able to track the spectrum of extended objects andtheir transformations under higher-form symmetries.

Reflections on the Matter of 3d $\mathcal{N} = 1$ Vacua and Local $Spin(7)$ Compactifications