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

Abstract

We use Higgs bundles to study the 3d $\mathcal{N}=1$ vacua obtained from M-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 type along one-dimensional subspaces. There can be strong quantum corrections to the massless degrees of freedom in the low energy effective field theory, but topologically robust quantities such as "parity" anomalies are still calculable. We show how geometric reflections of the compactification space descend to 3d reflections and discrete symmetries. The "parity" anomalies of the effective field theory descend from topological data of the compactification. The geometric perspective also allows us to track various perturbative 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 of our analysis is that we are able to track the spectrum of extended objects and their transformations under higher-form symmetries.

v3: 85 pages, 4 figures, typos corrected, clarifications added

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• Generalized Symmetries in F-theory and the Topology of Elliptic Fibrations
• SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds
• IIB Explored: Reflection 7-Branes
• Anomalies of Generalized Symmetries from Solitonic Defects
• Super-Spin Chains for 6D SCFTs
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• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories