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

TL;DR

This work develops a geometric framework for 3d $\mathcal{N}=1$ vacua from M-theory on local Spin(7) spaces using Higgs bundles arising from 7d SYM on an ADE-laden four-manifold. By analyzing parity and higher-form symmetries, it shows how robust topological data constrain quantum corrections even when holomorphy is absent, and it provides concrete profiles for localized and bulk matter, as well as gluing constructions (connected sums) that build richer 3d theories, including edge-mode analogues of topological insulators and GUT-like sectors. The authors derive how 10d/11d discrete symmetries descend to 7d, 4d, and 3d theories, compute parity anomalies $\nu_{\mathsf{R}}$ and $\nu_{\mathsf{R}H}$, and discuss possible flux and instanton-induced corrections to Chern–Simons terms, all within a controlled top-down setup. The results offer a principled route to constrain infrared dynamics and guide explicit model-building in 3d Spin(7) compactifications, with potential links to F-theory on Spin(7) backgrounds and related dualities.

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.

Figures (4)

• Figure 1: Depiction of a connected sum construction of $M_4$ in terms of three summands of the form $S^1 \times M^{(i)}_3$. Each of the three blobs represents a four-manifold $M^{(i)}_3\times S^1$ for $M^{(i)}_{3}$ a three-manifold. The dots indicate localized matter on circles, where the solid blue dots represent localized chiral matter in $\mathbf{R}$ and the open red dots represent localized chiral matter in $\overline{\mathbf{R}}$ from the point of view of the PW system on the three-manifold $M^{(i)}_3$.
• Figure 2: Matter content of the 3d theory when $M_4=S^1 \times M_3$. The theory has $\mathcal{N}=2$ supersymmetry but the quiver is expressed in terms of $\mathcal{N}=1$ fields where double lines connected to the gauged node are adjoint fields, single lines denote fields in $\mathbf{R}$ and its conjugate, and no lines denote singlets.
• Figure 3: An illustration of our gluing construction of Kähler manifolds.
• Figure 4: Depiction of the connected sum construction used in the section. The separate blue and red dots indicate localized modes (on circles) with opposite chiralities/Hessians.