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.
Highlights
Geometric engineering provides a promising way to recast difficult questions in quantum field theory (QFT) in terms of the geometry of extra dimensions in string theory
We provide some explicit constructions of well-known 3D theories, including those which arise as edge modes of 4D topological insulators, and 3D N 1⁄4 1 analogs of grand unified theories
As another general class of examples, we show how to take a chiral 4D N 1⁄4 1 system engineered from the PantevWijnholt (PW) system compactified on a further S1 and glue it to its reflected image, resulting in a reflection symmetric 3D N 1⁄4 1 theory
Summary
Geometric engineering provides a promising way to recast difficult questions in quantum field theory (QFT) in terms of the geometry of extra dimensions in string theory. Performing a general analysis of localized zero modes, we show that they generically reside on codimension-3 subspaces inside of M4 This is very much in line with what we would have gotten from the circle reduction of M-theory on a local G2 space, as obtained from a three-manifold M3 of ADE singularities, where chiral matter is localized at points of the three-manifold We shall be interested in the action of reflections for the 3D theory in a Euclidean spacetime, R3i d∶ xi → −xi; and R3i d∶ xj → xj for j ≠ i; ð1:1Þ and the corresponding transformations on our physical fields This turns out to be a bit subtle because the physical content of our 3D system descends from a higher-dimensional starting point, so reflections in three dimensions may end up being composed with other internal reflection symmetries. Appendix G reviews the various 3D parity anomalies discussed in the bulk of this paper and how they arise as the phase ambiguity of a 3D theory’s partition function after placing it on an nonorientable Pinþ manifold
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