$G_2$-Manifolds from 4d N=1 Theories, Part I: Domain Walls

TL;DR

This paper constructs new $G_2$-holonomy manifolds in M-theory that geometrize 4d N=1 duality domain walls of 5d theories, linking complex geometry with supersymmetric field theory transitions.

Contribution

It introduces a geometric method to realize 4d N=1 domain walls within $G_2$-holonomy manifolds derived from Calabi-Yau three-folds fibering over a line.

Findings

Constructed $G_2$-manifolds from Calabi-Yau fibrations passing through singularities.

Realized domain walls in 5d theories as geometric transitions in $G_2$-holonomy spaces.

Extended the construction to include domain walls in 5d SQCD and dual theories.

Abstract

We propose new $G_2$-holonomy manifolds, which geometrize the Gaiotto-Kim 4d N=1 duality domain walls of 5d N=1 theories. These domain walls interpolate between different extended Coulomb branch phases of a given 5d superconformal field theory. Our starting point is the geometric realization of such a 5d superconformal field theory and its extended Coulomb branch in terms of M-theory on a non-compact singular Calabi-Yau three-fold and its K\"ahler cone. We construct the 7-manifold that realizes the domain wall in M-theory by fibering the Calabi-Yau three-fold over a real line, whilst varying its K\"ahler parameters as prescribed by the domain wall construction. In particular this requires the Calabi-Yau fiber to pass through a canonical singularity at the locus of the domain wall. Due to the 4d N=1 supersymmetry that is preserved on the domain wall, we expect the resulting 7-manifold to have holonomy $G_2$. Indeed, for simple domain wall theories, this construction results in 7-manifolds, which are known to admit torsion-free $G_2$-holonomy metrics. We develop several generalizations to new 7-manifolds, which realize domain walls in 5d SQCD theories and walls between 5d theories which are UV-dual.