G₂-manifolds from 4d N=1 theories, part I: Domain walls

We propose new G₂-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 Kähler 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 Kähler 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 G₂. Indeed, for simple domain wall theories, this construction results in 7-manifolds, which are known to admit torsion-free G₂-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.