M theory and Singularities of Exceptional Holonomy Manifolds

TL;DR

Acharya and Gukov develop a cohesive framework linking M$ theory$ on $G_2$- and Spin$(7)$-holonomy manifolds to four- and three-dimensional gauge theories with minimal supersymmetry. By combining compactification geometry, dualities with the heterotic string and type IIA string theory, and local models of singularities, they show how non-Abelian gauge symmetry, chiral fermions, and nonperturbative phenomena such as confinement and mass gaps arise from singularities and topology-changing transitions. Their analysis employs Hitchin's flow, Kronheimer's hyper-Kähler quotient construction, and brane/flux dualities to connect geometric transitions with field-theoretic phase structure, providing a principled route to engineering realistic gauge dynamics in a gravitational context. The work thereby bridges detailed geometric constructions with the IR dynamics of strongly coupled gauge theories in minimally supersymmetric settings, highlighting both the potential and the challenges of realizing phenomenology from exceptional holonomy spaces.

Abstract

M theory compactifications on G_2 holonomy manifolds, whilst supersymmetric, require singularities in order to obtain non-Abelian gauge groups, chiral fermions and other properties necessary for a realistic model of particle physics. We review recent progress in understanding the physics of such singularities. Our main aim is to describe the techniques which have been used to develop our understanding of M theory physics near these singularities. In parallel, we also describe similar sorts of singularities in Spin(7) holonomy manifolds which correspond to the properties of three dimensional field theories. As an application, we review how various aspects of strongly coupled gauge theories, such as confinement, mass gap and non-perturbative phase transitions may be given a simple explanation in M theory.

Figures (13)

• Figure 1: Parallel transport of a vector $\vec{v}$ along a closed path on the manifold $X$.
• Figure 2: A cartoon representing Joyce orbifold $T^n/\Gamma$ with $\mathbb{C}^2 / \mathbb{Z}_2$ orbifold points.
• Figure 3: A cone over a compact space $Y$.
• Figure 4: A non-compact space $X$ can be viewed as a foliation by principal orbits $Y = G/K$. The non-trivial cycle in $X$ correspond to the degenerate orbit $G/H$, where $G \supset H \supset K.$
• Figure 5: A cartoon representing Taub-NUT space as a circle fibration over a 3-plane.