The Geometry and Physics of M-theory on $G_2$ -Holonomy Manifolds

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Aims: Characterizing M-theory compactifications on $G_2$ manifolds, and using these to obtain phenomenologically interesting 4-dimensional gauge theories.Novelty of research methodology: Utilizing new constructions of $G_2$ holonomy manifolds obtained recently in the mathematics literature. Exploring their implications in String Compactifications and connection to moduli spaces of gauge theories. String theory and M-theory are unique frameworks to study gauge theory and gravity in a fully consistent quantum theory. Nevertheless, much of the initial questions, motivating these theories have remained unanswered: what is the relevance for 4d physics? In recent years an approach to systematically characterize string theory vacua has emerged. On general grounds properties of the 4d physics are encoded in so-called compactification geometries: string/M-theory are defined in higher dimensions (10 or 11) and to obtain a 4d theory, the remaining dimensions are extended on a compact geometry. The properties of this encode most of the data that determine a 4d gauge theory: the gauge group, matter fields, couplings, and most importantly, symmetries. A key property, which ensures that this framework is robust against small fluctuations is the existence of supersymmetry in the resulting 4d theory. Supersymmetry, a symmetry between the bosonic and fermionic fields of a theory, has profound implications, in that it imposes constraints on the holonomies of the compactification geometries. In the case of M-theory, which is an 11d theory, the compactification geometry has to be 7-dimensional, and to preserve supersymmetry, has to have reduced holonomy $G_2$ (instead of $SO(7)$). The project gives a connection between results in pure mathematics, specifically differential geometry, and applications in mathematical physics, such as string theory and field theory. The project is timely, in that recently a large class of $G_2$ manifolds have been constructed by mathematicians, the so-called "twisted connected sum construction", and the implications of these geometries, as well as generalizations of these constructions are yet to be uncovered. One central goal of the PhD project will be to explore how singular limits of such $G_2$ manifolds can be constructed, as these will be of key importance in applications to M-theory. The goal is to modify the twisted connected sum constructions to include singularities that give rise to chiral matter in the four-dimensional compactification. One tool that will be used heavily is the duality to heterotic and F-theory compactifications, as obtained recently by Braun and Schafer-Nameki.This project falls within the EPSRC Mathematical Physics research area.