Field theory dynamics from M-theory on special holonomy manifolds

This thesis is concerned with M-theory compactifications on non-compact manifolds with special holonomy, which are the basic geometric objects at the centre of geometric engineering of super-symmetric quantum field theories in various dimensions. We begin by reviewing how special holonomy manifolds arise in the context of Kaluza Klein compactification of superstring/M-theory and the mathematical concept of holonomy. Moreover, we are considering various non-compact spaces such as four-dimensional gravitational instantons, the six-dimensional Calabi-Yau conifold, and the seven-dimensional G2-holonomy Bryant-Salamon space in the context of superstring/M-theory compactification. We focus on M-theory in the background of non-compact Calabi-Yau two-folds, as well as the Bryant-Salamon spin bundle over a three-sphere, with codimension four ADE type singularities and their low-energy physics interpretation. We then turn to construct infinitely many new non-compact G2 spaces as circle fibrations over particular discrete quotients of the conifold. The quotients of the conifold are known as hyperconifolds. In constructing the G2 conifolds, we rely on a recent theorem of Foscolo-Haskins-Nordstro¨m that realises a topological constraint for having complete G2-holonomy spaces from circle bundles over Calabi-Yau cones. In particular, we aim to generalise the M-theory flop transition that appears in the compactification on quotients of the Bryant-Salamon space. On the other side of the M-theory flop, we find codimension four Abelian singularities that can be realised as a set of special unitary non-Abelian gauge theories equipped with non-trivial Wilson lines. We see that in many cases, an ultraviolet perturbative gauge theory appears to have an infrared dual described by a smooth G2-holonomy background in M-theory, and the number of abelian U(1) factors on both sides matches. In addition, we can determine the number of L2-normalisable harmonic two-forms of the M-theory compactification on the new G2-conifolds. In the final part of the thesis, we study the gauging of a discrete Z3 symmetry in the five-dimensional superconformal TN theories. We argue that this leads to an infinite sequence of five-dimensional superconformal theories with either E₆ x SU(N) or SU(3) x SU(N) global symmetry group. In the M-theory realisation of TN theories as residing at the origin in the Calabi-Yau orbifolds C³/(ZN x ZN), we identify the Z₃ symmetry geometrically and the new theories arise from M-theory compactification on the non-Abelian trihedral orbifolds of C3, namely (C³/ZN x ZN)/Z₃. Moreover, we examine the non-Abelian orbifolds through their Type IIB dual description as (p, q) five-brane webs in the background of seven-branes, where the E₆ symmetry is manifest.