Abstract
This thesis is concerned with Mtheory compactiﬁcations on noncompact manifolds with special holonomy, which are the basic geometric objects at the centre of geometric engineering of supersymmetric quantum ﬁeld theories in various dimensions.We begin by reviewing how special holonomy manifolds arise in the context of Kaluza Klein compactiﬁcation of superstring/Mtheory and the mathematical concept of holonomy. Moreover, we are considering various noncompact spaces such as fourdimensional gravitational instantons, the sixdimensional CalabiYau conifold, and the sevendimensional G2holonomy BryantSalamon space in the context of superstring/Mtheory compactiﬁcation. We focus on Mtheory in the background of noncompact CalabiYau twofolds, as well as the BryantSalamon spin bundle over a threesphere, with codimension four ADE type singularities and their lowenergy physics interpretation.
We then turn to construct inﬁnitely many new noncompact 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 FoscoloHaskinsNordstro¨m that realises a topological constraint for having complete G2holonomy spaces from circle bundles over CalabiYau cones. In particular, we aim to generalise the Mtheory ﬂop transition that appears in the compactiﬁcation on quotients of the BryantSalamon space. On the other side of the Mtheory ﬂop, we ﬁnd codimension four Abelian singularities that can be realised as a set of special unitary nonAbelian gauge theories equipped with nontrivial Wilson lines. We see that in many cases, an ultraviolet perturbative gauge theory appears to have an infrared dual described by a smooth G2holonomy background in Mtheory, and the number of abelian U(1) factors on both sides matches. In addition, we can determine the number of L2normalisable harmonic twoforms of the Mtheory compactiﬁcation on the new G2conifolds.
In the ﬁnal part of the thesis, we study the gauging of a discrete Z3 symmetry in the ﬁvedimensional superconformal TN theories. We argue that this leads to an inﬁnite sequence of ﬁvedimensional superconformal theories with either E6 x SU(N) or SU(3) x SU(N) global symmetry group. In the Mtheory realisation of TN theories as residing at the origin in the CalabiYau orbifolds C3/(ZN x ZN), we identify the Z3 symmetry geometrically and the new theories arise from Mtheory compactiﬁcation on the nonAbelian trihedral orbifolds of C3, namely (C3/ZN x ZN)/Z3. Moreover, we examine the nonAbelian orbifolds through their Type IIB dual description as (p, q) ﬁvebrane webs in the background of sevenbranes, where the E6 symmetry is manifest.
Date of Award  1 Aug 2022 

Original language  English 
Awarding Institution 

Supervisor  Bobby Acharya (Supervisor) 