The object of my thesis is to investigate projectively related metrics, that is, metrics whose Levi-Civita connections admit exactly the same family of unparametrised geodesics on 4-dimensional manifolds with positive de nite or neutral (+;+;;) signatures. The general idea is to study the relationship between projectively related metrics and the holonomy types of each metric. The main technique presented in the work requires a certain classification of the curvature map which has been developed by G. S. Hall and D. P. Lonie in the case of Lorentz signature. In chapter 1, some of the background theory will be given. This will include an introduction to bivector algebra, a revision of the Riemann curvature tensor and holonomy theory and, in particular, the fundamental equations for projective related metrics. A brief historical and bibliographical review is also given. The subsequent chapter gives the details of projective related metrics of positive definite signature. In x2.1, the structure of so(4) is described with an emphasis on the canonical decomposition of bivectors and then the subalgebras of so(4) follow. In x2.2., the problem of projective related metrics can be solved case by case decided by holonomy types. In many of these cases, the connections are found to be necessarily equal. A few cases with nontrivial projectively related metrics have been found by only in the rather special case of curvature class D, and the metrics are given in the appendices. An extension of this method to spaces of neutral signature (+;+;;) is made in chapter 3. The rst part of the chapter discusses the algebraic structure of a 4-dimensional vector space with such a metric. In contrast to metrics of the other two signatures (positive definite and Lorentz), this metric admits totally null planes. The structure of the Lie algebra so(2; 2) can be described through the action on totally null planes. The classification of all subalgebras of so(2,2) is then obtained in terms of self-dual and anti-self-dual bivectors. In most holonomy types and curvature classes, the problem has only trivial solutions. Nontrivial projectively related metrics can be found for four holonomy types with curvature class D and two holonomy types with curvature class A.

Realistic four-dimensional model building from string theory has been a focus of the string theory community ever since its inception. Toroidal orientifold constructions have emerged as a technically simple class of candidate models. Novel ingredients, such as background fluxes, have been discovered and intensely studied over the past few years. They allow for a (partial) solution of several long standing problems associated with model building in this framework. In this thesis, I summarize progress that has been made in toroidal orientifold constructions in type IIA string theory.This includes a detailed discussion of moduli stabilization and (non-) supersymmetric AdS and Minkowski vacua. Furthermore I commence a systematic study of generalized NSNS, i.e., metric and non-geometric, fluxes. The emergence of novel D-terms is presented in detail. While most of the discussion applies to generic orientifolds of T⁶, most features are exemplified by and studied in terms of a certain orientifold of T⁶/ℤ₄ owing to its somewhat richer structure compared to simpler models studied before. It is also briefly reported on efforts of finding de Sitter vacua and inflation in this class of models. / text

by Leung Wai-Man Raymond. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves [1]-[3] (2nd gp.)). / Chapter CHAPTER 0 --- Introduction --- p.1 / Chapter CHAPTER 1 --- Einstein-Hermitian Vector Bundles / Chapter 1.1 --- Preliminaries on Einstein-Hermitian structures --- p.4 / Chapter 1.2 --- Conformal invariance --- p.7 / Chapter 1.3 --- A Chern number inequality --- p.9 / Chapter CHAPTER 2 --- Stable Vector Bundles / Chapter 2.1 --- Coherent analytic sheaves --- p.12 / Chapter 2.2 --- "Torsion-free, reflexive and normal coherent analytic sheaves" --- p.18 / Chapter 2.3 --- Determinant bundles --- p.22 / Chapter 2.4 --- Stable vector bundles --- p.27 / Chapter 2.5 --- Stability of Einstein-Hermitian vector bundles --- p.32 / Chapter CHAPTER 3 --- Existence of Einstein-Hermitian connection on stable vector bundle over a compact Riemann Surface --- p.34 / Chapter CHAPTER 4 --- Existence of Einstein-Hermitian metric on stable vector bundle over a projective algebraic manifold / Chapter 4.1 --- Solution of the evolution equation for finite time --- p.45 / Chapter 4.2 --- Convergence of solution for infinite time --- p.53 / APPENDIX / Chapter I. --- A vanishing theorem of Bochner type and its consequences --- p.67 / Chapter II. --- Uhlenbeck's results on connections with Lp bounds on curvature --- p.69 / REFERENCE

Chan, Kai Leung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 106-109). / Abstracts in English and Chinese. / Chapter 0 --- Introduction --- p.7 / Chapter I --- Background Scenery --- p.10 / Chapter 1 --- Seiberg-Witten Invariants --- p.11 / Chapter 1.1 --- Preliminaries --- p.11 / Chapter 1.2 --- Construction of Seiberg-Witten Invariants --- p.17 / Chapter 1.2.1 --- Seiberg-Witten Equations and the Moduli Space --- p.17 / Chapter 1.2.2 --- Seiberg-Witten Invariants --- p.19 / Chapter 1.2.3 --- Remarks --- p.20 / Chapter 1.2.4 --- Seiberg-Witten Invariants for b+2= 1 --- p.22 / Chapter 1.3 --- Important Results of Seiberg-Witten Invariants --- p.23 / Chapter 1.3.1 --- Manifolds Admit Positive Scalar Metrics --- p.23 / Chapter 1.3.2 --- Connected Sums --- p.25 / Chapter 1.3.3 --- Kahler Surfaces --- p.27 / Chapter 1.3.4 --- Symplectic Manifolds --- p.30 / Chapter 2 --- Intersection Forms --- p.32 / Chapter 2.1 --- Intersection Forms of 4-manifolds --- p.32 / Chapter 2.2 --- Classification Theorem --- p.34 / Chapter 2.3 --- Review: Van Kampen's Theorem --- p.35 / Chapter 3 --- Kirby Calculus --- p.37 / Chapter 3.1 --- Review on Handle Decompositions --- p.37 / Chapter 3.1.1 --- Constructions --- p.38 / Chapter 3.1.2 --- Handle Slides and Cancellations --- p.42 / Chapter 3.1.3 --- Calculation of Homology Groups --- p.44 / Chapter 3.2 --- Kirby Diagrams --- p.45 / Chapter 3.2.1 --- Constructions --- p.45 / Chapter 3.2.2 --- Handle Slides and Cancellations --- p.50 / Chapter 3.2.3 --- Dotted Notation for 1-handles --- p.56 / Chapter 3.3 --- 3-Manifolcis: As Boundaries of 4-Manifolds --- p.60 / Chapter 3.3.1 --- Introduction --- p.60 / Chapter 3.3.2 --- Lens spaces --- p.62 / Chapter 3.4 --- Linear Plumbing --- p.63 / Chapter 3.5 --- Rational Blowdown --- p.65 / Chapter II --- Examples of Exotic Structures --- p.71 / Chapter 4 --- mCP2#kCP2 --- p.72 / Chapter 4.1 --- Introduction --- p.72 / Chapter 4.2 --- Example: CP2#7CP2 --- p.73 / Chapter 4.3 --- Progress of Researches --- p.85 / Chapter 5 --- Gluing Results in Seiberg-Witten Theory --- p.89 / Chapter 5.1 --- Revisit of Seiberg-Witten Invariants --- p.89 / Chapter 5.2 --- Fiber Sums and its Generalization --- p.91 / Chapter 5.3 --- Logarithmic transformations and its Generalization --- p.93 / Chapter 5.4 --- Knot Theory and Alexander Polynomials --- p.96 / Chapter 5.5 --- Main Theorem --- p.102 / Bibliography --- p.106

Two special classes of solutions to the Yang-Mills equations are studied in this thesis; Hermitian-Einstein connections on holomorphic bundles over Kahler manifolds, and self-dual connections on bundles over Riemannian 4-manifolds. We give a new proof of a theorem of Narasimhan and Seshadri, which characterizes those holomorphic bundles over an algebraic curve admitting projectively flat connections, and describe a conjecture of Hitchin and Kobayashi that would extend this to Hermitian-Einstein connections over any smooth projective variety. This conjecture is proved to be true for the simplest interesting case: bundles of rank 2 over ℙ². Moduli spaces of self-dual connections are studied from the point of view of differential topology, For bundles of Chern class -1 over a simply connected 4-manifold this moduli space can be compactified in a straightforward way and is, in a generic sense, an orientable manifold with quotient singularities. Applying the theory of cobordism to this moduli space we deduce that there are severe constraints on the matrices which can be realised by the intersection pairing on the second homology group of a smooth 4-manifold.

Thesis (Ph.D.)--McMaster University (Canada), 1995. / Source: Dissertation Abstracts International, Volume: 57-10, Section: B, page: 6295. Adviser: I. Hambleton.

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

The spin^c-invariants for a compact smooth simply-connected oriented four-manifold, as defined by Pidstrigach and Tyurin, are studied in this thesis. Unlike the Donaldson polynomial invariants, they are defined by cutting down the moduli space M' of '1-instantons', which is the subspace of the moduli space M of anti-self-dual connections parametrizing coupled (spin^c) Dirac operators with non-trivial kernel. Our main goal is to study the relationship between these spin^c-invariants and the Donaldson polynomial invariants. The 'jumping subset' M' defined a cohomology class P of M which is given by the generalised Porteous formula. When the index l of the coupled Dirac operator is 1, the two smooth invariants are the same by definition. When l = 0 (or when M is compact), the spin^c-invariants are expressable as a Donaldson polynomial evaluating the 'Porteous class' P. Our main results concern the first two non-trivial cases l = -1 and -2, when the generalised Porteous formula can not be applied directly. Using cut-and-paste arguments to the moduli space M, we show that for the former case the spin^c-invariants and the contracted Donaldson invariants differ by a correction term. It is the number of points in the immediate lower stratum of the Uhlenbeck compactification times a universal 'linking invariant' on S⁴, which is obtained by computing an example (the K3 surface). The case when l = -2 and dimM = 8 is a parametrized version of the l = -1 situation and the correction term, which involves the same 'linking invariant', is obtained from a suitable obstruction theory.

Thesis (Ph.D.)--McMaster University, 2003. / Advisor: Ian Hambleton. Includes bibliographical references (leaves 109-112). Also available via World Wide Web.