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The instanton solutions to the Yang-Mills equations have a vast range of practical applications in field theories including gravitation and electro-magnetism. Solutions to Maxwell's equations, for example, are abelian gauge instantons on Minkowski space. Since these discoveries, a generalised theory of instantons has been emerging for manifolds with special holonomy. Beginning with connections and curvature on complex vector bundles, this thesis provides some of the essential background for studying moduli spaces of instantons. Manifolds with exceptional holonomy are special types of seven and eight dimensional manifolds whose holonomy group is contained in G2 and Spin(7), respectively. Focusing on the G2 case, instantons on G2 manifolds are defined to be solutions to an analogue of the four dimensional anti-self-dual equations. These connections are known as Donaldson-Thomas connections and a couple of examples are noted.

The instanton solutions to the Yang-Mills equations have a vast range of practical applications in field theories including gravitation and electro-magnetism. Solutions to Maxwell's equations, for example, are abelian gauge instantons on Minkowski space. Since these discoveries, a generalised theory of instantons has been emerging for manifolds with special holonomy. Beginning with connections and curvature on complex vector bundles, this thesis provides some of the essential background for studying moduli spaces of instantons. Manifolds with exceptional holonomy are special types of seven and eight dimensional manifolds whose holonomy group is contained in G2 and Spin(7), respectively. Focusing on the G2 case, instantons on G2 manifolds are defined to be solutions to an analogue of the four dimensional anti-self-dual equations. These connections are known as Donaldson-Thomas connections and a couple of examples are noted.

Let (Mn, g) be a Riemannian manifold. Say K ! E ! M is a principal K-bundle with connection A. We define a natural evolution equation for the pair (g,A) combining the Ricci flow for g and the Yang-Mills flow for A which we dub Ricci Yang-Mills flow. We show that these equations are, up to di eomorphism equivalence, the gradient flow equations for a Riemannian functional on M. Associated to this energy functional is an entropy functional which is monotonically increasing in areas close to a developing singularity. This entropy functional is used to prove a non-collapsing theorem for certain solutions to Ricci Yang-Mills flow. We show that these equations, after an appropriate change of gauge, are equivalent to a strictly parabolic system, and hence prove general unique short-time existence of solutions. Furthermore we prove derivative estimates of Bernstein-Shi type. These can be used to find a complete obstruction to long-time existence, as well as to prove a compactness theorem for Ricci Yang Mills flow solutions. Our main result is a fairly general long-time existence and convergence theorem for volume-normalized solutions to Ricci Yang-Mills flow. The limiting pair (g,A) satisfies equations coupling the Einstein and Yang-Mills conditions on g and A respectively. Roughly these conditions are that the associated curvature FA must be large, and satisfy a certain “stability” condition determined by a quadratic action of FA on symmetric two-tensors.

Let M be a Riemannian manifold with metric g, and let P be a principal G-bundle over M having connection one-form a. One can define a modified version of the Ricci flow on P by fixing the size of the fiber. These equations are called the Ricci Yang-Mills flow, due to their coupling of the Ricci flow and the Yang-Mills heat flow. In this thesis, we derive the Ricci Yang-Mills flow and show that solutions exist for a short time and are unique. We study obstructions to the long-time existence of the flow and prove a compactness theorem for pointed solutions. We represent the Ricci Yang-Mills flow as a gradient flow and derive monotonicity formulas that can be used to study breather and soliton solutions. Finally, we use maximal regularity theory and ideas of Simonett concerning the asymptotic behavior of abstract quasilinear parabolic partial differential equations to study the stability of the Ricci Yang-Mills flow in dimension 2 at Einstein Yang-Mills metrics. / text

Quantum Field Theory (QFT) provides the essential background for formulating the standard model of elementary particles and, moreover, practically all other theories attempting to explore the physical laws of nature at the sub-atomic level. One of the main observables in QFT are the scattering amplitudes, physical quantities which encode the information of the scattering process of particles. Accordingly, having authentic, well-defined and feasible prescriptions for the calculations of amplitudes is of huge importance to theoretical physicists. Actual calculations show that the text-book prescription, the Feynman method, besides in general being very cumbersome also hides some of the beautiful mathematical features of amplitudes. The last decade has seen tremendous efforts and achievements to improve such calculations, particularly in supersymmetric gauge theories, which have also led to better understanding of QFT itself. Among the known physically and mathematically interesting quantum field theories is perturbative gravity and its supersymmetric version, N=8 supergravity- much less understood than gauge theory. Following the developments in gauge theory, this dissertation mainly aims at exploring scattering amplitudes in gravity as a quantum field theory, using the modern approaches to QFT. The goal is not only to improve our understanding of gravity amplitudes by applying part of the known modern methods of calculations to it but also to introduce and develop new ones.

The one-loop effective action for N = 2 and N = 4 supersymmetric Yang-Mills theories are computed to order F5; and F6 respectively by the use of heat kernel techniques in N = 1 superspace. The computations are carried out via the introduction of a new method for computing DeWitt-Seeley coefficients in the coincidence limit. To order F5, the bosonic components of both N = 2 and N = 4 supersymmetric Yang-Mills theories are extracted and compared with the existing literature. For N = 4 super Yang-Mills theories the F5 terms are found to be consistent with the non-Abelian Born-Infeld action computed to this order by superstring methods and various other means of computing deformations of supersymmetric Yang-Mills theory. The result proved to be the final piece of a puzzle, leaving little doubt that there exists a unique deformation of maximally symmetric super Yang-Mills theories at this order. The F6 terms will be of importance for comparison with superstring calculations, including direct tests of the AdS/CFT conjecture. The bosonic components of N = 2 supersymmetric Yang-Mills are also shown to be consistent with existing literature, and will be of importance for testing of generalizations of the AdS/CFT conjecture.

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

Thesis (Ph. D.)--Duke University, 2007. / Includes bibliographical references.