Thesis (Ph. D.)--University of Chicago, Dept. of Physics, June 1999. / Includes bibliographical references. Also available on the Internet.

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

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.

Thesis (Ph. D.)--Ohio State University, 2005. / Title from first page of PDF file. Document formatted into pages; contains x, 94 p.; also includes graphics Includes bibliographical references (p. 52-54). Available online via OhioLINK's ETD Center

Thesis (Ph. D.)--University of Washington, 2006. / Vita. Includes bibliographical references (p. 65-70).

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 190-199).

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