Bibliography: leaves 66-70. / v, 70 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / "The goal of this thesis is to calculate the Seiberg-Witten invariant for complex surfaces, X, with odd first betti number, that is complex surfaces that do not admit a Kahler metric". / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1997

Bibliography: p. 135-138. / 140 p. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1998?

This thesis is concerned with so-called higher derivative terms which arise in low energy approximations to certain physical models. In particular, the aim is to investigate the role that such terms play in low energy N=2 supersymmetric gauge theories in 4 dimensions, with gauge group SU(2).Chapter one serves as an introduction to the notions of supersymmetry and superfields. The problem of constructing an effective action which describes the low energy dynamics is introduced, and the construction of the Wilsonian action in terms of light and heavy modes is developed. The concept on a derivative expansion is also described. Chapter two introduces N=2 supersymmetric gauge theories with spontaneous symmetry breaking. It is observed that such systems always have a Bogomolnyi bound, and the consequences are discussed. We then develop a derivative expansion of this system in terms of N=2 superfields, drawing particular attention to the next-to- leading order derivative term (that is, those with 4 derivatives/8 fermions). The duality properties of such a term are reviewed, and their impact on the mass formula discussed. Conclusions are drawn as to their influence on the results of Seiberg and Witten. Chapter three deals with a non-renormalisation theorem for the next-to-leading order higher derivative term proposed by Dine and Seiberg. This states that instanton contributions to such a term in massless N=2 SU(N(_c)) gauge theories vanish when the number of flavours N(_f) = 2N(_c). We prove this result using the ADHM formalism for multi-instantons in the case N(_c) = 2.

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

Orientador: Luiz Agostinho Ferreira / Mestre

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Let M=P(E) be a ruled surface. We introduce metrics of finite volume on M whose singularities are parametrized by a parabolic structure over E. Then, we generalise results of Burns--de Bartolomeis and LeBrun, by showing that the existence of a singular Kahler metric of finite volume and constant non positive scalar curvature on M is equivalent to the parabolic polystability of E; moreover these metrics all come from finite volume quotients of $H^2 \times CP^1$. In order to prove the theorem, we must produce a solution of Seiberg-Witten equations for a singular metric g. We use orbifold compactifications $\overline M$ on which we approximate g by a sequence of smooth metrics; the desired solution for g is obtained as the limit of a sequence of Seiberg-Witten solutions for these smooth metrics.

Thesis (Ph.D.)--Michigan State University. Dept. of Mathematics, 2008. / Title from PDF t.p. (viewed on July 6, 2009) Includes bibliographical references (p. 72-74). Also issued in print.

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<p>In this thesis, we give two constructions for Riemannian metrics on Seiberg-Witten moduli spaces. Both these constructions are naturally induced from the L2-metric on the configuration space. The construction of the so called quotient L2-metric is very similar to the one construction of an L2-metric on Yang-Mills moduli spaces as given by Groisser and Parker. To construct a Riemannian metric on the total space of the Seiberg-Witten bundle in a similar way, we define the reduced gauge group as a subgroup of the gauge group. We show, that the quotient of the premoduli space by the reduced gauge group is isomorphic as a U(1)-bundle to the quotient of the premoduli space by the based gauge group. The total space of this new representation of the Seiberg-Witten bundle carries a natural quotient L2-metric, and the bundle projection is a Riemannian submersion with respect to these metrics. We compute explicit formulae for the sectional curvature of the moduli space in terms of Green operators of the elliptic complex associated with a monopole. Further, we construct a Riemannian metric on the cobordism between moduli spaces for different perturbations. The second construction of a Riemannian metric on the moduli space uses a canonical global gauge fixing, which represents the total space of the Seiberg-Witten bundle as a finite dimensional submanifold of the configuration space.</p> <p>We consider the Seiberg-Witten moduli space on a simply connected K&auml;uhler surface. We show that the moduli space (when nonempty) is a complex projective space, if the perturbation does not admit reducible monpoles, and that the moduli space consists of a single point otherwise. The Seiberg-Witten bundle can then be identified with the Hopf fibration. On the complex projective plane with a special Spin-C structure, our Riemannian metrics on the moduli space are Fubini-Study metrics. Correspondingly, the metrics on the total space of the Seiberg-Witten bundle are Berger metrics. We show that the diameter of the moduli space shrinks to 0 when the perturbation approaches the wall of reducible perturbations. Finally we show, that the quotient L2-metric on the Seiberg-Witten moduli space on a K&auml;hler surface is a K&auml;hler metric.</p> / <p>In dieser Dissertationsschrift geben wir zwei Konstruktionen Riemannscher Metriken auf Seiberg-Witten-Modulr&auml;umen an. Beide Metriken werden in nat&uuml;rlicher Weise durch die L2-Metrik des Konfiguartionsraumes induziert. Die Konstruktion der sogenannten Quotienten-L2-Metrik entspricht der durch Groisser und Parker angegebenen Konstruktion einer L2-Metrik auf Yang-Mills-Modulr&auml;umen. Zur Konstruktion einer Quotienten-Metrik auf dem Totalraum des Seiberg-Witten-B&uuml;ndels f&uuml;hren wir die sogenannte reduzierte Eichgruppe ein. Wir zeigen, dass der Quotient des Pr&auml;modulraumes nach der reduzierten Eichgruppe als U(1)-B&uuml;ndel isomorph ist zu dem Quotienten nach der basierten Eichgruppe. Dadurch tr&auml;gt der Totalraum des Seiberg-Witten B&uuml;ndels eine nat&uuml;rliche Quotienten-L2-Metrik, bzgl. derer die B&uuml;ndelprojektion eine Riemannsche Submersion ist. Wir berechnen explizite Formeln f&uuml;r die Schnittr&uuml;mmung des Modulraumes in Ausdr&uuml;cken der Green-Operatoren des zu einem Monopol geh&ouml;rigen elliptischen Komplexes. Ferner konstruieren wir eine Riemannsche Metrik auf dem Kobordismus zwischen Modulr&auml;umen zu verschiedenen St&ouml;rungen. Die zweite Konstruktion einer Riemannschen Metrik auf Seiberg-Witten-Modulr&auml;umen benutzt eine kanonische globale Eichfixierung, verm&ouml;ge derer der Totalraum des Seiberg-Witten-B&uuml;ndels als endlich-dimensionale Untermannigfaltigkeit des Konfigurationsraumes dargestellt werden kann.</p> <p>Wir betrachten speziell die Seiberg-Witten-Modulr&auml;ume auf einfach zusammenh&auml;ngenden K&auml;hler-Mannigfaltigkeiten. Wir zeigen, dass der Seiberg-Witten-Modulraum (falls nicht-leer) im irreduziblen Fall ein komplex projektiver Raum its und im reduziblen Fall aus einem einzelnen Punkt besteht. Das Seiberg-Witten-B&uuml;ndel l&auml;&szlig;t sich mit der Hopf-Faserung identifizieren. Die L2-Metrik des Modulraumes auf der komplex projektiven Fl&auml;che CP2 (mit einer speziellen Spin-C-Struktur) ist die Fubini-Study-Metrik; entsprechend sind die Metriken auf dem Totalraum Berger-Metriken. Wir zeigen, dass der Durchmesser des Modulraumes gegen 0 konvergiert, wenn die St&ouml;rung sich dem reduziblen Fall n&auml;hert. Schlie&szlig;lich zeigen wir, dass die Quotienten-L2-Metrik auf dem Seiberg-Witten-Modulraum einer K&auml;hlerfl&auml;che eine K&auml;hler-Metrik ist.</p>