The 4-manifold S1 x S3, when endowed with the structure of a certain complex Hopf surface, is an example of a principal elliptic fibration. We use this structure to study the moduli spaces of anti-self-dual connections (instantons) on SU(2) bundles over S1 x S3. Chapter 1 is introductory. We define Buchdahl's notion of stability and outline the correspondence between instantons and stable holomorphic SL(2,C) bundles over S1 x S3. In Chapter 2 we study holomorphic line and SL(2, C) bundles over a general principal elliptic surface using an extension of the ‘graph’ invariant introduced by Braam and Hurtubise. We prove some auxiliary results needed in later chapters and introduce a stratification of the moduli space. In Chapter 3 we construct elements of one of the strata using the ‘Serre construction’ of algebraic geometry and deduce a structure result for the charge 1 case. Chapter 4 applies the results of the previous chapters in the construction of monopoles on the solid torus with a hyperbolic metric. We recover easily a result of Braam and Hurtubise. In Chapter 5 we adapt a construction of Friedman to describe a method of construction for elements of the remaining strata of the moduli spaces over the Hopf surface. In the charge 1 case we again determine the diffeomorphism type of the stratum completely. Combined with the results of Chapter 3 we deduce the natural action of S1 x S3 on the charge 1 moduli space is free. In Chapter 6 we study the charge 1 instanton moduli spaces over secondary Hopf surfaces diffeomorphic to the product of S1 and a Lens space. Chapter 7 considers twistorial methods and their application in the construction of explicit solutions. We define an invariant of an instanton, the spectral surface, which is a 2-dimensional analogue of Hitchin’s spectral curve. We use it to deduce that methods of Atiyah and Ward fail to generate a full family of charge 1 solutions. Finally we show how the spectral surface can be used in a sheaf theoretic construction of the ‘missing’ solutions.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:316503 |
| Date | January 1992 |
| Creators | Stevenson, David |
| Publisher | University of Warwick |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://wrap.warwick.ac.uk/109472/ |
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