Stevenson, David (1992) Yang-Mills instantons over Hopf surfaces. PhD thesis, University of Warwick.

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Abstract

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.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Library of Congress Subject Headings (LCSH):	Yang-Mills theory, Surfaces, Geometry, Algebraic, Vector spaces, Discrete groups
Official Date:	1992
Dates:	Date Event 1992 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Rawnsley, John H. (John Howard), 1947- ; Reid, Miles (Miles A.) ; Hitchin, N. J. (Nigel J.), 1946-
Sponsors:	Science and Engineering Research Council (Great Britain)
Extent:	xiii, 108 leaves : illustrations
Language:	eng

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