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Title: The geometry of calorons
Author: Nye, Thomas M. W.
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 2001
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Calorons, or periodic instantons, are anti-self-dual (ASD) connections on S1 x?3, and form an intermediate case between instantons (ASD connections on ?4) and monopoles (translation invariant instantons). Complete constrictions of instantons and monopoles have been found: there is a complete construction of instantons from algebraic data, the ADHM construction, due to Atiyah and others; while Nahm gave a construction of monopoles from solutions to a system of ODEs known as Nahm's equation. Both these constructions can be thought of as generalizations of a correspondence between ASD connections on the 4-torus, and ASD connections over the dual 4-torus, originally due to Mukai and Braam-Baal. This correspondence, often called the 'Nahm transform', is invertible and the inverse of the transform is the transform itself. Given an ASD connection on the 4-tours it is defined in terms of the kernel of a family of Dirac operators parameterized by the dual torus. The aim of this thesis is to generalize the Nahm transform to the caloron case. In particular, our approach is via analysis of these families of Dirac operators rather than via twistor theory. We start by exploring topological aspects of calorons, and boundary conditions. These are needed to ensure that the Dirac operators that define the Nahn transform are Fredholm. Our main innovation is to regard ?3 as the interior of the closed 3-ball B3, and to stipulate fixed behaviour on the boundary, rather than imposing asymptotic boundary conditions. The boundary conditions for calorons can be stated as follows: give a bundle on S1 x B3 we fix some gauge f on the boundary, and we require that in the gauge f, a U (n)caloron must resemble the pull-back of a U (n) monopole. There is a topological obstruction to extending f to the interior of S1 x B3, which we call the 'instanton charge' of the caloron.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available