Calderbank, David M. J. (1995) Geometrical aspects of spinor and twistor analysis. PhD thesis, University of Warwick.

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Abstract

This work is concerned with two examples of the interactions between differential
geometry and analysis, both related to spinors. The first example is the Dirac operator
on conformal spin manifolds with boundary. I aim to demonstrate that the analysis of the
Dirac operator is a natural generalisation of complex analysis to manifolds of arbitrary
dimension, by providing, as far as possible, elementary proofs of the main analytical
results about the boundary behaviour of solutions to the Dirac equation. I emphasise
throughout the conformal invariance of the theory, and also the usefulness of the Clifford
algebra formalism. The main result is that there is a conformally invariant Hilbert space
of boundary values of harmonic spinors, and that the pointwise evaluation map defines a
conformally invariant metric on the interior. Along the way, many results from complex
analysis are generalised to arbitrary (Riemannian or conformal spin) manifolds, such as
the Cauchy integral formula, the Plemelj formula, and the L2-boundedness of the Hilbert
transform.
The second example concerns the geometry of the twistor operator and the analysis
of differential operators arising in twistor theory. I study the differential equations on
a complex quadric induced by holomorphic vector bundles on its twistor space. In 4
dimensions, there is already a beautiful example of such a relationship, the Ward correspondence
between holomorphic vector bundles trivial on twistor lines, and self-dual
connections. There are many generalisations of twistor theory to higher dimensions, but
it is not clear how best to generalise the Ward correspondence. Consequently, I focus on
6 dimensional geometry, and one possible generalisation proposed by Atiyah and Hitchin,
and investigated by Manin and Minh. I study a number of differential equations produced
by this 6 dimensional twistor construction, with a view to reconstructing the holomorphic
vector bundle on the twistor space from these equations. While this aim has not been
realised, some progress has been made.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Spinor analysis, Dirac equation, Manifolds (Mathematics), Twistor theory
Official Date:	December 1995
Dates:	Date Event December 1995 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Hitchin, N. J. (Nigel J.), 1946-
Sponsors:	Engineering and Physical Sciences Research Council (EPSRC) ; St. John's College (University of Cambridge)
Extent:	132 p.
Language:	eng

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