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Title: Solutions of Dirac equations on certain non compact manifolds
Author: Baldwin, Peter John
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 1999
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This thesis is in two parts. Part one concerns a smooth Riemannian manifold Mn with a codimension two submanifold Σn-2 whose complement is spin. A removable singularities theorem is proven for square summable harmonic spinors associated to spin structures on Mn - Σn-2 which extend to Mn. It is shown that there may be a non-trivial finite dimensional vector space of L2 harmonic spinors, with interesting new behaviour, associated to a non-extending spin structure on Mn - Σn-2. Bounds are obtained for the dimension of this null space for any spin structure on a punctured Riemann surface. These bounds are seen to be sharp in certain cases. An index theorem for Dirac operators associated to non-extending spin structures on M2n - Σ2n-2 is proved when Σ2n-2 has trivial normal bundle. This index is computable in terms of characteristic classes of M2n and Σ2n-2. It does not depend upon the smooth metric on M2n. Reducing this index theorem modulo two gives a generalisation of the work of Esnault, Seade and Viehweg to the smooth category. An index theorem is conjectured for non-extending spin structure on M2n - Σ2n-2 when Σ2n-2 has non-trivial normal bundle. Evidence from a number of sources is presented for this. In part two the author proves the uniqueness of the Sen form on the two monopole moduli space. This was conjectured by Sen in 1994 motivated by physical arguments from S-duality in non-abelian gauge theory.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral