Oliver Nash
Published 2006-10-09Version 1
This thesis was motivated by a desire to understand the natural geometry of hyperbolic monopole moduli spaces. We take two approaches. Firstly we develop the twistor theory of singular hyperbolic monopoles and use it to study the geometry of their charge 1 moduli spaces. After this we introduce a new way to study the moduli spaces of both Euclidean and hyperbolic monopoles by applying Kodaira's deformation theory to the spectral curve. We obtain new results in both the Euclidean and hyperbolic cases. In particular we prove new cohomology vanishing theorems and find that the hyperbolic monopole moduli space appears to carry a new type of geometry whose complexification is similar to the complexification of hyperk\"ahler geometry but with different reality conditions.
Comments: 95 pages, author's DPhil thesis
A pictorial introduction to differential geometry, leading to Maxwell's equations as three pictures
The complexifications of pseudo-Riemannian manifolds and anti-Kaehler geometry
Differential geometry of $\mathsf{SO}^\ast(2n)$-type structures
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