Construction of Hyperkähler Metrics for Complex Adjoint Orbits

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Declaration The work in this thesis is, to the best of my knowledge, original, except where attributed to others. 4
Summary We study hyperkähler metrics on the orbits of the adjoint action of a complex semisimple Lie group on its Lie algebra. We use spectral curve methods to describe how to compute a Kähler potential for hyperkähler metrics on SL(k, C) orbits. This essentially reduces the problem to linear algebra. We carry out the calculations in the case k = 2. We develop a twistor theory for complex adjoint orbits, give a character- isation of the twistor lines, and derive an expression for the metric. 5
Page 1 and 2: Construction of Hyperkähler Metric
Page 3 and 4: Contents Acknowledgements 3 Declara
Page 5: Acknowledgements I would like to th
Page 9 and 10: tion to certain regular lines. We c
Page 11 and 12: Chapter 2 Preliminaries The materia
Page 13 and 14: 2.2 Nahm’s Equations and Spectral
Page 19 and 20: 2.4 Twistor Spaces 17 (iv) If the c
Page 21 and 22: Chapter 3 Spectral Curve and the K
Page 23 and 24: 3.1 The Spectral Curve 21 for smoot
Page 25 and 26: 3.1 The Spectral Curve 23 the secti
Page 27 and 28: 3.2 Regularity 25 Remark It is not
Page 29 and 30: 3.3 Regular Nahm Solutions and Line
Page 47 and 48: 3.4 The Kähler Potential for Integ
Page 49 and 50: 3.4 The Kähler Potential for Integ
Page 51 and 52: 3.5 Example: Kähler Potential for
Page 57 and 58:
4.1 The Kronheimer-Biquard Moduli S
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Page 63 and 64:
Page 65 and 66:
4.2 The Twistor Space 63 Here A(0,
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4.2 The Twistor Space 65 Remark. As
Page 69 and 70:
4.3 Regular Sections of g c ⊗ O(2
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4.3 Regular Sections of g c ⊗ O(2
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4.4 Regular Twistor Lines 71 projec
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4.4 Regular Twistor Lines 73 It fol
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4.4 Regular Twistor Lines 75 Lemma
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4.4 Regular Twistor Lines 77 as req
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4.5 Construction of the Metric 79 d
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4.5 Construction of the Metric 81
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4.5 Construction of the Metric 83 f
Page 87 and 88:
4.5 Construction of the Metric 85 S
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4.5 Construction of the Metric 87 I
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4.5 Construction of the Metric 89 [
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Construction of Hyperkähler Metrics for Complex Adjoint Orbits

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