Construction of Hyperkähler Metrics for Complex Adjoint Orbits

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the twistor space for Kronheimer’s metric, which motivates us to construct another, “smaller” twistor space containing the regular twistor lines. In Section 4.3 we recall how to obtain regular lines from line bundles on a spectral curve. Section 4.4 has as its main result a characterization of regular twistor lines. Finally in Section 4.5 we discuss the construction of the metric. 8
Chapter 2 Preliminaries The material in this chapter is not original; we just collect some facts that we shall refer to later, and give the appropriate references. 2.1 Conventions Throughout this thesis, we shall adopt the following notations: G is a com- pact connected semisimple Lie group, T a fixed maximal torus, g and t their Lie algebras; we denote their complexifications by G c , T c , g c , t c , respec- tively. Our objects of interest here are complex semisimple adjoint orbits. We recall that any semisimple element of the semisimple Lie algebra g c be- longs to some Cartan subalgebra, and since any two such subalgebras are conjugate it will suffice to consider adjoint orbits of elements ξ ∈ t c . In fact, for technical reasons we shall often restrict ourselves to orbits of elements of t under the action of the complex group G c . Thus, if ξ ∈ t, we denote by Oξ its G c -adjoint orbit. We shall also suppose that ξ is a regular element, which means that the centraliser of ξ in g c is just t c ; we shall say that the corre- sponding complex orbit is regular. We recall that the Killing form on g c is non-degenerate and G c -invariant and hence allows us to identify adjoint and coadjoint complex orbits; we shall use this identification at our convenience, 9
Page 1 and 2: Construction of Hyperkähler Metric
Page 3 and 4: Contents Acknowledgements 3 Declara
Page 5 and 6: Acknowledgements I would like to th
Page 7 and 8: Summary We study hyperkähler metri
Page 9: tion to certain regular lines. We c
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
Page 59 and 60: 4.1 The Kronheimer-Biquard Moduli S
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4.1 The Kronheimer-Biquard Moduli S
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4.1 The Kronheimer-Biquard Moduli S
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4.2 The Twistor Space 63 Here A(0,
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4.2 The Twistor Space 65 Remark. As
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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
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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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