Construction of Hyperkähler Metrics for Complex Adjoint Orbits

More documents

Recommendations

Info

2.4 Twistor Spaces 18 of (2.3), by µ = (µ2 + iµ3) + 2ζµ1 − ζ 2 (µ2 − iµ3).
Chapter 3 Spectral Curve and the Kähler Potential for Complex Semisimple Orbits The purpose of this chapter is to investigate the Kronheimer hyperkähler structure on complex semisimple adjoint orbits by means of spectral curve methods. These ideas rely on the description of Kronheimer’s moduli space as a space of solutions to Nahm’s equations, together with the fact that we may interpret Nahm’s equations as an equation in Lax form - a fact which has been widely investigated in the literature of magnetic monopoles, where Nahm’s equations arise in some sort of duality to the Bogomolny equations (see [Hit83]). In the situation which concerns us here, however, the solutions to Nahm’s equations must be defined on a half-line, and the boundary conditions are of a different nature from the ones that arise in the construction of monopoles. In particular, there is a unique spectral curve associated to an adjoint orbit; it can be described explicitly as a union of rational curves lying in the holomorphic tangent bundle of the projective line. On the other hand, whereas in th As an application, we shall consider the computation of the Kähler po- 19
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 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: 2.4 Twistor Spaces 17 (iv) If the c
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 65 and 66: 4.2 The Twistor Space 63 Here A(0,
Page 67 and 68: 4.2 The Twistor Space 65 Remark. As
Page 69 and 70: 4.3 Regular Sections of g c ⊗ O(2
Page 71 and 72:
4.3 Regular Sections of g c ⊗ O(2
Page 73 and 74:
4.4 Regular Twistor Lines 71 projec
Page 75 and 76:
4.4 Regular Twistor Lines 73 It fol
Page 77 and 78:
4.4 Regular Twistor Lines 75 Lemma
Page 79 and 80:
4.4 Regular Twistor Lines 77 as req
Page 81 and 82:
4.5 Construction of the Metric 79 d
Page 83 and 84:
4.5 Construction of the Metric 81
Page 85 and 86:
4.5 Construction of the Metric 83 f
Page 87 and 88:
4.5 Construction of the Metric 85 S
Page 89 and 90:
4.5 Construction of the Metric 87 I
Page 91 and 92:
4.5 Construction of the Metric 89 [
show all

Construction of Hyperkähler Metrics for Complex Adjoint Orbits

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?