Construction of Hyperkähler Metrics for Complex Adjoint Orbits

3.4 The Kähler Potential for Integral Orbits 46
We also recall from [Don92] how one may identify Kronheimer’s moduli
space M(ξ) with a submanifold of X , when ξ is an integral element of t in the
sense that exp(ξ) = 1. This correspondence is given as follows. If (T1, T2, T3)
is a solution to Nahm’s equations defined over the half-line (−∞, 0] and such
that as s → −∞
T1 → 0, T2 → 0, T3 → −Ad(g0)(ξ) for g0 ∈ G,
then we construct a rotation-invariant solution to Hitchin’s equations over
the disc, by
Ar = 0,
Aθ = T3(log r),
Φ = (T1 + iT2)(log r)z −1 dz. (3.37)
Donaldson observes that the apparent singularity at 0 is removable in this
integral case. Furthermore, the circle action on X induced by rotations of
the disc preserves the hyperkähler structure of X , so that M(ξ) inherits a
hyperkähler metric, which is in fact the same as Kronheimer’s metric.
Finally, recall that one has another circle action on X defined by
(A, Φ) ↦→ (A, e iθ Φ). (3.38)
It preserves the metric g as well as the complex structure I. It does not
preserve J and K, but acts on ω2 + iω3 as e iθ (ω2 + iω3).
As the circle action (3.38) above preserves the Kähler form ω1, we consider
the associated moment map µ : X → i R. It is computed as
µ(A, Φ) = − 1
Φ 2L
2= −1 Tr(ΦΦ
2 2
∗ );
see [Hit87a]. Writing Φ = ψ dz, Φ ∗ = ψ ∗ dz, where ψ : D → g c , we may
rewrite µ as
− 1
Tr(ψψ
2 D
∗
) dz ∧ dz = −i Tr(ψψ
D
∗ ) dx dy.
D