Construction of Hyperkähler Metrics for Complex Adjoint Orbits

3.3 Regular Nahm Solutions and Line bundles on S 28
We recall that the solution is obtained in the following two steps:
(i) Letting Vs = H 0 (S, ML s (k − 1)), one obtains endomorphisms Ã0(s),
Ã1(s), Ã2(s) of Vs, determined by requiring that
for all σ ∈ Vs.
(η + Ã0(s) + ζ Ã1(s) + ζ 2 Ã2(s))σ = 0 (3.6)
(ii) In order to arrive at A(s, ζ), one defines a vector bundle V over
(−∞, 0] whose fibre over s is Vs, and then one constructs a connection
on V which is used to trivialise V by taking a basis of covariant constant
sections. The matrices of the endomorphims Ã0(s), Ã1(s), Ã2(s) with re-
spect to this basis, denoted by A0(s), A1(s), A2(s), are such that A(s, ζ) =
A0(s) + A1(s)ζ + A2(s)ζ 2 corresponds to a solution to Nahm’s equations
defined on (−∞, 0].
In fact, for our purposes it will be enough to describe A(s, ζ) up to con-
jugation; note that if we take the matrices of the endomorphisms Ã0(s),
Ã1(s), Ã2(s) with respect to any basis of Vs, for each s ∈ (−∞, 0], then we
obtain a k × k matrix [ Ã(s, ζ)] which is of the form g(s)A(s, ζ)g(s)−1 with
g(s) ∈ GL(k, C). We shall see that one may find a suitable basis for which
the coefficients of [ Ã(s, ζ)] are rational functions in eλ1s , . . . , e λks .
Let us discuss next the requirement that the Nahm solutions obtained by
the method described in (i) and (ii) should be suk-valued.
3.3.1 Reality
The method of producing solutions to Nahm’s equations out of line bundles
on the spectral curve gives us, in general, only gl(k, C)−valued solutions.
In order to obtain suk−valued solutions, we must impose further conditions
on the line bundle. In [Hit83] it is shown how one can obtain skew-adjoint