Construction of Hyperkähler Metrics for Complex Adjoint Orbits

3.3 Regular Nahm Solutions and Line bundles on S 38
The coefficients of this linear system are clearly, as functions of s, rational
functions of e λ1s , . . . , e λks . Since dim H 0 (S; ML s (k−1)) = k, we may describe
such a section σ by means of specifying k “free unknows” ci,l, the others being
expressed in terms of these as linear combinations involving coefficients which
are rational functions of e λ1s , . . . , e λks .
Example : The case k = 3. As an illustration of the previous
discussion, consider now the SL(3, C) adjoint orbit of the element ξ = i
2 z,
where z = diag(λ1, λ2, λ3), with λ1, λ2, λ3 ∈ R, λ1 + λ2 + λ3 = 0, λ1 > λ2 >
λ3. Thus we are concerned with describing H 0 (S; ML s (2)).
Let us keep the notation of the previous section. We have
so that
We may then write
for i = 1, 2, 3, so that
q1 = a, q2 = b + cζ + dζ 2 ,
q + 1 = a, q − 1 = 0,
q + 2 = c + dζ, q − 2 = b ˜ ζ.
fi = exp(λia + λ 2 i (c + dζ)) exp(λis)(αi + βiζ + γiζ 2 ),
˜fi = exp(−λ 2 i b ˜ ζ)(αi ˜ ζ 2 + βi ˜ ζ + γi),
fi(0) = exp(λi(a + s) + λ 2 i c)αi,
˜fi(0) = γi,
and the 0-th order conditions
are simply
f1(0) = f2(0) = f3(0),
˜f1(0) = ˜ f2(0) = ˜ f3(0),
αi = exp((λ1 − λi)(s + a) + (λ 2 1 − λ 2 i )c)α1, (3.26)
γi = γ1, (3.27)