Construction of Hyperkähler Metrics for Complex Adjoint Orbits

2.2 Nahm’s Equations and Spectral Curves 14
flow on the Jacobian of S; more precisely, ker(η + A(s, ζ) T ) ∗ ∼ = NL s , where
N has degree k(k − 1); see [HM89].
Another observation we would like to make concerns the construction
of solutions to Nahm’s equation which are skew-adjoint. Again in [Hit83]
only the flows L s are treated, however the discussion extends immediately
to more general flows ML s as long as M is real, in a sense which we shall
describe soon. The important point to note is that the reality of M together
with the reality of the spectral curve, defines an anti-linear isomorphism
σ : H 0 (S, ML s (k −1)) → H 0 (S, M ∗ L −s (k −1)), which allows us to reproduce
the arguments in [Hit83]. We briefly recall them: first, using σ, one constructs
an hermitian inner product on V ; it turns out that the connection ∇ preserves
this product, so that by trivialising V with the connection one obtains skew-
adjoint matrices with respect to that hermitian inner product. Note that in
general this inner product is not positive-definite; we shall come back to this
question in Chapter 3. On the ot We now describe the reality condition on
line bundles.
Recall that T P 1 has a natural real structure, by which we mean an anti-
holomorphic involution. It is defined in terms of (η, ζ)-coordinates, by
τ : T P 1 −→ T P 1
(η, ζ) ↦−→ (− η
ζ 2 , −ζ−1 ). (2.2)
Suppose the spectral curve S is real, that is to say, invariant under τ. We
now consider reality on Jac(S). If M is a line bundle of degree 0 on S given
by the transition function
exp
k−1
ηi
qi(ζ) ,
ζi then we say that M is real, and write M ∈ Jac R (S), if
i=1
qi −ζ −1 = qi(ζ).