Construction of Hyperkähler Metrics for Complex Adjoint Orbits

2.2 Nahm’s Equations and Spectral Curves 10
without further notice. Finally, recall that g is the fixed-point set of
2.2 Nahm’s Equations and Spectral Curves
Consider a triple of k × k matrices T1(s), T2(s), T3(s), which we shall suppose
to be smoothly defined for s ∈ (−∞, 0], as this is the case we shall be dealing
with later on. They are said to satisfy Nahm’s equations if
dT1
ds = [T2, T3], dT2
ds = [T3, T1], dT3
ds = [T1, T2].
Although these are non-linear equations, they can in a certain sense be lin-
earised in the Jacobian of an algebraic curve - the spectral curve, which lies in
T P 1 . What we shall recall in this section is the inverse process, namely how
to get solutions to Nahm’s equations from line bundles on the spectral curve.
This is done by using the method described in [Hit83], slightly adapted to
our situation. In [Hit83], one is concerned with describing monopoles, and
they turn out to be determined by their spectral curve S and a “fixed” flow
of line bundles L s on S- more precisely, L s is the restriction to S of the line
bundle on T P 1 defined by the transition function exp(sη/ζ) with respect to
the usual open covering of T P 1 . One may then characterize those algebraic
curves in T P 1 which arise as spectral curves of a monopole.
In the situation which will concern us later on, we have a picture which
is somewhat dual to the one mentioned above. We want to fix the curve S
and we are led to consider more general flows on the Jacobian. These flows
still have the same direction, determined by the cocycle [η/ζ], but we now
allow the “origin” M to vary. In the monopole case, the boundary conditions
force the origin to be the trivial bundle, but we shall be interested in our
situation in boundary conditions which are quite different and allow us this
extra freedom. We must however impose a “non-singularity” condition on