I'm trying to understand the paper by Hitchin called ''Polygons and gravitons", Polygons and gravitons - INSPIRE-HEP. I'm stuck at page 471. At this point, he does some computations and obtains a metric:
and is defined implicitly by
We have that are complex coordinates. He proceeds by saying:
If we return to the form of the metric (4.4), and the description of the space of real quadratic polynomials as Euclidean 3-space with the metric given by the discriminant, the we obtain the metric which describes the gravitationals multi-instantons of Gibbons and Hawking:where and .
The form of the metric (4.4) mentioned isThe problem I have is that I don't know how to go from the first metric to the metric of Gibbons and Hawking. Is a change of variables? Which one?
PS: Maybe this helps to understand (4.4),