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:

$\displaystyle $$\gamma dz d\bar{z}+\gamma^{-1}\left(\dfrac{2dy}{y}+\bar{\delta}dz \right)\left(\dfrac{2d\bar{y}}{\bar{y}}+\delta d\bar{z} \right)$$$

where

$\displaystyle \gamma=\sum \dfrac{1}{\Delta_i}=\sum \dfrac{1}{\sqrt{(b-b_i)^2+|\bar{z}+a_i|^2}}$

$\displaystyle \delta=\sum \dfrac{(b-b_i)-\Delta_i}{\Delta_i(\bar{z}+a_i)}$

and $\displaystyle b$is defined implicitly by $\displaystyle $$\prod ((b-b_i)+\Delta_i)=y\bar{y}$$$

We have that $\displaystyle $z,y$$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:$\displaystyle $$\gamma d\vec{x} d\vec{x}+\gamma^{-1}(d\tau+ \vec{\omega}d\vec{x})$$$

where $\displaystyle \gamma=\sum \frac{1}{|x-x_i|}$ and $\displaystyle curl $\omega=$grad $\gamma$$.

The form of the metric (4.4) mentioned is$\displaystyle $$\gamma^2(b'^2+a'\bar{a}')+\left(Im\left(\frac{2A '}{A}-\delta a' \right) \right)^2$$$

The 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),

$\displaystyle $$A\bar{A}=\prod((b-b_i)+\Delta_i)$$$