I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471.

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At this point, he does some computations and obtains a metric:

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


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

and $b$ is defined implicitly by

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

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

where $\gamma=\sum \frac{1}{|x-x_i|}$ and curl $\omega=$grad $\gamma$. The form of the metric (4.4) mentioned is

$$\gamma^2(b'^2+a'\bar{a}')+\left(Im\left(\frac{2A'}{A}-\delta a' \right) \right)^2$$

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),



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