Gravitational instantons metric (change variables)

Question

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric:

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

where

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

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),

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

Answer 1

The equation 4.4 in the article gives the conformal structure. The calculations that follow that equation are to determine the metric within that conformal structure. After this is done and returning to equation 4.4, one may then see that the metric is actually $$ \gamma(b'^2 + a' \bar{a}') + \gamma^{-1}\left(\operatorname{Im}\left(\frac{2A'}{A} - \delta a'\right)\right)^2. $$ This is already starting to look like the Gibbons-Hawking ansatz! I think the change of coordinate formulas are $a = x + i y$, $z = b$ and $A = e^{i \frac{\tau}{2}}$. It remains only to check that $\nabla \times \omega = \nabla(\gamma)$, which is a calculation that looks doable (though I don't feel like doing it right now).

Reading Hitchin becomes a bit easier after some time and experience, but I feel like one needs some guidance at the beginning. I was fortunate to have had Claude LeBrun as a mentor, for a while. Even so, twistor theory is hard, though it gets easier after some time.