I am trying to read and understand this paper by Monteiro, Stark-Muchao, and Wikeley about self dual yang mills and self dual gravity.

In the introduction to this paper, they review a way to formulate both theories using a single scalar degree of freedom. They give an explicit derivation for self dual yang mills but not for self dual gravity. I am trying to find an elementary derivation for self dual gravity along the same lines. The paper they cite does not provide a proof in any way I understand.

For self dual yang mills, $A_\mu$ is a $\mathfrak{g}$-valued 1-form and the field strength tensor is $$ F_{\mu \nu} = \partial_\mu A_\nu + \partial_\nu A_\mu + [A_\mu, A_\nu] $$ and the self duality equation is $$ F_{\mu \nu} = \frac{i}{2} \epsilon_{\mu \nu \alpha \beta} F^{\alpha \beta}. $$

If one defines a set of coordinates $$ u = \frac{t - z}{\sqrt{2}}, \quad v = \frac{t + z}{\sqrt{2}}, \quad w = \frac{x + i y}{\sqrt{2}}, \quad \bar{w} = \frac{x - i y}{\sqrt{2}} $$ the metric and $\Box$ become $$ ds^2 = 2 (- du dv + dw d \bar{w}), \hspace{2 cm} \Box = 2( -\partial_u \partial_v + \partial_w \partial_{\bar{w}} ). $$ The self duality equation then reduces to

$$F_{uw} = 0, \qquad F_{u v } = F_{w \bar{w}}, \qquad F_{v \bar{w} } = 0. $$

We can pick a gauge where we can set $A_u = 0$, then the first two of these equations give us $$ F_{u w} = 0 \implies \partial_u A_{w} = 0 \implies A_{w} = 0 $$ $$ F_{uv} = F_{w \bar{w}} \implies \partial_u A_v = \partial_{w} A_{\bar{w}}. $$ The second equation is solved by \begin{equation} A_v = \partial_{w} \Phi, \hspace{1 cm} A_{\bar{w}} = \partial_u \Phi. \end{equation} The only remaining equation is then \begin{equation} 0 = F_{v \bar{w}} = \Box \Phi - [\partial_u \Phi, \partial_w \Phi ]. \end{equation}

This equation gives the equation of motion for self dual yang mills in terms of a $\mathfrak{g}$-valued scalar $\Phi$.

The equation for self-dual gravity is $$ R_{\mu \nu \rho \sigma} = \frac{i}{2} \epsilon_{\mu \nu \alpha \beta} R^{\alpha \beta}_{\quad \rho \sigma}. $$

According to the paper, using "steps analogous to" those above for self dual yang mills show that one can always write a metric in the form $$ ds^2 = -2 du dv + 2 dw d \bar{w} + \partial_w^2 \phi dv^2 + \partial_u^2 \phi d \bar{w}^2 + 2 \partial_u \partial_v \phi $$

where $\phi$ is a single scalar degree of freedom with equation of motion

$$ 0 = \Box \phi - [ \partial_u \phi, \partial_w \phi ]_{ P.B.} $$

where we have defined the Poisson bracket

$$ [ f, g ]_{P.B.} = \frac{\partial f}{\partial u} \frac{\partial g}{\partial w} - \frac{\partial g}{\partial u} \frac{\partial f}{\partial w} . $$

My question is for someone to essentially point me to a place where I can find a derivation of these equations (that one can always write the metric in the above form, and that the equation of motion for $\phi$ is given above) from first principles, where by "first principles" I mean the self duality equation $R_{\mu \nu \rho \sigma} = \frac{i}{2} \epsilon_{\mu \nu \alpha \beta} R^{\alpha \beta}_{\quad \rho \sigma}$.

For some reason I am having a lot of difficulty finding a source which does this and assumes no background knowledge in areas of mathematics I do not understand. I would also accept an answer which gives a "roadmap" of (digestible) sources necessary to understand the derivation of these equations where no step is left out along the way.

I am trying to read and understand this paper by Monteiro, Stark-Muchao, and Wikeley about self-dual yang-mills and self-dual gravity.

In the introduction to this paper, they review a way to formulate both theories using a single scalar degree of freedom. They give an explicit derivation for self dual yang mills but not for self dual gravity. I am trying to find an elementary derivation for self dual gravity along the same lines. The paper they cite does not provide a proof in any way I understand.

For self-dual yang-mills, $A_\mu$ is a $\mathfrak{g}$-valued 1-form and the field strength tensor is $$ F_{\mu \nu} = \partial_\mu A_\nu + \partial_\nu A_\mu + [A_\mu, A_\nu] $$ and the self duality equation is $$ F_{\mu \nu} = \frac{i}{2} \epsilon_{\mu \nu \alpha \beta} F^{\alpha \beta}. $$

If one defines a set of coordinates $$ u = \frac{t - z}{\sqrt{2}}, \quad v = \frac{t + z}{\sqrt{2}}, \quad w = \frac{x + i y}{\sqrt{2}}, \quad \bar{w} = \frac{x - i y}{\sqrt{2}} $$ the metric and $\Box$ become $$ ds^2 = 2 (- du dv + dw d \bar{w}), \hspace{2 cm} \Box = 2( -\partial_u \partial_v + \partial_w \partial_{\bar{w}} ). $$ The self-duality equation then reduces to

$$F_{uw} = 0, \qquad F_{u v } = F_{w \bar{w}}, \qquad F_{v \bar{w} } = 0. $$

We can pick a gauge where we can set $A_u = 0$, then the first two of these equations give us $$ F_{u w} = 0 \implies \partial_u A_{w} = 0 \implies A_{w} = 0 $$ $$ F_{uv} = F_{w \bar{w}} \implies \partial_u A_v = \partial_{w} A_{\bar{w}}. $$ The second equation is solved by \begin{equation} A_v = \partial_{w} \Phi, \hspace{1 cm} A_{\bar{w}} = \partial_u \Phi. \end{equation} The only remaining equation is then \begin{equation} 0 = F_{v \bar{w}} = \Box \Phi - [\partial_u \Phi, \partial_w \Phi ]. \end{equation}

This equation gives the equation of motion for self-dual yang-mills in terms of a $\mathfrak{g}$-valued scalar $\Phi$.

The equation for self-dual gravity is $$ R_{\mu \nu \rho \sigma} = \frac{i}{2} \epsilon_{\mu \nu \alpha \beta} R^{\alpha \beta}_{\quad \rho \sigma}. $$

According to the paper, using "steps analogous to" those above for self-dual yang mills show that one can always write a metric in the form $$ ds^2 = -2 du dv + 2 dw d \bar{w} + \partial_w^2 \phi dv^2 + \partial_u^2 \phi d \bar{w}^2 + 2 \partial_u \partial_v \phi $$

where $\phi$ is a single scalar degree of freedom with equation of motion

$$ 0 = \Box \phi - [ \partial_u \phi, \partial_w \phi ]_{ P.B.} $$

where we have defined the Poisson bracket

$$ [ f, g ]_{P.B.} = \frac{\partial f}{\partial u} \frac{\partial g}{\partial w} - \frac{\partial g}{\partial u} \frac{\partial f}{\partial w} . $$

My question is for someone to essentially point me to a place where I can find a derivation of these equations (that one can always write the metric in the above form, and that the equation of motion for $\phi$ is given above) from first principles, where by "first principles" I mean the self duality equation $R_{\mu \nu \rho \sigma} = \frac{i}{2} \epsilon_{\mu \nu \alpha \beta} R^{\alpha \beta}_{\quad \rho \sigma}$.

For some reason I am having a lot of difficulty finding a source which does this and assumes no background knowledge in areas of mathematics I do not understand. I would also accept an answer which gives a "roadmap" of (digestible) sources necessary to understand the derivation of these equations where no step is left out along the way.