In my mathematical meanderings, I've come across a few references to conformal transformations being representable with spinors. Now generally I think of a conformal transformation in a Riemanninan space with metric as being of the from:
$$g_{\mu\nu}\longrightarrow\lambda^{2}g_{\mu\nu}$$
Where $\lambda$ may be a function of the coordinates. In general I understand you can represent a conformal transformation as a composition of this dilation (our $\lambda)$ with a rotation (lets call that $R$) which in component form looks like:
$$\tilde{g}_{\alpha\beta}=\lambda R_{\alpha}^{\mu}g_{\mu\nu}R_{\beta}^{\nu}\lambda=\lambda R^{T}gR\lambda$$
Where the latter term is written in matrix form. So how do we get from here to a spinor representation of our transformation??? Note I'm just looking at conformal transformations of $R^{3,1}$ (with a point at infinity to compactify it).
I'm thinking something like:
$$\tilde{g}_{\mu\nu}=\bar{s}g_{\mu\nu}s$$
Where $s$ is the spinor and the bar denotes it's complex conjugate but I'm not sure that it actually has the required properties.