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edits and edit discussion on the entry conformal compactification is going on here
Thanks. It seems that’s a better place to engage Willie than here, but I may copy over any insights if they are warranted.
I added that the Klein quadric conformally compactifies complexified $\mathbb{R}^{3,1}$, with a sketchy reference as really I have to go offline now. (Lecturing in 8.5 hours and need to sleep first…)
I have added the brief remark that the example of complexified Minkowski spacetimes conformally compactifying to the Klein quadric is key in the twistor correspondence.
And thereby this discussion connects to the solitonic M5-dicussion here. Is that the reason why you are looking into conformal compactification at the moment? Still building that explicit String 2-bundle for Christian?
I added the equation, and the reference to Klein’s thesis.
Thanks! (we should say that your edit is at Klein quadric)
Whoops, thanks.
Added link to Klein quadric to published version of Klein’s thesis (is Springer hiding results from Google? I could only get it by searching the journal website itself, once I discovered where it was published) in German, and a link to a pdf of the English translation, not sure where that pdf is from, searching its title gives only that one result.
I added to conformal compactification the actual formula for the embedding $\mathbb{R}^{n,1}\hookrightarrow \mathbb{R}^{n+1,2}\setminus \{0\}$. In fact this is a diffeomorphism with the intersection of the real points of the Klein quadric with a particular hyperplane (the original Minkowski space times one of the null planes in the extra $\mathbb{R}^{1,1}$ factor – there’s another disjoint embedding using the other null plane). I need to calculate the correct normalisation factor that means this will land in $S^n\times S^1$ instead, but not now.
Well, as a first stab, I think with the normalising factor the map $\mathbb{R}^{n,1} \to S^n\times S^1 \subset \mathbb{R}^{n,1}\times\mathbb{R}^{1,1}$ looks like
$\frac{1}{\sqrt{q(x,t)^2 + 2(|x|^2+t^2)+1}}(2x,2t,1-q(x,t),-1-q(x,t))$in the notation of conformal compactification.
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