This article appeared recently on the arxiv about the twistor $\mathbf{P}^1$ and its role in number theory, algebraic geometry, and mathematical physics

- Woit, Peter.
*Notes on the Twistor $\mathbf{P}^1$*(arXiv:2202.02657)

Isn’t it the standard expression for the Grassmannians? But there is much room to expand on this. Feel free to remove that line and start afresh.

]]>Something is wrong in the part of the entry. The entry incorrectly says

The original *twistor correspondence* (Penrose 67) is the correspondence

The right-hand side is plainly wrong entrywise: all the quotients should be quotients of SL-s by the appropriate parabolics (block-triangular), which are therefore bigger than the diagonal block SL-s written. This way even the dimensions do not fit.

]]>Added also two references on *palatial* twistor theory, a new hope of Penrose, related to quantization.

Added also two references on *palatial* twistor theory, a new hope of Penrose, related to quantization.

Added monographs

- L. J. Mason, N. M. J. Woodhouse,
*Integrability, self-duality and twistor theory*, Oxford Univ. Press 1996 - R. S. Ward, R. O. Wells, Jr.
*Twistor geometry and field theory*, Cambridge Univ. Press 1990

Corrected a subscript.

]]>Sorry for the silly typos, fixed now. Thanks for catching them. I’ll try to catch the next train home to get some sleep.

]]>To discuss twistor space for Minkowski spacetime, it is useful to work more generally with $d$-dimensional Minkowski spacetime for $d \in \{3,4,6,8\}$.

Do you mean 10 rather than 8?

Then

…vectors in Minkowski spacetime in $d = \{2,3,4,10\}$…

why those dimensions?

For $d = 10$ there is no elegant statement like this, due to the non-associativity of the octonions

There’s plenty of discussion about why there can’t be an $\mathbb{O} P^3$ here. Enough associativity for $\mathbb{O} P^2$, but not projective space. Does the lack of octonion spinors tell us anything, or is it just an inconvenience?

]]>added an actual section *twistor space* with discussion of how $\mathbb{K}P^3$ encodes light-like geodesics in Minkowski spacetime. Did this in the generality that $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$, hence for Minkowski spacetime of total dimensions 3,4, and 6, in order to amplify the algebraic pattern.

added a section *twistors for 4d Minkowski spacetime*
with basics on the actual original definition and motivation for twistors.

(from looking around I gather this is now the only discussion on the web that comes out right away with admitting what a twistor actually *is*, conceptually :)

started something at *twistor space*