Let us consider the Bateman or Whittaker's pioneering examples of a Penrose transform. Starting from a holomorphic function on an open subset of twistor space, they constructed a solution to the Laplace equation (in one case in dimension $4$ and in the other case in dimension $3$).

Fine, except it is not clear what their holomorphic function represents. Does it represent an element of $H^0(Z,O(k))$ for some $k$, or for instance $H^1(Z, O(k))$ for some $k$? Of course by now, these issues have been sorted out. But to be honest, I don't understand them a 100%.

So my question is this: if I were say presented with such an example of a Penrose transform, how do I figure out:

1) the degree of the sheaf cohomology group ($H^0$, $H^1$,...)
2) the degree $k$ of the line bundle
3) what it corresponds to on the "spacetime" side?

Can you perhaps give me the general idea please?


1 Answer 1


I found the explanation in Huggett and Tod's book, "Introduction to Twistor theory", to be very clear and down to earth (it was recommended to me by D. Calderbank, and I thank him for this reference). Using homogeneous coordinates on (projective) twistor space, a la Penrose, and using the integral formula, one can then answer all my questions. For instance, the degree of the bundle is essentially the degree of homogeneity one needs for the expression to be well defined on the twistor lines, and so on. I highly recommend this book as an introduction to the Penrose transform.

Somehow, when I read first about the Penrose transform, it was in the very well written paper by Eastwood, Penrose and Wells. However, it was a little high tech for me at the time, and it hid somehow how someone may have "discovered" the Penrose transform, so to speak, similar to the pioneering work of Bateman and Whittaker, and later on, independently, by Penrose (the integral formula).


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