Penrose transform

Remarks

A class in $H^1$ is represented by a function $f(\omega, \pi)$, holomorphic on the intersection of two open sets and homogeneous of degree $-2h-2$. The field is the contour integral over a loop in the line $L_x$: $$\phi_{A'_1 \dots A'_{2h}}(x) = \frac{1}{2\pi i} \oint \pi_{A'_1} \cdots \pi_{A'_{2h}} \, f(iX\pi, \pi) \, \pi_{B'} \, d\pi^{B'}.$$ The integrand is homogeneous of degree $0$ in $\pi$, so the integral is well defined on the projective line. Differentiating under the integral sign produces factors $\pi_{A'}$, and contracting two of them gives $\pi^{A'}\pi_{A'} = 0$; this is why the field equations hold. The hard part of the proof is that the map is bijective, which uses the vanishing of suitable cohomology on the fibres of the correspondence. Example: for $h = 1$ the fields are the solutions of the source-free Maxwell equations of one duality type, self-dual or anti-self-dual according to the convention. The theorem holds for more general pairs of open sets, with conditions on the topology of the fibres; the forward tube is the classical case.

Going deeper

For a student

The transform is the main example of the dictionary of twistor theory, in which differential equations on spacetime become cohomology on twistor space. The degree $-2h-2$ of homogeneity encodes the helicity, and fields of the other sign of helicity correspond to the cohomology of sheaves $\mathcal{O}(k)$ with other values of $k$. Related transforms describe self-dual Yang–Mills fields through holomorphic vector bundles on twistor space, by the Ward correspondence.