Penrose transform

$$ \newcommand{\CP}{\mathbb{CP}} \newcommand{\Gr}{\operatorname{Gr}} $$ A construction from complex integral geometry, its definition very much resembling that of the Radon transform in real integral geometry. It was introduced by R. Penrose in the context of twistor theory [a4] but many mathematicians have introduced transforms which may now be viewed in the same framework.

In its most general formulation, one starts with a correspondence between two spaces $Z$ and $X$. In many cases $Z$ will be a complex manifold but it can also be a CR-manifold or, indeed, a manifold with an involutive or formally integrable structure so that it makes sense to say that a smooth submanifold of $Z$ is complex. The important aspect of the correspondence is that $X$ parameterizes certain compact complex submanifolds of $Z$ (called cycles).

In the classical case, $Z = \CP_3$ and $X = \Gr_2(\C^4)$ with the obvious incidence relation. Thus, a point $x\in X$ gives a complex projective line $L_x$ in $Z$. This correspondence also has a real form, namely the fibration $\pi : \CP_3 \to S^4$. It can be defined by regarding $S^4$ as quaternionic projective $1$-space with $\pi$ taking a complex line to its quaternionic span, having chosen an identification $\C^4 \equiv \H^2$. See [a3] for the Penrose transform in this real setting. There are several examples from representation theory, the prototype being due to W. Schmid [a5] with $Z = G/T$ and $X = G/K$. Here, $G$ is a semi-simple Lie group with maximal torus $T$ and maximal compact subgroup $K$ assumed to have the same rank as $G$ (cf. Lie group, semi-simple). In complex geometry there is also the Andreotti–Norguet transform [a1], with $Z = \CP_n \setminus \CP_{n-p-1}$ and with as cycles the linearly embedded subspaces $\CP_p$.

In all of these transforms, one starts with a cohomology class $\omega$ on $Z$. If $Z$ is complex, then it is a Dolbeault class (cf. Differential form), if $Z$ is CR, then it is a $\overline{\partial}_b$-cohomology class (cf. de Rham cohomology), and so on. Often, the coefficients are twisted by a holomorphic or CR-bundle on $Z$. In its simplest form, the Penrose transform simply restricts $\omega$ to the cycles. Since each cycle is a compact complex manifold, the restricted cohomology is a finite-dimensional vector space. Supposing that the dimension of this vector space is constant as one varies the cycle, this gives a vector bundle on $X$ and the restriction of $\omega$ to each cycle gives a section of this bundle.

In the classical case, or its real form, suppose that one considers a Dolbeault cohomology class $\omega$ of degree one with coefficients in $H^{-2}$, where $H$ is the hyperplane bundle on $\CP_3$. Since $H^1(\CP_1, \mathcal{O}(-2)$ is one-dimensional, the resulting bundle on $\Gr_2(\C^4)$ or $S^4$ is a line bundle. The Penrose transform of $\omega$ is the section $x \mapsto \omega|_{L_x}$ in the complex case, or $x \mapsto \omega|_{\pi^{-1}(X)}$ in the real case. For the Andreotti–Norguet transform one takes as coefficients $\Omega^1$, the bundle of holomorphic one-forms. Then, in view of the canonical isomorphism $H^1(\CP_1,\Omega^1) = \C$, the transform gives rise to a function.

In any case, there are preferred local trivializations of the bundles involved and one can ask whether the resulting function is general. It will be holomorphic in the complex case and smooth in the real case, but there are further conditions. Unlike the analogous Radon transform, these conditions apply locally, so that, for example, there results an isomorphism

$$ H^1(\pi^{-1}(U), \mathcal{O}(-2)) \xrightarrow{\ \simeq\ } \{ \text{Smooth $\phi$ on $U$} : \Delta \phi = 0\} $$

for any open subset $U \subset S^4$, where $\Delta$ is the conformally invariant Laplacian on $S^4$ (cf. Laplace operator). Similarly, the Penrose transform interprets $H^1(\pi^{-1}(U), \mathcal{O}(-3))$ as solutions of the Dirac equation on $U$. The range of the Andreotti–Norguet transform consists of functions in the kernel of the Paneitz operator, a conformally invariant fourth-order operator. The range of Schmid's transform, when the coefficients are in a line bundle induced from an anti-dominant integral weight sufficiently far from the walls and cohomology is taken in degree equal to the complex dimension of the cycles, is the kernel of the Schmid operator.

In several cases, these transforms have appeared much earlier as integral transforms of holomorphic functions. For example, on this level, the Penrose description of harmonic functions is due to H. Bateman in 1904. A proper understanding, however, arises only when these holomorphic functions are viewed as Čech cocycles representing a cohomology class. See [a2] for further discussion.

There is a "non-linear" version of the Penrose transform, due to R.S. Ward [a6] and known as the Ward transform. In the classical case, it identifies certain holomorphic vector bundles on $\CP_3$ with solutions of the self-dual Yang–Mills equations (cf. Yang–Mills field) on $S^4$.

References

[a1]	A. Andreotti, F. Norguet, "La convexité holomorphe dans l'espace analytique des cycles d'une variété algébrique" Ann. Sci. Norm. Sup. Pisa , 21 (1967) pp. 31–82 Zbl 0176.04001
[a2]	M.G. Eastwood, "Introduction to Penrose transform" , The Penrose Transform and Analytic Cohomology in Representation Theory , Contemp. Math. , 154 , Amer. Math. Soc. (1993) pp. 71–75
[a3]	N.J. Hitchin, "Linear field equations on self-dual spaces" Proc. R. Soc. Lond. , A370 (1980) pp. 173–191
[a4]	R. Penrose, "On the twistor description of massless fields" , Complex manifold techniques in theoretical physics , Res. Notes Math. , 32 , Pitman (1979) pp. 55–91
[a5]	W. Schmid, "Homogeneous complex manifolds and representations of semisimple Lie groups" , Representation Theory and Harmonic Analysis on Semisimple Lie Groups , Math. Surveys Monogr. , 31 , Amer. Math. Soc. (1989) pp. 223–286 (PhD Univ. Calif. Berkeley, 1967)
[a6]	R.S. Ward, "On self-dual gauge fields" Phys. Lett. , A61 (1977) pp. 81–82