View source for Penrose transform

A construction from complex [[Integral geometry|integral geometry]], its definition very much resembling that of the [[Radon transform|Radon transform]] in real integral geometry. It was introduced by R. Penrose in the context of twistor theory [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p1201001.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p1201002.png" />. In many cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p1201003.png" /> will be a [[Complex manifold|complex manifold]] but it can also be a [[CR-manifold|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p1201004.png" /> is complex. The important aspect of the correspondence is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p1201005.png" /> parameterizes certain compact complex submanifolds of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p1201006.png" /> (called cycles).

In the classical case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p1201007.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p1201008.png" /> with the obvious incidence relation. Thus, a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p1201009.png" /> gives a complex projective line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010010.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010011.png" />. This correspondence also has a real form, namely the [[Fibration|fibration]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010012.png" />. It can be defined by regarding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010013.png" /> as quaternionic projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010014.png" />-space with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010015.png" /> taking a complex line to its quaternionic span, having chosen an identification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010016.png" />. See [[#References|[a3]]] for the Penrose transform in this real setting. There are several examples from representation theory, the prototype being due to W. Schmid [[#References|[a5]]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010018.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010019.png" /> is a semi-simple Lie group with maximal torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010020.png" /> and maximal compact subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010021.png" /> assumed to have the same rank as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010022.png" /> (cf. [[Lie group, semi-simple|Lie group, semi-simple]]). In complex geometry there is also the Andreotti–Norguet transform [[#References|[a1]]], with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010023.png" /> and with as cycles the linearly embedded subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010024.png" />.

In all of these transforms, one starts with a [[Cohomology|cohomology]] class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010025.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010027.png" /> is complex, then it is a Dolbeault class (cf. [[Differential form|Differential form]]), if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010028.png" /> is CR, then it is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010030.png" />-cohomology class (cf. [[De Rham cohomology|de Rham cohomology]]), and so on. Often, the coefficients are twisted by a holomorphic or CR-bundle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010031.png" />. In its simplest form, the Penrose transform simply restricts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010032.png" /> to the cycles. Since each cycle is a compact complex manifold, the restricted cohomology is a finite-dimensional [[Vector space|vector space]]. Supposing that the dimension of this vector space is constant as one varies the cycle, this gives a [[Vector bundle|vector bundle]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010033.png" /> and the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010034.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010035.png" /> of degree one with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010037.png" /> is the hyperplane bundle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010038.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010039.png" /> is one-dimensional, the resulting bundle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010040.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010041.png" /> is a line bundle. The Penrose transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010042.png" /> is the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010043.png" /> in the complex case, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010044.png" /> in the real case. For the Andreotti–Norguet transform one takes as coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010045.png" />, the bundle of holomorphic one-forms. Then, in view of the canonical isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010046.png" />, 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010047.png" /></td> </tr></table>

for any open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010049.png" /> is the conformally invariant Laplacian on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010050.png" /> (cf. [[Laplace operator|Laplace operator]]). Similarly, the Penrose transform interprets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010051.png" /> as solutions of the [[Dirac equation|Dirac equation]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120100/p12010052.png" />. 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 [[#References|[a2]]] for further discussion.

There is a  "non-linear"  version of the Penrose transform, due to R.S. Ward [[#References|[a6]]] and known as the Ward transform. In the classical case, it identifies certain holomorphic vector bundles on $\CC P_3$ with solutions of the self-dual Yang–Mills equations (cf. [[Yang–Mills field|Yang–Mills field]]) on $S^4$.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  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}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.J. Hitchin,   "Linear field equations on self-dual spaces"  ''Proc. R. Soc. Lond.'' , '''A370'''  (1980)  pp. 173–191</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Penrose,   "On the twistor description of massless fields" , ''Complex manifold techniques in theoretical physics'' , ''Res. Notes Math.'' , '''32''' , Pitman  (1979)  pp. 55–91</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R.S. Ward,   "On self-dual gauge fields"  ''Phys. Lett.'' , '''A61'''  (1977)  pp. 81–82</TD></TR>
</table>