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In [[theoretical physics]], the '''Penrose transform''', introduced by {{harvs|txt|authorlink=Roger Penrose|first=Roger |last=Penrose|year1=1967|year2=1968|year3=1969}}, is a complex analogue of the [[Radon transform]] that relates massless fields on spacetime, or more precisely the space of solutions to massless [[field equation]]s, to [[sheaf cohomology]] groups on [[complex projective space]].  The projective space in question is the [[twistor space]], a geometrical space naturally associated to the original spacetime, and the twistor transform is also geometrically natural in the sense of [[integral geometry]].  The Penrose transform is a major component of classical [[twistor theory]].

==Overview==

Abstractly, the Penrose transform operates on a double [[fibration]] of a space ''Y'', over two spaces ''X'' and ''Z''

:<math>Z\xleftarrow{\eta} Y \xrightarrow{\tau} X.</math>

In the classical Penrose transform, ''Y'' is the [[spin bundle]], ''X'' is a compactified and complexified form of [[Minkowski space]] (which as a [[complex manifold]] is <math>\mathbf{Gr}(2,4)</math>) and ''Z'' is the twistor space (which is <math>\mathbb{P}^3</math>).  More general examples come from double fibrations of the form

:<math>G/H_1\xleftarrow{\eta} G/(H_1\cap H_2) \xrightarrow{\tau} G/H_2</math>
where ''G'' is a complex semisimple [[Lie group]] and ''H''<sub>1</sub> and ''H''<sub>2</sub> are [[Borel subgroup|parabolic subgroups]].

The Penrose transform operates in two stages.  First, one [[pullback|pulls back]] the sheaf cohomology groups ''H''<sup>''r''</sup>(''Z'','''F''') to the sheaf cohomology ''H''<sup>''r''</sup>(''Y'',η<sup>&minus;1</sup>'''F''') on ''Y''; in many cases where the Penrose transform is of interest, this pullback turns out to be an isomorphism.  One then pushes the resulting cohomology classes down to ''X''; that is, one investigates the [[direct image]] of a cohomology class by means of the [[Leray spectral sequence]].  The resulting direct image is then interpreted in terms of differential equations.  In the case of the classical 
Penrose transform, the resulting differential equations are precisely the massless field equations for a given spin.

==Example==

The classical example is given as follows
*The "twistor space" ''Z'' is complex projective 3-space '''CP'''<sup>3</sup>, which is also the [[Grassmannian]] Gr<sub>1</sub>('''C'''<sup>4</sup>) of lines in 4-dimensional complex space.
*''X'' = Gr<sub>2</sub>('''C'''<sup>4</sup>), the Grassmannian of 2-planes in 4-dimensional complex space. This is a [[Compactification (mathematics)|compactification]] of complex Minkowski space.
*''Y'' is the [[flag manifold]] whose elements correspond to a line in a plane of '''C'''<sup>4</sup>.
*''G'' is the group  SL<sub>4</sub>('''C''') and ''H''<sub>1</sub> and ''H''<sub>2</sub> are the parabolic subgroups fixing a line or a plane containing this line.

The maps from ''Y'' to ''X'' and ''Z'' are the natural projections.

Using spinor index notation, the Penrose transform gives a bijection between solutions to the spin <math>\pm n/2</math> massless field equation
<math display = block>\partial_A\,^{A_1'}\phi_{A_1'A_2'\cdots A_n'} = 0</math>
and the first sheaf cohomology group <math>H^1(\mathbb{P}^1, \mathcal{O}(\pm n-2))</math>, where <math>\mathbb{P}^1</math> is the [[Riemann sphere]], <math>\mathcal{O}(k)</math> are the usual [[holomorphic line bundle]]s over projective space, and the [[sheaf (mathematics)|sheaves]] under consideration are the sheaves of [[section (fiber bundle)|sections]] of <math>\mathcal{O}(k)</math>.<ref name="dunajski">{{cite book |last1=Dunajski |first1=Maciej |title=Solitons, instantons, and twistors |date=2010 |publisher=Oxford University Press |location=Oxford |isbn=9780198570639 |pages=145–146}}</ref>

==Penrose–Ward transform<!--'Penrose–Ward transform' redirects here-->==

The '''Penrose–Ward transform'''<!--boldface per WP:R#PLA--> is a nonlinear modification of the Penrose transform, introduced by {{harvtxt|Ward|1977}}, that (among other things) relates [[holomorphic vector bundle]]s on 3-dimensional complex projective space '''CP'''<sup>3</sup> to solutions of the [[self-dual Yang–Mills equations]] on '''S'''<sup>4</sup>.
{{harvtxt|Atiyah|Ward|1977}} used this to describe instantons in terms of  algebraic vector bundles on complex projective 3-space and {{harvtxt|Atiyah|1979}} explained how this could be used to classify instantons on a 4-sphere.

==See also==
* [[Twistor correspondence]]

==References==
{{Reflist}}
{{refbegin|2|indent=y}}
*{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Ward | first2=R. S. | author2-link=Richard S. Ward | title=Instantons and algebraic geometry | doi=10.1007/BF01626514 | publisher=Springer Berlin / Heidelberg | mr=0494098 | year=1977 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=55 | issue=2 | pages=117–124|bibcode = 1977CMaPh..55..117A | url=http://projecteuclid.org/euclid.cmp/1103900980 }}
*{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | title=Geometry of Yang-Mills fields | publisher=Scuola Normale Superiore Pisa, Pisa | series=Lezioni Fermiane | isbn=978-88-7642-303-1 | mr=554924 | year=1979}}
*{{Citation | last1=Baston | first1=Robert J. | last2=Eastwood | first2=Michael G. | title=The Penrose transform | publisher=The Clarendon Press Oxford University Press | series=Oxford Mathematical Monographs | isbn=978-0-19-853565-2 | mr=1038279 | year=1989}}.
*{{Citation | last1=Eastwood | first1=Michael | editor1-last=Eastwood | editor1-first=Michael | editor2-last=Wolf | editor2-first=Joseph | editor3-last=Zierau. | editor3-first=Roger | title=The Penrose transform and analytic cohomology in representation theory (South Hadley, MA, 1992) | chapter-url=https://www.ams.org/bookstore?fn=20&arg1=conmseries&ikey=CONM-154 | publisher=Amer. Math. Soc. | location=Providence, R.I. | series=Contemp. Math. | isbn=978-0-8218-5176-0  | mr=1246377 | year=1993 | volume=154 | chapter=Introduction to Penrose transform | pages=71–75}}
*{{eom|title=Penrose transform|first=M.G.|last= Eastwood|oldid=12831}}
*{{Citation | last=David | first=Liana | title=The Penrose transform and its applications|publisher=[[University of Edinburgh]]|year=2001|url=https://www.era.lib.ed.ac.uk/bitstream/handle/1842/16970/David2001_redux.pdf?sequence=1&isAllowed=y}}; Doctor of Philosophy thesis.
*{{Citation | last1=Penrose | first1=Roger | author1-link=Roger Penrose | title=Twistor algebra | url=http://link.aip.org/link/JMAPAQ/v8/i2/p345/s1 | doi=10.1063/1.1705200 | mr=0216828 | year=1967 | journal=[[Journal of Mathematical Physics]] | issn=0022-2488 | volume=8 | issue=2 | pages=345–366 | bibcode=1967JMP.....8..345P | url-status=dead | archive-url=https://archive.today/20130112095407/http://link.aip.org/link/JMAPAQ/v8/i2/p345/s1 | archive-date=2013-01-12 }}
*{{Citation | last1=Penrose | first1=Roger | author1-link=Roger Penrose | title=Twistor quantisation and curved space-time | doi=10.1007/BF00668831 | publisher=Springer Netherlands |year=1968 | journal=International Journal of Theoretical Physics | issn=0020-7748 | volume=1 | issue=1 | pages=61–99|bibcode = 1968IJTP....1...61P }}
*{{Citation | last1=Penrose | first1=Roger | author1-link=Roger Penrose | title=Solutions of the Zero-Rest-Mass Equations | url=http://link.aip.org/link/JMAPAQ/v10/i1/p38/s1 | archive-url=https://archive.today/20130112125501/http://link.aip.org/link/JMAPAQ/v10/i1/p38/s1 | url-status=dead | archive-date=2013-01-12 | doi=10.1063/1.1664756 | year=1969 | journal=[[Journal of Mathematical Physics]] | issn=0022-2488 | volume=10 | issue=1 | pages=38–39 | bibcode=1969JMP....10...38P }}
*{{Citation | last1=Penrose | first1=Roger | author1-link=Roger Penrose | last2=Rindler | first2=Wolfgang | author2-link=Wolfgang Rindler | title=Spinors and space-time. Vol. 2 | publisher=[[Cambridge University Press]] | series=Cambridge Monographs on Mathematical Physics | isbn=978-0-521-25267-6 | mr=838301 | year=1986}}.
*{{Citation | last1=Ward | first1=R. S. | author1-link=Richard S. Ward | title=On self-dual gauge fields | doi=10.1016/0375-9601(77)90842-8 | mr=0443823 | year=1977 | journal=Physics Letters A | issn=0375-9601 | volume=61 | issue=2 | pages=81–82|bibcode = 1977PhLA...61...81W }}
{{refend}}

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