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2012-10-15T18:52:32Z

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In [[mathematical 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 field]]s on spacetime to [[sheaf cohomology|cohomology]] of [[sheaf (mathematics)|sheaves]] 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]] and ''Z'' is the twistor space. More generally 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''1 and ''H''2 are parabolic subgroups.

The Penrose transform operates in two stages. First, one [[pullback|pulls back]] the sheaf cohomology groups ''H''''r''(''Z'','''F''') to the sheaf cohomology ''H''''r''(''Y'',η−1'''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'''3, which is also the Grassmannian Gr1('''C'''4) of lines in 4-dimensional complex space.
*''X'' = Gr2('''C'''4), the Grassmannian of 2-planes in 4-dimensional complex space. This is a compactification of complex Minkowski space.
*''Y'' is the flag manifold whose elements correspond to a line in a plane of '''C'''4.
*''G'' is the group SL4('''C''') and ''H''1 and ''H''2 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.

==Penrose–Ward transform==

The Penrose–Ward transform is a non-linear modification of the Penrose transform, introduced by {{harvtxt|Ward|1977}}, that (among other things) relates holomorphic vector bundles on 3-dimensional complex projective space '''CP'''3 to solutions of the [[self-dual Yang–Mills equations]] on '''S'''4.
{{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.

==References==

*{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Ward | first2=R. S. | title=Instantons and algebraic geometry | doi=10.1007/BF01626514 | publisher=Springer Berlin / Heidelberg | DUPLICATE DATA: doi=10.1007/BF01626514 | mr=0494098 | year=1977 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=55 | pages=117–124|bibcode = 1977CMaPh..55..117A }}
*{{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) | url=http://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|id=P/p120100|first=M.G.|last= Eastwood}}
*{{Citation | last=David | first=Liana | title=The Penrose transform and its applications|publisher=[[University of Edinburgh]]|year=2001|url=http://www.maths.ed.ac.uk/pg/thesis/david.pdf}}; 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 | pages=345–366|bibcode = 1967JMP.....8..345P }}
*{{Citation | last1=Penrose | first1=Roger | author1-link=Roger Penrose | title=Twistor quantisation and curved space-time | doi=10.1007/BF00668831 | publisher=Springer Netherlands | DUPLICATE DATA: doi=10.1007/BF00668831 | year=1968 | journal=International Journal of Theoretical Physics | issn=0020-7748 | volume=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 | 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. | 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 }}

{{Topics of twistor theory}}

[[Category:Integral geometry]]

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