Penrose transform

In theoretical physics, the Penrose transform, introduced by Roger Penrose (1967, 1968, 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 equations, 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

${\displaystyle Z{\xleftarrow {\eta }}Y{\xrightarrow {\tau }}X.}$

In the classical Penrose transform, Y is the

is ${\displaystyle \mathbf {Gr} (2,4)}$) and Z is the twistor space (which is ${\displaystyle \mathbb {P} ^{3}}$). More generally examples come from double fibrations of the form

${\displaystyle G/H_{1}{\xleftarrow {\eta }}G/(H_{1}\cap H_{2}){\xrightarrow {\tau }}G/H_{2}}$

where G is a complex semisimple Lie group and H₁ and H₂ are parabolic subgroups.

The Penrose transform operates in two stages. First, one

. 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 CP³, which is also the Grassmannian Gr₁(C⁴) of lines in 4-dimensional complex space.
• X = Gr₂(C⁴), 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 C⁴.
• G is the group SL₄(C) and H₁ and H₂ 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 ${\displaystyle \pm n/2}$ massless field equation

and the first sheaf cohomology group ${\displaystyle H^{1}(\mathbb {P} ^{1},{\mathcal {O}}(\pm n-2))}$, where ${\displaystyle \mathbb {P} ^{1}}$ is the Riemann sphere, ${\displaystyle {\mathcal {O}}(k)}$ are the usual of ${\displaystyle {\mathcal {O}}(k)}$.[1]

Penrose–Ward transform

The Penrose–Ward transform is a nonlinear modification of the Penrose transform, introduced by

on S⁴. Atiyah & Ward (1977) used this to describe instantons in terms of algebraic vector bundles on complex projective 3-space and Atiyah (1979) explained how this could be used to classify instantons on a 4-sphere.

See also

• Twistor correspondence

References

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  MR 1246377
• Eastwood, M.G. (2001) [1994], "Penrose transform", Encyclopedia of Mathematics, EMS Press
• David, Liana (2001), The Penrose transform and its applications (PDF), University of Edinburgh; Doctor of Philosophy thesis.
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