Twistor correspondence

In mathematical physics, the twistor correspondence (also known as Penrose–Ward correspondence) is a bijection between instantons on complexified Minkowski space and holomorphic vector bundles on twistor space, which as a complex manifold is  P 3 {\displaystyle \mathbb {P} ^{3}} , or complex projective 3-space. Twistor space was introduced by Roger Penrose, while Richard Ward formulated the correspondence between instantons and vector bundles on twistor space.

## Statement

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There is a bijection between

1. Gauge equivalence classes of anti-self dual Yang–Mills (ASDYM) connections on complexified Minkowski space  M C ≅ C 4 {\displaystyle M_{\mathbb {C} }\cong \mathbb {C} ^{4}} with gauge group G L ( n , C ) {\displaystyle \mathrm {GL} (n,\mathbb {C} )} (the complex general linear group)
2. Holomorphic rank n vector bundles E {\displaystyle E} over projective twistor space P T ≅ P 3 − P 1 {\displaystyle {\mathcal {PT}}\cong \mathbb {P} ^{3}-\mathbb {P} ^{1}} which are trivial on each degree one section of  P T -> P 1 {\displaystyle {\mathcal {PT}}\rightarrow \mathbb {P} ^{1}} .[1][2]

where  P n {\displaystyle \mathbb {P} ^{n}} is the complex projective space of dimension  n {\displaystyle n} .

## Applications

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### ADHM construction

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On the anti-self dual Yang–Mills side, the solutions, known as instantons, extend to solutions on compactified Euclidean 4-space. On the twistor side, the vector bundles extend from  P T {\displaystyle {\mathcal {PT}}} to  P 3 {\displaystyle \mathbb {P} ^{3}} , and the reality condition on the ASDYM side corresponds to a reality structure on the algebraic bundles on the twistor side. Holomorphic vector bundles over  P 3 {\displaystyle \mathbb {P} ^{3}} have been extensively studied in the field of algebraic geometry, and all relevant bundles can be generated by the monad construction[3] also known as the ADHM construction, hence giving a classification of instantons.

## References

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1. ^ Dunajski, Maciej (2010). Solitons, instantons, and twistors. Oxford: Oxford University Press. ISBN 9780198570622.
2. ^ Ward, R.S. (April 1977). "On self-dual gauge fields". Physics Letters A. 61 (2): 81–82\. doi:10.1016/0375-9601(77)90842-8.81-82&rft.date=1977-04&rft_id=info:doi/10.1016/0375-9601(77)90842-8&rft.aulast=Ward&rft.aufirst=R.S.&rfr_id=info:sid/en.wikipedia.org:Twistor+correspondence" class="Z3988">
3. ^ Atiyah, M.F.; Hitchin, N.J.; Drinfeld, V.G.; Manin, Yu.I. (March 1978). "Construction of instantons". Physics Letters A. 65 (3): 185–187\. doi:10.1016/0375-9601(78)90141-X.185-187&rft.date=1978-03&rft_id=info:doi/10.1016/0375-9601(78)90141-X&rft.aulast=Atiyah&rft.aufirst=M.F.&rft.au=Hitchin,+N.J.&rft.au=Drinfeld,+V.G.&rft.au=Manin,+Yu.I.&rfr_id=info:sid/en.wikipedia.org:Twistor+correspondence" class="Z3988">

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Topics of twistor theory

Objectives|

| Principles|

* Background independence

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Final objective|

* Quantum gravity
* Theory of everything

Mathematical concepts|

| Twistors|

* Penrose transform
* Twistor space

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Physical concepts|

* Twistor string theory
* Twistor correspondence
* Twistor theory

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