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 , 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   with gauge group   (the complex general linear group)
  2. Holomorphic rank n vector bundles   over projective twistor space   which are trivial on each degree one section of  .[1][2]

where   is the complex projective space of dimension  .

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   to  , 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   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.
  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.