Twistor correspondence


In mathematical physics, the twistor 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

There is a bijection between
  1. Gauge equivalence classes of anti-self dual Yang–Mills connections on complexified Minkowski space with gauge group
  2. Holomorphic rank n vector bundles over projective twistor space which are trivial on each degree one section of.
where is the complex projective space of dimension.

Applications

ADHM construction

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 also known as the ADHM construction, hence giving a classification of instantons.