Twistor correspondence
In
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
- Gauge equivalence classesof anti-self dual Yang–Mills (ASDYM) connections on complexified Minkowski space withgauge group(the complex general linear group)
- 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
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[3] also known as the ADHM construction, hence giving a classification of instantons.
References
- ISBN 9780198570622.
- .
- .