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 There is a bijection between
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) 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}} . where P n {\displaystyle \mathbb {P} ^{n}} is the complex projective space of dimension n {\displaystyle n} .
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 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 also known as the ADHM construction, hence giving a classification of instantons.
References