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 $\mathbb {P} ^{3}$, 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

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

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

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 ${\mathcal {PT}}$ to $\mathbb {P} ^{3}$, 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 $\mathbb {P} ^{3}$ 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.