Twistor correspondence

There is a bijection between

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

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

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 ${\displaystyle {\mathcal {PT}}}$ to ${\displaystyle \mathbb {P} ^{3}}$, 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 ${\displaystyle \mathbb {P} ^{3}}$ have been extensively studied in the field of algebraic geometry, and all relevant bundles can be generated by the monad construction^[3] also known as the ADHM construction, hence giving a classification of instantons.