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 where $$\mathbb{P}^n$$ is the complex projective space of dimension $$n$$.
 * 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 $$\mathcal{PT} \rightarrow \mathbb{P}^1$$.

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.