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

There is a bijection between

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 also known as the ADHM construction, hence giving a classification of instantons.