Noncommutative Instantons and Twistor Transform

TL;DR

This paper explores the relationship between noncommutative instantons, algebraic bundles, and the twistor transform, revealing deep geometric correspondences and extending instanton constructions to more general noncommutative spaces.

Contribution

It establishes one-to-one correspondences between noncommutative bundles, sheaves, and ADHM data, and interprets the modified ADHM construction via a noncommutative twistor transform.

Findings

Moduli space of noncommutative framed bundles has a natural hyperkahler metric.

This moduli space is isomorphic to the moduli space of sheaves on the commutative plane.

Complex structures on these moduli spaces are related by an SO(3) rotation.

Abstract

Recently N.Nekrasov and A.Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of the 4-dimensional real affine space. In this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces. We exhibit one-to-one correspondences between three classes of objects: framed bundles on a noncommutative projective plane, certain complexes of sheaves on a noncommutative 3-dimensional projective space, and the modified ADHM data. The modified ADHM construction itself is interpreted in terms of a noncommutative version of the twistor transform. We also prove that the moduli space of framed bundles on the noncommutative projective plane has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative projective plane. The natural complex structures on the two moduli spaces do not coincide but are related by an SO(3) rotation. Finally, we propose a construction of instantons on a more general noncommutative R^4 than the one considered by Nekrasov and Schwarz (a q-deformed R^4).