Noncommutative instantons and twistor transform

Recently N. Nekrasov and A. Schwarz proposed a modification of the ADHM con struction of instantons which produces instantons on a noncommutative defor mation of R-4. In this paper we study the relation between their constructi on and algebraic bundles on noncommutative projective spaces. We exhibit on e-to-one correspondences between three classes of objects: framed bundles o n a noncommutative P-2, certain complexes of sheaves on a noncommutative P- 3, and the modified ADHM data. The modified ADHM construction itself is int erpreted in terms of a noncommutative version of the twistor transform. We also prove that the moduli space of framed bundles on the noncommutative P- 2 has a natural hyperkahler metric and is isomorphic as a hyperkahler manif old to the moduli space of framed torsion free sheaves on the commutative P -2. 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 o f instantons on a more general noncommutative R4 than the one considered by Nekrasov and Schwarz (a q-deformed R-4).