Noncommutative Instantons and Twistor Transform

Details Noncommutative Instantons and Twistor Transform Noncommutative Instantons and Twistor Transform

2000-02-23

arxiv.org/abs/hep-th/0002193v3

Commun.Math.Phys.221:385-432,2001

Physics

High Energy Physics

High Energy Physics - Theory

56 pages, AMS-Latex. Dedicated to A.N.Tyurin on his 60th birthday. v2: A section about instantons on a q-deformed R^4 and the

Scientific paper

10.1007/PL00005576

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).

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