Communications in Mathematical Physics

, Volume 221, Issue 2, pp 385–432 | Cite as

Noncommutative Instantons and Twistor Transform

  • Anton Kapustin
  • Alexander Kuznetsov
  • Dmitri Orlov


Recently N. Nekrasov and A. Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of ℝ4. 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 ℙ2, certain complexes of sheaves on a noncommutative ℙ3, 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 ℙ2 has a natural hyperkähler metric and is isomorphic as a hyperkähler manifold to the moduli space of framed torsion free sheaves on the commutative ℙ2. 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 ℝ4 than the one considered by Nekrasov and Schwarz (a q-deformed ℝ4).


Manifold Modulus Space Projective Space Natural Complex Free Sheave 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Anton Kapustin
    • 1
  • Alexander Kuznetsov
    • 2
  • Dmitri Orlov
    • 3
  1. 1.School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA.¶E-mail: kapustin@ias.eduUS
  2. 2.Institute for Problems of Information Transmission, Russian Academy of Sciences, 19 Bolshoi Karetnyi, Moscow 101447, Russia. E-mail:; akuznet@ias.eduRU
  3. 3.Algebra Section, Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkin str., GSP-1, Moscow 117966, Russia. E-mail: orlov@mi.ras.ruRU

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