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Noncommutative Instantons and Twistor Transform

Kapustin, Anton and Kuznetsov, Alexander and Orlov, Dmitri (2001) Noncommutative Instantons and Twistor Transform. Communications in Mathematical Physics, 221 (2). pp. 385-432. ISSN 0010-3616. doi:10.1007/PL00005576.

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Recently N. Nekrasov and A. Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of ℝ^4. 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 ℙ^2, certain complexes of sheaves on a noncommutative ℙ^3, 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 ℙ^2 has a natural hyperkähler metric and is isomorphic as a hyperkähler manifold 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 of instantons on a more general noncommutative ℝ^4 than the one considered by Nekrasov and Schwarz (a q-deformed ℝ^4).

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Kapustin, Anton0000-0003-3903-5158
Orlov, Dmitri0000-0002-2230-457X
Additional Information:© 2001 Springer-Verlag. Received: 3 May 2000. Accepted: 3 April 2001. Dedicated to A.N. Tyurin on his 60th birthday. Supported by DOE grant DE-FG02-90ER4054442. Supported by NSF grant DMS97-29992 and RFFI grants 99-01-01144, 99-01-01204. Supported by NSF grant DMS97-29992 and RFFI grant 99-01-01144. We are grateful to A. Beilinson, V. Ginzburg, L. Katzarkov, N. Nekrasov, T. Pantev, and A.Yekutieli for useful discussions and to L. Le Bruyn for bringing to our attention Ref. [21].We also wish to thank the Institute for Advanced Study, Princeton, NJ, for a very stimulating atmosphere. Communicated by A. Connes.
Funding AgencyGrant Number
Department of Energy (DOE)DE-FG02-90ER4054442
Russian Foundation for Fundamental Investigations (RFFI)99-01-01144
Russian Foundation for Fundamental Investigations (RFFI)99-01-01204
Issue or Number:2
Record Number:CaltechAUTHORS:20160506-072313941
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:66707
Deposited By: Ruth Sustaita
Deposited On:06 May 2016 20:29
Last Modified:11 Nov 2021 00:01

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