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. https://resolver.caltech.edu/CaltechAUTHORS:20160506-072313941
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Abstract
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).
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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. | ||||||||||||
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Issue or Number: | 2 | ||||||||||||
DOI: | 10.1007/PL00005576 | ||||||||||||
Record Number: | CaltechAUTHORS:20160506-072313941 | ||||||||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20160506-072313941 | ||||||||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||
ID Code: | 66707 | ||||||||||||
Collection: | CaltechAUTHORS | ||||||||||||
Deposited By: | Ruth Sustaita | ||||||||||||
Deposited On: | 06 May 2016 20:29 | ||||||||||||
Last Modified: | 11 Nov 2021 00:01 |
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