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Advances in Mathematical Physics
Volume 2010, Article ID 270694, 28 pages
Review Article

Recent Developments in Instantons in Noncommutative 𝟒

Department of General Education, Kushiro National College of Technology Otanoshike-Nishi 2-32-1, Kushiro 084-0916, Japan

Received 24 January 2010; Accepted 20 April 2010

Academic Editor: K. B. Sinha

Copyright © 2010 Akifumi Sako. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We review recent developments in noncommutative deformations of instantons in 4 . In the operator formalism, we study how to make noncommutative instantons by using the ADHM method, and we review the relation between topological charges and noncommutativity. In the ADHM methods, there exist instantons whose commutative limits are singular. We review smooth noncommutative deformations of instantons, spinor zero-modes, the Green's functions, and the ADHM constructions from commutative ones that have no singularities. It is found that the instanton charges of these noncommutative instanton solutions coincide with the instanton charges of commutative instantons before noncommutative deformation. These smooth deformations are the latest developments in noncommutative gauge theories, and we can extend the procedure to other types of solitons. As an example, vortex deformations are studied.