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