Abstract
We review recent developments in noncommutative deformations of instantons in ℝ4. 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.
Highlights
Instantons in commutative space are one of the most important objects for nonperturbative analysis
We have reviewed developments for the last dozen years in NC instantons in R4
A lot of kinds of NC instanton solutions have been made by the ADHM method
Summary
Instantons in commutative space are one of the most important objects for nonperturbative analysis. Using these ADHM data, we can construct instanton 6–17. The NC ADHM method clarifies some important features, for example, topological charge, index theorems, Green’s functions, and so on. We study the NC instanton charge, an index theorem, and the correspondence relation with the ADHM construction for the smooth NC deformations of instantons 19.
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