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Sako, Akifumi

Recent developments in instantons in noncommutative \(\mathbb R^4\). (English) Zbl 1201.81103

Adv. Math. Phys. 2010, Article ID 270694, 28 p. (2010).
Summary: We review recent developments in noncommutative deformations of instantons in \(\mathbb R^4\). 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.

Cited in 5 Documents

MSC:

81T75 Noncommutative geometry methods in quantum field theory
81R60 Noncommutative geometry in quantum theory
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
35J08 Green’s functions for elliptic equations
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\textit{A. Sako}, Adv. Math. Phys. 2010, Article ID 270694, 28 p. (2010; Zbl 1201.81103)
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References:

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