Are vortex numbers preserved?

Journal of Geometry and Physics

2008

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We construct instanton solutions on noncommutative Euclidean 4-space which are deformations of instanton solutions on commutative Euclidean 4-space. We show that the instanton numbers of these noncommutative instanton solutions coincide with the commutative solutions and conjecture that the instanton number in R 4 is preserved for general noncommutative deformations. We also study noncommutative deformation of instanton solutions on a T 4 with twisted boundary conditions.

Section: Introductionmentioning

confidence: 99%

Noncommutative deformation of instantons

Maeda

Journal of Geometry and Physics

2008

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“…(See, for example, [7,46,47].) Another method to construct noncommutative instantons as smooth deformations of commutative instantons was provided in [48,49,50]. The correspondence between the smooth deformation and the ADHM construction are discussed in [51].…”

Section: B Noncommutative U (1) Instanton In the Fock Spacementioning

confidence: 99%

Hermitian-Einstein metrics from noncommutative U(1) instantons

Hara

Journal of Mathematical Physics

Yang

2019

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We show that Hermitian-Einstein metrics can be locally constructed by a map from (anti-)self-dual two-forms on Euclidean R 4 to symmetric two-tensors introduced in [1]. This correspondence is valid not only for a commutative space but also for a noncommutative space. We choose U (1) instantons on a noncommutative C 2 as the self-dual two-form, from which we derive a family of Hermitian-Einstein metrics. We also discuss the condition when the metric becomes Kähler.

“…g (l) l . As we see in [12] and Appendix C, g † ⋆ g = I is equivalent to an infinite hierarchy of algebraic equations which we can solve recursively starting with the 0 term. The Laplacian is defined by [15].…”

Section: Notations Definitions and Known Factsmentioning

confidence: 99%

“…On the other hand, we have constructed previously new noncommutative deformations of solitons in gauge theories. These deformations smoothly connect a commutative soliton to a noncommutative soliton [1,12,13]. In the following, we call these smooth noncommutative deformed instantons SNCD instantons for short.…”

Section: Introductionmentioning

confidence: 99%

Noncommutative deformation of spinor zero mode and Atiyah-Drinfeld-Hitchin-Manin construction

Maeda

Journal of Mathematical Physics

2012

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A method to construct noncommutative instantons as deformations from commutative instantons was provided in [1]. Using this noncommutative deformed instanton, we investigate the spinor zero modes of the Dirac operator in a noncommutative instanton background on noncommutative R 4 , and we modify the index of the Dirac operator on the noncommutative space slightly and show that the number of the zero mode of the Dirac operator is preserved under the noncommutative deformation. We prove the existence of the Green's function associated with instantons on noncommutative R 4 , as a smooth deformation of the commutative case. The feature of the zero modes of the Dirac operator and the Green's function derives noncommutative ADHM equations which coincide with the ones introduced by Nekrasov and Schwarz. We show a one-to-one correspondence between the instantons on noncommutative R 4 and ADHM data. An example of a noncommutative instanton and a spinor zero mode are also given.