Calculation of the Pontrjagin class for U(1) instantons on non-commutative Bbb R<sup>4</sup>

Ishikawa, Takeshi; Kuroki, Shinichiro; Sako, Akifumi

doi:10.1088/1126-6708/2002/08/028

Cited by 17 publications

(30 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Noncommutative instantons are labeled by topological charge called instanton number. The topological number of the noncommutative instanton is studied in [46,54,55,56,57]. It is shown that the topological number coincides with the dimension of a vector space appearing in the ADHM construction.…”

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

confidence: 99%

Hermitian-Einstein metrics from noncommutative U(1) instantons

Hara

Sako

Yang

2019

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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

confidence: 99%

Hermitian-Einstein metrics from noncommutative U(1) instantons

Hara

Sako

Yang

2019

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Geometrical origin of instanton number of NC instantons is also discussed in e.g. 45,72,73,86,120,145,150,159,161 . For comprehensive discussion on ADHM construction, see e.g.…”

Section: Comments On Instanton Moduli Spacesmentioning

confidence: 99%

Noncommutative Solitons and Integrable Systems

Hamanaka

2005

Noncommutative Geometry and Physics

View full text Add to dashboard Cite

We review recent developments of soliton theories and integrable systems on noncommutative spaces. The former part is a brief review of noncommutative gauge theories focusing on ADHM construction of noncommutative instantons. The latter part is a report on recent results of existence of infinite conserved densities and exact multi-soliton solutions for noncommutative Gelfand-Dickey hierarchies. Some examples of noncommutative Ward's conjecture are also presented. Finally, we discuss future directions on noncommutative Sato's theories and twistor theories.

show abstract

“…The method is to put a cut-off only for the initial and final states, to define the trace operation for finite matrices, i.e. the intermediate states are not restricted by the cut-off, (see [3,4] for details). Using such methods we can estimate the effect of infinite dimension, like the shift operator, by finite size computation.…”

Section: Moyal Plane Case In θ → ∞mentioning

confidence: 99%

“…For example, the Euler number of a noncommutative torus is independent of the noncommutative parameter θ and it is defined as a topological invariant by the difference of K 0 and K 1 . Another example is the possibility to define the instanton number (the integral of the first Pontrjagin class) as an integer for Moyal space [3,4], and this fact implies that the instanton number has some kind of "topological" nature even for the case that the base manifold is noncommutative space. (Here, we call Moyal space noncommutative Euclidian space whose commutation relations of the coordinates are given by [x µ , x ν ] = iθ µν , where θ µν is an anti-symmetric constant matrix.)…”

Section: Introductionmentioning

confidence: 99%

Noncommutative Cohomological Field Theories and Topological Aspects of Matrix Models

Sako

2005

Noncommutative Geometry and Physics

Self Cite

View full text Add to dashboard Cite

We study topological aspects of matrix models and noncommutative cohomological field theories (N.C.CohFT). N.C.CohFT have symmetry under an arbitrary infinitesimal deformation with noncommutative parameter θ. This fact implies that N.C.CohFT are topologically less sensitive than K-theory, but the classification of manifolds by N.C.CohFT opens the possibility to get a new view point for global characterization of noncommutative manifolds. To investigate the properties of N.C.CohFT, we construct some models whose fixed point loci are given by sets of projection operators. In particular, the partition function on the Moyal plane is calculated by using a matrix model. The moduli space of the matrix model is a union of Grassman manifolds. The partition function of the matrix model is calculated using the Euler number of the Grassman manifold. Identifying the N.C.CohFT with the matrix model, we obtain the partition function of the N.C.CohFT. To check the independence of the noncommutative parameters, we also study the moduli space in the large θ limit and for finite θ, for the case of the Moyal plane. If the partition function of N.C.CohFT is topological in the sense of noncommutative geometry, then this should reveal some relation with K-theory. Therefore we investigate certain models of CohFT and N.C.CohFT from the point of view of K-theory. Our observations give us an analogy between CohFT and N.C.CohFT in connection with K-theory. Furthermore, we verify for the Moyal plane and noncommutative torus cases that our partition functions are invariant under those deformations which do not change the K-theory. Finally, we discuss the noncommutative cohomological Yang-Mills theory.

show abstract

Calculation of the Pontrjagin class for U(1) instantons on non-commutative Bbb R⁴

Cited by 17 publications

References 19 publications

Hermitian-Einstein metrics from noncommutative U(1) instantons

Hermitian-Einstein metrics from noncommutative U(1) instantons

Noncommutative Solitons and Integrable Systems

Noncommutative Cohomological Field Theories and Topological Aspects of Matrix Models

Contact Info

Product

Resources

About

Calculation of the Pontrjagin class for U(1) instantons on non-commutative Bbb R4

Cited by 17 publications

References 19 publications

Hermitian-Einstein metrics from noncommutative U(1) instantons

Hermitian-Einstein metrics from noncommutative U(1) instantons

Noncommutative Solitons and Integrable Systems

Noncommutative Cohomological Field Theories and Topological Aspects of Matrix Models

Contact Info

Product

Resources

About

Calculation of the Pontrjagin class for U(1) instantons on non-commutative Bbb R⁴