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Research on the Field Theory in the Noncommutative Space obtained through the Deformation Quantization and its Application

变形量子化非交换空间场论研究及其应用

基本信息

批准号：
13640256
负责人：
WATAMURA Satoshi
金额：
$ 2.11万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640256/
关键词：
Noncommutative geometry Quantum Geometry Fuzzy Sphere Fuzzy CPn Projective Module Noncommutative Monpole Matrix Theory 非可換射影空間非可換インスタント Nahm duality ADHM構成非可換インスタントン

项目摘要

In the string theory it becomes very important to understand the field theory on the noncommutative geometry, since it appears as an effective theory of the D-brane. The essence of the field theory over the noncommutative geometry is considered to appear in the curved space. However, unlike the analysis of the instanton solution over (flat) four-dimensional Euclidian space, the study of the theory on the curved noncommutative space is not so well understood. One origin of this is the lack of typical examples.Our previous research about the fuzzy sphere gives many interesting aspects from various points of view : It appears as a D-brane configuration in string theory. However, it is also interesting as a regularized theory since it has a chiral fermion structure similar to the lattice case. It is therefore important to find more examples.In our recent investigation, we have constructed noncommutative CPn by generalizing the fuzzy sphere. The line bundle was constructed in terms of projective modules and we analyzed its topological structure. The differential calculus is defined in a similar way as in the fuzzy sphere case and the noncommutative analogue of the Chern character of the noncommutative CPn is given. In this context it is necessary to understand the meaning of the subsphere in order to perform the integration. It turns out that a certain class of subsphere remains as a subsphere in the commutative limit, and in this case we can evaluate the first Chern number. (Published as hep-th 0404130.)

在弦理论中，理解非对易几何上的场论变得非常重要，因为它是D-膜的有效理论。非对易几何上场论的本质被认为是出现在弯曲空间中。然而，不同于（平坦）四维欧氏空间上的瞬子解的分析，弯曲非对易空间上的理论研究还不是很好理解。我们以前对模糊球的研究从不同的角度给出了许多有趣的方面：它在弦理论中表现为D-膜构型。然而，它也是一个有趣的正则化理论，因为它有一个手征费米子结构类似于晶格的情况。在我们最近的研究中，我们通过推广模糊球构造了非交换CPn。利用投射模构造了线丛，并分析了它的拓扑结构。微分定义在一个类似的方式，在模糊球的情况下和非交换模拟的陈特征的非交换CPn。在这种情况下，为了执行集成，有必要理解子领域的含义。结果表明，某一类子球面在交换极限中保持为子球面，在这种情况下，我们可以计算第一个陈数。（出版编号hep-th 0404130）