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

在弦理论中，理解非交换几何的场论是非常重要的，因为它是d膜的一个有效理论。非交换几何上的场论的本质被认为出现在弯曲空间中。然而，与（平坦的）四维欧几里得空间上的瞬态解的分析不同，弯曲非交换空间上的理论研究还不是很清楚。其中一个原因是缺乏典型的例子。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，它作为一个正则化理论也很有趣，因为它具有类似于晶格情况的手性费米子结构。因此，找到更多的例子是很重要的。在我们最近的研究中，我们通过推广模糊球构造了非交换的CPn。用射影模构造了线束，并分析了其拓扑结构。用与模糊球情形类似的方法定义了微分学，并给出了非交换CPn的chen性质的非交换类比。在这种情况下，为了执行集成，有必要理解子域的含义。证明了某一类子球在交换极限下仍然是子球，在这种情况下我们可以求出第一个陈恩数。（发布号：help -th 0404130）

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.)