The construction of the field theory on the noncommutative spaces and its application

本研究的目的是建立非对易空间上的场论并分析其物理性质。在过去的4年里，在本补助金的支持下，我们一直在特别调查非交换领域的场论作为一个具体的例子，非交换弯曲空间。通过量子化2-球面上的代数定义了非交换球面。作为第一步，在这个方向上，我们已经建立了非交换微分代数根据的方法由Connes（文件1），并作为其应用，我们制定了标量场的非交换球（文件2）。我们的主要兴趣是在非对易球上定义规范理论。然而，为了这个目的，我们不得不进一步分析微分的详细结构。文3成功地建立了U（1）规范理论，并分析了它的交换极限。特别是在规范场论中，我们得到了一个新的项，这个新的项的结构在对易情形下是不对应的，它为我们在非对易球面上对规范场论进行分类提供了一个工具。最近，人们发现在反对称张量场的背景下，包在环面上的D-膜具有非对易结构。弦理论与非对易几何的关系是一个有趣的问题。从这个角度出发，我们研究了群流形中的D-膜，并构造了SU（2）流形中的边界态（论文5）。SU（2）中D-膜的有效理论是由上述非对易球上的规范理论给出的，我们目前正在研究这些关系。

The aim of this research is to construct the field theory over the non-commutative space and to analyze its physical properties. In the last 4 years research supported by the present Grant-in-Aid we have been especially investigating the field theory on the non-commutative sphere as a concrete example of a non-commutative curved space. The non-commutative sphere is defined by quantizing the algebra over the 2-sphere. As a first step in this direction we have constructed the non-commutative differential algebra according to the approach given by Connes (paper 1), and as its application we formulated the scalar field on the non-commutative sphere (paper 2). Our main interest is to define the gauge theory on the non-commutative sphere. However, for this end we had to analyze further detailed structure of the differential calculus. In paper 3 we succeeded to formulated the U (1) gauge theory, and we have analyzed its commutative limit. Especially in gauge theory we obtained a new term and the structure of this new term, which does not have a correspondence in the commutative case, gives us a tool to classify the gauge theory on the non-commutative sphere.Recently, it is found that the D-brane which wraps around the torus possesses a non-commutative structure in the background of an antisymmetric tensor field. It is an interesting problem to investigate the ralation between string theory and non-commutative geometry. From this point of view we investigated the D-brane in the group manifold and constructed the boundary state in the SU (2) manifold (paper 5). The effective theory of this D-brane in SU (2) is given by the gauge thoery on the non-commutative sphere described above and we are presently researching those relations.