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

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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.)