Noncommutative cohomological field theories and topological aspects of matrix models

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 0 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.