Noncommutative Cohomological Field Theories and Topological Aspects of Matrix models

Abstract

We study topological aspects of matrix models and noncommutative cohomological field theories (N.C.CohFT). N.C.CohFT have symmetry under the arbitrary infinitesimal noncommutative parameter $θ$ deformation. This fact implies that N.C.CohFT possess a less sensitive topological property than K-theory, but the classification of manifolds by N.C.CohFT has a possibility to give a new view point of global characterization of noncommutative manifolds. To investigate properties of N.C.CohFT, we construct some models whose fixed point loci are given by sets of projection operators. Particularly, 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 get 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 $θ$ limit and the finite $θ$, for the Moyal plane case. If the partition function of N.C.CohFT is topological in the sense of the noncommutative geometry, then it should have some relation with K-theory. Therefore we investigate certain models of CohFT and N.C.CohFT from the point of view of K-theory. These observations give us an analogy between CohFT and N.C.CohFT in connection with K-theory. Furthermore, we verify it for the Moyal plane and noncommutative torus cases that our partition functions are invariant under the those deformations which do not change the K-theory. Finally, we discuss the noncommutative cohomological Yang-Mills theory.