String Theory, Matrix Models and Noncommutative Geometry

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Superstring theory has been actively studied as a candidate for the unified theory of all the fundamental interactions including gravity. However, the fundamental formulation of the theory is not so well established. The perturbation theory is understood fairly well, but the construction of the field theory of strings is far from being completed. Because of this, we do not understand the fundamental symmetry of the theory and it is difficult to construct the model of elementary particles using this theory. As a fundamental formulation of string theory, matrix models are string theory which are not obvious when we consider the theory as a theory of strings. In this research, we studied the matrix models from the following points of view.Firstly we studied the membrane theory. The strong coupling limit of Type IIA string theory is considered to be described by a theory called M-theory. M-theory can be described as a theory of membranes. Recently, it is proposed that the membrane theory can be described as a matrix quantum mechanics. In this research we studied the matrix quantum mechanics and discuss the worldvolume symmetry of the membranes.Another topic we studied is the noncritical string theory. Noncritical string theory is a string theory in a spacetime with dimension less than 1.Such a string theory can be exactly solved by matrix model approach. It is used as a toy model for the critical ones, especially useful in studying the nonperturbative aspects of string theory. The nonperturbative effects in noncritical string theory are regarded as due to D-instantons. In this research we calculated the chemical potential of these instantons and showed that its value is a universal quantity independent of the method of regularization.