String Theory, Matrix Models and Noncommutative Geometry

超弦理论作为包括引力在内的所有基本相互作用的统一理论的候选者，已经得到了积极的研究。然而，这一理论的基本表述并没有得到很好的确立。微扰理论已经被理解得很好了，但是弦的场论的构建还远远没有完成。正因为如此，我们不了解该理论的基本对称性，很难用该理论来构建基本粒子模型。作为弦论的基本形式，矩阵模型是弦理论，当我们把它看作弦理论时，它并不明显。在本研究中，我们从以下几个角度对矩阵模型进行了研究。首先，研究了膜理论。IIA型弦理论的强耦合极限可用一个称为M理论的理论来描述。M理论可以描述为一种膜理论。最近，有人提出膜理论可以用矩阵量子力学来描述。在这项研究中，我们研究了矩阵量子力学，并讨论了膜的世界体积对称性。我们研究的另一个主题是非临界弦理论。非临界弦理论是一种维度小于1的时空中的弦理论，这种弦理论可以用矩阵模型方法精确求解。它被用作关键问题的玩具模型，特别是在研究弦理论的非微扰方面时。非临界弦理论中的非微扰效应被认为是由D瞬子引起的。在这项研究中，我们计算了这些瞬子的化学势，并表明它的值是一个与正则化方法无关的普适量。

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.