Studies in Matrix Model and Spacetime Structure

超弦理论是粒子物理学中包括引力在内的统一理论的候选者，在这个理论中，我们可以在同一个立足点上研究时空和物质。本文基于超弦理论的非微扰定义--矩阵模型，研究了时空结构及其物质实现，在本研究期间所取得的主要成果概括为以下三个方面.研究了矩阵模型中的时空结构与超引力理论中的时空结构之间的关系。通过研究矩阵模型中的Wilson圈算符，系统地构造了各种改变超引力理论背景场位形的顶点算符。然后，利用这些顶点算子，分析了矩阵模型中超引力多重态对应的场（如引力场）的凝聚.我们在矩阵模型和非对易几何中研究了物质的手征结构。应用格点场论中的Ginsparg-Wilson关系，给出了定义GW Dirac算子的一般规定，并给出了一般非对易几何中拓扑不变量和指数的公式.我们构造了拓扑非平凡的组态，分析了它们的动力学，并在超弦理论和矩阵模型中发展了对手征费米子在我们时空中的实现机制的理解.我们发现黑洞的Hawking辐射可以用场论中的反常来理解，我们也将其应用于各种黑洞，如旋转黑洞和高维黑洞。这些结果可望应用于黑洞熵等物理量的研究。

Superstring theory is a candidate for the unified theory including gravity in particle physics, the theory where we can study spacetime and matter at the same footing. Based on the matrix models which are a nonperturbative definition of superstring theory, we study the spacetime structure and the matter realization on it. The main results that we have obtained during the term of this grant are summarized in the following three categories.1. We studied the relation between the spacetime structure in the matrix model and that in the supergravity theory. By investigating the Wilson loop operator in the matrix model, we constructed systematically various vertex operators that alter the background field configurations in the supergravity theory. Then, by using these vertex operators, we analyzed the condensation of the fields corresponding to the supergravity multiplets, such as the gravitational field, in the matrix model.2. We studied the chiral structure of matter in matrix model and noncommutative geometry. By applying the Ginsparg-Wilson relation developed in the lattice field theory, we gave the general prescription to define the GW Dirac operator and to formulate the topological invariant and the index in general noncommutative geometries. We constructed topologically nontrivial configurations, analyzed their dynamics, and developed the understanding of the realization mechanism of chiral fermion on our spacetime in superstring theory and matrix model.3. We found that the Hawking radiation from the black holes can be understood by using the anomaly in the field theory, which we also applied to various black holes, such as the rotating black holes and the higher dimensional black holes. These results are expected to be applied to the study of various thermodynamical quantities such as black hole entropy.