Quantum Theory of Extended Objects and Unification Theory

该研究项目的目的是研究作为统一理论候选对象的扩展对象的量子理论。我们专注于构建膜的量子化理论，同时深入理解弦模型。我们进行了I）膜的数学性质的分析和II）它们的物理性质的分析，例如质谱和量子效应。I）在构建膜的量子理论时，理解模型的规范对称性是至关重要的。我们知道，保面积的超同态是作为剩余规范不变性而产生的。这类同态最近被证明可以用一个称为W_* 代数（W_n代数的n->* 极限）的无限维代数来表示。关于W_* 代数的超对称扩张，我们得到了以下结果：a）W代数的超对称推广的构造b）超W代数的超对称可积系统的构造c）基于李超代数的超KP族的推导d）将结果a）-c）推广到N=2对称的情形，这些结果不仅为分析超膜提供了有用的方法，而且为处理超弦的紧化提供了有用的框架II）Weyl不变性在弦的量子理论中起着至关重要的作用.我们构造了一个具有Weyl不变性的膜模型。我们现在正在研究这个理论中的量子效应。这项研究的结果也将对三维重力场产生影响。

The purpose of this research project was to study the quantum theory of extended objects as candidates for the unified theory. We focused on constructing a quantized theory of membranes and at the same time on deeper understanding of string models. We carried out both I) analysis of the mathematical properties of membranes and II) analysis of their physical properties such as the mass spectrum and quantum effects.I) In constructing quantum theory of membranes it is crucial to understand the gauge symmetries of the model. It has been known that the area preserving diffeomorphisms arises as the residual gauge invariance. This type of diffeomorphisms has recently been shown to be expressed in terms of an infinite dimensional algebra called W_* algebra (n->* limit of W_n algebra). Towards the supersymmetric extension of W_* algebra we have obtained the following results :a) Construction of Supersymmetric Generalization of W Algebrasb) Construction of Supersymmetric Integrable Systems Associated with Super W Algebrasc) Derivation of Super KP Hierarchy Based on Lie Superalgebrasd) The Results a) -c) were Generalized to the N=2 Symmetry Case.These results not only provide useful methods for analyzing supermembranes but also give a useful framework to deal with compactification of superstrings.II) Weyl invariance Plays a crucial role in quantum theory of strings. We have constructed a membrane model with Weyl invariance. We are now in the course of studying quantum effects in this theory. The result of this investigation is expected to have implications on three-dimensional gravity as wel.