Representation Theories of Superconformal Algebras and Superstring

Virasoro和Kac-Moody代数的表示理论最初是从基本粒子物理学的研究中发展起来的，它本身就是一个非常有趣的研究课题。目前的项目一直在进行，特别是集中在这样一个事实，即表示论的结果在超弦理论的建设和它们的紧化的一般调查，以及临界现象的可解模型的研究中发挥着重要的关键作用。具体地说，通过给出在超荷N的超共形代数的Verma模中构造零态的一般方法（Kato-Matsuda方法），分析了表示论的数学方面.作为我们1990-1991年研究的结果，我们成功地得到了N = 4超共形代数的SU（2）Kac-Moody代数的库仑气体表示，它允许包含代数的非幺正表示. 1992年，我们 关于我们 进一步发展了这项研究，利用前两年获得的结果找到了电荷屏蔽算子。我们构造了N = 4代数上的顶点算子，从而成功地确定了N = 4代数上的整组屏蔽算子，这些屏蔽算子是发展N = 4代数的完备表示理论的关键工具.我们的结果为实现N = 4 Kac行列式公式的严格证明提供了基础。我们还研究了N = 4表示的BRS上同调。我们进一步发展了我们对二维量子引力和可解模型的研究，重点是这些主题背后的二维对称性的预期共同特征。我们与物理学和数学研究人员进行了积极的接触，以便深入了解这个问题，并实现我们的项目目标。我们所获得的结果已发表在国际期刊上，并汇总在我们的科研资助（C）项目报告中。少

The representation theories of the Virasoro and Kac-Moody algebras originally developed from the study of elementary particle physics,is itself a very interesting topic of study. The present project has been pursued, particularly focusing on the fact that the results of the representation theories play an important key role in the general investigation of the construction of superstring theories and their compactification as well as the study of solvable models for critical phenomena. To be concrete, the mathematical aspects of the representation theories have been analyzed by providing the generic method (the Kato-Matsuda method) of constructing null states in the Verma modules of superconformal algebras with supercharge N. As the results of our investigation in 1990-1991, we succeeded in obtaining the Coulomb gas representations of the N = 4 superconformal algebras with the SU(2) Kac-Moody algebra, which allow for incorporating non-unitary representations of the algebras. In 1992 we … More futher developed the study, finding the charge screenign operators using the results obtained in the first two years. We constructed vertex operators in the N = 4 algebras and thereby suceeded in identifying the whole set of the screening operators which are the key tools for developing the complete representation theories of the N = 4 algebras. Our results provide the basis for achieving the rigorous proof of the N = 4 Kac determinant formulae. Also we have been pursuing the BRS cohomology of the N = 4 representations. We further developed our study on 2 dimesional quantum gravity and slovable models with an emphasis on the expected common features of 2 dimensional symmetry behind these topics. We made an active contact with researchers in physics and mathematics in order to have our deep understanding of the problem and also to achieve our goal of the project. Our obtained results have been published in the international juornals, and also put together in the Report of our project under Grant-in-Aid for Scientific Research (C). Less