Representation Theories of Superconformal Algebras and Superstring

点击翻译按钮获取中文摘要

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