Renormalization Group and Migdal-Kadanoff Transformation in Gauge and Spin Systems

重整化群中的尺度变换由Migdal-Kadanoff变换解析实现，是研究临界现象的一个重要的一般性概念。最近关于共形代数和超共形代数（包括Virasoro代数和Kac-Moody代数）的表示理论的研究可以看作是发展了局部尺度变换的一般理论。因此，共形对称的主题与本项目的主题密切相关。为了理解超弦理论的深刻结构，进行这种数学研究也很重要，最近人们对超弦理论进行了深入的研究，希望得到包括引力在内的“万物理论”（TOE）。在本项目中，我们在二维场论中实现共形对称的共形代数的表示理论方面取得了进展。对称性包括在规范系统和自旋系统中研究的尺度变换。特别地，我们提出了在共形代数的Verma模中构造零态的一般方法。我们还将我们的方法推广到具有非零超荷N（= 0,1,2）的超共形代数。因此，我们给出了N=0, 1, 2的零态的一般表达式，同时给出了每个N的Kac行列式的推导证明。用与本方法相同的方法和策略尝试具有更高N的超共形代数的数学分析是未来项目的主题。对于N=4情况的分析是一个迫切需要研究的问题，以便得到一个满意的超弦紧化。在本项目下取得的大部分成果都在下面列出的五篇论文中报告。其中之一，即最后一个，包含了我们的研究项目的全面说明，以及它自己的新结果。

Scale transformation in renormalization group as is analytically realized by the Migdal-Kadanoff transformation is an important general concept for studying critical phenomena. Recent study of the representation theory of conformal and superconformal algebras including the Virasoro and Kac-Moody algebras can be regarded as developing a general theory of local scale transformation. Hence, the subject of conformal symmetry is closely connected with the theme of the present project. It is also important to pursue this mathematical investigation for understanding the profound structure of superstring theory, which has recently been under intensive study with the hope of getting "Theory of Everything" (TOE) including gravity.In the present project we have made a progress in formulating the representation theory of conformal algebras which realize conformal symmetry in two dimensional field theories. The symmetry includes scale transformation as studied in gauge and spin systems. In particular, we have presented a general method of constructing null states in Verma modules of conformal algebras. We have also extended our method to superconformal algebras with nonzero supercharge N(=0, 1, 2). Consequently, we have given a generic expression of null states for N=0, 1, 2, and at the same time presented a proof of deriving Kac determinants for each N.It is the theme of future project to attempt the mathematical analysis of superconformal algebras with higher N by the same methology and strategy of the present approach. The analysis of the N=4 case is an urgent subject to be investigated for developing a satisfactory compactification of superstrings.Most of the achievements performed under the present project are reported in the five papers listed below. One of them, the last one, contains a comprehensive account of our research project performed as well as present new results of its own.