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

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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.