Conformal Supergravity and String Model

众所周知，弦模型是包括引力在内的所有基本相互作用的统一理论的最有前途的候选者。该模型的特点是在时空中用弦画出的二维世界薄片具有超共形对称性。共形超重力就是这种对称性的规范理论。最近，在弦的紧化和二维统计系统中的临界现象的背景下研究超共形代数引起了人们的极大兴趣。在本研究项目的第一年，我们得到了N=4个超共形代数的超空间公式。我们构造了对表示理论起重要作用的筛选算子。第二年，我们计算了表示有限变换反常的超Schwarzian导数，它在超黎曼曲面的研究中具有重要意义。在过去的一年里，我们研究了超共形场理论的超临界行为。我们将N=0 Sine Gordon理论作为一个扰动系统推广到N=1和N=2的超对称情形。除了得到超Virasoro算子中多项式的普通守恒荷外，我们还得到了顶点算子型的量子守恒荷。我们用微扰理论研究了N=1和N=2 Sine-Gordon理论的经典守恒律和量子守恒律。在超弦理论中，时空N=1的超对称性导致了N=2的世界薄层对称性。在这方面，N=2孤子理论引起了人们的极大兴趣。顶点算子型附加荷的存在与S矩阵的结构密切相关。在研究过程中，与东京大学和KEK等人的讨论帮助很大。我们在期刊上发表了我们在本研究中获得的成果，并在各研究所的几个讲习班、JPS会议和研讨会上展示了结果。

It has been widely known that the string model is the most promising candidate for the unified theory of all fundamental interactions including gravity. This model is characterized by the superconformal symmetry in the two-dimensional world sheet drawn by strings in space-time. The conformal supergravity is the gauge theory of this symmetry. Recently, there has been much interest in studying the superconformal algebras in the context of the compactification of strings and the critical phenomena in two-dimensional statistical systems. In the first year of the present research project, we obtained the superspace formulation of N=4 superconformal algebras. We constructed the screening operators which play the important role for the representation theory. In the second year, we calculated the super Schwarzian derivative that represents the anomalies of finite transformations and is important in the study of super Riemann surfaces. In the last year, we investigated off-critical behaviors of superconformal field theories. We extended N=0 sine Gordon theory as a perturbed system to the N=1 and N=2 supersymmetric cases. We obtained quantum conserved charges of vertex oparator type in addition to the ordinary conserved charges of polynomials in super-Virasoro operators. We studied conservation laws at the classical level as well as at the quantum level based on the perturbation theory for N=1 and N=2 sine-Gordon theories. In superstring theory, the space-time N=1 supersymmetry leads to the N=2 world-sheet symmetry. In this respect, N=2 soliton theories attract much interest. The existence of extra charges of vertex operator type are closely related to the structures of S matrices. During the course of research, the discussions with the people at University of Tokyo and KEK and so on were of great help. We published what we obtained in the present research in the journals and presented the results at several workshops, JPS meetings and seminars in various institues.