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最有希望的候选者。该模型的特点是时空中弦绘制的二维世界图具有超共形对称性。共形超引力就是这种对称的规范理论。近年来，在弦的紧化和二维统计系统的临界现象的背景下研究超共形代数引起了人们极大的兴趣。在本课题的第一年，我们得到了N=4个超共形代数的超空间公式。我们构造了对表征理论起重要作用的筛选算子。在第二年，我们计算了超级Schwarzian导数，它表示有限变换的异常，在超级黎曼曲面的研究中很重要。在过去的一年里，我们研究了超共形场理论的非临界行为。我们将N=0正弦Gordon理论作为摄动系统推广到N=1和N=2超对称情况。除了在超virasoro算子中多项式的普通守恒电荷外，我们还得到了顶点算子类型的量子守恒电荷。我们在N=1和N=2正弦戈登理论的微扰理论基础上，研究了经典水平和量子水平上的守恒定律。在超弦理论中，时空N=1的超对称性导致了N=2的世界表对称性。在这方面，N=2孤子理论引起了人们的极大兴趣。顶点算子型附加电荷的存在性与S矩阵的结构密切相关。在研究过程中，与东京大学、KEK等人的讨论给了我很大的帮助。我们在期刊上发表了我们在目前的研究中获得的成果，并在不同机构的几个研讨会，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.