Conformal invariance and superstring theory

超弦理论是包括引力在内的统一理论中最有希望的候选者。超弦理论最基本的原理是二维世界片的共形不变性。共形不变性在统计物理中对二维二级相变的分类中也起着至关重要的作用。矩阵模型的最新进展从离散理论的角度提供了一种强有力的二维引力场的非微扰处理方法。基于Liouville场论的连续统方法是一种补充，在揭示物理内容方面特别有用，我们利用连续统方法研究了耦合到二维引力场的共形场论。我们成功地分析了具有单位中心电荷的共形物质的配分函数和关联函数。从弦理论的观点来看，这种情况特别有趣，因为它允许通常意义上的时空解释。我们发现，算符乘积展开在揭示与共形物质耦合的二维引力场的奇异性结构方面是非常有用的。我们也在研究物理状态的耦合，使用面积保持的对称性。

Superstring Theory is the most promising candidate for a unified theory including gravity. The most basic principle underlying the superstring theory is the conformal invariance in the two-dimensional world sheet. The conformal invariance also plays a crucial role in classifying the second order phase transition in statistical physics in two demensions. Recent progress in matrix models provides a powerful non perturbative treatment of two-dimensional gravity form the viewpoint of discretized theory. Continuum approach based on the Liouville field theory is complementary and particularly useful in revealing physical content.We have studied the conformal field theory coupled to two-dimensional gravity using the continuum approach. We have succeeded to analyze the partition function and correlation functions in the case of conformal matter with the unit central charge. This case is especially interesting from the viewpoint of string therory, since it allows a space-time interpretalon in the usual sense. We have found that the operator product expansion is very useful in unraveling the singularity structure of the two-dimensional gravity coulpled with the conformal matter. We are also studying the coupling of physical states using the symmetry of area-preserving diffeomorphisms.