Quantum field theory with gravity and anomalies

本课题的主要课题是异常、拓扑和黎曼曲面的相关问题，近年来受到了广泛的关注。对于本研究项目的具体结果，K. Fujikawa研究了非临界弦的详细行为。通过将非临界弦理论看作具有异常的二维量子引力，证明了自由非临界弦的BRS对称性是很自然地定义的。然而，由于与黎曼-洛克定理（或鬼数异常）相关的拓扑约束，弦相互作用相当复杂。木村提出了约束系统的一般量子处理，这与量子引力密切相关。这一配方的应用将留给未来。细谷研究了量子引力中与拓扑变化有关的各种问题。具体来说，他在路径积分形式主义中分析了宇宙常数由于婴儿宇宙而消失的机制。与此同时，他发现了一个新的类瞬态引力解。对于三维量子引力，他使用Teichmuller参数作为动力学变量，给出了一个很好的规范公式。上原继续研究超弦和超粒子。特别是，他成功地提出了处理超黎曼曲面变形的一般公式。这一分析在数学上很有趣，有望为超弦动力学提供新的见解。

The main subjects of this research project, namely, the problems related to anomalies, topology and Riemann surfaces received much attention recently.As for the specific results of the present research project, K. Fujikawa studied the detailed behavior of non-critical strings. By regarding the non- critical string theory as a 2-dimensional quantum gravity with anomalies, it was shown that the BRS symmetry is quite naturally defined for a free non- critical string. The string interactions are, however, rather involved due to the topological constraints related to the Riemann-Roch theorem (or ghost number anomaly).T. Kimura formulated a general quantum treatment of constrained systems, which are closely related to the quantum gravity. The applications of this formulation are left for the future.A. Hosoya studied the various problems related to the topology changes in quantum gravity. To be specific, he analyzed the vanishing mechanism of the cosmological constant due to baby universes in the path integral formalism. At the same time, he discovered a new instanton-like gravitational solution. As for 3-dimensional quantum gravity, he gave a nice canonical formulation by using the Teichmuller parameters as dynamical variables.S. Uehara continued the study of super-strings and super-particle. In particular, he succeeded in the general formulation to deal with the deformations of super-Riemann surfaces. This analysis, which is mathematically interesting, is hoped to provide a new insight into the dynamics of super- strings.