Theory of Two-Dimensional Black Hole

拓扑场论是理解弦理论背后的几何原理的一种尝试，目前正处于积极的研究之中。已知N=2超对称非线性σ -模型可以适当地扭曲并转化为拓扑场理论拓扑σ -模型。拓扑sigma模型的配分函数由目标空间上全纯映射的和给出，一般难以求值。Eguchi和S.K.Yang研究了任意Riemann曲面上的拓扑CP^1模型，并证明了系统的可积结构可以用一维Toda层次来描述，其配分函数可以用一些大n矩阵积分来精确表示。c=1弦理论最初是作为一维矩阵量子力学引入的，并通过集体坐标、自由费米子等方法得到了广泛的研究。最近提出了一种将c=1弦理论解释为拓扑场论的方法，并引入了朗道-金兹堡描述。Eguchi和H.Kanno一起研究了这一建议，并证明了该理论的可积结构是由Toda格层给出的，并施加了一个特殊的约束条件。Eguchi与y.a yamada和s.k.l yang在分析流动方程的高阶修正的基础上，研究了拓扑弦理论的高格结构。证明了拓扑弦理论的一些模型在g=0和1时是一致的，但不能推广到g<或更大的属

Topolgical field theory is an attempt at understanding the geometrical principles behind the string theory and is currently under active investigations. It is known that an N=2 supersymmetric non-linear sigma-model can be suitably twisted and converted into a topological field theory, topological sigma-model. The partition function of the topological sigma-model is given by the sum over holomorphic maps onto some target space and in general difficult to evaluate. Eguchi, together with S.K.Yang, studied the topological CP^1 model on an arbitrary Riemann surfaces and have shown that the integrable structure of the system is described by the 1-dimensional Toda hierarchy and its partition function can be precisely represented in terms of some large-N matrix integral.The c=1 string theory was originally introduced as an 1-dimensional matrix quantum mechanics and has further been studied extensively using the moethod of collective coordinates, free fermions etc.. Recently an interpretation of the c=1 string theory as a topological field theory has been proposed and a Landau-Ginzburg description has been introduced. Eguchi, together with H.Kanno, studied this proposal and have shown that the integrable structrue of the theory is given by the Toda lattice hierarchy with a special constraint condition being imposed.Eguchi, togetehr with Y.Yamada and S.K.Yang, studied the higher genus structure of the topological string theory based on the analysis of the higher order corrections to the flow equations. It was shown that there are models of topological string theories which are cosistent at genus g=0 and 1 but can not be extended into higher genera g<greater than or