随着物理学中超弦理论和M理论的不断发展，镜像对称成为近20年来最热门的前沿课题之一。本项目将主要研究开Calabi-Yau三维流形的所有亏格的镜像对称。.众所周知，对于紧致三流形，高亏格不变量的计算多年来举步维艰，高亏格镜像对称的数学理论也一直未能建立。相对来说，开三流形在近十年来有了较大进展：对于A模型即Gromov-Witten不变量，已经建立起系统的理论；对于B模型，也发展出多个高亏格的物理模型。然而，这些模型大多缺乏严格的数学理论，也容易陷入极其复杂的递归或者组合计算，无法直接读取出各种不变量的闭公式。此外，各种模型之间的对偶性，大多也并不清楚。.本项目的主要目标，是通过对几个例子的深入分析，结合本人已有的一些成果，建立起一套新的场论及相关的计算工具。作为一个重要应用，希望得到一种数学上清晰且能够直接计算的局部B模型场理论，并最终建立起相应的所有亏格的局部镜像对称。

Since the development of superstring theory and M theory, mirror symmetry becomes one of the most important topics in mathematics in recent 20 years. In this program, our main battlefield is the local Calabi-Yau 3-folds. We focus on the problems of all genus local mirror symmetry. .Definitely we have made more rapid progress on local Calabi-Yau geometries in recent 10 years. On the A-side there are well-developed techniques and computation methods, while on the B-side several new computable physical models are developed. However, we do not have well-defined mathematical theory for most of these models, and we have hardly any direct computations to get closed formula. Also it is not clear how A-model and these different B-models are connected. .In this project, we want to establish a new kind of cohomological field theory, by studying several known examples and correspondences. As an application, we hope to give a mathematically well-defined B-model description for local Calabi-Yau geometries and deduce the corresponding all genus mirror symmetry.