School of Natural Sciences, Institute for Advanced Study,
School of Natural Sciences, Institute for Advanced Study,
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1991
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这些笔记致力于勾勒出一些与镜像对称有关的标准事实及其应用如何在拓扑场论的背景下自然地被理解。如果X是Calabi-Yau流形，通常的控制黎曼曲面Σ到X的映射的非线性Sigma模型可以用两种方式扭曲以给出拓扑场论，我称之为A模型和B模型。镜像对称将(一个Calabi-Yau流形的)A模型与(其镜像的)B模型联系起来。通过对有理曲线的计数和对微分形式周期的计算，可以分别计算A和B模型的相关函数。这可以被证明为弱耦合的结果(如在这些注释的§3-4中)，或者通过Feynman路径积分的一种不动点定理(见§5)。扭曲模型的关联函数与物理的非扭曲模型的某些矩阵元素重合，如§6所述--即那些确定超势的元素。在拓扑场论的背景下，西格玛模型的传统模空间可以被加厚到扩展的模空间，如§7所示，这可能是理解模空间之间仍然神秘的“镜像映射”的自然框架。
These notes are devoted to sketching how some of the standard facts relevant to mirror symmetry and its applications can be naturally understood in the context of topological field theory. If X is a Calabi-Yau manifold, the usual nonlinear sigma model governing maps of a Riemann surface Σ to X can be twisted in two ways to give topological field theories, which I call the A model and the B model. Mirror symmetry relates the A model (of one Calabi-Yau manifold) to the B model (of its mirror). The correlation functions of the A and B models can be computed, respectively, by counting rational curves and by calculating periods of differential forms. This can be proved as a consequence of a reduction to weak coupling (as in §3-4 of these notes) or by a sort of fixed point theorem for the Feynman path integral (see §5). The correlation functions of the twisted models coincide, as explained in §6, with certain matrix elements of the physical, untwisted model – namely those that determine the superpotential. The conventional moduli spaces of sigma models can be thickened, in the context of topological field theory, to extended moduli spaces, indicated in §7, which are probably the natural framework for understanding the still mysterious " mirror map " between moduli spaces.