Topological String Theory and N=2 Supersymmetry

二维拓扑Sigma模型：详细分析了拓扑CP^1模型的矩阵积分表示，阐明了该模型的可积结构，并在此基础上提出了Landau-Ginzburg公式。用矩阵模型预测的瞬时和与用几何方法计算的结果完全一致。我们还证明了拓扑CP^n和Grassmannian sigma模型中的相关函数可以直接使用对数型有效作用来评估。4D N=2超对称规范理论：利用Picard-Fuchs方程精确计算了N=2超对称QCD的库仑分支的低能有效作用（预势）。我们还得到了N=2超对称QCD的前势所遵从的一个基本恒等式。我们的推导是基于最近提出的与N=2规范理论有关的孤子方程。一维量子临界现象：我们借助于精确有限尺寸谱和边界共形场理论，确定了近藤问题中的各种临界指数。用同样的方法，我们计算了Calogero-Sutherland模型中有边界的相关函数的渐近形式。我们还考虑了分数阶不相容统计量量子系统的高自旋扩展。

2D Topological Sigma Model : Analyzing in detail the matrix integral representation of the topological CP^1 model, we clarify the integrable structure of the model, based on which we develop the Landau-Ginzburg formulation. The instanton sum predicted by the matrix model is in complete agreement with that obtained by a geometrical method. We also show that correlation functions in the topological CP^n as well as Grassmannian sigma models are directly evaluated using the logarithmic-type effective action.4D N=2 Supersymmetric Gauge Theory : We calculate exactly the low-energy effective action (prepotential) for the Coulomb branch of N=2 supersymmetric QCD by making use of the Picard-Fuchs equations. We also derive a basic identity obeyed by the prepotential of N=2 supersymmetric QCD.Our derivation is based on recently proposed soliton equations in connection with N=2 gauge theories.1D Quantum Critical Phenomena : We determine various critical exponents in the Kondo problem with the aid of the exact finite-size spectrum and boundary conformal field theory. In a similar manner we compute the asymptotic forms of correlation functions in the Calogero-Sutherland model with boundaries. We also consider the higher-spin extension of the quantum system of fractional exclusion statistics.