LOW DIMENSIONAL FIELD THEORIES AND THEIR APPLICATIONS

本项目研究了低维物理系统的基本问题，如量子霍尔效应场论、量子引力、超弦和超对称规范理论等，取得了以下成果：(1)基于冯·诺依曼格子上的Hartree Foch理论，建立了量子霍尔系统的可压缩气态。它们具有各向异性的密度调制、负压和可压缩性。负压使量子霍尔效应具有不同寻常的稳定性。提出了将分数霍尔效应与霍夫施塔特蝴蝶相联系的广义磁通态。(2)将Chern-Simons理论推广到偶数维，并一致地将其量子化为一个无限可约约束系统。(3)分析了违反超导体的时间反转现象，证明了伪Chern-Simons项的有效作用。(4)在晶格上定义了Chern-Simons引力和BF引力。(5)证明了D粒子的共形不变性，得到了基于哈密顿-雅可比方法的新公式。(6)发展了一种应用共形不变性计算临界点附近物理量的新方法。(7)推导了量子信息论中纠缠熵的普遍不等式。建立了保角映射与Toda Heireches和二维引力的普遍联系。

In this project, fundamental problems of low dimensional physical systems such as field theory of quantum Hall effects, quantum gravity, super string, and super symmetric gauge theory and others are investigated and the following results have been obtained:(1)Compressible gas states of quantum Hall system have been established based on Hartree Foch theory on von Neumann lattice. They have anisotropic density moduration, nagative pressure and compressibility. Negative pressure leads quantum Hall effect to have unusual stability. Generalized flux state which connects fractional Hall effect with Hofstadter butterfly was proposed.(2)Chern-Simons theory was generalized to even dimensions and was quantized as an infinity reducible constrained system consistently.(3)Time reversal violating superconductors was analyzed and was shown to have pseudo Chern-Simons term in effective action. Experiment for its observation was also proposed.(4)Chern-Simons gravity and BF gravity were defined on lattice.(5)Conformal invariance of D-particle was shown and a new formulation based on Hamiltonian-Jacobi method was achieved.(6)A new technique to compute physical quantities at around critical point where conformal invariance was applied was developed.(7)Universal inequarities for entanglement entropy in quantum information theory was derived. A universal connection of conformal mapping with Toda heirechies and two-dimensional gravity was established.