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、超对称规范理论等，取得了以下成果：（1）基于HartreeFoch理论建立了vonNeumann格点上量子霍尔系统的可压缩气体态。它们具有各向异性的密度调制、负压力和可压缩性。负压使量子霍尔效应具有异常的稳定性。提出了将分数霍尔效应与霍夫施塔特蝴蝶效应相结合的广义磁通态。（2）将Chern-Simons理论推广到偶数维，并一致量子化为无穷可约约束系统。（3）分析了时间反转破坏超导体，证明了在时间反转破坏超导体中存在有效的赝Chern-Simons项。并提出了观测的实验方案。（4）在格上定义了Chern-Simons引力和BF引力。（5）证明了D粒子的共形不变性，并基于Hamiltonian-Jacobi方法得到了一个新的公式。(6)A提出了一种利用共形不变性计算临界点附近物理量的新方法。（7）导出了量子信息论中纠缠熵的普适不等式。建立了保角映射与户田继承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.