Quantum Field Theories in Low Dimensions and Their Application

在粒子物理和凝聚态物理中，许多重要的物理现象和处理方法都是非微扰性质的，很难求解。同样的问题也出现在低维的物理系统中。我们打算发展低维量子场论的物理和数学方法，并追求它们的应用。具体而言，我们计划研究以下问题：1）可解格点模型/有边界可积场论的构造及其在凝聚态物理中的应用; 2）其他新类型的可积场论的构造; 3）二维量子引力的非微扰方法; 4）拓扑引力; 5）W_* 和W_N代数的表示理论及其在凝聚态物理中的应用; 6）纽结理论在统计力学中的应用; 7）QCD和非线性信号的非微扰方法 ...更多信息 8）超对称性和反常在物理学中的应用我们对上述大部分问题的研究都取得了进展。问题1）我们构造了几种可解格点模型和可积场论[8，17]，研究了半直线上系统的稳定性问题[18]。韩国团队（Nam和他的合作者）与Inami和Sasaki在这些问题上有共同的兴趣，他们交换了意见。作为边界CFT凝聚态物理的一个应用，在Kondo问题中导出了有限尺度标度谱[20]。问题5）：W_* 代数表示论研究取得进展[11-14]，从W_N代数表示论的观点研究了Calogero-Sutherland型模型[15，16，19]。Nam加入了日本队（Odake，Matsuo）的讨论。问题7）：这个问题主要由韩国队（Park和他的合作者）进行; Inami加入了讨论[7]。少

Many of the important physical phenomena and the methods to deal with them in particle physics and condensed matter physics are of nonperturbative nature and are difficult to solve. The same type of problems appears in physical systems in low dimensions. They are more easily accessible to rigorous methods.We intended to develop physical and mathematical methods in quantum field theories in low dimensions and to pursue their applications. Specifically we planed to investigate the following problems :1) Construction of solvable lattice models/integrable field theories with boundaries and their application to condensed matter physics.2) Construction of other new types of integrable field theories.3) Non-perturbative methods for 2-dimensional quantum gravity.4) Topological gravity.5) Representation theories of W_* and W_N algebras and their application to condensed matter physics.6) Application of the knot theory to statistical mechanics.7) Nonperturbative methods in QCD and nonlinear sigm … More a model in low dimensions.8) Application of supersymmetry and anomalies to physics.We have made progress in the research of most of the problems written above. We give below only a few of them.Problems 1) We have constructed a few kinds of solvable lattice models and integrable field theories [8,17 of the references] and studied the stability problems in systems on a half line [18]. The Korean team (Nam and his collaborators) has a common interest with lnami and Sasaki on these problems and they have exchnged ideas. As an applicatication of the boundary CFT condensed matter physics, finite-size scaling spectrum in the Kondo problem has been derived [20].Problems 5) : Progress has been made in the representation theoretic studies of W_* algebras [11-14], The Calogero-Sutherland type models has been studied from a representation theoretic viepoint of W_N algebras [15,16,19]. Nam has joined the discussion of the Japanese team (Odake, Matsuo).Problem 7) : This problem has been pursued mainly by the Korean team (Park and his collaborators) ; Inami has joined the discussion [7]. Less