Quantum Field Theories in Low Dimensions and Their Application

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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