Project/Area Number  08304001 
Research Category 
GrantinAid for Scientific Research (A)

Section  一般 
Research Field 
Algebra

Research Institution  KYOTO UNIVERSITY 
Principal Investigator 
MIWA Tetsuji KYOTO UNIVERSITY Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (10027386)

CoInvestigator(Kenkyūbuntansha) 
山田 泰彦 神戸大学, 理学部, 助教授 (00202383)
梁 成吉 筑波大学, 物理学系, 教授 (70201118)
中屋敷 厚 九州大学, 大学院・数理学研究科, 助教授 (10237456)
SHIRAISHI Junichi The University of Tokyo, The Institute for Solid State Physics, Assistant, 物性研究所, 助手 (20272536)
KUNIBA Atsuo The University of Tokyo, Graduate School of Arts and Sciences, Assistant Profess, 大学院・総合文化研究科, 助教授 (70211886)
HASEGAWA Koji Tohoku University, Graduate School of Science, Lecturer, 理学研究科, 講師 (30208483)
KONNO Hitoshi Hiroshima University, Faculty of Integrated Arts and Sciences, Assistant Profess, 助教授 (00291477)
OKADO Masato Osaka University, Faculty of Engineering Sciences, Assistant Professor, 大学院・基礎工学研究科, 助教授 (70221843)

Project Fiscal Year 
1996 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥10,000,000 (Direct Cost : ¥10,000,000)
Fiscal Year 1998 : ¥2,200,000 (Direct Cost : ¥2,200,000)
Fiscal Year 1997 : ¥3,900,000 (Direct Cost : ¥3,900,000)
Fiscal Year 1996 : ¥3,900,000 (Direct Cost : ¥3,900,000)

Keywords  solvable lattice models / quantum froups / vertex operators / bosonization / difference analogue of the KZ equation / crystal / 可解格子模型 / 頂点作用素 / 自由場表示 / 差分KZ方程式 / 結晶基底 / 格子模型 / GKO構造 / 楕円的量子群 / 可解模型 / ヘッケ環 / Demazure結晶 / KnizhnikZamolodchikov方程式 / Kostka多項式 / エネルギー関数 / XXZ模型 / screening作用素 / トロイダル代数 
Research Abstract 
The main theme of this project is the symmetry approach to solvable lattice models. As for this problem, In the case of elliptic models, which had not been solved In the symmetric approach, a bosonization of the vertex operators were obtained by Shiraishi and Odake using the representation theory of quaslHopf algebra, in particular, the twist of quantum groups. By Miwa and Konno, in the trigonometric limit (with q =1) of this model, an integral formula is obtained for the difference analogue of the KnnizhnikZamolodchhikov equation. Hasegawa constructed Ruijsenaars' commuting difference operators by using the intertwining vectors In the elliptic model. The theory of crystal is a key In the connection between solvable lattice models and combinatorics. As for this, Miwa, Okado, Kuniba, Yamada (Yasuhiko) found that the set of inhomogeneous paths give the crystal of tensor products of integrable highest weight representations. Several interesting examples of nonperfect crystals and the
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corresponding paths are also studied. As for nonsolvable models, Matsui showed that the matrix product states which represent the ground states correspond to the representations of the Kuntz algebra. The toroidal algebra is important because it governs the symmetry of the vertex operators. Mild found an automorphism of this algebra which connects two affine quantum algebras inside thereof. Solvable models In quantum field theory Is another main subject. Kawahigashi developed the method for calculating modular invariant quantities in the twodimensional conformal field theory. It is also important to apply techniques developed in the twodimensional solvable models to the problems in string theory and the fourdimensional gauge theory. As for this, Nakatsu showed that in the adiabatic limit the tau function of the Toda lattice gives the effective action of the N=2 supersymmetric YangMills theory. Kanno and Yang obtained a generalization of the DonaldsonWitten Invariant for the four dimensional manifolds. Kato analysed the condition for a matrix model to be understood as quantum gravity theory In a curved spacetime. The developments In the past three years Include (1) the symmetry method for the solvable lattice models of elliptic type ; (2) the method of representation theory for the mixed spin chains ; (3) the discovery of new links between lattice models and combinatorics ; (4) the quantization of several geometric invariants ; (5) the algebraic approach connecting the operator algebras and the conformal field theory. Less
