Mathematics of Symmetry

对称数学

• 批准号: 08304001
• 负责人: MIWA Tetsuji
• 金额: $ 6.4万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 1996
• 资助国家: 日本
• 起止时间: 1996 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08304001/
• 关键词: solvable lattice models; quantum froups; vertex operators; bosonization; difference analogue of the KZ equation; crystal; GKO構造; 楕円的量子群; 可解模型; ヘッケ環; Demazure結晶; Knizhnik-Zamolodchikov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

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 quasl-Hopf 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 Knnizhnik-Zamolodchhikov 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 non-perfect crystals and the … More corresponding paths are also studied.As for non-solvable 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 two-dimensional conformal field theory. It is also important to apply techniques developed in the two-dimensional solvable models to the problems in string theory and the four-dimensional 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 super-symmetric Yang-Mills theory. Kanno and Yang obtained a generalization of the Donaldson-Witten Invariant for the four dimensional manifolds. Kato analysed the condition for a matrix model to be understood as quantum gravity theory In a curved space-time.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

这个项目的主要主题是对称的方法来解决晶格模型。对于这个问题，Shiraishi和Odake利用拟Hopf代数的表示理论，特别是量子群的扭曲，在椭圆模型中得到了顶点算符的玻色化，这在对称方法中还没有得到解决。由Miwa和Konno，在三角极限（与|Q| = 1），得到了Knnizhnik-Zamolodchhikov方程的差分模拟的积分公式。长谷川利用椭圆模型中的交织向量构造了Ruijsenaars交换差分算子，晶体理论是可解格模型与组合数学联系的关键。对此，Miwa，Okado，Kuniba，Yamada（Yasuhiko）发现非齐次路径的集合给出了可积最高权表示的张量积的晶体。几个有趣的非完美晶体的例子， ...更多信息 对于不可解的模型，Matsui证明了表示基态的矩阵乘积态对应于Kuntz代数的表示，而环面代数是重要的，因为它决定了顶点算子的对称性. Mild发现了这个代数的一个自同构，它连接了其中的两个仿射量子代数。量子场论中的可解模型是另一个主要课题。Kawahigashi发展了计算二维共形场论中模不变量的方法。同样重要的是，将二维可解模型中的技术应用于弦理论和四维规范理论中的问题。对此，中津指出，在绝热极限下，户田格点的τ函数给出了N = 2超对称杨-米尔斯理论的有效作用。Kanno和Yang得到了四维流形的Donaldson-Witten不变量的推广。加藤分析了弯曲时空中矩阵模型被理解为量子引力理论的条件，近三年来的进展包括：（1）椭圆型可解格点模型的对称性方法，（2）混合自旋链的表示论方法，（3）格点模型与组合学之间新联系的发现，（4）格点模型的量子引力理论，（5）格点模型的量子引力理论，（6）格点模型的量子引力理论。（4）几种几何不变量的量子化;（5）连接算子代数和共形场论的代数方法。少

项目成果

Nakatsu,T.: "Toward second-quantization of D5 branes" Inter.J.Mod.Phys.A. (to appear). (1997)

Nakatsu,T.：“迈向 D5 膜的第二量子化”Inter.J.Mod.Phys.A。

Konno, H.: "An Elliptic algebra U9,P(S12) and the Fusion RSOS Model" Communications in Mathematical Physics. 195. 373-403 (1998)

Konno, H.：“椭圆代数 U9,P(S12) 和融合 RSOS 模型”数学物理通信。

Hasegawa, K.: "Ruijsenaars' commuting difference operators as commuting transfer matrices" Comm.Math.Phys.187. 289-325 (1997)

Hasekawa, K.：“Ruijsenaars 的通勤差分算子作为通勤转移矩阵”Comm.Math.Phys.187。

Hasegawa,K.: "Ruijsenaars' commuting difference operators as commuting transfer matrices" Comm.Math.Phys. 187. 289-325 (1997)

Hasekawa,K.：“Ruijsenaars 的通勤差分算子作为通勤转移矩阵”Comm.Math.Phys。

Kuniba, A: "Demazure modules and perfect crystals (with Misra, K.C.Okado, M.and Uchiyama, J.)" Commun.Math.Phys.192. 555-567 (1998)

Kuniba, A：“Demazure 模块和完美晶体（与 Misra、K.C.Okado, M. 和 Uchiyama, J.）”Commun.Math.Phys.192。