Mathematics of Symmetry

这个项目的主题是对称的方法来解决晶格模型。针对这一问题，Shiraishi和Odake利用准hopf代数的表示理论，特别是量子群的扭转，得到了在对称方法下尚未得到求解的椭圆模型的顶点算子的波色化。Miwa和Konno在该模型的三角极限（|q |=1）下，得到了Knnizhnik-Zamolodchhikov方程差分模拟的积分公式。Hasegawa利用椭圆模型中的缠结向量构造了rujsenaars交换差分算子。晶体理论是连接可解晶格模型和组合学的关键。对此，Miwa, Okado, Kuniba, Yamada （Yasuhiko）发现，非齐次路径集给出了可积最高权表示的张量积晶体。本文还研究了几个有趣的非完美晶体和相应的路径。对于不可解模型，Matsui证明了表示基态的矩阵积态对应于Kuntz代数的表示。环面代数很重要，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。可解模型在量子场论中是另一个主要课题。Kawahigashi发展了二维共形场理论中模不变量的计算方法。将二维可解模型中发展起来的技术应用于弦理论和四维规范理论中的问题也很重要。对此，Nakatsu证明了在绝热极限下，Toda晶格的tau函数给出了N=2超对称Yang-Mills理论的有效作用。Kanno和Yang对四维流形的Donaldson-Witten不变量进行了推广。加藤分析了矩阵模型在弯曲时空中被理解为量子引力理论的条件。近三年来的进展包括：(1)椭圆型可解晶格模型的对称方法；(2)混合自旋链的表示理论方法；(3)发现格模型与组合学之间的新联系；(4)若干几何不变量的量化；(5)连接算子代数与共形场论的代数方法。少

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