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Algebraic Analysis of Quantum Integrable Systems

量子可积系统的代数分析

基本信息

批准号：
12440039
负责人：
JIMBO Michio
金额：
$ 3.2万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

项目摘要

Solvable Lattice Models We constructed a free field realization of the ABF models in regime II and the higher spin face models, and a a new type of free field realization for the eight-vertex model at a special value of the parameters.Theory of crystals and applications We established an isomorphism between crystals constructed from inhomogeneous paths and crystals for tensor products of integrable highest weight modules of quantum affine algebras, thereby obtaining fermionic character formulas in many cases. We studied the Bethe equation in the limit q = 0 and derived a new representation for weight multiplicities for tensor products of Kirillov-Reshetikhin modules. As another application, we constructed soliton cellular automata from affine crystals, and described time evlolutions and scattering rules in terms of combinatorial R. We further derived a piecewise-linear formula for the latter and showed that its de-ultra-discretizatiohn affords the non-autonomous KP equations.Conformal field theory We considered a vertex operator algebra associated with the Virasoro minimal series M(3, p), derived a fermionic formula for the character of the subspace generated by an abelian current, and obtained a monomial basis thereof. We introduced a differential ideal of symmetric polynomials spanned by Jack polynomials with a negative rational value of the parameter. In a special case it coincides with the set of all correlation functions for the current mentioned above. We also obtained a bosonic character formula for a similar subspace of integrable representations of s1.

我们构造了ABF模型在状态II和高自旋面模型中的自由场实现，以及八顶点模型在特殊参数值下的一种新型自由场实现。我们建立了由非齐次路径构成的晶体与量子仿射代数可积最高权模张量积晶体之间的同构关系，从而得到了许多情况下的费米子特征公式。研究了极限q = 0条件下的Bethe方程，导出了Kirillov-Reshetikhin模张量积权重的新表示。作为另一个应用，我们从仿射晶体中构造了孤子元胞自动机，并用组合r描述了时间演化和散射规则。我们进一步推导了后者的分段线性公式，并证明了它的去超离散性提供了非自治KP方程。考虑了与Virasoro最小级数M（3, p）相关的顶点算子代数，导出了由阿贝尔电流生成的子空间特征的费米子公式，并得到了其单项式基。我们引入了由参数为负有理的杰克多项式张成的对称多项式的微分理想。在特殊情况下，它与上述电流的所有相关函数的集合一致。我们还得到了s1的可积表示的类似子空间的玻色子特征公式。