Structure and symmetries of integrable systems

本研究项目的主要成果如下：Jimbo推动了晶格模型状态空间的研究。Ising和RSOS型模型与顶点型模型的表述方式大致相同，并推导出相关性的差分方程。将该理论推广到有边界的自旋链，导出了真空态、能量和磁化强度等物理量。对于临界系统，得到了相关系数的积分公式。在表示理论中，在量子仿射代数上的一级模上构造了一个新的0级作用。Shiota用类似于Kontsevich的矩阵积分构造了[P，Q] =P的非平凡解。武井表明，使用精确的WKB分析，painlevel I方程可以作为一个简单拐点附近的标准形式。他还利用多尺度分析构造了一个通解。对于GLq (n)的Capelli恒等式中出现的微分算子，Umeda用经典的q-差分算子研究了一种新的q-模拟算子的构造。这开辟了与gelfand型超几何方程的联系。上野给出了曲线模空间上共形块向量束上的投影平面连接的一个具体构造。Kono研究了紧单李群的自由环群，发现了G在闭环群上的伴随作用的模p上同调与G的积分上同调之间的联系。

The major outcome of the present research project is as follows.1.Jimbo pushed forward the study of space of states of lattice models. Ising and RSOS type models were formulated in much the same way as for the vertex type models, and difference equations for correlations were derived. The theory was extended to spin chains with a boundary, and physical quantities were derived including vacuum states, energy and magnetization. For critical systems an integral formula for correlations was found. In representation theory, a new level 0 action was constructed on a level one module over the quantum affine algebra.2.Shiota constructed a notrivial solution to [P,Q] =P using a matrix integral similar to Kontsevich's. Takei showed, usuing the exact WKB analysis, that the Painleve I equation can be taken as a standard form near a simple turning point. He also constructed a general solution using the multiple scale anaysis.3.Umeda studied a new construction of a q-analog of differential operators, which appear in the Capelli identity for GLq (n) , in terms of classical q-difference operators. This opened up a connection with Gelfand-type hypergeometric equations.4.Ueno gave a concrete construction of projective flat connections on the vector bundles of conformal blocks over the moduli space of curves. Kono studied the free loop group of a compact simple Lie group, and found a connection between the mod p cohomology of the adjoint action of G on the closed loop group, and the integral cohomology of G.