Diflenence equation versions of integrable systems and geometric structures in the background

首先构造了椭圆曲线上的扭曲Wess-Zumino-维滕（WZW）模型，并给出了椭圆Gaudin模型的代数几何解释：椭圆曲线上的扭曲WZW模型是一种共形场论，它具有椭圆曲线上的非平凡平坦李代数丛作为规范对称。该模型的共形块所满足的线性微分方程，即椭圆Knizhnik-Zamolodchikov方程，其系数等于Belavin和Drinfeld的椭圆经典γ矩阵，椭圆Gaudin模型是作为某个自旋链模型的准经典极限而引入的量子可积系统.椭圆Gaudin模型的可换哈密顿量也用椭圆经典伽玛矩阵描述，实际上椭圆Gaudin模型在临界能级上可以与椭圆曲线上的扭曲WZW模型等同，因此二阶椭圆Gaudin哈密顿量的母函数可以由能量的Ward-Takahashi恒等式导出。第二，他构造了Knizhnik-Zamolodchikov-Bernard（KZB）方程解的可积表示，KZB方程是一个联络型线性微分方程，其系数由动力学椭圆经典gamma方程描述，它是一个线性微分方程，它的解是一个线性微分方程的解。算子，并可以确定的方程所满足的共形块的WZW模型定义在一个家庭的对一个点椭圆曲线和平坦的李代数丛。将Wakimoto模理论应用于方程的后一种解释，得到了方程解的可积表示，所得积分公式可看作是多元超几何函数的椭圆函数形式.

First the investigator constructed a twisted Wess-Zumino-Witten (WZW) model on elliptic curves and found an algebro-geometric interpretation of the elliptic Gaudin model.The twisted WZW model on elliptic curves is a conformal field theory which possesses certain non-trivial flat Lie algebra bundles on elliptic curves as gauge symmetry. Coefficients of the linear differential equations satisfied by conformal blocks of the model, the elliptic Knizhnik-Zamolodchikov equations, are equal to the elliptic classical gamma-matrices of Belavin and Drinfeld.The elliptic Gaudin model is the quantum integrable system introduced as a quasi-classical limit of a certain spin chain model. The commuting Hamiltonians of the model are also described by the elliptic classical gamma-matrices.In fact the elliptic Gaudin model can be identified with the twisted WZW model on elliptic curves at the critical level and hence the generating function of second-order elliptic Gaudin Hamiltonians can be derived from the Ward-Takahashi identity of the energy-momentum tensor defined by the Sugawara construction.Second he constructed integrable representations of solutions of Knizhnik-Zamolodchikov-Bernard (KZB) equations from the Wakimoto modules over an affine Lie algebra.The KZB equation is a linear differential equation of connection type with coefficients described by the dynamical elliptic classical gamma-operators and can be identified with the equation satisfied by the conformal blocks of the WZW model defined on a family of pairs of a pointed elliptic curve and a flat Lie algebra bundle. Applying the theory of the Wakimoto modules to the latter interpretation of the equation, we can obtain integrable representations of solutions of it. The integral formulas can be regarded as elliptic function versions of hypergeometric functions of several variables.