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

首先，研究者在椭圆曲线上构造了扭曲的Wess-Zumino-Witten （WZW）模型，并对椭圆Gaudin模型进行了代数-几何解释。椭圆曲线上的扭曲WZW模型是一个共形场论，它在椭圆曲线上具有一定的非平凡平坦李代数束作为规范对称。椭圆型Knizhnik-Zamolodchikov方程的共形块所满足的线性微分方程的系数等于Belavin和Drinfeld的椭圆型经典伽马矩阵。椭圆型Gaudin模型是作为某自旋链模型的准经典极限引入的量子可积系统。该模型的交换哈密顿量也用椭圆型经典伽玛矩阵来描述。事实上，椭圆型Gaudin模型可以在椭圆曲线的临界能级上用扭曲的WZW模型识别，因此二阶椭圆型Gaudin hamilton的生成函数可以由Sugawara构造定义的能量动量张量的Ward-Takahashi恒等式导出。其次，从仿射李代数上的Wakimoto模构造了Knizhnik-Zamolodchikov-Bernard （KZB）方程解的可积表示。KZB方程是一个用动态椭圆经典伽马算子描述系数的连接型线性微分方程，它可以用定义在尖椭圆曲线和平坦李代数束的对族上的WZW模型的共形块所满足的方程来识别。将Wakimoto模的理论应用到方程的后一种解释中，我们可以得到方程解的可积表示。积分公式可以看作是多变量超几何函数的椭圆函数版本。

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.