与复曲线相联系的可交换微分和差分算子，
Commuting differential and difference operators associated to complex curves, I
摘要：
通过曲线X上共形块的Cartan元引入twists，对应于一个李代数g.我们证明了这些扭曲定义了全纯函数，具有类似θ的行为，在其雅可比矩阵J（X）^r的副本的乘积上。然后，我们参数化的共形块的扭曲的相关函数的幂零电流，在费金-Stoyanovsky的精神。我们根据这些相关函数计算Sugawara张量的作用量。其结果是一族作用于X的雅可比矩阵的乘积J（X）^r \times \prod_{i} S^{n_i}X上的形式的算子，其具有对称幂。这些算子通过雅可比方向上的微分以及对称幂的变量中的求值和留数来起作用。它们用来表示KZB联络，我们将其视为具有标记a-圈的曲线空间上的联络。这些运算符在电平很关键时也有意义。在这种情况下，他们彼此通勤。我们解释他们的联系与贝林森-德林费尔德运营商。在本文的续集中，我们将做一个类似的计算在q变形的情况下，取代列入的包络代数的李代数的正则函数在X上的函数的正式盘上的一些列入准霍普夫代数，这是介绍了工作中的一个我们和V. Rubtsov。其结果是一个交换家庭的差异评估运营商，这可能被视为在理性的情况下，作为Bethe anomaly制定的qKZ运营商。
英文摘要：
We introduce twists by Cartan elements of conformal blocks on a curve X, corresponding to a Lie algebra g. We show that these twists define holomorphic functions, with theta-like behaviour, on a product of copies of its Jacobian J(X)^r. We then parametrise the conformal blocks by their twisted correlation functions of nilpotent currents, in the spirit of Feigin-Stoyanovsky. We compute the action of the Sugawara tensor in terms of these correlation functions. The result is a family of operators acting on forms on the product J(X)^r \times \prod_{i} S^{n_i}X of the Jacobian of X with its symmetric powers. These operators act by differentiation in the Jacobian directions, and evaluations and residues in the variables of the symmetric powers. They serve to express the KZB connection, which we view as a connection on the space of curves with marked a-cycles. These operators also make sense when the level is critical. In that case, they commute with each other. We explain their connection with the Beilinson-Drinfeld operators. In a sequel to this paper, we will do a similar computation in the q-deformed case, replacing the inclusion of the enveloping algebras of the Lie algebra of regular functions on X in that of functions on the formal disc by some inclusion of quasi-Hopf algebras, which were introduced in work of one of us and V. Rubtsov. The outcome is a commuting family of difference-evaluation operators, which may be viewed in the rational case as the Bethe ansatz formulation of the qKZ operators.
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