Commuting differential and difference operators associated to complex curves, I

Abstract

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.