Complex Divisors on Algebraic Curves and Some Applications to String Theory ∗

This talk presents some new notions of the theory of complex algebraic curves which have appeared as algebraic tools in string theory (see [4] for more details). In a sense, we have materialized non-existed complex powers of invertible sheaves on algebraic curves introduced at the level of the Atiyah algebras of invertible sheaves by Beilinson and Schechtman [1]. The Atiyah algebra AL in the case of an invertible sheaf L over a complete complex algebraic curve X is just the sheaf of differential operators of order ≤ 1 on L. There takes place the exact sequence 0 → O → AL → T → 0, where O is the structural sheaf and T is the tangent sheaf of X. The diagram