  The usual Laurent expansion of the analytic tensors on the complex plane is
generalized to any closed and orientable Riemann surface represented as an
affine algebraic curve. As an application, the operator formalism for the $b-c$
systems is developed. The physical states are expressed by means of creation
and annihilation operators as in the complex plane and the correlation
functions are evaluated starting from simple normal ordering rules. The Hilbert
space of the theory exhibits an interesting internal structure, being splitted
into $n$ ($n$ is the number of branches of the curve) independent Hilbert
spaces. Exploiting the operator formalism a large collection of explicit
formulas of string theory is derived.
