OPERATOR-FORMALISM ON GENERAL ALGEBRAIC-CURVES

The usual Laurent expansion of the analytic tensors on the complex pla ne is generalized to any closed and orientable Riemann surface represe nted as an affine algebraic curve, As an application, the operator for malism for the b - c systems is developed. The physical states are exp ressed by means of creation and annihilation operators as in the compl ex plane and the correlation functions are evaluated starting from sim ple normal ordering rules. The Hilbert space of the theory exhibits an interesting internal structure, being splitted into n (n is the numbe r of branches of the curve) independent Hilbert spaces. In this way we are able to realize new kinds of conformal field theories at genus ze ro with symmetry group Vir(n) x G, Vir being the Virasoro group and G denoting a discrete and nonabelian crystallographic group. Exploiting the operator formalism a large collection of explicit formulas of stri ng theory is derived. Finally, we develop as an important byproduct ne w methods in order to handle differential equations related to monodro my, like the Riemann monodromy problem.