Operator Formalism on the $Z_n$ Symmetric Algebraic Curves

In this work, the following conjectures are proven in the case of a Riemann surface with abelian group of symmetry: a) The $b-c$ systems on a Riemann surface $M$ are equivalent to a multivalued field theory on the complex plane if $M$ is represented as an algebraic curve; b) the amplitudes of the $b-c$ systems on a Riemann surface $M$ with discrete group of symmetry can be derived from the operator product expansions on the complex plane of an holonomic quantum field theory a la Sato, Jimbo and Miwa. To this purpose, the solutions of the Riemann-Hilbert problem on an algebraic curve with abelian monodromy group obtained by Zamolodchikov, Knizhnik and Bershadskii-Radul are used in order to expand the $b-c$ fields in a Fourier-like basis. The amplitudes of the $b-c$ systems on the Riemann surface are then recovered exploiting simple normal ordering rules on the complex plane.