Operator Formalism for Bosonic Beta-Gamma Fields on General Algebraic Curves

An operator formalism for bosonic $\beta-\gamma$ systems on arbitrary algebraic curves is introduced. The classical degrees of freedom are identified and their commutation relations are postulated. The explicit realization of the algebra formed by the fields is given in a Hilbert space equipped with a bilinear form. The construction is based on the "gaussian" representation for $\beta-\gamma$ systems on the complex sphere [Alvarez-Gaum\' e et al, Nucl. Phys. B 311 (1988) 333]. Detailed computations are provided for the two and four points correlation functions.