String multi-loop corrections to equations of motion

The recently developed operator formalism on Riemann surfaces is used to study nonlinear sigma models on higher genus surfaces. Short-distance divergences due to small handles are evaluated with handle operators which implement the higher genus periodicity in the operator formalism. Self-energy corrections from multi-loop amplitudes are explicitly shown to exponentiate and to give loop corrections to anomalous dimensions. The Schottky parametrization is found to be useful in extracting the short-distance divergences. (orig.)