classical correlation functions of liouville vertex operators on riemann surfaces with genus g>1

The classical correlation function of liouville vertex operators on a riemann surface with genus g>1 is related to the on-shell value of the liouville action functional on the same riemann surface but with the insertion of conical points at the location of those operators. in this work, using the results of [1,2], we study the appropriate classical liouville action on a riemann orbisurface using the schottky global coordinates. we also study the first and second variation formulas for this action on the schottky deformation space and show that this classical liouville action is a kähler potential for a special combination of weil-petersson metric and takhtajan-zograf metrics which appears in the local index theorem for riemann orbisurfaces [2].