A Polyakov action on Riemann surfaces. Pt. 2

The model independent study of the Polyakov action is continued on an arbitrary compact surface without boundary of genus larger than 2 as the general solution of the relevant conformal Ward identity. A general formula for the Polyakov action and an explicit calculation of the energy-momentum tensor density is provided. The general geometric setting of the construction is described in detail. It is shown that the Polyakov action defines a distribution of finite dimensional directions in the holomorphic tangent bundle of the manifold of Beltrami differentials. It is further argued that motions parallel to such distribution correspond to Polyakov's SL(2,C) symmetry transformations. Owing to the existence of renormalization ambiguities on a topologically non-trivial surface, the energy-momentum tensor needs not be invariant under the full SL(2,C) symmetry. The residual SL(2,C) symmetry is characterized geometrically. (author) 31 refs