  We present analytical implementation of conformal field theory on a compact
Riemann surface. We consider statistical fields constructed from background
charge modifications of the Gaussian free field and derive Ward identities
which represent the Lie derivative operators in terms of the Virasoro fields
and the puncture operators associated with the background charges. As
applications, we derive Eguchi-Ooguri's version of Ward's equations and certain
types of BPZ equations on a torus.
