THE NONCRITICAL BOSONIC AND SPINNING STRING IN BACKGROUND FIELDS

Using the Ward-Takahashi identify of the B.R.S.T current of the bosoni c and spinning string in background fields, we incorporate the Liouvil le action in the formal path integral construction of the partition fu nction. This construction is possible at one-loop order with respect t o string and Liouville fields and for a background fields satisfying t he equations of motion. In these conditions, the results of the standa rd perturbative treatment show that the central charge term of the ope rator product expansion of two stress tensors vanishes at d < 26 for t he bosonic string and at d < 10 for the spinning string.