ON THE INVARIANCE OF THE STRING PARTITION-FUNCTION UNDER DUALITY

We consider the N = 1 supersymmetric sigma-model and we examine the tr ansformation properties of the partition function under target-space d uality. Contrary to what one would expect, we find that it is not, in general, invariant. In fact, besides the dilaton shift emerging from t he Jacobian of the duality transformation of the bosonic part, there a lso exist a Jacobian for the fermionic part since fermions also transf orm under the duality process. The latter is just the parity of the sp in structure of the word sheet and since it cannot be compensated, the dual theory is not equivalent to the original one.