Covariant calculation of the partition function of the two-dimensional sigma model on compact two-surfaces

Motivated by string theory connection, a covariant procedure for perturbative calculation of the partition function of the two-dimensional generalized $\sigma$-model is considered. The importance of a consistent regularization of the measure in the path integral is emphasized. The partition function is computed for a number of specific 2-manifolds: sphere, disk and torus.