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

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