Anomalies of the products of vertex functions in the superstring theories

We obtain the two-dimensional anomalous transforms of the products of several vertex functions (of Neveu-Schwarz sector) in the path integral formulation of closed superstrings: Type II super and heterotic strings. The products of vertex functions are free from local anomalies except for Weyl anomalies. In the presence of background fields this kind of Weyl anomalies leads to interaction terms in the background field equations of motion, which are derived by the requirement of the cancellation of Weyl anomalies. (orig.)