RNS and Pure Spinors Equivalence for Type I Tree Level Amplitudes Involving up to Four Fermions

In this paper we give a proof of the equivalence between the RNS and Pure Spinor formalism for Type I tree level amplitudes involving up to four fermions. This result have been obtained previously for amplitudes involving only closed or open amplitudes and here we extend it to include Type I amplitudes. For this we first prove the cyclic symmetry of Type I superstring in a way that it is also valid for the Pure Spinor formalism. The technique used is applied to simplify the calculation of the tree level three point amplitude previously computed. As a byproduct, we are able to calculate the gauge anomaly amplitude of type I Superstrings in this formalism.