On RR couplings on D-branes at order O(α′2)

Recently, it has been found that there are couplings of the RR field strength F-(p) and the B-field strength H on the world volume of D-p-branes at order O(alpha'(2)). These couplings which have both world-volume and transverse indices, are invariant under the linear T-duality transformations. Consistency with the nonlinear T-duality indicates that the RR field strength F-(p) in these couplings should be replaced by F-(p) = dC((p-1)) where C = e(B)C. This replacement, however, produces some non-gauge invariant terms. On the other hand, the nonlinear terms are invariant under the linear T-duality transformations at the level of two B-fields. This allows one to remove some of the nonlinear terms in F-(p). We fix this by comparing the nonlinear couplings with the S-matrix element of one RR and two NSNS vertex operators. Our results indicate that in the expansion of F-(p) one should keep only the B-field gauge invariant terms, e. g., B Lambda dC((p-3)) where both indices of B-field lie along the brane. Moreover, in this case one should replace B with B + 2 pi alpha' f to have the B-field gauge invariance.