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

Abstract

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 $\mathcal{O}\left( {{\alpha^{{\prime}2}}} \right)$. 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 ${\mathcal{F}^{(p)}} = d{\mathcal{C}^{\left( {p - 1} \right)}}$ where $\mathcal{C} = {e^B}C$. 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 ${\mathcal{F}^{(p)}}$. 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 ${\mathcal{F}^{(p)}}$ one should keep only the B-field gauge invariant terms, e.g., B ∧ dC ^(p−3) where both indices of B-field lie along the brane. Moreover, in this case one should replace B with B + 2πα′f to have the B-field gauge invariance.