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 Dp-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.