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 O(α ′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 nongauge 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 ∧ 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.0

Duality constraints on effective actions

“…Using T-duality and S-duality Ward identities, all four-point S-matrix elements of NSNS and RR vertex operators have been found and confirmed by explicit calculations [34,54,55]. The disk-level S-matrix element of two closed string vertex operators at four-momentum level in spacetime are given by the couplings (57), (64) and (65). It has been shown in [29,63,32] that these couplings satisfy the duality Ward identities, i.e., the couplings of two RR two-form and the couplings of two Kalb-Ramond can be written as [32] A s tree ∼…”

“…However, it is very hard to calculate such S-matrix element and to extract its four-momentum contact terms. See [64,65,66,67,68] for the calculations of the disk-level S-matrix element of one specific RR and two NSNS vertex operators from which some of the couplings of one RR and two NSNS fields at order α ′2 have been extracted. The disk-level S-matrix element of one arbitrary RR and two NSNS vertex operators have been calculated in [69].…”

“…Since the kinematic factor for O p -planes are exactly those for D p -branes, the two closed string couplings at order α ′2 are exactly given by (57), (64) and (65) in which the second fundamental form is set to zero, i.e., geodesic embedding. In fact the gravity part of the second fundamental form (63) has one index along the i-directions, as a result, it is projected out under the Z 2 transformation.…”

“…In fact the gravity part of the second fundamental form (63) has one index along the i-directions, as a result, it is projected out under the Z 2 transformation. There are many couplings of massless closed string fields at order α ′2 which do not appear in (57), (64) and (65). These couplings have also genus and non-perturbative corrections.…”

Off-shell D-brane action at orderα′2in flat spacetime

“…Either through its field strength H or through the replacement F → F + B. The massless poles in the Smatrix which are produced by the gauge fields, should be combined with some of the contact terms which are produced by F + B terms, to be able to rewrite the S-matrix element in terms of H [25,26]. Then the H-contact terms of the S-matrix element are the H-couplings in the effective action.…”