On T-duality of R 2 -corrections to DBI action at all orders of gauge field

Recently, it has been observed that in a t-duality invariant world-volume theory in flat spacetime, all orders of gauge field strength and all orders of the d-brane velocity appear in two specific matrices. using these two matrices, we construct the world-volume couplings of two massless nsns states at order α ′2 and all orders of the velocity and the gauge field strength, by requiring them to be invariant under the linear t-duality. the standard extension f → f + p[b], then produces all orders of the pull-back of b-field into the action. we compare the resulting couplings for zero velocity and gauge field strength, with the α ′2 terms of the disk-level s-matrix element of two massless nsns vertex operators in the presence of a constant background b-field. we have found an exact agreement.