On T-duality of R2-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 alpha'(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 alpha'(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.