‎It is known that D$_p$-brane effective action at the leading order of $\\alpha\'$ in flat space-time which is given by DBI action‎, ‎transforms to D$_{p-1}$-brane effective action under standard T-duality transformations of the open string gauge bosons and transverse scalar fields‎. ‎Extending this duality to order $\\alpha\'$‎, ‎one may find corrections to the DBI action which include the second fundamental form $\\Omega$ and the covariant derivative of gauge field strength $DF$‎, ‎as well as the corrections to the T-duality transformations‎. ‎Using this idea‎, ‎up to two parameters‎, ‎we have found all 81 covariant couplings of $D FD F$ and $\\Omega\\Omega$ with zero‎, ‎two‎, ‎four and six $F$\'s‎. ‎The four gauge field couplings that the T-duality constraint fixes are consistent with the known couplings in the literature‎.

Keywords