Transformation of the Dirac-Born-Infeld action under the Seiberg-Witten map

We explicitly evaluate the disk S-matrix elements of one closed string and an arbitrary number of open string states in the presence of a large background B-flux. From this calculation, we show that in the world-volume action of D-branes in terms of non-commutative fields, the closed string fields must be treated as functionals of the non-commutative gauge fields. We also find the generalized multiplication rule *_N between N open string fields on the world-volume of the D-brane. In particular, this result indicates that the difference between the familiar * and the *_N product is just some total derivative terms. We show that the *_N product and the dependence of the closed string fields on the non-commutative gauge fields emerge also from transforming the ordinary Dirac-Born-Infeld action(including the closed string fields) under the Seiberg-Witten map. We then conjecture a non-commutative DBI action for the transformation of the commutative DBI action under the SW map.