On S-duality of D₃-brane S-matrix

There is a conjecture in the literature that indicates the tree-level S-matrix elements of graviton become symmetric under the $SL(2,Z)$ transformation after including the loops and the nonperturbative effects. Using the Ward identity corresponding to the global S-duality transformations, this conjecture can be extended to other S-matrix elements as well. While the $SL(2,Z)$ transformation on the background dilaton is nonlinear, the Ward identity dictates the S-duality transformations on external states should be the linearized $SL(2,R)$ transformations. We examine in details various S-matrix elements involving massless closed string and/or open string vertex operators on the world volume of D$_3$-brane in favor of this conjecture.