On S-duality of D$_3$-brane S-matrix

TL;DR

This paper investigates the S-duality symmetry of D3-brane S-matrix elements, showing that linearized S-duality transformations govern external states and supporting the conjecture of S-matrix invariance under SL(2,Z).

Contribution

It provides a detailed analysis of S-matrix elements on D3-branes, confirming the role of linearized S-duality transformations in the context of the conjectured SL(2,Z) symmetry.

Findings

S-matrix elements are consistent with linearized S-duality transformations.

Ward identities support the extension of S-duality to external states.

The analysis reinforces the conjecture of SL(2,Z) invariance of D3-brane S-matrix.

Abstract

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.