S-duality of S-matrix

TL;DR

This paper explores the invariance of D-brane S-matrix elements under S-duality, proposing an $SL(2,Z)$-covariant framework that relates different branes and strings, and derives consistent effective actions.

Contribution

It extends the understanding of S-matrix invariance under S-duality to D-branes and fundamental strings, proposing an $SL(2,Z)$-covariant formulation and deriving related effective actions.

Findings

S-matrix elements on D3-brane are compatible with S-duality.

S-matrix elements on D1 and D5-branes extend to $SL(2,Z)$-covariant form.

F1-string and NS5-brane S-matrix elements relate to D1 and D5-branes via S-duality.

Abstract

Recently, it has been observed that the kinematic factor of the disk-level S-matrix element of two RR two-forms and the disk-level S-matrix element of two B-fields on the world volume of D$_3$-brane are compatible with the standard rules of S-duality. Inspired by this observation, we speculate that the S-matrix elements on the world volume of D$_3$-brane are invariant under the $SL(2,Z)$ transformation. Compatibility with S-duality requires the S-matrix elements on the world volume of D$_1$-brane and D$_5$-brane to be extended to the $SL(2,Z)$-covariant form. In particular, this implies the S-matrix elements on the world volume of F$_1$-string and NS$_5$-brane at strong coupling to be related by S-duality to the disk-level S-matrix elements of D$_1$-string and D$_5$-brane, respectively. %The S-matrix elements then dictate that both F$_1$-string and NS$_5$-brane have D$_1$-string excitations. The contact terms of these S-matrix elements at order $O(\alpha'^0)$ produce a Born-Infeld and Chern-Simons type effective action for both F$_1$-string and NS$_5$-brane. They are consistent with the $SL(2,R)$-covariant action appears in the literature.