$SL(2,Z)$ Self-duality of Super D3-brane Action on $AdS_5 \times S^5$

TL;DR

This paper demonstrates that the supersymmetric D3-brane action in the $AdS_5 imes S^5$ background exhibits $SL(2,Z)$ self-duality, extending known flat space dualities to curved backgrounds and suggesting their general applicability.

Contribution

It proves the $SL(2,Z)$ invariance of the D3-brane action in $AdS_5 imes S^5$, showing duality transformations are valid even in curved space-time backgrounds.

Findings

D3-brane action is self-dual under $SL(2,Z)$ transformations.

Duality relations in flat space extend to curved backgrounds.

Background geometry does not break the $SL(2,Z)$ duality symmetry.

Abstract

It is shown that a supersymmetric and $\kappa$-symmetric D3-brane action on $AdS_5 \times S^5$ is mapped into itself by a duality transformaion, thereby verifying the $SL(2,Z)$ invariance of the D3-brane action in the $AdS_5 \times S^5$ background as in the flat background. To this end, we fix the $\kappa$-symmetry in a gauge which simplifies the classical action in order to perform an SO(2) rotation of the N=2 spinor index in a manifest way, though this may not be necessary. This situation is the same as the case of a super D-string on $AdS_5 \times S^5$ where it was shown that the super D-string action is transformed to a form of the IIB Green-Schwarz superstring action with the $SL(2,Z)$ covariant tension in the $AdS_5 \times S^5$ background through a duality transformation. These results strongly suggest that various duality relations originally found in the flat background may be independent of background geometry, in other words, the duality transformations in string and p-brane theories may exist even in general curved space-time.