A note on the quartic effective action of type IIB superstring

TL;DR

The paper reformulates the quartic effective action of type IIB supergravity at order $\alpha'^3$ in a manifest $SL(2,\mathbb{R})$ covariant form, enabling a direct comparison with a longstanding $SL(2,\mathbb{Z})$ invariant conjecture. It expresses the action in terms of covariant field strengths $P$ and $G$, clarifying the $U(1)$ charge structure and distinguishing operators beyond the standard $t_8t_8$ tensor. The authors show clear discrepancies with the Khuri proposal at tree level, arguing that Khuri's $SL(2,\mathbb{Z})$-invariant Lagrangian cannot be correct, and outline how the full $SL(2,\mathbb{Z})$ extension might be pursued via prior methods. The work also provides a pathway to reconstruct all-order $\alpha'$ dependence via a right-acting operator and sets the stage for completing the quartic action, including RR sector effects, in a covariant framework.

Abstract

We recast the result of our recent computation of the quartic action of ten-dimensional IIB supergravity in a manifestly SL(2,R)-covariant form. This affords us a critical assessment of an earlier conjecture in the literature.