On the exact quartic effective action for the type IIB superstring

TL;DR

This work constructs an explicit $SL(2,\mathbb{Z})$-invariant ten-dimensional four-point effective action for Type IIB superstring, incorporating the graviton, dilaton, axion, and antisymmetric fields while excluding the self-dual four-form. The authors show that perturbative NS-NS corrections terminate at one loop and that non-perturbative contributions arise from D-instantons, encapsulated by non-holomorphic modular functions $f_k(\tau,\bar{\tau})$ that weight higher-derivative terms such as $R^4$. The action is built by replacing simple derivatives with covariant objects $P_M$, $G_{KMN}$ and employing $f_k(\tau,\bar{\tau})$ to restore $SL(2,\mathbb{Z})$ invariance, ensuring consistency with known tree-level amplitudes and recent compactification checks. A key consistency check uses a Type II/ K3 result, and the paper outlines how further tests (e.g., determining coefficients $c_k$) could solidify the proposal. Overall, the work provides a coherent, duality-respecting framework for the exact four-point effective action in Type IIB theory with controlled perturbative and non-perturbative structure.

Abstract

We propose a four-point effective action for the graviton, antisymmetric two-forms, dilaton and axion of type IIB superstring in ten dimensions. It is explicitly SL(2,Z)-invariant and reproduces the known tree-level results. Perturbatively, it has only one-loop corrections for the NS-NS sector, generalizing the non-renormalization theorem of the R^4 term. Finally, the non-perturbative corrections are of the expected form, namely, they can be interpreted as arising from single D-instantons of multiple charge.