Recursion Relations from Space-time Supersymmetry

TL;DR

The paper develops a SUSY- and duality-based framework to derive recursion relations among protected higher-derivative couplings in the type IIB string theory effective action, showing that couplings of the form $\hat{G}^{2k}\lambda^{16}$ satisfy coupled Poisson equations on the moduli space with sources from lower-order terms. This structure implies a perturbative non-renormalization property: each coupling receives only a finite number of perturbative contributions, complemented by non-perturbative corrections, and persists consistently to large $k$ through a richer modular-form content. For small $k$, the equations reduce to Laplace eigenvalue problems with known solutions (e.g., $f^{(0,0)}_0 = E_{3/2}$ and $f^{(0,0)}_2 = E_{5/2}$), while at higher orders the framework predicts multiple modular forms per weight due to distinct space-time structures and source terms. Overall, the work reveals a deep interplay between modular forms, SL$(2,\mathbb{Z})$ duality, and space-time supersymmetry, providing a non-perturbative organization of protected couplings and guiding expectations for their perturbative spectra and instanton contributions.

Abstract

We describe a method for obtaining relations between higher derivative interactions in supersymmetric effective actions. The method extends to all orders in the momentum expansion. As an application, we consider the string coupling dependence of the \hat{G}^{2k} λ^{16} interaction in type IIB string theory. Using supersymmetry, we show that each of these interactions satisfies a Poisson equation on the moduli space with sources determined by lower momentum interactions. We argue that these protected couplings are only renormalized by a finite number of string loops together with non-perturbative terms. Finally, we explore some consequences of the Poisson equation for low values of k.