R^4, purified

TL;DR

This paper derives the complete four-point, tree-level effective action for type II superstrings to all orders in $α'$ using the pure-spinor formalism, including fermions. It reveals a quartic factorization ${\cal L}_{\text{SUGRA}}={\cal L}_{\text{SYM}}\otimes\widetilde{{\cal L}}_{\text{SYM}}$ and derives a fully covariant closed-string vertex operator to all orders in $\θ$, grounded in Kawai–Lewellen–Tye relations between open and closed strings. The closed-string action is organized into NS-NS and RR sectors with a torsionful connection $\widehat{R}$ and modified RR strengths $\widehat{F}$, producing amplitudes that factorize as $K_{SS}\otimes\tilde{K}_{SS}$ and yield terms like $t_8t_8\widehat{R}^4$, $(∂F)^2R^2$, and $(∂F)^4$, plus fermionic couplings. The results confirm and extend previous partial analyses, connect to the linearized superfield predictions, and provide a covariant framework for higher-derivative corrections with potential implications for black-hole physics and M-theory extensions.

Abstract

We derive, using the pure-spinor formalism, the complete -- including the fermions -- four-point effective action of both type II superstrings to all orders in $α'$, at tree level in string loops. We find that, in the quartic-field approximation, the supergravity Lagrangian can be thought of as the tensor product, in a suitable sense, of two copies of the superYang-Mills Lagrangian in ten dimensions. The NS-NS three-form enters the supergravity Lagrangian through a modified connection with torsion. As a byproduct, we derive the complete, i.e. to all orders in the $θ$-expansion, closed-string vertex operator in a flat target-space background.