Inverse of the String Theory KLT Kernel

TL;DR

The paper extends the KLT framework to string theory by introducing α'-corrected bi-adjoint amplitudes m_{α'}(β|β~) that yield a compact, exact diagrammatic representation using sine and tangent propagators. It develops separate diagrammatic rules for off-diagonal and diagonal cases, revealing a structured α' dependence and a potential Catalan-number pattern in higher-point diagonal amplitudes. By treating m_{α'} as the inverse KLT kernel, the authors derive string KLT relations for open and closed strings, showcase explicit n=4–6 examples (recovering Virasoro-Shapiro among others), and derive soft-limit behavior consistent with gravity. They further demonstrate basis expansions of open-string amplitudes in BCJ-like bases via m_{α'}, and discuss connections to CHY, Z-objects, and broader algebraic structures, pointing to rich future directions. Overall, the work provides a practical and conceptually illuminating framework for α'-corrected amplitude relations and basis transformations in string theory.

Abstract

The field theory Kawai-Lewellen-Tye (KLT) kernel, which relates scattering amplitudes of gravitons and gluons, turns out to be the inverse of a matrix whose components are bi-adjoint scalar partial amplitudes. In this note we propose an analogous construction for the string theory KLT kernel. We present simple diagrammatic rules for the computation of the $α'$-corrected bi-adjoint scalar amplitudes that are exact in $α'$. We find compact expressions in terms of graphs, where the standard Feynman propagators $1/p^2$ are replaced by either $1/\sin (πα' p^2/2)$ or $1/\tan (πα' p^2/2)$, as determined by a recursive procedure. We demonstrate how the same object can be used to conveniently expand open string partial amplitudes in a BCJ basis.