Effective action of bosonic string theory at order $α'^3$

TL;DR

The paper determines the classical bosonic string effective action at order $\alpha'^3$ by enforcing a higher-derivative extension of T-duality (Buscher rules) on circle-reduced backgrounds, in both Meissner and Metsaev-Tseytlin schemes. T-duality fixes all couplings up to a single parameter, which is then fixed by matching the four-graviton S-matrix's single-trace term to the string amplitude, yielding a final coefficient $a=\tfrac{1}{8}-\tfrac{9}{8}\zeta(3)$. The resulting gravity-only ($H=0$) sector agrees with the nonlinear sigma-model derivations of Riemann quartic terms, and, after field redefinitions to canonical form, the action reproduces known results and aligns with KLT predictions for higher-point amplitudes. The work also presents canonical forms of the $\alpha'^3$ couplings in both schemes, demonstrates consistency with sigma-model results, and discusses implications for DFT, heterotic extensions, and potential higher-order determinations.

Abstract

In this work, we derive the classical effective action of bosonic string theory at order $α'^{3}$ for the metric, Kalb-Ramond field, and dilaton by imposing a higher-derivative extension of the Buscher rules on the circular reduction of the minimal basis at this order, in the schemes where their corresponding actions at order $α'$ are the Meissner and the Metsaev-Tseytlin schemes. We find that T-duality fixes all coupling constants in terms of the known overall factor at order $α'$ and a single remaining parameter. This final parameter is determined by matching the single-trace term $\Tr(εεεε)$ in the four-graviton S-matrix element which lacks a massless pole, with the corresponding string theory amplitude. Our results for the Riemann quartic terms are in full agreement with those obtained from the nonlinear sigma-model approach.