Dimensional reduction of the M-theory Chern-Simons term at order $\ell_p^6$

Abstract: The dimensional reduction of M-theory couplings at order $\ell_p^6$ is known to produce one-loop $\alpha'^3$ corrections in type IIA string theory. In this paper, we perform the Kaluza-Klein reduction of the M-theory Chern-Simons coupling $t_8\epsilon_{11} A R^4$ at this order. By meticulously accounting for non-gauge-invariant total derivative terms, we derive the complete set of corresponding one-loop, gauge-invariant couplings in the type IIA effective action. Our results not only reproduce the standard Chern-Simons term $t_8\epsilon_{10} B R^4$ which is gauge invariant up to total derivatives, but also unveil a new set of gauge-invariant couplings involving RR and NS-NS field strengths. To validate our findings, we test their consistency under string dualities. We dimensionally reduce the derived type IIA couplings on a K3 manifold and show that the resulting one-loop $\alpha'$ corrections in six dimensions transform under S-duality into the dimensional reduction of the tree-level heterotic string Chern-Simons coupling $H_{\mu\nu\alpha} \Omega^{\mu\nu\alpha}$ on $T^4$. This non-trivial agreement provides strong evidence for the correctness of both the M-theory Chern-Simons term and its reduction to type IIA.