The circle compactification of M-theory is dual to type IIA string theory, requiring that the dimensional reduction of the M-theory couplings \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\((t_8 t_8 - \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\frac{1}{4} \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\epsilon_8 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\epsilon_8) R^4\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\) must reproduce the type IIA one-loop effective action at order \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\(\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\alpha\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'^3\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\), including contributions from the metric, dilaton, and RR one-form. Through compactification, we obtain 1,276 couplings involving Riemann, Ricci, and Ricci scalar tensors, along with first and second derivatives of the dilaton and RR one-form. By employing field redefinitions, we reduce these to a basis of 359 independent couplings. Crucially, we observe that the dilaton cannot be entirely removed from the couplings via field redefinitions, even in the pure metric-dilaton sector. We validate our results by showing exact agreement between all four-field couplings and the corresponding string-theory S-matrix elements in the string frame. Further, upon compactifying on K3, we demonstrate that the resulting six-dimensional \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\(\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\alpha\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\) couplings at one-loop level transform under S-duality into the tree-level \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\(\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\alpha\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\) couplings of heterotic string theory on \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\(T^4\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\). This match necessitates carefully chosen field redefinitions for both the type IIA (on K3) and heterotic (on \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\(T^4\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\)) sectors, providing a stringent test of the duality.

Keywords