Type IIA supergravity at one loop: $α'^3$ terms in the metric-dilaton-RR one-form sector

TL;DR

This work derives the Type IIA one-loop $\alpha'^3$ corrections in the metric-dilaton-RR one-form sector by dimensional reduction of the M-theory eight-derivative $ (t_8 t_8 - \frac{1}{4!}\epsilon_{11}\epsilon_{11})R^4$ term on a circle. Starting from 1,276 couplings in 10D, the authors construct a minimal basis of 377 eight-derivative dilaton-gravity-RR couplings, of which 359 are non-zero; crucially, the dilaton cannot be removed by field redefinitions, even in the pure gravity-dilaton sector, and the four-field couplings reproduce known string-frame S-matrix elements. Upon reducing on K3, the resulting six-dimensional one-loop $\alpha'$ terms map under S-duality to the tree-level $\alpha'$ couplings of heterotic theory on $T^4$, provided appropriate field redefinitions are employed in both theories. This yields a stringent test of IIA/heterotic duality in the presence of higher-derivative corrections and clarifies how dilaton and RR couplings behave under duality at the one-loop level. Overall, the paper demonstrates a consistent, duality-respecting structure for $\alpha'^3$ corrections across dimensions and string theories, enriching our understanding of M-theory reductions and their holographic and phenomenological implications.

Abstract

The circle compactification of M-theory is dual to type IIA string theory, requiring that the dimensional reduction of the M-theory couplings \((t_8 t_8 - \frac{1}{4} ε_8 ε_8) R^4\) must reproduce the type IIA one-loop effective action at order \(α'^3\), 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 \(α'\) couplings at one-loop level transform under S-duality into the tree-level \(α'\) 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.