The emergence of inherently 9-dimensional one-loop effective action from T-duality

TL;DR

This work shows that T-duality of circle-compactified ten-dimensional string theories naturally produces inherently nine-dimensional, loop-level couplings that cannot be obtained by simple KK reduction alone. By explicitly reducing the 10D IIA one-loop Chern–Simons term and pure gravity couplings to 9D and applying T-duality, the author constructs the corresponding IIB nine-dimensional actions in the small-radius limit and proves their invariance under S-duality with zero RR fields. A key finding is that the radius expansion of these couplings truncates to two distinct limits (large- and small-radius) with no interpolating corrections, and that a K3 reduction to five dimensions yields results consistent with heterotic on T^5, supporting a precise radius-dilaton mapping. Collectively, the results illuminate how duality symmetries constrain the structure of nine-dimensional effective actions and their M-/F-theory connections, providing a robust cross-check via heterotic dualities.

Abstract

Recent studies suggest that applying the Buscher rules to the dimensional reduction of ten-dimensional, one-loop effective actions generate "purely stringy" couplings in nine dimensions that cannot be lifted to a local, covariant form in ten dimensions. We investigate this phenomenon at order $α'^3$ in type IIA string theory. By computing the circular reduction of the one-loop Chern-Simons term and pure-gravity couplings in type IIA theory and applying the T-duality transformation to the resulting couplings, we derive their counterparts in the type IIB effective action. We demonstrate that the resulting nine-dimensional type IIB couplings are invariant under S-duality without requiring contributions from the tree-level effective action or non-perturbative effects. As a consistency check, we show that the nine-dimensional type IIB couplings, when reduced on a K3 surface, reproduce the known heterotic string couplings on \( T^5 \) at order \( α' \), via the duality between the two theories.