The Ten-dimensional Effective Action of Strongly Coupled Heterotic String Theory

TL;DR

This work derives the ten-dimensional effective action for the strongly coupled $E_8\times E_8$ heterotic string from M-theory on $S^1/Z_2$, showing that heavy Kaluza-Klein modes—generated by orbifold-fixed-plane sources—must be integrated out to achieve a consistent reduction. The authors develop a controlled procedure to incorporate these modes, then compute the parity-odd quartic invariants in 10D, including the $R^4$, $R^2F^2$, and $F^4$ terms and the Green-Schwarz mechanism, obtaining results that precisely match the one-loop weakly coupled theory. They demonstrate that anomaly cancellation and SUSY fixed the structure and coefficients of these invariants (with $c'=c^2$) and that the non-renormalization of these terms helps explain the identical form of the 4D effective actions across strong and weak coupling limits, modulo parameter differences. Overall, the paper provides a systematic, boundary-aware dimensional-reduction framework for generating higher-derivative, supersymmetric invariants in the heterotic context.

Abstract

We derive the ten-dimensional effective action of the strongly coupled heterotic string as the low energy limit of M-theory on S^1/Z_2. In contrast to a conventional dimensional reduction, it is necessary to integrate out nontrivial heavy modes which arise from the sources located on the orbifold fixed hyperplanes. This procedure, characteristic of theories with dynamical boundaries, is illustrated by a simple example. Using this method, we determine a complete set of R^4, F^2R^2, and F^4 terms and the corresponding Chern-Simons and Green-Schwarz terms in ten dimensions. As required by anomaly cancelation and supersymmetry, these terms are found to exactly coincide with their weakly coupled one-loop counterparts.