The Supersymmetric Effective Action of the Heterotic String in Ten Dimensions

TL;DR

The paper tackles the problem of supersymmetrizing the quartic corrections in the ten-dimensional heterotic string effective action, focusing on $R^4$-type terms. Using a component-field Noether method, it derives two independent locally supersymmetric invariants, $I_1$ and $I_2$, whose bosonic sectors are combinations of the known tree-level and one-loop $R^4$ structures expressed via $X, Y_1, Y_2, Z$, and shows that the dilaton prefactor must vanish ($y=0$). A corresponding Yang–Mills invariant emerges, generalizing to $(R^2+ ext{tr}F^2)^2$, while the tree-level $\zeta(3)R^4$ term cannot be supersymmetrized within the considered field content. The results align with Green–Schwarz anomaly-cancellation structure through $B\wedge R\wedge R\wedge R\wedge R$ (and $B\wedge F^4$) terms and illuminate how SUSY constraints connect bosonic amplitudes to higher-derivative counterterms in the heterotic string. Overall, the work clarifies which quartic curvature terms admit supersymmetric completion and how these invariants interplay with anomaly cancellation and YM couplings.

Abstract

We construct the supersymmetric completion of quartic $R+R^4$-actions in the ten-dimensional effective action of the heterotic string. Two invariants, of which the bosonic parts are known from one-loop string amplitude calculations, are obtained. One of these invariants can be generalized to an $R+F^2+F^4$-invariant for supersymmetric Yang-Mills theory coupled to supergravity. Supersymmetry requires the presence of $B\wedge R\wedge R\wedge R\wedge R$-terms, ($B\wedge F\wedge F\wedge F\wedge F$ for Yang-Mills) which correspond to counterterms in the Green-Schwarz anomaly cancellation. Within the context of our calculation the $ζ(3)R^4$-term from the tree-level string effective action does not allow supersymmetrization.