On Anomaly-free Supergravity as an Effective String Theory

The equations of motion of anomaly-free supergravity are shown to fulfil (to all orders in $\alpha'$) a differential condition corresponding to the one relating the Weyl anomaly coefficients for a non-linear sigma model representing a (heterotic) string propagating in a non-trivial background. This supports the possibility that anomaly-free supergravity could provide the complete massless effective theory for the heterotic string.