Heterotic moduli, the double extension and the alpha'^2 metric

We compute the heterotic moduli-space metric through ${\alpha }^{\mathrm{\prime }2}$ for backgrounds admitting a smooth ${\alpha }^{\mathrm{\prime }}\to 0$ limit. The Kaehler potential is unchanged to this order, but the metric receives corrections from the deformation of the Hull connection which mixes complex structure and hermitian moduli. We discuss this and clarify the roles of the extension bundle, F-terms, the D-terms and the string-derived moduli space metric.