  In this paper we study compactifications of the ${\cal N}=2$ heterotic
$E_8\times E_8$ string on $(K3\times T^2)/\mathbb{Z}_3$ with various gauge
backgrounds and calculate the topological couplings in the effective
supergravity action that arise from one-loop amplitudes. We then identify
candidates for dual type IIA compactifications on Calabi-Yau threefolds and
compare the heterotic results with the corresponding topological string
amplitudes. We find that the dual Calabi-Yau geometries are $K3$ fibrations
that are also genus one fibered with three-sections. Moreover, we show that the
intersection form on the polarization lattice of the $K3$ fibration has to be
three times the intersection form on the Narain lattice $\Gamma^{1,1}$.
