Heterotic strings on $(K3\times T^2)/\mathbb{Z}_3$ and their dual Calabi-Yau threefolds

TL;DR

This paper explores heterotic string compactifications on a specific orbifold of K3 and T^2, identifies dual Calabi-Yau threefolds, and compares topological couplings and string amplitudes to establish dualities.

Contribution

It provides a detailed analysis of heterotic compactifications on $(K3 imes T^2)/bZ_3$, identifies dual Calabi-Yau geometries, and relates their topological string amplitudes to heterotic one-loop results.

Findings

Dual Calabi-Yau threefolds are $K3$ fibrations with genus one fibers and three-sections.

The intersection form on the polarization lattice is three times the Narain lattice form.

Heterotic and type IIA topological amplitudes match under the duality.

Abstract

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 $\Gamma^{1,1}$.