F-theory duals of singular heterotic K3 mode

TL;DR

This paper constructs F-theory duals of singular heterotic K3 models, revealing detailed geometric and gauge structure correspondences, especially for orbifolds with specific singularities, and clarifies their interrelations.

Contribution

It provides a detailed Weierstrass description of heterotic orbifold models and uses it to determine F-theory Calabi-Yau structures matching gauge and matter spectra.

Findings

Weierstrass models for heterotic orbifolds with specific singularities.

Correlation between orbifold order and Dynkin diagram multiplicities.

Explicit construction of F-theory duals matching heterotic gauge groups.

Abstract

We study F-theory duals of singular heterotic K3 models that correspond to abelian toroidal orbifolds $T^4/Z_N$. While our focus is on the standard embedding, we also comment on models with Wilson lines and more general gauge embeddings. In the process of constructing the duals, we work out a Weierstrass description of the heterotic toroidal orbifold models, which exhibit singularities of Kodaira type $I_0^*, IV^*, III^*$, and $II^*$. This construction unveils properties like the instanton number per fixed point and a correlation between the orbifold order and the multiplicities in the Dynkin diagram. The results from the Weierstrass description are then used to restrict the complex structure of the F-theory Calabi-Yau threefold such that the gauge group and the matter spectrum of the heterotic theories are reproduced. We also comment on previous approaches that have been employed to construct the duality and point out the differences to our case. Our results show explicitly how the various orbifold models are connected and described in F-theory.