## Abstract

We study aspects of Heterotic/F-theory duality for compactifications with Abelian discrete gauge symmetries. We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with *n*-sections, associated with the Tate-Shafarevich group \( {\mathbb{Z}}_n \). Such models are obtained by studying first a specific toric set-up whose associated Heterotic vector bundle has structure group \( {\mathbb{Z}}_n \). By employing a conjectured Heterotic/F-theory mirror symmetry we construct dual geometries of these original toric models, where in the stable degeneration limit we obtain a discrete gauge symmetry of order two and three, for compactifications to six dimensions. We provide explicit constructions of mirror-pairs for symmetric examples with \( {\mathbb{Z}}_2 \) and \( {\mathbb{Z}}_3 \), in six dimensions. The Heterotic models with symmetric discrete symmetries are related in field theory to a Higgsing of Heterotic models with two symmetric abelian U(1) gauge factors, where due to the Stückelberg mechanism only a diagonal U(1) factor remains massless, and thus after Higgsing only a diagonal discrete symmetry of order *n* is present in the Heterotic models and detected via Heterotic/F-theory duality. These constructions also provide further evidence for the conjectured mirror symmetry in Heterotic/F-theory at the level of fibrations with torsional sections and those with multi-sections.

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Cvetič, M., Grassi, A. & Poretschkin, M. Discrete symmetries in Heterotic/F-theory duality and mirror symmetry.
*J. High Energ. Phys.* **2017**, 156 (2017). https://doi.org/10.1007/JHEP06(2017)156

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DOI: https://doi.org/10.1007/JHEP06(2017)156

### Keywords

- Discrete Symmetries
- F-theory
- String Duality
- Superstrings and Heterotic Strings