Discrete Symmetries in Heterotic/F-theory Duality and Mirror Symmetry

TL;DR

This paper explores the duality between Heterotic and F-theory compactifications with discrete gauge symmetries, constructing mirror pairs and analyzing their geometric and field-theoretic properties to support a conjectured mirror symmetry.

Contribution

It introduces explicit mirror constructions for models with Z_2 and Z_3 discrete symmetries, linking geometric dualities to field theory Higgsing processes.

Findings

Constructed mirror pairs with Z_2 and Z_3 symmetries in six dimensions.

Linked discrete symmetries to Higgsing of U(1) factors via the Stuckelberg mechanism.

Provided evidence for the conjectured Heterotic/F-theory mirror symmetry involving fibrations with torsional and multi-sections.

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 Z_n. Such models are obtained by studying first a specific toric set-up whose associated Heterotic vector bundle has structure group 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 Z_2 and 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 Stuckelberg 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.