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

TL;DR

The paper investigates discrete Abelian gauge symmetries within Heterotic/F-theory duality by framing F-theory on genus-one fibrations with n-sections through the Tate-Shafarevich group $Z_n$, linked to heterotic bundles with structure group $Z_n$. Employing a conjectured Heterotic/F-theory mirror symmetry, it constructs dual toric geometries whose stable degeneration yields discrete symmetries of order 2 and 3 in six dimensions, with explicit $Z_2$ and $Z_3$ mirror pairs; the heterotic side interprets these as Higgsed symmetric U(1) models where the Stückelberg mechanism leaves only a diagonal discrete symmetry. The work connects multi-section and torsion phenomena under mirror symmetry, providing evidence for a broader Heterotic/F-theory mirror relation and offering concrete geometric realizations of discrete gauge structure in both theories. These results illuminate how discrete symmetries arise from geometric data and how mirror symmetry exchanges fiber data between torsion and multi-section realizations, with potential implications for constructing and interpreting dual vacua in string theory.

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.

Figures (2)

• Figure 1: The polytope on the left shows the ambient space whose associated hypersurface leads to the $\mathbb{Z}_2$-geometry. The polytope on the right provides the ambient space of the geometry with gauge symmetry $(($E$_7$$\times$ SU(2)$)/\mathbb{Z}_2)^2$. The zero plane along which the symplectic cut is performed is marked by the black points. The yellow and blue points give the affine Dynkin diagram of E$_7$. The latter are inherited by the half K3 surfaces $X_2^\pm$, respectively. The purple point corresponds to an SU(2) gauge group which appears in both half K3 surfaces $X_2^\pm$ after the stable degeneration limit. Orange points mark inner points of the facets. Finally, beige-coloured points are on the invisible facets of the polytope.
• Figure 2: The polytope on the left shows the ambient space whose associated hypersurface leads to the $\mathbb{Z}_3$-geometry. The polytope on the right provides the ambient space with gauge symmetry $(($E$_6$$\times$ SU(3)$)/\mathbb{Z}_3)^2$. The zero plane along which the symplectic cut is performed is marked by the black points. The yellow and blue points give the affine Dynkin diagram of E$_6$. The latter are inherited by the half K3 surfaces $\chi^\pm$, respectively. Beige-coloured points are on the invisible facets of the polytope. In particular, the two points on the invisible edge correspond to the Dynkin diagram of SU(3) which is inherited by both half K3 surfaces. Finally, orange points mark inner points of the facets and the purple point marks the inner point of the polytope.