Toric geometry and F-theory/Heterotic Duality in Four Dimensions

TL;DR

This paper explores the geometric relationship between toric Calabi-Yau fourfolds in F-theory and heterotic string compactifications, enabling the extraction of heterotic bundle data and revealing geometric constraints and topology transitions.

Contribution

It provides a method to read heterotic vector bundle data from toric fourfolds and investigates how fourfold geometry constrains heterotic bundles and topology changes.

Findings

Constructed Calabi-Yau fourfolds for three-generation GUT models.

Identified geometric restrictions on heterotic vector bundles.

Evidence of topology-changing extremal transitions related to fivebranes.

Abstract

We study, as hypersurfaces in toric varieties, elliptic Calabi-Yau fourfolds for F-theory compactifications dual to E8xE8 heterotic strings compactified to four dimensions on elliptic Calabi-Yau threefolds with some choice of vector bundle. We describe how to read off the vector bundle data for the heterotic compactification from the toric data of the fourfold. This map allows us to construct, for example, Calabi-Yau fourfolds corresponding to three generation models with unbroken GUT groups. We also find that the geometry of the Calabi-Yau fourfold restricts the heterotic vector bundle data in a manner related to the stability of these bundles. Finally, we study Calabi-Yau fourfolds corresponding to heterotic models with fivebranes wrapping curves in the base of the Calabi-Yau threefolds. We find evidence of a topology changing extremal transition on the fourfold side which corresponds, on the heterotic side, to fivebranes wrapping different curves in the same homology class in the base.