Toric Calabi-Yau Fourfolds, Duality Between N=1 Theories and Divisors that Contribute to the Superpotential

TL;DR

The paper develops a toric framework to study F-theory on Calabi–Yau fourfolds and its heterotic duals, reading heterotic data from reflexive polyhedra and constructing examples with specified gauge groups. It analyzes the computation of the F-theory superpotential, emphasizing Witten's condition $\chi({\cal O}_D)=1$ for divisors to contribute and the special status of dissident divisors associated with short roots of non-simply laced groups, where $\chi({\cal O}_D)>1$ indicates non-general moduli and $\chi({\cal O}_D)\le 0$ signals instabilities. The work connects local-to-global aspects, discusses the role of mirror symmetry, and highlights how toric data encode fibration structures and divisor contributions to non-perturbative dynamics in the dual heterotic theory. This framework advances understanding of $(0,2)$ heterotic compactifications and the landscape of fourfolds with rich gauge and superpotential structures. The findings have implications for constructing explicit compactifications with controlled non-perturbative effects and for exploring the moduli dependence of superpotentials in F-theory/heterotic duality.

Abstract

We study issues related to F-theory on Calabi-Yau fourfolds and its duality to heterotic theory for Calabi-Yau threefolds. We discuss principally fourfolds that are described by reflexive polyhedra and show how to read off some of the data for the heterotic theory from the polyhedron. We give a procedure for constructing examples with given gauge groups and describe some of these examples in detail. Interesting features arise when the local pieces are fitted into a global manifold. An important issue is how to compute the superpotential explicitly. Witten has shown that the condition for a divisor to contribute to the superpotential is that it have arithmetic genus 1. Divisors associated with the short roots of non-simply laced gauge groups do not always satisfy this condition while the divisors associated to all other roots do. For such a `dissident' divisor we distinguish cases for which the arithmetic genus is greater than unity corresponding to an X that is not general in moduli (in the toric case this corresponds to the existence of non-toric parameters). In these cases the `dissident' divisor D does not remain an effective divisor for general complex structure. If however the arithmetic genus is less than or equal to 0, then the divisor is general in moduli and there is a genuine instability.