Elevating the Free-Fermion $Z_2\times Z_2$ Orbifold Model to a Compactification of F-Theory

TL;DR

This paper investigates elliptic fibrations of specific Calabi-Yau three-folds related to free-fermion models, highlighting puzzles in F-theory compactifications and proposing potential resolutions involving localized matter at singular points.

Contribution

It extends the understanding of F-theory compactifications on orbifold Calabi-Yau three-folds and addresses anomaly cancellation issues in these models.

Findings

Identified singular points where the elliptic fibration is ill-defined.

Proposed the existence of localized matter to cancel gravitational anomalies.

Connected free-fermion models with F-theory compactifications.

Abstract

We study the elliptic fibrations of some Calabi-Yau three-folds, including the $Z_2\times Z_2$ orbifold with $(h_{1,1},h_{2,1})=(27,3)$, which is equivalent to the common framework of realistic free-fermion models, as well as related orbifold models with $(h_{1,1},h_{2,1})=(51,3)$ and (31,7). However, two related puzzles arise when one considers the $(h_{1,1},h_{2,1})=(27,3)$ model as an F-theory compactification to six dimensions. The condition for the vanishing of the gravitational anomaly is not satisfied, suggesting that the F-theory compactification does not make sense, and the elliptic fibration is well defined everywhere except at four singular points in the base. We speculate on the possible existence of N=1 tensor and hypermultiplets at these points which would cancel the gravitational anomaly in this case.