$\frac{1}{2}$Calabi-Yau 4-folds and four-dimensional F-theory on Calabi-Yau 4-folds with U(1) factors

TL;DR

This paper introduces a new class of rational elliptic 4-folds called $rac{1}{2}$Calabi-Yau 4-folds, enabling the construction of 4D F-theory models with multiple U(1) gauge factors and positive Mordell-Weil ranks.

Contribution

The paper presents the concept of $rac{1}{2}$Calabi-Yau 4-folds and their use in systematically building elliptically fibered Calabi-Yau 4-folds with various U(1) factors.

Findings

Constructed elliptically fibered Calabi-Yau 4-folds with 1 to 6 U(1) factors.

Discovered the property that the sum of singularity rank and Mordell-Weil rank equals six.

Proposed geometric conditions for the base 3-fold to identify models with heterotic duals.

Abstract

In this study, four-dimensional $N=1$ F-theory models with multiple U(1) gauge group factors are constructed. A class of rational elliptic 4-folds, which we call as "$\frac{1}{2}$Calabi-Yau 4-folds," is introduced, and we construct the elliptically fibered 4-folds by utilizing them. This yields a novel approach for building families of elliptically fibered Calabi-Yau 4-folds with positive Mordell-Weil ranks. The introduced $\frac{1}{2}$Calabi-Yau 4-folds possess the characteristic property wherein the sum of the ranks of the singularity type and the Mordell-Weil group is always equal to six. This interesting property enables us to construct the elliptically fibered Calabi-Yau 4-folds of various positive Mordell-Weil ranks. From one to six U(1) factors form in four-dimensional F-theory on the resulting Calabi-Yau 4-folds. We also propose the geometric condition on the base 3-fold of the built Calabi-Yau 4-folds that allows four-dimensional F-theory models that have heterotic duals to be distinguished from those that do not.