F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds

TL;DR

This paper explores the structure of the Mordell-Weil group in elliptically fibered Calabi-Yau threefolds within F-theory, revealing new models with U(1) gauge symmetry and matter charges, using anomaly equations as key tools.

Contribution

It studies Calabi-Yau threefolds with Mordell-Weil rank one, generating new F-theory models with matter charges 1 and 2, and analyzes their anomaly relations.

Findings

Generated F-theory models with U(1) gauge symmetry and matter charges 1 and 2.

Analyzed the relation between Neron-Tate height and intersection numbers.

Provided insights into the Mordell-Weil group structure in Calabi-Yau threefolds.

Abstract

The Mordell-Weil group of an elliptically fibered Calabi-Yau threefold X contains information about the abelian sector of the six-dimensional theory obtained by compactifying F-theory on X. After examining features of the abelian anomaly coefficient matrix and U(1) charge quantization conditions of general F-theory vacua, we study Calabi-Yau threefolds with Mordell-Weil rank-one as a first step towards understanding the features of the Mordell-Weil group of threefolds in more detail. In particular, we generate an interesting class of F-theory models with U(1) gauge symmetry that have matter with both charges 1 and 2. The anomaly equations --- which relate the Neron-Tate height of a section to intersection numbers between the section and fibral rational curves of the manifold --- serve as an important tool in our analysis.