Tate's algorithm and F-theory

TL;DR

This paper critically reexamines Tate's algorithm for elliptic fibrations in F-theory, revealing limitations and global obstructions in the traditional Tate forms, which impact model-building and landscape exploration.

Contribution

It identifies implicit assumptions in Tate's original derivation, proposes a new ansatz, and demonstrates global obstructions to Tate forms in F-theory models.

Findings

Tate forms do not always hold without additional divisibility assumptions.

Global obstructions can prevent the existence of Tate forms across the entire base.

Implications for F-theory model-building and vacua landscape exploration.

Abstract

The "Tate forms" for elliptically fibered Calabi-Yau manifolds are reconsidered in order to determine their general validity. We point out that there were some implicit assumptions made in the original derivation of these "Tate forms" from the Tate algorithm. By a careful analysis of the Tate algorithm itself, we deduce that the "Tate forms" (without any futher divisiblity assumptions) do not hold in some instances and have to be replaced by a new type of ansatz. Furthermore, we give examples in which the existence of a "Tate form" can be globally obstructed, i.e., the change of coordinates does not extend globally to sections of the entire base of the elliptic fibration. These results have implications both for model-building and for the exploration of the landscape of F-theory vacua.