Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tate's algorithm and F-theory

Sheldon Katz, David R. Morrison, Sakura Schäfer-Nameki, James Sully

TL;DR

<3-5 sentence high-level summary> This work reexamines the Tate forms used in the Tate algorithm for elliptically fibered Calabi-Yau manifolds in F-theory, identifying hidden assumptions and showing that Tate forms are not universally valid in all bases or for all singularity types. By analyzing the coordinate changes required by the algorithm, the authors introduce a new ansatz for certain odd $I_m$ cases with monodromy and establish an induction framework that handles large ranks, while also showing that global obstructions can prevent a Tate form from existing on the entire base. They provide explicit obstruction examples and discuss how these results constrain model building, particularly in SU($5$) GUT scenarios, and invite broader exploration of the F-theory landscape using locally Tate-form data. These findings sharpen the link between Kodaira/Tate classifications and physically realized gauge sectors, highlighting when local normal forms suffice and when new structures are needed to capture global geometry.

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.

Tate's algorithm and F-theory

TL;DR

<3-5 sentence high-level summary> This work reexamines the Tate forms used in the Tate algorithm for elliptically fibered Calabi-Yau manifolds in F-theory, identifying hidden assumptions and showing that Tate forms are not universally valid in all bases or for all singularity types. By analyzing the coordinate changes required by the algorithm, the authors introduce a new ansatz for certain odd cases with monodromy and establish an induction framework that handles large ranks, while also showing that global obstructions can prevent a Tate form from existing on the entire base. They provide explicit obstruction examples and discuss how these results constrain model building, particularly in SU() GUT scenarios, and invite broader exploration of the F-theory landscape using locally Tate-form data. These findings sharpen the link between Kodaira/Tate classifications and physically realized gauge sectors, highlighting when local normal forms suffice and when new structures are needed to capture global geometry.

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.

Paper Structure

This paper contains 26 sections, 4 theorems, 104 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Normal forms for Weierstrass equations
  3. Global obstructions to Tate form: An example
  4. The $I_m$ case
  5. Step 1
  6. Step 2
  7. Step 2 without monodromy
  8. Step 3
  9. Step 4
  10. Step 4 without monodromy
  11. Induction
  12. Case without monodromy
  13. Outliers
  14. Summary for $I_m$
  15. The $II$, $III$, and $IV$ cases
  16. ...and 11 more sections

Key Result

Lemma 1

If $(4f_0^3+27g_0^2)|_S=0$ then possibly after shrinking $U$, there exists a function $u_0$ on $U$ such that $f_0|_S=-\frac{1}{3}u_0^2|_S$ and $g_0|_S=\frac{2}{27}u_0^3|_S$, i.e., $f_0=-\frac{1}{3}u_0^2+O(z)$ and $g_0=\frac{2}{27}u_0^3+O(z)$.

Theorems & Definitions (4)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4