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Welcome to FTheoryTools

Overview

FTheoryTools is a computational toolkit within the OSCAR computer algebra system, designed to assist researchers in working with F-theory models. It focuses on automating and simplifying calculations involving singular elliptic fibrations—key geometric objects in F-theory phenomenology.

While the module is tailored for string theorists, it is equally accessible to mathematicians interested in the rich geometry of singular fibrations, even if they are not familiar with F-theory itself.

This page is meant for end users of OSCAR, including students and researchers in mathematics and the natural sciences. No background in string theory or theoretical physics is assumed beyond what is needed to understand the geometry of elliptic fibrations. We encourage interested readers to consult the exposition in Weigand 2018 for more background information.

Why Use FTheoryTools?

F-theory encodes physical data into the structure of singular elliptic fibrations. In these models:

  • The type and location of singularities relate directly to gauge groups and matter content.
  • Smooth fibrations typically yield trivial physics and are therefore less interesting for physical applications.

To analyze the geometry effectively, model builders look for a crepant resolution of the singular space—one that preserves the Calabi-Yau condition and retains physical meaning. These resolutions are challenging to compute, especially in higher codimension or for non-toric singularities.

FTheoryTools aims to:

  • Automate and streamline the crepant resolution process,
  • Make computations more reproducible,
  • Extract physically relevant features from the resolved space,
  • Offer a modular and extensible framework for researchers in both physics and mathematics.

Key Features

Constructing Elliptic Fibrations

FTheoryTools supports construction of elliptic fibrations via:

All of these represent singular elliptic fibrations, so many operations and properties are shared across them. This shared functionality is documented at Functionality for all F-theory models.

Fibrations can be defined over various base spaces:

  • Families of abstract bases,
  • Toric varieties (best supported),
  • (Planned) General schemes and varieties.

Physically relevant cases often have base dimension 1, 2, or 3, but FTheoryTools is not limited to these.

General Blowups (Beyond Toric)

FTheoryTools enables blowups on arbitrary loci—not just toric centers. This allows users to work with a wider class of singularities, including those without a known toric resolution.

Literature Models

FTheoryTools includes a curated database of well-known F-theory models from the literature. These models are stored using the MaRDI file format, a JSON-based format aligned with FAIR data principles:

  • Findability
  • Accessibility
  • Interoperability
  • Reusability

Learn more about the format:

Current literature models include:

More information: Literature constructions.

$G_4$-Flux Enumeration

FTheoryTools supports the enumeration of vertical $G_4$-fluxes—important for understanding chiral spectra in F-theory.

Example: Taylor, Wang 2015

  • 101 toric rays
  • 198 maximal cones
  • Defining equation with 355,785 monomials
  • 206 toric blowups + 3 smoothness blowups

The computed flux space: $\mathbb{Z}^{224} \times \mathbb{Q}^{127}$.

Details are available in BMT25. See also: G4-Fluxes.

Tutorials

Explore example-driven tutorials at: https://www.oscar-system.org/tutorials/FTheoryTools/

These walk through:

  • Building models
  • Performing blowups
  • Extracting physical/geometric information

Getting Started

  1. Install OSCAR: Installation instructions
  2. Update regularly: Stay current with new features via the upgrade guide

Project Status

FTheoryTools is an experimental module. Most features are well-tested for toric models, with active development underway for:

  • Support for general base families and schemes
  • Line bundle cohomology for vector-like spectra
  • Support for terminal/non-minimal singularities
  • Base blowup automation

Contact & Community

For questions, suggestions, or collaboration:

Community platforms:

Acknowledgements

We thank Mirjam Cvetič and Mohab Safey El Din for valuable discussions.

The authors are thankful for the support offered by the TU-Nachwuchsring. This work was supported by the SFB-TRR 195 Symbolic Tools in Mathematics and their Application of the German Research Foundation (DFG). Martin Bies acknowledges financial support from the \emph{Forschungsinitiative des Landes Rheinland-Pfalz} through the project SymbTools – Symbolic Tools in Mathematics and their Application. Andrew P. Turner acknowledges funding from DOE (HEP) Award DE-SC0013528 and NSF grant PHY-2014086.