Literature Models

In the landscape of F-theory model building, many constructions introduced over the years remain relevant and influential. However, revisiting or building upon these models can be challenging—especially when prior computations must be repeated manually due to evolving mathematical techniques or notation. This not only hinders efficiency but can also deter newcomers from engaging deeply with the field.

To overcome these obstacles, we introduce the concept of a literature model: a well-established F-theory construction from the research literature, made accessible in a fully computational format.

What is a Literature Model?

A Computational Interface to the Literature

The LiteratureModels framework provides a curated database of such models, drawn from influential F-theory papers. Each entry is preprocessed to automate core calculations, store topological and geometric data, and support further exploration. Rather than repeating extensive algebraic or topological derivations, researchers can immediately load a literature model and apply the full power of FTheoryTools to study or extend it.

This offers several advantages:

Rapid access to existing constructions: Retrieve previously published models without manually reproducing their setup.
Integrated results: Access known data, such as intersection numbers, crepant resolutions, or gauge groups.
Targeted search: Filter models by desired features (e.g., gauge symmetry, fiber type).
Seamless analysis: Load a model and perform further computations using the broader functionality of FTheoryTools.

The current database includes a wide range of examples, including models:

This collection is actively expanding.

Interoperable and Reproducible Model Storage

All literature models are stored in a structured and platform-independent format based on JSON. The framework is transitioning toward the MaRDI file format, a standardized data format developed as part of the Mathematics Research Data Initiative (MaRDI). This format ensures that models built in FTheoryTools can be used across a wide range of computer algebra systems, including Oscar, Sage, Macaulay, and others.

The benefits of this include:

Interoperability: Models can be exported, shared, and loaded across different systems.
Data compression: Efficient storage of complex models (via subtree reduction).
Machine-readability: Makes it easy to contribute new models or verify existing ones.
Transparency: Published models can be peer-reviewed in both mathematical and computational terms.

We envision literature models not only as a computational resource, but also as a new standard for reproducibility in F-theory research. Authors can supplement future publications with accompanying data files, making it easier for others to explore, verify, and build upon their work. This aligns with growing community efforts—such as those within MaRDI—to ensure that mathematical software and data are as verifiable and accessible as the results they support.

Constructing Literature Models

The constructor for literature models is necessarily complex, reflecting the wide range of constructions explored in the F-theory literature. While some keyword arguments are self-explanatory—for example, the string doi uniquely identifies a publication—others are more technical or use terminology that may be non-standard. A prominent example is model_sections, which plays a key role in defining the geometry of the model.

We defer a detailed discussion of these technical inputs (and outputs) to the next section. The examples in the constructor below serve as a high-level overview of how literature models are initialized in practice:

literature_model — Method

literature_model(; doi::String="", arxiv_id::String="", version::String="", equation::String="", model_parameters::Dict{String,<:Any} = Dict{String,Any}(), base_space::FTheorySpace = affine_space(NormalToricVariety, 0), model_sections::Dict{String, <:Any} = Dict{String,Any}(), defining_classes::Dict{String, <:Any} = Dict{String,Any}(), completeness_check::Bool = true)

Return a model from the F-theory literature identified by bibliographic metadata such as arXiv ID, DOI, or equation reference. Many such models are well-known in the field—e.g., the U(1)-restricted SU(5)-GUT model Krause, Mayrhofer, Weigand 2011—and are stored in a curated database accessible through this method.

To identify and retrieve a model from the database, supply one or more of the following optional arguments:

doi: A string representing the DOI of the publication in which the model was introduced.
arxiv_id: A string specifying the arXiv identifier of the preprint version or the journal print version.
version: A string specifying the arXiv version (e.g. "v1").
equation: A string indicating the equation label used to define the model within the source publication.

The method attempts to match the given identifiers with a unique entry in the database. If no match is found, or if the information is ambiguous (i.e., multiple matches exist), an error is raised.

Some literature models require additional input to be uniquely determined and constructed. For such cases, more optional arguments must be provided:

model_parameters: A dictionary specifying parameters needed to fix a model within a family, e.g. Dict("k" => 5).
defining_classes: A dictionary specifying divisor classes necessary to fully define the geometry.
base_space: Optionally specify a concrete base over which the model should be constructed. Currently, only toric base spaces are supported.
completeness_check: Set this to false to skip time consuming completeness checks of the base geometry to gain more performance.

See the Literature Models documentation page for a deeper discussion of these fields.

First, notice how you can create the global Tate model from Krause, Mayrhofer, Weigand 2011 over an unspecified base:

Examples

julia> t = literature_model(arxiv_id = "1109.3454", equation = "3.1")
Assuming that the first row of the given grading is the grading under Kbar

Global Tate model over a not fully specified base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

julia> coordinate_ring(ambient_space(t))
Multivariate polynomial ring in 8 variables over QQ graded by
  w -> [0 1 0]
  a1 -> [1 0 0]
  a21 -> [2 -1 0]
  a32 -> [3 -2 0]
  a43 -> [4 -3 0]
  x -> [2 0 2]
  y -> [3 0 3]
  z -> [0 0 1]

Certainly, this is also possible over a concrete base, provided that the defining classes are provided:

Examples

julia> B3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> w = torusinvariant_prime_divisors(B3)[1]
Torus-invariant, prime divisor on a normal toric variety

julia> t = literature_model(arxiv_id = "1109.3454", equation = "3.1", base_space = B3, defining_classes = Dict("w" => w), completeness_check = false)
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!

Global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

Certainly, this can be mimiced for Weierstrass models, e.g. Morrison, Park 2012:

Examples

julia> B2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> b = torusinvariant_prime_divisors(B2)[1]
Torus-invariant, prime divisor on a normal toric variety

julia> w = literature_model(arxiv_id = "1208.2695", equation = "B.19", base_space = B2, defining_classes = Dict("b" => b), completeness_check = false)
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!

Weierstrass model over a concrete base -- U(1) Weierstrass model based on arXiv paper 1208.2695 Eq. (B.19)

julia> length(singular_loci(w))
1

For convenience, we also support a simplified constructor. Instead of the meta data of the article, this constructor accepts an integer, which specifies the position of this model in our database. Here is how this can be used to create the model Morrison, Park 2012:

Examples

julia> B2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> b = torusinvariant_prime_divisors(B2)[1]
Torus-invariant, prime divisor on a normal toric variety

julia> w = literature_model(3, base_space = B2, defining_classes = Dict("b" => b), completeness_check = false)
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!

Weierstrass model over a concrete base -- U(1) Weierstrass model based on arXiv paper 1208.2695 Eq. (B.19)

julia> length(singular_loci(w))
1

Finally, let us showcase that we also support hypersurface model constructions from the literature. For this, we stay with Morrison, Park 2012, but this time create this model as a hypersurface model. This works as follows:

Examples

julia> h = literature_model(arxiv_id = "1208.2695", equation = "B.5")
Assuming that the first row of the given grading is the grading under Kbar

Hypersurface model over a not fully specified base

julia> explicit_model_sections(h)
Dict{String, MPolyRingElem} with 5 entries:
  "c2" => c2
  "c1" => c1
  "c3" => c3
  "b"  => b
  "c0" => c0

julia> B2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> b = torusinvariant_prime_divisors(B2)[1]
Torus-invariant, prime divisor on a normal toric variety

julia> h2 = literature_model(arxiv_id = "1208.2695", equation = "B.5", base_space = B2, defining_classes = Dict("b" => b))
Construction over concrete base may lead to singularity enhancement. Consider computing singular_loci. However, this may take time!

Hypersurface model over a concrete base

julia> hypersurface_equation_parametrization(h2)
b*w*v^2 - c0*u^4 - c1*u^3*v - c2*u^2*v^2 - c3*u*v^3 + w^2

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.