Literature constructions

su5_tate_model_over_arbitrary_3d_base()

Return the SU(5) Tate model over an arbitrary 3-dimensional base space. For more details see e.g. [Wei18] and references therein.

julia> tm = su5_tate_model_over_arbitrary_3d_base()
Assuming that the first row of the given grading is the grading under Kbar

Global Tate model over a not fully specified base

julia> v = ambient_space(tm)
Family of spaces of dimension d = 5

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)

Many models have been created in the F-theory literature. A significant number of them have even been given specific names, for instance the "U(1)-restricted SU(5)-GUT model". This method has access to a database, from which it can look up such literature models.

Currently, you can provide any combination of the following optional arguments to the method literature_model:

• doi: A string representing the DOI of the publication that

introduced the model in question.

• equation: A string representing the number of the equation that introduced

the model in question. For papers, that were posted on the arXiv, we can instead of the doi also provide the following:

• arxiv_id: A string that represents the arXiv identifier of the paper that

introduced the model in question.

• version: A string representing the version of the arXiv upload.

The method literature_model attempts to find a model in our database for which the provided data matches the information in our record. If no such model can be found, or multiple models exist with information matching the provided information, then an error is raised.

Some literature models require additional parameters to specified to single out a model from a family of models. Such parameters can be provided using the optional argument model_parameters, which should be a dictionary such as Dict("k" => 5).

Further, some literature models require the specification of one or more divisor classes that define the model. This information can be provided using the optional argument defining_classes, which should be a dictionary such as Dict("w" => w), where w is a divisor, such as that provided by torusinvariant_prime_divisors.

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> v = ambient_space(t)
Family of spaces of dimension d = 5

julia> coordinate_ring(v)
Multivariate polynomial ring in 8 variables w, a1, a21, a32, ..., z
  over rational field

It is also possible to construct a literature model over a particular base. Currently, this feature is only supported for toric base spaces.

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> t2 = 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)

julia> length(singular_loci(t2))
2

Of course, this is also possible for Weierstrass models.

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.

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

Similarly, also hypersurface models are supported:

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

We can also create the model with the largest number of F-theory vacua. This happens by executing the line h = literature_model(arxiv_id = "1511.03209"). The first time this line is executed, it downloads .mrdi-files from zenodo, which encode pre-computed results regarding this model and one of its resolutions. Thereby, you can create this F-theory model including a lot of advanced information (e.g. more than 10.000.000 intersection numbers and explicit descriptions for the G4-fluxes on this space) within just a couple of minutes. For comparison, one a personal computer we expect that the computation of one resolution of this model takes about three to four hours. Identifying also all $G_4$-fluxes (vertical, well-quantized and modelled by pullbacks from the toric ambient space) will likely take a few hours more. So, this infrastructure provides a very stark performence improvement.

arxiv_id(m::AbstractFTheoryModel)

Return the arxiv_id of the preprint that introduced the given model. If no arxiv_id is known, an error is raised.

julia> m = 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> arxiv_id(m)
"1109.3454"

arxiv_doi(m::AbstractFTheoryModel)

Return the arxiv_doi of the preprint that introduced the given model. If no arxiv_doi is known, an error is raised.

julia> m = 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> arxiv_doi(m)
"10.48550/arXiv.1109.3454"

arxiv_link(m::AbstractFTheoryModel)

Return the arxiv_link (formatted as string) to the arXiv version of the paper that introduced the given model. If no arxiv_link is known, an error is raised.

julia> m = 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> arxiv_link(m)
"https://arxiv.org/abs/1109.3454v2"

arxiv_model_equation_number(m::AbstractFTheoryModel)

Return the arxiv_model_equation_number in which the given model was introduced in the arXiv preprint in our record. If no arxiv_model_equation_number is known, an error is raised.

julia> m = 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> arxiv_model_equation_number(m)
"3.1"

arxiv_model_page(m::AbstractFTheoryModel)

Return the arxiv_model_page on which the given model was introduced in the arXiv preprint in our record. If no arxiv_model_page is known, an error is raised.

julia> m = 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> arxiv_model_page(m)
"10"

arxiv_model_section(m::AbstractFTheoryModel)

Return the arxiv_model_section in which the given model was introduced in the arXiv preprint in our record. If no arxiv_model_section is known, an error is raised.

julia> m = 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> arxiv_model_section(m)
"3"

arxiv_version(m::AbstractFTheoryModel)

Return the arxiv_version of the arXiv preprint that introduced the given model. If no arxiv_version is known, an error is raised.

julia> m = 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> arxiv_version(m)
"2"

associated_literature_models(m::AbstractFTheoryModel)

Return a list of the unique identifiers of any associated_literature_models of the given model. These are models that are introduced in the same paper as the given model, but that are distinct from the given model. If no associated_literature_models are known, an error is raised.

julia> m = literature_model(arxiv_id = "1212.2949", equation = "3.2", model_parameters = Dict("k" => 5))
Assuming that the first row of the given grading is the grading under Kbar

Global Tate model over a not fully specified base -- SU(11) Tate model with parameter values (k = 5) based on arXiv paper 1212.2949 Eq. (3.2)

julia> associated_literature_models(m)
6-element Vector{String}:
 "1212_2949-2"
 "1212_2949-3"
 "1212_2949-4"
 "1212_2949-5"
 "1212_2949-6"
 "1212_2949-7"

generating_sections(m::AbstractFTheoryModel)

Return a list of the known Mordell–Weil generating sections of the given model. If no generating sections are known, an error is raised.

julia> m = 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> generating_sections(m)
1-element Vector{Vector{QQMPolyRingElem}}:
 [0, 0, 1]

journal_doi(m::AbstractFTheoryModel)

Return the journal_doi of the publication that introduced the given model. If no journal_doi is known, an error is raised.

julia> m = 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> journal_doi(m)
"10.1016/j.nuclphysb.2011.12.013"

journal_link(m::AbstractFTheoryModel)

Return the journal_link (formatted as string) to the published version of the paper that introduced the given model. If no journal_link is known, an error is raised.

julia> m = 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> journal_link(m)
"https://www.sciencedirect.com/science/article/pii/S0550321311007115"

journal_model_equation_number(m::AbstractFTheoryModel)

Return the journal_model_equation_number in which the given model was introduced in the published paper in our record. If no journal_model_equation_number is known, an error is raised.

julia> m = 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> journal_model_equation_number(m)
"3.1"

journal_model_page(m::AbstractFTheoryModel)

Return the journal_model_page on which the given model was introduced in the published paper in our record. If no journal_model_page is known, an error is raised.

julia> m = 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> journal_model_page(m)
"9"

journal_model_section(m::AbstractFTheoryModel)

Return the journal_model_section in which the given model was introduced in the published paper in our record. If no journal_model_section is known, an error is raised.

julia> m = 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> journal_model_section(m)
"3"

journal_name(m::AbstractFTheoryModel)

Return the journal_name of the published paper in which the given model was introduced. If no journal_name are known, an error is raised.

julia> m = 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> journal_name(m)
"Nucl. Phys. B"

journal_pages(m::AbstractFTheoryModel)

Return the journal_pages of the published paper in which the given model was introduced. If no journal_pages are known, an error is raised.

julia> m = 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> journal_pages(m)
"1–47"

journal_report_numbers(m::AbstractFTheoryModel)

Return the journal_report_numbers of the published paper in which the given model was introduced. If no journal_report_numbers is known, an error is raised.

julia> m = literature_model(arxiv_id = "1507.05954", equation = "A.1")
Assuming that the first row of the given grading is the grading under Kbar

Weierstrass model over a not fully specified base -- U(1)xU(1) Weierstrass model based on arXiv paper 1507.05954 Eq. (A.1)

julia> journal_report_numbers(m)
3-element Vector{String}:
 "UPR-1274-T"
 "CERN-PH-TH-2015-157"
 "MIT-CTP-4678"

journal_volume(m::AbstractFTheoryModel)

Return the journal_volume of the published paper in which the given model was introduced. If no journal_volume are known, an error is raised.

julia> m = 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> journal_volume(m)
"858"

journal_year(m::AbstractFTheoryModel)

Return the journal_year of the published paper in which the given model was introduced. If no journal_year is known, an error is raised.

julia> m = 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> journal_year(m)
"2012"

literature_identifier(m::AbstractFTheoryModel)

Return the literature_identifier of the given mode, which is a unique string that distinguishes the model from all others in the literature model database. If no literature_identifier is known, an error is raised.

julia> m = 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> literature_identifier(m)
"1109_3454"

model_parameters(m::AbstractFTheoryModel)

Return the model_parameters of the given model. If no model_parameters are known, an error is raised.

julia> m = literature_model(arxiv_id = "1212.2949", equation = "3.2", model_parameters = Dict("k" => 5))
Assuming that the first row of the given grading is the grading under Kbar

Global Tate model over a not fully specified base -- SU(11) Tate model with parameter values (k = 5) based on arXiv paper 1212.2949 Eq. (3.2)

julia> model_parameters(m)
Dict{String, Int64} with 1 entry:
  "k" => 5

paper_authors(m::AbstractFTheoryModel)

Return the paper_authors of the paper that introduced the given model. If no paper_authors are known, an error is raised.

julia> m = 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> paper_authors(m)
3-element Vector{String}:
 "Sven Krause"
 "Christoph Mayrhofer"
 "Timo Weigand"

paper_buzzwords(m::AbstractFTheoryModel)

Return the paper_buzzwords of the paper that introduced the given model. If no paper_buzzwords are known, an error is raised.

julia> m = 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> paper_buzzwords(m)
4-element Vector{String}:
 "GUT model"
 "Tate"
 "U(1)"
 "SU(5)"

paper_description(m::AbstractFTheoryModel)

Return the paper_description of the paper that introduced the given model. If no paper_description is known, an error is raised.

julia> m = 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> paper_description(m)
"SU(5)xU(1) restricted Tate model"

paper_title(m::AbstractFTheoryModel)

Return the paper_title of the arXiv preprint that introduced the given model. If no paper_title is known, an error is raised.

julia> m = 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> paper_title(m)
"\$G_4\$ flux, chiral matter and singularity resolution in F-theory compactifications"

birational_literature_models(m::AbstractFTheoryModel)

Return a list of the unique identifiers of birational_literature_models of the given model. These are either other presentations (Weierstrass, Tate, ...) of the given model, or other version of the same model from a different paper in the literature. If no birational_literature_models are known, an error is raised.

julia> m = literature_model(arxiv_id = "1507.05954", equation = "A.1")
Assuming that the first row of the given grading is the grading under Kbar

Weierstrass model over a not fully specified base -- U(1)xU(1) Weierstrass model based on arXiv paper 1507.05954 Eq. (A.1)

julia> birational_literature_models(m)
1-element Vector{String}:
 "1507_05954-1"

resolutions(m::AbstractFTheoryModel)

Return the list of all known resolutions for the given model. If no resolutions are known, an error is raised.

julia> m = 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> resolutions(m)
1-element Vector{Tuple{Vector{Vector{String}}, Vector{String}}}:
 ([["x", "y", "w"], ["y", "e1"], ["x", "e4"], ["y", "e2"], ["x", "y"]], ["e1", "e4", "e2", "e3", "s"])

resolution_generating_sections(m::AbstractFTheoryModel)

Return a list of lists of known Mordell–Weil generating sections for the given model after each known resolution. Each element of the outer list corresponds to a known resolution (in the same order), and each element of the list associated to a given resolution corresponds to a known generating section (in the same order). If no resolution generating sections are known, an error is raised.

julia> m = 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> resolution_generating_sections(m)
1-element Vector{Vector{Vector{Vector{QQMPolyRingElem}}}}:
 [[[0, 0, 1], [0, 0, 1], [0, 1], [0, 1], [0, 1], [a32, -a43]]]

resolution_zero_sections(m::AbstractFTheoryModel)

Return a list of known Mordell–Weil zero sections for the given model after each known resolution. Each element of the list corresponds to a known resolution (in the same order). If no resolution zero sections are known, an error is raised.

julia> m = 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> resolution_zero_sections(m)
1-element Vector{Vector{Vector{QQMPolyRingElem}}}:
 [[1, 1, 0], [1, 1, w], [1, 1], [1, 1], [1, 1], [1, 1]]

weighted_resolutions(m::AbstractFTheoryModel)

Return the list of all known weighted resolutions for the given model. If no weighted resolutions are known, an error is raised.

julia> m = 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> weighted_resolutions(m)
1-element Vector{Tuple{Vector{Tuple{Vector{String}, Vector{Int64}}}, Vector{String}}}:
 ([(["x", "y", "w"], [1, 1, 1]), (["x", "y", "w"], [1, 2, 1]), (["x", "y", "w"], [2, 2, 1]), (["x", "y", "w"], [2, 3, 1]), (["x", "y"], [1, 1])], ["e1", "e4", "e2", "e3", "s"])

weighted_resolution_generating_sections(m::AbstractFTheoryModel)

Return a list of lists of known Mordell–Weil generating sections for the given model after each known weighted resolution. Each element of the outer list corresponds to a known weighted resolution (in the same order), and each element of the list associated to a given weighted resolution corresponds to a known generating section (in the same order). If no weighted resolution generating sections are known, an error is raised.

julia> m = 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> weighted_resolution_generating_sections(m)
1-element Vector{Vector{Vector{Vector{QQMPolyRingElem}}}}:
 [[[0, 0, 1], [0, 0, 1], [0, 0, 1], [0, 0, 1], [0, 0, 1], [a32, -a43]]]

weighted_resolution_zero_sections(m::AbstractFTheoryModel)

Return a list of known Mordell–Weil zero sections for the given model after each known weighted resolution. Each element of the list corresponds to a known weighted resolution (in the same order). If no weighted resolution zero sections are known, an error is raised.

julia> m = 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> weighted_resolution_zero_sections(m)
1-element Vector{Vector{Vector{QQMPolyRingElem}}}:
 [[1, 1, 0], [1, 1, w], [1, 1, w], [1, 1, w], [1, 1, w], [1, 1]]

zero_section(m::AbstractFTheoryModel)

Return the zero section of the given model. If no zero section is known, an error is raised. This information is not typically stored as an attribute for Weierstrass and global Tate models, whose zero sections are known.

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> zero_section(h)
3-element Vector{QQMPolyRingElem}:
 0
 1
 0

zero_section_class(m::AbstractFTheoryModel)

Return the zero section class of a model as a cohomology class in the toric ambient space. If no zero section class is known, an error is raised. This information is always available for Weierstrass and global Tate models, whose zero section classes are known.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> zero_section_class(qsm_model)
Cohomology class on a normal toric variety given by e2 + 2*u + 3*e4 + e1 - w

zero_section_index(m::AbstractFTheoryModel)

Return the index of the generator of the Cox ring of the ambient space, whose corresponding vanishing locus defines the zero section of a model. If no zero section class is known, an error is raised. This attribute is always set simultaneously with zerosectionclass. This information is always available for Weierstrass and global Tate models, whose zero section classes are known.

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

julia> Kbar = anticanonical_divisor_class(B3)
Divisor class on a normal toric variety

julia> foah15_B3 = literature_model(arxiv_id = "1408.4808", equation = "3.190", type = "hypersurface", base_space = B3, defining_classes = Dict("s7" => Kbar, "s9" => Kbar))
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> zero_section_index(foah15_B3)
5

exceptional_classes(m::AbstractFTheoryModel)

Return the cohomology classes of the exceptional toric divisors of a model as a vector of cohomology classes in the toric ambient space. This information is only supported for models over a concrete base that is a normal toric variety, but is always available in this case. After a toric blow up this information is updated.

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

julia> Kbar = anticanonical_divisor_class(B3)
Divisor class on a normal toric variety

julia> foah11_B3 = literature_model(arxiv_id = "1408.4808", equation = "3.142", type = "hypersurface", base_space = B3, defining_classes = Dict("s7" => Kbar, "s9" => Kbar))
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> exceptional_classes(foah11_B3)
4-element Vector{CohomologyClass}:
 Cohomology class on a normal toric variety given by e1
 Cohomology class on a normal toric variety given by e2
 Cohomology class on a normal toric variety given by e3
 Cohomology class on a normal toric variety given by e4

exceptional_divisor_indices(m::AbstractFTheoryModel)

Return the indices of the generators of the Cox ring of the ambient space which correspond to exceptional divisors. This information is only supported for models over a concrete base that is a normal toric variety, but is always available in this case. After a toric blow up this information is updated.

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

julia> Kbar = anticanonical_divisor_class(B3)
Divisor class on a normal toric variety

julia> foah11_B3 = literature_model(arxiv_id = "1408.4808", equation = "3.142", type = "hypersurface", base_space = B3, defining_classes = Dict("s7" => Kbar, "s9" => Kbar))
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> exceptional_divisor_indices(foah11_B3)
4-element Vector{Int64}:
  8
  9
 10
 11

torsion_sections(m::AbstractFTheoryModel)

Return the torsion sections of the given model. If no torsion sections are known, an error is raised.

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

julia> Kbar = anticanonical_divisor_class(B3)
Divisor class on a normal toric variety

julia> foah15_B3 = literature_model(arxiv_id = "1408.4808", equation = "3.190", type = "hypersurface", base_space = B3, defining_classes = Dict("s7" => Kbar, "s9" => Kbar))
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> length(torsion_sections(foah15_B3))
1

add_resolution(m::AbstractFTheoryModel, centers::Vector{Vector{String}}, exceptionals::Vector{String})

Add a known resolution for a model.

julia> m = 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> add_resolution(m, [["x", "y"], ["y", "s", "w"], ["s", "e4"], ["s", "e3"], ["s", "e1"]], ["s", "w", "e3", "e1", "e2"])

julia> length(resolutions(m))
2

resolve(m::AbstractFTheoryModel, index::Int)

Resolve a model with the index-th resolution that is known.

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)

julia> t2 = resolve(t, 1)
Partially resolved global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

julia> cox_ring(ambient_space(t2))
Multivariate polynomial ring in 12 variables over QQ graded by
  x1 -> [1 0 0 0 0 0 0]
  x2 -> [0 1 0 0 0 0 0]
  x3 -> [0 1 0 0 0 0 0]
  x4 -> [0 1 0 0 0 0 0]
  x -> [0 0 1 0 0 0 0]
  y -> [0 0 0 1 0 0 0]
  z -> [0 0 0 0 1 0 0]
  e1 -> [0 0 0 0 0 1 0]
  e4 -> [0 0 0 0 0 0 1]
  e2 -> [-1 -3 -1 1 -1 -1 0]
  e3 -> [0 4 1 -1 1 0 -1]
  s -> [2 6 -1 0 2 1 1]

julia> w2 = 2 * torusinvariant_prime_divisors(B3)[1]
Torus-invariant, non-prime divisor on a normal toric variety

julia> t3 = literature_model(arxiv_id = "1109.3454", equation = "3.1", base_space = B3, defining_classes = Dict("w" => w2), 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)

julia> t4 = resolve(t3, 1)
Partially resolved global Tate model over a concrete base -- SU(5)xU(1) restricted Tate model based on arXiv paper 1109.3454 Eq. (3.1)

vertices(m::AbstractFTheoryModel)

This method returns the vertices of the polytope the the base of the F-theory QSM is build from. Note that those vertices are normalized according to the Polymake standard to rational numbers.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> vertices(qsm_model)
4-element Vector{Vector{QQFieldElem}}:
 [-1, -1, -1]
 [1, -1//2, -1//2]
 [-1, 2, -1]
 [-1, -1, 5]

polytope_index(m::AbstractFTheoryModel)

Of the 3-dimensional reflexive polytope that the base of this F-theory model is build from, this method returns the index within the Kreuzer-Skarke list.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> polytope_index(qsm_model)
4

has_quick_triangulation(m::AbstractFTheoryModel)

For a 3-dimensional reflexive polytope in the Kreuzer-Skarke list, the list of full (sometimes also called fine), regular, star triangulations can be extremely large. Consequently, one may wonder if the triangulations can be enumerated in a somewhat reasonable time (say 5 minutes on a personal computer). This method tries to provide an answer to this. It returns true if one should expect a timely response to the attempt to enumerate all (full, regular, star) triangulations. Otherwise, this method returns false.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> has_quick_triangulation(qsm_model)
true

max_lattice_pts_in_facet(m::AbstractFTheoryModel)

In order to enumerate the number of full, regular, star triangulations of a 3-dimensional reflexive polytope, it is possible to first find the corresponding triangulations of all facets of the polytope [HT17]. A first indication for the complexity of this triangulation task is the maximum number of lattice points in a facet of the polytope in question. This method returns this maximal number of lattice points.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> max_lattice_pts_in_facet(qsm_model)
16

estimated_number_of_triangulations(m::AbstractFTheoryModel)

This method returns an estimate for the number of full, regular, star triangulations of the 3-dimensional reflexive polytope, those triangulations define the possible base spaces of the F-theory QSM in question.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> estimated_number_of_triangulations(qsm_model)
212533333333

kbar3(m::AbstractFTheoryModel)

Let Kbar denote the anticanonical class of the 3-dimensional base space of the F-theory QSM. Of ample importance is the triple intersection number of Kbar, i.e. Kbar * Kbar * Kbar. This method returns this intersection number.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> kbar3(qsm_model)
6

hodge_h11(m::AbstractFTheoryModel)

This methods return the Hodge number h11 of the elliptically fibered 4-fold that defined the F-theory QSM in question.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> hodge_h11(qsm_model)
31

hodge_h12(m::AbstractFTheoryModel)

This methods return the Hodge number h12 of the elliptically fibered 4-fold that defined the F-theory QSM in question.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> hodge_h12(qsm_model)
10

hodge_h13(m::AbstractFTheoryModel)

This methods return the Hodge number h13 of the elliptically fibered 4-fold that defined the F-theory QSM in question.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> hodge_h13(qsm_model)
34

hodge_h22(m::AbstractFTheoryModel)

This methods return the Hodge number h22 of the elliptically fibered 4-fold that defined the F-theory QSM in question.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> hodge_h22(qsm_model)
284

genera_of_ci_curves(m::AbstractFTheoryModel)

This methods return the genera of the Ci curves. Recall that Ci = V(xi, s), where xi is a homogeneous coordinate of the 3-dimensional toric base space B3 of the QSM hypersurface model in question, and s is a generic section of the anticanonical bundle of B3. Consequently, we may use the coordinates xi as labels for the curves Ci.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> keys_list = collect(keys(genera_of_ci_curves(qsm_model)));

julia> my_key = only(filter(k -> string(k) == "x7", keys_list))
x7

julia> genera_of_ci_curves(qsm_model)[my_key]
0

degrees_of_kbar_restrictions_to_ci_curves(m::AbstractFTheoryModel)

The anticanonical divisor of the 3-dimensional toric base space B3 of the QSM hypersurface model in question can be restricted to the Ci curves. The result of this operation is a line bundle. This method returns the degree of this line bundle for every Ci curve.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> keys_list = collect(keys(degrees_of_kbar_restrictions_to_ci_curves(qsm_model)));

julia> my_key = only(filter(k -> string(k) == "x7", keys_list))
x7

julia> degrees_of_kbar_restrictions_to_ci_curves(qsm_model)[my_key]
0

topological_intersection_numbers_among_ci_curves(m::AbstractFTheoryModel)

The topological intersection numbers among Ci curves are also of ample importance. This method returns those intersection numbers in the form of a matrix.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> n_rows(topological_intersection_numbers_among_ci_curves(qsm_model))
29

julia> n_columns(topological_intersection_numbers_among_ci_curves(qsm_model))
29

indices_of_trivial_ci_curves(m::AbstractFTheoryModel)

Some of the Ci curves are trivial, in that V(xi, s) is the empty set. This method returns the vector of all indices of trivial Ci curves. That is, should V(x23, s) be the empty set, then 23 will be included in the returned list.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> indices_of_trivial_ci_curves(qsm_model)
10-element Vector{Int64}:
 23
 22
 18
 19
 20
 26
 10
 11
 12
 15

topological_intersection_numbers_among_nontrivial_ci_curves(m::AbstractFTheoryModel)

The topological intersection numbers among the non-trivial Ci curves are used frequently. This method returns those intersection numbers in the form of a matrix.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> n_rows(topological_intersection_numbers_among_nontrivial_ci_curves(qsm_model))
19

julia> n_columns(topological_intersection_numbers_among_nontrivial_ci_curves(qsm_model))
19

dual_graph(m::AbstractFTheoryModel)

This method returns the dual graph of the QSM model in question. Note that no labels are (currently) attached to the vertices/nodes or edges. To understand/read this graph correctly, please use the methods listed below.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> dual_graph(qsm_model)
Undirected graph with 21 nodes and the following edges:
(5, 1)(6, 5)(7, 6)(8, 7)(9, 4)(9, 8)(10, 1)(11, 4)(12, 3)(12, 10)(13, 3)(13, 11)(14, 1)(15, 4)(16, 3)(17, 3)(18, 2)(18, 14)(19, 2)(19, 15)(20, 2)(20, 16)(21, 2)(21, 17)

components_of_dual_graph(m::AbstractFTheoryModel)

This method returns a vector with labels for each node/vertex of the dual graph of the QSM model in question. Those labels allow to understand the geometric origin of the node/vertex.

Specifically, recall that those nodes are associated to the Ci-curves, which are in turn given by Ci = V(xi, s). xi is a homogeneous coordinate of the 3-dimensional toric base space B3 of the QSM in question, and s is a generic section of the anticanonical bundle of B3.

Only non-trivial Ci = V(xi, s) correspond to vertices/nodes of the dual graph.

If Ci = V(xi, s) is irreducible and corresponds to the k-th component, then the label "Ci" appears at position k of the vector returned by this method. However, if Ci = V(xi, s) is reducible, then we introduce the labels Ci-0, Ci-1, Ci-2 etc. for those irreducible components of Ci. If Ci-0 corresponds to the k-th components of the dual graph, then the label "Ci-0" appears at position k of the vector returned by this method.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> components_of_dual_graph(qsm_model)
21-element Vector{String}:
 "C0"
 "C1"
 "C2"
 "C3"
 "C4"
 "C5"
 "C6"
 "C7"
 "C8"
 "C9"
 ⋮
 "C16"
 "C17"
 "C21"
 "C24-0"
 "C24-1"
 "C25"
 "C27"
 "C28-0"
 "C28-1"

degrees_of_kbar_restrictions_to_components_of_dual_graph(m::AbstractFTheoryModel)

The anticanonical bundle of the toric 3-dimensional base space of the F-theory QSM in question can be restricted to the (geometric counterparts of the) nodes/vertices of the dual graph. The result is a line bundle for each node/vertex. This method returns a vector with the degrees of these line bundles.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> degrees_of_kbar_restrictions_to_components_of_dual_graph(qsm_model)["C28-1"]
0

genera_of_components_of_dual_graph(m::AbstractFTheoryModel)

This methods returns a vector with the genera of the (geometric counterparts of the) nodes/vertices of the dual graph.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> genera_of_components_of_dual_graph(qsm_model)["C28-1"]
0

simplified_dual_graph(m::AbstractFTheoryModel)

This method returns the simplified dual graph of the QSM model in question. Note that no labels are (currently) attached to the vertices/nodes or edges. To understand/read this graph correctly, please use the methods listed below.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> simplified_dual_graph(qsm_model)
Undirected graph with 4 nodes and the following edges:
(2, 1)(3, 1)(3, 2)(4, 1)(4, 2)(4, 3)

components_of_simplified_dual_graph(m::AbstractFTheoryModel)

This method returns a vector with labels for each node/vertex of the simplified dual graph. Otherwise, works identical to components_of_dual_graph(m::AbstractFTheoryModel).

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> components_of_simplified_dual_graph(qsm_model)
4-element Vector{String}:
 "C0"
 "C1"
 "C2"
 "C3"

degrees_of_kbar_restrictions_to_components_of_simplified_dual_graph(m::AbstractFTheoryModel)

Same as degrees_of_kbar_restrictions_to_components_of_dual_graph(m::AbstractFTheoryModel), but for the simplified dual graph.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> degrees_of_kbar_restrictions_to_components_of_simplified_dual_graph(qsm_model)["C2"]
2

genera_of_components_of_simplified_dual_graph(m::AbstractFTheoryModel)

This methods returns a vector with the genera of the (geometric counterparts of the) nodes/vertices of the dual graph.

julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
Hypersurface model over a concrete base

julia> genera_of_components_of_simplified_dual_graph(qsm_model)["C2"]
0