The Quadrillion F-Theory Standard Models (QSMs)

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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.

Examples

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).

Examples

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.

Examples

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.

Examples

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