Functionality for all F-theory models

family_of_spaces(coordinate_ring::MPolyRing, grading::Matrix{Int64}, dim::Int)

Return a family of spaces that is (currently) used to build families of F-theory models, defined by using a family of base spaces for an elliptic fibration. It is specified by the following data:

• A polynomial ring. This may be thought of as the coordinate ring of the generic member in this family of spaces.
• A grading for this polynomial ring. This may be thought of as specifying certain line bundles on the generic member in this family of spaces. Of particular interest for F-theory is always the canonical bundle. For this reason, the first row is always thought of as grading the coordinate ring with regard to the canonical bundle. Or put differently, if a variable shares the weight 1 in the first row, then this means that we should think of this coordinate as a section of 1 times the canonical bundle.
• An integer, which specifies the dimension of the generic member within this family of spaces.

Note that the coordinate ring can have strictly more variables than the dimension. This is a desired feature for most, if not all, F-theory literature constructions.

julia> coord_ring, _ = QQ[:f, :g, :Kbar, :u];

julia> grading = [4 6 1 0; 0 0 0 1]
2×4 Matrix{Int64}:
 4  6  1  0
 0  0  0  1

julia> d = 3
3

julia> f = family_of_spaces(coord_ring, grading, d)
A family of spaces of dimension d = 3

coordinate_ring(f::FamilyOfSpaces)

Return the coordinate ring of a generic member of the family of spaces.

julia> ring, (f, g, Kbar, u) = QQ[:f, :g, :Kbar, :u]
(Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[f, g, Kbar, u])

julia> grading = [4 6 1 0; 0 0 0 1]
2×4 Matrix{Int64}:
 4  6  1  0
 0  0  0  1

julia> d = 3
3

julia> f = family_of_spaces(ring, grading, d)
A family of spaces of dimension d = 3

julia> coordinate_ring(f)
Multivariate polynomial ring in 4 variables f, g, Kbar, u
  over rational field

weights(f::FamilyOfSpaces)

Return the grading of the coordinate ring of a generic member of the family of spaces.

julia> ring, (f, g, Kbar, u) = QQ[:f, :g, :Kbar, :u]
(Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[f, g, Kbar, u])

julia> grading = [4 6 1 0; 0 0 0 1]
2×4 Matrix{Int64}:
 4  6  1  0
 0  0  0  1

julia> d = 3
3

julia> f = family_of_spaces(ring, grading, d)
A family of spaces of dimension d = 3

julia> weights(f)
2×4 Matrix{Int64}:
 4  6  1  0
 0  0  0  1

dim(f::FamilyOfSpaces)

Return the dimension of the generic member of the family of spaces.

julia> ring, (f, g, Kbar, u) = QQ[:f, :g, :Kbar, :u]
(Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[f, g, Kbar, u])

julia> grading = [4 6 1 0; 0 0 0 1]
2×4 Matrix{Int64}:
 4  6  1  0
 0  0  0  1

julia> d = 3
3

julia> f = family_of_spaces(ring, grading, d)
A family of spaces of dimension d = 3

julia> dim(f)
3

irrelevant_ideal(f::FamilyOfSpaces)

Return the equivalent of the irrelevant ideal for the generic member of the family of spaces.

julia> coord_ring, (f, g, Kbar, u) = QQ[:f, :g, :Kbar, :u]
(Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[f, g, Kbar, u])

julia> grading = [4 6 1 0; 0 0 0 1]
2×4 Matrix{Int64}:
 4  6  1  0
 0  0  0  1

julia> d = 3
3

julia> f = family_of_spaces(coord_ring, grading, d)
A family of spaces of dimension d = 3

julia> irrelevant_ideal(f)
Ideal generated by
  u
  Kbar
  g
  f

ideal_of_linear_relations(f::FamilyOfSpaces)

Return the equivalent of the ideal of linear relations for the generic member of the family of spaces.

julia> coord_ring, (f, g, Kbar, u) = QQ[:f, :g, :Kbar, :u]
(Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[f, g, Kbar, u])

julia> grading = [4 6 1 0; 0 0 0 1]
2×4 Matrix{Int64}:
 4  6  1  0
 0  0  0  1

julia> f = family_of_spaces(coord_ring, grading, 3)
A family of spaces of dimension d = 3

julia> ideal_of_linear_relations(f)
Ideal generated by
  -5*f + 3*g + 2*Kbar
  -3*f + 2*g

ambient_space(m::AbstractFTheoryModel)

Return the ambient space of the F-theory 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> ambient_space(m)
A family of spaces of dimension d = 5

base_space(m::AbstractFTheoryModel)

Return the base space of the F-theory 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> base_space(m)
A family of spaces of dimension d = 3

fiber_ambient_space(m::AbstractFTheoryModel)

Return the fiber ambient space of an F-theory model.

julia> t = 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> fiber_ambient_space(t)
Normal toric variety

explicit_model_sections(m::AbstractFTheoryModel)

Return the model sections in explicit form, that is as polynomials of the base space coordinates.

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> explicit_model_sections(t)
Dict{String, QQMPolyRingElem} with 9 entries:
  "a6"  => 0
  "a21" => a21
  "a3"  => w^2*a32
  "w"   => w
  "a2"  => w*a21
  "a1"  => a1
  "a4"  => w^3*a43
  "a43" => a43
  "a32" => a32

defining_section_parametrization(m::AbstractFTheoryModel)

Return the model sections in explicit form, that is as polynomials of the base space coordinates.

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> defining_section_parametrization(t)
Dict{String, MPolyRingElem} with 4 entries:
  "a6" => 0
  "a3" => w^2*a32
  "a2" => w*a21
  "a4" => w^3*a43

classes_of_model_sections(m::AbstractFTheoryModel)

Return the divisor classes of all model sections.

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> classes_of_model_sections(t)
Dict{String, ToricDivisorClass} with 9 entries:
  "a21" => Divisor class on a normal toric variety
  "a6"  => Divisor class on a normal toric variety
  "a3"  => Divisor class on a normal toric variety
  "w"   => Divisor class on a normal toric variety
  "a2"  => Divisor class on a normal toric variety
  "a1"  => Divisor class on a normal toric variety
  "a43" => Divisor class on a normal toric variety
  "a4"  => Divisor class on a normal toric variety
  "a32" => Divisor class on a normal toric variety

defining_classes(m::AbstractFTheoryModel)

Return the defining divisor classes of the model in question.

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> defining_classes(t)
Dict{String, ToricDivisorClass} with 2 entries:
  "w"    => Divisor class on a normal toric variety
  "Kbar" => Divisor class on a normal toric variety

gauge_algebra(m::AbstractFTheoryModel)

Return the gauge algebra of the given model. If no gauge algebra is known, an error is raised. This information is typically available for all models, however.

julia> t = literature_model(arxiv_id = "1408.4808", equation = "3.190", type = "hypersurface")
Assuming that the first row of the given grading is the grading under Kbar

Hypersurface model over a not fully specified base

julia> gauge_algebra(t)
Direct sum Lie algebra
  of dimension 13
with summands
  sl_2
  sl_2
  sl_2
  sl_2
  linear Lie algebra
over field of algebraic numbers

global_gauge_quotients(m::AbstractFTheoryModel)

Return list of lists of matrices, where each list of matrices corresponds to a gauge factor of the same index given by gauge_algebra(m). These matrices are elements of the center of the corresponding gauge factor and quotienting by them replicates the action of some discrete group on the center of the lie algebra. This list combined with gauge_algebra(m) completely determines the gauge group of the model. If no gauge quotients are known, an error is raised.

julia> t = literature_model(arxiv_id = "1408.4808", equation = "3.190", type = "hypersurface")
Assuming that the first row of the given grading is the grading under Kbar

Hypersurface model over a not fully specified base

julia> global_gauge_quotients(t)
5-element Vector{Vector{String}}:
 ["-identity_matrix(C,2)", "-identity_matrix(C,2)"]
 ["-identity_matrix(C,2)"]
 ["-identity_matrix(C,2)"]
 ["-identity_matrix(C,2)", "-identity_matrix(C,2)"]
 ["-identity_matrix(C,1)"]

chern_class(m::AbstractFTheoryModel, k::Int; check::Bool = true)

If the elliptically fibered n-fold $Y_n$ underlying the F-theory model in question is given as a hypersurface in a toric ambient space, we can compute a cohomology class $h$ on the toric ambient space $X_\Sigma$, such that its restriction to $Y_n$ is the k-th Chern class $c_k$ of the tangent bundle of $Y_n$. If those assumptions are satisfied, this method returns this very cohomology class $h$, otherwise it raises an error.

The theory guarantees that the implemented algorithm works for toric ambient spaces which are smooth and complete. The check for completeness can be very time consuming. This check can be switched off by setting the optional argument check to the value false, as demonstrated below.

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

julia> h = chern_class(qsm_model, 4; check = false);

julia> is_trivial(h)
false

chern_classes(m::AbstractFTheoryModel; check::Bool = true)

If the elliptically fibered n-fold $Y_n$ underlying the F-theory model in question is given as a hypersurface in a toric ambient space, we can compute a cohomology class $h$ on the toric ambient space $X_\Sigma$, such that its restriction to $Y_n$ is the k-th Chern class $c_k$ of the tangent bundle of $Y_n$. If those assumptions are satisfied, this method returns a vector with the cohomology classes corresponding to all non-trivial Chern classes $c_k$ of $Y_n$. Otherwise, this methods raises an error.

As of right now, this method is computationally expensive for involved toric ambient spaces, such as in the example below.

The theory guarantees that the implemented algorithm works for toric ambient spaces which are simplicial and complete. The check for completeness can be very time consuming. This check can be switched off by setting the optional argument check to the value false, as demonstrated below.

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

julia> h = chern_classes(qsm_model; check = false);

julia> is_one(polynomial(h[1]))
true

julia> is_trivial(h[2])
true

julia> is_trivial(h[3])
false

euler_characteristic(m::AbstractFTheoryModel; check::Bool = true)

If the elliptically fibered n-fold $Y_n$ underlying the F-theory model in question is given as a hypersurface in a toric ambient space, we can compute the Euler characteristic. If this assumptions is satisfied, this method returns the Euler characteristic, otherwise it raises an error.

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

julia> h = euler_characteristic(qsm_model; check = false)
378

julia> h = euler_characteristic(qsm_model; check = false)
378

is_base_space_fully_specified(m::AbstractFTheoryModel)

Return true if the F-theory model has a concrete base space and false otherwise.

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> is_base_space_fully_specified(t)
false

is_calabi_yau(m::AbstractFTheoryModel; check::Bool = true)

Verify if the first Chern class of the tangent bundle of the F-theory geometry $Y_n$ vanishes. If so, this confirms that this geometry is indeed Calabi-Yau, as required by the reasoning of F-theory.

The implemented algorithm works for hypersurface, Weierstrass and global Tate models, which are defined in a toric ambient space. It expresses $c_1(Y_n)$ as the restriction of a cohomology class $h$ on the toric ambient space. This in turn requires that the toric ambient space is simplicial and complete. We provide a switch to turn off these computationally very demanding checks. This is demonstrated in the example below.

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

julia> is_calabi_yau(qsm_model, check = false)
true

is_partially_resolved(m::AbstractFTheoryModel)

Return true if resolution techniques were applied to the F-theory model, thereby potentially resolving its singularities. Otherwise, return false.

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> is_partially_resolved(t)
false

julia> t2 = blow_up(t, ["x", "y", "x1"]; coordinate_name = "e1")
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> is_partially_resolved(t2)
true

verify_euler_characteristic_from_hodge_numbers(m::AbstractFTheoryModel; check::Bool = true)

Verify if the Euler characteristic, as computed from integrating the 4-th Chern class, agrees with the results obtained from using the alternating sum of the Hodge numbers. If so, this method returns true. However, should information be missing, (e.g. some Hodge numbers), or the dimension of the F-theory model differ form 4, then this method raises an error.

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

julia> verify_euler_characteristic_from_hodge_numbers(qsm_model, check = false)
true

blow_up(m::AbstractFTheoryModel, ideal_gens::Vector{String}; coordinate_name::String = "e")

Resolve an F-theory model by blowing up a locus in the ambient space.

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)

julia> blow_up(t, ["x", "y", "x1"]; coordinate_name = "e1")
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)

Here is an example for a Weierstrass model.

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> blow_up(w, ["x", "y", "x1"]; coordinate_name = "e1")
Partially resolved Weierstrass model over a concrete base -- U(1) Weierstrass model based on arXiv paper 1208.2695 Eq. (B.19)

blow_up(m::AbstractFTheoryModel, I::MPolyIdeal; coordinate_name::String = "e")

Resolve an F-theory model by blowing up a locus in the ambient space.

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)

julia> x1, x2, x3, x4, x, y, z = gens(cox_ring(ambient_space(t)))
7-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
 x1
 x2
 x3
 x4
 x
 y
 z

julia> blow_up(t, ideal([x, y, x1]); coordinate_name = "e1")
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)

blow_up(m::AbstractFTheoryModel, I::AbsIdealSheaf; coordinate_name::String = "e")

Resolve an F-theory model by blowing up a locus in the ambient space. For this method, the blowup center is encoded by an ideal sheaf.

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)

julia> x1, x2, x3, x4, x, y, z = gens(cox_ring(ambient_space(t)))
7-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
 x1
 x2
 x3
 x4
 x
 y
 z

julia> blowup_center = ideal_sheaf(ambient_space(t), ideal([x, y, x1]))
Sheaf of ideals
  on normal, simplicial toric variety
with restrictions
   1: Ideal (x_5_1, x_4_1, x_1_1)
   2: Ideal (1)
   3: Ideal (x_5_3, x_4_3, x_1_3)
   4: Ideal (x_5_4, x_4_4, x_1_4)
   5: Ideal (1)
   6: Ideal (1)
   7: Ideal (1)
   8: Ideal (1)
   9: Ideal (1)
  10: Ideal (1)
  11: Ideal (1)
  12: Ideal (1)

julia> blow_up(t, blowup_center; coordinate_name = "e1")
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)

tune(m::AbstractFTheoryModel, p::MPolyRingElem; completeness_check::Bool = true)

Tune an F-theory model by replacing the hypersurface equation by a custom (polynomial) equation. The latter can be any type of polynomial: a Tate polynomial, a Weierstrass polynomial or a general polynomial. We do not conduct checks to tell which type the provided polynomial is. Consequently, this tuning will always return a hypersurface model.

Note that there is less functionality for hypersurface models than for Weierstrass or Tate models. For instance, singular_loci can (currently) not be computed for hypersurface models.

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)

julia> x1, x2, x3, x4, x, y, z = gens(parent(tate_polynomial(t)))
7-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
 x1
 x2
 x3
 x4
 x
 y
 z

julia> new_tate_polynomial = x^3 - y^2 - x * y * z * x4^4
-x4^4*x*y*z + x^3 - y^2

julia> tuned_t = tune(t, new_tate_polynomial)
Hypersurface model over a concrete base

julia> hypersurface_equation(tuned_t) == new_tate_polynomial
true

julia> base_space(tuned_t) == base_space(t)
true

put_over_concrete_base(m::AbstractFTheoryModel, concrete_data::Dict{String, <:Any}; completeness_check::Bool = true)

Put an F-theory model defined over a family of spaces over a concrete base.

Currently, this functionality is limited to Tate and Weierstrass models.

Examples

julia> t = literature_model(arxiv_id = "1109.3454", equation = "3.1", completeness_check = false)
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> B3 = projective_space(NormalToricVariety, 3)
Normal toric variety

julia> w_bundle = toric_line_bundle(torusinvariant_prime_divisors(B3)[1])
Toric line bundle on a normal toric variety

julia> kbar = anticanonical_bundle(B3)
Toric line bundle on a normal toric variety

julia> w = generic_section(w_bundle);

julia> a21 = generic_section(kbar^2 * w_bundle^(-1));

julia> a32 = generic_section(kbar^3 * w_bundle^(-2));

julia> a43 = generic_section(kbar^4 * w_bundle^(-3));

julia> t2 = put_over_concrete_base(t, Dict("base" => B3, "w" => w, "a21" => a21, "a32" => a32, "a43" => a43), completeness_check = false)
Global Tate model over a concrete base