From 8fa5603bd6a9ebc7bc5ed564ae41f8370d27507a Mon Sep 17 00:00:00 2001
From: Martin Bies <MBies87@googlemail.com>
Date: Wed, 24 Jul 2024 09:10:53 +0100
Subject: [PATCH 1/6] [ToricVarieties] Introduce flags to switch off time
 consuming checks

---
 .../ToricVarieties/CohomologyClasses.md        |  4 ++--
 .../CohomologyClasses/methods.jl               | 15 +++++++++++----
 .../CohomologyClasses/special_attributes.jl    | 18 ++++++++++++------
 3 files changed, 25 insertions(+), 12 deletions(-)

diff --git a/docs/src/AlgebraicGeometry/ToricVarieties/CohomologyClasses.md b/docs/src/AlgebraicGeometry/ToricVarieties/CohomologyClasses.md
index 26a3e87818e5..5c05bbefb755 100644
--- a/docs/src/AlgebraicGeometry/ToricVarieties/CohomologyClasses.md
+++ b/docs/src/AlgebraicGeometry/ToricVarieties/CohomologyClasses.md
@@ -54,14 +54,14 @@ polynomial(ring::MPolyQuoRing, c::CohomologyClass)
 ## Methods
 
 ```@docs
-integrate(c::CohomologyClass)
+integrate(c::CohomologyClass; check::Bool = true)
 ```
 
 
 ## Special attributes of toric varieties
 
 ```@docs
-cohomology_ring(v::NormalToricVarietyType)
+cohomology_ring(v::NormalToricVarietyType; check::Bool = true)
 volume_form(v::NormalToricVariety)
 intersection_form(v::NormalToricVariety)
 ```
diff --git a/src/AlgebraicGeometry/ToricVarieties/CohomologyClasses/methods.jl b/src/AlgebraicGeometry/ToricVarieties/CohomologyClasses/methods.jl
index dfa115c6e579..23eca4350f2f 100644
--- a/src/AlgebraicGeometry/ToricVarieties/CohomologyClasses/methods.jl
+++ b/src/AlgebraicGeometry/ToricVarieties/CohomologyClasses/methods.jl
@@ -1,9 +1,14 @@
 @doc raw"""
-    integrate(c::CohomologyClass)
+    integrate(c::CohomologyClass; check::Bool = true)
 
 Integrate the cohomolgy class `c` over the normal
 toric variety `toric_variety(c)`.
 
+The theory underlying this method requires that the toric variety
+in question is simplicial and complete. The check of completeness
+may take a long time to complete. If desired, this can be switched
+off by setting the optional argument `check` to the value `false`.
+
 # Examples
 ```jldoctest
 julia> dP3 = del_pezzo_surface(NormalToricVariety, 3)
@@ -88,10 +93,12 @@ julia> integrate(cohomology_class(anticanonical_divisor_class(X))^3)
 62
 ```
 """
-function integrate(c::CohomologyClass)
+function integrate(c::CohomologyClass; check::Bool = true)
     # can only integrate if the variety is simplicial, complete
-    @req is_simplicial(toric_variety(c)) && is_complete(toric_variety(c)) "Integration only supported over complete and simplicial toric varieties"
-    
+    if check
+      @req is_simplicial(toric_variety(c)) && is_complete(toric_variety(c)) "Integration only supported over complete and simplicial toric varieties"
+    end
+
     # if the intersection form is known, we can use it
     if has_attribute(toric_variety(c), :_intersection_form_via_exponents)
         intersection_dict = _intersection_form_via_exponents(toric_variety(c))
diff --git a/src/AlgebraicGeometry/ToricVarieties/CohomologyClasses/special_attributes.jl b/src/AlgebraicGeometry/ToricVarieties/CohomologyClasses/special_attributes.jl
index 0892d9a927a1..7a75886b86e4 100644
--- a/src/AlgebraicGeometry/ToricVarieties/CohomologyClasses/special_attributes.jl
+++ b/src/AlgebraicGeometry/ToricVarieties/CohomologyClasses/special_attributes.jl
@@ -3,7 +3,7 @@
 ########################
 
 @doc raw"""
-    cohomology_ring(v::NormalToricVarietyType)
+    cohomology_ring(v::NormalToricVarietyType; check::Bool = true)
 
 Return the cohomology ring of the simplicial and complete toric variety `v`.
 
@@ -15,12 +15,18 @@ julia> ngens(cohomology_ring(p2))
 3
 ```
 """
-@attr MPolyQuoRing function cohomology_ring(v::NormalToricVarietyType)
+function cohomology_ring(v::NormalToricVarietyType; check::Bool = true)
+  if has_attribute(v, :cohomology_ring)
+    return get_attribute(v, :cohomology_ring)
+  end
+  if check
     @req is_simplicial(v) && is_complete(v) "The cohomology ring is only supported for simplicial and complete toric varieties"
-    R, _ = graded_polynomial_ring(coefficient_ring(v), coordinate_names(v), cached = false)
-    linear_relations = ideal_of_linear_relations(R, v)
-    stanley_reisner = stanley_reisner_ideal(R, v)
-    return quo(R, linear_relations + stanley_reisner)[1]
+  end
+  R, _ = graded_polynomial_ring(coefficient_ring(v), coordinate_names(v), cached = false)
+  linear_relations = ideal_of_linear_relations(R, v)
+  stanley_reisner = stanley_reisner_ideal(R, v)
+  set_attribute!(v, :cohomology_ring, quo(R, linear_relations + stanley_reisner)[1])
+  return get_attribute(v, :cohomology_ring)
 end
 
 

From dee1685afaeef9e0f0f6e1e40d401bde436bec16 Mon Sep 17 00:00:00 2001
From: Martin Bies <MBies87@googlemail.com>
Date: Wed, 24 Jul 2024 09:19:04 +0100
Subject: [PATCH 2/6] [FTheoryTools] Add c1 and c2 of hypersurface, Weierstrass
 and Tate models

---
 .../FTheoryTools/docs/src/generalities.md     |   2 +
 .../src/AbstractFTheoryModels/attributes.jl   | 128 ++++++++++++++++++
 experimental/FTheoryTools/src/exports.jl      |   2 +
 3 files changed, 132 insertions(+)

diff --git a/experimental/FTheoryTools/docs/src/generalities.md b/experimental/FTheoryTools/docs/src/generalities.md
index 1c63c013e62c..cb8ebd06d2a2 100644
--- a/experimental/FTheoryTools/docs/src/generalities.md
+++ b/experimental/FTheoryTools/docs/src/generalities.md
@@ -59,6 +59,8 @@ classes_of_model_sections(m::AbstractFTheoryModel)
 defining_classes(m::AbstractFTheoryModel)
 gauge_algebra(m::AbstractFTheoryModel)
 global_gauge_quotients(m::AbstractFTheoryModel)
+chern_class_c1(m::AbstractFTheoryModel; check::Bool = true)
+chern_class_c2(m::AbstractFTheoryModel; check::Bool = true)
 ```
 
 
diff --git a/experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl b/experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl
index fbedf3c50c73..73748aa63e03 100644
--- a/experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl
+++ b/experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl
@@ -1096,6 +1096,134 @@ function global_gauge_quotients(m::AbstractFTheoryModel)
 end
 
 
+@doc raw"""
+    chern_class_c1(m::AbstractFTheoryModel)
+
+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 first Chern class
+$c_1$ of the tangent bundle of $Y_n$. If those assumptions are satisfied, this method returns
+this very cohomology class $h$, otherwise it raises and error.
+
+Note that $Y_n$ is a Calabi-Yau variety precisely if $c_1$ is trivial. Thus, the restriction
+of $h$ to the hypersurface $Y_n$ must be trivial. Upon closer inspection, in the given toric
+setting, this is equivalent to $h$ being trivial.
+
+The computation of the cohomology ring verifies if the toric variety is simplicial and
+complete. The check for it to be complete can be very time consuming. This can be switched
+off by setting the optional argument `check` to the value `falue`, as in the example below.
+
+```jldoctest
+julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
+Hypersurface model over a concrete base
+
+julia> h = chern_class_c1(qsm_model; check = false)
+Cohomology class on a normal toric variety given by 0
+
+julia> is_trivial(h)
+true
+```
+"""
+function chern_class_c1(m::AbstractFTheoryModel; check::Bool = true)
+  @req (m isa WeierstrassModel || m isa GlobalTateModel || m isa HypersurfaceModel) "First Chern class of F-theory model supported for Weierstrass, global Tate and hypersurface models only"
+  @req base_space(m) isa NormalToricVariety "First Chern class of F-theory model currently supported only for toric base"
+  @req ambient_space(m) isa NormalToricVariety "Firfalsest Chern class of F-theory model currently supported only for toric ambient space"
+
+  # Check if the answer is known
+  if has_attribute(m, :chern_class_c1)
+    return get_attribute(m, :chern_class_c1)
+  end
+
+  # Trigger potential short-cut computation of cohomology ring
+  cohomology_ring(ambient_space(m); check)
+
+  # Compute the cohomology class corresponding to the hypersurface equation
+  if m isa WeierstrassModel
+    cl = toric_divisor_class(ambient_space(m), degree(weierstrass_polynomial(m)))
+  end
+  if m isa GlobalTateModel
+    cl = toric_divisor_class(ambient_space(m), degree(tate_polynomial(m)))
+  end
+  if m isa HypersurfaceModel
+    cl = toric_divisor_class(ambient_space(m), degree(hypersurface_equation(m)))
+  end
+  cy = cohomology_class(cl)
+
+  # Compute and set h
+  c1_t_ambient = cohomology_class(anticanonical_divisor(ambient_space(m)))
+  set_attribute!(m, :chern_class_c1, c1_t_ambient - cy)
+  return c1_t_ambient - cy
+end
+
+
+@doc raw"""
+    chern_class_c2(m::AbstractFTheoryModel)
+
+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 2nd Chern class
+$c_2$ of the tangent bundle of $Y_n$. If those assumptions are satisfied, this method returns
+this very cohomology class $h$, otherwise it raises and error.
+
+As of right now, this method is computationally quite expensive for involved toric varieties,
+such as in the example below. Therefore, think carefully if you truly want to compute this quantity.
+
+The computation of the cohomology ring verifies if the toric variety is simplicial and
+complete. The check for it to be complete can be very time consuming. This can be switched
+off by setting the optional argument `check` to the value `falue`, as in the example below.
+
+```jldoctest
+julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
+Hypersurface model over a concrete base
+
+julia> h = chern_class_c2(qsm_model; check = false)
+Cohomology class on a normal toric variety given by x5*x32 + 2*x5*x33 + 3*x5*x34 + 2*x5*x35 + x5*x36 + x6*x32 + 2*x6*x33 + 3*x6*x34 + 2*x6*x35 + x6*x36 + x7*x32 + 2*x7*x33 + 3*x7*x34 + 2*x7*x35 + x7*x36 + 13*x8*x23 + 11//2*x10*x23 - 6*x15*x26 + x15*x32 + 2*x15*x33 + 3*x15*x34 + 2*x15*x35 + x15*x36 - 4*x16^2 + x16*x32 + 2*x16*x33 + 3*x16*x34 + 2*x16*x35 + x16*x36 - 52*x17*x24 + 2*x17*x32 + 4*x17*x33 + 6*x17*x34 + 4*x17*x35 + 2*x17*x36 - 24*x18^2 + 7*x18*x25 + 2*x18*x32 + 4*x18*x33 + 6*x18*x34 + 4*x18*x35 + 2*x18*x36 + 3*x19*x32 + 6*x19*x33 + 9*x19*x34 + 6*x19*x35 + 3*x19*x36 + 20*x20^2 + 200//3*x20*x21 - 14*x20*x25 + 2*x20*x32 + 4*x20*x33 + 6*x20*x34 + 4*x20*x35 + 2*x20*x36 + 28//3*x21^2 - 163//3*x21*x24 + 2*x21*x32 + 4*x21*x33 + 6*x21*x34 + 4*x21*x35 + 2*x21*x36 + 21*x22^2 + 11//2*x22*x23 + 32*x22*x25 + 15//2*x23^2 - 17*x24^2 - 10*x24*x27 + x24*x32 + 2*x24*x33 + 3*x24*x34 + 2*x24*x35 + x24*x36 + 7*x25^2 + x25*x32 + 2*x25*x33 + 3*x25*x34 + 2*x25*x35 + x25*x36 + 2*x26^2 + 4*x26*x27 - 4*x26*x28 + 5*x27^2 + x27*x32 + 2*x27*x33 + 3*x27*x34 + 2*x27*x35 + x27*x36 - 3*x28^2 - 6*x29^2 + 7*x35*x36
+
+julia> is_trivial(h)
+false
+```
+"""
+function chern_class_c2(m::AbstractFTheoryModel; check::Bool = true)
+  @req (m isa WeierstrassModel || m isa GlobalTateModel || m isa HypersurfaceModel) "First Chern class of F-theory model supported for Weierstrass, global Tate and hypersurface models only"
+  @req base_space(m) isa NormalToricVariety "First Chern class of F-theory model currently supported only for toric base"
+  @req ambient_space(m) isa NormalToricVariety "First Chern class of F-theory model currently supported only for toric ambient space"
+
+  # Check if the answer is known
+  if has_attribute(m, :chern_class_c2)
+    return get_attribute(m, :chern_class_c2)
+  end
+
+  # Trigger potential short-cut computation of cohomology ring
+  cohomology_ring(ambient_space(m); check)
+
+  # Compute the cohomology class corresponding to the hypersurface equation
+  if m isa WeierstrassModel
+    cl = toric_divisor_class(ambient_space(m), degree(weierstrass_polynomial(m)))
+  end
+  if m isa GlobalTateModel
+    cl = toric_divisor_class(ambient_space(m), degree(tate_polynomial(m)))
+  end
+  if m isa HypersurfaceModel
+    cl = toric_divisor_class(ambient_space(m), degree(hypersurface_equation(m)))
+  end
+  cy = cohomology_class(cl)
+
+  # Compute sum of products of cohomolgy classes of torus invariant prime divisors
+  c_ds = [polynomial(cohomology_class(d)) for d in torusinvariant_prime_divisors(ambient_space(m))]
+  c2_t_ambient = zero(cohomology_ring(ambient_space(m)))
+  for i in 1:length(c_ds)
+    for j in i+1:length(c_ds)
+      c2_t_ambient = c2_t_ambient + c_ds[i]*c_ds[j]
+    end
+  end
+  c2_t_ambient = cohomology_class(ambient_space(m), c2_t_ambient)
+
+  # Compute and set h
+  h = c2_t_ambient - cy * chern_class_c1(m; check = check)
+  set_attribute!(m, :chern_class_c2, h)
+  return h
+end
+
 
 ##########################################
 ### (4) Attributes specially for the QSMs
diff --git a/experimental/FTheoryTools/src/exports.jl b/experimental/FTheoryTools/src/exports.jl
index a277d429f29c..f0183af7f515 100644
--- a/experimental/FTheoryTools/src/exports.jl
+++ b/experimental/FTheoryTools/src/exports.jl
@@ -35,6 +35,8 @@ export associated_literature_models
 export base_space
 export blow_up
 export calabi_yau_hypersurface
+export chern_class_c1
+export chern_class_c2
 export classes_of_model_sections
 export components_of_dual_graph
 export components_of_simplified_dual_graph

From daed9f34c32d544c4b6ac7f7340c501c45a3ee02 Mon Sep 17 00:00:00 2001
From: Martin Bies <MBies87@googlemail.com>
Date: Wed, 24 Jul 2024 09:26:53 +0100
Subject: [PATCH 3/6] [FTheoryTools] First implementation of G4-fluxes

---
 docs/oscar_references.bib                     |  13 ++
 experimental/FTheoryTools/docs/doc.main       |   3 +-
 experimental/FTheoryTools/docs/src/g4.md      |  35 +++++
 experimental/FTheoryTools/src/FTheoryTools.jl |   4 +
 .../FTheoryTools/src/G4Fluxes/attributes.jl   |  48 +++++++
 .../FTheoryTools/src/G4Fluxes/constructors.jl | 120 ++++++++++++++++++
 .../FTheoryTools/src/G4Fluxes/properties.jl   |  69 ++++++++++
 experimental/FTheoryTools/src/exports.jl      |   3 +
 experimental/FTheoryTools/src/types.jl        |  11 ++
 9 files changed, 305 insertions(+), 1 deletion(-)
 create mode 100644 experimental/FTheoryTools/docs/src/g4.md
 create mode 100644 experimental/FTheoryTools/src/G4Fluxes/attributes.jl
 create mode 100644 experimental/FTheoryTools/src/G4Fluxes/constructors.jl
 create mode 100644 experimental/FTheoryTools/src/G4Fluxes/properties.jl

diff --git a/docs/oscar_references.bib b/docs/oscar_references.bib
index 47fbde9fe0e0..e7963429c25b 100644
--- a/docs/oscar_references.bib
+++ b/docs/oscar_references.bib
@@ -2238,6 +2238,19 @@ @Article{Wit88
   reportnumber  = {IASSNS-HEP-88/7}
 }
 
+@Article{Wit97,
+  author        = {Witten, Edward},
+  title         = {{On flux quantization in M theory and the effective action}},
+  journal       = {J. Geom. Phys.},
+  volume        = {22},
+  pages         = {1--13},
+  year          = {1997},
+  doi           = {10.1016/S0393-0440(96)00042-3},
+  eprint        = {hep-th/9609122},
+  archiveprefix = {arXiv},
+  reportnumber  = {IASSNS-HEP-96-96}
+}
+
 @Article{Yam18,
   author        = {Yamagishi, Ryo},
   title         = {On smoothness of minimal models of quotient singularities by finite subgroups of ${\rm SL}_n(\mathbb
diff --git a/experimental/FTheoryTools/docs/doc.main b/experimental/FTheoryTools/docs/doc.main
index a7e330011f80..7f322b33fc73 100644
--- a/experimental/FTheoryTools/docs/doc.main
+++ b/experimental/FTheoryTools/docs/doc.main
@@ -5,6 +5,7 @@
       "weierstrass.md",
       "tate.md",
       "hypersurface.md",
-      "literature.md"
+      "literature.md",
+      "g4.md"
    ],
 ]
diff --git a/experimental/FTheoryTools/docs/src/g4.md b/experimental/FTheoryTools/docs/src/g4.md
new file mode 100644
index 000000000000..a6c5c8095da5
--- /dev/null
+++ b/experimental/FTheoryTools/docs/src/g4.md
@@ -0,0 +1,35 @@
+```@meta
+CurrentModule = Oscar
+```
+
+# G4-Fluxes
+
+$G_4$-fluxes are at the heart of F-theory model building.
+
+
+## Constructors
+
+We currently support the following constructor:
+```@docs
+g4_flux(model::AbstractFTheoryModel, class::CohomologyClass)
+```
+
+
+## Attributes
+
+We currently support the following attributes:
+```@docs
+model(gf::G4Flux)
+cohomology_class(gf::G4Flux)
+```
+
+
+## Properties
+
+We currently support the following properties:
+```@docs
+passes_elementary_quantization_checks(gf::G4Flux)
+```
+
+
+## Methods
diff --git a/experimental/FTheoryTools/src/FTheoryTools.jl b/experimental/FTheoryTools/src/FTheoryTools.jl
index 3774867fb0aa..d8d4cccfa42f 100644
--- a/experimental/FTheoryTools/src/FTheoryTools.jl
+++ b/experimental/FTheoryTools/src/FTheoryTools.jl
@@ -26,6 +26,10 @@ include("standard_constructions.jl")
 include("LiteratureModels/constructors.jl")
 include("LiteratureModels/create_index.jl")
 
+include("G4Fluxes/constructors.jl")
+include("G4Fluxes/attributes.jl")
+include("G4Fluxes/properties.jl")
+
 include("Serialization/tate_models.jl")
 include("Serialization/weierstrass_models.jl")
 include("Serialization/hypersurface_models.jl")
diff --git a/experimental/FTheoryTools/src/G4Fluxes/attributes.jl b/experimental/FTheoryTools/src/G4Fluxes/attributes.jl
new file mode 100644
index 000000000000..332d9b4a31de
--- /dev/null
+++ b/experimental/FTheoryTools/src/G4Fluxes/attributes.jl
@@ -0,0 +1,48 @@
+#####################################################
+# 1 Basic attributes
+#####################################################
+
+@doc raw"""
+    model(gf::G4Flux)
+
+Return the F-theory model for which this $G_4$-flux candidate is defined.
+
+```jldoctest
+julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
+Hypersurface model over a concrete base
+
+julia> cohomology_ring(ambient_space(qsm_model), check = false);
+
+julia> g4_class = cohomology_class(anticanonical_divisor_class(ambient_space(qsm_model)))^2;
+
+julia> g4f = g4_flux(qsm_model, g4_class, check = false)
+G4-flux candidate lacking elementary quantization checks
+
+julia> model(g4f)
+Hypersurface model over a concrete base
+```
+"""
+model(gf::G4Flux) = gf.model
+
+
+@doc raw"""
+    cohomology_class(gf::G4Flux)
+
+Return the cohomology class which defines the $G_4$-flux candidate.
+
+```jldoctest
+julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
+Hypersurface model over a concrete base
+
+julia> cohomology_ring(ambient_space(qsm_model), check = false);
+
+julia> g4_class = cohomology_class(anticanonical_divisor_class(ambient_space(qsm_model)))^2;
+
+julia> g4f = g4_flux(qsm_model, g4_class, check = false)
+G4-flux candidate lacking elementary quantization checks
+
+julia> cohomology_class(g4f) == g4_class
+true
+```
+"""
+cohomology_class(gf::G4Flux) = gf.class
diff --git a/experimental/FTheoryTools/src/G4Fluxes/constructors.jl b/experimental/FTheoryTools/src/G4Fluxes/constructors.jl
new file mode 100644
index 000000000000..b09296337e83
--- /dev/null
+++ b/experimental/FTheoryTools/src/G4Fluxes/constructors.jl
@@ -0,0 +1,120 @@
+################
+# 1: Constructor
+################
+
+@doc raw"""
+    g4_flux(model::AbstractFTheoryModel, class::CohomologyClass)
+
+Construct a G4-flux candidate on an F-theory model. This functionality is
+currently limited to
+- Weierstrass models,
+- global Tate models,
+- hypersurface models.
+Furthermore, our functionality requires a concrete geometry. That is,
+the base space as well as the ambient space must be toric varieties.
+In the toric ambient space $X_\Sigma$, the elliptically fibered space $Y$
+that defines the F-theory model, is given by a hypersurface (cut out by
+the Weierstrass, Tate or hypersurface polynomial, respectively).
+
+In this setting, we assume that a $G_4$-flux candidate is represented by a
+cohomology class $h$ in $H^{(2,2)} (X_\Sigma)$. The actual $G_4$-flux candidate
+is then obtained by restricting $h$ to $Y$.
+
+It is worth recalling that the $G_4$-flux candidate is subject to the quantization
+condition $G_4 + \frac{1}{2} c_2(Y) \in H^{/2,2)}( Y_, \mathbb{Z})$ (see [Wit97](@cite)).
+This condition is very hard to verify. However, it is relatively easy to gather
+evidence for this condition to be satisfied/show that it is violated. To this end, let
+$D_1$, $D_2$ be two toric divisors in $X_\Sigma$, then the topological intersection number
+$\left[ h|_Y \right] \cdot \left[ P \right] \cdot \left[ D_1 \right] \cdot \left[ D_2 \right]$
+must be an integer. Even this rather elementary check can be computationally expensive.
+Users can therefore decide to skip this check upon construction by setting the parameter
+`check` to the value `false`.
+
+Another bottleneck can be the computation of the cohomology ring, which is necessary to
+work with cohomology classes on the toric ambient space, which in turn define the G4-flux,
+as explained above. The reason for this is, that by employing the theory explained in
+[CLS11](@cite), we can only work out the cohomology ring of simpicial and complete (i.e. compact)
+toric varieties. However, checking if a toric variety is complete (i.e. compact) can take
+a long time. If the geometry in question is involved and you already know that the variety
+is simplicial and complete, then we recommend to trigger the computation of the cohomology
+ring with `check = false`. This will avoid this time consuming test.
+
+An example is in order.
+
+# Examples
+```jldoctest
+julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
+Hypersurface model over a concrete base
+
+julia> cohomology_ring(ambient_space(qsm_model), check = false);
+
+julia> g4_class = cohomology_class(anticanonical_divisor_class(ambient_space(qsm_model)))^2;
+
+julia> g4f = g4_flux(qsm_model, g4_class)
+G4-flux candidate
+
+julia> g4f2 = g4_flux(qsm_model, g4_class, check = false)
+G4-flux candidate lacking elementary quantization checks
+```
+"""
+function g4_flux(m::AbstractFTheoryModel, g4_class::CohomologyClass; check::Bool = true)
+  @req (m isa WeierstrassModel || m isa GlobalTateModel || m isa HypersurfaceModel) "G4-fluxes only supported for Weierstrass, global Tate and hypersurface models"
+  @req base_space(m) isa NormalToricVariety "G4-flux currently supported only for toric base"
+  @req ambient_space(m) isa NormalToricVariety "G4-flux currently supported only for toric ambient space"
+  g4_candidate = G4Flux(m, g4_class)
+  if check && !passes_elementary_quantization_checks(g4_candidate)
+    error("Given G4-flux candidate found to violate quantization condition")
+  end
+  return g4_candidate
+end
+
+
+################################################
+# 2: Equality and hash
+################################################
+
+function Base.:(==)(gf1::G4Flux, gf2::G4Flux)
+  # G4-fluxes can only be equal if they are defined for identically the same model
+  model(gf1) !== model(gf2) && return false
+
+  # Currently, can only decide equality for Weierstrass, global Tate and hypersurface models
+  if (m isa WeierstrassModel || m isa GlobalTateModel || m isa HypersurfaceModel) == false
+    error("Can currently only decide equality of G4-fluxes for Weierstrass, global Tate and hypersurface models")
+  end
+
+  # Compute the cohomology class corresponding to the hypersurface equation
+  if m isa WeierstrassModel
+    cl = toric_divisor_class(ambient_space(m), degree(weierstrass_polynomial(m)))
+  end
+  if m isa GlobalTateModel
+    cl = toric_divisor_class(ambient_space(m), degree(tate_polynomial(m)))
+  end
+  if m isa HypersurfaceModel
+    cl = toric_divisor_class(ambient_space(m), degree(hypersurface_equation(m)))
+  end
+  cy = cohomology_class(cl)
+
+  # Now can return the result
+  return cy * cohomology_class(gf1) == cy * cohomology_class(gf2)
+
+end
+
+function Base.hash(gf::G4Flux, h::UInt)
+  b = 0x92bd6ac4f87d834e % UInt
+  h = hash(model(gf), h)
+  h = hash(cohomology_class(gf), h)
+  return xor(h, b)
+end
+
+
+################################################
+# 3: Display
+################################################
+
+function Base.show(io::IO, g4::G4Flux)
+  properties_string = String["G4-flux candidate"]
+  if !has_attribute(g4, :passes_elementary_quantization_checks)
+    push!(properties_string, "lacking elementary quantization checks")
+  end
+  join(io, properties_string, " ")
+end
diff --git a/experimental/FTheoryTools/src/G4Fluxes/properties.jl b/experimental/FTheoryTools/src/G4Fluxes/properties.jl
new file mode 100644
index 000000000000..e4c6f9142f46
--- /dev/null
+++ b/experimental/FTheoryTools/src/G4Fluxes/properties.jl
@@ -0,0 +1,69 @@
+#####################################################
+# 1 Basic properties
+#####################################################
+
+@doc raw"""
+    passes_elementary_quantization_checks(gf::G4Flux)
+
+G4-fluxes are subject to the quantization condition
+[Wit97](@cite) $G_4 + \frac{1}{2} c_2(Y) \in H^{(2,2)}(Y, \mathbb{Z})$.
+It is hard to verify that this condition is met. However,
+we can execute a number of simple consistency checks, by
+verifying that $\int_{Y}{G_4 \wedge [D_1] \wedge [D_2]} \in \mathbb{Z}$
+for any two toric divisors $D_1$, $D_2$. If all of these
+simple consistency checks are met, this method will return
+`true` and otherwise `false`.
+
+It is worth mentioning that currently (July 2024), we only
+support this check for $G_4$-fluxes defined on Weierstrass,
+global Tate and hypersurface models. If this condition is not
+met, this method will return an error.
+
+```jldoctest
+julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
+Hypersurface model over a concrete base
+
+julia> cohomology_ring(ambient_space(qsm_model), check = false);
+
+julia> g4_class = cohomology_class(anticanonical_divisor_class(ambient_space(qsm_model)))^2;
+
+julia> g4 = g4_flux(qsm_model, g4_class, check = false)
+G4-flux candidate lacking elementary quantization checks
+
+julia> passes_elementary_quantization_checks(g4)
+true
+```
+"""
+@attr Bool function passes_elementary_quantization_checks(g4::G4Flux)
+  m = model(g4)
+  @req (m isa WeierstrassModel || m isa GlobalTateModel || m isa HypersurfaceModel) "Elementary quantization checks for  G4-fluxes only supported for Weierstrass, global Tate and hypersurface models"
+  @req base_space(m) isa NormalToricVariety "Elementary quantization checks for G4-flux currently supported only for toric base"
+  @req ambient_space(m) isa NormalToricVariety "Elementary quantization checks for G4-flux currently supported only for toric ambient space"
+
+  # Compute the cohomology class corresponding to the hypersurface equation
+  if m isa WeierstrassModel
+    cl = toric_divisor_class(ambient_space(m), degree(weierstrass_polynomial(m)))
+  end
+  if m isa GlobalTateModel
+    cl = toric_divisor_class(ambient_space(m), degree(tate_polynomial(m)))
+  end
+  if m isa HypersurfaceModel
+    cl = toric_divisor_class(ambient_space(m), degree(hypersurface_equation(m)))
+  end
+  cy = polynomial(cohomology_class(cl))
+
+  # Now check quantization condition G4 + 1/2 c2 is integral.
+  c_ds = [polynomial(cohomology_class(d)) for d in torusinvariant_prime_divisors(ambient_space(m))]
+
+  # explicitly switched off an expensive test in the following line
+  twist_g4 = polynomial(cohomology_class(g4) + 1//2 * chern_class_c2(m; check = false))
+
+  # now execute elementary checks of the quantization condition
+  for i in 1:length(c_ds)
+    for j in i:length(c_ds)
+      numb = integrate(cohomology_class(ambient_space(m), twist_g4 * cy * c_ds[i] * c_ds[j]); check = false)
+      !is_integer(numb) && return false
+    end
+  end
+  return true
+end
diff --git a/experimental/FTheoryTools/src/exports.jl b/experimental/FTheoryTools/src/exports.jl
index f0183af7f515..4c2931d4e508 100644
--- a/experimental/FTheoryTools/src/exports.jl
+++ b/experimental/FTheoryTools/src/exports.jl
@@ -53,6 +53,7 @@ export estimated_number_of_triangulations
 export explicit_model_sections
 export family_of_spaces
 export fiber_ambient_space
+export g4_flux
 export gauge_algebra
 export genera_of_ci_curves
 export genera_of_components_of_dual_graph
@@ -123,6 +124,7 @@ export kbar3
 export literature_identifier
 export literature_model
 export max_lattice_pts_in_facet
+export model
 export model_description
 export model_index
 export model_parameters
@@ -130,6 +132,7 @@ export paper_authors
 export paper_buzzwords
 export paper_description
 export paper_title
+export passes_elementary_quantization_checks
 export polytope_index
 export put_over_concrete_base
 export birational_literature_models
diff --git a/experimental/FTheoryTools/src/types.jl b/experimental/FTheoryTools/src/types.jl
index 4f696f158928..8b67d0e3585f 100644
--- a/experimental/FTheoryTools/src/types.jl
+++ b/experimental/FTheoryTools/src/types.jl
@@ -226,3 +226,14 @@ struct QSMModel
   genus_of_components_of_simplified_dual_graph::Dict{String, Int64}
   
 end
+
+
+################################################
+# 4: The julia type for G4-fluxes
+################################################
+
+@attributes mutable struct G4Flux
+  model::AbstractFTheoryModel
+  class::CohomologyClass
+  G4Flux(model::AbstractFTheoryModel, class::CohomologyClass) = new(model, class)
+end

From e6171e0535bf3c97e3aa0642fe6dd7d850b92da6 Mon Sep 17 00:00:00 2001
From: Martin Bies <HereAround@users.noreply.github.com>
Date: Thu, 25 Jul 2024 22:39:37 +0100
Subject: [PATCH 4/6] [FTheoryTools] Suppress unnessary output

Co-authored-by: Lars Kastner <lkastner@users.noreply.github.com>
---
 .../FTheoryTools/src/AbstractFTheoryModels/attributes.jl       | 3 +--
 1 file changed, 1 insertion(+), 2 deletions(-)

diff --git a/experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl b/experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl
index 73748aa63e03..7dae457dcf7c 100644
--- a/experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl
+++ b/experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl
@@ -1176,8 +1176,7 @@ off by setting the optional argument `check` to the value `falue`, as in the exa
 julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
 Hypersurface model over a concrete base
 
-julia> h = chern_class_c2(qsm_model; check = false)
-Cohomology class on a normal toric variety given by x5*x32 + 2*x5*x33 + 3*x5*x34 + 2*x5*x35 + x5*x36 + x6*x32 + 2*x6*x33 + 3*x6*x34 + 2*x6*x35 + x6*x36 + x7*x32 + 2*x7*x33 + 3*x7*x34 + 2*x7*x35 + x7*x36 + 13*x8*x23 + 11//2*x10*x23 - 6*x15*x26 + x15*x32 + 2*x15*x33 + 3*x15*x34 + 2*x15*x35 + x15*x36 - 4*x16^2 + x16*x32 + 2*x16*x33 + 3*x16*x34 + 2*x16*x35 + x16*x36 - 52*x17*x24 + 2*x17*x32 + 4*x17*x33 + 6*x17*x34 + 4*x17*x35 + 2*x17*x36 - 24*x18^2 + 7*x18*x25 + 2*x18*x32 + 4*x18*x33 + 6*x18*x34 + 4*x18*x35 + 2*x18*x36 + 3*x19*x32 + 6*x19*x33 + 9*x19*x34 + 6*x19*x35 + 3*x19*x36 + 20*x20^2 + 200//3*x20*x21 - 14*x20*x25 + 2*x20*x32 + 4*x20*x33 + 6*x20*x34 + 4*x20*x35 + 2*x20*x36 + 28//3*x21^2 - 163//3*x21*x24 + 2*x21*x32 + 4*x21*x33 + 6*x21*x34 + 4*x21*x35 + 2*x21*x36 + 21*x22^2 + 11//2*x22*x23 + 32*x22*x25 + 15//2*x23^2 - 17*x24^2 - 10*x24*x27 + x24*x32 + 2*x24*x33 + 3*x24*x34 + 2*x24*x35 + x24*x36 + 7*x25^2 + x25*x32 + 2*x25*x33 + 3*x25*x34 + 2*x25*x35 + x25*x36 + 2*x26^2 + 4*x26*x27 - 4*x26*x28 + 5*x27^2 + x27*x32 + 2*x27*x33 + 3*x27*x34 + 2*x27*x35 + x27*x36 - 3*x28^2 - 6*x29^2 + 7*x35*x36
+julia> h = chern_class_c2(qsm_model; check = false);
 
 julia> is_trivial(h)
 false

From 5db0c97a38828f5cb6370bdec00e43cc19c2d7d7 Mon Sep 17 00:00:00 2001
From: Martin Bies <MBies87@googlemail.com>
Date: Fri, 26 Jul 2024 10:08:12 +0100
Subject: [PATCH 5/6] [FTheoryTools] Code improvement

Co-authored-by: Andrew P Turner <apturner@mac.com>
---
 .../FTheoryTools/src/AbstractFTheoryModels/attributes.jl | 9 ++-------
 1 file changed, 2 insertions(+), 7 deletions(-)

diff --git a/experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl b/experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl
index 7dae457dcf7c..8314a30dd268 100644
--- a/experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl
+++ b/experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl
@@ -1152,7 +1152,7 @@ function chern_class_c1(m::AbstractFTheoryModel; check::Bool = true)
   # Compute and set h
   c1_t_ambient = cohomology_class(anticanonical_divisor(ambient_space(m)))
   set_attribute!(m, :chern_class_c1, c1_t_ambient - cy)
-  return c1_t_ambient - cy
+  return get_attribute(m, :chern_class_c1)
 end
 
 
@@ -1209,12 +1209,7 @@ function chern_class_c2(m::AbstractFTheoryModel; check::Bool = true)
 
   # Compute sum of products of cohomolgy classes of torus invariant prime divisors
   c_ds = [polynomial(cohomology_class(d)) for d in torusinvariant_prime_divisors(ambient_space(m))]
-  c2_t_ambient = zero(cohomology_ring(ambient_space(m)))
-  for i in 1:length(c_ds)
-    for j in i+1:length(c_ds)
-      c2_t_ambient = c2_t_ambient + c_ds[i]*c_ds[j]
-    end
-  end
+  c2_t_ambient = sum(c_ds[i]*c_ds[j] for i in 1:length(c_ds)-1 for j in i+1:length(c_ds))
   c2_t_ambient = cohomology_class(ambient_space(m), c2_t_ambient)
 
   # Compute and set h

From 8ffaef415526f6158e5d0755821acad5b8b4aa3a Mon Sep 17 00:00:00 2001
From: Martin Bies <MBies87@googlemail.com>
Date: Fri, 26 Jul 2024 10:09:23 +0100
Subject: [PATCH 6/6] [FTheoryTools] Wording change

Co-authored-by: Andrew P Turner <apturner@mac.com>
---
 .../src/AbstractFTheoryModels/attributes.jl          | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl b/experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl
index 8314a30dd268..71645ef9318f 100644
--- a/experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl
+++ b/experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl
@@ -1111,7 +1111,7 @@ setting, this is equivalent to $h$ being trivial.
 
 The computation of the cohomology ring verifies if the toric variety is simplicial and
 complete. The check for it to be complete can be very time consuming. This can be switched
-off by setting the optional argument `check` to the value `falue`, as in the example below.
+off by setting the optional argument `check` to the value `false`, as in the example below.
 
 ```jldoctest
 julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
@@ -1127,7 +1127,7 @@ true
 function chern_class_c1(m::AbstractFTheoryModel; check::Bool = true)
   @req (m isa WeierstrassModel || m isa GlobalTateModel || m isa HypersurfaceModel) "First Chern class of F-theory model supported for Weierstrass, global Tate and hypersurface models only"
   @req base_space(m) isa NormalToricVariety "First Chern class of F-theory model currently supported only for toric base"
-  @req ambient_space(m) isa NormalToricVariety "Firfalsest Chern class of F-theory model currently supported only for toric ambient space"
+  @req ambient_space(m) isa NormalToricVariety "First Chern class of F-theory model currently supported only for toric ambient space"
 
   # Check if the answer is known
   if has_attribute(m, :chern_class_c1)
@@ -1170,7 +1170,7 @@ such as in the example below. Therefore, think carefully if you truly want to co
 
 The computation of the cohomology ring verifies if the toric variety is simplicial and
 complete. The check for it to be complete can be very time consuming. This can be switched
-off by setting the optional argument `check` to the value `falue`, as in the example below.
+off by setting the optional argument `check` to the value `false`, as in the example below.
 
 ```jldoctest
 julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
@@ -1183,9 +1183,9 @@ false
 ```
 """
 function chern_class_c2(m::AbstractFTheoryModel; check::Bool = true)
-  @req (m isa WeierstrassModel || m isa GlobalTateModel || m isa HypersurfaceModel) "First Chern class of F-theory model supported for Weierstrass, global Tate and hypersurface models only"
-  @req base_space(m) isa NormalToricVariety "First Chern class of F-theory model currently supported only for toric base"
-  @req ambient_space(m) isa NormalToricVariety "First Chern class of F-theory model currently supported only for toric ambient space"
+  @req (m isa WeierstrassModel || m isa GlobalTateModel || m isa HypersurfaceModel) "Second Chern class of F-theory model supported for Weierstrass, global Tate and hypersurface models only"
+  @req base_space(m) isa NormalToricVariety "Second Chern class of F-theory model currently supported only for toric base"
+  @req ambient_space(m) isa NormalToricVariety "Second Chern class of F-theory model currently supported only for toric ambient space"
 
   # Check if the answer is known
   if has_attribute(m, :chern_class_c2)