[FTheoryTools] Compute well-quantized and vertical G4-fluxes

docs/oscar_references.bib

@@ -565,6 +565,20 @@ @Article{CMT04
   doi           = {10.1090/S0025-5718-03-01582-5}
 }
 
+@Article{CS12,
+  author        = {Collinucci, Andres and Savelli, Raffaele},
+  title         = {{On Flux Quantization in F-Theory}},
+  journal       = {JHEP},
+  volume        = {02},
+  pages         = {015},
+  year          = {2012},
+  doi           = {10.1007/JHEP02(2012)015},
+  eprint        = {1011.6388},
+  archiveprefix = {arXiv},
+  primaryclass  = {hep-th},
+  reportnumber  = {MAD-TH-10-09, LMU-ASC-100-10}
+}
+
 @Book{CS99,
   author        = {Conway, J. H. and Sloane, N. J. A.},
   title         = {Sphere packings, lattices and groups},

experimental/FTheoryTools/docs/src/g4.md

@@ -74,3 +74,20 @@ replace an involved algebraic cycle.
 ambient_space_models_of_g4_fluxes(m::AbstractFTheoryModel; check::Bool = true)
 basis_of_h22(v::NormalToricVariety; check::Bool = true)
 ```
+
+Among the $G_4$-flux candidates, the physics is interested in the well-quantized fluxes. That is,
+those cohomology classes which integrate to an integer against any other 2-cycle in the elliptic
+4-fold $\widehat{Y}_4$. Even in theory, this is a hard task. In practice, one therefore focuses
+on consistency checks. In the case at hand, we can integrate any ambient space $G_4$-flux candidate
+against a pair of (algebraic cycles associated to) toric divisors. If for any two toric divisors,
+the result is an integer, then this $G_4$-flux candidate passes a rather non-trivial and necessary
+test. The following method identifies all $G_4$-flux ambient space candidates which pass this test.
+Please note that this method may take a long time to execute for involved geometries $\widehat{Y}_4$.
+
+```@docs
+well_quantized_ambient_space_models_of_g4_fluxes(m::AbstractFTheoryModel; check::Bool = true)
+```
+Similarly, we have a method for all vertical and well-quantized ambient space $G_4$-flux candidates:
+```@docs
+well_quantized_and_vertical_ambient_space_models_of_g4_fluxes(m::AbstractFTheoryModel; check::Bool = true)
+```

experimental/FTheoryTools/src/FTheoryTools.jl

@@ -29,6 +29,8 @@ include("LiteratureModels/create_index.jl")
 include("G4Fluxes/constructors.jl")
 include("G4Fluxes/attributes.jl")
 include("G4Fluxes/properties.jl")
+include("G4Fluxes/auxiliary.jl")
+include("G4Fluxes/special-intersection-theory.jl")
 include("G4Fluxes/special_attributes.jl")
 
 include("Serialization/tate_models.jl")

experimental/FTheoryTools/src/G4Fluxes/auxiliary.jl

@@ -0,0 +1,292 @@
+@doc raw"""
+    basis_of_h22(v::NormalToricVariety; check::Bool = true)::Vector{CohomologyClass}
+
+By virtue of Theorem 12.4.1 in [CLS11](@cite), one can compute a monomial
+basis of $H^4(X, \mathbb{Q})$ for a simplicial, complete toric variety $X$
+by truncating its cohomology ring to degree $2$. Inspired by this, this
+method identifies a basis of $H^{(2,2)}(X, \mathbb{Q})$ by multiplying
+pairs of cohomology classes associated with toric coordinates.
+
+By definition, $H^{(2,2)}(X, \mathbb{Q})$ is a subset of $H^{4}(X, \mathbb{Q})$.
+However, by Theorem 9.3.2 in [CLS11](@cite), for complete and simplicial
+toric varieties and $p \neq q$ it holds $H^{(p,q)}(X, \mathbb{Q}) = 0$. It follows
+that for such varieties $H^{(2,2)}(X, \mathbb{Q}) = H^4(X, \mathbb{Q})$ and the
+vector space dimension of those spaces agrees with the Betti number $b_4(X)$.
+
+Note that it can be computationally very demanding to check if a toric variety
+$X$ is complete (and simplicial). The optional argument `check` can be set
+to `false` to skip these tests.
+
+# Examples
+```jldoctest
+julia> Y1 = hirzebruch_surface(NormalToricVariety, 2)
+Normal toric variety
+
+julia> Y2 = hirzebruch_surface(NormalToricVariety, 2)
+Normal toric variety
+
+julia> Y = Y1 * Y2
+Normal toric variety
+
+julia> h22_basis = basis_of_h22(Y, check = false)
+6-element Vector{CohomologyClass}:
+ Cohomology class on a normal toric variety given by xx2*yx2
+ Cohomology class on a normal toric variety given by xt2*yt2
+ Cohomology class on a normal toric variety given by xx2*yt2
+ Cohomology class on a normal toric variety given by xt2*yx2
+ Cohomology class on a normal toric variety given by yx2^2
+ Cohomology class on a normal toric variety given by xx2^2
+
+julia> betti_number(Y, 4) == length(h22_basis)
+true
+```
+"""
+function basis_of_h22(v::NormalToricVariety; check::Bool = true)::Vector{CohomologyClass}
+
+  # (0) Some initial checks
+  if check
+    @req is_complete(v) "Computation of basis of H22 is currently only supported for complete toric varieties"
+    @req is_simplicial(v) "Computation of basis of H22 is currently only supported for simplicial toric varieties"
+  end
+  if dim(v) < 4
+    set_attribute!(v, :basis_of_h22, Vector{CohomologyClass}())
+  end
+  if has_attribute(v, :basis_of_h22)
+    return get_attribute(v, :basis_of_h22)
+  end
+
+  # (1) Prepare some data of the variety
+  mnf = Oscar._minimal_nonfaces(v)
+  ignored_sets = Set([Tuple(sort(Vector{Int}(Polymake.row(mnf, i)))) for i in 1:Polymake.nrows(mnf)])
+
+  # (2) Prepare the linear relations
+  N_lin_rel, my_mat = rref(transpose(matrix(QQ, rays(v))))
+  @req N_lin_rel == nrows(my_mat) "Cannot remove as many variables as there are linear relations - weird!"
+  bad_positions = [findfirst(!iszero, row) for row in eachrow(my_mat)]
+  lin_rels = Dict{Int, Vector{QQFieldElem}}()
+  for k in 1:nrows(my_mat)
+    my_relation = (-1) * my_mat[k, :]
+    my_relation[bad_positions[k]] = 0
+    @req all(k -> k == 0, my_relation[bad_positions]) "Inconsistency!"
+    lin_rels[bad_positions[k]] = my_relation
+  end
+
+  # (3) Prepare a list of those variables that we keep, a.k.a. a basis of H^(1,1)
+  good_positions = setdiff(1:n_rays(v), bad_positions)
+  n_good_positions = length(good_positions)
+
+  # (4) Make a list of all quadratic elements in the cohomology ring, which are not generators of the SR-ideal.
+  N_filtered_quadratic_elements = 0
+  dict_of_filtered_quadratic_elements = Dict{Tuple{Int64, Int64}, Int64}()
+  for k in 1:n_good_positions
+    for l in k:n_good_positions
+      my_tuple = (min(good_positions[k], good_positions[l]), max(good_positions[k], good_positions[l]))
+      if !(my_tuple in ignored_sets)
+        N_filtered_quadratic_elements += 1
+        dict_of_filtered_quadratic_elements[my_tuple] = N_filtered_quadratic_elements
+      end
+    end
+  end
+
+  # (5) We only care about the SR-ideal gens of degree 2. Above, we took care of all relations,
+  # (5) for which both variables are not replaced by one of the linear relations. So, let us identify
+  # (5) all remaining relations of the SR-ideal, and apply the linear relations to them.
+  remaining_relations = Vector{Vector{QQFieldElem}}()
+  for my_tuple in ignored_sets
+
+    # The generator must have degree 2 and at least one variable is to be replaced
+    if length(my_tuple) == 2 && (my_tuple[1] in bad_positions || my_tuple[2] in bad_positions)
+
+      # Represent first variable by list of coefficients, after plugging in the linear relation
+      var1 = zeros(QQ, ncols(my_mat))
+      var1[my_tuple[1]] = 1
+      if my_tuple[1] in bad_positions
+        var1 = lin_rels[my_tuple[1]]
+      end
+
+      # Represent second variable by list of coefficients, after plugging in the linear relation
+      var2 = zeros(QQ, ncols(my_mat))
+      var2[my_tuple[2]] = 1
+      if my_tuple[2] in bad_positions
+        var2 = lin_rels[my_tuple[2]]
+      end
+
+      # Compute the product of the two variables, which represents the new relation
+      prod = zeros(QQ, N_filtered_quadratic_elements)
+      for k in 1:length(var1)
+        if var1[k] != 0
+          for l in 1:length(var2)
+            if var2[l] != 0
+              my_tuple = (min(k, l), max(k, l))
+              if haskey(dict_of_filtered_quadratic_elements, my_tuple)
+                prod[dict_of_filtered_quadratic_elements[my_tuple]] += var1[k] * var2[l]
+              end
+            end
+          end
+        end
+      end
+
+      # Remember the result
+      push!(remaining_relations, prod)
+
+    end
+
+  end
+
+  # (9) Identify variables that we can remove with the remaining relations
+  new_good_positions = 1:N_filtered_quadratic_elements
+  if length(remaining_relations) != 0
+    remaining_relations_matrix = matrix(QQ, remaining_relations)
+    r, new_mat = rref(remaining_relations_matrix)
+    @req r == nrows(remaining_relations_matrix) "Cannot remove a variable via linear relations - weird!"
+    new_bad_positions = [findfirst(!iszero, row) for row in eachrow(new_mat)]
+    new_good_positions = setdiff(1:N_filtered_quadratic_elements, new_bad_positions)
+  end
+
+  # (10) Return the basis elements in terms of cohomology classes
+  S = cohomology_ring(v, check = check)
+  c_ds = [k.f for k in gens(S)]
+  final_list_of_tuples = []
+  for (key, value) in dict_of_filtered_quadratic_elements
+    if value in new_good_positions
+      push!(final_list_of_tuples, key)
+    end
+  end
+  basis_of_h22 = [cohomology_class(v, MPolyQuoRingElem(c_ds[my_tuple[1]]*c_ds[my_tuple[2]], S)) for my_tuple in final_list_of_tuples]
+  set_attribute!(v, :basis_of_h22, basis_of_h22)
+  set_attribute!(v, :basis_of_h22_indices, final_list_of_tuples)
+  return basis_of_h22
+
+end
+
+
+# The following is an internal function, that is being used to identify all well-quantized G4-fluxes.
+# Let G4 in H^(2,2)(toric_ambient_space) a G4-flux ambient space candidate, i.e. the physically
+# truly relevant quantity is the restriction of G4 to a hypersurface V(pt) in the toric_ambient_space.
+# To tell if this candidate is well-quantized, we execute a necessary check, namely we verify that
+# the integral of G4 * [pt] * [d1] * [d2] over X_Sigma is an integer for any two toric divisors d1, d2.
+# While we wish to execute this test for any two toric divisors d1, d2 some pairs can be ignored.
+# Say, because their intersection locus is trivial because of the SR-ideal, or because their intersection
+# has empty intersection with the hypersurface. The following method identifies the remaining pairs of
+# toric divisors d1, d2 that we must consider.
+
+function _ambient_space_divisor_pairs_to_be_considered(m::AbstractFTheoryModel)::Vector{Tuple{Int64, Int64}}
+
+  if has_attribute(m, :_ambient_space_divisor_pairs_to_be_considered)
+    return get_attribute(m, :_ambient_space_divisor_pairs_to_be_considered)
+  end
+
+  gS = gens(cox_ring(ambient_space(m)))
+  mnf = Oscar._minimal_nonfaces(ambient_space(m))
+  ignored_sets = Set([Tuple(sort(Vector{Int}(Polymake.row(mnf, i)))) for i in 1:Polymake.nrows(mnf)])
+
+  list_of_elements = Vector{Tuple{Int64, Int64}}()
+  for k in 1:n_rays(ambient_space(m))
+    for l in k:n_rays(ambient_space(m))
+
+      # V(x_k, x_l) = emptyset?
+      (k,l) in ignored_sets && continue
+
+      # Simplify the hypersurface polynomial by setting relevant variables to zero.
+      # If all coefficients of this new polynomial add to sum, then we keep this generator.
+      new_p_hyper = divrem(hypersurface_equation(m), gS[k])[2]
+      if k != l
+        new_p_hyper = divrem(new_p_hyper, gS[l])[2]
+      end
+      if sum(coefficients(new_p_hyper)) == 0
+        push!(list_of_elements, (k,l))
+        continue
+      end
+
+      # Determine remaining variables, after scaling "away" others.
+      remaining_vars_list = Set(1:length(gS))
+      for my_exps in ignored_sets
+        len_my_exps = length(my_exps)
+        inter_len = count(idx -> idx in [k,l], my_exps)
+        if (len_my_exps == 2 && inter_len == 1) || (len_my_exps == 3 && inter_len == 2)
+          delete!(remaining_vars_list, my_exps[findfirst(idx -> !(idx in [k,l]), my_exps)])
+        end
+      end
+      remaining_vars_list = collect(remaining_vars_list)
+
+      # If one monomial of `new_p_hyper` has unset positions (a.k.a. new_p_hyper is not constant upon
+      # scaling the remaining variables), then keep this generator.
+      for exps in exponents(new_p_hyper)
+        if any(x -> x != 0, exps[remaining_vars_list])
+          push!(list_of_elements, (k,l))
+          break
+        end
+      end
+
+    end
+  end
+
+  # Remember this result as attribute and return the findings.
+  set_attribute!(m, :_ambient_space_divisor_pairs_to_be_considered, list_of_elements)
+  return list_of_elements
+
+end
+
+
+# The following is an internal function, that is being used to identify all well-quantized and
+# vertical G4-flux ambient candidates. For this, we look for pairs of (pushforwards of) base divisors,
+# s.t. their common zero locus does not intersect the CY hypersurface trivially.
+# This method makes a pre-selection of such base divisor pairs. "Pre" means that we execute a sufficient,
+# but not necessary, check to tell if a pair of base divisors restricts trivially.
+
+function _ambient_space_base_divisor_pairs_to_be_considered(m::AbstractFTheoryModel)::Vector{Tuple{Int64, Int64}}
+
+  if has_attribute(m, :_ambient_space_base_divisor_pairs_to_be_considered)
+    return get_attribute(m, :_ambient_space_base_divisor_pairs_to_be_considered)
+  end
+
+  gS = gens(cox_ring(ambient_space(m)))
+  mnf = Oscar._minimal_nonfaces(ambient_space(m))
+  ignored_sets = Set([Tuple(sort(Vector{Int}(Polymake.row(mnf, i)))) for i in 1:Polymake.nrows(mnf)])
+
+  list_of_elements = Vector{Tuple{Int64, Int64}}()
+  for k in 1:n_rays(base_space(m))
+    for l in k:n_rays(base_space(m))
+
+      # V(x_k, x_l) = emptyset?
+      (k,l) in ignored_sets && continue
+
+      # Simplify the hypersurface polynomial by setting relevant variables to zero.
+      # If all coefficients of this new polynomial add to sum, then we keep this generator.
+      new_p_hyper = divrem(hypersurface_equation(m), gS[k])[2]
+      if k != l
+        new_p_hyper = divrem(new_p_hyper, gS[l])[2]
+      end
+      if sum(coefficients(new_p_hyper)) == 0
+        push!(list_of_elements, (k,l))
+        continue
+      end
+
+      # Determine remaining variables, after scaling "away" others.
+      remaining_vars_list = Set(1:length(gS))
+      for my_exps in ignored_sets
+        len_my_exps = length(my_exps)
+        inter_len = count(idx -> idx in [k,l], my_exps)
+        if (len_my_exps == 2 && inter_len == 1) || (len_my_exps == 3 && inter_len == 2)
+          delete!(remaining_vars_list, my_exps[findfirst(idx -> !(idx in [k,l]), my_exps)])
+        end
+      end
+      remaining_vars_list = collect(remaining_vars_list)
+
+      # If one monomial of `new_p_hyper` has unset positions (a.k.a. new_p_hyper is not constant upon
+      # scaling the remaining variables), then keep this generator.
+      for exps in exponents(new_p_hyper)
+        if any(x -> x != 0, exps[remaining_vars_list])
+          push!(list_of_elements, (k,l))
+          break
+        end
+      end
+
+    end
+  end
+
+  # Remember this result as attribute and return the findings.
+  set_attribute!(m, :_ambient_space_base_divisor_pairs_to_be_considered, list_of_elements)
+  return list_of_elements
+
+end