[FTheoryTools] Add support for all vertical, well-quantized G4s that do not break the non-Abelian gauge group

Author: HereAround

experimental/FTheoryTools/docs/src/g4.md

@@ -87,7 +87,10 @@ Please note that this method may take a long time to execute for involved geomet
 ```@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:
+Similarly, we have a method for all vertical and well-quantized ambient space $G_4$-flux candidates,
+and the vertical and well-quantized ambient space $G_4$-flux candidates, that do not break the non-Abelian
+gauge group.
 ```@docs
 well_quantized_and_vertical_ambient_space_models_of_g4_fluxes(m::AbstractFTheoryModel; check::Bool = true)
+well_quantized_and_vertical_and_no_non_abelian_gauge_group_breaking_ambient_space_models_of_g4_fluxes(m::AbstractFTheoryModel; check::Bool = true)::Tuple{QQMatrix, QQMatrix}
 ```

experimental/FTheoryTools/src/G4Fluxes/attributes.jl

@@ -20,6 +20,7 @@ G4-flux candidate
   - Elementary quantization checks: not executed
   - Tadpole cancellation check: not executed
   - Verticality checks: not executed
+  - Non-Abelian gauge group: breaking pattern not analyzed  
 
 julia> model(g4f)
 Hypersurface model over a concrete base
@@ -46,6 +47,7 @@ G4-flux candidate
   - Elementary quantization checks: not executed
   - Tadpole cancellation check: not executed
   - Verticality checks: not executed
+  - Non-Abelian gauge group: breaking pattern not analyzed
 
 julia> cohomology_class(g4f) == g4_class
 true

experimental/FTheoryTools/src/G4Fluxes/constructors.jl

@@ -55,12 +55,14 @@ G4-flux candidate
   - Elementary quantization checks: satisfied
   - Tadpole cancellation check: not executed
   - Verticality checks: not executed
+  - Non-Abelian gauge group: breaking pattern not analyzed
 
 julia> g4f2 = g4_flux(qsm_model, g4_class, check = false)
 G4-flux candidate
   - Elementary quantization checks: not executed
   - Tadpole cancellation check: not executed
   - Verticality checks: not executed
+  - Non-Abelian gauge group: breaking pattern not analyzed
 ```
 """
 function g4_flux(m::AbstractFTheoryModel, g4_class::CohomologyClass; check::Bool = true)
@@ -153,6 +155,17 @@ function Base.show(io::IO, g4::G4Flux)
     push!(properties_string, "  - Verticality checks: not executed")
   end
 
+  # Check for non-Abelian gauge group breaking
+  if has_attribute(g4, :breaks_non_abelian_gauge_group)
+    if breaks_non_abelian_gauge_group(g4)
+      push!(properties_string, "  - Non-Abelian gauge group: broken")
+    else
+      push!(properties_string, "  - Non-Abelian gauge group: not broken")
+    end
+  else
+    push!(properties_string, "  - Non-Abelian gauge group: breaking pattern not analyzed")
+  end
+
   # Print each line separately, to avoid extra line break at the end
   for (i, line) in enumerate(properties_string)
     if i == length(properties_string)

experimental/FTheoryTools/src/G4Fluxes/properties.jl

@@ -32,6 +32,7 @@ G4-flux candidate
   - Elementary quantization checks: not executed
   - Tadpole cancellation check: not executed
   - Verticality checks: not executed
+  - Non-Abelian gauge group: breaking pattern not analyzed
 
 julia> passes_elementary_quantization_checks(g4)
 true
@@ -41,6 +42,7 @@ G4-flux candidate
   - Elementary quantization checks: satisfied
   - Tadpole cancellation check: not executed
   - Verticality checks: not executed
+  - Non-Abelian gauge group: breaking pattern not analyzed
 ```
 """
 @attr Bool function passes_elementary_quantization_checks(g4::G4Flux)
@@ -97,6 +99,7 @@ G4-flux candidate
   - Elementary quantization checks: not executed
   - Tadpole cancellation check: not executed
   - Verticality checks: not executed
+  - Non-Abelian gauge group: breaking pattern not analyzed
 
 julia> passes_verticality_checks(g4)
 true
@@ -106,6 +109,7 @@ G4-flux candidate
   - Elementary quantization checks: not executed
   - Tadpole cancellation check: not executed
   - Verticality checks: satisfied
+  - Non-Abelian gauge group: breaking pattern not analyzed
 ```
 """
 @attr Bool function passes_verticality_checks(g4::G4Flux)
@@ -160,6 +164,7 @@ G4-flux candidate
   - Elementary quantization checks: not executed
   - Tadpole cancellation check: not executed
   - Verticality checks: not executed
+  - Non-Abelian gauge group: breaking pattern not analyzed
 
 julia> passes_tadpole_cancellation_check(g4)
 true
@@ -169,11 +174,12 @@ G4-flux candidate
   - Elementary quantization checks: not executed
   - Tadpole cancellation check: satisfied
   - Verticality checks: not executed
+  - Non-Abelian gauge group: breaking pattern not analyzed
 ```
 """
 @attr Bool function passes_tadpole_cancellation_check(g4::G4Flux)
   m = model(g4)
-  @req (m isa WeierstrassModel || m isa GlobalTateModel || m isa HypersurfaceModel) "Tadpole cancellation checks for  G4-fluxes only supported for Weierstrass, global Tate and hypersurface models"
+  @req (m isa WeierstrassModel || m isa GlobalTateModel || m isa HypersurfaceModel) "Tadpole cancellation checks for G4-fluxes only supported for Weierstrass, global Tate and hypersurface models"
   @req base_space(m) isa NormalToricVariety "Tadpole cancellation checks for G4-flux currently supported only for toric base"
   @req ambient_space(m) isa NormalToricVariety "Tadpole cancellation checks for G4-flux currently supported only for toric ambient space"
 
@@ -187,3 +193,71 @@ G4-flux candidate
   end
   return false
 end
+
+
+@doc raw"""
+    breaks_non_abelian_gauge_group(gf::G4Flux)
+
+G4-fluxes may break the non-Abelian gauge group (cf. [Wei18](@cite)).
+This function verifies if this is the case for the given G4-flux.
+If it does not break any non-Abelian gauge factor, we return 
+`true` and otherwise `false`
+
+```jldoctest; setup = :(Oscar.LazyArtifacts.ensure_artifact_installed("QSMDB", Oscar.LazyArtifacts.find_artifacts_toml(Oscar.oscardir)))
+julia> qsm_model = literature_model(arxiv_id = "1903.00009", model_parameters = Dict("k" => 4))
+Hypersurface model over a concrete base
+
+julia> divs = torusinvariant_prime_divisors(ambient_space(qsm_model));
+
+julia> e1 = cohomology_class(divs[35]);e2 = cohomology_class(divs[32]);e4 = cohomology_class(divs[34]);
+
+julia> u = cohomology_class(divs[33]);v = cohomology_class(divs[30]);pb_Kbar = cohomology_class(sum([divs[k] for k in 1:29]));
+
+julia> g4_class = (-3) // kbar3(qsm_model) * (5 * e1 * e4 + pb_Kbar * (-3 * e1 - 2 * e2 - 6 * e4 + pb_Kbar - 4 * u + v));
+
+julia> g4 = g4_flux(qsm_model, g4_class, check = false)
+G4-flux candidate
+  - Elementary quantization checks: not executed
+  - Tadpole cancellation check: not executed
+  - Verticality checks: not executed
+  - Non-Abelian gauge group: breaking pattern not analyzed
+
+julia> breaks_non_abelian_gauge_group(g4)
+false
+
+julia> g4
+G4-flux candidate
+  - Elementary quantization checks: not executed
+  - Tadpole cancellation check: not executed
+  - Verticality checks: satisfied
+  - Non-Abelian gauge group: not broken
+```
+"""
+@attr Bool function breaks_non_abelian_gauge_group(g4::G4Flux)
+  m = model(g4)
+  @req (m isa WeierstrassModel || m isa GlobalTateModel || m isa HypersurfaceModel) "Checks for breaking non-Abelian gauge group factors only supported for Weierstrass, global Tate and hypersurface models"
+  @req base_space(m) isa NormalToricVariety "Checks for breaking non-Abelian gauge group factors currently supported only for toric base"
+  @req ambient_space(m) isa NormalToricVariety "Checks for breaking non-Abelian gauge group factors currently supported only for toric ambient space"
+  
+  # Compute the cohomology class corresponding to the hypersurface equation
+  cy = polynomial(cohomology_class(toric_divisor_class(ambient_space(m), degree(hypersurface_equation(m)))))
+
+  # Identify the cohomology classes of all base divisors
+  n = length(gens(cox_ring(base_space(m))))
+  c_ds = [polynomial(cohomology_class(d)) for d in torusinvariant_prime_divisors(ambient_space(m))[1:n]]
+
+  # Identify the cohomology classes of all exceptional divisors
+  gS = gens(cox_ring(ambient_space(m)))
+  exceptional_divisor_positions = findall(x -> occursin(r"^e\d+$", x), string.(gS))
+  exceptional_divisors = [torusinvariant_prime_divisors(ambient_space(m))[k] for k in exceptional_divisor_positions];
+  c_ei = [polynomial(cohomology_class(d)) for d in exceptional_divisors]
+
+  # now execute the checks if any non-Abelian gauge group factor is broken
+  for i in 1:n
+    for j in 1:length(exceptional_divisors)
+      numb = integrate(cohomology_class(ambient_space(m), polynomial(cohomology_class(g4)) * c_ds[i] * c_ei[j] * cy); check = false)
+      numb!=0 && return true
+    end
+  end
+  return false
+end