Added D3-tadpole cancellation check for G4-fluxes

docs/oscar_references.bib

@@ -1490,6 +1490,19 @@ @Article{KMSS11
   reportnumber  = {UCSB-MATH-2011-09, IPMU11-0107, NSF-KITP-11-110, KCL-MTH-11-13}
 }
 
+@Article{KMW12,
+  author        = {Krause, Sven and Mayrhofer, Christoph and Weigand, Timo},
+  title         = {Gauge fluxes in F-theory and type IIB orientifolds},
+  journal       = {Journal of High Energy Physics},
+  volume        = {2012},
+  number        = {8},
+  publisher     = {Springer Science and Business Media LLC},
+  year          = {2012},
+  month         = {8},
+  doi           = {10.1007/jhep08(2012)119},
+  eprint        = {1202.3138}
+}
+
 @Article{KO14,
   author        = {Kelleher, Jerome and O'Sullivan, Barry},
   title         = {Generating All Partitions: A Comparison Of Two Encodings},

experimental/FTheoryTools/src/AbstractFTheoryModels/attributes.jl

@@ -1234,6 +1234,9 @@ 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
 ```
@@ -1245,7 +1248,7 @@ function euler_characteristic(m::AbstractFTheoryModel; check::Bool = true)
 
   # Check if the answer is known
   if has_attribute(m, :euler_characteristic)
-    return get_attribute(m, :euler_characteristic)::CohomologyClass
+    return get_attribute(m, :euler_characteristic)::Int
   end
 
   # Trigger potential short-cut computation of cohomology ring
@@ -1255,7 +1258,7 @@ function euler_characteristic(m::AbstractFTheoryModel; check::Bool = true)
   cy = cohomology_class(toric_divisor_class(ambient_space(m), degree(hypersurface_equation(m))))
 
   # Compute the Euler characteristic
-  h = integrate(chern_class(m, 4; check) * cy; check)
+  h = Int(integrate(chern_class(m, 4; check) * cy; check))
   set_attribute!(m, :euler_characteristic, h)
   return h
 end

experimental/FTheoryTools/src/G4Fluxes/properties.jl

@@ -63,7 +63,7 @@ end
 @doc raw"""
     passes_verticality_checks(gf::G4Flux)
 
-G4-fluxes are subject to verticality conditions described in [Lin16](@cite).
+G4-fluxes are subject to verticality conditions described in [Lin16](@cite) and [KMW12](@cite).
 It is hard to verify that these condition are met. However,
 we can execute a number of simple consistency checks, by
 verifying that $\int_{Y}{G_4 \wedge [D_1] \wedge [zero section]} = 0$ and $\int_{Y}{G_4 \wedge [D_1] \wedge [D_2]} = 0$
@@ -92,9 +92,9 @@ true
 """
 @attr Bool function passes_verticality_checks(g4::G4Flux)
   m = model(g4)
-  @req (m isa WeierstrassModel || m isa GlobalTateModel || m isa HypersurfaceModel) "Verticality checks for  G4-fluxes only supported for Weierstrass, global Tate and hypersurface models"
-  @req base_space(m) isa NormalToricVariety "Verticality checks for G4-flux currently supported only for toric base"
-  @req ambient_space(m) isa NormalToricVariety "Verticality checks for G4-flux currently supported only for toric ambient space"
+  @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"
   @req has_zero_section(m) "For verticality checks, a model zero section needs to be specified"
 
   # Compute the cohomology class corresponding to the hypersurface equation
@@ -117,3 +117,46 @@ true
   end
   return true
 end
+
+
+@doc raw"""
+    passes_tadpole_cancellation_check(gf::G4Flux)
+
+G4-fluxes are subject to the D3-tadpole cancellation condition described in [Wei18](@cite) and [KMW12](@cite).
+This check verifies that $euler_characteristic(Y)/24 - 1/2 * \int_{Y}{G_4 \wedge G_4}$ is a non-negative integer.
+
+```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 lacking elementary quantization checks
+
+julia> passes_tadpole_cancellation_check(g4)
+true
+```
+"""
+@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 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"
+
+  # Compute the cohomology class corresponding to the hypersurface equation
+  cy = polynomial(cohomology_class(toric_divisor_class(ambient_space(m), degree(hypersurface_equation(m)))))
+
+  # Now check if the D3-tadpole cancellation condition holds
+  numb = euler_characteristic(m; check = false)/24 - 1/2*integrate(cohomology_class(ambient_space(m), polynomial(cohomology_class(g4)) * polynomial(cohomology_class(g4)) * cy); check = false)
+  if numb > 0 && is_integer(numb)
+    return true
+  end
+  return false
+end

experimental/FTheoryTools/src/exports.jl

@@ -136,6 +136,7 @@ export paper_buzzwords
 export paper_description
 export paper_title
 export passes_elementary_quantization_checks
+export passes_tadpole_cancellation_check
 export passes_verticality_checks
 export polytope_index
 export put_over_concrete_base