Merge pull request #11 from HEPLean/AnomalyCancellation/PlaneMax
feat: Add theorem about max charges
This commit is contained in:
Joseph Tooby-Smith
GitHub
commit
1f573395cc
6 changed files with 774 additions and 2 deletions
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@ -36,11 +36,14 @@ import HepLean.AnomalyCancellation.SMNu.Basic
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import HepLean.AnomalyCancellation.SMNu.FamilyMaps
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import HepLean.AnomalyCancellation.SMNu.NoGrav.Basic
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import HepLean.AnomalyCancellation.SMNu.Ordinary.Basic
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import HepLean.AnomalyCancellation.SMNu.Ordinary.DimSevenPlane
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import HepLean.AnomalyCancellation.SMNu.Ordinary.FamilyMaps
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import HepLean.AnomalyCancellation.SMNu.Permutations
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import HepLean.AnomalyCancellation.SMNu.PlusU1.BMinusL
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import HepLean.AnomalyCancellation.SMNu.PlusU1.Basic
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import HepLean.AnomalyCancellation.SMNu.PlusU1.BoundPlaneDim
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import HepLean.AnomalyCancellation.SMNu.PlusU1.FamilyMaps
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import HepLean.AnomalyCancellation.SMNu.PlusU1.HyperCharge
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import HepLean.AnomalyCancellation.SMNu.PlusU1.PlaneNonSols
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import HepLean.AnomalyCancellation.SMNu.PlusU1.QuadSol
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import HepLean.AnomalyCancellation.SMNu.PlusU1.QuadSolToSol
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@ -4,6 +4,8 @@ Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.AnomalyCancellation.LinearMaps
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import Mathlib.Algebra.Module.Basic
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import Mathlib.LinearAlgebra.FiniteDimensional
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/-!
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# Basic set up for anomaly cancellation conditions
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@ -57,6 +59,9 @@ instance chargesModule (χ : ACCSystemCharges) : Module ℚ χ.charges :=
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instance ChargesAddCommGroup (χ : ACCSystemCharges) : AddCommGroup χ.charges :=
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Module.addCommMonoidToAddCommGroup ℚ
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instance (χ : ACCSystemCharges) : Module.Finite ℚ χ.charges :=
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FiniteDimensional.finiteDimensional_pi ℚ
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end ACCSystemCharges
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/-- The type of charges plus the linear ACCs. -/
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@ -230,6 +235,10 @@ lemma Sols.ext {χ : ACCSystem} {S T : χ.Sols} (h : S.val = T.val) :
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cases' S
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simp_all only
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/-- We say a charge S is a solution if it extends to a solution. -/
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def isSolution (χ : ACCSystem) (S : χ.charges) : Prop :=
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∃ (sol : χ.Sols), sol.val = S
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/-- An instance giving the properties and structures to define an action of `ℚ` on `Sols`. -/
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instance solsMulAction (χ : ACCSystem) : MulAction ℚ χ.Sols where
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smul a S := ⟨a • S.toQuadSols , by
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@ -5,6 +5,7 @@ Authors: Joseph Tooby-Smith
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-/
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import Mathlib.Tactic.Polyrith
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import Mathlib.Algebra.Module.LinearMap.Basic
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import Mathlib.Algebra.BigOperators.Fin
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/-!
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# Linear maps
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@ -107,6 +108,7 @@ class IsSymmetric {V : Type} [AddCommMonoid V] [Module ℚ V] (f : V →ₗ[ℚ]
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swap : ∀ S T, f S T = f T S
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namespace BiLinearSymm
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open BigOperators
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variable {V : Type} [AddCommMonoid V] [Module ℚ V]
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@ -147,6 +149,30 @@ lemma map_add₂ (f : BiLinearSymm V) (S : V) (T1 T2 : V) :
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f (S, T1 + T2) = f (S, T1) + f (S, T2) := by
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rw [f.swap, f.map_add₁, f.swap T1 S, f.swap T2 S]
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/-- Fixing the second input vectors, the resulting linear map. -/
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def toLinear₁ (f : BiLinearSymm V) (T : V) : V →ₗ[ℚ] ℚ where
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toFun S := f (S, T)
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map_add' S1 S2 := by
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simp only [f.map_add₁]
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map_smul' a S := by
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simp only [f.map_smul₁]
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rfl
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lemma toLinear₁_apply (f : BiLinearSymm V) (S T : V) : f (S, T) = f.toLinear₁ T S := rfl
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lemma map_sum₁ {n : ℕ} (f : BiLinearSymm V) (S : Fin n → V) (T : V) :
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f (∑ i, S i, T) = ∑ i, f (S i, T) := by
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rw [f.toLinear₁_apply]
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rw [map_sum]
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rfl
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lemma map_sum₂ {n : ℕ} (f : BiLinearSymm V) (S : Fin n → V) (T : V) :
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f ( T, ∑ i, S i) = ∑ i, f (T, S i) := by
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rw [swap, map_sum₁]
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apply Fintype.sum_congr
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intro i
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rw [swap]
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/-- The homogenous quadratic equation obtainable from a bilinear function. -/
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@[simps!]
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@ -214,6 +240,11 @@ instance instFun : FunLike (TriLinear V) (V × V × V) ℚ where
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simp_all
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end TriLinear
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/-- The structure of a symmetric trilinear function. -/
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structure TriLinearSymm' (V : Type) [AddCommMonoid V] [Module ℚ V] extends
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V →ₗ[ℚ] V →ₗ[ℚ] V →ₗ[ℚ] ℚ where
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swap₁' : ∀ S T L, toFun S T L = toFun T S L
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swap₂' : ∀ S T L, toFun S T L = toFun S L T
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/-- The structure of a symmetric trilinear function. -/
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structure TriLinearSymm (V : Type) [AddCommMonoid V] [Module ℚ V] where
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@ -225,7 +256,7 @@ structure TriLinearSymm (V : Type) [AddCommMonoid V] [Module ℚ V] where
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swap₂' : ∀ S T L, toFun (S, T, L) = toFun (S, L, T)
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namespace TriLinearSymm
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open BigOperators
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variable {V : Type} [AddCommMonoid V] [Module ℚ V]
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instance instFun : FunLike (TriLinearSymm V) (V × V × V) ℚ where
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@ -235,7 +266,6 @@ instance instFun : FunLike (TriLinearSymm V) (V × V × V) ℚ where
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cases g
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simp_all
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lemma toFun_eq_coe (f : TriLinearSymm V) : f.toFun = f := rfl
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lemma swap₁ (f : TriLinearSymm V) (S T L : V) : f (S, T, L) = f (T, S, L) :=
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@ -271,6 +301,49 @@ lemma map_add₃ (f : TriLinearSymm V) (S T L1 L2 : V) :
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f (S, T, L1 + L2) = f (S, T, L1) + f (S, T, L2) := by
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rw [f.swap₃, f.map_add₁, f.swap₃, f.swap₃ L2 T S]
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/-- Fixing the second and third input vectors, the resulting linear map. -/
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def toLinear₁ (f : TriLinearSymm V) (T L : V) : V →ₗ[ℚ] ℚ where
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toFun S := f (S, T, L)
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map_add' S1 S2 := by
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simp only [f.map_add₁]
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map_smul' a S := by
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simp only [f.map_smul₁]
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rfl
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lemma toLinear₁_apply (f : TriLinearSymm V) (S T L : V) : f (S, T, L) = f.toLinear₁ T L S := rfl
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lemma map_sum₁ {n : ℕ} (f : TriLinearSymm V) (S : Fin n → V) (T : V) (L : V) :
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f (∑ i, S i, T, L) = ∑ i, f (S i, T, L) := by
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rw [f.toLinear₁_apply]
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rw [map_sum]
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rfl
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lemma map_sum₂ {n : ℕ} (f : TriLinearSymm V) (S : Fin n → V) (T : V) (L : V) :
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f ( T, ∑ i, S i, L) = ∑ i, f (T, S i, L) := by
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rw [swap₁, map_sum₁]
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apply Fintype.sum_congr
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intro i
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rw [swap₁]
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lemma map_sum₃ {n : ℕ} (f : TriLinearSymm V) (S : Fin n → V) (T : V) (L : V) :
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f ( T, L, ∑ i, S i) = ∑ i, f (T, L, S i) := by
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rw [swap₃, map_sum₁]
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apply Fintype.sum_congr
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intro i
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rw [swap₃]
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lemma map_sum₁₂₃ {n1 n2 n3 : ℕ} (f : TriLinearSymm V) (S : Fin n1 → V)
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(T : Fin n2 → V) (L : Fin n3 → V) :
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f (∑ i, S i, ∑ i, T i, ∑ i, L i) = ∑ i, ∑ k, ∑ l, f (S i, T k, L l) := by
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rw [map_sum₁]
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apply Fintype.sum_congr
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intro i
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rw [map_sum₂]
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apply Fintype.sum_congr
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intro i
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rw [map_sum₃]
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/-- The homogenous cubic equation obtainable from a symmetric trilinear function. -/
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@[simps!]
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def toCubic {charges : Type} [AddCommMonoid charges] [Module ℚ charges]
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349
HepLean/AnomalyCancellation/SMNu/Ordinary/DimSevenPlane.lean
Normal file
349
HepLean/AnomalyCancellation/SMNu/Ordinary/DimSevenPlane.lean
Normal file
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@ -0,0 +1,349 @@
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.AnomalyCancellation.SMNu.Ordinary.Basic
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/-!
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# Dimension 7 plane
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We work here in the three family case.
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We give an example of a 7 dimensional plane on which every point satisfies the ACCs.
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The main result of this file is `seven_dim_plane_exists` which states that there exists a
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7 dimensional plane of charges on which every point satisfies the ACCs.
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-/
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namespace SMRHN
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namespace SM
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open SMνCharges
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open SMνACCs
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open BigOperators
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namespace PlaneSeven
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₀ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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match s, i with
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| 0, 0 => 1
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| 0, 1 => - 1
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| _, _ => 0
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)
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lemma B₀_cubic (S T : (SM 3).charges) : cubeTriLin (B₀, S, T) =
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6 * (S (0 : Fin 18) * T (0 : Fin 18) - S (1 : Fin 18) * T (1 : Fin 18)) := by
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simp [Fin.sum_univ_three, B₀, Fin.divNat, Fin.modNat, finProdFinEquiv]
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ring
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₁ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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match s, i with
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| 1, 0 => 1
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| 1, 1 => - 1
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| _, _ => 0
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)
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lemma B₁_cubic (S T : (SM 3).charges) : cubeTriLin (B₁, S, T) =
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3 * (S (3 : Fin 18) * T (3 : Fin 18) - S (4 : Fin 18) * T (4 : Fin 18)) := by
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simp [Fin.sum_univ_three, B₁, Fin.divNat, Fin.modNat, finProdFinEquiv]
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ring
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₂ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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match s, i with
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| 2, 0 => 1
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| 2, 1 => - 1
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| _, _ => 0
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)
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lemma B₂_cubic (S T : (SM 3).charges) : cubeTriLin (B₂, S, T) =
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3 * (S (6 : Fin 18) * T (6 : Fin 18) - S (7 : Fin 18) * T (7 : Fin 18)) := by
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simp [Fin.sum_univ_three, B₂, Fin.divNat, Fin.modNat, finProdFinEquiv]
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ring
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₃ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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match s, i with
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| 3, 0 => 1
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| 3, 1 => - 1
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| _, _ => 0
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)
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lemma B₃_cubic (S T : (SM 3).charges) : cubeTriLin (B₃, S, T) =
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2 * (S (9 : Fin 18) * T (9 : Fin 18) - S (10 : Fin 18) * T (10 : Fin 18)) := by
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simp [Fin.sum_univ_three, B₃, Fin.divNat, Fin.modNat, finProdFinEquiv]
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ring_nf
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rfl
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₄ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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match s, i with
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| 4, 0 => 1
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| 4, 1 => - 1
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| _, _ => 0
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)
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lemma B₄_cubic (S T : (SM 3).charges) : cubeTriLin (B₄, S, T) =
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(S (12 : Fin 18) * T (12 : Fin 18) - S (13 : Fin 18) * T (13 : Fin 18)) := by
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simp [Fin.sum_univ_three, B₄, Fin.divNat, Fin.modNat, finProdFinEquiv]
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ring_nf
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rfl
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₅ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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match s, i with
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| 5, 0 => 1
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| 5, 1 => - 1
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| _, _ => 0
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)
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lemma B₅_cubic (S T : (SM 3).charges) : cubeTriLin (B₅, S, T) =
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(S (15 : Fin 18) * T (15 : Fin 18) - S (16 : Fin 18) * T (16 : Fin 18)) := by
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simp [Fin.sum_univ_three, B₅, Fin.divNat, Fin.modNat, finProdFinEquiv]
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ring_nf
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rfl
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₆ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
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match s, i with
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| 1, 2 => 1
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| 2, 2 => -1
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| _, _ => 0
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)
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lemma B₆_cubic (S T : (SM 3).charges) : cubeTriLin (B₆, S, T) =
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3* (S (5 : Fin 18) * T (5 : Fin 18) - S (8 : Fin 18) * T (8 : Fin 18)) := by
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simp [Fin.sum_univ_three, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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ring_nf
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/-- The charge assignments forming a basis of the plane. -/
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@[simp]
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def B : Fin 7 → (SM 3).charges := fun i =>
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match i with
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| 0 => B₀
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| 1 => B₁
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| 2 => B₂
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| 3 => B₃
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| 4 => B₄
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| 5 => B₅
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| 6 => B₆
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lemma B₀_Bi_cubic {i : Fin 7} (hi : 0 ≠ i) (S : (SM 3).charges) :
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cubeTriLin (B 0, B i, S) = 0 := by
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change cubeTriLin (B₀, B i, S) = 0
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rw [B₀_cubic]
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fin_cases i <;>
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₁_Bi_cubic {i : Fin 7} (hi : 1 ≠ i) (S : (SM 3).charges) :
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cubeTriLin (B 1, B i, S) = 0 := by
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change cubeTriLin (B₁, B i, S) = 0
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rw [B₁_cubic]
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fin_cases i <;>
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₂_Bi_cubic {i : Fin 7} (hi : 2 ≠ i) (S : (SM 3).charges) :
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cubeTriLin (B 2, B i, S) = 0 := by
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change cubeTriLin (B₂, B i, S) = 0
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rw [B₂_cubic]
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fin_cases i <;>
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₃_Bi_cubic {i : Fin 7} (hi : 3 ≠ i) (S : (SM 3).charges) :
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cubeTriLin (B 3, B i, S) = 0 := by
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change cubeTriLin (B₃, B i, S) = 0
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rw [B₃_cubic]
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fin_cases i <;>
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₄_Bi_cubic {i : Fin 7} (hi : 4 ≠ i) (S : (SM 3).charges) :
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cubeTriLin (B 4, B i, S) = 0 := by
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change cubeTriLin (B₄, B i, S) = 0
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rw [B₄_cubic]
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fin_cases i <;>
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₅_Bi_cubic {i : Fin 7} (hi : 5 ≠ i) (S : (SM 3).charges) :
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cubeTriLin (B 5, B i, S) = 0 := by
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change cubeTriLin (B₅, B i, S) = 0
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rw [B₅_cubic]
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fin_cases i <;>
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₆_Bi_cubic {i : Fin 7} (hi : 6 ≠ i) (S : (SM 3).charges) :
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cubeTriLin (B 6, B i, S) = 0 := by
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change cubeTriLin (B₆, B i, S) = 0
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rw [B₆_cubic]
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fin_cases i <;>
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simp at hi <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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|
||||
lemma Bi_Bj_ne_cubic {i j : Fin 7} (h : i ≠ j) (S : (SM 3).charges) :
|
||||
cubeTriLin (B i, B j, S) = 0 := by
|
||||
fin_cases i
|
||||
exact B₀_Bi_cubic h S
|
||||
exact B₁_Bi_cubic h S
|
||||
exact B₂_Bi_cubic h S
|
||||
exact B₃_Bi_cubic h S
|
||||
exact B₄_Bi_cubic h S
|
||||
exact B₅_Bi_cubic h S
|
||||
exact B₆_Bi_cubic h S
|
||||
|
||||
lemma B₀_B₀_Bi_cubic {i : Fin 7} :
|
||||
cubeTriLin (B 0, B 0, B i) = 0 := by
|
||||
change cubeTriLin (B₀, B₀, B i) = 0
|
||||
rw [B₀_cubic]
|
||||
fin_cases i <;>
|
||||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
|
||||
lemma B₁_B₁_Bi_cubic {i : Fin 7} :
|
||||
cubeTriLin (B 1, B 1, B i) = 0 := by
|
||||
change cubeTriLin (B₁, B₁, B i) = 0
|
||||
rw [B₁_cubic]
|
||||
fin_cases i <;>
|
||||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
|
||||
lemma B₂_B₂_Bi_cubic {i : Fin 7} :
|
||||
cubeTriLin (B 2, B 2, B i) = 0 := by
|
||||
change cubeTriLin (B₂, B₂, B i) = 0
|
||||
rw [B₂_cubic]
|
||||
fin_cases i <;>
|
||||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
|
||||
lemma B₃_B₃_Bi_cubic {i : Fin 7} :
|
||||
cubeTriLin (B 3, B 3, B i) = 0 := by
|
||||
change cubeTriLin (B₃, B₃, B i) = 0
|
||||
rw [B₃_cubic]
|
||||
fin_cases i <;>
|
||||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
|
||||
lemma B₄_B₄_Bi_cubic {i : Fin 7} :
|
||||
cubeTriLin (B 4, B 4, B i) = 0 := by
|
||||
change cubeTriLin (B₄, B₄, B i) = 0
|
||||
rw [B₄_cubic]
|
||||
fin_cases i <;>
|
||||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
|
||||
lemma B₅_B₅_Bi_cubic {i : Fin 7} :
|
||||
cubeTriLin (B 5, B 5, B i) = 0 := by
|
||||
change cubeTriLin (B₅, B₅, B i) = 0
|
||||
rw [B₅_cubic]
|
||||
fin_cases i <;>
|
||||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
|
||||
lemma B₆_B₆_Bi_cubic {i : Fin 7} :
|
||||
cubeTriLin (B 6, B 6, B i) = 0 := by
|
||||
change cubeTriLin (B₆, B₆, B i) = 0
|
||||
rw [B₆_cubic]
|
||||
fin_cases i <;>
|
||||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||||
|
||||
|
||||
lemma Bi_Bi_Bj_cubic (i j : Fin 7) :
|
||||
cubeTriLin (B i, B i, B j) = 0 := by
|
||||
fin_cases i
|
||||
exact B₀_B₀_Bi_cubic
|
||||
exact B₁_B₁_Bi_cubic
|
||||
exact B₂_B₂_Bi_cubic
|
||||
exact B₃_B₃_Bi_cubic
|
||||
exact B₄_B₄_Bi_cubic
|
||||
exact B₅_B₅_Bi_cubic
|
||||
exact B₆_B₆_Bi_cubic
|
||||
|
||||
lemma Bi_Bj_Bk_cubic (i j k : Fin 7) :
|
||||
cubeTriLin (B i, B j, B k) = 0 := by
|
||||
by_cases hij : i ≠ j
|
||||
exact Bi_Bj_ne_cubic hij (B k)
|
||||
simp at hij
|
||||
rw [hij]
|
||||
exact Bi_Bi_Bj_cubic j k
|
||||
|
||||
theorem B_in_accCube (f : Fin 7 → ℚ) : accCube (∑ i, f i • B i) = 0 := by
|
||||
change cubeTriLin _ = 0
|
||||
rw [cubeTriLin.map_sum₁₂₃]
|
||||
apply Fintype.sum_eq_zero
|
||||
intro i
|
||||
apply Fintype.sum_eq_zero
|
||||
intro k
|
||||
apply Fintype.sum_eq_zero
|
||||
intro l
|
||||
rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂, cubeTriLin.map_smul₃]
|
||||
rw [Bi_Bj_Bk_cubic]
|
||||
simp
|
||||
|
||||
lemma B_sum_is_sol (f : Fin 7 → ℚ) : (SM 3).isSolution (∑ i, f i • B i) := by
|
||||
let X := chargeToAF (∑ i, f i • B i) (by
|
||||
rw [map_sum]
|
||||
apply Fintype.sum_eq_zero
|
||||
intro i
|
||||
rw [map_smul]
|
||||
have h : accGrav (B i) = 0 := by
|
||||
fin_cases i <;> rfl
|
||||
rw [h]
|
||||
simp)
|
||||
(by
|
||||
rw [map_sum]
|
||||
apply Fintype.sum_eq_zero
|
||||
intro i
|
||||
rw [map_smul]
|
||||
have h : accSU2 (B i) = 0 := by
|
||||
fin_cases i <;> rfl
|
||||
rw [h]
|
||||
simp)
|
||||
(by
|
||||
rw [map_sum]
|
||||
apply Fintype.sum_eq_zero
|
||||
intro i
|
||||
rw [map_smul]
|
||||
have h : accSU3 (B i) = 0 := by
|
||||
fin_cases i <;> rfl
|
||||
rw [h]
|
||||
simp)
|
||||
(B_in_accCube f)
|
||||
use X
|
||||
rfl
|
||||
|
||||
theorem basis_linear_independent : LinearIndependent ℚ B := by
|
||||
apply Fintype.linearIndependent_iff.mpr
|
||||
intro f h
|
||||
have h0 := congrFun h (0 : Fin 18)
|
||||
have h1 := congrFun h (3 : Fin 18)
|
||||
have h2 := congrFun h (6 : Fin 18)
|
||||
have h3 := congrFun h (9 : Fin 18)
|
||||
have h4 := congrFun h (12 : Fin 18)
|
||||
have h5 := congrFun h (15 : Fin 18)
|
||||
have h6 := congrFun h (5 : Fin 18)
|
||||
rw [@Fin.sum_univ_seven] at h0 h1 h2 h3 h4 h5 h6
|
||||
simp [HSMul.hSMul] at h0 h1 h2 h3 h4 h5 h6
|
||||
rw [B₀, B₁, B₂, B₃, B₄, B₅, B₆] at h0 h1 h2 h3 h4 h5 h6
|
||||
simp [Fin.divNat, Fin.modNat] at h0 h1 h2 h3 h4 h5 h6
|
||||
intro i
|
||||
match i with
|
||||
| 0 => exact h0
|
||||
| 1 => exact h1
|
||||
| 2 => exact h2
|
||||
| 3 => exact h3
|
||||
| 4 => exact h4
|
||||
| 5 => exact h5
|
||||
| 6 => exact h6
|
||||
|
||||
end PlaneSeven
|
||||
|
||||
theorem seven_dim_plane_exists : ∃ (B : Fin 7 → (SM 3).charges),
|
||||
LinearIndependent ℚ B ∧ ∀ (f : Fin 7 → ℚ), (SM 3).isSolution (∑ i, f i • B i) := by
|
||||
use PlaneSeven.B
|
||||
apply And.intro
|
||||
exact PlaneSeven.basis_linear_independent
|
||||
exact PlaneSeven.B_sum_is_sol
|
||||
|
||||
|
||||
end SM
|
||||
end SMRHN
|
||||
73
HepLean/AnomalyCancellation/SMNu/PlusU1/BoundPlaneDim.lean
Normal file
73
HepLean/AnomalyCancellation/SMNu/PlusU1/BoundPlaneDim.lean
Normal file
|
|
@ -0,0 +1,73 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.AnomalyCancellation.SMNu.PlusU1.Basic
|
||||
import HepLean.AnomalyCancellation.SMNu.PlusU1.FamilyMaps
|
||||
import HepLean.AnomalyCancellation.SMNu.PlusU1.PlaneNonSols
|
||||
/-!
|
||||
# Bound on plane dimension
|
||||
|
||||
We place an upper bound on the dimension of a plane of charges on which every point is a solution.
|
||||
The upper bound is 7, proven in the theorem `plane_exists_dim_le_7`.
|
||||
|
||||
-/
|
||||
universe v u
|
||||
|
||||
namespace SMRHN
|
||||
namespace PlusU1
|
||||
|
||||
open SMνCharges
|
||||
open SMνACCs
|
||||
open BigOperators
|
||||
|
||||
/-- A proposition which is true if for a given `n` a plane of charges of dimension `n` exists
|
||||
in which each point is a solution. -/
|
||||
def existsPlane (n : ℕ) : Prop := ∃ (B : Fin n → (PlusU1 3).charges),
|
||||
LinearIndependent ℚ B ∧ ∀ (f : Fin n → ℚ), (PlusU1 3).isSolution (∑ i, f i • B i)
|
||||
|
||||
lemma exists_plane_exists_basis {n : ℕ} (hE : existsPlane n) :
|
||||
∃ (B : Fin 11 ⊕ Fin n → (PlusU1 3).charges), LinearIndependent ℚ B := by
|
||||
obtain ⟨E, hE1, hE2⟩ := hE
|
||||
obtain ⟨B, hB1, hB2⟩ := eleven_dim_plane_of_no_sols_exists
|
||||
let Y := Sum.elim B E
|
||||
use Y
|
||||
apply Fintype.linearIndependent_iff.mpr
|
||||
intro g hg
|
||||
rw [@Fintype.sum_sum_type] at hg
|
||||
rw [@add_eq_zero_iff_eq_neg] at hg
|
||||
rw [← @Finset.sum_neg_distrib] at hg
|
||||
have h1 : ∑ x : Fin n, -(g (Sum.inr x) • Y (Sum.inr x)) =
|
||||
∑ x : Fin n, (-g (Sum.inr x)) • Y (Sum.inr x):= by
|
||||
apply Finset.sum_congr
|
||||
simp only
|
||||
intro i _
|
||||
simp
|
||||
rw [h1] at hg
|
||||
have h2 : ∑ a₁ : Fin 11, g (Sum.inl a₁) • Y (Sum.inl a₁) = 0 := by
|
||||
apply hB2
|
||||
erw [hg]
|
||||
exact hE2 fun i => -g (Sum.inr i)
|
||||
rw [Fintype.linearIndependent_iff] at hB1 hE1
|
||||
have h3 : ∀ i, g (Sum.inl i) = 0 := hB1 (fun i => (g (Sum.inl i))) h2
|
||||
rw [h2] at hg
|
||||
have h4 : ∀ i, - g (Sum.inr i) = 0 := hE1 (fun i => (- g (Sum.inr i))) hg.symm
|
||||
simp at h4
|
||||
intro i
|
||||
match i with
|
||||
| Sum.inl i => exact h3 i
|
||||
| Sum.inr i => exact h4 i
|
||||
|
||||
|
||||
theorem plane_exists_dim_le_7 {n : ℕ} (hn : existsPlane n) : n ≤ 7 := by
|
||||
obtain ⟨B, hB⟩ := exists_plane_exists_basis hn
|
||||
have h1 := LinearIndependent.fintype_card_le_finrank hB
|
||||
simp at h1
|
||||
rw [show FiniteDimensional.finrank ℚ (PlusU1 3).charges = 18 from
|
||||
FiniteDimensional.finrank_fin_fun ℚ] at h1
|
||||
exact Nat.le_of_add_le_add_left h1
|
||||
|
||||
|
||||
end PlusU1
|
||||
end SMRHN
|
||||
265
HepLean/AnomalyCancellation/SMNu/PlusU1/PlaneNonSols.lean
Normal file
265
HepLean/AnomalyCancellation/SMNu/PlusU1/PlaneNonSols.lean
Normal file
|
|
@ -0,0 +1,265 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.AnomalyCancellation.SMNu.PlusU1.Basic
|
||||
import HepLean.AnomalyCancellation.SMNu.PlusU1.FamilyMaps
|
||||
/-!
|
||||
# Plane of non-solutions
|
||||
|
||||
Working in the three family case, we show that there exists an eleven dimensional plane in the
|
||||
vector space of charges on which there are no solutions.
|
||||
|
||||
The main result of this file is `eleven_dim_plane_of_no_sols_exists`, which states that
|
||||
an 11 dimensional plane of charges exists on which there are no solutions except the origin.
|
||||
-/
|
||||
|
||||
universe v u
|
||||
|
||||
namespace SMRHN
|
||||
namespace PlusU1
|
||||
|
||||
open SMνCharges
|
||||
open SMνACCs
|
||||
open BigOperators
|
||||
|
||||
namespace ElevenPlane
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₀ : (PlusU1 3).charges := ![1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₁ : (PlusU1 3).charges := ![0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₂ : (PlusU1 3).charges := ![0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₃ : (PlusU1 3).charges := ![0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₄ : (PlusU1 3).charges := ![0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₅ : (PlusU1 3).charges := ![0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₆ : (PlusU1 3).charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₇ : (PlusU1 3).charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₈ : (PlusU1 3).charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₉ : (PlusU1 3).charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0]
|
||||
|
||||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||||
def B₁₀ : (PlusU1 3).charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
|
||||
|
||||
/-- The charge assignment forming a basis of the plane. -/
|
||||
def B : Fin 11 → (PlusU1 3).charges := fun i =>
|
||||
match i with
|
||||
| 0 => B₀
|
||||
| 1 => B₁
|
||||
| 2 => B₂
|
||||
| 3 => B₃
|
||||
| 4 => B₄
|
||||
| 5 => B₅
|
||||
| 6 => B₆
|
||||
| 7 => B₇
|
||||
| 8 => B₈
|
||||
| 9 => B₉
|
||||
| 10 => B₁₀
|
||||
|
||||
lemma Bi_Bj_quad {i j : Fin 11} (hi : i ≠ j) : quadBiLin (B i, B j) = 0 := by
|
||||
fin_cases i <;> fin_cases j
|
||||
any_goals rfl
|
||||
all_goals simp at hi
|
||||
|
||||
lemma Bi_sum_quad (i : Fin 11) (f : Fin 11 → ℚ) :
|
||||
quadBiLin (B i, ∑ k, f k • B k) = f i * quadBiLin (B i, B i) := by
|
||||
rw [quadBiLin.map_sum₂]
|
||||
rw [Fintype.sum_eq_single i]
|
||||
rw [quadBiLin.map_smul₂]
|
||||
intro k hij
|
||||
rw [quadBiLin.map_smul₂, Bi_Bj_quad hij.symm]
|
||||
simp
|
||||
|
||||
/-- The coefficents of the quadratic equation in our basis. -/
|
||||
@[simp]
|
||||
def quadCoeff : Fin 11 → ℚ := ![1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0]
|
||||
|
||||
lemma quadCoeff_eq_bilinear (i : Fin 11) : quadCoeff i = quadBiLin (B i, B i) := by
|
||||
fin_cases i
|
||||
all_goals rfl
|
||||
|
||||
lemma on_accQuad (f : Fin 11 → ℚ) :
|
||||
accQuad (∑ i, f i • B i) = ∑ i, quadCoeff i * (f i)^2 := by
|
||||
change quadBiLin _ = _
|
||||
rw [quadBiLin.map_sum₁]
|
||||
apply Fintype.sum_congr
|
||||
intro i
|
||||
rw [quadBiLin.map_smul₁, Bi_sum_quad, quadCoeff_eq_bilinear]
|
||||
ring
|
||||
|
||||
|
||||
lemma isSolution_quadCoeff_f_sq_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i))
|
||||
(k : Fin 11) : quadCoeff k * (f k)^2 = 0 := by
|
||||
obtain ⟨S, hS⟩ := hS
|
||||
have hQ := quadSol S.1
|
||||
rw [hS, on_accQuad] at hQ
|
||||
rw [Fintype.sum_eq_zero_iff_of_nonneg] at hQ
|
||||
exact congrFun hQ k
|
||||
intro i
|
||||
simp only [Pi.zero_apply, quadCoeff]
|
||||
rw [mul_nonneg_iff]
|
||||
apply Or.inl
|
||||
apply And.intro
|
||||
fin_cases i <;> rfl
|
||||
exact sq_nonneg (f i)
|
||||
|
||||
lemma isSolution_f0 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 0 = 0 := by
|
||||
simpa using (isSolution_quadCoeff_f_sq_zero f hS 0)
|
||||
|
||||
lemma isSolution_f1 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 1 = 0 := by
|
||||
simpa using (isSolution_quadCoeff_f_sq_zero f hS 1)
|
||||
|
||||
lemma isSolution_f2 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 2 = 0 := by
|
||||
simpa using (isSolution_quadCoeff_f_sq_zero f hS 2)
|
||||
|
||||
lemma isSolution_f3 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 3 = 0 := by
|
||||
simpa using (isSolution_quadCoeff_f_sq_zero f hS 3)
|
||||
|
||||
lemma isSolution_f4 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 4 = 0 := by
|
||||
simpa using (isSolution_quadCoeff_f_sq_zero f hS 4)
|
||||
|
||||
lemma isSolution_f5 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 5 = 0 := by
|
||||
have h := isSolution_quadCoeff_f_sq_zero f hS 5
|
||||
rw [mul_eq_zero] at h
|
||||
change 1 = 0 ∨ _ = _ at h
|
||||
simpa using h
|
||||
|
||||
lemma isSolution_f6 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 6 = 0 := by
|
||||
have h := isSolution_quadCoeff_f_sq_zero f hS 6
|
||||
rw [mul_eq_zero] at h
|
||||
change 1 = 0 ∨ _ = _ at h
|
||||
simpa using h
|
||||
|
||||
lemma isSolution_f7 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 7 = 0 := by
|
||||
have h := isSolution_quadCoeff_f_sq_zero f hS 7
|
||||
rw [mul_eq_zero] at h
|
||||
change 1 = 0 ∨ _ = _ at h
|
||||
simpa using h
|
||||
|
||||
lemma isSolution_f8 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) : f 8 = 0 := by
|
||||
have h := isSolution_quadCoeff_f_sq_zero f hS 8
|
||||
rw [mul_eq_zero] at h
|
||||
change 1 = 0 ∨ _ = _ at h
|
||||
simpa using h
|
||||
|
||||
lemma isSolution_sum_part (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) :
|
||||
∑ i, f i • B i = f 9 • B₉ + f 10 • B₁₀ := by
|
||||
rw [Fin.sum_univ_castSucc, Fin.sum_univ_castSucc, Fin.sum_univ_castSucc, Fin.sum_univ_eight]
|
||||
change f 0 • B 0 + f 1 • B 1 + f 2 • B 2 + f 3 • B 3 + f 4 • B 4 + f 5 • B 5 + f 6 • B 6 +
|
||||
f 7 • B 7 + f 8 • B 8 + f 9 • B 9 + f 10 • B 10 = f 9 • B₉ + f 10 • B₁₀
|
||||
rw [isSolution_f0 f hS, isSolution_f1 f hS, isSolution_f2 f hS, isSolution_f3 f hS,
|
||||
isSolution_f4 f hS, isSolution_f5 f hS,
|
||||
isSolution_f6 f hS, isSolution_f7 f hS, isSolution_f8 f hS]
|
||||
simp
|
||||
rfl
|
||||
|
||||
lemma isSolution_grav (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) :
|
||||
f 10 = - 3 * f 9 := by
|
||||
have hx := isSolution_sum_part f hS
|
||||
obtain ⟨S, hS'⟩ := hS
|
||||
have hg := gravSol S.toLinSols
|
||||
rw [hS'] at hg
|
||||
rw [hx] at hg
|
||||
rw [accGrav.map_add, accGrav.map_smul, accGrav.map_smul] at hg
|
||||
rw [show accGrav B₉ = 3 by rfl] at hg
|
||||
rw [show accGrav B₁₀ = 1 by rfl] at hg
|
||||
simp at hg
|
||||
linear_combination hg
|
||||
|
||||
lemma isSolution_sum_part' (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) :
|
||||
∑ i, f i • B i = f 9 • B₉ + (- 3 * f 9) • B₁₀ := by
|
||||
rw [isSolution_sum_part f hS]
|
||||
rw [isSolution_grav f hS]
|
||||
|
||||
lemma isSolution_f9 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) :
|
||||
f 9 = 0 := by
|
||||
have hx := isSolution_sum_part' f hS
|
||||
obtain ⟨S, hS'⟩ := hS
|
||||
have hc := cubeSol S
|
||||
rw [hS'] at hc
|
||||
rw [hx] at hc
|
||||
change cubeTriLin.toCubic _ = _ at hc
|
||||
rw [cubeTriLin.toCubic_add] at hc
|
||||
erw [accCube.map_smul] at hc
|
||||
erw [accCube.map_smul (- 3 * f 9) B₁₀] at hc
|
||||
rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₁, cubeTriLin.map_smul₂, cubeTriLin.map_smul₂,
|
||||
cubeTriLin.map_smul₃, cubeTriLin.map_smul₃] at hc
|
||||
rw [show accCube B₉ = 9 by rfl] at hc
|
||||
rw [show accCube B₁₀ = 1 by rfl] at hc
|
||||
rw [show cubeTriLin (B₉, B₉, B₁₀) = 0 by rfl] at hc
|
||||
rw [show cubeTriLin (B₁₀, B₁₀, B₉) = 0 by rfl] at hc
|
||||
simp at hc
|
||||
have h1 : f 9 ^ 3 * 9 + (-(3 * f 9)) ^ 3 = - 18 * f 9 ^ 3 := by
|
||||
ring
|
||||
rw [h1] at hc
|
||||
simpa using hc
|
||||
|
||||
lemma isSolution_f10 (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) :
|
||||
f 10 = 0 := by
|
||||
rw [isSolution_grav f hS, isSolution_f9 f hS]
|
||||
simp
|
||||
|
||||
lemma isSolution_f_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i))
|
||||
(k : Fin 11) : f k = 0 := by
|
||||
fin_cases k
|
||||
exact isSolution_f0 f hS
|
||||
exact isSolution_f1 f hS
|
||||
exact isSolution_f2 f hS
|
||||
exact isSolution_f3 f hS
|
||||
exact isSolution_f4 f hS
|
||||
exact isSolution_f5 f hS
|
||||
exact isSolution_f6 f hS
|
||||
exact isSolution_f7 f hS
|
||||
exact isSolution_f8 f hS
|
||||
exact isSolution_f9 f hS
|
||||
exact isSolution_f10 f hS
|
||||
|
||||
|
||||
lemma isSolution_only_if_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).isSolution (∑ i, f i • B i)) :
|
||||
∑ i, f i • B i = 0 := by
|
||||
rw [isSolution_sum_part f hS]
|
||||
rw [isSolution_grav f hS]
|
||||
rw [isSolution_f9 f hS]
|
||||
simp
|
||||
|
||||
|
||||
theorem basis_linear_independent : LinearIndependent ℚ B := by
|
||||
apply Fintype.linearIndependent_iff.mpr
|
||||
intro f h
|
||||
let X : (PlusU1 3).Sols := chargeToAF 0 (by rfl) (by rfl) (by rfl) (by rfl) (by rfl) (by rfl)
|
||||
have hS : (PlusU1 3).isSolution (∑ i, f i • B i) := by
|
||||
use X
|
||||
rw [h]
|
||||
rfl
|
||||
exact isSolution_f_zero f hS
|
||||
|
||||
end ElevenPlane
|
||||
|
||||
theorem eleven_dim_plane_of_no_sols_exists : ∃ (B : Fin 11 → (PlusU1 3).charges),
|
||||
LinearIndependent ℚ B ∧
|
||||
∀ (f : Fin 11 → ℚ), (PlusU1 3).isSolution (∑ i, f i • B i) → ∑ i, f i • B i = 0 := by
|
||||
use ElevenPlane.B
|
||||
apply And.intro
|
||||
exact ElevenPlane.basis_linear_independent
|
||||
exact ElevenPlane.isSolution_only_if_zero
|
||||
|
||||
end PlusU1
|
||||
end SMRHN
|
||||
Loading…
Add table
Add a link
Reference in a new issue