349 lines
10 KiB
Text
349 lines
10 KiB
Text
/-
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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) :
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cubeTriLin (B i) (B j) S = 0 := by
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fin_cases i
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exact B₀_Bi_cubic h S
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exact B₁_Bi_cubic h S
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exact B₂_Bi_cubic h S
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exact B₃_Bi_cubic h S
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exact B₄_Bi_cubic h S
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exact B₅_Bi_cubic h S
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exact B₆_Bi_cubic h S
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lemma B₀_B₀_Bi_cubic {i : Fin 7} :
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cubeTriLin (B 0) (B 0) (B i) = 0 := by
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change cubeTriLin (B₀) (B₀) (B i) = 0
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rw [B₀_cubic]
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fin_cases i <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₁_B₁_Bi_cubic {i : Fin 7} :
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cubeTriLin (B 1) (B 1) (B i) = 0 := by
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change cubeTriLin (B₁) (B₁) (B i) = 0
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rw [B₁_cubic]
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fin_cases i <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₂_B₂_Bi_cubic {i : Fin 7} :
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cubeTriLin (B 2) (B 2) (B i) = 0 := by
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change cubeTriLin (B₂) (B₂) (B i) = 0
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rw [B₂_cubic]
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fin_cases i <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₃_B₃_Bi_cubic {i : Fin 7} :
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cubeTriLin (B 3) (B 3) (B i) = 0 := by
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change cubeTriLin (B₃) (B₃) (B i) = 0
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rw [B₃_cubic]
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fin_cases i <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₄_B₄_Bi_cubic {i : Fin 7} :
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cubeTriLin (B 4) (B 4) (B i) = 0 := by
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change cubeTriLin (B₄) (B₄) (B i) = 0
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rw [B₄_cubic]
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fin_cases i <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₅_B₅_Bi_cubic {i : Fin 7} :
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cubeTriLin (B 5) (B 5) (B i) = 0 := by
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change cubeTriLin (B₅) (B₅) (B i) = 0
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rw [B₅_cubic]
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fin_cases i <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma B₆_B₆_Bi_cubic {i : Fin 7} :
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cubeTriLin (B 6) (B 6) (B i) = 0 := by
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change cubeTriLin (B₆) (B₆) (B i) = 0
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rw [B₆_cubic]
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fin_cases i <;>
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simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
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lemma Bi_Bi_Bj_cubic (i j : Fin 7) :
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cubeTriLin (B i) (B i) (B j) = 0 := by
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fin_cases i
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exact B₀_B₀_Bi_cubic
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exact B₁_B₁_Bi_cubic
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exact B₂_B₂_Bi_cubic
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exact B₃_B₃_Bi_cubic
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exact B₄_B₄_Bi_cubic
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exact B₅_B₅_Bi_cubic
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exact B₆_B₆_Bi_cubic
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lemma Bi_Bj_Bk_cubic (i j k : Fin 7) :
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cubeTriLin (B i) (B j) (B k) = 0 := by
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by_cases hij : i ≠ j
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exact Bi_Bj_ne_cubic hij (B k)
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simp at hij
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rw [hij]
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exact Bi_Bi_Bj_cubic j k
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theorem B_in_accCube (f : Fin 7 → ℚ) : accCube (∑ i, f i • B i) = 0 := by
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change cubeTriLin _ _ _ = 0
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rw [cubeTriLin.map_sum₁₂₃]
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apply Fintype.sum_eq_zero
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intro i
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apply Fintype.sum_eq_zero
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intro k
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apply Fintype.sum_eq_zero
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intro l
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rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂, cubeTriLin.map_smul₃]
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rw [Bi_Bj_Bk_cubic]
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simp
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lemma B_sum_is_sol (f : Fin 7 → ℚ) : (SM 3).IsSolution (∑ i, f i • B i) := by
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let X := chargeToAF (∑ i, f i • B i) (by
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rw [map_sum]
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apply Fintype.sum_eq_zero
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intro i
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rw [map_smul]
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have h : accGrav (B i) = 0 := by
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fin_cases i <;> rfl
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rw [h]
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simp)
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(by
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rw [map_sum]
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apply Fintype.sum_eq_zero
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intro i
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rw [map_smul]
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have h : accSU2 (B i) = 0 := by
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fin_cases i <;> rfl
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rw [h]
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simp)
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(by
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rw [map_sum]
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apply Fintype.sum_eq_zero
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intro i
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rw [map_smul]
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have h : accSU3 (B i) = 0 := by
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fin_cases i <;> rfl
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rw [h]
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simp)
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(B_in_accCube f)
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use X
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rfl
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theorem basis_linear_independent : LinearIndependent ℚ B := by
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apply Fintype.linearIndependent_iff.mpr
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intro f h
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have h0 := congrFun h (0 : Fin 18)
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have h1 := congrFun h (3 : Fin 18)
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have h2 := congrFun h (6 : Fin 18)
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have h3 := congrFun h (9 : Fin 18)
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have h4 := congrFun h (12 : Fin 18)
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have h5 := congrFun h (15 : Fin 18)
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have h6 := congrFun h (5 : Fin 18)
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rw [@Fin.sum_univ_seven] at h0 h1 h2 h3 h4 h5 h6
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simp [HSMul.hSMul] at h0 h1 h2 h3 h4 h5 h6
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rw [B₀, B₁, B₂, B₃, B₄, B₅, B₆] at h0 h1 h2 h3 h4 h5 h6
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simp [Fin.divNat, Fin.modNat] at h0 h1 h2 h3 h4 h5 h6
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intro i
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match i with
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| 0 => exact h0
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| 1 => exact h1
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| 2 => exact h2
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| 3 => exact h3
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| 4 => exact h4
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| 5 => exact h5
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| 6 => exact h6
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end PlaneSeven
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theorem seven_dim_plane_exists : ∃ (B : Fin 7 → (SM 3).Charges),
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LinearIndependent ℚ B ∧ ∀ (f : Fin 7 → ℚ), (SM 3).IsSolution (∑ i, f i • B i) := by
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use PlaneSeven.B
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apply And.intro
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exact PlaneSeven.basis_linear_independent
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exact PlaneSeven.B_sum_is_sol
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end SM
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end SMRHN
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