349 lines
10 KiB
Text
349 lines
10 KiB
Text
/-
|
||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||
Released under Apache 2.0 license.
|
||
Authors: Joseph Tooby-Smith
|
||
-/
|
||
import HepLean.AnomalyCancellation.SMNu.Ordinary.Basic
|
||
/-!
|
||
# Dimension 7 plane
|
||
|
||
We work here in the three family case.
|
||
We give an example of a 7 dimensional plane on which every point satisfies the ACCs.
|
||
|
||
The main result of this file is `seven_dim_plane_exists` which states that there exists a
|
||
7 dimensional plane of charges on which every point satisfies the ACCs.
|
||
|
||
-/
|
||
|
||
namespace SMRHN
|
||
namespace SM
|
||
open SMνCharges
|
||
open SMνACCs
|
||
open BigOperators
|
||
|
||
namespace PlaneSeven
|
||
|
||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||
def B₀ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||
match s, i with
|
||
| 0, 0 => 1
|
||
| 0, 1 => - 1
|
||
| _, _ => 0
|
||
)
|
||
|
||
lemma B₀_cubic (S T : (SM 3).charges) : cubeTriLin (B₀, S, T) =
|
||
6 * (S (0 : Fin 18) * T (0 : Fin 18) - S (1 : Fin 18) * T (1 : Fin 18)) := by
|
||
simp [Fin.sum_univ_three, B₀, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||
ring
|
||
|
||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||
def B₁ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||
match s, i with
|
||
| 1, 0 => 1
|
||
| 1, 1 => - 1
|
||
| _, _ => 0
|
||
)
|
||
|
||
lemma B₁_cubic (S T : (SM 3).charges) : cubeTriLin (B₁, S, T) =
|
||
3 * (S (3 : Fin 18) * T (3 : Fin 18) - S (4 : Fin 18) * T (4 : Fin 18)) := by
|
||
simp [Fin.sum_univ_three, B₁, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||
ring
|
||
|
||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||
def B₂ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||
match s, i with
|
||
| 2, 0 => 1
|
||
| 2, 1 => - 1
|
||
| _, _ => 0
|
||
)
|
||
|
||
lemma B₂_cubic (S T : (SM 3).charges) : cubeTriLin (B₂, S, T) =
|
||
3 * (S (6 : Fin 18) * T (6 : Fin 18) - S (7 : Fin 18) * T (7 : Fin 18)) := by
|
||
simp [Fin.sum_univ_three, B₂, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||
ring
|
||
|
||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||
def B₃ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||
match s, i with
|
||
| 3, 0 => 1
|
||
| 3, 1 => - 1
|
||
| _, _ => 0
|
||
)
|
||
|
||
lemma B₃_cubic (S T : (SM 3).charges) : cubeTriLin (B₃, S, T) =
|
||
2 * (S (9 : Fin 18) * T (9 : Fin 18) - S (10 : Fin 18) * T (10 : Fin 18)) := by
|
||
simp [Fin.sum_univ_three, B₃, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||
ring_nf
|
||
rfl
|
||
|
||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||
def B₄ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||
match s, i with
|
||
| 4, 0 => 1
|
||
| 4, 1 => - 1
|
||
| _, _ => 0
|
||
)
|
||
|
||
lemma B₄_cubic (S T : (SM 3).charges) : cubeTriLin (B₄, S, T) =
|
||
(S (12 : Fin 18) * T (12 : Fin 18) - S (13 : Fin 18) * T (13 : Fin 18)) := by
|
||
simp [Fin.sum_univ_three, B₄, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||
ring_nf
|
||
rfl
|
||
|
||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||
def B₅ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||
match s, i with
|
||
| 5, 0 => 1
|
||
| 5, 1 => - 1
|
||
| _, _ => 0
|
||
)
|
||
|
||
lemma B₅_cubic (S T : (SM 3).charges) : cubeTriLin (B₅, S, T) =
|
||
(S (15 : Fin 18) * T (15 : Fin 18) - S (16 : Fin 18) * T (16 : Fin 18)) := by
|
||
simp [Fin.sum_univ_three, B₅, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||
ring_nf
|
||
rfl
|
||
|
||
/-- A charge assignment forming one of the basis elements of the plane. -/
|
||
def B₆ : (SM 3).charges := toSpeciesEquiv.invFun ( fun s => fun i =>
|
||
match s, i with
|
||
| 1, 2 => 1
|
||
| 2, 2 => -1
|
||
| _, _ => 0
|
||
)
|
||
|
||
lemma B₆_cubic (S T : (SM 3).charges) : cubeTriLin (B₆, S, T) =
|
||
3* (S (5 : Fin 18) * T (5 : Fin 18) - S (8 : Fin 18) * T (8 : Fin 18)) := by
|
||
simp [Fin.sum_univ_three, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||
ring_nf
|
||
|
||
/-- The charge assignments forming a basis of the plane. -/
|
||
@[simp]
|
||
def B : Fin 7 → (SM 3).charges := fun i =>
|
||
match i with
|
||
| 0 => B₀
|
||
| 1 => B₁
|
||
| 2 => B₂
|
||
| 3 => B₃
|
||
| 4 => B₄
|
||
| 5 => B₅
|
||
| 6 => B₆
|
||
|
||
lemma B₀_Bi_cubic {i : Fin 7} (hi : 0 ≠ i) (S : (SM 3).charges) :
|
||
cubeTriLin (B 0, B i, S) = 0 := by
|
||
change cubeTriLin (B₀, B i, S) = 0
|
||
rw [B₀_cubic]
|
||
fin_cases i <;>
|
||
simp at hi <;>
|
||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||
|
||
|
||
lemma B₁_Bi_cubic {i : Fin 7} (hi : 1 ≠ i) (S : (SM 3).charges) :
|
||
cubeTriLin (B 1, B i, S) = 0 := by
|
||
change cubeTriLin (B₁, B i, S) = 0
|
||
rw [B₁_cubic]
|
||
fin_cases i <;>
|
||
simp at hi <;>
|
||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||
|
||
lemma B₂_Bi_cubic {i : Fin 7} (hi : 2 ≠ i) (S : (SM 3).charges) :
|
||
cubeTriLin (B 2, B i, S) = 0 := by
|
||
change cubeTriLin (B₂, B i, S) = 0
|
||
rw [B₂_cubic]
|
||
fin_cases i <;>
|
||
simp at hi <;>
|
||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||
|
||
lemma B₃_Bi_cubic {i : Fin 7} (hi : 3 ≠ i) (S : (SM 3).charges) :
|
||
cubeTriLin (B 3, B i, S) = 0 := by
|
||
change cubeTriLin (B₃, B i, S) = 0
|
||
rw [B₃_cubic]
|
||
fin_cases i <;>
|
||
simp at hi <;>
|
||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||
|
||
lemma B₄_Bi_cubic {i : Fin 7} (hi : 4 ≠ i) (S : (SM 3).charges) :
|
||
cubeTriLin (B 4, B i, S) = 0 := by
|
||
change cubeTriLin (B₄, B i, S) = 0
|
||
rw [B₄_cubic]
|
||
fin_cases i <;>
|
||
simp at hi <;>
|
||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||
|
||
lemma B₅_Bi_cubic {i : Fin 7} (hi : 5 ≠ i) (S : (SM 3).charges) :
|
||
cubeTriLin (B 5, B i, S) = 0 := by
|
||
change cubeTriLin (B₅, B i, S) = 0
|
||
rw [B₅_cubic]
|
||
fin_cases i <;>
|
||
simp at hi <;>
|
||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||
|
||
lemma B₆_Bi_cubic {i : Fin 7} (hi : 6 ≠ i) (S : (SM 3).charges) :
|
||
cubeTriLin (B 6, B i, S) = 0 := by
|
||
change cubeTriLin (B₆, B i, S) = 0
|
||
rw [B₆_cubic]
|
||
fin_cases i <;>
|
||
simp at hi <;>
|
||
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
|
||
|
||
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
|