264 lines
9.5 KiB
Text
264 lines
9.5 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.PlusU1.Basic
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/-!
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# Plane of non-solutions
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Working in the three family case, we show that there exists an eleven dimensional plane in the
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vector space of charges on which there are no solutions.
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The main result of this file is `eleven_dim_plane_of_no_sols_exists`, which states that
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an 11 dimensional plane of charges exists on which there are no solutions except the origin.
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-/
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universe v u
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namespace SMRHN
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namespace PlusU1
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open SMνCharges
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open SMνACCs
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open BigOperators
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namespace ElevenPlane
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₀ : (PlusU1 3).Charges := ![1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₁ : (PlusU1 3).Charges := ![0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₂ : (PlusU1 3).Charges := ![0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₃ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₄ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₅ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₆ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0]
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₇ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0]
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₈ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₉ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0]
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/-- A charge assignment forming one of the basis elements of the plane. -/
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def B₁₀ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
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/-- The charge assignment forming a basis of the plane. -/
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def B : Fin 11 → (PlusU1 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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| 7 => B₇
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| 8 => B₈
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| 9 => B₉
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| 10 => B₁₀
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lemma Bi_Bj_quad {i j : Fin 11} (hi : i ≠ j) : quadBiLin (B i) (B j) = 0 := by
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fin_cases i <;> fin_cases j
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any_goals rfl
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all_goals simp at hi
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lemma Bi_sum_quad (i : Fin 11) (f : Fin 11 → ℚ) :
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quadBiLin (B i) (∑ k, f k • B k) = f i * quadBiLin (B i) (B i) := by
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rw [quadBiLin.map_sum₂]
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rw [Fintype.sum_eq_single i]
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rw [quadBiLin.map_smul₂]
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intro k hij
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rw [quadBiLin.map_smul₂, Bi_Bj_quad hij.symm]
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simp
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/-- The coefficents of the quadratic equation in our basis. -/
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@[simp]
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def quadCoeff : Fin 11 → ℚ := ![1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0]
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lemma quadCoeff_eq_bilinear (i : Fin 11) : quadCoeff i = quadBiLin (B i) (B i) := by
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fin_cases i
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all_goals rfl
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lemma on_accQuad (f : Fin 11 → ℚ) :
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accQuad (∑ i, f i • B i) = ∑ i, quadCoeff i * (f i)^2 := by
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change quadBiLin _ _ = _
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rw [quadBiLin.map_sum₁]
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apply Fintype.sum_congr
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intro i
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rw [quadBiLin.map_smul₁, Bi_sum_quad, quadCoeff_eq_bilinear]
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ring
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lemma isSolution_quadCoeff_f_sq_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i))
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(k : Fin 11) : quadCoeff k * (f k)^2 = 0 := by
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obtain ⟨S, hS⟩ := hS
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have hQ := quadSol S.1
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rw [hS, on_accQuad] at hQ
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rw [Fintype.sum_eq_zero_iff_of_nonneg] at hQ
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exact congrFun hQ k
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intro i
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simp only [Pi.zero_apply, quadCoeff]
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rw [mul_nonneg_iff]
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apply Or.inl
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apply And.intro
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fin_cases i <;> rfl
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exact sq_nonneg (f i)
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lemma isSolution_f0 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 0 = 0 := by
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simpa using (isSolution_quadCoeff_f_sq_zero f hS 0)
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lemma isSolution_f1 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 1 = 0 := by
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simpa using (isSolution_quadCoeff_f_sq_zero f hS 1)
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lemma isSolution_f2 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 2 = 0 := by
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simpa using (isSolution_quadCoeff_f_sq_zero f hS 2)
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lemma isSolution_f3 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 3 = 0 := by
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simpa using (isSolution_quadCoeff_f_sq_zero f hS 3)
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lemma isSolution_f4 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 4 = 0 := by
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simpa using (isSolution_quadCoeff_f_sq_zero f hS 4)
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lemma isSolution_f5 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 5 = 0 := by
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have h := isSolution_quadCoeff_f_sq_zero f hS 5
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rw [mul_eq_zero] at h
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change 1 = 0 ∨ _ = _ at h
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simpa using h
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lemma isSolution_f6 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 6 = 0 := by
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have h := isSolution_quadCoeff_f_sq_zero f hS 6
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rw [mul_eq_zero] at h
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change 1 = 0 ∨ _ = _ at h
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simpa using h
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lemma isSolution_f7 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 7 = 0 := by
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have h := isSolution_quadCoeff_f_sq_zero f hS 7
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rw [mul_eq_zero] at h
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change 1 = 0 ∨ _ = _ at h
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simpa using h
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lemma isSolution_f8 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) : f 8 = 0 := by
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have h := isSolution_quadCoeff_f_sq_zero f hS 8
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rw [mul_eq_zero] at h
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change 1 = 0 ∨ _ = _ at h
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simpa using h
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lemma isSolution_sum_part (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) :
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∑ i, f i • B i = f 9 • B₉ + f 10 • B₁₀ := by
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rw [Fin.sum_univ_castSucc, Fin.sum_univ_castSucc, Fin.sum_univ_castSucc, Fin.sum_univ_eight]
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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 +
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f 7 • B 7 + f 8 • B 8 + f 9 • B 9 + f 10 • B 10 = f 9 • B₉ + f 10 • B₁₀
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rw [isSolution_f0 f hS, isSolution_f1 f hS, isSolution_f2 f hS, isSolution_f3 f hS,
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isSolution_f4 f hS, isSolution_f5 f hS,
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isSolution_f6 f hS, isSolution_f7 f hS, isSolution_f8 f hS]
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simp
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rfl
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lemma isSolution_grav (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) :
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f 10 = - 3 * f 9 := by
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have hx := isSolution_sum_part f hS
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obtain ⟨S, hS'⟩ := hS
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have hg := gravSol S.toLinSols
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rw [hS'] at hg
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rw [hx] at hg
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rw [accGrav.map_add, accGrav.map_smul, accGrav.map_smul] at hg
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rw [show accGrav B₉ = 3 by rfl] at hg
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rw [show accGrav B₁₀ = 1 by rfl] at hg
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simp at hg
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linear_combination hg
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lemma isSolution_sum_part' (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) :
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∑ i, f i • B i = f 9 • B₉ + (- 3 * f 9) • B₁₀ := by
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rw [isSolution_sum_part f hS]
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rw [isSolution_grav f hS]
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lemma isSolution_f9 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) :
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f 9 = 0 := by
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have hx := isSolution_sum_part' f hS
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obtain ⟨S, hS'⟩ := hS
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have hc := cubeSol S
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rw [hS'] at hc
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rw [hx] at hc
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change cubeTriLin.toCubic _ = _ at hc
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rw [cubeTriLin.toCubic_add] at hc
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erw [accCube.map_smul] at hc
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erw [accCube.map_smul (- 3 * f 9) B₁₀] at hc
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rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₁, cubeTriLin.map_smul₂, cubeTriLin.map_smul₂,
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cubeTriLin.map_smul₃, cubeTriLin.map_smul₃] at hc
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rw [show accCube B₉ = 9 by rfl] at hc
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rw [show accCube B₁₀ = 1 by rfl] at hc
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rw [show cubeTriLin B₉ B₉ B₁₀ = 0 by rfl] at hc
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rw [show cubeTriLin B₁₀ B₁₀ B₉ = 0 by rfl] at hc
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simp at hc
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have h1 : f 9 ^ 3 * 9 + (-(3 * f 9)) ^ 3 = - 18 * f 9 ^ 3 := by
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ring
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rw [h1] at hc
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simpa using hc
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lemma isSolution_f10 (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) :
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f 10 = 0 := by
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rw [isSolution_grav f hS, isSolution_f9 f hS]
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simp
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lemma isSolution_f_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i))
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(k : Fin 11) : f k = 0 := by
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fin_cases k
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exact isSolution_f0 f hS
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exact isSolution_f1 f hS
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exact isSolution_f2 f hS
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exact isSolution_f3 f hS
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exact isSolution_f4 f hS
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exact isSolution_f5 f hS
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exact isSolution_f6 f hS
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exact isSolution_f7 f hS
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exact isSolution_f8 f hS
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exact isSolution_f9 f hS
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exact isSolution_f10 f hS
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lemma isSolution_only_if_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) :
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∑ i, f i • B i = 0 := by
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rw [isSolution_sum_part f hS]
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rw [isSolution_grav f hS]
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rw [isSolution_f9 f hS]
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simp
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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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let X : (PlusU1 3).Sols := chargeToAF 0 (by rfl) (by rfl) (by rfl) (by rfl) (by rfl) (by rfl)
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have hS : (PlusU1 3).IsSolution (∑ i, f i • B i) := by
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use X
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rw [h]
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rfl
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exact isSolution_f_zero f hS
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end ElevenPlane
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theorem eleven_dim_plane_of_no_sols_exists : ∃ (B : Fin 11 → (PlusU1 3).Charges),
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LinearIndependent ℚ B ∧
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∀ (f : Fin 11 → ℚ), (PlusU1 3).IsSolution (∑ i, f i • B i) → ∑ i, f i • B i = 0 := by
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use ElevenPlane.B
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apply And.intro
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exact ElevenPlane.basis_linear_independent
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exact ElevenPlane.isSolution_only_if_zero
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end PlusU1
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end SMRHN
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