/- Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved. Released under Apache 2.0 license. Authors: Joseph Tooby-Smith -/ import HepLean.AnomalyCancellation.PureU1.Basic import HepLean.AnomalyCancellation.PureU1.Permutations import HepLean.AnomalyCancellation.PureU1.VectorLike import HepLean.AnomalyCancellation.PureU1.ConstAbs import HepLean.AnomalyCancellation.PureU1.LineInPlaneCond import HepLean.AnomalyCancellation.PureU1.Odd.BasisLinear import Mathlib.Tactic.Polyrith import Mathlib.RepresentationTheory.Basic /-! # Line In Cubic Odd case We say that a linear solution satisfies the `lineInCubic` property if the line through that point and through the two different planes formed by the basis of `LinSols` lies in the cubic. We show that for a solution all its permutations satisfy this property, then the charge must be zero. The main reference for this file is: - https://arxiv.org/pdf/1912.04804.pdf -/ namespace PureU1 namespace Odd open BigOperators variable {n : ℕ} open VectorLikeOddPlane /-- A property on `LinSols`, satisfied if every point on the line between the two planes in the basis through that point is in the cubic. -/ def LineInCubic (S : (PureU1 (2 * n + 1)).LinSols) : Prop := ∀ (g f : Fin n → ℚ) (_ : S.val = Pa g f) (a b : ℚ) , accCube (2 * n + 1) (a • P g + b • P! f) = 0 lemma lineInCubic_expand {S : (PureU1 (2 * n + 1)).LinSols} (h : LineInCubic S) : ∀ (g : Fin n → ℚ) (f : Fin n → ℚ) (_ : S.val = P g + P! f) (a b : ℚ) , 3 * a * b * (a * accCubeTriLinSymm (P g) (P g) (P! f) + b * accCubeTriLinSymm (P! f) (P! f) (P g)) = 0 := by intro g f hS a b have h1 := h g f hS a b change accCubeTriLinSymm.toCubic (a • P g + b • P! f) = 0 at h1 simp only [TriLinearSymm.toCubic_add] at h1 simp only [HomogeneousCubic.map_smul, accCubeTriLinSymm.map_smul₁, accCubeTriLinSymm.map_smul₂, accCubeTriLinSymm.map_smul₃] at h1 erw [P_accCube, P!_accCube] at h1 rw [← h1] ring lemma line_in_cubic_P_P_P! {S : (PureU1 (2 * n + 1)).LinSols} (h : LineInCubic S) : ∀ (g : Fin n → ℚ) (f : Fin n → ℚ) (_ : S.val = P g + P! f), accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by intro g f hS linear_combination 2 / 3 * (lineInCubic_expand h g f hS 1 1 ) - (lineInCubic_expand h g f hS 1 2 ) / 6 /-- We say a `LinSol` satisfies `lineInCubicPerm` if all its permutations satisfy `lineInCubic`. -/ def LineInCubicPerm (S : (PureU1 (2 * n + 1)).LinSols) : Prop := ∀ (M : (FamilyPermutations (2 * n + 1)).group ), LineInCubic ((FamilyPermutations (2 * n + 1)).linSolRep M S) /-- If `lineInCubicPerm S` then `lineInCubic S`. -/ lemma lineInCubicPerm_self {S : (PureU1 (2 * n + 1)).LinSols} (hS : LineInCubicPerm S) : LineInCubic S := hS 1 /-- If `lineInCubicPerm S` then `lineInCubicPerm (M S)` for all permutations `M`. -/ lemma lineInCubicPerm_permute {S : (PureU1 (2 * n + 1)).LinSols} (hS : LineInCubicPerm S) (M' : (FamilyPermutations (2 * n + 1)).group) : LineInCubicPerm ((FamilyPermutations (2 * n + 1)).linSolRep M' S) := by rw [LineInCubicPerm] intro M have ht : ((FamilyPermutations (2 * n + 1)).linSolRep M) ((FamilyPermutations (2 * n + 1)).linSolRep M' S) = (FamilyPermutations (2 * n + 1)).linSolRep (M * M') S := by simp [(FamilyPermutations (2 * n.succ)).linSolRep.map_mul'] rw [ht] exact hS (M * M') lemma lineInCubicPerm_swap {S : (PureU1 (2 * n.succ + 1)).LinSols} (LIC : LineInCubicPerm S) : ∀ (j : Fin n.succ) (g f : Fin n.succ → ℚ) (_ : S.val = Pa g f) , (S.val (δ!₂ j) - S.val (δ!₁ j)) * accCubeTriLinSymm (P g) (P g) (basis!AsCharges j) = 0 := by intro j g f h let S' := (FamilyPermutations (2 * n.succ + 1)).linSolRep (pairSwap (δ!₁ j) (δ!₂ j)) S have hSS' : ((FamilyPermutations (2 * n.succ + 1)).linSolRep (pairSwap (δ!₁ j) (δ!₂ j))) S = S' := rfl obtain ⟨g', f', hall⟩ := span_basis_swap! j hSS' g f h have h1 := line_in_cubic_P_P_P! (lineInCubicPerm_self LIC) g f h have h2 := line_in_cubic_P_P_P! (lineInCubicPerm_self (lineInCubicPerm_permute LIC (pairSwap (δ!₁ j) (δ!₂ j)))) g' f' hall.1 rw [hall.2.1, hall.2.2] at h2 rw [accCubeTriLinSymm.map_add₃, h1, accCubeTriLinSymm.map_smul₃] at h2 simpa using h2 lemma P_P_P!_accCube' {S : (PureU1 (2 * n.succ.succ + 1)).LinSols} (f g : Fin n.succ.succ → ℚ) (hS : S.val = Pa f g) : accCubeTriLinSymm (P f) (P f) (basis!AsCharges 0) = (S.val (δ!₁ 0) + S.val (δ!₂ 0)) * (2 * S.val δ!₃ + S.val (δ!₁ 0) + S.val (δ!₂ 0)) := by rw [P_P_P!_accCube f 0] rw [← Pa_δa₁ f g] rw [← hS] have ht : δ!₁ (0 : Fin n.succ.succ) = δ₁ 1 := by simp [δ!₁, δ₁] rw [Fin.ext_iff] simp nth_rewrite 1 [ht] rw [P_δ₁] have h1 := Pa_δa₁ f g have h4 := Pa_δa₄ f g 0 have h2 := Pa_δa₂ f g 0 rw [← hS] at h1 h2 h4 simp at h2 have h5 : f 1 = S.val (δa₂ 0) + S.val δa₁ + S.val (δa₄ 0):= by linear_combination -(1 * h1) - 1 * h4 - 1 * h2 rw [h5] rw [δa₄_δ!₂] have h0 : (δa₂ (0 : Fin n.succ)) = δ!₁ 0 := by rw [δa₂_δ!₁] simp rw [h0, δa₁_δ!₃] ring lemma lineInCubicPerm_last_cond {S : (PureU1 (2 * n.succ.succ+1)).LinSols} (LIC : LineInCubicPerm S) : LineInPlaneProp ((S.val (δ!₂ 0)), ((S.val (δ!₁ 0)), (S.val δ!₃))) := by obtain ⟨g, f, hfg⟩ := span_basis S have h1 := lineInCubicPerm_swap LIC 0 g f hfg rw [P_P_P!_accCube' g f hfg] at h1 simp at h1 cases h1 <;> rename_i h1 apply Or.inl linear_combination h1 cases h1 <;> rename_i h1 apply Or.inr apply Or.inl linear_combination h1 apply Or.inr apply Or.inr linear_combination h1 lemma lineInCubicPerm_last_perm {S : (PureU1 (2 * n.succ.succ + 1)).LinSols} (LIC : LineInCubicPerm S) : LineInPlaneCond S := by refine @Prop_three (2 * n.succ.succ + 1) LineInPlaneProp S (δ!₂ 0) (δ!₁ 0) δ!₃ ?_ ?_ ?_ ?_ simp [Fin.ext_iff, δ!₂, δ!₁] simp [Fin.ext_iff, δ!₂, δ!₃] simp [Fin.ext_iff, δ!₁, δ!₃] intro M exact lineInCubicPerm_last_cond (lineInCubicPerm_permute LIC M) lemma lineInCubicPerm_constAbs {S : (PureU1 (2 * n.succ.succ + 1)).LinSols} (LIC : LineInCubicPerm S) : ConstAbs S.val := linesInPlane_constAbs (lineInCubicPerm_last_perm LIC) theorem lineInCubicPerm_zero {S : (PureU1 (2 * n.succ.succ + 1)).LinSols} (LIC : LineInCubicPerm S) : S = 0 := ConstAbs.boundary_value_odd S (lineInCubicPerm_constAbs LIC) end Odd end PureU1