171 lines
4.5 KiB
Text
171 lines
4.5 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 as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.AnomalyCancellation.Basic
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import Mathlib.Algebra.Module.Equiv
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import Mathlib.Algebra.BigOperators.Ring
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import Mathlib.Algebra.BigOperators.Fin
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/-!
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# Pure U(1) ACC system.
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We define the anomaly cancellation conditions for a pure U(1) gauge theory with `n` fermions.
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-/
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universe v u
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open Nat
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open Finset
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namespace PureU1
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open BigOperators
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/-- The vector space of charges. -/
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@[simps!]
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def PureU1Charges (n : ℕ) : ACCSystemCharges := ACCSystemChargesMk n
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open BigOperators in
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/-- The gravitational anomaly. -/
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def accGrav (n : ℕ) : ((PureU1Charges n).Charges →ₗ[ℚ] ℚ) where
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toFun S := ∑ i : Fin n, S i
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map_add' S T := Finset.sum_add_distrib
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map_smul' a S := by
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simp [HSMul.hSMul, SMul.smul]
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rw [← Finset.mul_sum]
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/-- The symmetric trilinear form used to define the cubic anomaly. -/
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@[simps!]
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def accCubeTriLinSymm {n : ℕ} : TriLinearSymm (PureU1Charges n).Charges := TriLinearSymm.mk₃
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(fun S => ∑ i, S.1 i * S.2.1 i * S.2.2 i)
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(by
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intro a S L T
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simp [HSMul.hSMul]
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rw [Finset.mul_sum]
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apply Fintype.sum_congr
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intro i
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ring)
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(by
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intro S L T R
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simp only [PureU1Charges_numberCharges, ACCSystemCharges.chargesAddCommMonoid_add]
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rw [← Finset.sum_add_distrib]
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apply Fintype.sum_congr
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intro i
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ring)
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(by
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intro S L T
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simp only [PureU1Charges_numberCharges]
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apply Fintype.sum_congr
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intro i
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ring)
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(by
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intro S L T
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simp only [PureU1Charges_numberCharges]
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apply Fintype.sum_congr
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intro i
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ring)
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/-- The cubic anomaly equation. -/
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@[simp]
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def accCube (n : ℕ) : HomogeneousCubic ((PureU1Charges n).Charges) :=
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(accCubeTriLinSymm).toCubic
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lemma accCube_explicit (n : ℕ) (S : (PureU1Charges n).Charges) :
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accCube n S = ∑ i : Fin n, S i ^ 3:= by
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rw [accCube, TriLinearSymm.toCubic]
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change accCubeTriLinSymm S S S = _
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rw [accCubeTriLinSymm]
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simp only [PureU1Charges_numberCharges, TriLinearSymm.mk₃_toFun_apply_apply]
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apply Finset.sum_congr
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simp only
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ring_nf
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simp
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end PureU1
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/-- The ACC system for a pure $U(1)$ gauge theory with $n$ fermions. -/
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@[simps!]
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def PureU1 (n : ℕ) : ACCSystem where
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numberLinear := 1
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linearACCs := fun i =>
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match i with
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| 0 => PureU1.accGrav n
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numberQuadratic := 0
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quadraticACCs := Fin.elim0
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cubicACC := PureU1.accCube n
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/-- An equivalence of vector spaces of charges when the number of fermions is equal. -/
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def pureU1EqCharges {n m : ℕ} (h : n = m) :
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(PureU1 n).Charges ≃ₗ[ℚ] (PureU1 m).Charges where
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toFun f := f ∘ Fin.cast h.symm
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invFun f := f ∘ Fin.cast h
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map_add' _ _ := rfl
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map_smul' _ _:= rfl
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left_inv _ := rfl
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right_inv _ := rfl
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open BigOperators
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lemma pureU1_linear {n : ℕ} (S : (PureU1 n.succ).LinSols) :
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∑ i, S.val i = 0 := by
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have hS := S.linearSol
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simp at hS
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exact hS 0
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lemma pureU1_cube {n : ℕ} (S : (PureU1 n.succ).Sols) :
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∑ i, (S.val i) ^ 3 = 0 := by
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have hS := S.cubicSol
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erw [PureU1.accCube_explicit] at hS
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exact hS
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lemma pureU1_last {n : ℕ} (S : (PureU1 n.succ).LinSols) :
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S.val (Fin.last n) = - ∑ i : Fin n, S.val i.castSucc := by
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have hS := pureU1_linear S
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simp at hS
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rw [Fin.sum_univ_castSucc] at hS
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linear_combination hS
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lemma pureU1_anomalyFree_ext {n : ℕ} {S T : (PureU1 n.succ).LinSols}
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(h : ∀ (i : Fin n), S.val i.castSucc = T.val i.castSucc) : S = T := by
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apply ACCSystemLinear.LinSols.ext
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funext i
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by_cases hi : i ≠ Fin.last n
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have hiCast : ∃ j : Fin n, j.castSucc = i := by
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exact Fin.exists_castSucc_eq.mpr hi
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obtain ⟨j, hj⟩ := hiCast
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rw [← hj]
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exact h j
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simp at hi
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rw [hi]
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rw [pureU1_last, pureU1_last]
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simp only [neg_inj]
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apply Finset.sum_congr
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simp only
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intro j _
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exact h j
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namespace PureU1
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lemma sum_of_charges {n : ℕ} (f : Fin k → (PureU1 n).Charges) (j : Fin n) :
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(∑ i : Fin k, (f i)) j = ∑ i : Fin k, (f i) j := by
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induction k
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simp
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rfl
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rename_i k hl
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rw [Fin.sum_univ_castSucc, Fin.sum_univ_castSucc]
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have hlt := hl (f ∘ Fin.castSucc)
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erw [← hlt]
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simp
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lemma sum_of_anomaly_free_linear {n : ℕ} (f : Fin k → (PureU1 n).LinSols) (j : Fin n) :
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(∑ i : Fin k, (f i)).1 j = (∑ i : Fin k, (f i).1 j) := by
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induction k
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simp
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rfl
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rename_i k hl
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rw [Fin.sum_univ_castSucc, Fin.sum_univ_castSucc]
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have hlt := hl (f ∘ Fin.castSucc)
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erw [← hlt]
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simp
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end PureU1
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