2024-04-17 14:25:17 -04:00
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/-
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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.SM.Basic
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import HepLean.AnomalyCancellation.SM.NoGrav.Basic
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import HepLean.AnomalyCancellation.SM.NoGrav.One.LinearParameterization
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/-!
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# Lemmas for 1 family SM Accs
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The main result of this file is the conclusion of this paper:
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https://arxiv.org/abs/1907.00514
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That eveery solution to the ACCs without gravity satifies for free the gravitational anomaly.
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-/
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2024-04-17 14:49:59 -04:00
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universe v u
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2024-04-17 14:25:17 -04:00
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namespace SM
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namespace SMNoGrav
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namespace One
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open SMCharges
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open SMACCs
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open BigOperators
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lemma E_zero_iff_Q_zero {S : (SMNoGrav 1).Sols} : Q S.val (0 : Fin 1) = 0 ↔
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E S.val (0 : Fin 1) = 0 := by
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let S' := linearParameters.bijection.symm S.1.1
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have hC := cubeSol S
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have hS' := congrArg (fun S => S.val) (linearParameters.bijection.right_inv S.1.1)
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change S'.asCharges = S.val at hS'
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rw [← hS'] at hC
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apply Iff.intro
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intro hQ
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exact S'.cubic_zero_Q'_zero hC hQ
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intro hE
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exact S'.cubic_zero_E'_zero hC hE
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lemma accGrav_Q_zero {S : (SMNoGrav 1).Sols} (hQ : Q S.val (0 : Fin 1) = 0) :
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accGrav S.val = 0 := by
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rw [accGrav]
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simp only [SMSpecies_numberCharges, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
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Finset.sum_singleton, LinearMap.coe_mk, AddHom.coe_mk]
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erw [hQ, E_zero_iff_Q_zero.mp hQ]
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have h1 := SU2Sol S.1.1
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have h2 := SU3Sol S.1.1
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simp only [accSU2, SMSpecies_numberCharges, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
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Finset.sum_singleton, LinearMap.coe_mk, AddHom.coe_mk, accSU3] at h1 h2
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erw [hQ] at h1 h2
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simp_all
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linear_combination 3 * h2
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2024-05-08 15:42:54 -04:00
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lemma accGrav_Q_neq_zero {S : (SMNoGrav 1).Sols} (hQ : Q S.val (0 : Fin 1) ≠ 0)
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(FLTThree : FermatLastTheoremWith ℚ 3) :
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accGrav S.val = 0 := by
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have hE := E_zero_iff_Q_zero.mpr.mt hQ
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let S' := linearParametersQENeqZero.bijection.symm ⟨S.1.1, And.intro hQ hE⟩
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have hC := cubeSol S
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have hS' := congrArg (fun S => S.val.val)
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(linearParametersQENeqZero.bijection.right_inv ⟨S.1.1, And.intro hQ hE⟩)
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change _ = S.val at hS'
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rw [← hS'] at hC
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rw [← hS']
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2024-05-08 15:42:54 -04:00
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exact S'.grav_of_cubic hC FLTThree
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2024-04-17 14:25:17 -04:00
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/-- Any solution to the ACCs without gravity satifies the gravitational ACC. -/
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theorem accGravSatisfied {S : (SMNoGrav 1).Sols} (FLTThree : FermatLastTheoremWith ℚ 3) :
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accGrav S.val = 0 := by
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2024-04-17 14:25:17 -04:00
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by_cases hQ : Q S.val (0 : Fin 1)= 0
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exact accGrav_Q_zero hQ
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2024-05-08 15:42:54 -04:00
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exact accGrav_Q_neq_zero hQ FLTThree
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2024-04-17 14:25:17 -04:00
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end One
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end SMNoGrav
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end SM
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